Subgroups — #24918719

Order 10624.

332 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	#19128307	C₂	1	1	yes	maximal, characteristic
3	#19128111	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128067	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128048	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128028	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128027	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128021	C₂	1	1	yes	maximal, characteristic, complemented
9	#19128016	C₂	1	1	yes	maximal, characteristic, complemented
10	#19128004	C₂	1	1	yes	maximal, characteristic, complemented
11	#19128000	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127990	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127980	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127760	C₂	1	1	yes	maximal, characteristic, complemented
15	#499×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	#14581792	C₂²	1	1	yes	characteristic
17	#14581791	—	2	1	no	—
18	#14581791	C₂²	1	1	yes	characteristic
19	#14581789	C₂²	1	1	yes	characteristic
20	#14581783	C₂²	1	1	yes	characteristic
21	#14581782	—	2	1	no	—
22	#14581777	—	2	1	no	—
23	#14581777	—	2	1	no	—
24	#14581772	—	2	1	no	—
25	#14581770	—	2	1	no	—
26	#14581767	—	2	1	no	—
27	#14581767	—	2	1	no	—
28	#14581767	C₂²	1	1	yes	characteristic
29	#14581767	C₂²	1	1	yes	characteristic
30	#14581754	C₂²	1	1	yes	characteristic
31	#14581754	C₂²	1	1	yes	characteristic
32	#14581750	C₂²	1	1	yes	characteristic
33	#14581749	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
35	#14581725	C₂²	1	1	yes	characteristic
36	#14581725	C₂²	1	1	yes	characteristic
37	#14581723	C₂²	1	1	yes	characteristic
38	#14581723	C₂²	1	1	yes	characteristic
39	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
41	#14581714	C₂²	1	1	yes	characteristic
42	#14581714	C₂²	1	1	yes	characteristic
43	#14581714	C₂²	1	1	yes	characteristic
44	#14581714	C₂²	1	1	yes	characteristic
45	#14581712	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	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	#14581703	C₂²	1	1	yes	characteristic
52	#125×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	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×D₃₃₂	—	2	1	no	—
62	C₂×D₃₃₂	—	2	1	no	—
63	C₂×D₃₃₂	—	2	1	no	—
64	C₂×D₃₃₂	—	2	1	no	—
65	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
66	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
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	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₃	—	2	1	no	—
74	C₄⋊Dic₈₃	—	2	1	no	—
75	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
77	Dic₈₃⋊C₄	—	2	1	no	—
78	Dic₈₃⋊C₄	—	2	1	no	—
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
81	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
82	(C₂×D₈₃)⋊C₄	—	2	1	no	—
83	(C₂×D₈₃)⋊C₄	—	2	1	no	—
84	(C₂×D₈₃)⋊C₄	—	2	1	no	—
85	(C₂×D₈₃)⋊C₄	—	2	1	no	—
86	(C₂×D₈₃)⋊C₄	—	2	1	no	—
87	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
88	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
89	C₄×Dic₈₃	—	2	1	no	—
90	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
91	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₈₃	—	2	1	no	—
94	C₄⋊C₄×C₈₃	—	2	1	no	—
95	C₄⋊C₄×C₈₃	—	2	1	no	—
96	C₄⋊C₄×C₈₃	—	2	1	no	—
97	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
101	C₄²×C₈₃	C₂³	1	1	yes	characteristic
102	C₂²×D₈₃	—	2	1	no	—
103	C₂²×D₈₃	—	2	1	no	—
104	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₈₃	—	2	1	no	—
106	C₂×C₄×C₈₃	—	2	1	no	—
107	C₂×C₄×C₈₃	—	2	1	no	—
108	C₂×C₄×C₈₃	—	2	1	no	—
109	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
112	C₂×Dic₈₃	—	2	1	no	—
113	C₂×Dic₈₃	—	2	1	no	—
114	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
117	D₃₃₂	—	2	1	no	—
118	D₃₃₂	—	2	1	no	—
119	D₃₃₂	—	2	1	no	—
120	D₃₃₂	—	2	1	no	—
121	D₃₃₂	—	8	1	no	—
122	D₃₃₂	—	8	1	no	—
123	D₃₃₂	—	8	1	no	—
124	D₃₃₂	—	8	1	no	—
125	C₄×D₈₃	—	2	1	no	—
126	C₄×D₈₃	—	2	1	no	—
127	C₄×D₈₃	—	8	1	no	—
128	C₄×D₈₃	—	8	1	no	—
129	C₄×D₈₃	—	8	1	no	—
130	Dic₁₆₆	—	8	1	no	—
131	Dic₁₆₆	—	8	1	no	—
132	C₈×C₈₃	—	2	1	no	—
133	C₈×C₈₃	—	2	1	no	—
134	C₈₃⋊C₈	—	2	1	no	—
135	C₈₃⋊C₈	—	2	1	no	—
136	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₈₃	—	2	1	no	—
138	C₂×D₈₃	—	2	1	no	—
139	C₂×D₈₃	—	2	1	no	—
140	C₂×D₈₃	—	4	1	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	C₂×D₈₃	—	4	1	no	—
146	C₄×C₈₃	—	2	1	no	—
147	C₄×C₈₃	—	2	1	no	—
148	C₄×C₈₃	—	4	1	no	—
149	C₄×C₈₃	—	4	1	no	—
150	C₄×C₈₃	—	4	1	no	—
151	C₄×C₈₃	—	4	1	no	—
152	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
153	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₈₃	—	2	1	no	—
155	Dic₈₃	—	2	1	no	—
156	Dic₈₃	—	2	1	no	—
157	Dic₈₃	—	4	1	no	—
158	Dic₈₃	—	4	1	no	—
159	C₂×C₈₃	#536	1	1	yes	characteristic
160	C₂×C₈₃	#576	1	1	yes	characteristic
161	C₂×C₈₃	#577	1	1	yes	characteristic
162	D₈₃	—	4	1	no	—
163	D₈₃	—	8	1	no	—
164	D₈₃	—	8	1	no	—
165	#3241	—	83	1	no	maximal, Sylow
166	C₈₃	#3241	1	1	yes	characteristic, complemented, Sylow
167	#544	—	83	1	no	—
168	#540	—	83	1	no	—
169	#501	—	83	1	no	—
170	#499	D₈₃	1	1	yes	characteristic, complemented
171	#490	—	83	1	no	—
172	#486	—	83	1	no	—
173	#482	—	83	1	no	—
174	#482	—	83	1	no	—
175	#481	—	83	1	no	—
176	#480	—	83	1	no	—
177	#478	—	83	1	no	—
178	#478	—	83	1	no	—
179	#476	—	83	1	no	—
180	#474	—	83	1	no	—
181	#433	—	83	1	no	—
182	C₄⋊Q₈	—	83	1	no	—
183	C₄⋊Q₈	—	83	1	no	—
184	C₄⋊Q₈	—	166	1	no	—
185	#127	—	83	1	no	—
186	#125	—	83	1	no	—
187	#125	—	166	1	no	—
188	#125	C₂×D₈₃	1	1	yes	characteristic
189	#125	C₂×D₈₃	1	1	yes	characteristic
190	#123	—	166	1	no	—
191	#123	—	166	1	no	—
192	C₂²⋊Q₈	—	166	1	no	—
193	C₂²⋊Q₈	—	166	1	no	—
194	#121	—	83	1	no	—
195	#121	—	83	1	no	—
196	#121	—	166	1	no	—
197	#121	—	166	1	no	—
198	C₄×D₄	—	83	1	no	—
199	C₄×D₄	—	83	1	no	—
200	#117	—	83	1	no	—
201	#117	—	83	1	no	—
202	C₂×C₄⋊C₄	—	83	1	no	—
203	#107	—	83	1	no	—
204	#107	—	83	1	no	—
205	#107	C₂×D₈₃	1	1	yes	characteristic
206	#107	C₂×D₈₃	1	1	yes	characteristic
207	#106	—	83	1	no	—
208	#106	—	83	1	no	—
209	#106	C₂×D₈₃	1	1	yes	characteristic
210	#106	C₂×D₈₃	1	1	yes	characteristic
211	C₄⋊C₈	—	83	1	no	—
212	C₄⋊C₈	—	83	1	no	—
213	#102	—	83	1	no	—
214	#102	—	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	C₂²⋊C₈	—	83	1	no	—
222	C₂²⋊C₈	—	83	1	no	—
223	#97	—	83	1	no	—
224	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
225	C₂×Q₈	—	166	1	no	—
226	C₂×Q₈	—	166	1	no	—
227	C₂×D₄	—	83	1	no	—
228	C₂×D₄	—	83	1	no	—
229	C₂×D₄	—	166	1	no	—
230	C₂×D₄	—	166	1	no	—
231	C₂×D₄	—	166	1	no	—
232	C₂×D₄	—	166	1	no	—
233	C₂²×C₄	—	83	1	no	—
234	C₂²×C₄	—	166	1	no	—
235	C₂²×C₄	—	166	1	no	—
236	C₂²×C₄	—	166	1	no	—
237	C₂×C₈	—	83	1	no	—
238	C₂×C₈	—	83	1	no	—
239	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	83	1	no	—
246	C₄⋊C₄	—	83	1	no	—
247	C₄⋊C₄	—	83	1	no	—
248	C₄⋊C₄	—	83	1	no	—
249	C₄⋊C₄	—	166	1	no	—
250	C₄⋊C₄	—	166	1	no	—
251	C₄⋊C₄	—	166	1	no	—
252	C₄⋊C₄	—	166	1	no	—
253	C₄⋊C₄	—	166	1	no	—
254	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
258	C₂²⋊C₄	—	83	1	no	—
259	C₂²⋊C₄	—	83	1	no	—
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	C₄²	—	83	1	no	—
266	C₄²	—	166	1	no	—
267	C₄²	C₂²×D₈₃	1	1	yes	characteristic
268	C₂³	—	83	1	no	—
269	C₂³	—	166	1	no	—
270	C₂³	—	166	1	no	—
271	Q₈	—	664	1	no	—
272	Q₈	—	664	1	no	—
273	D₄	—	664	1	no	—
274	D₄	—	664	1	no	—
275	D₄	—	664	1	no	—
276	D₄	—	664	1	no	—
277	D₄	—	166	1	no	—
278	D₄	—	166	1	no	—
279	D₄	—	166	1	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	1	no	—
287	C₂×C₄	—	83	1	no	—
288	C₂×C₄	—	664	1	no	—
289	C₂×C₄	—	664	1	no	—
290	C₂×C₄	—	664	1	no	—
291	C₂×C₄	—	166	1	no	—
292	C₂×C₄	—	166	1	no	—
293	C₂×C₄	—	166	1	no	—
294	C₂×C₄	—	166	1	no	—
295	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
296	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	166	1	no	—
301	C₈	—	166	1	no	—
302	C₂²	—	332	1	no	—
303	C₂²	—	332	1	no	—
304	C₂²	—	332	1	no	—
305	C₂²	—	332	1	no	—
306	C₂²	—	332	1	no	—
307	C₂²	—	332	1	no	—
308	C₂²	—	166	1	no	—
309	C₂²	—	166	1	no	—
310	C₂²	—	166	1	no	—
311	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	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	—	166	1	no	—
321	C₄	—	166	1	no	—
322	C₄	—	166	1	no	—
323	C₄	#14581840	1	1	yes	characteristic
324	C₄	#14581840	1	1	yes	characteristic
325	C₂	—	332	1	no	—
326	C₂	—	664	1	no	—
327	C₂	—	664	1	no	—
328	C₂	#19128276	1	1	yes	characteristic
329	C₂	#19128503	1	1	yes	characteristic
330	C₂	#19128508	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#499×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128307	C₂
hypercenter	#499	D₈₃
hypocenter	C₈₃	#3241
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
1. #19128331	C₂	semidirect
3. #19128111	C₂	semidirect
4. #19128067	C₂	semidirect
5. #19128048	C₂	semidirect
6. #19128028	C₂	semidirect
7. #19128027	C₂	semidirect
8. #19128021	C₂	semidirect
9. #19128016	C₂	semidirect
10. #19128004	C₂	semidirect
11. #19128000	C₂	semidirect
12. #19127990	C₂	semidirect
13. #19127980	C₂	semidirect
14. #19127760	C₂	semidirect
15. #499×C₈₃	C₂	semidirect
166. C₈₃	#3241	semidirect
170. #499	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
#19128307	2
#19128111	2
#19128067	2
#19128048	2
#19128028	2
#19128027	2
#19128021	2
#19128016	2
#19128004	2
#19128000	2
#19127990	2
#19127980	2
#19127760	2
#499×C₈₃	2
#3241	83