Subgroups — #35233122

Order 21184.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₆₆₂	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₃₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₆₂	C₄	1	1	yes	characteristic
5	C₂×C₃₃₁⋊C₈	—	2	1	no	—
6	C₄⋊Dic₃₃₁	C₄	1	1	yes	characteristic
7	Dic₃₃₁⋊C₄	—	2	1	no	—
8	C₄×Dic₃₃₁	—	2	1	no	—
9	C₂×C₈×C₃₃₁	—	2	1	no	—
10	C₄²×C₃₃₁	C₂²	1	1	yes	characteristic
11	Dic₆₆₂	—	2	1	no	—
12	Dic₆₆₂	C₈	1	2	yes	complemented
13	C₂×Dic₃₃₁	—	2	1	no	—
14	C₂×Dic₃₃₁	—	2	1	no	—
15	C₃₃₁⋊C₈	—	2	2	no	—
16	C₂×C₄×C₃₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₃₃₁	D₄	1	1	yes	characteristic
18	C₂×C₄×C₃₃₁	D₄	1	1	yes	characteristic
19	C₈×C₃₃₁	—	4	1	no	—
20	C₂²×C₃₃₁	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₃₃₁	—	2	1	no	—
22	C₄×C₃₃₁	C₂×C₈	1	1	yes	characteristic
23	C₄×C₃₃₁	M₁₆	1	1	yes	characteristic
24	C₄×C₃₃₁	SD₁₆	1	1	yes	characteristic
25	C₄×C₃₃₁	Q₁₆	1	1	yes	characteristic
26	Dic₃₃₁	—	2	2	no	—
27	Dic₃₃₁	—	4	1	no	—
28	C₂×C₃₃₁	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₃₃₁	#103	1	1	yes	characteristic
30	C₂×C₃₃₁	#104	1	1	yes	characteristic
31	C₃₃₁	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈⋊C₈	—	331	1	no	maximal, Sylow
33	C₄×Q₈	—	331	1	no	—
34	C₄⋊C₈	D₃₃₁	1	1	yes	characteristic
35	C₄×C₈	—	331	1	no	—
36	C₂×Q₈	—	331	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	662	1	no	—
39	C₄⋊C₄	—	331	1	no	—
40	C₄⋊C₄	—	662	1	no	—
41	C₄²	—	662	1	no	—
42	C₄²	C₂×D₃₃₁	1	1	yes	characteristic
43	Q₈	—	331	2	no	—
44	Q₈	—	662	1	no	—
45	C₂×C₄	—	662	1	no	—
46	C₂×C₄	—	662	1	no	—
47	C₂×C₄	C₄×D₃₃₁	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₃₃₁	1	1	yes	characteristic
49	C₂×C₄	D₁₃₂₄	1	1	yes	characteristic
50	C₈	—	4	1	no	—
51	C₈	—	662	2	no	—
52	C₂²	(C₂×D₃₃₁)⋊C₄	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	1324	1	no	—
55	C₄	—	662	2	no	—
56	C₄	C₈×D₃₃₁	1	1	yes	characteristic
57	C₄	#19127019	1	1	yes	characteristic
58	C₄	#19127024	1	1	yes	characteristic
59	C₄	#19127026	1	1	yes	characteristic
60	C₂	(C₂×D₃₃₁)⋊C₈	1	1	yes	characteristic
61	C₂	#24902531	1	1	yes	characteristic
62	C₂	#24902534	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₃₃₁
commutator	C₄×C₃₃₁	C₂×C₈
Frattini	C₄²	C₂×D₃₃₁
Fitting	C₄⋊C₈×C₃₃₁	C₂
socle, prime radical	C₂²×C₃₃₁	C₂²⋊C₄
cosocle	C₄²×C₃₃₁	C₂²
hypercenter	C₄⋊C₈	D₃₃₁
hypocenter	C₃₃₁	Q₈⋊C₈
LCS	G ▷ C₄×C₃₃₁ ▷ C₂×C₃₃₁ ▷ C₃₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
12. Dic₆₆₂	C₈	semidirect
31. C₃₃₁	Q₈⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₆₆₂	2
C₄×C₃₃₁⋊C₈	2
C₄⋊C₈×C₃₃₁	2
Q₈⋊C₈	331