Subgroups — #20288417

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	#16811334	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811328	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811327	C₂	1	2	yes	maximal, complemented
4	#16811323	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811320	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811318	C₂	1	2	yes	maximal, complemented
7	#16811313	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811310	C₂	1	2	yes	maximal, complemented
9	#16811308	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811273	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811272	C₂	1	2	yes	maximal, complemented
12	#16811250	C₂	1	1	yes	maximal, characteristic, complemented
13	#16811219	C₂	1	2	yes	maximal, complemented
14	#16811216	C₂	1	2	yes	maximal, complemented
15	#16811202	C₂	1	1	yes	maximal, characteristic, complemented
16	#16811133	C₂	1	1	yes	maximal, characteristic, complemented
17	#16811132	C₂	1	1	yes	maximal, characteristic, complemented
18	#16811131	C₂	1	2	yes	maximal, complemented
19	#16811130	C₂	1	2	yes	maximal, complemented
20	#16811104	C₂	1	1	yes	maximal, characteristic, complemented
21	#16811102	C₂	1	1	yes	maximal, characteristic, complemented
22	C₂×C₄⋊C₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
23	C₅₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11719945	C₂²	1	1	yes	characteristic
25	C₂×#11719947	C₂²	1	1	yes	characteristic
26	C₂×#11719947	C₂²	1	2	yes	—
27	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
29	#13133216	C₂²	1	1	yes	characteristic, complemented
30	#13133215	C₂²	1	2	yes	complemented
31	#13133212	C₂²	1	2	yes	complemented
32	#13133210	C₂²	1	2	yes	complemented
33	#13133210	C₂²	1	2	yes	complemented
34	#13133210	C₂²	1	2	yes	complemented
35	#13133209	C₂²	1	1	yes	characteristic, complemented
36	#13133208	C₂²	1	1	yes	characteristic, complemented
37	#13133208	C₂²	1	2	yes	complemented
38	#13133207	C₂²	1	2	yes	complemented
39	#13133207	C₂²	1	2	yes	complemented
40	#13133207	C₂²	1	2	yes	complemented
41	#13133207	C₂²	1	2	yes	complemented
42	#13133202	C₂²	1	1	yes	characteristic
43	#13133200	C₂²	1	2	yes	—
44	#13133200	C₂²	1	2	yes	complemented
45	#13133199	C₂²	1	1	yes	characteristic
46	#13133199	C₂²	1	2	yes	—
47	#13133199	C₂²	1	2	yes	complemented
48	#13133198	C₂²	1	2	yes	—
49	#13133198	C₂²	1	2	yes	—
50	#13133198	C₂²	1	2	yes	complemented
51	#13133197	C₂²	1	1	yes	characteristic
52	#13133197	C₂²	1	1	yes	characteristic, complemented
53	#13133196	C₂²	1	1	yes	characteristic
54	#13133196	C₂²	1	2	yes	complemented
55	#13133196	C₂²	1	1	yes	characteristic, complemented
56	#13133195	C₂²	1	1	yes	characteristic
57	#13133195	C₂²	1	2	yes	—
58	#13133195	C₂²	1	2	yes	—
59	#13133194	C₂²	1	1	yes	characteristic
60	#13133194	C₂²	1	2	yes	complemented
61	#13133194	C₂²	1	2	yes	complemented
62	#13133193	C₂²	1	2	yes	—
63	#13133193	C₂²	1	2	yes	—
64	#13133193	C₂²	1	2	yes	complemented
65	#13133190	C₂²	1	1	yes	characteristic
66	#13133190	C₂²	1	2	yes	—
67	#13133190	C₂²	1	2	yes	—
68	#13133189	C₂²	1	1	yes	characteristic
69	#13133189	C₂²	1	2	yes	complemented
70	#13133188	C₂²	1	2	yes	—
71	#13133188	C₂²	1	1	yes	characteristic
72	#13133188	C₂²	1	2	yes	—
73	#13133188	C₂²	1	1	yes	characteristic, complemented
74	#13133187	C₂²	1	1	yes	characteristic
75	#13133185	C₂²	1	1	yes	characteristic
76	#13133185	C₂²	1	2	yes	—
77	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
78	#13133182	C₂²	1	1	yes	characteristic, complemented
79	#13133182	C₂²	1	2	yes	complemented
80	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
81	D₄×Dic₅₃	C₂²	1	2	yes	—
82	D₄×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
83	#13133179	C₂²	1	2	yes	—
84	#13133179	C₂²	1	2	yes	—
85	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
86	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
87	C₄×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
88	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
89	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
90	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	—
91	C₄×C₂²⋊D₅₃	C₂²	1	2	yes	complemented
92	#13133177	C₂²	1	2	yes	—
93	#13133177	C₂²	1	2	yes	—
94	#13133177	C₂²	1	2	yes	—
95	#13133176	C₂²	1	2	yes	—
96	#13133176	C₂²	1	1	yes	characteristic, complemented
97	#13133175	C₂²	1	1	yes	characteristic
98	#13133175	C₂²	1	1	yes	characteristic, complemented
99	#13133174	C₂²	1	1	yes	characteristic
100	#13133173	C₂²	1	2	yes	—
101	#13133173	C₂²	1	2	yes	complemented
102	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
103	C₂×#11719919	C₂²	1	2	yes	—
104	C₂×#11719919	C₂²	1	2	yes	complemented
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	1	yes	characteristic
110	C₄⋊C₄×D₅₃	C₂²	1	2	yes	—
111	C₄⋊C₄×D₅₃	C₂²	1	1	yes	characteristic, complemented
112	C₂×#41×C₅₃	C₂²	1	2	yes	—
113	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
114	#123×C₅₃	C₂²	1	2	yes	—
115	#123×C₅₃	C₂²	1	2	yes	complemented
116	C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
117	#121×C₅₃	C₂²	1	1	yes	characteristic
118	#121×C₅₃	C₂²	1	1	yes	characteristic, complemented
119	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
120	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
121	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
122	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
123	C₂³×D₅₃	C₂³	1	1	yes	characteristic
124	#41×C₅₃	—	4	4	no	—
125	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
126	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
127	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
128	C₂×D₄×C₅₃	C₂³	1	2	yes	—
129	C₂×D₄×C₅₃	C₂³	1	2	yes	—
130	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
132	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
133	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
134	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
135	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
136	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
137	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
138	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
139	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
140	C₂×C₂²⋊D₅₃	C₂³	1	1	yes	characteristic
141	C₂×C₂²⋊D₅₃	C₂³	1	2	yes	—
142	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
143	C₂²×Dic₅₃	C₂³	1	2	yes	—
144	C₂²×Dic₅₃	C₂³	1	2	yes	—
145	C₂²×Dic₅₃	C₂³	1	2	yes	—
146	#11719947	—	2	4	no	—
147	#11719947	—	4	4	no	—
148	#11719945	—	2	4	no	—
149	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
150	C₂×D₂₁₂	C₂³	1	1	yes	characteristic
151	C₂×C₄×D₅₃	—	2	4	no	—
152	C₂×C₄×D₅₃	—	4	4	no	—
153	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
154	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
155	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
156	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
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	1	yes	characteristic
170	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
171	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
172	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
173	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
174	C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
175	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
176	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
177	C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
178	C₄²×C₅₃	C₂³	1	1	yes	characteristic
179	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
180	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
181	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
182	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
183	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
184	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
185	C₂²⋊Dic₅₃	C₂³	1	1	yes	characteristic
186	C₂²⋊Dic₅₃	C₂³	1	2	yes	—
187	#11719921	C₂³	1	1	yes	characteristic
188	#11719921	C₂³	1	1	yes	characteristic
189	#11719921	C₂³	1	2	yes	—
190	#11719921	C₂³	1	2	yes	—
191	#11719921	C₂³	1	1	yes	characteristic
192	#11719921	C₂³	1	2	yes	—
193	#11719921	C₂³	1	1	yes	characteristic
194	#11719921	C₂³	1	1	yes	characteristic
195	#11719921	C₂³	1	2	yes	—
196	#11719921	C₂³	1	1	yes	characteristic
197	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
198	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
199	C₄⋊Dic₅₃	C₂³	1	2	yes	—
200	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
201	C₄⋊Dic₅₃	C₂³	1	2	yes	—
202	C₄⋊Dic₅₃	C₂³	1	2	yes	—
203	C₄⋊Dic₅₃	C₂³	1	2	yes	—
204	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
205	#11719919	C₂³	1	2	yes	—
206	#11719919	C₂³	1	2	yes	—
207	#11719919	C₂³	1	2	yes	—
208	#11719919	C₂³	1	2	yes	—
209	#11719919	C₂³	1	2	yes	—
210	#11719919	C₂³	1	1	yes	characteristic
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	2	yes	—
220	#11719919	C₂³	1	1	yes	characteristic
221	#11719919	C₂³	1	1	yes	characteristic
222	C₄×Dic₅₃	C₂³	1	2	yes	—
223	C₄×Dic₅₃	C₂³	1	2	yes	—
224	C₄×Dic₅₃	C₂³	1	2	yes	—
225	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
226	C₄×Dic₅₃	C₂³	1	2	yes	—
227	C₄×Dic₅₃	C₂³	1	2	yes	—
228	C₂³×C₅₃	C₂⁴	1	2	yes	—
229	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
230	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
231	C₂²×D₅₃	—	2	2	no	—
232	C₂²×D₅₃	—	2	2	no	—
233	C₂²×D₅₃	—	2	2	no	—
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	—	4	2	no	—
238	D₄×C₅₃	—	2	4	no	—
239	D₄×C₅₃	—	2	2	no	—
240	D₄×C₅₃	—	2	2	no	—
241	D₄×C₅₃	—	4	2	no	—
242	C₂×C₄×C₅₃	—	2	4	no	—
243	C₂×C₄×C₅₃	—	2	2	no	—
244	C₂×C₄×C₅₃	—	2	2	no	—
245	C₂×C₄×C₅₃	—	4	2	no	—
246	C₂×C₄×C₅₃	—	4	2	no	—
247	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
248	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
249	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
250	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
251	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
253	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
254	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
255	C₂²⋊D₅₃	—	2	2	no	—
256	C₂²⋊D₅₃	—	2	2	no	—
257	C₂²⋊D₅₃	—	2	4	no	—
258	C₂²⋊D₅₃	—	2	4	no	—
259	C₂²⋊D₅₃	—	2	2	no	—
260	C₂²⋊D₅₃	—	2	2	no	—
261	C₂²⋊D₅₃	—	4	2	no	—
262	C₂×Dic₅₃	—	2	4	no	—
263	C₂×Dic₅₃	—	2	2	no	—
264	C₂×Dic₅₃	—	2	4	no	—
265	C₂×Dic₅₃	—	4	2	no	—
266	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
267	C₂×Dic₅₃	C₂⁴	1	2	yes	—
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	2	no	—
277	C₄×D₅₃	—	2	2	no	—
278	C₄×D₅₃	—	2	2	no	—
279	C₄×D₅₃	—	2	2	no	—
280	C₄×D₅₃	—	2	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	—	2	2	no	—
289	Dic₁₀₆	—	4	2	no	—
290	C₂²×C₅₃	—	2	1	no	—
291	C₂²×C₅₃	—	2	1	no	—
292	C₂²×C₅₃	—	2	2	no	—
293	C₂²×C₅₃	—	2	1	no	—
294	C₂²×C₅₃	—	2	2	no	—
295	C₂²×C₅₃	—	2	1	no	—
296	C₂²×C₅₃	—	2	2	no	—
297	C₂²×C₅₃	—	2	1	no	—
298	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
299	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
300	C₂×D₅₃	—	2	8	no	—
301	C₂×D₅₃	—	2	1	no	—
302	C₂×D₅₃	—	2	1	no	—
303	C₂×D₅₃	—	2	1	no	—
304	C₂×D₅₃	—	2	1	no	—
305	C₂×D₅₃	—	2	1	no	—
306	C₂×D₅₃	—	2	1	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	2	no	—
314	C₄×C₅₃	—	2	2	no	—
315	C₄×C₅₃	—	2	1	no	—
316	C₄×C₅₃	—	2	1	no	—
317	C₄×C₅₃	C₂×#41	1	2	yes	—
318	Dic₅₃	—	2	1	no	—
319	Dic₅₃	—	2	2	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	2	no	—
330	C₂×C₅₃	—	4	1	no	—
331	C₂×C₅₃	C₂²×#41	1	1	yes	characteristic
332	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
333	C₂×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
334	D₅₃	—	2	2	no	—
335	D₅₃	—	2	2	no	—
336	D₅₃	—	4	1	no	—
337	#554	—	53	1	no	—
338	#554	—	53	2	no	—
339	#554	—	53	1	no	—
340	#553	—	53	2	no	—
341	#553	—	53	1	no	—
342	#553	—	53	1	no	—
343	#552	—	53	2	no	—
344	#552	—	53	1	no	—
345	#552	—	53	1	no	—
346	#547	—	53	2	no	—
347	#547	—	53	1	no	—
348	#547	D₅₃	1	1	yes	characteristic, complemented
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	2	no	—
355	#525	—	53	1	no	—
356	#525	—	53	1	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	C₂×D₅₃	1	2	yes	—
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	C₂×D₅₃	1	1	yes	characteristic, complemented
370	#126	—	53	2	no	—
371	#126	—	53	2	no	—
372	#126	—	53	2	no	—
373	#126	—	53	2	no	—
374	#125	—	53	2	no	—
375	#125	—	53	2	no	—
376	#125	—	53	2	no	—
377	#125	—	53	2	no	—
378	#125	—	53	2	no	—
379	#125	—	53	1	no	—
380	#125	—	53	1	no	—
381	#123	—	53	2	no	—
382	#123	—	53	2	no	—
383	#123	—	53	2	no	—
384	#123	—	53	2	no	—
385	#123	—	53	2	no	—
386	#123	—	53	1	no	—
387	#123	—	53	1	no	—
388	#123	—	53	2	no	—
389	#123	—	53	2	no	—
390	#123	C₂×D₅₃	1	2	yes	—
391	#123	C₂×D₅₃	1	2	yes	complemented
392	C₂²⋊Q₈	—	53	2	no	—
393	C₂²⋊Q₈	—	53	2	no	—
394	C₂²⋊Q₈	—	53	2	no	—
395	C₂²⋊Q₈	—	53	1	no	—
396	C₂²⋊Q₈	—	53	1	no	—
397	C₂²⋊Q₈	—	53	2	no	—
398	C₂²⋊Q₈	—	53	2	no	—
399	C₂²⋊Q₈	—	53	1	no	—
400	C₂²⋊Q₈	—	53	2	no	—
401	C₂²⋊Q₈	—	53	1	no	—
402	C₂²⋊Q₈	—	53	1	no	—
403	C₂²⋊Q₈	—	53	1	no	—
404	C₂²⋊Q₈	—	53	2	no	—
405	C₂²⋊Q₈	—	53	2	no	—
406	C₂²⋊Q₈	C₂×D₅₃	1	2	yes	complemented
407	#121	—	53	2	no	—
408	#121	—	53	1	no	—
409	#121	—	53	2	no	—
410	#121	—	53	1	no	—
411	#121	—	53	1	no	—
412	#121	—	53	2	no	—
413	#121	—	53	2	no	—
414	#121	—	53	2	no	—
415	#121	—	53	1	no	—
416	#121	C₂×D₅₃	1	1	yes	characteristic
417	#121	C₂×D₅₃	1	1	yes	characteristic, complemented
418	#120	—	53	2	no	—
419	#120	—	53	1	no	—
420	#120	—	53	1	no	—
421	C₄×Q₈	—	53	2	no	—
422	C₄×Q₈	—	53	1	no	—
423	C₄×Q₈	—	53	1	no	—
424	C₄×D₄	—	53	1	no	—
425	C₄×D₄	—	53	2	no	—
426	C₄×D₄	—	53	2	no	—
427	C₄×D₄	—	53	2	no	—
428	C₄×D₄	—	53	2	no	—
429	C₄×D₄	—	53	1	no	—
430	C₄×D₄	—	53	2	no	—
431	C₄×D₄	—	53	2	no	—
432	C₄×D₄	—	53	1	no	—
433	C₄×D₄	—	53	2	no	—
434	C₄×D₄	—	53	2	no	—
435	C₄×D₄	—	53	2	no	—
436	C₄×D₄	—	53	2	no	—
437	C₄×D₄	—	53	2	no	—
438	C₄×D₄	—	53	1	no	—
439	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
440	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
441	#117	—	53	1	no	—
442	#117	—	53	2	no	—
443	#117	—	53	1	no	—
444	#117	—	53	1	no	—
445	#117	—	53	2	no	—
446	#117	—	53	2	no	—
447	#117	—	53	1	no	—
448	C₂×C₄⋊C₄	—	53	2	no	—
449	C₂×C₄⋊C₄	—	53	1	no	—
450	C₂×C₄⋊C₄	—	53	2	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	1	no	—
458	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
459	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
460	C₂⁴	—	53	1	no	—
461	#41	—	4	4	no	—
462	#41	—	106	4	no	—
463	#41	—	106	4	no	—
464	#41	—	212	4	no	—
465	C₂×Q₈	—	53	1	no	—
466	C₂×Q₈	—	53	1	no	—
467	C₂×Q₈	—	53	2	no	—
468	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
469	C₂×D₄	—	53	1	no	—
470	C₂×D₄	—	53	2	no	—
471	C₂×D₄	—	53	1	no	—
472	C₂×D₄	—	53	2	no	—
473	C₂×D₄	—	53	1	no	—
474	C₂×D₄	—	53	1	no	—
475	C₂×D₄	—	53	2	no	—
476	C₂×D₄	—	53	1	no	—
477	C₂×D₄	—	53	1	no	—
478	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
479	C₂×D₄	C₂²×D₅₃	1	2	yes	—
480	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
481	C₂×D₄	C₂²×D₅₃	1	2	yes	—
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	2	no	—
487	C₂²×C₄	—	53	1	no	—
488	C₂²×C₄	—	53	1	no	—
489	C₂²×C₄	—	53	1	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	—	106	4	no	—
497	C₂²×C₄	—	212	4	no	—
498	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
499	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
500	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
501	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
502	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
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	2	no	—
512	C₄⋊C₄	—	53	2	no	—
513	C₄⋊C₄	—	53	2	no	—
514	C₄⋊C₄	—	53	1	no	—
515	C₄⋊C₄	—	53	1	no	—
516	C₄⋊C₄	—	53	2	no	—
517	C₄⋊C₄	—	53	2	no	—
518	C₄⋊C₄	—	53	2	no	—
519	C₄⋊C₄	—	53	2	no	—
520	C₄⋊C₄	—	53	1	no	—
521	C₄⋊C₄	—	53	2	no	—
522	C₄⋊C₄	—	53	2	no	—
523	C₄⋊C₄	—	53	2	no	—
524	C₄⋊C₄	—	53	2	no	—
525	C₄⋊C₄	—	53	2	no	—
526	C₄⋊C₄	—	53	1	no	—
527	C₄⋊C₄	—	53	1	no	—
528	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
529	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
530	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
531	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
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	2	no	—
537	C₂²⋊C₄	—	53	2	no	—
538	C₂²⋊C₄	—	53	2	no	—
539	C₂²⋊C₄	—	53	2	no	—
540	C₂²⋊C₄	—	53	2	no	—
541	C₂²⋊C₄	—	53	1	no	—
542	C₂²⋊C₄	—	53	2	no	—
543	C₂²⋊C₄	—	53	2	no	—
544	C₂²⋊C₄	—	53	1	no	—
545	C₂²⋊C₄	—	53	1	no	—
546	C₂²⋊C₄	—	53	2	no	—
547	C₂²⋊C₄	—	53	2	no	—
548	C₂²⋊C₄	—	53	1	no	—
549	C₂²⋊C₄	—	53	1	no	—
550	C₂²⋊C₄	—	53	2	no	—
551	C₂²⋊C₄	—	53	1	no	—
552	C₂²⋊C₄	—	53	1	no	—
553	C₂²⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
554	C₂²⋊C₄	C₂²×D₅₃	1	2	yes	—
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	2	no	—
562	C₄²	—	53	2	no	—
563	C₄²	—	53	1	no	—
564	C₄²	C₂²×D₅₃	1	1	yes	characteristic
565	C₂³	—	53	1	no	—
566	C₂³	—	53	1	no	—
567	C₂³	—	53	1	no	—
568	C₂³	—	106	2	no	—
569	C₂³	—	106	2	no	—
570	C₂³	—	106	2	no	—
571	C₂³	C₂³×D₅₃	1	1	yes	characteristic
572	C₂³	C₂³×D₅₃	1	2	yes	—
573	C₂³	C₂³×D₅₃	1	1	yes	characteristic
574	Q₈	—	4	2	no	—
575	Q₈	—	106	2	no	—
576	Q₈	—	106	2	no	—
577	Q₈	—	212	2	no	—
578	D₄	—	2	2	no	—
579	D₄	—	2	4	no	—
580	D₄	—	2	2	no	—
581	D₄	—	4	2	no	—
582	D₄	—	106	2	no	—
583	D₄	—	106	2	no	—
584	D₄	—	106	2	no	—
585	D₄	—	106	2	no	—
586	D₄	—	106	4	no	—
587	D₄	—	106	2	no	—
588	D₄	—	106	4	no	—
589	D₄	—	106	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	—	4	2	no	—
595	C₂×C₄	—	4	2	no	—
596	C₂×C₄	—	53	1	no	—
597	C₂×C₄	—	53	4	no	—
598	C₂×C₄	—	53	1	no	—
599	C₂×C₄	—	53	2	no	—
600	C₂×C₄	—	53	1	no	—
601	C₂×C₄	—	53	2	no	—
602	C₂×C₄	—	53	2	no	—
603	C₂×C₄	—	53	1	no	—
604	C₂×C₄	—	53	1	no	—
605	C₂×C₄	—	53	2	no	—
606	C₂×C₄	—	106	2	no	—
607	C₂×C₄	—	106	2	no	—
608	C₂×C₄	—	106	2	no	—
609	C₂×C₄	—	106	4	no	—
610	C₂×C₄	—	106	4	no	—
611	C₂×C₄	—	106	2	no	—
612	C₂×C₄	—	106	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	—	212	2	no	—
619	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
620	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
621	C₂×C₄	C₂³×D₅₃	1	2	yes	—
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	2	no	—
631	C₂²	—	2	1	no	—
632	C₂²	—	2	2	no	—
633	C₂²	—	2	1	no	—
634	C₂²	—	2	2	no	—
635	C₂²	—	53	2	no	—
636	C₂²	—	53	2	no	—
637	C₂²	—	106	8	no	—
638	C₂²	—	106	1	no	—
639	C₂²	—	106	1	no	—
640	C₂²	—	106	1	no	—
641	C₂²	—	106	1	no	—
642	C₂²	—	106	1	no	—
643	C₂²	—	106	1	no	—
644	C₂²	—	106	1	no	—
645	C₂²	#13133262	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	1	no	—
652	C₄	—	2	2	no	—
653	C₄	—	2	1	no	—
654	C₄	—	53	2	no	—
655	C₄	—	106	1	no	—
656	C₄	—	106	2	no	—
657	C₄	—	106	1	no	—
658	C₄	—	106	2	no	—
659	C₄	—	106	2	no	—
660	C₄	—	106	1	no	—
661	C₄	—	106	1	no	—
662	C₄	—	106	2	no	—
663	C₄	#41×D₅₃	1	2	yes	—
664	C₂	—	2	2	no	—
665	C₂	—	4	1	no	—
666	C₂	—	4	2	no	—
667	C₂	—	106	2	no	—
668	C₂	—	106	2	no	—
669	C₂	—	212	1	no	—
670	C₂	C₂×#41×D₅₃	1	1	yes	characteristic
671	C₂	C₂×#13133262	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×#547	C₂
hypercenter	#547	D₅₃
hypocenter	C₅₃	#3492
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #547

Complements

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

N	H ≅ G/N	type
1. #16811334	C₂	semidirect
2. #16811328	C₂	semidirect
3. #16811327	C₂	semidirect
4. #16811323	C₂	semidirect
5. #16811320	C₂	semidirect
6. #16811318	C₂	semidirect
7. #16811313	C₂	semidirect
8. #16811310	C₂	semidirect
9. #16811308	C₂	semidirect
10. #16811273	C₂	semidirect
11. #16811272	C₂	semidirect
12. #16811250	C₂	semidirect
13. #16811219	C₂	semidirect
14. #16811216	C₂	semidirect
15. #16811202	C₂	semidirect
16. #16811133	C₂	semidirect
17. #16811132	C₂	semidirect
18. #16811131	C₂	semidirect
19. #16811130	C₂	semidirect
20. #16811104	C₂	semidirect
21. #16811102	C₂	semidirect
22. C₂×C₄⋊C₄×D₅₃	C₂	semidirect
23. C₅₃×#547	C₂	semidirect
29. #13133216	C₂²	semidirect
30. #13133215	C₂²	semidirect
31. #13133212	C₂²	semidirect
32. #13133210	C₂²	semidirect
33. #13133210	C₂²	semidirect
34. #13133210	C₂²	semidirect
35. #13133209	C₂²	semidirect
36. #13133208	C₂²	semidirect
37. #13133208	C₂²	semidirect
38. #13133207	C₂²	semidirect
39. #13133207	C₂²	semidirect
40. #13133207	C₂²	semidirect
41. #13133207	C₂²	semidirect
44. #13133200	C₂²	semidirect
47. #13133199	C₂²	semidirect
50. #13133198	C₂²	semidirect
52. #13133197	C₂²	semidirect
54. #13133196	C₂²	semidirect
55. #13133196	C₂²	semidirect
60. #13133194	C₂²	semidirect
61. #13133194	C₂²	semidirect
64. #13133193	C₂²	semidirect
69. #13133189	C₂²	semidirect
73. #13133188	C₂²	semidirect
77. C₄×Dic₁₀₆	C₂²	semidirect
78. #13133182	C₂²	semidirect
79. #13133182	C₂²	semidirect
82. D₄×Dic₅₃	C₂²	semidirect
91. C₄×C₂²⋊D₅₃	C₂²	semidirect
96. #13133176	C₂²	semidirect
98. #13133175	C₂²	semidirect
101. #13133173	C₂²	semidirect
104. C₂×#11719919	C₂²	semidirect
111. C₄⋊C₄×D₅₃	C₂²	semidirect
113. C₄⋊Q₈×C₅₃	C₂²	semidirect
115. #123×C₅₃	C₂²	semidirect
116. C₂²⋊Q₈×C₅₃	C₂²	semidirect
118. #121×C₅₃	C₂²	semidirect
120. C₄×D₄×C₅₃	C₂²	semidirect
122. C₂×C₄⋊C₄×C₅₃	C₂²	semidirect
286. C₄×D₅₃	#41	semidirect
348. #547	D₅₃	semidirect
360. C₅₃	#3492	semidirect
369. C₄⋊Q₈	C₂×D₅₃	semidirect
391. #123	C₂×D₅₃	semidirect
406. C₂²⋊Q₈	C₂×D₅₃	semidirect
417. #121	C₂×D₅₃	semidirect
440. C₄×D₄	C₂×D₅₃	semidirect
459. C₂×C₄⋊C₄	C₂×D₅₃	semidirect