Subgroups — #20288940

Order 6784.

1134 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811370	C₂	1	2	yes	maximal, complemented
2	#16811367	C₂	1	2	yes	maximal, complemented
3	#16811334	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811333	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811285	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811281	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811278	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811276	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811268	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811266	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811266	C₂	1	2	yes	maximal, complemented
12	#16811265	C₂	1	2	yes	maximal, complemented
13	#16811248	C₂	1	2	yes	maximal, complemented
14	#16811244	C₂	1	1	yes	maximal, characteristic, complemented
15	#16811214	C₂	1	2	yes	maximal, complemented
16	#16811212	C₂	1	2	yes	maximal, complemented
17	#16811203	C₂	1	2	yes	maximal, complemented
18	#16811203	C₂	1	1	yes	maximal, characteristic, complemented
19	#16811202	C₂	1	1	yes	maximal, characteristic, complemented
20	#16811135	C₂	1	1	yes	maximal, characteristic, complemented
21	C₂²⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
22	#121×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
23	C₅₃×#542	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11719949	C₂²	1	2	yes	complemented
25	C₂×#11719949	C₂²	1	1	yes	characteristic, complemented
26	C₂×#11719945	C₂²	1	2	yes	complemented
27	C₂×#11719945	C₂²	1	1	yes	characteristic, complemented
28	C₂×#11719947	C₂²	1	1	yes	characteristic, complemented
29	C₂×Q₈×D₅₃	C₂²	1	1	yes	characteristic, complemented
30	C₂×D₄×D₅₃	C₂²	1	2	yes	—
31	C₂×D₄×D₅₃	C₂²	1	1	yes	characteristic
32	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
33	#13133217	C₂²	1	1	yes	characteristic, complemented
34	#13133212	C₂²	1	2	yes	complemented
35	#13133209	C₂²	1	2	yes	complemented
36	#13133208	C₂²	1	2	yes	complemented
37	#13133205	C₂²	1	1	yes	characteristic, complemented
38	#13133205	C₂²	1	2	yes	complemented
39	#13133203	C₂²	1	2	yes	complemented
40	#13133203	C₂²	1	2	yes	complemented
41	#13133202	C₂²	1	2	yes	complemented
42	#13133202	C₂²	1	2	yes	complemented
43	#13133201	C₂²	1	2	yes	complemented
44	#13133201	C₂²	1	1	yes	characteristic, complemented
45	#13133201	C₂²	1	1	yes	characteristic, complemented
46	#13133199	C₂²	1	2	yes	complemented
47	#13133199	C₂²	1	2	yes	complemented
48	#13133199	C₂²	1	2	yes	complemented
49	#13133199	C₂²	1	2	yes	complemented
50	#13133197	C₂²	1	1	yes	characteristic
51	#13133197	C₂²	1	1	yes	characteristic
52	#13133197	C₂²	1	1	yes	characteristic, complemented
53	#13133197	C₂²	1	2	yes	complemented
54	#13133197	C₂²	1	1	yes	characteristic, complemented
55	#13133196	C₂²	1	2	yes	complemented
56	#13133196	C₂²	1	1	yes	characteristic, complemented
57	#13133195	C₂²	1	2	yes	—
58	#13133195	C₂²	1	2	yes	—
59	#13133195	C₂²	1	2	yes	complemented
60	#13133195	C₂²	1	2	yes	complemented
61	#13133195	C₂²	1	2	yes	complemented
62	#13133194	C₂²	1	2	yes	complemented
63	#13133194	C₂²	1	2	yes	complemented
64	#13133194	C₂²	1	2	yes	complemented
65	#13133193	C₂²	1	1	yes	characteristic
66	#13133193	C₂²	1	2	yes	—
67	#13133193	C₂²	1	2	yes	complemented
68	#13133193	C₂²	1	1	yes	characteristic, complemented
69	#13133193	C₂²	1	2	yes	complemented
70	#13133192	C₂²	1	1	yes	characteristic
71	#13133192	C₂²	1	1	yes	characteristic
72	#13133192	C₂²	1	2	yes	complemented
73	#13133192	C₂²	1	1	yes	characteristic, complemented
74	#13133192	C₂²	1	1	yes	characteristic, complemented
75	#13133192	C₂²	1	2	yes	complemented
76	#13133191	C₂²	1	1	yes	characteristic
77	#13133191	C₂²	1	2	yes	complemented
78	#13133190	C₂²	1	2	yes	—
79	#13133190	C₂²	1	2	yes	complemented
80	#13133190	C₂²	1	2	yes	complemented
81	#13133190	C₂²	1	2	yes	complemented
82	#13133190	C₂²	1	2	yes	complemented
83	#13133190	C₂²	1	2	yes	complemented
84	#13133189	C₂²	1	2	yes	—
85	#13133189	C₂²	1	2	yes	complemented
86	#13133188	C₂²	1	1	yes	characteristic
87	#13133188	C₂²	1	1	yes	characteristic, complemented
88	#13133188	C₂²	1	2	yes	complemented
89	#13133187	C₂²	1	1	yes	characteristic
90	#13133187	C₂²	1	2	yes	—
91	#13133187	C₂²	1	1	yes	characteristic
92	#13133187	C₂²	1	2	yes	—
93	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
94	C₄×D₂₁₂	C₂²	1	2	yes	—
95	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
96	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
97	C₄×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
98	C₄×D₂₁₂	C₂²	1	2	yes	complemented
99	C₄×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
100	C₄×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
101	#13133179	C₂²	1	2	yes	—
102	#13133177	C₂²	1	2	yes	complemented
103	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
104	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
105	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
106	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
107	C₄⋊C₄×D₅₃	C₂²	1	2	yes	complemented
108	C₂×#11719921	C₂²	1	2	yes	—
109	C₂²⋊C₄×D₅₃	C₂²	1	2	yes	—
110	C₂²⋊C₄×D₅₃	C₂²	1	2	yes	—
111	C₂²⋊C₄×D₅₃	C₂²	1	2	yes	—
112	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
113	#126×C₅₃	C₂²	1	2	yes	complemented
114	#124×C₅₃	C₂²	1	1	yes	characteristic, complemented
115	#124×C₅₃	C₂²	1	2	yes	complemented
116	#123×C₅₃	C₂²	1	2	yes	complemented
117	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
118	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
119	#121×C₅₃	C₂²	1	1	yes	characteristic
120	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
121	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
122	#117×C₅₃	C₂²	1	1	yes	characteristic, complemented
123	C₂³×D₅₃	C₂³	1	1	yes	characteristic
124	C₂³×D₅₃	C₂³	1	2	yes	—
125	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
126	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
127	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
128	C₂×D₄×C₅₃	C₂³	1	2	yes	—
129	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
131	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
132	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
133	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
134	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
135	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
136	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
137	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
138	C₂²×Dic₅₃	C₂³	1	2	yes	—
139	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
140	#11719949	—	4	4	no	—
141	#11719949	—	4	2	no	—
142	Q₈×D₅₃	—	4	2	no	—
143	#11719947	—	4	2	no	—
144	D₄×D₅₃	—	4	2	no	—
145	D₄×D₅₃	—	4	4	no	—
146	#11719945	—	4	2	no	—
147	#11719945	—	4	4	no	—
148	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
149	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
150	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
151	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
152	C₂×D₂₁₂	C₂³	1	2	yes	—
153	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
154	C₂×D₂₁₂	C₂³	1	2	yes	—
155	C₂×D₂₁₂	C₂³	1	2	yes	—
156	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
157	C₂×C₄×D₅₃	—	4	2	no	—
158	C₂×C₄×D₅₃	C₂³	1	2	yes	—
159	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
160	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
161	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
162	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
163	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
164	C₂×C₄×D₅₃	C₂³	1	2	yes	—
165	C₂×C₄×D₅₃	C₂³	1	2	yes	—
166	C₂×C₄×D₅₃	C₂³	1	2	yes	—
167	C₂×C₄×D₅₃	C₂³	1	2	yes	—
168	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
169	C₂×C₄×D₅₃	C₂³	1	2	yes	—
170	C₂×C₄×D₅₃	C₂³	1	2	yes	—
171	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
172	C₂×Dic₁₀₆	C₂³	1	2	yes	—
173	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
174	C₂×Dic₁₀₆	C₂³	1	2	yes	—
175	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
176	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
177	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
178	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
179	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
180	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
181	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
182	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
183	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
184	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
185	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
186	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
187	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
188	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
189	C₄²×C₅₃	C₂³	1	1	yes	characteristic
190	C₄²×C₅₃	C₂³	1	1	yes	characteristic
191	C₄²×C₅₃	C₂³	1	2	yes	—
192	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
193	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
194	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
195	#11719921	C₂³	1	1	yes	characteristic
196	#11719921	C₂³	1	2	yes	—
197	#11719921	C₂³	1	2	yes	—
198	#11719921	C₂³	1	2	yes	—
199	#11719921	C₂³	1	2	yes	—
200	#11719921	C₂³	1	1	yes	characteristic
201	#11719921	C₂³	1	2	yes	—
202	#11719921	C₂³	1	2	yes	—
203	#11719921	C₂³	1	1	yes	characteristic
204	#11719921	C₂³	1	2	yes	—
205	#11719921	C₂³	1	1	yes	characteristic
206	#11719921	C₂³	1	2	yes	—
207	#11719921	C₂³	1	1	yes	characteristic
208	#11719921	C₂³	1	1	yes	characteristic
209	#11719921	C₂³	1	2	yes	—
210	#11719921	C₂³	1	2	yes	—
211	#11719921	C₂³	1	1	yes	characteristic
212	#11719921	C₂³	1	1	yes	characteristic
213	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
214	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
215	C₄⋊Dic₅₃	C₂³	1	2	yes	—
216	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
217	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
218	C₄⋊Dic₅₃	C₂³	1	2	yes	—
219	C₄⋊Dic₅₃	C₂³	1	2	yes	—
220	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
221	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
222	#11719919	C₂³	1	2	yes	—
223	#11719919	C₂³	1	2	yes	—
224	#11719919	C₂³	1	2	yes	—
225	#11719919	C₂³	1	2	yes	—
226	#11719919	C₂³	1	2	yes	—
227	#11719919	C₂³	1	2	yes	—
228	#11719919	C₂³	1	2	yes	—
229	#11719919	C₂³	1	2	yes	—
230	C₄×Dic₅₃	C₂³	1	2	yes	—
231	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
232	C₂³×C₅₃	C₂⁴	1	2	yes	—
233	C₂²×D₅₃	—	2	2	no	—
234	C₂²×D₅₃	—	2	2	no	—
235	C₂²×D₅₃	—	2	4	no	—
236	C₂²×D₅₃	—	4	1	no	—
237	C₂²×D₅₃	—	4	2	no	—
238	C₂²×D₅₃	—	4	2	no	—
239	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
240	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
241	C₂²×D₅₃	C₂⁴	1	2	yes	—
242	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
243	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
244	Q₈×C₅₃	—	2	2	no	—
245	Q₈×C₅₃	—	4	2	no	—
246	D₄×C₅₃	—	2	2	no	—
247	D₄×C₅₃	—	4	2	no	—
248	C₂×C₄×C₅₃	—	2	2	no	—
249	C₂×C₄×C₅₃	—	4	2	no	—
250	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
251	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
253	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
254	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
255	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
256	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
257	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
258	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
259	C₂²⋊D₅₃	—	2	4	no	—
260	C₂²⋊D₅₃	—	4	1	no	—
261	C₂²⋊D₅₃	—	4	2	no	—
262	C₂²⋊D₅₃	—	4	1	no	—
263	C₂²⋊D₅₃	—	4	2	no	—
264	C₂²⋊D₅₃	—	4	2	no	—
265	C₂²⋊D₅₃	—	4	2	no	—
266	C₂×Dic₅₃	—	4	1	no	—
267	C₂×Dic₅₃	—	4	2	no	—
268	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
269	C₂×Dic₅₃	C₂⁴	1	2	yes	—
270	C₂×Dic₅₃	C₂⁴	1	2	yes	—
271	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
272	C₂×Dic₅₃	C₂⁴	1	2	yes	—
273	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
274	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
275	D₂₁₂	—	2	2	no	—
276	D₂₁₂	—	2	4	no	—
277	D₂₁₂	—	2	2	no	—
278	D₂₁₂	—	2	2	no	—
279	D₂₁₂	—	2	2	no	—
280	D₂₁₂	—	2	2	no	—
281	D₂₁₂	—	4	1	no	—
282	D₂₁₂	—	4	2	no	—
283	D₂₁₂	—	4	2	no	—
284	C₄×D₅₃	—	2	4	no	—
285	C₄×D₅₃	—	2	2	no	—
286	C₄×D₅₃	—	2	2	no	—
287	C₄×D₅₃	—	2	2	no	—
288	C₄×D₅₃	—	2	2	no	—
289	C₄×D₅₃	—	4	1	no	—
290	C₄×D₅₃	—	4	1	no	—
291	C₄×D₅₃	—	4	2	no	—
292	C₄×D₅₃	—	4	2	no	—
293	C₄×D₅₃	—	4	2	no	—
294	C₄×D₅₃	—	4	2	no	—
295	C₄×D₅₃	—	4	2	no	—
296	C₄×D₅₃	—	4	2	no	—
297	Dic₁₀₆	—	2	2	no	—
298	Dic₁₀₆	—	4	1	no	—
299	Dic₁₀₆	—	4	2	no	—
300	Dic₁₀₆	—	4	2	no	—
301	C₂²×C₅₃	—	2	1	no	—
302	C₂²×C₅₃	—	2	2	no	—
303	C₂²×C₅₃	—	2	2	no	—
304	C₂²×C₅₃	—	2	1	no	—
305	C₂²×C₅₃	—	2	1	no	—
306	C₂²×C₅₃	—	2	2	no	—
307	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
308	C₂×D₅₃	—	2	1	no	—
309	C₂×D₅₃	—	2	1	no	—
310	C₂×D₅₃	—	2	1	no	—
311	C₂×D₅₃	—	2	1	no	—
312	C₂×D₅₃	—	2	1	no	—
313	C₂×D₅₃	—	2	1	no	—
314	C₂×D₅₃	—	2	1	no	—
315	C₂×D₅₃	—	2	1	no	—
316	C₂×D₅₃	—	2	2	no	—
317	C₂×D₅₃	—	2	2	no	—
318	C₂×D₅₃	—	2	1	no	—
319	C₂×D₅₃	—	2	1	no	—
320	C₂×D₅₃	—	2	2	no	—
321	C₂×D₅₃	—	4	4	no	—
322	C₂×D₅₃	—	8	4	no	—
323	C₂×D₅₃	H(5,𝔽₂)	1	2	yes	—
324	C₂×D₅₃	2₋¹⁺⁴	1	2	yes	—
325	C₄×C₅₃	—	2	1	no	—
326	C₄×C₅₃	—	2	1	no	—
327	C₄×C₅₃	—	2	2	no	—
328	C₄×C₅₃	—	2	2	no	—
329	C₄×C₅₃	—	2	2	no	—
330	C₄×C₅₃	—	2	1	no	—
331	C₄×C₅₃	—	2	1	no	—
332	C₄×C₅₃	—	2	1	no	—
333	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
334	Dic₅₃	—	2	2	no	—
335	Dic₅₃	—	2	1	no	—
336	Dic₅₃	—	2	2	no	—
337	Dic₅₃	—	2	1	no	—
338	Dic₅₃	—	2	1	no	—
339	Dic₅₃	—	2	2	no	—
340	Dic₅₃	—	2	1	no	—
341	#3536	—	53	1	no	maximal, Sylow
342	C₂×C₅₃	—	4	1	no	—
343	C₂×C₅₃	—	4	2	no	—
344	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
345	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
346	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
347	D₅₃	—	2	2	no	—
348	D₅₃	—	2	2	no	—
349	D₅₃	—	4	1	no	—
350	D₅₃	—	4	1	no	—
351	D₅₃	—	4	2	no	—
352	#562	—	53	2	no	—
353	#562	—	53	2	no	—
354	#554	—	53	1	no	—
355	#554	—	53	1	no	—
356	#548	—	53	1	no	—
357	#548	—	53	1	no	—
358	#547	—	53	1	no	—
359	#547	—	53	1	no	—
360	#546	—	53	2	no	—
361	#546	—	53	1	no	—
362	#546	—	53	2	no	—
363	#546	—	53	1	no	—
364	#543	—	53	2	no	—
365	#542	—	53	1	no	—
366	#542	D₅₃	1	1	yes	characteristic, complemented
367	#539	—	53	2	no	—
368	#538	—	53	2	no	—
369	#537	—	53	2	no	—
370	#537	—	53	1	no	—
371	#537	—	53	1	no	—
372	#525	—	53	1	no	—
373	C₂×C₂²⋊Q₈	—	53	1	no	—
374	C₂×#121	—	53	1	no	—
375	C₅₃	#3536	1	1	yes	characteristic, complemented, Sylow
376	C₂×#41	—	53	2	no	—
377	C₂×#41	—	53	1	no	—
378	C₂×#41	—	53	1	no	—
379	C₂×#41	—	53	1	no	—
380	C₂×#41	—	53	2	no	—
381	C₂²×Q₈	—	53	1	no	—
382	C₂²×D₄	—	53	1	no	—
383	C₂²×D₄	—	53	2	no	—
384	C₂³×C₄	—	53	1	no	—
385	C₄⋊Q₈	—	53	1	no	—
386	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
387	#126	—	53	2	no	—
388	#126	C₂×D₅₃	1	2	yes	complemented
389	#125	—	53	2	no	—
390	#125	—	53	2	no	—
391	#124	—	53	2	no	—
392	#124	—	53	2	no	—
393	#124	—	53	2	no	—
394	#124	—	53	1	no	—
395	#124	C₂×D₅₃	1	1	yes	characteristic, complemented
396	#124	C₂×D₅₃	1	2	yes	complemented
397	#123	—	53	2	no	—
398	#123	—	53	2	no	—
399	#123	—	53	1	no	—
400	#123	—	53	1	no	—
401	#123	—	53	2	no	—
402	#123	—	53	2	no	—
403	#123	—	53	2	no	—
404	#123	—	53	2	no	—
405	#123	—	53	2	no	—
406	#123	C₂×D₅₃	1	2	yes	complemented
407	C₂²⋊Q₈	—	53	1	no	—
408	C₂²⋊Q₈	—	53	1	no	—
409	C₂²⋊Q₈	—	53	2	no	—
410	C₂²⋊Q₈	—	53	2	no	—
411	C₂²⋊Q₈	—	53	2	no	—
412	C₂²⋊Q₈	—	53	1	no	—
413	C₂²⋊Q₈	—	53	2	no	—
414	C₂²⋊Q₈	—	53	2	no	—
415	C₂²⋊Q₈	—	53	1	no	—
416	C₂²⋊Q₈	—	53	1	no	—
417	C₂²⋊Q₈	—	53	1	no	—
418	C₂²⋊Q₈	—	53	2	no	—
419	C₂²⋊Q₈	—	53	2	no	—
420	C₂²⋊Q₈	—	53	1	no	—
421	C₂²⋊Q₈	—	53	2	no	—
422	C₂²⋊Q₈	—	53	2	no	—
423	C₂²⋊Q₈	—	53	2	no	—
424	C₂²⋊Q₈	—	53	2	no	—
425	C₂²⋊Q₈	—	53	2	no	—
426	C₂²⋊Q₈	—	53	2	no	—
427	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
428	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
429	#121	—	53	1	no	—
430	#121	—	53	2	no	—
431	#121	—	53	1	no	—
432	#121	—	53	2	no	—
433	#121	—	53	2	no	—
434	#121	—	53	1	no	—
435	#121	—	53	2	no	—
436	#121	—	53	1	no	—
437	#121	—	53	1	no	—
438	#121	—	53	1	no	—
439	#121	—	53	2	no	—
440	#121	—	53	1	no	—
441	#121	—	53	2	no	—
442	#121	—	53	2	no	—
443	#121	—	53	2	no	—
444	#121	—	53	2	no	—
445	#121	—	53	2	no	—
446	#121	—	53	2	no	—
447	#121	—	53	2	no	—
448	#121	C₂×D₅₃	1	1	yes	characteristic
449	#120	—	53	1	no	—
450	#120	—	53	1	no	—
451	#120	—	53	2	no	—
452	#120	—	53	2	no	—
453	C₄×Q₈	—	53	1	no	—
454	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
455	C₄×D₄	—	53	2	no	—
456	C₄×D₄	—	53	2	no	—
457	C₄×D₄	—	53	1	no	—
458	C₄×D₄	—	53	1	no	—
459	C₄×D₄	—	53	1	no	—
460	C₄×D₄	—	53	1	no	—
461	C₄×D₄	—	53	2	no	—
462	C₄×D₄	—	53	2	no	—
463	C₄×D₄	—	53	1	no	—
464	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
465	#117	C₂×D₅₃	1	1	yes	characteristic, complemented
466	C₂×C₄⋊C₄	—	53	1	no	—
467	C₂×C₄⋊C₄	—	53	2	no	—
468	C₂×C₄⋊C₄	—	53	1	no	—
469	C₂×C₄⋊C₄	—	53	1	no	—
470	C₂×C₄⋊C₄	—	53	1	no	—
471	C₂×C₂²⋊C₄	—	53	2	no	—
472	C₂×C₂²⋊C₄	—	53	2	no	—
473	C₂×C₂²⋊C₄	—	53	2	no	—
474	C₂×C₂²⋊C₄	—	53	2	no	—
475	C₂⁴	—	53	1	no	—
476	C₂⁴	—	53	2	no	—
477	#41	—	212	2	no	—
478	#41	—	212	4	no	—
479	#41	—	212	2	no	—
480	#41	—	212	4	no	—
481	#41	—	212	2	no	—
482	C₂×Q₈	—	53	1	no	—
483	C₂×Q₈	—	53	1	no	—
484	C₂×Q₈	—	53	2	no	—
485	C₂×Q₈	—	53	2	no	—
486	C₂×Q₈	—	212	2	no	—
487	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
488	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
489	C₂×D₄	—	53	2	no	—
490	C₂×D₄	—	53	1	no	—
491	C₂×D₄	—	53	2	no	—
492	C₂×D₄	—	53	1	no	—
493	C₂×D₄	—	53	2	no	—
494	C₂×D₄	—	53	2	no	—
495	C₂×D₄	—	53	2	no	—
496	C₂×D₄	—	53	1	no	—
497	C₂×D₄	—	53	1	no	—
498	C₂×D₄	—	53	1	no	—
499	C₂×D₄	—	53	1	no	—
500	C₂×D₄	—	53	2	no	—
501	C₂×D₄	—	53	1	no	—
502	C₂×D₄	—	53	2	no	—
503	C₂×D₄	—	53	2	no	—
504	C₂×D₄	—	53	1	no	—
505	C₂×D₄	—	212	2	no	—
506	C₂×D₄	—	212	4	no	—
507	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
508	C₂×D₄	C₂²×D₅₃	1	2	yes	—
509	C₂²×C₄	—	53	2	no	—
510	C₂²×C₄	—	53	1	no	—
511	C₂²×C₄	—	53	2	no	—
512	C₂²×C₄	—	53	1	no	—
513	C₂²×C₄	—	53	1	no	—
514	C₂²×C₄	—	53	1	no	—
515	C₂²×C₄	—	53	1	no	—
516	C₂²×C₄	—	53	1	no	—
517	C₂²×C₄	—	53	2	no	—
518	C₂²×C₄	—	53	2	no	—
519	C₂²×C₄	—	53	2	no	—
520	C₂²×C₄	—	53	2	no	—
521	C₂²×C₄	—	53	1	no	—
522	C₂²×C₄	—	53	2	no	—
523	C₂²×C₄	—	53	2	no	—
524	C₂²×C₄	—	212	2	no	—
525	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
526	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
527	C₄⋊C₄	—	53	1	no	—
528	C₄⋊C₄	—	53	1	no	—
529	C₄⋊C₄	—	53	2	no	—
530	C₄⋊C₄	—	53	2	no	—
531	C₄⋊C₄	—	53	2	no	—
532	C₄⋊C₄	—	53	2	no	—
533	C₄⋊C₄	—	53	2	no	—
534	C₄⋊C₄	—	53	2	no	—
535	C₄⋊C₄	—	53	2	no	—
536	C₄⋊C₄	—	53	2	no	—
537	C₄⋊C₄	—	53	2	no	—
538	C₄⋊C₄	—	53	1	no	—
539	C₄⋊C₄	—	53	1	no	—
540	C₄⋊C₄	—	53	1	no	—
541	C₄⋊C₄	—	53	1	no	—
542	C₄⋊C₄	—	53	2	no	—
543	C₄⋊C₄	—	53	2	no	—
544	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
545	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
546	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
547	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
548	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
549	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
550	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
551	C₂²⋊C₄	—	53	2	no	—
552	C₂²⋊C₄	—	53	2	no	—
553	C₂²⋊C₄	—	53	2	no	—
554	C₂²⋊C₄	—	53	2	no	—
555	C₂²⋊C₄	—	53	1	no	—
556	C₂²⋊C₄	—	53	1	no	—
557	C₂²⋊C₄	—	53	2	no	—
558	C₂²⋊C₄	—	53	2	no	—
559	C₂²⋊C₄	—	53	2	no	—
560	C₂²⋊C₄	—	53	2	no	—
561	C₂²⋊C₄	—	53	2	no	—
562	C₂²⋊C₄	—	53	1	no	—
563	C₂²⋊C₄	—	53	1	no	—
564	C₂²⋊C₄	—	53	2	no	—
565	C₂²⋊C₄	—	53	2	no	—
566	C₂²⋊C₄	—	53	1	no	—
567	C₂²⋊C₄	—	53	1	no	—
568	C₂²⋊C₄	—	53	2	no	—
569	C₂²⋊C₄	—	53	2	no	—
570	C₂²⋊C₄	—	53	1	no	—
571	C₂²⋊C₄	—	53	1	no	—
572	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
573	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
574	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
575	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
576	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
577	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
578	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
579	C₄²	—	53	2	no	—
580	C₄²	C₂²×D₅₃	1	1	yes	characteristic
581	C₄²	C₂²×D₅₃	1	1	yes	characteristic
582	C₄²	C₂²×D₅₃	1	2	yes	—
583	C₂³	—	53	1	no	—
584	C₂³	—	53	1	no	—
585	C₂³	—	53	2	no	—
586	C₂³	—	53	1	no	—
587	C₂³	—	53	1	no	—
588	C₂³	—	106	4	no	—
589	C₂³	—	106	2	no	—
590	C₂³	—	106	2	no	—
591	C₂³	—	212	2	no	—
592	C₂³	—	212	2	no	—
593	C₂³	—	212	1	no	—
594	C₂³	C₂³×D₅₃	1	2	yes	—
595	C₂³	C₂³×D₅₃	1	1	yes	characteristic
596	Q₈	—	2	2	no	—
597	Q₈	—	4	2	no	—
598	Q₈	—	106	2	no	—
599	Q₈	—	212	2	no	—
600	Q₈	—	212	2	no	—
601	Q₈	—	212	1	no	—
602	D₄	—	2	2	no	—
603	D₄	—	4	2	no	—
604	D₄	—	106	2	no	—
605	D₄	—	106	2	no	—
606	D₄	—	106	4	no	—
607	D₄	—	106	2	no	—
608	D₄	—	106	2	no	—
609	D₄	—	106	4	no	—
610	D₄	—	106	2	no	—
611	D₄	—	212	2	no	—
612	D₄	—	212	1	no	—
613	D₄	—	212	2	no	—
614	D₄	—	212	2	no	—
615	D₄	—	212	2	no	—
616	D₄	—	212	2	no	—
617	D₄	—	212	1	no	—
618	D₄	—	212	1	no	—
619	D₄	—	212	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	4	2	no	—
622	C₂×C₄	—	53	2	no	—
623	C₂×C₄	—	53	2	no	—
624	C₂×C₄	—	53	1	no	—
625	C₂×C₄	—	53	2	no	—
626	C₂×C₄	—	53	1	no	—
627	C₂×C₄	—	53	1	no	—
628	C₂×C₄	—	53	1	no	—
629	C₂×C₄	—	106	4	no	—
630	C₂×C₄	—	106	2	no	—
631	C₂×C₄	—	106	2	no	—
632	C₂×C₄	—	106	2	no	—
633	C₂×C₄	—	106	2	no	—
634	C₂×C₄	—	212	1	no	—
635	C₂×C₄	—	212	1	no	—
636	C₂×C₄	—	212	2	no	—
637	C₂×C₄	—	212	2	no	—
638	C₂×C₄	—	212	2	no	—
639	C₂×C₄	—	212	2	no	—
640	C₂×C₄	—	212	2	no	—
641	C₂×C₄	—	212	2	no	—
642	C₂×C₄	—	212	1	no	—
643	C₂×C₄	—	212	2	no	—
644	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
645	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
646	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
647	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
648	C₂×C₄	C₂³×D₅₃	1	2	yes	—
649	C₂×C₄	C₂³×D₅₃	1	2	yes	—
650	C₂×C₄	C₂³×D₅₃	1	2	yes	—
651	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
652	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
653	C₂²	—	2	2	no	—
654	C₂²	—	2	1	no	—
655	C₂²	—	2	1	no	—
656	C₂²	—	2	2	no	—
657	C₂²	—	2	2	no	—
658	C₂²	—	2	1	no	—
659	C₂²	—	53	2	no	—
660	C₂²	—	53	2	no	—
661	C₂²	—	106	1	no	—
662	C₂²	—	106	1	no	—
663	C₂²	—	106	1	no	—
664	C₂²	—	106	1	no	—
665	C₂²	—	106	1	no	—
666	C₂²	—	106	1	no	—
667	C₂²	—	106	2	no	—
668	C₂²	—	106	1	no	—
669	C₂²	—	106	1	no	—
670	C₂²	—	106	2	no	—
671	C₂²	—	106	1	no	—
672	C₂²	—	106	1	no	—
673	C₂²	—	106	2	no	—
674	C₂²	—	424	4	no	—
675	C₂²	—	212	4	no	—
676	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
677	C₄	—	2	1	no	—
678	C₄	—	2	2	no	—
679	C₄	—	2	2	no	—
680	C₄	—	2	1	no	—
681	C₄	—	2	1	no	—
682	C₄	—	2	2	no	—
683	C₄	—	2	1	no	—
684	C₄	—	2	1	no	—
685	C₄	—	106	2	no	—
686	C₄	—	106	2	no	—
687	C₄	—	106	1	no	—
688	C₄	—	106	1	no	—
689	C₄	—	106	1	no	—
690	C₄	—	106	1	no	—
691	C₄	—	106	2	no	—
692	C₄	#13133263	1	2	yes	—
693	C₂	—	4	2	no	—
694	C₂	—	4	1	no	—
695	C₂	—	106	2	no	—
696	C₂	—	106	2	no	—
697	C₂	—	212	2	no	—
698	C₂	—	212	1	no	—
699	C₂	—	212	1	no	—
700	C₂	H(5,𝔽₂)×D₅₃	1	1	yes	characteristic
701	C₂	C₂×#13133263	1	1	yes	characteristic
702	C₂	2₋¹⁺⁴×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#542	C₂
hypercenter	#542	D₅₃
hypocenter	C₅₃	#3536
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #542

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #16811370	C₂	semidirect
2. #16811367	C₂	semidirect
3. #16811334	C₂	semidirect
4. #16811333	C₂	semidirect
5. #16811285	C₂	semidirect
6. #16811281	C₂	semidirect
7. #16811278	C₂	semidirect
8. #16811276	C₂	semidirect
9. #16811268	C₂	semidirect
10. #16811266	C₂	semidirect
11. #16811266	C₂	semidirect
12. #16811265	C₂	semidirect
13. #16811248	C₂	semidirect
14. #16811244	C₂	semidirect
15. #16811214	C₂	semidirect
16. #16811212	C₂	semidirect
17. #16811203	C₂	semidirect
18. #16811203	C₂	semidirect
19. #16811202	C₂	semidirect
20. #16811135	C₂	semidirect
21. C₂²⋊Q₈×D₅₃	C₂	semidirect
22. #121×D₅₃	C₂	semidirect
23. C₅₃×#542	C₂	semidirect
24. C₂×#11719949	C₂²	semidirect
25. C₂×#11719949	C₂²	semidirect
26. C₂×#11719945	C₂²	semidirect
27. C₂×#11719945	C₂²	semidirect
28. C₂×#11719947	C₂²	semidirect
29. C₂×Q₈×D₅₃	C₂²	semidirect
33. #13133217	C₂²	semidirect
34. #13133212	C₂²	semidirect
35. #13133209	C₂²	semidirect
36. #13133208	C₂²	semidirect
37. #13133205	C₂²	semidirect
38. #13133205	C₂²	semidirect
39. #13133203	C₂²	semidirect
40. #13133203	C₂²	semidirect
41. #13133202	C₂²	semidirect
42. #13133202	C₂²	semidirect
43. #13133201	C₂²	semidirect
44. #13133201	C₂²	semidirect
45. #13133201	C₂²	semidirect
46. #13133199	C₂²	semidirect
47. #13133199	C₂²	semidirect
48. #13133199	C₂²	semidirect
49. #13133199	C₂²	semidirect
52. #13133197	C₂²	semidirect
53. #13133197	C₂²	semidirect
54. #13133197	C₂²	semidirect
55. #13133196	C₂²	semidirect
56. #13133196	C₂²	semidirect
59. #13133195	C₂²	semidirect
60. #13133195	C₂²	semidirect
61. #13133195	C₂²	semidirect
62. #13133194	C₂²	semidirect
63. #13133194	C₂²	semidirect
64. #13133194	C₂²	semidirect
67. #13133193	C₂²	semidirect
68. #13133193	C₂²	semidirect
69. #13133193	C₂²	semidirect
72. #13133192	C₂²	semidirect
73. #13133192	C₂²	semidirect
74. #13133192	C₂²	semidirect
75. #13133192	C₂²	semidirect
77. #13133191	C₂²	semidirect
79. #13133190	C₂²	semidirect
80. #13133190	C₂²	semidirect
81. #13133190	C₂²	semidirect
82. #13133190	C₂²	semidirect
83. #13133190	C₂²	semidirect
85. #13133189	C₂²	semidirect
87. #13133188	C₂²	semidirect
88. #13133188	C₂²	semidirect
93. C₄×Dic₁₀₆	C₂²	semidirect
97. C₄×D₂₁₂	C₂²	semidirect
98. C₄×D₂₁₂	C₂²	semidirect
99. C₄×D₂₁₂	C₂²	semidirect
100. C₄×D₂₁₂	C₂²	semidirect
102. #13133177	C₂²	semidirect
104. C₄⋊C₄×D₅₃	C₂²	semidirect
105. C₄⋊C₄×D₅₃	C₂²	semidirect
106. C₄⋊C₄×D₅₃	C₂²	semidirect
107. C₄⋊C₄×D₅₃	C₂²	semidirect
112. C₄⋊Q₈×C₅₃	C₂²	semidirect
113. #126×C₅₃	C₂²	semidirect
114. #124×C₅₃	C₂²	semidirect
115. #124×C₅₃	C₂²	semidirect
116. #123×C₅₃	C₂²	semidirect
117. C₂²⋊Q₈×C₅₃	C₂²	semidirect
118. C₂²⋊Q₈×C₅₃	C₂²	semidirect
120. C₄×Q₈×C₅₃	C₂²	semidirect
121. C₄×D₄×C₅₃	C₂²	semidirect
122. #117×C₅₃	C₂²	semidirect
366. #542	D₅₃	semidirect
375. C₅₃	#3536	semidirect
386. C₄⋊Q₈	C₂×D₅₃	semidirect
388. #126	C₂×D₅₃	semidirect
395. #124	C₂×D₅₃	semidirect
396. #124	C₂×D₅₃	semidirect
406. #123	C₂×D₅₃	semidirect
427. C₂²⋊Q₈	C₂×D₅₃	semidirect
428. C₂²⋊Q₈	C₂×D₅₃	semidirect
454. C₄×Q₈	C₂×D₅₃	semidirect
464. C₄×D₄	C₂×D₅₃	semidirect
465. #117	C₂×D₅₃	semidirect