Subgroups — #24920299

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	#19128429	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128069	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128060	C₂	1	1	yes	maximal, characteristic
4	#19128055	C₂	1	1	yes	maximal, characteristic
5	#19128008	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128001	C₂	1	1	yes	maximal, characteristic
7	#19127996	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127920	C₂	1	1	yes	maximal, characteristic
9	#19127920	C₂	1	1	yes	maximal, characteristic
10	#19127908	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127832	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127806	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127801	C₂	1	1	yes	maximal, characteristic
14	C₄×#13050580	C₂	1	1	yes	maximal, characteristic, complemented
15	#556×C₈₃	C₂	1	1	yes	maximal, characteristic
16	C₂×#13050580	C₂²	1	1	yes	characteristic
17	C₂×#13050580	C₂²	1	1	yes	characteristic
18	C₂×#13050578	C₂²	1	1	yes	characteristic
19	#14581792	C₂²	1	1	yes	characteristic
20	#14581790	—	2	1	no	—
21	#14581784	C₂²	1	1	yes	characteristic
22	#14581784	C₂²	1	1	yes	characteristic
23	#14581782	—	2	1	no	—
24	#14581782	—	2	1	no	—
25	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
28	#14581757	—	2	1	no	—
29	#14581725	C₂²	1	1	yes	characteristic
30	#14581725	C₂²	1	1	yes	characteristic
31	#14581723	C₂²	1	1	yes	characteristic
32	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
33	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
34	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
35	#14581716	C₂²	1	1	yes	characteristic
36	#14581716	C₂²	1	1	yes	characteristic
37	#14581716	C₂²	1	1	yes	characteristic
38	#14581716	C₂²	1	1	yes	characteristic
39	#14581716	C₂²	1	1	yes	characteristic
40	#14581716	C₂²	1	1	yes	characteristic
41	#14581715	C₂²	1	1	yes	characteristic
42	#14581713	C₂²	1	1	yes	characteristic
43	#14581713	C₂²	1	1	yes	characteristic
44	#14581713	C₂²	1	1	yes	characteristic
45	#14581703	C₂²	1	1	yes	characteristic
46	#14581703	C₂²	1	1	yes	characteristic
47	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
48	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
49	#126×C₈₃	—	2	1	no	—
50	#126×C₈₃	—	2	1	no	—
51	#124×C₈₃	C₂²	1	1	yes	characteristic
52	#124×C₈₃	C₂²	1	1	yes	characteristic
53	C₂²⋊Q₈×C₈₃	—	2	1	no	—
54	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
57	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
58	#117×C₈₃	—	2	1	no	—
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	#13050580	—	2	2	no	—
62	#13050580	—	2	2	no	—
63	#13050578	—	2	2	no	—
64	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
65	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
66	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₈₃	—	2	1	no	—
69	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	C₄×Dic₈₃	—	2	1	no	—
80	C₄×Dic₈₃	—	2	1	no	—
81	C₄×Dic₈₃	—	2	1	no	—
82	C₂×Q₈×C₈₃	D₄	1	1	yes	characteristic
83	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
84	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
85	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₈₃	—	2	1	no	—
87	C₄⋊C₄×C₈₃	—	2	1	no	—
88	C₄⋊C₄×C₈₃	—	2	1	no	—
89	C₄⋊C₄×C₈₃	—	2	1	no	—
90	C₄⋊C₄×C₈₃	—	2	1	no	—
91	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
92	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
93	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
94	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
95	C₂²⋊C₄×C₈₃	—	2	1	no	—
96	C₂²⋊C₄×C₈₃	—	2	1	no	—
97	C₂²⋊C₄×C₈₃	—	2	1	no	—
98	C₂²⋊C₄×C₈₃	—	2	1	no	—
99	C₂²⋊C₄×C₈₃	—	2	1	no	—
100	C₄²×C₈₃	—	2	1	no	—
101	C₄²×C₈₃	—	2	1	no	—
102	C₄²×C₈₃	C₂³	1	1	yes	characteristic
103	C₄²×C₈₃	C₂³	1	1	yes	characteristic
104	C₄²×C₈₃	C₂³	1	1	yes	characteristic
105	C₂³×C₈₃	—	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	—	4	1	no	—
111	D₄×C₈₃	—	2	1	no	—
112	D₄×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	—	4	2	no	—
117	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
124	C₂×Dic₈₃	—	2	1	no	—
125	C₂×Dic₈₃	—	2	1	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	—	2	1	no	—
128	Dic₁₆₆	—	2	1	no	—
129	Dic₁₆₆	—	4	2	no	—
130	Dic₁₆₆	#41	1	2	yes	—
131	C₈₃⋊C₈	—	2	1	no	—
132	C₈₃⋊C₈	—	2	1	no	—
133	C₈₃⋊C₈	—	2	1	no	—
134	C₈₃⋊C₈	—	2	1	no	—
135	C₂²×C₈₃	—	4	1	no	—
136	C₂²×C₈₃	—	4	1	no	—
137	C₂²×C₈₃	—	4	1	no	—
138	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₈₃	—	2	1	no	—
140	C₄×C₈₃	—	2	1	no	—
141	C₄×C₈₃	—	2	1	no	—
142	C₄×C₈₃	—	2	1	no	—
143	C₄×C₈₃	—	2	1	no	—
144	C₄×C₈₃	—	2	1	no	—
145	C₄×C₈₃	—	4	1	no	—
146	C₄×C₈₃	—	4	1	no	—
147	C₄×C₈₃	—	4	1	no	—
148	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
149	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
150	Dic₈₃	—	2	2	no	—
151	Dic₈₃	—	4	1	no	—
152	Dic₈₃	—	4	1	no	—
153	Dic₈₃	—	4	1	no	—
154	C₂×C₈₃	—	8	1	no	—
155	C₂×C₈₃	#548	1	1	yes	characteristic
156	C₂×C₈₃	#575	1	1	yes	characteristic
157	C₂×C₈₃	#578	1	1	yes	characteristic
158	#3377	—	83	1	no	maximal, Sylow
159	C₈₃	#3377	1	1	yes	characteristic, complemented, Sylow
160	#557	—	83	1	no	—
161	#556	D₈₃	1	1	yes	characteristic
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	—	83	1	no	—
171	#446	—	83	1	no	—
172	#441	—	83	1	no	—
173	#440	—	83	1	no	—
174	C₄×Q₁₆	—	83	1	no	—
175	C₂×Q₁₆	—	83	1	no	—
176	C₂×Q₁₆	—	83	1	no	—
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	—	2	1	no	—
182	#126	—	2	1	no	—
183	#125	—	83	1	no	—
184	#125	—	83	1	no	—
185	#125	—	166	1	no	—
186	#125	—	166	1	no	—
187	#124	C₂×D₈₃	1	1	yes	characteristic
188	#124	C₂×D₈₃	1	1	yes	characteristic
189	C₂²⋊Q₈	—	2	1	no	—
190	C₄×Q₈	—	83	1	no	—
191	C₄×Q₈	—	83	1	no	—
192	C₄×Q₈	—	83	1	no	—
193	C₄×Q₈	—	166	1	no	—
194	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
195	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
196	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
197	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
198	#117	—	2	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	—	83	1	no	—
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	—	83	1	no	—
211	#103	—	83	1	no	—
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	C₂²×D₈₃	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	Q₁₆	—	166	2	no	—
226	Q₁₆	—	166	2	no	—
227	SD₁₆	—	166	2	no	—
228	C₂×C₈	—	83	1	no	—
229	C₂×C₈	—	83	1	no	—
230	C₂×C₈	—	83	1	no	—
231	C₂×C₈	—	83	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	83	1	no	—
237	C₄⋊C₄	—	83	1	no	—
238	C₄⋊C₄	—	83	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	—	166	1	no	—
245	C₄⋊C₄	—	166	1	no	—
246	C₄⋊C₄	—	166	1	no	—
247	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
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	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	166	1	no	—
259	C₄²	—	166	1	no	—
260	C₄²	—	166	1	no	—
261	C₄²	C₂²×D₈₃	1	1	yes	characteristic
262	C₄²	C₂²×D₈₃	1	1	yes	characteristic
263	C₄²	C₂²×D₈₃	1	1	yes	characteristic
264	C₂³	—	2	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	4	1	no	—
270	Q₈	—	332	2	no	—
271	Q₈	—	83	2	no	—
272	Q₈	—	166	1	no	—
273	D₄	—	2	1	no	—
274	D₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	166	1	no	—
280	C₂×C₄	—	166	1	no	—
281	C₂×C₄	—	166	1	no	—
282	C₂×C₄	—	166	1	no	—
283	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
284	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
285	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
286	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
287	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
288	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
290	C₈	—	166	1	no	—
291	C₈	—	166	1	no	—
292	C₈	—	166	1	no	—
293	C₈	—	166	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	332	1	no	—
308	C₄	—	332	1	no	—
309	C₄	—	332	1	no	—
310	C₄	—	166	2	no	—
311	C₄	#41×D₈₃	1	1	yes	characteristic
312	C₄	#14581839	1	1	yes	characteristic
313	C₂	—	8	1	no	—
314	C₂	#19128358	1	1	yes	characteristic
315	C₂	#19128499	1	1	yes	characteristic
316	C₂	#19128516	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#556×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×D₄×C₈₃	C₂³
hypercenter	#556	D₈₃
hypocenter	C₈₃	#3377
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #556

Complements

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

N	H ≅ G/N	type
1. #19128429	C₂	semidirect
2. #19128069	C₂	semidirect
5. #19128008	C₂	semidirect
7. #19127996	C₂	semidirect
10. #19127908	C₂	semidirect
11. #19127832	C₂	semidirect
12. #19127806	C₂	semidirect
14. C₄×#13050580	C₂	semidirect
159. C₈₃	#3377	semidirect

Maximal subgroups

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

subgroup	index
#19128429	2
#19128069	2
#19128060	2
#19128055	2
#19128008	2
#19128001	2
#19127996	2
#19127920	2
#19127908	2
#19127832	2
#19127806	2
#19127801	2
C₄×#13050580	2
#556×C₈₃	2
#3377	83