Subgroups — #24909321

Order 10624.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₆₆₄	C₂	1	1	yes	maximal, characteristic
2	#19127795	C₂	1	2	yes	maximal
3	#19127504	C₂	1	2	yes	maximal
4	C₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic
5	#366×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₃₂	C₂²	1	2	yes	—
7	#14581757	—	2	2	no	—
8	Dic₆₆₄	C₄	1	4	yes	complemented
9	C₂×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
10	#14581715	—	2	2	no	—
11	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
13	#107×C₈₃	C₂²	1	2	yes	—
14	C₂×Dic₁₆₆	—	2	2	no	—
15	Dic₃₃₂	—	2	2	no	—
16	Dic₃₃₂	C₂×C₄	1	4	yes	—
17	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
18	Dic₈₃⋊C₄	—	4	2	no	—
19	C₄×Dic₈₃	—	4	2	no	—
20	C₄×Dic₈₃	D₄	1	1	yes	characteristic
21	C₈₃⋊C₁₆	—	2	1	no	—
22	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₈₃	—	2	2	no	—
24	C₁₆×C₈₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₈₃	—	4	2	no	—
26	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₈₃	—	4	2	no	—
28	C₂×Dic₈₃	D₈	1	1	yes	characteristic
29	C₂×Dic₈₃	D₈	1	1	yes	characteristic
30	Dic₁₆₆	—	2	4	no	—
31	Dic₁₆₆	—	4	2	no	—
32	C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₈₃	#41	1	1	yes	characteristic
34	C₈₃⋊C₈	—	2	1	no	—
35	C₂²×C₈₃	C₂×D₈	1	1	yes	characteristic
36	C₄×C₈₃	—	8	2	no	—
37	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
38	C₄×C₈₃	#135	1	1	yes	characteristic
39	Dic₈₃	—	2	1	no	—
40	Dic₈₃	—	4	4	no	—
41	Dic₈₃	Q₃₂	1	2	yes	complemented
42	C₂×C₈₃	C₄×D₈	1	1	yes	characteristic
43	C₂×C₈₃	C₂×Q₃₂	1	1	yes	characteristic
44	C₂×C₈₃	#508	1	1	yes	characteristic
45	C₄×Q₃₂	—	83	1	no	maximal, Sylow
46	C₈₃	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
47	C₂×Q₃₂	—	83	1	no	—
48	C₄×Q₁₆	—	83	2	no	—
49	#366	D₈₃	1	1	yes	characteristic
50	#358	—	83	2	no	—
51	C₄×C₁₆	—	83	1	no	—
52	C₂×Q₁₆	—	83	2	no	—
53	C₄×Q₈	—	166	2	no	—
54	Q₃₂	—	83	4	no	—
55	C₂×C₁₆	—	83	1	no	—
56	C₂×C₁₆	C₂×D₈₃	1	1	yes	characteristic
57	#107	C₂×D₈₃	1	2	yes	—
58	#103	—	166	2	no	—
59	C₄×C₈	—	83	1	no	—
60	C₂×Q₈	—	166	2	no	—
61	Q₁₆	—	83	4	no	—
62	Q₁₆	—	166	2	no	—
63	C₂×C₈	—	83	1	no	—
64	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	332	2	no	—
67	C₄²	—	332	2	no	—
68	C₄²	—	83	1	no	—
69	C₁₆	—	166	1	no	—
70	C₁₆	C₄×D₈₃	1	2	yes	—
71	Q₈	—	332	2	no	—
72	Q₈	—	166	4	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	332	2	no	—
75	C₂×C₄	—	83	1	no	—
76	C₂×C₄	—	83	1	no	—
77	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
78	C₈	—	166	1	no	—
79	C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
80	C₈	#13050591	1	1	yes	characteristic
81	C₂²	D₈×D₈₃	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	332	4	no	—
84	C₄	—	83	2	no	—
85	C₄	—	166	1	no	—
86	C₄	#14581754	1	1	yes	characteristic
87	C₄	#14581815	1	1	yes	characteristic
88	C₂	#19127783	1	1	yes	characteristic
89	C₂	Q₃₂×D₈₃	1	1	yes	characteristic
90	C₂	#19128130	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₈₃
commutator	C₈×C₈₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₈₃
Fitting	#366×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₂×D₈
cosocle	C₂×C₈×C₈₃	C₂³
hypercenter	#366	D₈₃
hypocenter	C₈₃	C₄×Q₃₂
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
8. Dic₆₆₄	C₄	semidirect
41. Dic₈₃	Q₃₂	semidirect
46. C₈₃	C₄×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₆₆₄	2
#19127795	2
#19127504	2
C₁₆×Dic₈₃	2
#366×C₈₃	2
C₄×Q₃₂	83