Subgroups — #24918724

Order 10624.

308 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128328	C₂	1	1	yes	maximal, characteristic
2	#19128312	C₂	1	1	yes	maximal, characteristic
3	#19128107	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128103	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128069	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128041	C₂	1	1	yes	maximal, characteristic
7	#19128040	C₂	1	1	yes	maximal, characteristic
8	#19128035	C₂	1	1	yes	maximal, characteristic
9	#19128030	C₂	1	1	yes	maximal, characteristic
10	#19128010	C₂	1	1	yes	maximal, characteristic, complemented
11	#19128006	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127994	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127984	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127760	C₂	1	1	yes	maximal, characteristic
15	#487×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	#14581791	C₂²	1	1	yes	characteristic
17	#14581790	—	2	1	no	—
18	#14581787	—	2	1	no	—
19	#14581787	—	2	1	no	—
20	#14581784	C₂²	1	1	yes	characteristic
21	#14581784	C₂²	1	1	yes	characteristic
22	#14581783	C₂²	1	1	yes	characteristic
23	#14581782	—	2	1	no	—
24	#14581782	—	2	1	no	—
25	#14581782	—	2	1	no	—
26	#14581770	—	2	1	no	—
27	#14581768	—	2	1	no	—
28	#14581768	C₂²	1	1	yes	characteristic
29	#14581768	C₂²	1	1	yes	characteristic
30	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
31	Q₈×Dic₈₃	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	#14581726	C₂²	1	1	yes	characteristic
36	#14581726	C₂²	1	1	yes	characteristic
37	#14581725	C₂²	1	1	yes	characteristic
38	#14581725	C₂²	1	1	yes	characteristic
39	#14581724	C₂²	1	1	yes	characteristic
40	#14581724	C₂²	1	1	yes	characteristic
41	#14581723	C₂²	1	1	yes	characteristic
42	#14581723	C₂²	1	1	yes	characteristic
43	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
45	#14581716	C₂²	1	1	yes	characteristic
46	#14581716	C₂²	1	1	yes	characteristic
47	#14581716	C₂²	1	1	yes	characteristic
48	#14581716	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	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
54	#103×C₈₃	C₂²	1	1	yes	characteristic
55	#103×C₈₃	C₂²	1	1	yes	characteristic
56	#103×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×C₄×D₈₃	—	2	1	no	—
61	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
62	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
63	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
64	C₄⋊Dic₈₃	—	2	1	no	—
65	C₄⋊Dic₈₃	—	2	1	no	—
66	C₄⋊Dic₈₃	—	2	1	no	—
67	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
71	Dic₈₃⋊C₄	—	2	1	no	—
72	Dic₈₃⋊C₄	—	2	1	no	—
73	Dic₈₃⋊C₄	—	2	1	no	—
74	Dic₈₃⋊C₄	—	2	1	no	—
75	Dic₈₃⋊C₄	—	2	1	no	—
76	Dic₈₃⋊C₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
86	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
87	C₄×Dic₈₃	—	2	1	no	—
88	C₄×Dic₈₃	—	2	1	no	—
89	C₄×Dic₈₃	—	2	1	no	—
90	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
91	C₂×Q₈×C₈₃	—	2	1	no	—
92	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
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	C₄²×C₈₃	C₂³	1	1	yes	characteristic
102	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
103	Q₈×C₈₃	—	2	1	no	—
104	Q₈×C₈₃	—	2	1	no	—
105	Q₈×C₈₃	—	2	1	no	—
106	Q₈×C₈₃	—	2	1	no	—
107	Q₈×C₈₃	—	8	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	—	2	1	no	—
112	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₈₃	—	2	1	no	—
116	C₂×Dic₈₃	—	2	1	no	—
117	C₂×Dic₈₃	—	2	1	no	—
118	C₂×Dic₈₃	—	2	1	no	—
119	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
122	C₄×D₈₃	—	2	1	no	—
123	C₄×D₈₃	—	2	1	no	—
124	C₄×D₈₃	—	8	1	no	—
125	Dic₁₆₆	—	8	1	no	—
126	C₈×C₈₃	—	2	1	no	—
127	C₈×C₈₃	—	2	1	no	—
128	C₈₃⋊C₈	—	2	1	no	—
129	C₈₃⋊C₈	—	2	1	no	—
130	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×D₈₃	—	2	1	no	—
132	C₂×D₈₃	—	2	1	no	—
133	C₂×D₈₃	—	2	1	no	—
134	C₄×C₈₃	—	2	1	no	—
135	C₄×C₈₃	—	2	1	no	—
136	C₄×C₈₃	—	4	1	no	—
137	C₄×C₈₃	—	4	1	no	—
138	C₄×C₈₃	—	4	1	no	—
139	C₄×C₈₃	—	4	1	no	—
140	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
141	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
142	Dic₈₃	—	2	1	no	—
143	Dic₈₃	—	2	1	no	—
144	Dic₈₃	—	2	1	no	—
145	Dic₈₃	—	4	1	no	—
146	Dic₈₃	—	4	1	no	—
147	Dic₈₃	—	4	1	no	—
148	Dic₈₃	—	4	1	no	—
149	C₂×C₈₃	#536	1	1	yes	characteristic
150	C₂×C₈₃	#577	1	1	yes	characteristic
151	C₂×C₈₃	#578	1	1	yes	characteristic
152	D₈₃	—	4	1	no	—
153	#3242	—	83	1	no	maximal, Sylow
154	C₈₃	#3242	1	1	yes	characteristic, complemented, Sylow
155	#544	—	83	1	no	—
156	#541	—	83	1	no	—
157	#501	—	83	1	no	—
158	#499	—	83	1	no	—
159	#491	—	83	1	no	—
160	#487	D₈₃	1	1	yes	characteristic, complemented
161	#485	—	83	1	no	—
162	#485	—	83	1	no	—
163	#484	—	83	1	no	—
164	#483	—	83	1	no	—
165	#479	—	83	1	no	—
166	#479	—	83	1	no	—
167	#477	—	83	1	no	—
168	#475	—	83	1	no	—
169	#433	—	83	1	no	—
170	C₄⋊Q₈	—	83	1	no	—
171	C₄⋊Q₈	—	166	1	no	—
172	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
173	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
174	#126	—	166	1	no	—
175	#126	—	166	1	no	—
176	#125	—	83	1	no	—
177	#125	—	83	1	no	—
178	#125	—	83	1	no	—
179	#125	—	166	1	no	—
180	#125	—	166	1	no	—
181	#125	—	166	1	no	—
182	C₂²⋊Q₈	—	83	1	no	—
183	C₂²⋊Q₈	—	83	1	no	—
184	C₂²⋊Q₈	—	166	1	no	—
185	C₂²⋊Q₈	—	166	1	no	—
186	C₄×Q₈	—	83	1	no	—
187	C₄×Q₈	—	83	1	no	—
188	#117	—	83	1	no	—
189	#117	—	83	1	no	—
190	C₂×C₄⋊C₄	—	83	1	no	—
191	#107	—	83	1	no	—
192	#107	—	83	1	no	—
193	#107	—	83	1	no	—
194	#107	—	83	1	no	—
195	#106	—	83	1	no	—
196	#106	—	83	1	no	—
197	#106	—	83	1	no	—
198	#106	—	83	1	no	—
199	C₄⋊C₈	—	83	1	no	—
200	C₄⋊C₈	—	83	1	no	—
201	#103	—	83	1	no	—
202	#103	—	83	1	no	—
203	#103	—	83	1	no	—
204	#103	—	83	1	no	—
205	#103	C₂×D₈₃	1	1	yes	characteristic
206	#103	C₂×D₈₃	1	1	yes	characteristic
207	#103	C₂×D₈₃	1	1	yes	characteristic
208	#103	C₂×D₈₃	1	1	yes	characteristic
209	C₂²⋊C₈	—	83	1	no	—
210	C₂²⋊C₈	—	83	1	no	—
211	#97	—	83	1	no	—
212	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	—	166	1	no	—
215	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
216	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
217	C₂²×C₄	—	83	1	no	—
218	C₂²×C₄	—	166	1	no	—
219	C₂×C₈	—	83	1	no	—
220	C₂×C₈	—	83	1	no	—
221	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	83	1	no	—
227	C₄⋊C₄	—	83	1	no	—
228	C₄⋊C₄	—	83	1	no	—
229	C₄⋊C₄	—	83	1	no	—
230	C₄⋊C₄	—	83	1	no	—
231	C₄⋊C₄	—	83	1	no	—
232	C₄⋊C₄	—	166	1	no	—
233	C₄⋊C₄	—	166	1	no	—
234	C₄⋊C₄	—	166	1	no	—
235	C₄⋊C₄	—	166	1	no	—
236	C₄⋊C₄	—	166	1	no	—
237	C₄⋊C₄	—	166	1	no	—
238	C₄⋊C₄	—	166	1	no	—
239	C₄⋊C₄	—	166	1	no	—
240	C₄⋊C₄	—	166	1	no	—
241	C₄⋊C₄	—	166	1	no	—
242	C₄⋊C₄	—	166	1	no	—
243	C₄⋊C₄	—	166	1	no	—
244	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
245	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
246	C₂²⋊C₄	—	83	1	no	—
247	C₂²⋊C₄	—	83	1	no	—
248	C₂²⋊C₄	—	166	1	no	—
249	C₂²⋊C₄	—	166	1	no	—
250	C₂²⋊C₄	—	166	1	no	—
251	C₄²	—	83	1	no	—
252	C₄²	—	166	1	no	—
253	C₄²	—	166	1	no	—
254	C₄²	—	166	1	no	—
255	C₄²	C₂²×D₈₃	1	1	yes	characteristic
256	C₂³	—	83	1	no	—
257	Q₈	—	2	1	no	—
258	Q₈	—	2	1	no	—
259	Q₈	—	2	1	no	—
260	Q₈	—	2	1	no	—
261	Q₈	—	8	1	no	—
262	Q₈	—	664	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	83	1	no	—
268	C₂×C₄	—	83	1	no	—
269	C₂×C₄	—	83	1	no	—
270	C₂×C₄	—	664	1	no	—
271	C₂×C₄	—	166	1	no	—
272	C₂×C₄	—	166	1	no	—
273	C₂×C₄	—	166	1	no	—
274	C₂×C₄	—	166	1	no	—
275	C₂×C₄	—	166	1	no	—
276	C₂×C₄	—	166	1	no	—
277	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
278	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
279	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	166	1	no	—
283	C₈	—	166	1	no	—
284	C₂²	—	166	1	no	—
285	C₂²	—	166	1	no	—
286	C₂²	—	166	1	no	—
287	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	332	1	no	—
295	C₄	—	332	1	no	—
296	C₄	—	332	1	no	—
297	C₄	—	332	1	no	—
298	C₄	—	166	1	no	—
299	C₄	—	166	1	no	—
300	C₄	—	166	1	no	—
301	C₄	#14581839	1	1	yes	characteristic
302	C₄	#14581839	1	1	yes	characteristic
303	C₂	—	332	1	no	—
304	C₂	#19128271	1	1	yes	characteristic
305	C₂	#19128507	1	1	yes	characteristic
306	C₂	#19128514	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#487×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂²×D₈₃	C₂×D₄
hypercenter	#487	D₈₃
hypocenter	C₈₃	#3242
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

Complements

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

N	H ≅ G/N	type
3. #19128107	C₂	semidirect
4. #19128103	C₂	semidirect
5. #19128069	C₂	semidirect
10. #19128010	C₂	semidirect
11. #19128006	C₂	semidirect
12. #19127994	C₂	semidirect
13. #19127984	C₂	semidirect
15. #487×C₈₃	C₂	semidirect
154. C₈₃	#3242	semidirect
160. #487	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128328	2
#19128312	2
#19128107	2
#19128103	2
#19128069	2
#19128041	2
#19128040	2
#19128035	2
#19128030	2
#19128010	2
#19128006	2
#19127994	2
#19127984	2
#19127760	2
#487×C₈₃	2
#3242	83