Subgroups — #27212876

Order 13504.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₄₂₂	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₂₂	C₄	1	1	yes	characteristic
5	C₂×C₂₁₁⋊C₈	—	2	1	no	—
6	C₄⋊Dic₂₁₁	C₄	1	1	yes	characteristic
7	Dic₂₁₁⋊C₄	—	2	1	no	—
8	C₄×Dic₂₁₁	—	2	1	no	—
9	C₂×C₈×C₂₁₁	—	2	1	no	—
10	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
11	Dic₄₂₂	—	2	1	no	—
12	Dic₄₂₂	C₈	1	2	yes	complemented
13	C₂×Dic₂₁₁	—	2	1	no	—
14	C₂×Dic₂₁₁	—	2	1	no	—
15	C₂₁₁⋊C₈	—	2	2	no	—
16	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
19	C₈×C₂₁₁	—	4	1	no	—
20	C₂²×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₂₁₁	—	2	1	no	—
22	C₄×C₂₁₁	C₂×C₈	1	1	yes	characteristic
23	C₄×C₂₁₁	M₁₆	1	1	yes	characteristic
24	C₄×C₂₁₁	SD₁₆	1	1	yes	characteristic
25	C₄×C₂₁₁	Q₁₆	1	1	yes	characteristic
26	Dic₂₁₁	—	2	2	no	—
27	Dic₂₁₁	—	4	1	no	—
28	C₂×C₂₁₁	C₂²⋊C₈	1	1	yes	characteristic
29	C₂×C₂₁₁	#103	1	1	yes	characteristic
30	C₂×C₂₁₁	#104	1	1	yes	characteristic
31	C₂₁₁	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
32	Q₈⋊C₈	—	211	1	no	maximal, Sylow
33	C₄×Q₈	—	211	1	no	—
34	C₄⋊C₈	D₂₁₁	1	1	yes	characteristic
35	C₄×C₈	—	211	1	no	—
36	C₂×Q₈	—	211	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	422	1	no	—
39	C₄⋊C₄	—	422	1	no	—
40	C₄⋊C₄	—	211	1	no	—
41	C₄²	—	422	1	no	—
42	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
43	Q₈	—	422	1	no	—
44	Q₈	—	211	2	no	—
45	C₂×C₄	—	422	1	no	—
46	C₂×C₄	—	422	1	no	—
47	C₂×C₄	C₄×D₂₁₁	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
49	C₂×C₄	D₈₄₄	1	1	yes	characteristic
50	C₈	—	4	1	no	—
51	C₈	—	422	2	no	—
52	C₂²	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	844	1	no	—
55	C₄	—	422	2	no	—
56	C₄	C₈×D₂₁₁	1	1	yes	characteristic
57	C₄	#16809945	1	1	yes	characteristic
58	C₄	#16809950	1	1	yes	characteristic
59	C₄	#16809952	1	1	yes	characteristic
60	C₂	(C₂×D₂₁₁)⋊C₈	1	1	yes	characteristic
61	C₂	#20269900	1	1	yes	characteristic
62	C₂	#20269903	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₂₁₁
commutator	C₄×C₂₁₁	C₂×C₈
Frattini	C₄²	C₂×D₂₁₁
Fitting	C₄⋊C₈×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	C₂²⋊C₄
cosocle	C₄²×C₂₁₁	C₂²
hypercenter	C₄⋊C₈	D₂₁₁
hypocenter	C₂₁₁	Q₈⋊C₈
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
12. Dic₄₂₂	C₈	semidirect
31. C₂₁₁	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×Dic₄₂₂	2
C₄×C₂₁₁⋊C₈	2
C₄⋊C₈×C₂₁₁	2
Q₈⋊C₈	211