Subgroups — #35559096

Order 21248.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918003	C₂	1	1	yes	maximal, characteristic, complemented
2	#24916314	C₂	1	1	yes	maximal, characteristic
3	#24909623	C₂	1	2	yes	maximal, complemented
4	#24909619	C₂	1	2	yes	maximal, complemented
5	#24909609	C₂	1	2	yes	maximal, complemented
6	#24909599	C₂	1	2	yes	maximal, complemented
7	#24909252	C₂	1	1	yes	maximal, characteristic, complemented
8	#24909221	C₂	1	1	yes	maximal, characteristic, complemented
9	#24909178	C₂	1	1	yes	maximal, characteristic
10	C₂×#19127503	C₂	1	1	yes	maximal, characteristic
11	C₈₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₆₆₄	C₂²	1	1	yes	characteristic
13	#19128329	—	2	1	no	—
14	#19128175	—	2	1	no	—
15	C₂×#14581732	C₂²	1	1	yes	characteristic
16	#19128108	—	2	2	no	—
17	#19128105	C₂²	1	2	yes	complemented
18	#19128102	C₂²	1	2	yes	complemented
19	#19127812	—	2	2	no	—
20	#19127783	C₂²	1	2	yes	—
21	#19127783	C₂²	1	2	yes	—
22	#19127744	—	2	1	no	—
23	C₂×#14581726	C₂²	1	1	yes	characteristic
24	#19127705	—	2	1	no	—
25	#19127656	C₂²	1	1	yes	characteristic
26	#19127528	C₂²	1	2	yes	complemented
27	#19127526	C₂²	1	2	yes	complemented
28	Dic₈₃⋊C₁₆	C₂²	1	2	yes	complemented
29	Dic₈₃⋊C₁₆	C₂²	1	2	yes	complemented
30	#19127503	C₂²	1	2	yes	—
31	#19127503	C₂²	1	2	yes	—
32	#19127501	C₂²	1	2	yes	—
33	#19127501	C₂²	1	2	yes	—
34	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
35	#427×C₈₃	C₂²	1	1	yes	characteristic
36	#427×C₈₃	C₂²	1	1	yes	characteristic
37	#367×C₈₃	C₂²	1	1	yes	characteristic, complemented
38	#367×C₈₃	C₂²	1	1	yes	characteristic, complemented
39	#366×C₈₃	C₂²	1	1	yes	characteristic, complemented
40	#366×C₈₃	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₃₃₂	—	2	1	no	—
42	C₂×D₆₆₄	—	2	1	no	—
43	C₂×D₆₆₄	—	2	2	no	—
44	C₂×D₆₆₄	Q₈	1	4	yes	complemented
45	C₂×D₆₆₄	C₂³	1	2	yes	—
46	C₂×#13050574	D₄	1	1	yes	characteristic
47	#14581792	—	4	1	no	—
48	#14581791	—	2	2	no	—
49	#14581784	—	4	1	no	—
50	#14581783	—	2	2	no	—
51	#14581769	—	4	2	no	—
52	C₄×D₃₃₂	—	4	1	no	—
53	C₄×D₃₃₂	—	4	1	no	—
54	#14581754	—	2	2	no	—
55	#14581754	—	2	2	no	—
56	#14581751	D₄	1	1	yes	characteristic
57	C₂×C₄⋊Dic₈₃	—	2	1	no	—
58	C₂²⋊C₄×D₈₃	—	4	2	no	—
59	#14581732	—	2	2	no	—
60	C₂×C₈₃⋊C₁₆	C₂³	1	2	yes	—
61	#14581726	—	2	1	no	—
62	#14581726	C₂³	1	2	yes	—
63	#14581725	—	2	2	no	—
64	#14581723	—	2	2	no	—
65	#14581712	—	2	2	no	—
66	#14581712	—	2	2	no	—
67	#14581702	D₄	1	2	yes	—
68	C₈×Dic₈₃	C₂³	1	2	yes	—
69	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
70	#117×C₈₃	—	2	1	no	—
71	#117×C₈₃	—	2	1	no	—
72	C₂×C₁₆×C₈₃	—	2	2	no	—
73	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
75	#107×C₈₃	C₂³	1	1	yes	characteristic
76	#107×C₈₃	C₂³	1	1	yes	characteristic
77	#107×C₈₃	C₂³	1	1	yes	characteristic
78	#107×C₈₃	C₂³	1	1	yes	characteristic
79	#106×C₈₃	—	2	1	no	—
80	#106×C₈₃	—	2	1	no	—
81	C₂³×D₈₃	—	4	1	no	—
82	C₂×Dic₁₆₆	—	4	2	no	—
83	C₂×D₃₃₂	—	2	4	no	—
84	C₂×D₃₃₂	—	2	2	no	—
85	C₂×D₃₃₂	—	4	1	no	—
86	C₂×D₃₃₂	—	4	2	no	—
87	C₂²×Dic₈₃	—	4	1	no	—
88	C₂×C₄×D₈₃	—	4	2	no	—
89	C₂×C₄×D₈₃	—	4	2	no	—
90	D₆₆₄	—	2	4	no	—
91	D₆₆₄	—	2	2	no	—
92	D₆₆₄	C₂×Q₈	1	4	yes	—
93	#13050574	—	2	2	no	—
94	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
95	C₄⋊Dic₈₃	—	2	2	no	—
96	C₄⋊Dic₈₃	—	4	1	no	—
97	C₄⋊Dic₈₃	—	4	2	no	—
98	C₄⋊Dic₈₃	D₈	1	1	yes	characteristic
99	C₄⋊Dic₈₃	D₈	1	1	yes	characteristic
100	Dic₈₃⋊C₄	—	4	2	no	—
101	Dic₈₃⋊C₄	—	4	2	no	—
102	C₂²⋊Dic₈₃	D₈	1	2	yes	—
103	(C₂×D₈₃)⋊C₄	—	4	2	no	—
104	(C₂×D₈₃)⋊C₄	—	4	2	no	—
105	C₄×Dic₈₃	C₂×D₄	1	2	yes	—
106	C₈₃⋊C₁₆	—	2	2	no	—
107	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
110	C₂×C₈×C₈₃	#41	1	2	yes	—
111	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₈₃	—	2	1	no	—
113	C₄⋊C₄×C₈₃	—	2	1	no	—
114	C₄⋊C₄×C₈₃	—	2	1	no	—
115	C₄⋊C₄×C₈₃	—	2	1	no	—
116	C₂²⋊C₄×C₈₃	—	4	1	no	—
117	C₂²⋊C₄×C₈₃	—	4	1	no	—
118	C₄²×C₈₃	—	4	1	no	—
119	C₄²×C₈₃	—	4	1	no	—
120	C₁₆×C₈₃	—	2	1	no	—
121	C₁₆×C₈₃	—	2	1	no	—
122	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₈₃	—	4	2	no	—
124	C₂²×D₈₃	—	4	4	no	—
125	C₂²×D₈₃	—	16	1	no	—
126	C₂²×D₈₃	—	16	1	no	—
127	C₂×C₄×C₈₃	—	4	1	no	—
128	C₂×C₄×C₈₃	—	4	1	no	—
129	C₂×C₄×C₈₃	—	4	1	no	—
130	C₂×C₄×C₈₃	—	4	1	no	—
131	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₈₃	#137	1	2	yes	—
134	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₈₃	—	4	2	no	—
136	C₂×Dic₈₃	—	16	1	no	—
137	C₂×Dic₈₃	C₂×D₈	1	2	yes	—
138	C₂×Dic₈₃	C₂×D₈	1	2	yes	—
139	D₃₃₂	—	2	4	no	—
140	D₃₃₂	—	4	2	no	—
141	D₃₃₂	—	4	4	no	—
142	C₄×D₈₃	—	16	2	no	—
143	C₄×D₈₃	—	16	2	no	—
144	Dic₁₆₆	—	16	2	no	—
145	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
146	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
147	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
148	C₈₃⋊C₈	—	2	2	no	—
149	C₂²×C₈₃	—	2	1	no	—
150	C₂²×C₈₃	—	2	1	no	—
151	C₂²×C₈₃	#474	1	2	yes	—
152	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₈₃	—	4	4	no	—
154	C₂×D₈₃	—	8	2	no	—
155	C₂×D₈₃	—	8	2	no	—
156	C₂×D₈₃	—	16	4	no	—
157	C₄×C₈₃	—	8	1	no	—
158	C₄×C₈₃	—	8	1	no	—
159	C₄×C₈₃	—	8	1	no	—
160	C₄×C₈₃	—	8	1	no	—
161	C₄×C₈₃	#474	1	2	yes	—
162	C₄×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
163	C₄×C₈₃	C₂×#137	1	1	yes	characteristic
164	Dic₈₃	—	2	2	no	—
165	Dic₈₃	—	2	2	no	—
166	Dic₈₃	—	8	2	no	—
167	#33631	—	83	1	no	maximal, Sylow
168	C₂×C₈₃	—	2	2	no	—
169	C₂×C₈₃	C₂×#474	1	1	yes	characteristic
170	C₂×C₈₃	#3415	1	1	yes	characteristic
171	C₂×C₈₃	#3416	1	1	yes	characteristic
172	D₈₃	—	8	4	no	—
173	#3164	—	83	1	no	—
174	#2948	—	83	1	no	—
175	#2228	—	83	2	no	—
176	#2228	—	83	2	no	—
177	#2226	—	83	2	no	—
178	#2224	—	83	2	no	—
179	#2159	—	83	1	no	—
180	#2158	D₈₃	1	1	yes	characteristic, complemented
181	#2151	—	83	1	no	—
182	#2140	—	83	1	no	—
183	C₂×#357	—	83	1	no	—
184	C₈₃	#33631	1	1	yes	characteristic, complemented, Sylow
185	C₂²×D₈	—	83	1	no	—
186	#544	—	166	1	no	—
187	#518	—	166	1	no	—
188	C₂×M₃₂	—	83	1	no	—
189	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
190	#501	—	166	2	no	—
191	#500	—	83	2	no	—
192	#499	—	83	2	no	—
193	#442	—	166	2	no	—
194	C₄×D₈	—	83	2	no	—
195	C₄×D₈	—	83	2	no	—
196	#428	—	166	1	no	—
197	#427	C₂×D₈₃	1	1	yes	characteristic
198	#427	C₂×D₈₃	1	1	yes	characteristic
199	C₂×#107	—	83	1	no	—
200	#418	—	166	1	no	—
201	#405	—	83	1	no	—
202	#367	—	83	2	no	—
203	#367	C₂×D₈₃	1	1	yes	characteristic, complemented
204	#367	C₂×D₈₃	1	1	yes	characteristic, complemented
205	#366	—	83	2	no	—
206	#366	C₂×D₈₃	1	1	yes	characteristic, complemented
207	#366	C₂×D₈₃	1	1	yes	characteristic, complemented
208	C₄⋊C₁₆	—	83	2	no	—
209	C₄⋊C₁₆	—	83	2	no	—
210	#357	—	83	2	no	—
211	#357	—	83	2	no	—
212	#357	—	83	2	no	—
213	#357	—	83	2	no	—
214	C₂²×D₄	—	166	1	no	—
215	C₂×D₈	—	83	4	no	—
216	C₂×D₈	—	83	2	no	—
217	C₂×D₈	—	166	1	no	—
218	C₂×D₈	—	166	2	no	—
219	C₂×M₁₆	—	83	1	no	—
220	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
221	C₄⋊Q₈	—	332	1	no	—
222	C₄⋊Q₈	—	166	2	no	—
223	#125	—	332	1	no	—
224	#125	—	166	2	no	—
225	C₂²⋊Q₈	—	332	2	no	—
226	C₄×D₄	—	332	1	no	—
227	C₄×D₄	—	332	1	no	—
228	C₄×D₄	—	166	2	no	—
229	C₄×D₄	—	166	2	no	—
230	#117	—	2	1	no	—
231	#117	—	2	1	no	—
232	#117	—	83	1	no	—
233	C₂×C₄⋊C₄	—	166	1	no	—
234	C₂×C₂²⋊C₄	—	332	2	no	—
235	M₃₂	—	166	2	no	—
236	C₂×C₁₆	—	2	2	no	—
237	C₂×C₁₆	—	83	2	no	—
238	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
239	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
240	#107	—	83	2	no	—
241	#107	—	166	2	no	—
242	#107	—	166	1	no	—
243	#107	C₂²×D₈₃	1	1	yes	characteristic
244	#107	C₂²×D₈₃	1	1	yes	characteristic
245	#107	C₂²×D₈₃	1	1	yes	characteristic
246	#107	C₂²×D₈₃	1	1	yes	characteristic
247	#106	—	2	1	no	—
248	#106	—	2	1	no	—
249	#106	—	166	2	no	—
250	#102	—	166	2	no	—
251	#102	—	166	2	no	—
252	#97	—	83	2	no	—
253	C₄×C₈	—	83	2	no	—
254	C₂⁴	—	332	1	no	—
255	C₂×Q₈	—	332	2	no	—
256	C₂×D₄	—	332	1	no	—
257	C₂×D₄	—	332	2	no	—
258	C₂×D₄	—	166	4	no	—
259	C₂×D₄	—	166	2	no	—
260	C₂²×C₄	—	332	2	no	—
261	C₂²×C₄	—	332	2	no	—
262	C₂²×C₄	—	332	1	no	—
263	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
264	D₈	—	83	4	no	—
265	D₈	—	166	4	no	—
266	D₈	—	166	2	no	—
267	M₁₆	—	166	2	no	—
268	C₂×C₈	—	83	2	no	—
269	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
270	C₂×C₈	Q₈×D₈₃	1	2	yes	—
271	C₂×C₈	#13050590	1	2	yes	—
272	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	332	2	no	—
278	C₄⋊C₄	—	332	2	no	—
279	C₄⋊C₄	—	332	2	no	—
280	C₄⋊C₄	—	332	1	no	—
281	C₄⋊C₄	—	83	1	no	—
282	C₄⋊C₄	—	83	1	no	—
283	C₄⋊C₄	—	166	2	no	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	332	2	no	—
287	C₂²⋊C₄	—	332	2	no	—
288	C₂²⋊C₄	—	83	2	no	—
289	C₄²	—	4	1	no	—
290	C₄²	—	4	1	no	—
291	C₄²	—	83	2	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	2	1	no	—
294	C₁₆	—	166	2	no	—
295	C₂³	—	1328	1	no	—
296	C₂³	—	1328	1	no	—
297	C₂³	—	332	4	no	—
298	C₂³	—	332	2	no	—
299	C₂³	D₈×D₈₃	1	1	yes	characteristic
300	Q₈	—	1328	2	no	—
301	D₄	—	332	4	no	—
302	D₄	—	332	2	no	—
303	D₄	—	166	4	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	1328	2	no	—
309	C₂×C₄	—	1328	2	no	—
310	C₂×C₄	—	1328	1	no	—
311	C₂×C₄	—	332	2	no	—
312	C₂×C₄	—	83	2	no	—
313	C₂×C₄	—	83	2	no	—
314	C₂×C₄	#14581770	1	2	yes	—
315	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
316	C₂×C₄	#14581824	1	2	yes	—
317	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
318	C₈	—	166	2	no	—
319	C₈	#14581770	1	2	yes	—
320	C₈	C₂×Q₈×D₈₃	1	1	yes	characteristic
321	C₈	C₂×#13050590	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	1328	4	no	—
325	C₂²	—	332	4	no	—
326	C₂²	—	664	2	no	—
327	C₂²	—	664	2	no	—
328	C₂²	#19127980	1	2	yes	—
329	C₂²	C₂×D₈×D₈₃	1	1	yes	characteristic
330	C₄	—	8	1	no	—
331	C₄	—	8	1	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	664	2	no	—
335	C₄	—	166	2	no	—
336	C₄	—	166	2	no	—
337	C₄	#19127980	1	2	yes	—
338	C₄	C₂×#14581770	1	1	yes	characteristic
339	C₄	C₂×#14581824	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	664	4	no	—
342	C₂	C₂×#19127980	1	1	yes	characteristic
343	C₂	#24920601	1	1	yes	characteristic
344	C₂	#24920606	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₃
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#2158	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	C₂²×D₆₆₄	C₂²
hypercenter	#2158	D₈₃
hypocenter	C₈₃	#33631
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
1. #24918003	C₂	semidirect
3. #24909623	C₂	semidirect
4. #24909619	C₂	semidirect
5. #24909609	C₂	semidirect
6. #24909599	C₂	semidirect
7. #24909252	C₂	semidirect
8. #24909221	C₂	semidirect
11. C₈₃×#2158	C₂	semidirect
17. #19128105	C₂²	semidirect
18. #19128102	C₂²	semidirect
26. #19127528	C₂²	semidirect
27. #19127526	C₂²	semidirect
28. Dic₈₃⋊C₁₆	C₂²	semidirect
29. Dic₈₃⋊C₁₆	C₂²	semidirect
37. #367×C₈₃	C₂²	semidirect
38. #367×C₈₃	C₂²	semidirect
39. #366×C₈₃	C₂²	semidirect
40. #366×C₈₃	C₂²	semidirect
44. C₂×D₆₆₄	Q₈	semidirect
180. #2158	D₈₃	semidirect
184. C₈₃	#33631	semidirect
203. #367	C₂×D₈₃	semidirect
204. #367	C₂×D₈₃	semidirect
206. #366	C₂×D₈₃	semidirect
207. #366	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24918003	2
#24916314	2
#24909623	2
#24909619	2
#24909609	2
#24909599	2
#24909252	2
#24909221	2
#24909178	2
C₂×#19127503	2
C₈₃×#2158	2
#33631	83