Subgroups — #24920370

Order 10624.

340 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128431	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128407	C₂	1	1	yes	maximal, characteristic
3	#19128108	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128105	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128011	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128005	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127995	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127990	C₂	1	1	yes	maximal, characteristic
9	#19127980	C₂	1	1	yes	maximal, characteristic
10	#19127978	C₂	1	1	yes	maximal, characteristic
11	#19127829	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127803	C₂	1	1	yes	maximal, characteristic
13	#19127800	C₂	1	1	yes	maximal, characteristic
14	#19127788	C₂	1	1	yes	maximal, characteristic
15	#501×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050579	C₂²	1	1	yes	characteristic
17	#14581791	—	2	1	no	—
18	#14581791	C₂²	1	1	yes	characteristic
19	#14581791	C₂²	1	1	yes	characteristic
20	#14581791	C₂²	1	1	yes	characteristic
21	#14581790	—	2	1	no	—
22	#14581790	—	2	1	no	—
23	#14581783	C₂²	1	1	yes	characteristic
24	#14581772	—	2	1	no	—
25	#14581770	—	2	1	no	—
26	#14581770	—	2	1	no	—
27	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
29	#14581757	—	2	1	no	—
30	#14581757	C₂²	1	1	yes	characteristic
31	#14581757	C₂²	1	1	yes	characteristic
32	C₄×D₃₃₂	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	#14581726	C₂²	1	1	yes	characteristic
37	#14581725	C₂²	1	1	yes	characteristic
38	#14581725	C₂²	1	1	yes	characteristic
39	#14581725	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	#14581717	C₂²	1	1	yes	characteristic
45	#14581715	C₂²	1	1	yes	characteristic
46	#14581715	C₂²	1	1	yes	characteristic
47	#14581714	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	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
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	#13050579	Q₈	1	4	yes	complemented
69	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₈₃	—	2	1	no	—
72	C₄⋊Dic₈₃	D₄	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	—	2	1	no	—
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
83	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
84	(C₂×D₈₃)⋊C₄	—	2	1	no	—
85	(C₂×D₈₃)⋊C₄	—	2	1	no	—
86	(C₂×D₈₃)⋊C₄	—	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	—	2	1	no	—
93	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
94	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₈₃	—	2	1	no	—
96	C₄⋊C₄×C₈₃	—	2	1	no	—
97	C₄⋊C₄×C₈₃	—	2	1	no	—
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	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
102	C₄²×C₈₃	C₂³	1	1	yes	characteristic
103	C₂²×D₈₃	—	2	1	no	—
104	Q₈×C₈₃	—	8	1	no	—
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	—	2	1	no	—
115	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
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	D₃₃₂	—	2	1	no	—
120	D₃₃₂	C₂×Q₈	1	2	yes	—
121	C₄×D₈₃	—	4	2	no	—
122	C₄×D₈₃	—	8	1	no	—
123	C₄×D₈₃	—	8	1	no	—
124	Dic₁₆₆	—	2	1	no	—
125	Dic₁₆₆	—	2	2	no	—
126	Dic₁₆₆	—	8	1	no	—
127	Dic₁₆₆	—	8	1	no	—
128	Dic₁₆₆	—	8	1	no	—
129	Dic₁₆₆	C₂×Q₈	1	2	yes	—
130	C₈×C₈₃	—	2	1	no	—
131	C₈×C₈₃	C₂×Q₈	1	2	yes	—
132	C₈₃⋊C₈	—	2	1	no	—
133	C₈₃⋊C₈	—	2	1	no	—
134	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×D₈₃	—	2	2	no	—
136	C₂×D₈₃	—	4	1	no	—
137	C₂×D₈₃	—	4	1	no	—
138	C₄×C₈₃	—	2	1	no	—
139	C₄×C₈₃	—	2	1	no	—
140	C₄×C₈₃	—	4	1	no	—
141	C₄×C₈₃	—	4	1	no	—
142	C₄×C₈₃	—	4	1	no	—
143	C₄×C₈₃	—	4	1	no	—
144	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
145	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
146	Dic₈₃	—	2	2	no	—
147	Dic₈₃	—	2	1	no	—
148	Dic₈₃	—	2	1	no	—
149	Dic₈₃	—	2	1	no	—
150	Dic₈₃	—	4	1	no	—
151	Dic₈₃	—	4	1	no	—
152	Dic₈₃	#137	1	2	yes	—
153	C₂×C₈₃	D₄×Q₈	1	1	yes	characteristic
154	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
155	C₂×C₈₃	#576	1	1	yes	characteristic
156	D₈₃	—	4	2	no	—
157	#3385	—	83	1	no	maximal, Sylow
158	C₈₃	#3385	1	1	yes	characteristic, complemented, Sylow
159	Q₈²	—	83	1	no	—
160	#554	—	83	1	no	—
161	#501	—	83	1	no	—
162	#501	D₈₃	1	1	yes	characteristic, complemented
163	#500	—	83	1	no	—
164	#479	—	83	1	no	—
165	#479	—	83	1	no	—
166	#477	—	83	1	no	—
167	#476	—	83	1	no	—
168	#474	—	83	1	no	—
169	#474	—	83	1	no	—
170	#446	—	83	1	no	—
171	#440	—	83	1	no	—
172	#440	—	83	1	no	—
173	C₄×SD₁₆	—	83	1	no	—
174	C₂×SD₁₆	—	83	1	no	—
175	C₄⋊Q₈	—	83	1	no	—
176	C₄⋊Q₈	—	83	1	no	—
177	C₄⋊Q₈	—	83	1	no	—
178	C₄⋊Q₈	—	166	1	no	—
179	C₄⋊Q₈	—	166	1	no	—
180	C₄⋊Q₈	—	166	1	no	—
181	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
182	#125	—	83	1	no	—
183	#125	C₂×D₈₃	1	1	yes	characteristic
184	C₂²⋊Q₈	—	166	1	no	—
185	C₂²⋊Q₈	—	166	1	no	—
186	C₂²⋊Q₈	—	166	1	no	—
187	C₄×Q₈	—	83	1	no	—
188	C₄×Q₈	—	83	1	no	—
189	C₄×Q₈	—	83	1	no	—
190	C₄×Q₈	—	83	1	no	—
191	C₄×Q₈	—	166	1	no	—
192	C₄×D₄	—	83	1	no	—
193	C₄×D₄	—	83	1	no	—
194	C₄×D₄	—	83	1	no	—
195	C₂×C₄⋊C₄	—	166	1	no	—
196	#107	—	83	1	no	—
197	#107	—	83	1	no	—
198	#107	—	83	1	no	—
199	#107	—	83	1	no	—
200	#107	C₂×D₈₃	1	1	yes	characteristic
201	#107	C₂×D₈₃	1	1	yes	characteristic
202	#106	—	83	1	no	—
203	#106	C₂×D₈₃	1	1	yes	characteristic
204	#106	C₂×D₈₃	1	1	yes	characteristic
205	C₄⋊C₈	—	83	1	no	—
206	C₄⋊C₈	—	83	1	no	—
207	C₄⋊C₈	—	83	1	no	—
208	#103	—	83	1	no	—
209	#103	—	83	1	no	—
210	#103	—	83	1	no	—
211	#102	—	83	1	no	—
212	#102	—	83	1	no	—
213	#102	—	83	1	no	—
214	#97	—	83	1	no	—
215	#97	C₂×D₈₃	1	1	yes	characteristic
216	C₄×C₈	—	83	1	no	—
217	C₂×Q₈	—	2	1	no	—
218	C₂×Q₈	—	83	1	no	—
219	C₂×Q₈	—	83	1	no	—
220	C₂×Q₈	—	166	1	no	—
221	C₂×Q₈	—	166	1	no	—
222	C₂×Q₈	—	166	1	no	—
223	C₂×D₄	—	83	1	no	—
224	C₂²×C₄	—	166	1	no	—
225	C₂²×C₄	—	166	1	no	—
226	C₂²×C₄	—	166	1	no	—
227	SD₁₆	—	83	4	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	—	166	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	C₂²×D₈₃	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
251	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
252	C₂²⋊C₄	—	166	1	no	—
253	C₂²⋊C₄	—	166	1	no	—
254	C₂²⋊C₄	—	166	1	no	—
255	C₄²	—	83	1	no	—
256	C₄²	—	83	1	no	—
257	C₄²	—	166	1	no	—
258	C₄²	—	166	1	no	—
259	C₄²	—	166	1	no	—
260	C₄²	C₂²×D₈₃	1	1	yes	characteristic
261	C₂³	—	166	1	no	—
262	Q₈	—	8	1	no	—
263	Q₈	—	83	2	no	—
264	Q₈	—	664	1	no	—
265	Q₈	—	664	1	no	—
266	Q₈	—	664	1	no	—
267	Q₈	—	166	2	no	—
268	Q₈	—	166	1	no	—
269	D₄	—	83	2	no	—
270	D₄	—	166	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	332	2	no	—
276	C₂×C₄	—	83	1	no	—
277	C₂×C₄	—	83	1	no	—
278	C₂×C₄	—	83	1	no	—
279	C₂×C₄	—	83	1	no	—
280	C₂×C₄	—	664	1	no	—
281	C₂×C₄	—	664	1	no	—
282	C₂×C₄	—	166	1	no	—
283	C₂×C₄	—	166	1	no	—
284	C₂×C₄	—	166	1	no	—
285	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
286	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
287	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
288	C₈	—	2	1	no	—
289	C₈	—	166	1	no	—
290	C₈	—	166	1	no	—
291	C₈	Q₈×D₈₃	1	2	yes	—
292	C₂²	—	332	1	no	—
293	C₂²	—	332	1	no	—
294	C₂²	—	166	2	no	—
295	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	332	1	no	—
303	C₄	—	332	1	no	—
304	C₄	—	83	2	no	—
305	C₄	—	166	2	no	—
306	C₄	—	166	1	no	—
307	C₄	—	166	1	no	—
308	C₄	—	166	1	no	—
309	C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
310	C₄	#14581840	1	1	yes	characteristic
311	C₂	—	332	2	no	—
312	C₂	#19128366	1	1	yes	characteristic
313	C₂	#137×D₈₃	1	1	yes	characteristic
314	C₂	#19128505	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×D₃₃₂	C₂³
hypercenter	#501	D₈₃
hypocenter	C₈₃	#3385
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. #19128431	C₂	semidirect
3. #19128108	C₂	semidirect
4. #19128105	C₂	semidirect
5. #19128011	C₂	semidirect
6. #19128005	C₂	semidirect
7. #19127995	C₂	semidirect
11. #19127829	C₂	semidirect
15. #501×C₈₃	C₂	semidirect
68. #13050579	Q₈	semidirect
158. C₈₃	#3385	semidirect
162. #501	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128431	2
#19128407	2
#19128108	2
#19128105	2
#19128011	2
#19128005	2
#19127995	2
#19127990	2
#19127980	2
#19127978	2
#19127829	2
#19127803	2
#19127800	2
#19127788	2
#501×C₈₃	2
#3385	83