Subgroups — #24920385

Order 10624.

360 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	D₄×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
3	#19128110	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128109	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128105	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128003	C₂	1	1	yes	maximal, characteristic
7	#19128001	C₂	1	1	yes	maximal, characteristic
8	#19127996	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127994	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127984	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127979	C₂	1	1	yes	maximal, characteristic
12	#19127829	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127802	C₂	1	1	yes	maximal, characteristic
14	SD₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic
15	#440×C₈₃	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₁₆₆	—	2	1	no	—
17	#14581792	C₂²	1	1	yes	characteristic
18	#14581791	—	2	1	no	—
19	#14581791	C₂²	1	1	yes	characteristic
20	#14581791	C₂²	1	1	yes	characteristic
21	#14581791	C₂²	1	1	yes	characteristic
22	#14581784	—	2	1	no	—
23	#14581784	C₂²	1	1	yes	characteristic
24	#14581783	—	2	1	no	—
25	#14581783	C₂²	1	1	yes	characteristic
26	#14581771	—	2	1	no	—
27	#14581769	—	2	1	no	—
28	#14581769	—	2	1	no	—
29	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
30	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
31	C₄×Dic₁₆₆	—	2	1	no	—
32	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
33	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
34	D₄×Dic₈₃	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	#14581725	C₂²	1	1	yes	characteristic
40	#14581724	C₂²	1	1	yes	characteristic
41	#14581724	C₂²	1	1	yes	characteristic
42	#14581723	C₂²	1	1	yes	characteristic
43	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
46	#14581716	C₂²	1	1	yes	characteristic
47	#14581716	C₂²	1	1	yes	characteristic
48	#14581713	C₂²	1	1	yes	characteristic
49	#14581713	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
55	#107×C₈₃	C₂²	1	1	yes	characteristic
56	#103×C₈₃	C₂²	1	1	yes	characteristic
57	#102×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	—	2	1	no	—
63	C₂×Dic₁₆₆	—	4	4	no	—
64	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
65	C₂²×Dic₈₃	—	2	1	no	—
66	C₂²×Dic₈₃	—	2	1	no	—
67	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₈₃	—	2	1	no	—
70	C₄⋊Dic₈₃	—	2	1	no	—
71	C₄⋊Dic₈₃	—	2	1	no	—
72	C₄⋊Dic₈₃	—	2	1	no	—
73	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
74	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
85	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₈₃	—	2	1	no	—
87	C₂²⋊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×D₄×C₈₃	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₈₃	—	2	1	no	—
95	SD₁₆×C₈₃	Q₈	1	4	yes	complemented
96	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₈₃	—	2	1	no	—
102	C₄²×C₈₃	—	2	1	no	—
103	C₄²×C₈₃	C₂³	1	1	yes	characteristic
104	C₂³×C₈₃	—	2	1	no	—
105	Q₈×C₈₃	—	2	1	no	—
106	Q₈×C₈₃	C₂×Q₈	1	2	yes	complemented
107	D₄×C₈₃	—	2	1	no	—
108	D₄×C₈₃	C₂×Q₈	1	2	yes	—
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	—	4	2	no	—
113	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₈₃	—	2	1	no	—
117	C₂×Dic₈₃	—	2	1	no	—
118	C₂×Dic₈₃	—	2	1	no	—
119	C₂×Dic₈₃	—	2	1	no	—
120	C₂×Dic₈₃	—	4	2	no	—
121	C₂×Dic₈₃	—	4	2	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	Dic₁₆₆	—	2	2	no	—
127	Dic₁₆₆	—	4	2	no	—
128	Dic₁₆₆	—	4	2	no	—
129	Dic₁₆₆	—	4	2	no	—
130	Dic₁₆₆	—	4	2	no	—
131	C₈×C₈₃	—	2	1	no	—
132	C₈×C₈₃	C₂×Q₈	1	2	yes	complemented
133	C₈₃⋊C₈	—	2	1	no	—
134	C₈₃⋊C₈	—	2	1	no	—
135	C₂²×C₈₃	—	2	2	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	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	C₂²×Q₈	1	1	yes	characteristic
145	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
146	Dic₈₃	—	2	1	no	—
147	Dic₈₃	—	2	1	no	—
148	Dic₈₃	—	2	1	no	—
149	Dic₈₃	—	4	1	no	—
150	Dic₈₃	—	4	1	no	—
151	Dic₈₃	—	4	1	no	—
152	Dic₈₃	—	4	1	no	—
153	Dic₈₃	AGL(1,ℤ/8)	1	2	yes	—
154	C₂×C₈₃	—	4	2	no	—
155	C₂×C₈₃	D₄×Q₈	1	1	yes	characteristic
156	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₈₃	#578	1	1	yes	characteristic
158	#3386	—	83	1	no	maximal, Sylow
159	C₈₃	#3386	1	1	yes	characteristic, complemented, Sylow
160	#557	—	83	1	no	—
161	D₄×Q₈	—	83	1	no	—
162	#501	—	83	1	no	—
163	#501	—	83	1	no	—
164	#500	—	83	1	no	—
165	#478	—	83	1	no	—
166	#478	—	83	1	no	—
167	#477	—	83	1	no	—
168	#477	—	83	1	no	—
169	#475	—	83	1	no	—
170	#474	—	83	1	no	—
171	#446	—	83	1	no	—
172	#440	—	83	1	no	—
173	#440	D₈₃	1	1	yes	characteristic
174	C₄×SD₁₆	—	83	1	no	—
175	C₂²×Q₈	—	166	1	no	—
176	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
177	C₄⋊Q₈	—	83	1	no	—
178	C₄⋊Q₈	—	83	1	no	—
179	C₄⋊Q₈	—	83	1	no	—
180	C₄⋊Q₈	—	83	1	no	—
181	C₄⋊Q₈	—	166	1	no	—
182	#125	—	83	1	no	—
183	#125	—	83	1	no	—
184	#125	—	166	1	no	—
185	#125	—	166	1	no	—
186	C₂²⋊Q₈	—	166	1	no	—
187	C₂²⋊Q₈	—	166	1	no	—
188	C₂²⋊Q₈	—	166	1	no	—
189	C₄×Q₈	—	83	1	no	—
190	C₄×Q₈	—	83	1	no	—
191	C₄×Q₈	—	83	1	no	—
192	C₄×Q₈	—	166	1	no	—
193	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
194	C₄×D₄	—	83	1	no	—
195	C₄×D₄	—	83	1	no	—
196	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
197	#107	—	83	1	no	—
198	#107	—	83	1	no	—
199	#107	—	83	1	no	—
200	#107	—	83	1	no	—
201	#107	—	83	1	no	—
202	#107	C₂×D₈₃	1	1	yes	characteristic
203	#106	—	83	1	no	—
204	#106	—	83	1	no	—
205	#106	—	83	1	no	—
206	C₄⋊C₈	—	83	1	no	—
207	C₄⋊C₈	—	83	1	no	—
208	C₄⋊C₈	—	83	1	no	—
209	#103	—	83	1	no	—
210	#103	—	83	1	no	—
211	#103	C₂×D₈₃	1	1	yes	characteristic
212	#102	—	83	1	no	—
213	#102	—	83	1	no	—
214	#102	C₂×D₈₃	1	1	yes	characteristic
215	#97	—	83	1	no	—
216	#97	C₂×D₈₃	1	1	yes	characteristic
217	C₄×C₈	—	83	1	no	—
218	C₂×Q₈	—	332	4	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×Q₈	—	166	1	no	—
224	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
225	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	166	1	no	—
228	C₂²×C₄	—	166	1	no	—
229	SD₁₆	Dic₁₆₆	1	4	yes	complemented
230	C₂×C₈	—	83	1	no	—
231	C₂×C₈	—	83	1	no	—
232	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
233	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
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	—	83	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	—	166	1	no	—
251	C₄⋊C₄	—	166	1	no	—
252	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	166	1	no	—
256	C₂²⋊C₄	—	166	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	83	1	no	—
259	C₄²	—	83	1	no	—
260	C₄²	—	166	1	no	—
261	C₄²	—	166	1	no	—
262	C₄²	C₂²×D₈₃	1	1	yes	characteristic
263	C₂³	—	2	1	no	—
264	Q₈	—	2	1	no	—
265	Q₈	—	332	2	no	—
266	Q₈	—	332	2	no	—
267	Q₈	—	332	2	no	—
268	Q₈	—	332	2	no	—
269	Q₈	—	166	2	no	—
270	Q₈	C₂×Dic₁₆₆	1	2	yes	complemented
271	D₄	—	2	1	no	—
272	D₄	C₂×Dic₁₆₆	1	2	yes	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	332	2	no	—
278	C₂×C₄	—	332	2	no	—
279	C₂×C₄	—	83	1	no	—
280	C₂×C₄	—	83	1	no	—
281	C₂×C₄	—	83	1	no	—
282	C₂×C₄	—	83	1	no	—
283	C₂×C₄	—	166	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	—	166	1	no	—
292	C₈	—	166	1	no	—
293	C₈	C₂×Dic₁₆₆	1	2	yes	complemented
294	C₂²	—	2	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	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	—	332	1	no	—
304	C₄	—	332	1	no	—
305	C₄	—	332	1	no	—
306	C₄	—	332	1	no	—
307	C₄	—	83	2	no	—
308	C₄	—	166	1	no	—
309	C₄	—	166	1	no	—
310	C₄	—	166	1	no	—
311	C₄	C₂²×Dic₁₆₆	1	1	yes	characteristic
312	C₄	#14581839	1	1	yes	characteristic
313	C₂	—	4	2	no	—
314	C₂	D₄×Dic₁₆₆	1	1	yes	characteristic
315	C₂	AGL(1,ℤ/8)×D₈₃	1	1	yes	characteristic
316	C₂	#19128517	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#440×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×D₄×C₈₃	C₂³
hypercenter	#440	D₈₃
hypocenter	C₈₃	#3386
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #19128427	C₂	semidirect
3. #19128110	C₂	semidirect
4. #19128109	C₂	semidirect
5. #19128105	C₂	semidirect
8. #19127996	C₂	semidirect
9. #19127994	C₂	semidirect
10. #19127984	C₂	semidirect
12. #19127829	C₂	semidirect
95. SD₁₆×C₈₃	Q₈	semidirect
106. Q₈×C₈₃	C₂×Q₈	semidirect
132. C₈×C₈₃	C₂×Q₈	semidirect
159. C₈₃	#3386	semidirect
229. SD₁₆	Dic₁₆₆	semidirect
270. Q₈	C₂×Dic₁₆₆	semidirect
293. C₈	C₂×Dic₁₆₆	semidirect

Maximal subgroups

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

subgroup	index
#19128427	2
D₄×Dic₁₆₆	2
#19128110	2
#19128109	2
#19128105	2
#19128003	2
#19128001	2
#19127996	2
#19127994	2
#19127984	2
#19127979	2
#19127829	2
#19127802	2
SD₁₆×Dic₈₃	2
#440×C₈₃	2
#3386	83