Subgroups — #24920298

Order 10624.

332 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128427	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128425	C₂	1	1	yes	maximal, characteristic
3	#19128069	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128062	C₂	1	1	yes	maximal, characteristic
5	#19128057	C₂	1	1	yes	maximal, characteristic
6	#19128006	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128002	C₂	1	1	yes	maximal, characteristic
8	#19127994	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127925	C₂	1	1	yes	maximal, characteristic
10	#19127925	C₂	1	1	yes	maximal, characteristic
11	#19127830	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127808	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127799	C₂	1	1	yes	maximal, characteristic
14	Q₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#462×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050577	C₂²	1	1	yes	characteristic
17	#14581791	—	2	1	no	—
18	#14581791	C₂²	1	1	yes	characteristic
19	#14581787	—	2	1	no	—
20	#14581787	—	2	1	no	—
21	#14581784	—	2	1	no	—
22	#14581784	C₂²	1	1	yes	characteristic
23	#14581783	—	2	1	no	—
24	#14581783	C₂²	1	1	yes	characteristic
25	#14581781	C₂²	1	1	yes	characteristic
26	#14581781	C₂²	1	1	yes	characteristic
27	#14581772	—	2	1	no	—
28	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
29	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
30	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
31	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₆₆	—	2	1	no	—
33	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
34	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
35	#14581750	—	2	1	no	—
36	#14581726	C₂²	1	1	yes	characteristic
37	#14581725	C₂²	1	1	yes	characteristic
38	#14581724	C₂²	1	1	yes	characteristic
39	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
41	#14581716	C₂²	1	1	yes	characteristic
42	#14581716	C₂²	1	1	yes	characteristic
43	#14581716	C₂²	1	1	yes	characteristic
44	#14581716	C₂²	1	1	yes	characteristic
45	#14581716	C₂²	1	1	yes	characteristic
46	#14581714	C₂²	1	1	yes	characteristic
47	#14581714	C₂²	1	1	yes	characteristic
48	#14581712	C₂²	1	1	yes	characteristic
49	#14581703	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
55	C₄×Q₈×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	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
61	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₈₃	—	2	1	no	—
63	#13050577	—	2	2	no	—
64	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
65	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₈₃	—	2	1	no	—
67	C₄⋊Dic₈₃	—	2	1	no	—
68	C₄⋊Dic₈₃	—	2	1	no	—
69	C₄⋊Dic₈₃	—	2	1	no	—
70	C₄⋊Dic₈₃	—	2	1	no	—
71	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
72	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
80	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
81	(C₂×D₈₃)⋊C₄	—	2	1	no	—
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	C₄×Dic₈₃	—	2	1	no	—
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	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
95	Q₁₆×C₈₃	—	2	2	no	—
96	Q₁₆×C₈₃	—	2	2	no	—
97	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₃	—	2	1	no	—
100	C₄⋊C₄×C₈₃	—	2	1	no	—
101	C₄⋊C₄×C₈₃	—	2	1	no	—
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₄²×C₈₃	—	2	1	no	—
104	C₄²×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²×D₈₃	—	2	1	no	—
106	Q₈×C₈₃	—	2	1	no	—
107	Q₈×C₈₃	—	2	1	no	—
108	Q₈×C₈₃	—	2	1	no	—
109	Q₈×C₈₃	—	2	1	no	—
110	Q₈×C₈₃	—	2	1	no	—
111	Q₈×C₈₃	#41	1	2	yes	—
112	C₂×C₄×C₈₃	—	2	1	no	—
113	C₂×C₄×C₈₃	—	2	1	no	—
114	C₂×C₄×C₈₃	—	2	1	no	—
115	C₂×C₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
119	C₂×Dic₈₃	—	2	1	no	—
120	C₂×Dic₈₃	—	2	1	no	—
121	C₂×Dic₈₃	—	2	1	no	—
122	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
126	D₃₃₂	—	2	1	no	—
127	D₃₃₂	—	2	1	no	—
128	C₄×D₈₃	—	4	2	no	—
129	Dic₁₆₆	—	4	1	no	—
130	Dic₁₆₆	—	4	2	no	—
131	C₈×C₈₃	—	2	1	no	—
132	C₈×C₈₃	—	2	1	no	—
133	C₈₃⋊C₈	—	2	1	no	—
134	C₈₃⋊C₈	—	2	1	no	—
135	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×D₈₃	—	4	1	no	—
137	C₂×D₈₃	—	4	1	no	—
138	C₂×D₈₃	—	4	1	no	—
139	C₄×C₈₃	—	2	2	no	—
140	C₄×C₈₃	—	2	1	no	—
141	C₄×C₈₃	—	2	1	no	—
142	C₄×C₈₃	—	4	1	no	—
143	C₄×C₈₃	—	4	1	no	—
144	C₄×C₈₃	—	4	1	no	—
145	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
146	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
147	Dic₈₃	—	2	1	no	—
148	Dic₈₃	—	2	1	no	—
149	Dic₈₃	—	2	1	no	—
150	Dic₈₃	—	2	1	no	—
151	Dic₈₃	—	4	1	no	—
152	Dic₈₃	—	4	1	no	—
153	Dic₈₃	—	4	1	no	—
154	C₂×C₈₃	#548	1	1	yes	characteristic
155	C₂×C₈₃	#575	1	1	yes	characteristic
156	C₂×C₈₃	#578	1	1	yes	characteristic
157	D₈₃	—	8	1	no	—
158	#3377	—	83	1	no	maximal, Sylow
159	C₈₃	#3377	1	1	yes	characteristic, complemented, Sylow
160	#557	—	83	1	no	—
161	#556	—	83	1	no	—
162	#491	—	83	1	no	—
163	#489	—	83	1	no	—
164	#488	—	83	1	no	—
165	#479	—	83	1	no	—
166	#478	—	83	1	no	—
167	#477	—	83	1	no	—
168	#464	—	83	1	no	—
169	#464	—	83	1	no	—
170	#462	D₈₃	1	1	yes	characteristic, complemented
171	#446	—	83	1	no	—
172	#441	—	83	1	no	—
173	#440	—	83	1	no	—
174	C₄×Q₁₆	—	83	1	no	—
175	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
176	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
177	C₂×SD₁₆	—	83	1	no	—
178	C₄⋊Q₈	—	83	1	no	—
179	C₄⋊Q₈	—	166	1	no	—
180	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
181	#126	—	166	1	no	—
182	#126	—	166	1	no	—
183	#125	—	83	1	no	—
184	#125	—	83	1	no	—
185	#125	—	166	1	no	—
186	#125	—	166	1	no	—
187	#124	—	83	1	no	—
188	#124	—	83	1	no	—
189	C₂²⋊Q₈	—	166	1	no	—
190	C₄×Q₈	—	83	1	no	—
191	C₄×Q₈	—	83	1	no	—
192	C₄×Q₈	—	83	1	no	—
193	C₄×Q₈	—	83	1	no	—
194	C₄×Q₈	—	83	1	no	—
195	C₄×Q₈	—	166	1	no	—
196	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
197	C₄×D₄	—	83	1	no	—
198	#117	—	166	1	no	—
199	#107	—	83	1	no	—
200	#107	—	83	1	no	—
201	#106	—	83	1	no	—
202	C₄⋊C₈	—	83	1	no	—
203	C₄⋊C₈	—	83	1	no	—
204	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
205	#103	—	83	1	no	—
206	#103	—	83	1	no	—
207	#103	—	83	1	no	—
208	#103	—	83	1	no	—
209	#103	—	83	1	no	—
210	#103	C₂×D₈₃	1	1	yes	characteristic
211	#103	C₂×D₈₃	1	1	yes	characteristic
212	#102	—	83	1	no	—
213	#102	—	83	1	no	—
214	#102	—	83	1	no	—
215	#97	—	83	1	no	—
216	#97	—	83	1	no	—
217	C₄×C₈	—	83	1	no	—
218	C₂×Q₈	—	83	1	no	—
219	C₂×Q₈	—	166	1	no	—
220	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
221	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
222	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
223	C₂×D₄	—	83	1	no	—
224	C₂²×C₄	—	166	1	no	—
225	Q₁₆	—	2	2	no	—
226	Q₁₆	—	2	2	no	—
227	SD₁₆	—	166	2	no	—
228	C₂×C₈	—	83	1	no	—
229	C₂×C₈	—	83	1	no	—
230	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	83	1	no	—
236	C₄⋊C₄	—	83	1	no	—
237	C₄⋊C₄	—	83	1	no	—
238	C₄⋊C₄	—	83	1	no	—
239	C₄⋊C₄	—	83	1	no	—
240	C₄⋊C₄	—	83	1	no	—
241	C₄⋊C₄	—	166	1	no	—
242	C₄⋊C₄	—	166	1	no	—
243	C₄⋊C₄	—	166	1	no	—
244	C₄⋊C₄	—	166	1	no	—
245	C₄⋊C₄	—	166	1	no	—
246	C₄⋊C₄	—	166	1	no	—
247	C₄⋊C₄	—	166	1	no	—
248	C₄⋊C₄	—	166	1	no	—
249	C₄⋊C₄	—	166	1	no	—
250	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
251	C₂²⋊C₄	—	166	1	no	—
252	C₂²⋊C₄	—	166	1	no	—
253	C₂²⋊C₄	—	166	1	no	—
254	C₂²⋊C₄	—	166	1	no	—
255	C₂²⋊C₄	—	166	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	83	1	no	—
258	C₄²	—	83	1	no	—
259	C₄²	—	166	1	no	—
260	C₄²	—	166	1	no	—
261	C₄²	—	166	1	no	—
262	C₄²	—	166	1	no	—
263	C₄²	C₂²×D₈₃	1	1	yes	characteristic
264	C₂³	—	166	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	332	1	no	—
271	Q₈	—	332	2	no	—
272	Q₈	#13050589	1	2	yes	—
273	D₄	—	166	1	no	—
274	D₄	—	166	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	332	2	no	—
280	C₂×C₄	—	83	1	no	—
281	C₂×C₄	—	83	1	no	—
282	C₂×C₄	—	83	1	no	—
283	C₂×C₄	—	83	1	no	—
284	C₂×C₄	—	166	1	no	—
285	C₂×C₄	—	166	1	no	—
286	C₂×C₄	—	166	1	no	—
287	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
288	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	166	1	no	—
293	C₈	—	166	1	no	—
294	C₂²	—	332	1	no	—
295	C₂²	—	332	1	no	—
296	C₂²	—	332	1	no	—
297	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	332	1	no	—
305	C₄	—	332	1	no	—
306	C₄	—	332	1	no	—
307	C₄	—	166	1	no	—
308	C₄	—	166	1	no	—
309	C₄	—	166	1	no	—
310	C₄	—	166	1	no	—
311	C₄	C₂×#13050589	1	1	yes	characteristic
312	C₄	#14581839	1	1	yes	characteristic
313	C₂	—	664	1	no	—
314	C₂	#19128357	1	1	yes	characteristic
315	C₂	#19128500	1	1	yes	characteristic
316	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	#462×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×D₃₃₂	C₂³
hypercenter	#462	D₈₃
hypocenter	C₈₃	#3377
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

Complements

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

N	H ≅ G/N	type
1. #19128427	C₂	semidirect
3. #19128069	C₂	semidirect
6. #19128006	C₂	semidirect
8. #19127994	C₂	semidirect
11. #19127830	C₂	semidirect
12. #19127808	C₂	semidirect
14. Q₁₆×Dic₈₃	C₂	semidirect
15. #462×C₈₃	C₂	semidirect
159. C₈₃	#3377	semidirect
170. #462	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128427	2
#19128425	2
#19128069	2
#19128062	2
#19128057	2
#19128006	2
#19128002	2
#19127994	2
#19127925	2
#19127830	2
#19127808	2
#19127799	2
Q₁₆×Dic₈₃	2
#462×C₈₃	2
#3377	83