Subgroups — C₈×D₂₁₂ #16810690

Order 3392.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₅₃	C₂	1	2	yes	maximal, complemented
2	C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133129	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₁₂	C₄	1	1	yes	characteristic
7	C₂×C₄×D₅₃	C₂²	1	2	yes	—
8	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	#11719921	C₄	1	2	yes	—
12	C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
13	C₂×#88322	C₂²	1	2	yes	—
14	C₈×D₅₃	—	2	4	no	—
15	C₂²×D₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂×Dic₅₃	C₂×C₄	1	2	yes	—
20	D₂₁₂	C₈	1	4	yes	direct product factor
21	C₄×D₅₃	—	2	2	no	—
22	C₄×D₅₃	—	2	2	no	—
23	C₈×C₅₃	—	2	1	no	—
24	C₈×C₅₃	D₄	1	2	yes	complemented
25	#88322	—	2	2	no	—
26	C₂²×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₅₃	—	2	2	no	—
28	C₂×D₅₃	—	2	2	no	—
29	C₂×D₅₃	C₂×C₈	1	4	yes	complemented
30	C₄×C₅₃	—	2	1	no	—
31	C₄×C₅₃	C₂×C₈	1	2	yes	complemented
32	C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₅₃	#41	1	1	yes	characteristic
34	Dic₅₃	—	2	2	no	—
35	C₂×C₅₃	C₄×D₄	1	1	yes	characteristic
36	C₂×C₅₃	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₅₃	#131	1	1	yes	characteristic
38	D₅₃	—	2	4	no	—
39	C₈×D₄	—	53	1	no	maximal, Sylow
40	C₅₃	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₈	—	53	2	no	—
42	C₄×D₄	—	53	1	no	—
43	C₄⋊C₈	—	53	1	no	—
44	C₂²⋊C₈	—	53	2	no	—
45	C₄×C₈	D₅₃	1	1	yes	characteristic, complemented
46	C₂×D₄	—	53	1	no	—
47	C₂²×C₄	—	53	2	no	—
48	C₂×C₈	—	53	2	no	—
49	C₂×C₈	—	106	4	no	—
50	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
51	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
52	C₄⋊C₄	—	53	1	no	—
53	C₂²⋊C₄	—	53	2	no	—
54	C₄²	C₂×D₅₃	1	1	yes	characteristic
55	C₂³	—	53	2	no	—
56	D₄	—	53	4	no	—
57	C₂×C₄	—	53	2	no	—
58	C₂×C₄	—	106	2	no	—
59	C₂×C₄	—	106	2	no	—
60	C₂×C₄	C₄×D₅₃	1	1	yes	characteristic
61	C₂×C₄	C₄×D₅₃	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	106	2	no	—
65	C₈	D₂₁₂	1	2	yes	direct product factor
66	C₂²	—	53	4	no	—
67	C₂²	—	106	2	no	—
68	C₂²	—	106	2	no	—
69	C₂²	C₂×C₄×D₅₃	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	106	2	no	—
72	C₄	C₈×D₅₃	1	2	yes	complemented
73	C₄	C₂×D₂₁₂	1	1	yes	characteristic
74	C₄	#11719945	1	1	yes	characteristic
75	C₂	—	106	4	no	—
76	C₂	C₄×D₂₁₂	1	1	yes	characteristic
77	C₂	C₂×C₈×D₅₃	1	1	yes	characteristic
78	C₂	#13133223	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₅₃
Fitting	C₄×C₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄
prime radical	C₂×D₂₁₂	C₄
cosocle	C₂×C₄×C₅₃	C₂³
hypercenter	C₄×C₈	D₅₃
hypocenter	C₅₃	C₈×D₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×D₅₃	C₂	semidirect
3. C₄⋊#88322	C₂	semidirect
4. #13133129	C₂	semidirect
5. C₄×C₈×C₅₃	C₂	semidirect
20. D₂₁₂	C₈	direct (N × H)
24. C₈×C₅₃	D₄	semidirect
29. C₂×D₅₃	C₂×C₈	semidirect
31. C₄×C₅₃	C₂×C₈	semidirect
40. C₅₃	C₈×D₄	semidirect
45. C₄×C₈	D₅₃	semidirect
65. C₈	D₂₁₂	direct (N × H)
72. C₄	C₈×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₅₃	2
C₄×D₂₁₂	2
C₄⋊#88322	2
#13133129	2
C₄×C₈×C₅₃	2
C₈×D₄	53

Subgroups — C8×D212 #16810690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×D₂₁₂ #16810690