Subgroups — C₈×Dic₇₀₆ #36377912

Order 22592.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₇₀₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#15667725	C₂	1	1	yes	maximal, characteristic
3	#26014921	C₂	1	2	yes	maximal
4	C₈×Dic₃₅₃	C₂	1	2	yes	maximal
5	C₄×C₈×C₃₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₀₆	C₄	1	1	yes	characteristic
7	C₂×#15667725	C₂²	1	2	yes	—
8	C₄⋊Dic₃₅₃	C₄	1	1	yes	characteristic
9	#19150716	C₄	1	2	yes	—
10	C₄×Dic₃₅₃	C₂²	1	2	yes	—
11	C₂×C₈×C₃₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₃₅₃	C₂²	1	1	yes	characteristic
13	C₄²×C₃₅₃	C₂²	1	1	yes	characteristic
14	Dic₇₀₆	C₈	1	4	yes	direct product factor
15	C₂×Dic₃₅₃	C₂×C₄	1	2	yes	—
16	C₂×Dic₃₅₃	C₂×C₄	1	2	yes	—
17	#15667725	—	2	2	no	—
18	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
21	C₈×C₃₅₃	—	2	1	no	—
22	C₈×C₃₅₃	Q₈	1	2	yes	complemented
23	C₂²×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₃₅₃	—	2	1	no	—
25	C₄×C₃₅₃	C₂×C₈	1	2	yes	—
26	C₄×C₃₅₃	C₂×Q₈	1	1	yes	characteristic
27	C₄×C₃₅₃	#41	1	1	yes	characteristic
28	Dic₃₅₃	—	2	2	no	—
29	Dic₃₅₃	C₂×C₈	1	4	yes	—
30	C₂×C₃₅₃	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₃₅₃	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₃₅₃	#131	1	1	yes	characteristic
33	C₃₅₃	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
34	C₈×Q₈	—	353	1	no	maximal, Sylow
35	C₄×Q₈	—	353	1	no	—
36	C₄⋊C₈	—	353	2	no	—
37	C₄⋊C₈	—	353	1	no	—
38	C₄×C₈	—	353	2	no	—
39	C₄×C₈	D₃₅₃	1	1	yes	characteristic
40	C₂×Q₈	—	353	1	no	—
41	C₂×C₈	—	353	2	no	—
42	C₂×C₈	C₂×D₃₅₃	1	1	yes	characteristic
43	C₂×C₈	C₂×D₃₅₃	1	1	yes	characteristic
44	C₄⋊C₄	—	353	2	no	—
45	C₄⋊C₄	—	353	1	no	—
46	C₄²	—	353	2	no	—
47	C₄²	C₂×D₃₅₃	1	1	yes	characteristic
48	Q₈	—	353	4	no	—
49	C₂×C₄	—	353	2	no	—
50	C₂×C₄	—	353	2	no	—
51	C₂×C₄	C₄×D₃₅₃	1	1	yes	characteristic
52	C₂×C₄	C₄×D₃₅₃	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
54	C₈	—	2	1	no	—
55	C₈	—	706	2	no	—
56	C₈	Dic₇₀₆	1	2	yes	direct product factor
57	C₂²	C₂×C₄×D₃₅₃	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	353	4	no	—
60	C₄	—	706	2	no	—
61	C₄	C₈×D₃₅₃	1	2	yes	—
62	C₄	C₂×Dic₇₀₆	1	1	yes	characteristic
63	C₄	#19150737	1	1	yes	characteristic
64	C₂	C₄×Dic₇₀₆	1	1	yes	characteristic
65	C₂	C₂×C₈×D₃₅₃	1	1	yes	characteristic
66	C₂	#26014998	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₃
commutator	C₂×C₃₅₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₃₅₃
Fitting	C₄×C₈×C₃₅₃	C₂
socle, prime radical	C₂²×C₃₅₃	C₂²×C₄
cosocle	C₂×C₄×C₃₅₃	C₂³
hypercenter	C₄×C₈	D₃₅₃
hypocenter	C₃₅₃	C₈×Q₈
LCS	G ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
14. Dic₇₀₆	C₈	direct (N × H)
22. C₈×C₃₅₃	Q₈	semidirect
33. C₃₅₃	C₈×Q₈	semidirect
56. C₈	Dic₇₀₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×Dic₇₀₆	2
C₄⋊#15667725	2
#26014921	2
C₈×Dic₃₅₃	2
C₄×C₈×C₃₅₃	2
C₈×Q₈	353

Subgroups — C8×Dic706 #36377912

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×Dic₇₀₆ #36377912