Subgroups — #24920359

Order 10624.

374 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128407	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128366	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128109	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128098	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128004	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127998	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127990	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127990	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127988	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127980	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127829	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127814	C₂	1	1	yes	maximal, characteristic
13	#19127812	C₂	1	1	yes	maximal, characteristic
14	#19127783	C₂	1	1	yes	maximal, characteristic
15	#501×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₈₃	—	2	1	no	—
17	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
18	#14581791	C₂²	1	1	yes	characteristic
19	#14581791	C₂²	1	1	yes	characteristic
20	#14581791	C₂²	1	1	yes	characteristic
21	#14581783	C₂²	1	1	yes	characteristic
22	#14581772	—	2	1	no	—
23	#14581770	—	2	1	no	—
24	#14581770	—	2	1	no	—
25	#14581770	—	2	1	no	—
26	#14581770	—	2	1	no	—
27	#14581768	—	2	1	no	—
28	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
29	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
30	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
31	#14581754	C₂²	1	1	yes	characteristic
32	#14581754	C₂²	1	1	yes	characteristic
33	#14581754	C₂²	1	1	yes	characteristic
34	#14581754	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₈₃	—	2	1	no	—
36	#14581725	C₂²	1	1	yes	characteristic
37	#14581724	C₂²	1	1	yes	characteristic
38	#14581723	C₂²	1	1	yes	characteristic
39	#14581723	C₂²	1	1	yes	characteristic
40	#14581723	C₂²	1	1	yes	characteristic
41	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
44	#14581714	C₂²	1	1	yes	characteristic
45	#14581714	C₂²	1	1	yes	characteristic
46	#14581712	C₂²	1	1	yes	characteristic
47	#14581712	C₂²	1	1	yes	characteristic
48	#14581712	C₂²	1	1	yes	characteristic
49	#14581712	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	#125×C₈₃	C₂²	1	1	yes	characteristic
54	#107×C₈₃	C₂²	1	1	yes	characteristic
55	#107×C₈₃	C₂²	1	1	yes	characteristic
56	#106×C₈₃	C₂²	1	1	yes	characteristic
57	#106×C₈₃	C₂²	1	1	yes	characteristic
58	#97×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
63	Q₈×D₈₃	—	4	4	no	—
64	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
65	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	C₂×C₄×D₈₃	—	2	1	no	—
72	D₆₆₄	Q₈	1	4	yes	complemented
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₃	—	2	1	no	—
76	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
84	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
85	(C₂×D₈₃)⋊C₄	—	2	1	no	—
86	(C₂×D₈₃)⋊C₄	—	2	1	no	—
87	(C₂×D₈₃)⋊C₄	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
92	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₈₃	—	2	1	no	—
94	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₈₃	—	2	1	no	—
97	C₄⋊C₄×C₈₃	—	2	1	no	—
98	C₄⋊C₄×C₈₃	—	2	1	no	—
99	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₄²×C₈₃	C₂³	1	1	yes	characteristic
104	C₂²×D₈₃	—	2	1	no	—
105	C₂²×D₈₃	—	2	1	no	—
106	Q₈×C₈₃	—	4	2	no	—
107	C₂×C₄×C₈₃	—	2	1	no	—
108	C₂×C₄×C₈₃	—	2	1	no	—
109	C₂×C₄×C₈₃	—	2	1	no	—
110	C₂×C₄×C₈₃	—	2	1	no	—
111	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
114	C₂×Dic₈₃	—	2	1	no	—
115	C₂×Dic₈₃	—	2	1	no	—
116	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
120	D₃₃₂	—	2	1	no	—
121	D₃₃₂	—	2	1	no	—
122	D₃₃₂	C₂×Q₈	1	2	yes	—
123	D₃₃₂	C₂×Q₈	1	2	yes	complemented
124	C₄×D₈₃	—	4	2	no	—
125	C₄×D₈₃	—	4	2	no	—
126	C₄×D₈₃	—	4	2	no	—
127	C₄×D₈₃	—	4	2	no	—
128	C₄×D₈₃	—	8	1	no	—
129	C₄×D₈₃	—	8	1	no	—
130	Dic₁₆₆	—	2	2	no	—
131	Dic₁₆₆	—	4	2	no	—
132	Dic₁₆₆	—	4	2	no	—
133	Dic₁₆₆	—	8	1	no	—
134	C₈×C₈₃	—	2	1	no	—
135	C₈×C₈₃	C₂×Q₈	1	2	yes	complemented
136	C₈₃⋊C₈	—	2	1	no	—
137	C₈₃⋊C₈	—	2	1	no	—
138	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₈₃	—	2	2	no	—
140	C₂×D₈₃	—	2	2	no	—
141	C₂×D₈₃	—	4	1	no	—
142	C₂×D₈₃	—	4	1	no	—
143	C₂×D₈₃	—	4	1	no	—
144	C₂×D₈₃	—	4	1	no	—
145	C₄×C₈₃	—	2	1	no	—
146	C₄×C₈₃	—	2	1	no	—
147	C₄×C₈₃	—	4	1	no	—
148	C₄×C₈₃	—	4	1	no	—
149	C₄×C₈₃	—	4	1	no	—
150	C₄×C₈₃	—	4	1	no	—
151	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
152	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₈₃	—	2	1	no	—
154	Dic₈₃	—	2	1	no	—
155	Dic₈₃	—	2	1	no	—
156	Dic₈₃	—	4	1	no	—
157	Dic₈₃	—	4	1	no	—
158	Dic₈₃	AGL(1,ℤ/8)	1	2	yes	—
159	C₂×C₈₃	D₄×Q₈	1	1	yes	characteristic
160	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×C₈₃	#577	1	1	yes	characteristic
162	D₈₃	—	4	2	no	—
163	D₈₃	—	4	2	no	—
164	#3384	—	83	1	no	maximal, Sylow
165	C₈₃	#3384	1	1	yes	characteristic, complemented, Sylow
166	#554	—	83	1	no	—
167	D₄×Q₈	—	83	1	no	—
168	#501	—	83	1	no	—
169	#501	D₈₃	1	1	yes	characteristic, complemented
170	#498	—	83	1	no	—
171	#478	—	83	1	no	—
172	#478	—	83	1	no	—
173	#476	—	83	1	no	—
174	#476	—	83	1	no	—
175	#476	—	83	1	no	—
176	#474	—	83	1	no	—
177	#446	—	83	1	no	—
178	#442	—	83	1	no	—
179	#442	—	83	1	no	—
180	C₄×D₈	—	83	1	no	—
181	C₂²×Q₈	—	166	1	no	—
182	C₂×D₈	—	83	1	no	—
183	C₄⋊Q₈	—	83	1	no	—
184	C₄⋊Q₈	—	83	1	no	—
185	C₄⋊Q₈	—	83	1	no	—
186	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
187	#125	—	83	1	no	—
188	#125	C₂×D₈₃	1	1	yes	characteristic
189	C₂²⋊Q₈	—	166	1	no	—
190	C₂²⋊Q₈	—	166	1	no	—
191	C₂²⋊Q₈	—	166	1	no	—
192	C₂²⋊Q₈	—	166	1	no	—
193	C₂²⋊Q₈	—	166	1	no	—
194	C₂²⋊Q₈	—	166	1	no	—
195	C₄×Q₈	—	83	1	no	—
196	C₄×D₄	—	83	1	no	—
197	C₄×D₄	—	83	1	no	—
198	C₄×D₄	—	83	1	no	—
199	C₄×D₄	—	83	1	no	—
200	C₄×D₄	—	83	1	no	—
201	C₄×D₄	—	83	1	no	—
202	C₂×C₄⋊C₄	—	166	1	no	—
203	#107	—	83	1	no	—
204	#107	C₂×D₈₃	1	1	yes	characteristic
205	#107	C₂×D₈₃	1	1	yes	characteristic
206	#106	—	83	1	no	—
207	#106	—	83	1	no	—
208	#106	—	83	1	no	—
209	#106	—	83	1	no	—
210	#106	C₂×D₈₃	1	1	yes	characteristic
211	#106	C₂×D₈₃	1	1	yes	characteristic
212	C₄⋊C₈	—	83	1	no	—
213	C₄⋊C₈	—	83	1	no	—
214	C₄⋊C₈	—	83	1	no	—
215	#102	—	83	1	no	—
216	#102	—	83	1	no	—
217	#102	—	83	1	no	—
218	#102	—	83	1	no	—
219	#102	—	83	1	no	—
220	#102	—	83	1	no	—
221	#97	—	83	1	no	—
222	#97	C₂×D₈₃	1	1	yes	characteristic
223	C₄×C₈	—	83	1	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	332	4	no	—
226	C₂×Q₈	—	83	1	no	—
227	C₂×Q₈	—	166	1	no	—
228	C₂×Q₈	—	166	1	no	—
229	C₂×Q₈	—	166	1	no	—
230	C₂×D₄	—	83	1	no	—
231	C₂×D₄	—	83	1	no	—
232	C₂²×C₄	—	166	1	no	—
233	C₂²×C₄	—	166	1	no	—
234	C₂²×C₄	—	166	1	no	—
235	C₂²×C₄	—	166	1	no	—
236	C₂²×C₄	—	166	1	no	—
237	C₂²×C₄	—	166	1	no	—
238	D₈	—	83	4	no	—
239	C₂×C₈	—	83	1	no	—
240	C₂×C₈	—	83	1	no	—
241	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	83	1	no	—
247	C₄⋊C₄	—	83	1	no	—
248	C₄⋊C₄	—	83	1	no	—
249	C₄⋊C₄	—	83	1	no	—
250	C₄⋊C₄	—	83	1	no	—
251	C₄⋊C₄	—	166	1	no	—
252	C₄⋊C₄	—	166	1	no	—
253	C₄⋊C₄	—	166	1	no	—
254	C₄⋊C₄	—	166	1	no	—
255	C₄⋊C₄	—	166	1	no	—
256	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
260	C₂²⋊C₄	—	166	1	no	—
261	C₂²⋊C₄	—	166	1	no	—
262	C₂²⋊C₄	—	166	1	no	—
263	C₂²⋊C₄	—	166	1	no	—
264	C₂²⋊C₄	—	166	1	no	—
265	C₂²⋊C₄	—	166	1	no	—
266	C₄²	—	83	1	no	—
267	C₄²	—	83	1	no	—
268	C₄²	C₂²×D₈₃	1	1	yes	characteristic
269	C₂³	—	166	1	no	—
270	C₂³	—	166	1	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	332	2	no	—
273	Q₈	—	332	2	no	—
274	Q₈	—	664	1	no	—
275	Q₈	—	166	2	no	—
276	D₄	—	83	2	no	—
277	D₄	—	83	2	no	—
278	D₄	—	166	1	no	—
279	D₄	—	166	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	332	2	no	—
285	C₂×C₄	—	332	2	no	—
286	C₂×C₄	—	332	2	no	—
287	C₂×C₄	—	332	2	no	—
288	C₂×C₄	—	83	1	no	—
289	C₂×C₄	—	83	1	no	—
290	C₂×C₄	—	83	1	no	—
291	C₂×C₄	—	83	1	no	—
292	C₂×C₄	—	664	1	no	—
293	C₂×C₄	—	664	1	no	—
294	C₂×C₄	—	166	1	no	—
295	C₂×C₄	—	166	1	no	—
296	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
297	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	166	1	no	—
301	C₈	—	166	1	no	—
302	C₈	Q₈×D₈₃	1	2	yes	complemented
303	C₂²	—	332	1	no	—
304	C₂²	—	332	1	no	—
305	C₂²	—	332	1	no	—
306	C₂²	—	332	1	no	—
307	C₂²	—	166	2	no	—
308	C₂²	—	166	2	no	—
309	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	332	1	no	—
317	C₄	—	332	1	no	—
318	C₄	—	83	2	no	—
319	C₄	—	166	1	no	—
320	C₄	—	166	1	no	—
321	C₄	—	166	1	no	—
322	C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
323	C₄	#14581840	1	1	yes	characteristic
324	C₂	—	332	2	no	—
325	C₂	—	332	2	no	—
326	C₂	#19128366	1	1	yes	characteristic
327	C₂	AGL(1,ℤ/8)×D₈₃	1	1	yes	characteristic
328	C₂	#19128510	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#501×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×D₆₆₄	C₂²
hypercenter	#501	D₈₃
hypocenter	C₈₃	#3384
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
1. #19128407	C₂	semidirect
2. #19128366	C₂	semidirect
3. #19128109	C₂	semidirect
4. #19128098	C₂	semidirect
5. #19128004	C₂	semidirect
6. #19127998	C₂	semidirect
7. #19127990	C₂	semidirect
8. #19127990	C₂	semidirect
9. #19127988	C₂	semidirect
10. #19127980	C₂	semidirect
11. #19127829	C₂	semidirect
15. #501×C₈₃	C₂	semidirect
72. D₆₆₄	Q₈	semidirect
123. D₃₃₂	C₂×Q₈	semidirect
135. C₈×C₈₃	C₂×Q₈	semidirect
165. C₈₃	#3384	semidirect
169. #501	D₈₃	semidirect
302. C₈	Q₈×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128407	2
#19128366	2
#19128109	2
#19128098	2
#19128004	2
#19127998	2
#19127990	2
#19127988	2
#19127980	2
#19127829	2
#19127814	2
#19127812	2
#19127783	2
#501×C₈₃	2
#3384	83