Subgroups — #43246177

Order 28256.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₇₆₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₈₈₃	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Dic₈₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₈₈₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₈₈₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×Dic₈₈₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₄×C₈₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈₈₃	C₂²	1	1	yes	characteristic
9	Dic₁₇₆₆	—	2	2	no	—
10	C₂²⋊D₈₈₃	—	2	2	no	—
11	C₂×Dic₈₈₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₈₈₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₈₈₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈₈₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈₈₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈₈₃	—	2	1	no	—
18	C₂²×C₈₈₃	—	2	1	no	—
19	C₂²×C₈₈₃	—	2	1	no	—
20	C₂²×C₈₈₃	C₂³	1	1	yes	characteristic
21	C₂×D₈₈₃	—	2	1	no	—
22	C₂×D₈₈₃	—	2	1	no	—
23	C₂×D₈₈₃	—	2	1	no	—
24	C₄×C₈₈₃	—	2	1	no	—
25	C₄×C₈₈₃	—	2	1	no	—
26	Dic₈₈₃	—	2	1	no	—
27	Dic₈₈₃	—	2	1	no	—
28	Dic₈₈₃	D₄	1	2	yes	—
29	C₂×C₈₈₃	—	4	1	no	—
30	C₂×C₈₈₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₈₈₃	#41	1	1	yes	characteristic
32	C₂×C₈₈₃	#41	1	1	yes	characteristic
33	D₈₈₃	—	4	1	no	—
34	C₈₈₃	#124	1	1	yes	characteristic, complemented, Sylow
35	#124	—	883	1	no	maximal, Sylow
36	C₂×Q₈	—	883	1	no	—
37	C₂×D₄	—	883	1	no	—
38	C₂²⋊C₄	—	883	1	no	—
39	C₂²⋊C₄	—	883	1	no	—
40	C₂²⋊C₄	—	883	1	no	—
41	C₂²⋊C₄	D₈₈₃	1	1	yes	characteristic, complemented
42	C₄²	—	883	1	no	—
43	C₂³	—	883	1	no	—
44	C₂³	C₂×D₈₈₃	1	1	yes	characteristic
45	Q₈	—	1766	2	no	—
46	D₄	—	1766	2	no	—
47	C₂×C₄	—	883	1	no	—
48	C₂×C₄	—	883	1	no	—
49	C₂×C₄	—	883	1	no	—
50	C₂×C₄	C₂×D₈₈₃	1	1	yes	characteristic
51	C₂×C₄	C₂×D₈₈₃	1	1	yes	characteristic
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	1766	1	no	—
56	C₂²	—	1766	1	no	—
57	C₂²	—	1766	1	no	—
58	C₂²	C₂²×D₈₈₃	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	2	1	no	—
61	C₄	—	883	2	no	—
62	C₄	—	1766	1	no	—
63	C₄	—	1766	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	3532	1	no	—
66	C₂	D₄×D₈₈₃	1	1	yes	characteristic
67	C₂	#28381496	1	1	yes	characteristic
68	C₂	#28381497	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₈₈₃
commutator, socle, cosocle	C₂²×C₈₈₃	C₂³
Fitting	C₂²⋊C₄×C₈₈₃	C₂
prime radical	C₂×C₂²⋊D₈₈₃	C₂
hypercenter	C₂²⋊C₄	D₈₈₃
hypocenter	C₈₈₃	#124
LCS	G ▷ C₂²×C₈₈₃ ▷ C₈₈₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₇₆₆	C₂	semidirect
3. C₂²⋊Dic₈₈₃	C₂	semidirect
4. (C₂×D₈₈₃)⋊C₄	C₂	semidirect
5. (C₂×D₈₈₃)⋊C₄	C₂	semidirect
6. C₄×Dic₈₈₃	C₂	semidirect
7. C₂²⋊C₄×C₈₈₃	C₂	semidirect
34. C₈₈₃	#124	semidirect
41. C₂²⋊C₄	D₈₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₇₆₆	2
C₂×C₂²⋊D₈₈₃	2
C₂²⋊Dic₈₈₃	2
(C₂×D₈₈₃)⋊C₄	2
C₄×Dic₈₈₃	2
C₂²⋊C₄×C₈₈₃	2
#124	883