Subgroups — #15692232

Order 3232.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#13108053	C₂	1	1	yes	maximal, characteristic, complemented
5	#13108053	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
7	C₂²⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₁₀₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₀₁	—	2	2	no	—
12	C₂×Dic₁₀₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₀₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₀₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₀₁	C₂²	1	1	yes	characteristic
16	Dic₂₀₂	—	2	2	no	—
17	C₂²×C₁₀₁	—	2	1	no	—
18	C₂²×C₁₀₁	—	2	1	no	—
19	C₂²×C₁₀₁	Q₈	1	2	yes	complemented
20	C₂²×C₁₀₁	C₂³	1	1	yes	characteristic
21	C₄×C₁₀₁	—	2	1	no	—
22	C₄×C₁₀₁	—	2	1	no	—
23	Dic₁₀₁	—	2	1	no	—
24	Dic₁₀₁	—	2	1	no	—
25	Dic₁₀₁	—	2	1	no	—
26	Dic₁₀₁	D₄	1	2	yes	—
27	C₂×C₁₀₁	—	2	2	no	—
28	C₂×C₁₀₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₁₀₁	#41	1	1	yes	characteristic
31	C₁₀₁	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
32	C₂²⋊Q₈	—	101	1	no	maximal, Sylow
33	C₂×Q₈	—	101	1	no	—
34	C₂²×C₄	—	101	1	no	—
35	C₄⋊C₄	—	101	1	no	—
36	C₄⋊C₄	—	101	1	no	—
37	C₄⋊C₄	—	101	1	no	—
38	C₂²⋊C₄	—	101	1	no	—
39	C₂²⋊C₄	D₁₀₁	1	1	yes	characteristic
40	C₂³	C₂×D₁₀₁	1	1	yes	characteristic
41	Q₈	—	202	2	no	—
42	C₂×C₄	—	101	1	no	—
43	C₂×C₄	—	101	1	no	—
44	C₂×C₄	—	101	1	no	—
45	C₂×C₄	—	101	1	no	—
46	C₂×C₄	—	202	2	no	—
47	C₂×C₄	C₂×D₁₀₁	1	1	yes	characteristic
48	C₂×C₄	C₂×D₁₀₁	1	1	yes	characteristic
49	C₂²	—	2	1	no	—
50	C₂²	—	2	1	no	—
51	C₂²	Dic₂₀₂	1	2	yes	complemented
52	C₂²	C₂²×D₁₀₁	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	2	1	no	—
55	C₄	—	101	2	no	—
56	C₄	—	202	1	no	—
57	C₄	—	202	1	no	—
58	C₄	—	202	1	no	—
59	C₂	—	2	2	no	—
60	C₂	D₄×D₁₀₁	1	1	yes	characteristic
61	C₂	C₂×Dic₂₀₂	1	1	yes	characteristic
62	C₂	#13108073	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₁₀₁
commutator, socle, cosocle	C₂²×C₁₀₁	C₂³
Fitting	C₂²⋊C₄×C₁₀₁	C₂
prime radical	C₂³×C₁₀₁	C₂²
hypercenter	C₂²⋊C₄	D₁₀₁
hypocenter	C₁₀₁	C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₂₀₂	C₂	semidirect
3. C₄⋊Dic₁₀₁	C₂	semidirect
4. #13108053	C₂	semidirect
5. #13108053	C₂	semidirect
19. C₂²×C₁₀₁	Q₈	semidirect
31. C₁₀₁	C₂²⋊Q₈	semidirect
51. C₂²	Dic₂₀₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₀₂	2
C₂²×Dic₁₀₁	2
C₄⋊Dic₁₀₁	2
#13108053	2
C₂²⋊Dic₁₀₁	2
C₂²⋊C₄×C₁₀₁	2
C₂²⋊Q₈	101