Subgroups — #28292812

Order 13568.

7330 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289287	C₂	1	8	yes	maximal, complemented
2	#20289234	C₂	1	8	yes	maximal, complemented
3	#20289072	C₂	1	8	yes	maximal, complemented
4	#20288670	C₂	1	16	yes	maximal, complemented
5	#20288609	C₂	1	4	yes	maximal, complemented
6	#20288505	C₂	1	4	yes	maximal, complemented
7	#20288404	C₂	1	8	yes	maximal, complemented
8	#20288366	C₂	1	2	yes	maximal, complemented
9	#20288350	C₂	1	2	yes	maximal, complemented
10	#20288011	C₂	1	2	yes	maximal, complemented
11	C₂²×#124×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133265	C₂²	1	2	yes	complemented
13	C₂×#13133262	C₂²	1	2	yes	complemented
14	#16811373	C₂²	1	8	yes	complemented
15	#16811367	C₂²	1	16	yes	complemented
16	#16811365	C₂²	1	8	yes	complemented
17	#16811355	C₂²	1	16	yes	complemented
18	#16811342	C₂²	1	16	yes	complemented
19	#16811335	C₂²	1	16	yes	complemented
20	#16811303	C₂²	1	16	yes	complemented
21	#16811299	C₂²	1	32	yes	complemented
22	#16811299	C₂²	1	32	yes	complemented
23	Q₈×C₂²⋊D₅₃	C₂²	1	16	yes	complemented
24	#16811284	C₂²	1	16	yes	complemented
25	#16811281	C₂²	1	16	yes	complemented
26	#16811267	C₂²	1	16	yes	complemented
27	#16811266	C₂²	1	16	yes	complemented
28	#16811263	C₂²	1	32	yes	complemented
29	#16811263	C₂²	1	32	yes	complemented
30	D₄×C₂²⋊D₅₃	C₂²	1	16	yes	complemented
31	#16811249	C₂²	1	8	yes	complemented
32	#16811242	C₂²	1	16	yes	complemented
33	#16811237	C₂²	1	32	yes	complemented
34	#16811211	C₂²	1	16	yes	complemented
35	#16811211	C₂²	1	16	yes	complemented
36	#16811207	C₂²	1	32	yes	complemented
37	#16811197	C₂²	1	8	yes	complemented
38	#16811197	C₂²	1	4	yes	complemented
39	#16811193	C₂²	1	8	yes	complemented
40	#16811191	C₂²	1	8	yes	complemented
41	#16811190	C₂²	1	4	yes	—
42	#16811185	C₂²	1	8	yes	complemented
43	#16811182	C₂²	1	8	yes	complemented
44	#16811180	C₂²	1	4	yes	complemented
45	#16811180	C₂²	1	2	yes	complemented
46	#16811173	C₂²	1	4	yes	complemented
47	#16811173	C₂²	1	2	yes	complemented
48	#16811172	C₂²	1	8	yes	complemented
49	#16811157	C₂²	1	8	yes	complemented
50	#16811157	C₂²	1	8	yes	complemented
51	#16811157	C₂²	1	8	yes	complemented
52	#16811152	C₂²	1	16	yes	complemented
53	#124×D₅₃	C₂²	1	8	yes	complemented
54	#16811134	C₂²	1	4	yes	complemented
55	#16811134	C₂²	1	8	yes	complemented
56	#16811134	C₂²	1	8	yes	complemented
57	C₂×#13133186	C₂²	1	8	yes	complemented
58	#16811101	C₂²	1	2	yes	complemented
59	#16811098	C₂²	1	2	yes	complemented
60	#16811096	C₂²	1	16	yes	complemented
61	C₄×#11719945	C₂²	1	4	yes	complemented
62	C₄×#11719945	C₂²	1	8	yes	complemented
63	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
64	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
65	C₂×#124×C₅₃	C₂²	1	8	yes	complemented
66	C₂×#124×C₅₃	C₂²	1	4	yes	complemented
67	C₂×#124×C₅₃	C₂²	1	4	yes	complemented
68	C₂×#124×C₅₃	C₂²	1	8	yes	complemented
69	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
70	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
71	#13133265	—	4	8	no	—
72	#13133262	—	4	8	no	—
73	C₂×#11719949	C₂³	1	8	yes	complemented
74	C₂×#11719945	C₂³	1	4	yes	—
75	C₂×#11719945	C₂³	1	8	yes	—
76	C₂×#11719945	C₂³	1	8	yes	—
77	C₂×#11719947	C₂³	1	8	yes	—
78	C₂×Q₈×D₅₃	C₂³	1	8	yes	complemented
79	C₂²×C₂²⋊D₅₃	C₂³	1	16	yes	—
80	C₂²×C₂²⋊D₅₃	C₂³	1	8	yes	—
81	C₂×D₄×D₅₃	C₂³	1	8	yes	—
82	#13133217	C₂³	1	2	yes	complemented
83	#13133217	C₂³	1	4	yes	complemented
84	#13133215	C₂³	1	16	yes	complemented
85	#13133215	C₂³	1	8	yes	complemented
86	#13133214	C₂³	1	2	yes	complemented
87	#13133213	C₂³	1	8	yes	—
88	#13133211	C₂³	1	16	yes	complemented
89	#13133210	C₂³	1	32	yes	complemented
90	#13133210	C₂³	1	32	yes	complemented
91	#13133209	C₂³	1	8	yes	complemented
92	#13133209	C₂³	1	4	yes	complemented
93	#13133206	C₂³	1	16	yes	complemented
94	#13133206	C₂³	1	8	yes	complemented
95	#13133205	C₂³	1	8	yes	complemented
96	#13133205	C₂³	1	4	yes	complemented
97	#13133204	C₂³	1	8	yes	—
98	#13133203	C₂³	1	32	yes	complemented
99	#13133203	C₂³	1	32	yes	complemented
100	#13133201	C₂³	1	32	yes	complemented
101	#13133200	C₂³	1	16	yes	—
102	#13133200	C₂³	1	16	yes	—
103	#13133200	C₂³	1	8	yes	—
104	#13133199	C₂³	1	32	yes	complemented
105	#13133199	C₂³	1	32	yes	complemented
106	#13133198	C₂³	1	16	yes	—
107	#13133196	C₂³	1	8	yes	—
108	#13133196	C₂³	1	16	yes	—
109	#13133196	C₂³	1	8	yes	—
110	#13133196	C₂³	1	8	yes	—
111	#13133194	C₂³	1	32	yes	complemented
112	#13133194	C₂³	1	32	yes	complemented
113	#13133193	C₂³	1	16	yes	complemented
114	#13133193	C₂³	1	32	yes	complemented
115	#13133191	C₂³	1	8	yes	—
116	#13133191	C₂³	1	16	yes	—
117	#13133191	C₂³	1	8	yes	—
118	#13133191	C₂³	1	8	yes	—
119	#13133190	C₂³	1	32	yes	complemented
120	#13133190	C₂³	1	32	yes	complemented
121	#13133189	C₂³	1	32	yes	—
122	#13133188	C₂³	1	16	yes	—
123	#13133187	C₂³	1	32	yes	complemented
124	#13133186	C₂³	1	32	yes	—
125	#13133186	C₂³	1	16	yes	—
126	#13133186	C₂³	1	8	yes	—
127	#13133185	C₂³	1	16	yes	—
128	Q₈×Dic₅₃	C₂³	1	16	yes	complemented
129	C₄×Dic₁₀₆	C₂³	1	8	yes	complemented
130	C₄×Dic₁₀₆	C₂³	1	8	yes	complemented
131	C₄×Dic₁₀₆	C₂³	1	8	yes	complemented
132	C₄×D₂₁₂	C₂³	1	8	yes	complemented
133	C₄×D₂₁₂	C₂³	1	8	yes	complemented
134	C₄×D₂₁₂	C₂³	1	8	yes	complemented
135	D₄×Dic₅₃	C₂³	1	16	yes	—
136	C₄×C₂²⋊D₅₃	C₂³	1	16	yes	—
137	C₄×C₂²⋊D₅₃	C₂³	1	32	yes	—
138	C₄×C₂²⋊D₅₃	C₂³	1	16	yes	—
139	C₄×C₂²⋊D₅₃	C₂³	1	16	yes	—
140	#13133177	C₂³	1	32	yes	complemented
141	#13133177	C₂³	1	32	yes	complemented
142	#13133176	C₂³	1	8	yes	—
143	#13133176	C₂³	1	8	yes	—
144	#13133176	C₂³	1	4	yes	—
145	#13133174	C₂³	1	16	yes	complemented
146	#13133174	C₂³	1	8	yes	complemented
147	#13133173	C₂³	1	32	yes	complemented
148	C₂×C₂²⋊Dic₅₃	C₂³	1	16	yes	—
149	C₂×C₂²⋊Dic₅₃	C₂³	1	8	yes	—
150	C₂²⋊C₄×D₅₃	C₂³	1	32	yes	complemented
151	C₄²×D₅₃	C₂³	1	8	yes	complemented
152	C₂⁵×C₅₃	C₂³	1	2	yes	—
153	C₂²×Q₈×C₅₃	—	4	4	no	—
154	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
155	C₂²×Q₈×C₅₃	C₂³	1	8	yes	—
156	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
157	C₂²×D₄×C₅₃	—	4	4	no	—
158	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
159	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
160	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
161	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
162	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
163	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
164	#124×C₅₃	C₂³	1	16	yes	—
165	#124×C₅₃	C₂³	1	8	yes	—
166	#124×C₅₃	C₂³	1	8	yes	—
167	#124×C₅₃	C₂³	1	8	yes	—
168	#124×C₅₃	C₂³	1	8	yes	—
169	#124×C₅₃	C₂³	1	16	yes	—
170	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
171	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
172	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
173	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
174	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
175	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
176	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
177	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
178	C₂⁴×C₅₃	—	4	4	no	—
179	C₂⁴×C₅₃	—	4	8	no	—
180	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
181	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
182	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
183	C₂³×D₅₃	C₂⁴	1	8	yes	—
184	C₂×Q₈×C₅₃	—	2	8	no	—
185	C₂×Q₈×C₅₃	—	4	4	no	—
186	C₂×Q₈×C₅₃	—	4	16	no	—
187	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
188	C₂×Q₈×C₅₃	C₂⁴	1	8	yes	—
189	C₂×D₄×C₅₃	—	2	8	no	—
190	C₂×D₄×C₅₃	—	4	16	no	—
191	C₂×D₄×C₅₃	—	4	4	no	—
192	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
193	C₂×D₄×C₅₃	C₂⁴	1	8	yes	—
194	C₂²×C₄×C₅₃	—	2	4	no	—
195	C₂²×C₄×C₅₃	—	4	4	no	—
196	C₂²×C₄×C₅₃	—	4	4	no	—
197	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
198	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
199	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
200	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
201	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
203	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
204	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
205	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
206	C₂×C₂²⋊D₅₃	—	4	32	no	—
207	C₂×C₂²⋊D₅₃	—	4	16	no	—
208	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
209	C₂×C₂²⋊D₅₃	C₂⁴	1	8	yes	—
210	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
211	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
212	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
213	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
214	#11719949	—	4	16	no	—
215	Q₈×D₅₃	—	4	16	no	—
216	#11719947	—	4	16	no	—
217	D₄×D₅₃	—	4	16	no	—
218	#11719945	—	2	16	no	—
219	#11719945	—	4	16	no	—
220	#11719945	—	4	16	no	—
221	C₂×D₂₁₂	C₂⁴	1	4	yes	—
222	C₂×D₂₁₂	C₂⁴	1	8	yes	—
223	C₂×D₂₁₂	C₂⁴	1	8	yes	—
224	C₂×C₄×D₅₃	C₂⁴	1	8	yes	—
225	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
226	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
227	C₂×Dic₁₀₆	C₂⁴	1	4	yes	—
228	C₂×Dic₁₀₆	C₂⁴	1	8	yes	—
229	C₂×Dic₁₀₆	C₂⁴	1	8	yes	—
230	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
231	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
232	C₄²×C₅₃	C₂⁴	1	2	yes	—
233	C₄²×C₅₃	C₂⁴	1	2	yes	—
234	C₄²×C₅₃	C₂⁴	1	2	yes	—
235	C₄²×C₅₃	C₂⁴	1	4	yes	—
236	C₄²×C₅₃	C₂⁴	1	4	yes	—
237	C₄²×C₅₃	C₂⁴	1	2	yes	—
238	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
239	C₂²⋊Dic₅₃	C₂⁴	1	8	yes	—
240	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
241	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	—
242	#11719921	C₂⁴	1	16	yes	—
243	#11719921	C₂⁴	1	32	yes	—
244	#11719921	C₂⁴	1	16	yes	—
245	#11719921	C₂⁴	1	16	yes	—
246	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
247	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
248	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
249	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
250	#11719919	C₂⁴	1	16	yes	—
251	#11719919	C₂⁴	1	32	yes	—
252	#11719919	C₂⁴	1	16	yes	—
253	#11719919	C₂⁴	1	16	yes	—
254	C₄×Dic₅₃	C₂⁴	1	16	yes	—
255	C₄×Dic₅₃	C₂⁴	1	8	yes	—
256	C₄×Dic₅₃	C₂⁴	1	16	yes	—
257	C₂³×C₅₃	—	2	2	no	—
258	C₂³×C₅₃	—	2	4	no	—
259	C₂³×C₅₃	—	2	8	no	—
260	C₂³×C₅₃	—	2	16	no	—
261	C₂³×C₅₃	—	4	8	no	—
262	C₂³×C₅₃	—	4	8	no	—
263	C₂³×C₅₃	—	4	8	no	—
264	C₂³×C₅₃	—	8	16	no	—
265	C₂³×C₅₃	C₂⁵	1	8	yes	—
266	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
267	C₂³×C₅₃	C₂⁵	1	2	yes	—
268	C₂²×D₅₃	—	4	8	no	—
269	C₂²×D₅₃	—	4	16	no	—
270	C₂²×D₅₃	C₂⁵	1	8	yes	—
271	Q₈×C₅₃	—	2	16	no	—
272	Q₈×C₅₃	—	4	8	no	—
273	D₄×C₅₃	—	2	16	no	—
274	D₄×C₅₃	—	4	8	no	—
275	C₂×C₄×C₅₃	—	2	8	no	—
276	C₂×C₄×C₅₃	—	2	2	no	—
277	C₂×C₄×C₅₃	—	2	8	no	—
278	C₂×C₄×C₅₃	—	2	8	no	—
279	C₂×C₄×C₅₃	—	4	2	no	—
280	C₂×C₄×C₅₃	—	4	4	no	—
281	C₂×C₄×C₅₃	—	4	2	no	—
282	C₂×C₄×C₅₃	—	4	8	no	—
283	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
284	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
285	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
286	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
287	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
288	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
289	C₂²⋊D₅₃	—	2	16	no	—
290	C₂²⋊D₅₃	—	2	32	no	—
291	C₂²⋊D₅₃	—	4	16	no	—
292	C₂²⋊D₅₃	—	4	32	no	—
293	C₂×Dic₅₃	—	4	8	no	—
294	C₂×Dic₅₃	—	4	16	no	—
295	C₂×Dic₅₃	C₂⁵	1	8	yes	—
296	C₂×Dic₅₃	C₂⁵	1	16	yes	—
297	D₂₁₂	—	2	8	no	—
298	D₂₁₂	—	4	8	no	—
299	D₂₁₂	—	4	8	no	—
300	C₄×D₅₃	—	2	16	no	—
301	C₄×D₅₃	—	4	16	no	—
302	C₄×D₅₃	—	4	16	no	—
303	Dic₁₀₆	—	2	8	no	—
304	Dic₁₀₆	—	4	8	no	—
305	Dic₁₀₆	—	4	8	no	—
306	#62824	—	53	1	no	maximal, Sylow
307	C₂²×C₅₃	—	2	8	no	—
308	C₂²×C₅₃	—	2	2	no	—
309	C₂²×C₅₃	—	2	2	no	—
310	C₂²×C₅₃	—	2	2	no	—
311	C₂²×C₅₃	—	2	8	no	—
312	C₂²×C₅₃	—	2	16	no	—
313	C₂²×C₅₃	—	8	8	no	—
314	C₂²×C₅₃	—	8	16	no	—
315	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
316	C₂²×C₅₃	#585	1	4	yes	—
317	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
318	C₂×D₅₃	—	2	8	no	—
319	C₂×D₅₃	—	2	16	no	—
320	C₂×D₅₃	—	16	8	no	—
321	C₄×C₅₃	—	2	8	no	—
322	C₄×C₅₃	—	2	2	no	—
323	C₄×C₅₃	—	2	4	no	—
324	C₄×C₅₃	—	2	4	no	—
325	C₄×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
326	Dic₅₃	—	2	8	no	—
327	Dic₅₃	—	2	16	no	—
328	#3561	—	53	8	no	—
329	#3558	—	53	8	no	—
330	#3545	—	53	8	no	—
331	#3514	—	53	16	no	—
332	#3510	—	53	4	no	—
333	#3500	—	53	4	no	—
334	#3490	—	53	8	no	—
335	#3486	—	53	2	no	—
336	#3484	—	53	2	no	—
337	#3444	—	53	2	no	—
338	C₂²×#124	D₅₃	1	1	yes	characteristic, complemented
339	C₂×C₅₃	—	2	2	no	—
340	C₂×C₅₃	—	2	4	no	—
341	C₂×C₅₃	—	4	8	no	—
342	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
343	C₂×C₅₃	C₂×#585	1	2	yes	—
344	D₅₃	—	4	8	no	—
345	C₂×2₋¹⁺⁴	—	53	2	no	—
346	C₂×H(5,𝔽₂)	—	53	2	no	—
347	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
348	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
349	#563	—	53	8	no	—
350	#562	—	53	16	no	—
351	H(3,𝔽₄)	—	53	8	no	—
352	#559	—	53	16	no	—
353	#556	—	53	16	no	—
354	#555	—	53	16	no	—
355	#551	—	53	16	no	—
356	#551	—	53	32	no	—
357	#551	—	53	32	no	—
358	D₄×Q₈	—	53	16	no	—
359	#548	—	53	16	no	—
360	#548	—	53	16	no	—
361	#546	—	53	16	no	—
362	#546	—	53	16	no	—
363	#546	—	53	32	no	—
364	#546	—	53	32	no	—
365	D₄²	—	53	16	no	—
366	#543	—	53	8	no	—
367	#542	—	53	16	no	—
368	#542	—	53	32	no	—
369	#538	—	53	32	no	—
370	#538	—	53	16	no	—
371	#538	—	53	16	no	—
372	#536	—	53	4	no	—
373	#536	—	53	8	no	—
374	#536	—	53	8	no	—
375	#535	—	53	4	no	—
376	#535	—	53	8	no	—
377	#534	—	53	8	no	—
378	#534	—	53	8	no	—
379	#533	—	53	2	no	—
380	#533	—	53	4	no	—
381	#532	—	53	8	no	—
382	#532	—	53	2	no	—
383	#532	—	53	4	no	—
384	#529	—	53	16	no	—
385	#529	—	53	8	no	—
386	#529	—	53	8	no	—
387	#529	—	53	8	no	—
388	C₂×#124	—	53	8	no	—
389	C₂×#124	C₂×D₅₃	1	4	yes	complemented
390	C₂×#124	C₂×D₅₃	1	8	yes	complemented
391	C₂×#124	C₂×D₅₃	1	4	yes	complemented
392	C₂×#124	C₂×D₅₃	1	8	yes	complemented
393	#525	—	53	4	no	—
394	#525	—	53	8	no	—
395	#525	—	53	8	no	—
396	C₂×#120	—	53	8	no	—
397	#519	—	53	2	no	—
398	#518	—	53	2	no	—
399	#518	—	53	16	no	—
400	C₄×#41	—	53	4	no	—
401	C₄×#41	—	53	8	no	—
402	C₂²×C₂²⋊C₄	C₂×D₅₃	1	4	yes	complemented
403	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
404	C₅₃	#62824	1	1	yes	characteristic, complemented, Sylow
405	C₂⁵	C₂²×D₅₃	1	2	yes	—
406	2₋¹⁺⁴	—	212	8	no	—
407	H(5,𝔽₂)	—	212	8	no	—
408	C₂×#41	—	53	8	no	—
409	C₂×#41	—	53	8	no	—
410	C₂×#41	—	53	4	no	—
411	C₂×#41	—	53	8	no	—
412	C₂×#41	—	53	8	no	—
413	C₂²×Q₈	—	4	4	no	—
414	C₂²×Q₈	—	53	8	no	—
415	C₂²×Q₈	C₂²×D₅₃	1	2	yes	—
416	C₂²×Q₈	C₂²×D₅₃	1	2	yes	—
417	C₂²×Q₈	C₂²×D₅₃	1	8	yes	—
418	C₂²×D₄	—	4	4	no	—
419	C₂²×D₄	—	53	16	no	—
420	C₂²×D₄	—	53	8	no	—
421	C₂²×D₄	—	53	8	no	—
422	C₂²×D₄	C₂²×D₅₃	1	2	yes	—
423	C₂²×D₄	C₂²×D₅₃	1	2	yes	—
424	C₂²×D₄	C₂²×D₅₃	1	8	yes	—
425	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
426	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
427	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
428	C₄⋊Q₈	—	53	8	no	—
429	C₄⋊Q₈	—	53	16	no	—
430	C₄⋊Q₈	—	53	2	no	—
431	C₄⋊Q₈	—	53	4	no	—
432	#127	—	53	8	no	—
433	#127	—	53	2	no	—
434	#126	—	53	32	no	—
435	#126	—	53	32	no	—
436	#126	—	53	16	no	—
437	#125	—	53	4	no	—
438	#125	—	53	8	no	—
439	#124	—	53	8	no	—
440	#124	—	53	16	no	—
441	#124	—	53	8	no	—
442	#124	—	53	32	no	—
443	#124	—	53	32	no	—
444	#124	—	53	4	no	—
445	#124	—	53	8	no	—
446	#124	C₂²×D₅₃	1	8	yes	—
447	#124	C₂²×D₅₃	1	8	yes	—
448	#124	C₂²×D₅₃	1	16	yes	—
449	#124	C₂²×D₅₃	1	16	yes	—
450	#124	C₂²×D₅₃	1	8	yes	—
451	#124	C₂²×D₅₃	1	8	yes	—
452	#123	—	53	16	no	—
453	#123	—	53	8	no	—
454	#123	—	53	16	no	—
455	#123	—	53	16	no	—
456	#123	—	53	32	no	—
457	#123	—	53	32	no	—
458	#123	—	53	32	no	—
459	C₂²⋊Q₈	—	53	16	no	—
460	C₂²⋊Q₈	—	53	32	no	—
461	C₂²⋊Q₈	—	53	8	no	—
462	C₂²⋊Q₈	—	53	8	no	—
463	C₂²⋊Q₈	—	53	8	no	—
464	C₂²⋊Q₈	—	53	16	no	—
465	C₂²⋊Q₈	—	53	32	no	—
466	C₂²⋊Q₈	—	53	32	no	—
467	#121	—	53	16	no	—
468	#121	—	53	32	no	—
469	#121	—	53	8	no	—
470	#121	—	53	8	no	—
471	#121	—	53	8	no	—
472	#121	—	53	16	no	—
473	#121	—	53	32	no	—
474	#121	—	53	32	no	—
475	#120	—	53	8	no	—
476	#120	—	53	16	no	—
477	#120	—	53	32	no	—
478	#120	—	53	16	no	—
479	#120	—	53	32	no	—
480	C₄×Q₈	—	53	16	no	—
481	C₄×Q₈	—	53	8	no	—
482	C₄×Q₈	—	53	8	no	—
483	C₄×Q₈	—	53	8	no	—
484	C₄×D₄	—	53	16	no	—
485	C₄×D₄	—	53	16	no	—
486	C₄×D₄	—	53	16	no	—
487	C₄×D₄	—	53	16	no	—
488	C₄×D₄	—	53	32	no	—
489	C₄×D₄	—	53	32	no	—
490	C₄×D₄	—	53	32	no	—
491	C₄×D₄	—	53	8	no	—
492	C₄×D₄	—	53	8	no	—
493	C₄×D₄	—	53	8	no	—
494	#117	—	53	4	no	—
495	#117	—	53	8	no	—
496	#117	—	53	8	no	—
497	#117	—	53	32	no	—
498	#117	—	53	8	no	—
499	#117	—	53	16	no	—
500	C₂×C₂²⋊C₄	—	53	8	no	—
501	C₂×C₂²⋊C₄	—	53	16	no	—
502	C₂×C₂²⋊C₄	—	53	32	no	—
503	C₂×C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
504	C₂×C₂²⋊C₄	C₂²×D₅₃	1	8	yes	—
505	C₂×C₂²⋊C₄	C₂²×D₅₃	1	16	yes	—
506	C₂×C₂²⋊C₄	C₂²×D₅₃	1	16	yes	—
507	C₂×C₄²	—	53	8	no	—
508	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
509	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
510	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
511	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
512	C₂⁴	—	4	4	no	—
513	C₂⁴	—	4	8	no	—
514	C₂⁴	—	53	8	no	—
515	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
516	C₂⁴	C₂³×D₅₃	1	4	yes	—
517	C₂⁴	C₂³×D₅₃	1	8	yes	—
518	#41	—	106	16	no	—
519	#41	—	212	16	no	—
520	#41	—	212	16	no	—
521	#41	—	212	16	no	—
522	#41	—	212	16	no	—
523	C₂×Q₈	—	2	8	no	—
524	C₂×Q₈	—	4	4	no	—
525	C₂×Q₈	—	4	16	no	—
526	C₂×Q₈	—	53	4	no	—
527	C₂×Q₈	—	53	8	no	—
528	C₂×Q₈	—	53	8	no	—
529	C₂×Q₈	—	212	16	no	—
530	C₂×Q₈	C₂³×D₅₃	1	8	yes	—
531	C₂×Q₈	C₂³×D₅₃	1	8	yes	—
532	C₂×D₄	—	2	8	no	—
533	C₂×D₄	—	4	16	no	—
534	C₂×D₄	—	4	4	no	—
535	C₂×D₄	—	53	16	no	—
536	C₂×D₄	—	53	8	no	—
537	C₂×D₄	—	53	32	no	—
538	C₂×D₄	—	53	16	no	—
539	C₂×D₄	—	53	4	no	—
540	C₂×D₄	—	53	8	no	—
541	C₂×D₄	—	53	8	no	—
542	C₂×D₄	—	212	16	no	—
543	C₂×D₄	—	212	32	no	—
544	C₂×D₄	—	212	16	no	—
545	C₂×D₄	C₂³×D₅₃	1	8	yes	—
546	C₂×D₄	C₂³×D₅₃	1	8	yes	—
547	C₂²×C₄	—	2	4	no	—
548	C₂²×C₄	—	4	4	no	—
549	C₂²×C₄	—	4	4	no	—
550	C₂²×C₄	—	53	16	no	—
551	C₂²×C₄	—	53	8	no	—
552	C₂²×C₄	—	53	8	no	—
553	C₂²×C₄	—	53	16	no	—
554	C₂²×C₄	—	53	16	no	—
555	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
556	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
557	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
558	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
559	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
560	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
561	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
562	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
563	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
564	C₄⋊C₄	—	53	16	no	—
565	C₄⋊C₄	—	53	8	no	—
566	C₄⋊C₄	—	53	32	no	—
567	C₄⋊C₄	—	53	16	no	—
568	C₄⋊C₄	—	53	16	no	—
569	C₄⋊C₄	—	53	8	no	—
570	C₄⋊C₄	—	53	8	no	—
571	C₄⋊C₄	—	53	16	no	—
572	C₂²⋊C₄	—	53	8	no	—
573	C₂²⋊C₄	—	53	16	no	—
574	C₂²⋊C₄	—	53	32	no	—
575	C₂²⋊C₄	—	53	16	no	—
576	C₂²⋊C₄	—	53	16	no	—
577	C₂²⋊C₄	—	53	32	no	—
578	C₂²⋊C₄	—	53	16	no	—
579	C₂²⋊C₄	—	53	16	no	—
580	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
581	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
582	C₄²	—	53	16	no	—
583	C₄²	—	53	8	no	—
584	C₄²	—	53	16	no	—
585	C₄²	C₂³×D₅₃	1	2	yes	—
586	C₄²	C₂³×D₅₃	1	2	yes	—
587	C₄²	C₂³×D₅₃	1	2	yes	—
588	C₄²	C₂³×D₅₃	1	4	yes	—
589	C₄²	C₂³×D₅₃	1	4	yes	—
590	C₄²	C₂³×D₅₃	1	2	yes	—
591	C₂³	—	2	2	no	—
592	C₂³	—	2	4	no	—
593	C₂³	—	2	8	no	—
594	C₂³	—	2	16	no	—
595	C₂³	—	4	8	no	—
596	C₂³	—	4	8	no	—
597	C₂³	—	4	8	no	—
598	C₂³	—	8	16	no	—
599	C₂³	—	53	8	no	—
600	C₂³	—	212	8	no	—
601	C₂³	—	212	16	no	—
602	C₂³	C₂⁴×D₅₃	1	8	yes	—
603	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
604	C₂³	C₂⁴×D₅₃	1	2	yes	—
605	Q₈	—	2	16	no	—
606	Q₈	—	4	8	no	—
607	Q₈	—	106	8	no	—
608	Q₈	—	212	8	no	—
609	Q₈	—	212	8	no	—
610	D₄	—	2	16	no	—
611	D₄	—	4	8	no	—
612	D₄	—	106	16	no	—
613	D₄	—	106	32	no	—
614	D₄	—	106	8	no	—
615	D₄	—	212	8	no	—
616	D₄	—	212	8	no	—
617	D₄	—	212	32	no	—
618	D₄	—	212	16	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	8	no	—
621	C₂×C₄	—	2	8	no	—
622	C₂×C₄	—	2	8	no	—
623	C₂×C₄	—	4	2	no	—
624	C₂×C₄	—	4	4	no	—
625	C₂×C₄	—	4	2	no	—
626	C₂×C₄	—	4	8	no	—
627	C₂×C₄	—	53	16	no	—
628	C₂×C₄	—	53	8	no	—
629	C₂×C₄	—	106	16	no	—
630	C₂×C₄	—	212	16	no	—
631	C₂×C₄	—	212	16	no	—
632	C₂×C₄	—	212	16	no	—
633	C₂×C₄	—	212	8	no	—
634	C₂×C₄	#13133262	1	4	yes	—
635	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
636	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
637	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
638	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
639	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
640	C₂²	—	2	8	no	—
641	C₂²	—	2	2	no	—
642	C₂²	—	2	2	no	—
643	C₂²	—	2	2	no	—
644	C₂²	—	2	8	no	—
645	C₂²	—	2	16	no	—
646	C₂²	—	8	8	no	—
647	C₂²	—	8	16	no	—
648	C₂²	—	848	8	no	—
649	C₂²	—	106	8	no	—
650	C₂²	—	106	16	no	—
651	C₂²	C₂×#13133262	1	2	yes	—
652	C₂²	#16811459	1	4	yes	—
653	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
654	C₄	—	2	2	no	—
655	C₄	—	2	8	no	—
656	C₄	—	2	4	no	—
657	C₄	—	2	4	no	—
658	C₄	—	106	8	no	—
659	C₄	—	106	16	no	—
660	C₄	C₂×#13133262	1	4	yes	—
661	C₂	—	2	2	no	—
662	C₂	—	2	4	no	—
663	C₂	—	4	8	no	—
664	C₂	—	212	8	no	—
665	C₂	C₂²×#13133262	1	1	yes	characteristic
666	C₂	C₂×#16811459	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁶
Fitting	C₂²×#124×C₅₃	C₂
hypercenter	C₂²×#124	D₅₃
hypocenter	C₅₃	#62824
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×#124

Complements

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

N	H ≅ G/N	type
1. #20289287	C₂	semidirect
2. #20289234	C₂	semidirect
3. #20289072	C₂	semidirect
4. #20288670	C₂	semidirect
5. #20288609	C₂	semidirect
6. #20288505	C₂	semidirect
7. #20288404	C₂	semidirect
8. #20288366	C₂	semidirect
9. #20288350	C₂	semidirect
10. #20288011	C₂	semidirect
11. C₂²×#124×C₅₃	C₂	semidirect
12. C₂×#13133265	C₂²	semidirect
13. C₂×#13133262	C₂²	semidirect
14. #16811373	C₂²	semidirect
15. #16811367	C₂²	semidirect
16. #16811365	C₂²	semidirect
17. #16811355	C₂²	semidirect
18. #16811342	C₂²	semidirect
19. #16811335	C₂²	semidirect
20. #16811303	C₂²	semidirect
21. #16811299	C₂²	semidirect
22. #16811299	C₂²	semidirect
23. Q₈×C₂²⋊D₅₃	C₂²	semidirect
24. #16811284	C₂²	semidirect
25. #16811281	C₂²	semidirect
26. #16811267	C₂²	semidirect
27. #16811266	C₂²	semidirect
28. #16811263	C₂²	semidirect
29. #16811263	C₂²	semidirect
30. D₄×C₂²⋊D₅₃	C₂²	semidirect
31. #16811249	C₂²	semidirect
32. #16811242	C₂²	semidirect
33. #16811237	C₂²	semidirect
34. #16811211	C₂²	semidirect
35. #16811211	C₂²	semidirect
36. #16811207	C₂²	semidirect
37. #16811197	C₂²	semidirect
38. #16811197	C₂²	semidirect
39. #16811193	C₂²	semidirect
40. #16811191	C₂²	semidirect
42. #16811185	C₂²	semidirect
43. #16811182	C₂²	semidirect
44. #16811180	C₂²	semidirect
45. #16811180	C₂²	semidirect
46. #16811173	C₂²	semidirect
47. #16811173	C₂²	semidirect
48. #16811172	C₂²	semidirect
49. #16811157	C₂²	semidirect
50. #16811157	C₂²	semidirect
51. #16811157	C₂²	semidirect
52. #16811152	C₂²	semidirect
53. #124×D₅₃	C₂²	semidirect
54. #16811134	C₂²	semidirect
55. #16811134	C₂²	semidirect
56. #16811134	C₂²	semidirect
57. C₂×#13133186	C₂²	semidirect
58. #16811101	C₂²	semidirect
59. #16811098	C₂²	semidirect
60. #16811096	C₂²	semidirect
61. C₄×#11719945	C₂²	semidirect
62. C₄×#11719945	C₂²	semidirect
63. C₂³×Q₈×C₅₃	C₂²	semidirect
65. C₂×#124×C₅₃	C₂²	semidirect
66. C₂×#124×C₅₃	C₂²	semidirect
67. C₂×#124×C₅₃	C₂²	semidirect
68. C₂×#124×C₅₃	C₂²	semidirect
69. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
70. C₂²×C₄²×C₅₃	C₂²	semidirect
73. C₂×#11719949	C₂³	semidirect
78. C₂×Q₈×D₅₃	C₂³	semidirect
82. #13133217	C₂³	semidirect
83. #13133217	C₂³	semidirect
84. #13133215	C₂³	semidirect
85. #13133215	C₂³	semidirect
86. #13133214	C₂³	semidirect
88. #13133211	C₂³	semidirect
89. #13133210	C₂³	semidirect
90. #13133210	C₂³	semidirect
91. #13133209	C₂³	semidirect
92. #13133209	C₂³	semidirect
93. #13133206	C₂³	semidirect
94. #13133206	C₂³	semidirect
95. #13133205	C₂³	semidirect
96. #13133205	C₂³	semidirect
98. #13133203	C₂³	semidirect
99. #13133203	C₂³	semidirect
100. #13133201	C₂³	semidirect
104. #13133199	C₂³	semidirect
105. #13133199	C₂³	semidirect
111. #13133194	C₂³	semidirect
112. #13133194	C₂³	semidirect
113. #13133193	C₂³	semidirect
114. #13133193	C₂³	semidirect
119. #13133190	C₂³	semidirect
120. #13133190	C₂³	semidirect
123. #13133187	C₂³	semidirect
128. Q₈×Dic₅₃	C₂³	semidirect
129. C₄×Dic₁₀₆	C₂³	semidirect
130. C₄×Dic₁₀₆	C₂³	semidirect
131. C₄×Dic₁₀₆	C₂³	semidirect
132. C₄×D₂₁₂	C₂³	semidirect
133. C₄×D₂₁₂	C₂³	semidirect
134. C₄×D₂₁₂	C₂³	semidirect
140. #13133177	C₂³	semidirect
141. #13133177	C₂³	semidirect
145. #13133174	C₂³	semidirect
146. #13133174	C₂³	semidirect
147. #13133173	C₂³	semidirect
150. C₂²⋊C₄×D₅₃	C₂³	semidirect
151. C₄²×D₅₃	C₂³	semidirect
338. C₂²×#124	D₅₃	semidirect
347. C₂³×Q₈	C₂×D₅₃	semidirect
389. C₂×#124	C₂×D₅₃	semidirect
390. C₂×#124	C₂×D₅₃	semidirect
391. C₂×#124	C₂×D₅₃	semidirect
392. C₂×#124	C₂×D₅₃	semidirect
402. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
403. C₂²×C₄²	C₂×D₅₃	semidirect
404. C₅₃	#62824	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20289287	2
#20289234	2
#20289072	2
#20288670	2
#20288609	2
#20288505	2
#20288404	2
#20288366	2
#20288350	2
#20288011	2
C₂²×#124×C₅₃	2
#62824	53