Subgroups — #24917328

Order 10624.

376 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128331	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128183	C₂	1	1	yes	maximal, characteristic
3	#19128006	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128004	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127998	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127992	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127990	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127986	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127984	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127742	C₂	1	1	yes	maximal, characteristic, complemented
11	#106×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127726	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127701	C₂	1	1	yes	maximal, characteristic
14	#19127696	C₂	1	1	yes	maximal, characteristic
15	#479×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050591	C₂²	1	1	yes	characteristic
17	C₂×#13050573	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
19	#14581792	C₂²	1	1	yes	characteristic
20	#14581791	—	2	1	no	—
21	#14581791	C₂²	1	1	yes	characteristic
22	#14581783	C₂²	1	1	yes	characteristic
23	#14581782	—	2	1	no	—
24	#14581772	—	2	1	no	—
25	#14581770	—	2	1	no	—
26	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
27	C₄×D₃₃₂	—	2	1	no	—
28	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
29	#14581754	—	2	1	no	—
30	#14581754	C₂²	1	1	yes	characteristic
31	#14581750	C₂²	1	1	yes	characteristic
32	#14581749	—	2	1	no	—
33	#14581749	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₈₃	—	2	1	no	—
35	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
37	#14581725	C₂²	1	1	yes	characteristic
38	#14581724	C₂²	1	1	yes	characteristic
39	#14581724	C₂²	1	1	yes	characteristic
40	#14581723	C₂²	1	1	yes	characteristic
41	#14581723	C₂²	1	1	yes	characteristic
42	#14581723	C₂²	1	1	yes	characteristic
43	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
46	#14581716	C₂²	1	1	yes	characteristic
47	#14581716	C₂²	1	1	yes	characteristic
48	#14581714	C₂²	1	1	yes	characteristic
49	#14581714	C₂²	1	1	yes	characteristic
50	#14581712	C₂²	1	1	yes	characteristic
51	#14581712	C₂²	1	1	yes	characteristic
52	#125×C₈₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
54	#107×C₈₃	C₂²	1	1	yes	characteristic
55	#106×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	#103×C₈₃	C₂²	1	1	yes	characteristic
59	#13050591	—	2	2	no	—
60	#13050591	Q₈	1	4	yes	complemented
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×Dic₁₆₆	—	2	1	no	—
63	C₂×D₃₃₂	—	2	1	no	—
64	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₈₃	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
71	#13050573	—	2	2	no	—
72	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₈×D₈₃	—	2	2	no	—
75	C₄⋊Dic₈₃	—	2	1	no	—
76	C₄⋊Dic₈₃	—	2	1	no	—
77	C₄⋊Dic₈₃	—	2	1	no	—
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
86	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
91	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₃	—	2	1	no	—
99	C₄⋊C₄×C₈₃	—	2	1	no	—
100	C₄⋊C₄×C₈₃	—	2	1	no	—
101	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₄²×C₈₃	—	2	1	no	—
105	C₄²×C₈₃	C₂³	1	1	yes	characteristic
106	C₂²×D₈₃	—	2	1	no	—
107	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
108	Q₈×C₈₃	—	2	1	no	—
109	Q₈×C₈₃	C₂×Q₈	1	2	yes	—
110	C₂×C₄×C₈₃	—	2	1	no	—
111	C₂×C₄×C₈₃	—	2	1	no	—
112	C₂×C₄×C₈₃	—	2	1	no	—
113	C₂×C₄×C₈₃	—	2	1	no	—
114	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₈₃	—	2	1	no	—
118	C₂×Dic₈₃	—	2	1	no	—
119	C₂×Dic₈₃	—	2	1	no	—
120	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
123	D₃₃₂	—	2	4	no	—
124	D₃₃₂	—	2	1	no	—
125	D₃₃₂	C₂×Q₈	1	2	yes	—
126	C₄×D₈₃	—	2	4	no	—
127	C₄×D₈₃	—	8	1	no	—
128	C₄×D₈₃	—	8	1	no	—
129	C₄×D₈₃	—	8	1	no	—
130	C₄×D₈₃	—	8	1	no	—
131	C₄×D₈₃	C₂×Q₈	1	2	yes	—
132	C₄×D₈₃	#41	1	2	yes	—
133	Dic₁₆₆	—	8	1	no	—
134	Dic₁₆₆	—	8	1	no	—
135	C₈×C₈₃	—	2	1	no	—
136	C₈×C₈₃	—	2	1	no	—
137	C₈₃⋊C₈	—	2	1	no	—
138	C₈₃⋊C₈	—	2	1	no	—
139	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₈₃	—	2	1	no	—
141	C₂×D₈₃	—	2	1	no	—
142	C₂×D₈₃	—	2	2	no	—
143	C₂×D₈₃	—	4	1	no	—
144	C₂×D₈₃	—	4	1	no	—
145	C₂×D₈₃	C₂²⋊Q₈	1	2	yes	—
146	C₄×C₈₃	—	2	2	no	—
147	C₄×C₈₃	—	2	1	no	—
148	C₄×C₈₃	—	2	1	no	—
149	C₄×C₈₃	—	4	1	no	—
150	C₄×C₈₃	—	4	1	no	—
151	C₄×C₈₃	—	4	1	no	—
152	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
153	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
154	Dic₈₃	—	2	1	no	—
155	Dic₈₃	—	2	1	no	—
156	Dic₈₃	—	4	1	no	—
157	Dic₈₃	—	4	1	no	—
158	Dic₈₃	—	4	1	no	—
159	Dic₈₃	C₂²⋊Q₈	1	2	yes	—
160	C₂×C₈₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₂×C₈₃	#577	1	1	yes	characteristic
162	C₂×C₈₃	#577	1	1	yes	characteristic
163	D₈₃	—	2	2	no	—
164	D₈₃	—	4	2	no	—
165	#3084	—	83	1	no	maximal, Sylow
166	C₈₃	#3084	1	1	yes	characteristic, complemented, Sylow
167	#544	—	83	1	no	—
168	#520	—	83	1	no	—
169	#479	—	83	1	no	—
170	#479	D₈₃	1	1	yes	characteristic, complemented
171	#478	—	83	1	no	—
172	#478	—	83	1	no	—
173	#476	—	83	1	no	—
174	#476	—	83	1	no	—
175	#475	—	83	1	no	—
176	#475	—	83	1	no	—
177	#428	—	83	1	no	—
178	C₂×#106	—	83	1	no	—
179	#424	—	83	1	no	—
180	#417	—	83	1	no	—
181	#416	—	83	1	no	—
182	C₂×#41	—	83	1	no	—
183	C₂×M₁₆	—	83	1	no	—
184	C₂²×C₈	—	83	1	no	—
185	C₄⋊Q₈	—	83	1	no	—
186	C₄⋊Q₈	—	83	1	no	—
187	C₄⋊Q₈	—	166	1	no	—
188	#125	—	83	1	no	—
189	#125	—	166	1	no	—
190	#125	C₂×D₈₃	1	1	yes	characteristic
191	C₂²⋊Q₈	—	166	1	no	—
192	C₂²⋊Q₈	—	166	1	no	—
193	C₄×Q₈	—	83	1	no	—
194	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
195	C₄×D₄	—	83	1	no	—
196	C₄×D₄	—	83	1	no	—
197	C₄×D₄	—	166	1	no	—
198	C₄×D₄	—	166	1	no	—
199	#117	—	83	1	no	—
200	#117	—	83	1	no	—
201	#117	—	166	1	no	—
202	C₂×C₄⋊C₄	—	83	1	no	—
203	C₂×C₄⋊C₄	—	83	1	no	—
204	C₂×C₄⋊C₄	—	166	1	no	—
205	#107	—	83	1	no	—
206	#107	C₂×D₈₃	1	1	yes	characteristic
207	#106	—	83	1	no	—
208	#106	—	83	1	no	—
209	#106	—	83	1	no	—
210	#106	—	83	1	no	—
211	#106	—	83	1	no	—
212	#106	C₂×D₈₃	1	1	yes	characteristic
213	C₄⋊C₈	—	83	1	no	—
214	C₄⋊C₈	—	83	1	no	—
215	C₄⋊C₈	—	83	1	no	—
216	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
217	#103	—	83	1	no	—
218	#103	—	83	1	no	—
219	#103	C₂×D₈₃	1	1	yes	characteristic
220	#103	C₂×D₈₃	1	1	yes	characteristic
221	#102	—	83	1	no	—
222	#102	—	83	1	no	—
223	#102	—	83	1	no	—
224	#102	—	83	1	no	—
225	#41	—	83	4	no	—
226	#41	—	166	2	no	—
227	C₂×Q₈	—	166	1	no	—
228	C₂×Q₈	—	166	1	no	—
229	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
230	C₂×D₄	—	83	1	no	—
231	C₂×D₄	—	166	1	no	—
232	C₂²×C₄	—	83	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	M₁₆	—	166	2	no	—
239	C₂×C₈	—	83	1	no	—
240	C₂×C₈	—	83	1	no	—
241	C₂×C₈	—	166	2	no	—
242	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	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	—	83	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	—	166	1	no	—
257	C₄⋊C₄	—	166	1	no	—
258	C₄⋊C₄	—	166	1	no	—
259	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
262	C₂²⋊C₄	—	83	1	no	—
263	C₂²⋊C₄	—	83	1	no	—
264	C₂²⋊C₄	—	166	1	no	—
265	C₂²⋊C₄	—	166	1	no	—
266	C₂²⋊C₄	—	166	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	83	1	no	—
269	C₄²	—	166	1	no	—
270	C₄²	—	166	1	no	—
271	C₄²	C₂²×D₈₃	1	1	yes	characteristic
272	C₂³	—	83	1	no	—
273	C₂³	—	166	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	664	1	no	—
276	Q₈	—	664	1	no	—
277	Q₈	Q₈×D₈₃	1	2	yes	—
278	D₄	—	83	2	no	—
279	D₄	—	166	4	no	—
280	D₄	—	166	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	83	1	no	—
286	C₂×C₄	—	83	2	no	—
287	C₂×C₄	—	83	2	no	—
288	C₂×C₄	—	83	1	no	—
289	C₂×C₄	—	83	1	no	—
290	C₂×C₄	—	664	1	no	—
291	C₂×C₄	—	664	1	no	—
292	C₂×C₄	—	664	1	no	—
293	C₂×C₄	—	664	1	no	—
294	C₂×C₄	—	166	4	no	—
295	C₂×C₄	—	166	1	no	—
296	C₂×C₄	—	166	1	no	—
297	C₂×C₄	—	166	1	no	—
298	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
299	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	166	1	no	—
304	C₈	—	166	1	no	—
305	C₂²	—	332	1	no	—
306	C₂²	—	332	1	no	—
307	C₂²	—	83	2	no	—
308	C₂²	—	166	2	no	—
309	C₂²	—	166	1	no	—
310	C₂²	—	166	1	no	—
311	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	332	1	no	—
319	C₄	—	332	1	no	—
320	C₄	—	332	1	no	—
321	C₄	—	83	2	no	—
322	C₄	—	166	1	no	—
323	C₄	—	166	1	no	—
324	C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
325	C₄	#41×D₈₃	1	1	yes	characteristic
326	C₂	—	332	2	no	—
327	C₂	—	166	2	no	—
328	C₂	C₂²⋊Q₈×D₈₃	1	1	yes	characteristic
329	C₂	#19128508	1	1	yes	characteristic
330	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	#479×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×#13050591	C₂²
hypercenter	#479	D₈₃
hypocenter	C₈₃	#3084
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
1. #19128331	C₂	semidirect
3. #19128006	C₂	semidirect
4. #19128004	C₂	semidirect
5. #19127998	C₂	semidirect
6. #19127992	C₂	semidirect
7. #19127990	C₂	semidirect
8. #19127986	C₂	semidirect
9. #19127984	C₂	semidirect
10. #19127742	C₂	semidirect
11. #106×D₈₃	C₂	semidirect
12. #19127726	C₂	semidirect
15. #479×C₈₃	C₂	semidirect
60. #13050591	Q₈	semidirect
166. C₈₃	#3084	semidirect
170. #479	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128331	2
#19128183	2
#19128006	2
#19128004	2
#19127998	2
#19127992	2
#19127990	2
#19127986	2
#19127984	2
#19127742	2
#106×D₈₃	2
#19127726	2
#19127701	2
#19127696	2
#479×C₈₃	2
#3084	83