Subgroups — #20288415

Order 6784.

1124 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811333	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811331	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811329	C₂	1	2	yes	maximal, complemented
4	#16811326	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811324	C₂	1	2	yes	maximal, complemented
6	#16811317	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811315	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811306	C₂	1	2	yes	maximal, complemented
9	#16811277	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811274	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811271	C₂	1	2	yes	maximal, complemented
12	#16811253	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811221	C₂	1	2	yes	maximal, complemented
14	#16811214	C₂	1	2	yes	maximal, complemented
15	#16811203	C₂	1	1	yes	maximal, characteristic, complemented
16	#16811136	C₂	1	1	yes	maximal, characteristic, complemented
17	#16811135	C₂	1	2	yes	maximal, complemented
18	#16811132	C₂	1	1	yes	maximal, characteristic, complemented
19	#16811131	C₂	1	2	yes	maximal, complemented
20	#16811105	C₂	1	1	yes	maximal, characteristic, complemented
21	#16811103	C₂	1	1	yes	maximal, characteristic, complemented
22	C₂×C₄⋊C₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
23	C₅₃×#552	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11719949	C₂²	1	1	yes	characteristic
25	C₂×#11719945	C₂²	1	1	yes	characteristic
26	C₂×#11719945	C₂²	1	2	yes	—
27	C₂×#11719947	C₂²	1	2	yes	—
28	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
30	#13133217	C₂²	1	1	yes	characteristic, complemented
31	#13133216	C₂²	1	2	yes	complemented
32	#13133216	C₂²	1	1	yes	characteristic, complemented
33	#13133212	C₂²	1	2	yes	complemented
34	#13133212	C₂²	1	2	yes	complemented
35	#13133211	C₂²	1	2	yes	complemented
36	#13133209	C₂²	1	2	yes	complemented
37	#13133208	C₂²	1	1	yes	characteristic, complemented
38	#13133208	C₂²	1	2	yes	complemented
39	#13133207	C₂²	1	2	yes	complemented
40	#13133207	C₂²	1	2	yes	complemented
41	#13133202	C₂²	1	1	yes	characteristic
42	#13133202	C₂²	1	2	yes	—
43	#13133202	C₂²	1	2	yes	complemented
44	#13133202	C₂²	1	2	yes	complemented
45	#13133201	C₂²	1	2	yes	—
46	#13133201	C₂²	1	2	yes	—
47	#13133201	C₂²	1	2	yes	complemented
48	#13133200	C₂²	1	2	yes	—
49	#13133199	C₂²	1	1	yes	characteristic
50	#13133199	C₂²	1	2	yes	complemented
51	#13133197	C₂²	1	2	yes	—
52	#13133197	C₂²	1	2	yes	—
53	#13133197	C₂²	1	2	yes	complemented
54	#13133195	C₂²	1	1	yes	characteristic
55	#13133195	C₂²	1	2	yes	—
56	#13133195	C₂²	1	2	yes	—
57	#13133195	C₂²	1	2	yes	complemented
58	#13133195	C₂²	1	2	yes	complemented
59	#13133195	C₂²	1	2	yes	complemented
60	#13133194	C₂²	1	1	yes	characteristic
61	#13133194	C₂²	1	2	yes	complemented
62	#13133193	C₂²	1	1	yes	characteristic
63	#13133193	C₂²	1	1	yes	characteristic, complemented
64	#13133192	C₂²	1	1	yes	characteristic
65	#13133192	C₂²	1	2	yes	—
66	#13133192	C₂²	1	2	yes	—
67	#13133192	C₂²	1	1	yes	characteristic, complemented
68	#13133191	C₂²	1	1	yes	characteristic
69	#13133191	C₂²	1	1	yes	characteristic, complemented
70	#13133190	C₂²	1	2	yes	—
71	#13133190	C₂²	1	1	yes	characteristic
72	#13133190	C₂²	1	2	yes	—
73	#13133190	C₂²	1	2	yes	complemented
74	#13133189	C₂²	1	1	yes	characteristic
75	#13133187	C₂²	1	2	yes	—
76	#13133187	C₂²	1	1	yes	characteristic
77	#13133185	C₂²	1	1	yes	characteristic
78	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
79	C₄×Dic₁₀₆	C₂²	1	2	yes	complemented
80	C₄×D₂₁₂	C₂²	1	2	yes	—
81	C₄×D₂₁₂	C₂²	1	2	yes	—
82	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
83	C₄×D₂₁₂	C₂²	1	2	yes	—
84	C₄×D₂₁₂	C₂²	1	2	yes	—
85	C₄×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
86	#13133179	C₂²	1	2	yes	—
87	#13133179	C₂²	1	2	yes	—
88	#13133179	C₂²	1	2	yes	—
89	#13133179	C₂²	1	1	yes	characteristic, complemented
90	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
91	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
92	#13133177	C₂²	1	2	yes	—
93	#13133177	C₂²	1	2	yes	—
94	#13133177	C₂²	1	2	yes	—
95	#13133177	C₂²	1	2	yes	complemented
96	#13133175	C₂²	1	1	yes	characteristic
97	#13133175	C₂²	1	1	yes	characteristic, complemented
98	#13133174	C₂²	1	1	yes	characteristic
99	#13133174	C₂²	1	2	yes	complemented
100	#13133173	C₂²	1	2	yes	—
101	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
102	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
103	C₂×#11719919	C₂²	1	2	yes	—
104	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
105	C₄⋊C₄×D₅₃	C₂²	1	2	yes	—
106	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
107	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
108	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic
109	C₄⋊C₄×D₅₃	C₂²	1	2	yes	—
110	C₄⋊C₄×D₅₃	C₂²	1	2	yes	complemented
111	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
112	#126×C₅₃	C₂²	1	2	yes	complemented
113	#125×C₅₃	C₂²	1	1	yes	characteristic, complemented
114	#125×C₅₃	C₂²	1	2	yes	complemented
115	#123×C₅₃	C₂²	1	2	yes	—
116	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
117	C₂²⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
118	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
119	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
120	#117×C₅₃	C₂²	1	2	yes	—
121	#117×C₅₃	C₂²	1	1	yes	characteristic, complemented
122	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
123	C₂³×D₅₃	C₂³	1	1	yes	characteristic
124	C₂×Q₈×C₅₃	C₂³	1	1	yes	characteristic
125	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
127	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
129	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
130	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
131	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
132	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
133	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
134	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
135	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
136	C₂²×Dic₅₃	C₂³	1	2	yes	—
137	C₂²×Dic₅₃	C₂³	1	2	yes	—
138	#11719949	—	2	4	no	—
139	#11719947	—	4	4	no	—
140	#11719945	—	2	4	no	—
141	#11719945	—	4	4	no	—
142	C₂×D₂₁₂	C₂³	1	2	yes	—
143	C₂×D₂₁₂	C₂³	1	2	yes	—
144	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
145	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
146	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
147	C₂×D₂₁₂	C₂³	1	2	yes	—
148	C₂×C₄×D₅₃	—	2	4	no	—
149	C₂×C₄×D₅₃	—	4	4	no	—
150	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
151	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
152	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
153	C₂×C₄×D₅₃	C₂³	1	2	yes	—
154	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
155	C₂×C₄×D₅₃	C₂³	1	2	yes	—
156	C₂×C₄×D₅₃	C₂³	1	2	yes	—
157	C₂×C₄×D₅₃	C₂³	1	2	yes	—
158	C₂×C₄×D₅₃	C₂³	1	2	yes	—
159	C₂×C₄×D₅₃	C₂³	1	2	yes	—
160	C₂×C₄×D₅₃	C₂³	1	2	yes	—
161	C₂×C₄×D₅₃	C₂³	1	2	yes	—
162	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
163	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
164	C₂×Dic₁₀₆	C₂³	1	2	yes	—
165	C₂×Dic₁₀₆	C₂³	1	2	yes	—
166	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
167	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
168	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
169	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
170	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
171	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
172	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
173	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
174	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
175	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
176	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
177	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
178	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
179	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
180	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
181	C₄²×C₅₃	C₂³	1	2	yes	—
182	C₄²×C₅₃	C₂³	1	2	yes	—
183	C₄²×C₅₃	C₂³	1	1	yes	characteristic
184	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
185	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
186	#11719921	C₂³	1	1	yes	characteristic
187	#11719921	C₂³	1	1	yes	characteristic
188	#11719921	C₂³	1	2	yes	—
189	#11719921	C₂³	1	2	yes	—
190	#11719921	C₂³	1	2	yes	—
191	#11719921	C₂³	1	2	yes	—
192	#11719921	C₂³	1	2	yes	—
193	#11719921	C₂³	1	2	yes	—
194	#11719921	C₂³	1	1	yes	characteristic
195	#11719921	C₂³	1	1	yes	characteristic
196	#11719921	C₂³	1	1	yes	characteristic
197	#11719921	C₂³	1	2	yes	—
198	#11719921	C₂³	1	2	yes	—
199	#11719921	C₂³	1	1	yes	characteristic
200	#11719921	C₂³	1	2	yes	—
201	#11719921	C₂³	1	2	yes	—
202	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
203	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
204	C₄⋊Dic₅₃	C₂³	1	2	yes	—
205	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
206	C₄⋊Dic₅₃	C₂³	1	2	yes	—
207	C₄⋊Dic₅₃	C₂³	1	2	yes	—
208	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
209	C₄⋊Dic₅₃	C₂³	1	2	yes	—
210	C₄⋊Dic₅₃	C₂³	1	2	yes	—
211	#11719919	C₂³	1	2	yes	—
212	#11719919	C₂³	1	2	yes	—
213	#11719919	C₂³	1	2	yes	—
214	#11719919	C₂³	1	2	yes	—
215	#11719919	C₂³	1	1	yes	characteristic
216	#11719919	C₂³	1	2	yes	—
217	#11719919	C₂³	1	2	yes	—
218	#11719919	C₂³	1	2	yes	—
219	#11719919	C₂³	1	1	yes	characteristic
220	#11719919	C₂³	1	2	yes	—
221	#11719919	C₂³	1	2	yes	—
222	#11719919	C₂³	1	1	yes	characteristic
223	#11719919	C₂³	1	1	yes	characteristic
224	C₄×Dic₅₃	C₂³	1	2	yes	—
225	C₄×Dic₅₃	C₂³	1	2	yes	—
226	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
227	C₄×Dic₅₃	C₂³	1	2	yes	—
228	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
229	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
230	C₂²×D₅₃	—	2	2	no	—
231	C₂²×D₅₃	—	2	2	no	—
232	C₂²×D₅₃	—	2	2	no	—
233	C₂²×D₅₃	C₂⁴	1	2	yes	—
234	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
235	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
236	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
237	Q₈×C₅₃	—	2	2	no	—
238	D₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	2	4	no	—
240	C₂×C₄×C₅₃	—	2	2	no	—
241	C₂×C₄×C₅₃	—	2	2	no	—
242	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
243	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
244	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
245	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
246	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
247	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
248	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
249	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
250	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
251	C₂²⋊D₅₃	—	2	2	no	—
252	C₂²⋊D₅₃	—	2	4	no	—
253	C₂²⋊D₅₃	—	2	2	no	—
254	C₂²⋊D₅₃	—	2	2	no	—
255	C₂²⋊D₅₃	—	2	2	no	—
256	C₂²⋊D₅₃	—	4	2	no	—
257	C₂×Dic₅₃	—	2	4	no	—
258	C₂×Dic₅₃	—	2	2	no	—
259	C₂×Dic₅₃	—	4	2	no	—
260	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
261	C₂×Dic₅₃	C₂⁴	1	2	yes	—
262	C₂×Dic₅₃	C₂⁴	1	2	yes	—
263	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
264	C₂×Dic₅₃	C₂⁴	1	2	yes	—
265	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
266	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
267	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
268	D₂₁₂	—	2	2	no	—
269	D₂₁₂	—	2	4	no	—
270	D₂₁₂	—	2	4	no	—
271	D₂₁₂	—	2	2	no	—
272	D₂₁₂	—	2	2	no	—
273	D₂₁₂	—	4	2	no	—
274	C₄×D₅₃	—	2	2	no	—
275	C₄×D₅₃	—	2	2	no	—
276	C₄×D₅₃	—	2	2	no	—
277	C₄×D₅₃	—	2	2	no	—
278	C₄×D₅₃	—	2	4	no	—
279	C₄×D₅₃	—	4	2	no	—
280	C₄×D₅₃	—	4	2	no	—
281	C₄×D₅₃	—	4	2	no	—
282	C₄×D₅₃	—	4	2	no	—
283	C₄×D₅₃	—	4	2	no	—
284	C₄×D₅₃	—	4	2	no	—
285	C₄×D₅₃	—	4	2	no	—
286	C₄×D₅₃	#41	1	4	yes	complemented
287	Dic₁₀₆	—	2	2	no	—
288	Dic₁₀₆	—	4	2	no	—
289	Dic₁₀₆	—	4	2	no	—
290	C₂²×C₅₃	—	2	1	no	—
291	C₂²×C₅₃	—	2	1	no	—
292	C₂²×C₅₃	—	2	1	no	—
293	C₂²×C₅₃	—	2	1	no	—
294	C₂²×C₅₃	—	2	1	no	—
295	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
296	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
297	C₂×D₅₃	—	2	8	no	—
298	C₂×D₅₃	—	2	1	no	—
299	C₂×D₅₃	—	2	1	no	—
300	C₂×D₅₃	—	2	1	no	—
301	C₂×D₅₃	—	2	1	no	—
302	C₂×D₅₃	—	2	2	no	—
303	C₂×D₅₃	—	2	1	no	—
304	C₂×D₅₃	—	2	1	no	—
305	C₂×D₅₃	—	2	2	no	—
306	C₂×D₅₃	—	2	2	no	—
307	C₂×D₅₃	—	2	1	no	—
308	C₂×D₅₃	C₂×#41	1	2	yes	—
309	C₂×D₅₃	2₋¹⁺⁴	1	2	yes	—
310	C₄×C₅₃	—	2	1	no	—
311	C₄×C₅₃	—	2	1	no	—
312	C₄×C₅₃	—	2	2	no	—
313	C₄×C₅₃	—	2	1	no	—
314	C₄×C₅₃	—	2	2	no	—
315	C₄×C₅₃	—	2	2	no	—
316	C₄×C₅₃	—	2	2	no	—
317	C₄×C₅₃	—	2	1	no	—
318	C₄×C₅₃	C₂×#41	1	2	yes	—
319	Dic₅₃	—	2	1	no	—
320	Dic₅₃	—	2	1	no	—
321	Dic₅₃	—	2	1	no	—
322	Dic₅₃	—	2	2	no	—
323	Dic₅₃	—	2	2	no	—
324	Dic₅₃	—	2	2	no	—
325	Dic₅₃	—	2	1	no	—
326	Dic₅₃	C₂×#41	1	2	yes	—
327	#3492	—	53	1	no	maximal, Sylow
328	C₂×C₅₃	—	2	2	no	—
329	C₂×C₅₃	—	4	1	no	—
330	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
331	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
332	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
333	D₅₃	—	2	2	no	—
334	D₅₃	—	2	2	no	—
335	D₅₃	—	4	2	no	—
336	D₅₃	—	4	1	no	—
337	#554	—	53	2	no	—
338	#554	—	53	1	no	—
339	#554	—	53	1	no	—
340	#553	—	53	2	no	—
341	#553	—	53	1	no	—
342	#553	—	53	1	no	—
343	#552	—	53	1	no	—
344	#552	—	53	2	no	—
345	#552	D₅₃	1	1	yes	characteristic, complemented
346	#547	—	53	2	no	—
347	#547	—	53	1	no	—
348	#547	—	53	1	no	—
349	#544	—	53	1	no	—
350	#539	—	53	2	no	—
351	#539	—	53	2	no	—
352	#537	—	53	1	no	—
353	#525	—	53	2	no	—
354	#525	—	53	1	no	—
355	#525	—	53	1	no	—
356	#525	—	53	2	no	—
357	#520	—	53	1	no	—
358	#520	—	53	1	no	—
359	C₂²×C₄⋊C₄	—	53	1	no	—
360	C₅₃	#3492	1	1	yes	characteristic, complemented, Sylow
361	C₂×#41	—	53	1	no	—
362	C₂×#41	—	53	2	no	—
363	C₂×#41	—	53	1	no	—
364	C₂×#41	—	53	2	no	—
365	C₂³×C₄	—	53	1	no	—
366	C₂³×C₄	—	53	2	no	—
367	C₄⋊Q₈	—	53	2	no	—
368	C₄⋊Q₈	—	53	1	no	—
369	C₄⋊Q₈	—	53	1	no	—
370	#126	—	53	2	no	—
371	#126	—	53	2	no	—
372	#126	—	53	2	no	—
373	#126	C₂×D₅₃	1	2	yes	complemented
374	#125	—	53	2	no	—
375	#125	—	53	2	no	—
376	#125	—	53	2	no	—
377	#125	—	53	1	no	—
378	#125	—	53	2	no	—
379	#125	C₂×D₅₃	1	2	yes	complemented
380	#125	C₂×D₅₃	1	1	yes	characteristic, complemented
381	#123	—	53	2	no	—
382	#123	—	53	2	no	—
383	#123	—	53	1	no	—
384	#123	—	53	2	no	—
385	#123	—	53	2	no	—
386	#123	—	53	1	no	—
387	#123	—	53	2	no	—
388	#123	—	53	2	no	—
389	#123	—	53	2	no	—
390	#123	—	53	2	no	—
391	#123	C₂×D₅₃	1	2	yes	—
392	C₂²⋊Q₈	—	53	1	no	—
393	C₂²⋊Q₈	—	53	1	no	—
394	C₂²⋊Q₈	—	53	2	no	—
395	C₂²⋊Q₈	—	53	2	no	—
396	C₂²⋊Q₈	—	53	1	no	—
397	C₂²⋊Q₈	—	53	2	no	—
398	C₂²⋊Q₈	—	53	2	no	—
399	C₂²⋊Q₈	—	53	2	no	—
400	C₂²⋊Q₈	—	53	2	no	—
401	C₂²⋊Q₈	—	53	2	no	—
402	C₂²⋊Q₈	—	53	2	no	—
403	C₂²⋊Q₈	—	53	1	no	—
404	C₂²⋊Q₈	—	53	2	no	—
405	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
406	C₂²⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
407	#121	—	53	1	no	—
408	#121	—	53	1	no	—
409	#121	—	53	1	no	—
410	#121	—	53	2	no	—
411	#121	—	53	2	no	—
412	#121	—	53	1	no	—
413	#121	—	53	1	no	—
414	#121	—	53	2	no	—
415	#121	—	53	2	no	—
416	#121	—	53	1	no	—
417	#121	—	53	2	no	—
418	#120	—	53	1	no	—
419	#120	—	53	2	no	—
420	#120	—	53	1	no	—
421	C₄×Q₈	—	53	1	no	—
422	C₄×Q₈	—	53	2	no	—
423	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
424	C₄×D₄	—	53	1	no	—
425	C₄×D₄	—	53	1	no	—
426	C₄×D₄	—	53	2	no	—
427	C₄×D₄	—	53	1	no	—
428	C₄×D₄	—	53	2	no	—
429	C₄×D₄	—	53	2	no	—
430	C₄×D₄	—	53	2	no	—
431	C₄×D₄	—	53	2	no	—
432	C₄×D₄	—	53	2	no	—
433	C₄×D₄	—	53	2	no	—
434	C₄×D₄	—	53	2	no	—
435	C₄×D₄	—	53	2	no	—
436	C₄×D₄	—	53	1	no	—
437	C₄×D₄	—	53	2	no	—
438	C₄×D₄	—	53	1	no	—
439	C₄×D₄	—	53	2	no	—
440	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
441	#117	—	53	1	no	—
442	#117	—	53	1	no	—
443	#117	—	53	2	no	—
444	#117	—	53	1	no	—
445	#117	—	53	2	no	—
446	#117	C₂×D₅₃	1	2	yes	—
447	#117	C₂×D₅₃	1	1	yes	characteristic, complemented
448	C₂×C₄⋊C₄	—	53	2	no	—
449	C₂×C₄⋊C₄	—	53	1	no	—
450	C₂×C₄⋊C₄	—	53	1	no	—
451	C₂×C₄⋊C₄	—	53	1	no	—
452	C₂×C₄⋊C₄	—	53	1	no	—
453	C₂×C₄⋊C₄	—	53	2	no	—
454	C₂×C₄⋊C₄	—	53	2	no	—
455	C₂×C₄⋊C₄	—	53	1	no	—
456	C₂×C₄⋊C₄	—	53	1	no	—
457	C₂×C₄⋊C₄	—	53	2	no	—
458	C₂×C₄⋊C₄	—	53	1	no	—
459	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
460	C₂⁴	—	53	1	no	—
461	#41	—	106	4	no	—
462	#41	—	106	4	no	—
463	#41	—	212	4	no	—
464	#41	—	212	4	no	—
465	C₂×Q₈	—	53	1	no	—
466	C₂×Q₈	—	53	2	no	—
467	C₂×Q₈	—	53	2	no	—
468	C₂×Q₈	C₂²×D₅₃	1	1	yes	characteristic
469	C₂×D₄	—	53	1	no	—
470	C₂×D₄	—	53	1	no	—
471	C₂×D₄	—	53	1	no	—
472	C₂×D₄	—	53	2	no	—
473	C₂×D₄	—	53	1	no	—
474	C₂×D₄	—	53	2	no	—
475	C₂×D₄	—	53	2	no	—
476	C₂×D₄	—	53	1	no	—
477	C₂×D₄	—	53	2	no	—
478	C₂×D₄	—	53	1	no	—
479	C₂×D₄	—	53	1	no	—
480	C₂×D₄	—	53	2	no	—
481	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
482	C₂²×C₄	—	53	1	no	—
483	C₂²×C₄	—	53	2	no	—
484	C₂²×C₄	—	53	1	no	—
485	C₂²×C₄	—	53	2	no	—
486	C₂²×C₄	—	53	1	no	—
487	C₂²×C₄	—	53	1	no	—
488	C₂²×C₄	—	53	1	no	—
489	C₂²×C₄	—	53	2	no	—
490	C₂²×C₄	—	53	1	no	—
491	C₂²×C₄	—	53	2	no	—
492	C₂²×C₄	—	53	2	no	—
493	C₂²×C₄	—	53	2	no	—
494	C₂²×C₄	—	53	2	no	—
495	C₂²×C₄	—	53	2	no	—
496	C₂²×C₄	—	53	2	no	—
497	C₂²×C₄	—	53	2	no	—
498	C₂²×C₄	—	106	4	no	—
499	C₂²×C₄	—	212	4	no	—
500	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
501	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
502	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
503	C₄⋊C₄	—	53	2	no	—
504	C₄⋊C₄	—	53	1	no	—
505	C₄⋊C₄	—	53	2	no	—
506	C₄⋊C₄	—	53	1	no	—
507	C₄⋊C₄	—	53	2	no	—
508	C₄⋊C₄	—	53	2	no	—
509	C₄⋊C₄	—	53	2	no	—
510	C₄⋊C₄	—	53	1	no	—
511	C₄⋊C₄	—	53	1	no	—
512	C₄⋊C₄	—	53	2	no	—
513	C₄⋊C₄	—	53	2	no	—
514	C₄⋊C₄	—	53	2	no	—
515	C₄⋊C₄	—	53	1	no	—
516	C₄⋊C₄	—	53	2	no	—
517	C₄⋊C₄	—	53	2	no	—
518	C₄⋊C₄	—	53	1	no	—
519	C₄⋊C₄	—	53	2	no	—
520	C₄⋊C₄	—	53	2	no	—
521	C₄⋊C₄	—	53	2	no	—
522	C₄⋊C₄	—	53	2	no	—
523	C₄⋊C₄	—	53	1	no	—
524	C₄⋊C₄	—	53	1	no	—
525	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
526	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
527	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
528	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
529	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
530	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
531	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
532	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
533	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
534	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
535	C₂²⋊C₄	—	53	1	no	—
536	C₂²⋊C₄	—	53	1	no	—
537	C₂²⋊C₄	—	53	1	no	—
538	C₂²⋊C₄	—	53	2	no	—
539	C₂²⋊C₄	—	53	1	no	—
540	C₂²⋊C₄	—	53	2	no	—
541	C₂²⋊C₄	—	53	2	no	—
542	C₂²⋊C₄	—	53	2	no	—
543	C₂²⋊C₄	—	53	1	no	—
544	C₂²⋊C₄	—	53	2	no	—
545	C₂²⋊C₄	—	53	1	no	—
546	C₂²⋊C₄	—	53	2	no	—
547	C₂²⋊C₄	—	53	2	no	—
548	C₂²⋊C₄	—	53	2	no	—
549	C₂²⋊C₄	—	53	2	no	—
550	C₂²⋊C₄	—	53	1	no	—
551	C₂²⋊C₄	—	53	1	no	—
552	C₂²⋊C₄	—	53	2	no	—
553	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
554	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
555	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
556	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
557	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
558	C₄²	—	53	2	no	—
559	C₄²	—	53	2	no	—
560	C₄²	—	53	2	no	—
561	C₄²	—	53	1	no	—
562	C₄²	C₂²×D₅₃	1	2	yes	—
563	C₄²	C₂²×D₅₃	1	1	yes	characteristic
564	C₄²	C₂²×D₅₃	1	2	yes	—
565	C₂³	—	53	1	no	—
566	C₂³	—	53	1	no	—
567	C₂³	—	53	1	no	—
568	C₂³	—	53	2	no	—
569	C₂³	—	106	2	no	—
570	C₂³	—	106	2	no	—
571	C₂³	—	106	2	no	—
572	C₂³	C₂³×D₅₃	1	1	yes	characteristic
573	C₂³	C₂³×D₅₃	1	1	yes	characteristic
574	Q₈	—	2	2	no	—
575	Q₈	—	106	2	no	—
576	Q₈	—	212	2	no	—
577	Q₈	—	212	2	no	—
578	D₄	—	2	2	no	—
579	D₄	—	106	4	no	—
580	D₄	—	106	2	no	—
581	D₄	—	106	2	no	—
582	D₄	—	106	4	no	—
583	D₄	—	106	2	no	—
584	D₄	—	106	2	no	—
585	D₄	—	106	2	no	—
586	D₄	—	106	2	no	—
587	D₄	—	106	4	no	—
588	D₄	—	106	2	no	—
589	D₄	—	212	2	no	—
590	D₄	—	212	2	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	53	1	no	—
595	C₂×C₄	—	53	4	no	—
596	C₂×C₄	—	53	1	no	—
597	C₂×C₄	—	53	1	no	—
598	C₂×C₄	—	53	2	no	—
599	C₂×C₄	—	53	1	no	—
600	C₂×C₄	—	53	2	no	—
601	C₂×C₄	—	53	1	no	—
602	C₂×C₄	—	53	2	no	—
603	C₂×C₄	—	106	2	no	—
604	C₂×C₄	—	106	4	no	—
605	C₂×C₄	—	106	2	no	—
606	C₂×C₄	—	106	2	no	—
607	C₂×C₄	—	106	4	no	—
608	C₂×C₄	—	106	2	no	—
609	C₂×C₄	—	106	2	no	—
610	C₂×C₄	—	212	2	no	—
611	C₂×C₄	—	212	2	no	—
612	C₂×C₄	—	212	2	no	—
613	C₂×C₄	—	212	2	no	—
614	C₂×C₄	—	212	2	no	—
615	C₂×C₄	—	212	2	no	—
616	C₂×C₄	—	212	2	no	—
617	C₂×C₄	—	212	2	no	—
618	C₂×C₄	C₂³×D₅₃	1	2	yes	—
619	C₂×C₄	C₂³×D₅₃	1	2	yes	—
620	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
621	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
622	C₂×C₄	C₂³×D₅₃	1	2	yes	—
623	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
624	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
625	C₂×C₄	C₂³×D₅₃	1	2	yes	—
626	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
627	C₂²	—	2	1	no	—
628	C₂²	—	2	1	no	—
629	C₂²	—	2	1	no	—
630	C₂²	—	2	1	no	—
631	C₂²	—	2	1	no	—
632	C₂²	—	53	2	no	—
633	C₂²	—	53	2	no	—
634	C₂²	—	106	8	no	—
635	C₂²	—	106	1	no	—
636	C₂²	—	106	1	no	—
637	C₂²	—	106	1	no	—
638	C₂²	—	106	1	no	—
639	C₂²	—	106	2	no	—
640	C₂²	—	106	1	no	—
641	C₂²	—	106	1	no	—
642	C₂²	—	106	2	no	—
643	C₂²	—	106	1	no	—
644	C₂²	—	106	2	no	—
645	C₂²	#13133263	1	2	yes	—
646	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
647	C₄	—	2	1	no	—
648	C₄	—	2	1	no	—
649	C₄	—	2	2	no	—
650	C₄	—	2	2	no	—
651	C₄	—	2	2	no	—
652	C₄	—	2	1	no	—
653	C₄	—	2	2	no	—
654	C₄	—	2	1	no	—
655	C₄	—	53	2	no	—
656	C₄	—	106	1	no	—
657	C₄	—	106	1	no	—
658	C₄	—	106	2	no	—
659	C₄	—	106	2	no	—
660	C₄	—	106	1	no	—
661	C₄	—	106	2	no	—
662	C₄	—	106	1	no	—
663	C₄	#41×D₅₃	1	2	yes	—
664	C₂	—	2	2	no	—
665	C₂	—	4	1	no	—
666	C₂	—	106	2	no	—
667	C₂	—	106	2	no	—
668	C₂	—	212	2	no	—
669	C₂	—	212	1	no	—
670	C₂	C₂×#41×D₅₃	1	1	yes	characteristic
671	C₂	C₂×#13133263	1	1	yes	characteristic
672	C₂	2₋¹⁺⁴×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Fitting	C₅₃×#552	C₂
hypercenter	#552	D₅₃
hypocenter	C₅₃	#3492
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #552

Complements

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

N	H ≅ G/N	type
1. #16811333	C₂	semidirect
2. #16811331	C₂	semidirect
3. #16811329	C₂	semidirect
4. #16811326	C₂	semidirect
5. #16811324	C₂	semidirect
6. #16811317	C₂	semidirect
7. #16811315	C₂	semidirect
8. #16811306	C₂	semidirect
9. #16811277	C₂	semidirect
10. #16811274	C₂	semidirect
11. #16811271	C₂	semidirect
12. #16811253	C₂	semidirect
13. #16811221	C₂	semidirect
14. #16811214	C₂	semidirect
15. #16811203	C₂	semidirect
16. #16811136	C₂	semidirect
17. #16811135	C₂	semidirect
18. #16811132	C₂	semidirect
19. #16811131	C₂	semidirect
20. #16811105	C₂	semidirect
21. #16811103	C₂	semidirect
22. C₂×C₄⋊C₄×D₅₃	C₂	semidirect
23. C₅₃×#552	C₂	semidirect
30. #13133217	C₂²	semidirect
31. #13133216	C₂²	semidirect
32. #13133216	C₂²	semidirect
33. #13133212	C₂²	semidirect
34. #13133212	C₂²	semidirect
35. #13133211	C₂²	semidirect
36. #13133209	C₂²	semidirect
37. #13133208	C₂²	semidirect
38. #13133208	C₂²	semidirect
39. #13133207	C₂²	semidirect
40. #13133207	C₂²	semidirect
43. #13133202	C₂²	semidirect
44. #13133202	C₂²	semidirect
47. #13133201	C₂²	semidirect
50. #13133199	C₂²	semidirect
53. #13133197	C₂²	semidirect
57. #13133195	C₂²	semidirect
58. #13133195	C₂²	semidirect
59. #13133195	C₂²	semidirect
61. #13133194	C₂²	semidirect
63. #13133193	C₂²	semidirect
67. #13133192	C₂²	semidirect
69. #13133191	C₂²	semidirect
73. #13133190	C₂²	semidirect
78. Q₈×Dic₅₃	C₂²	semidirect
79. C₄×Dic₁₀₆	C₂²	semidirect
85. C₄×D₂₁₂	C₂²	semidirect
89. #13133179	C₂²	semidirect
95. #13133177	C₂²	semidirect
97. #13133175	C₂²	semidirect
99. #13133174	C₂²	semidirect
102. C₂×C₄⋊Dic₅₃	C₂²	semidirect
110. C₄⋊C₄×D₅₃	C₂²	semidirect
111. C₄⋊C₄×D₅₃	C₂²	semidirect
112. #126×C₅₃	C₂²	semidirect
113. #125×C₅₃	C₂²	semidirect
114. #125×C₅₃	C₂²	semidirect
117. C₂²⋊Q₈×C₅₃	C₂²	semidirect
118. C₄×Q₈×C₅₃	C₂²	semidirect
121. #117×C₅₃	C₂²	semidirect
286. C₄×D₅₃	#41	semidirect
345. #552	D₅₃	semidirect
360. C₅₃	#3492	semidirect
373. #126	C₂×D₅₃	semidirect
379. #125	C₂×D₅₃	semidirect
380. #125	C₂×D₅₃	semidirect
406. C₂²⋊Q₈	C₂×D₅₃	semidirect
423. C₄×Q₈	C₂×D₅₃	semidirect
447. #117	C₂×D₅₃	semidirect