Subgroups — #27213508

Order 13504.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269965	C₂	1	1	yes	maximal, characteristic
2	#20269900	C₂	1	2	yes	maximal, complemented
3	#20269897	C₂	1	2	yes	maximal
4	#20269888	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₂₂	C₂²	1	1	yes	characteristic
7	C₂×D₈₄₄	C₂²	1	1	yes	characteristic
8	C₂×C₂₁₁⋊C₈	C₂²	1	2	yes	—
9	(C₂×D₂₁₁)⋊C₄	—	2	1	no	—
10	(C₂×D₂₁₁)⋊C₄	—	2	1	no	—
11	C₂×Q₈×C₂₁₁	—	2	1	no	—
12	C₄⋊C₄×C₂₁₁	—	2	1	no	—
13	C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
14	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂²×D₂₁₁	—	2	1	no	—
16	Dic₄₂₂	—	2	2	no	—
17	D₈₄₄	—	2	2	no	—
18	C₂×Dic₂₁₁	—	2	1	no	—
19	C₂₁₁⋊C₈	—	2	2	no	—
20	Q₈×C₂₁₁	—	4	2	no	—
21	C₂×C₄×C₂₁₁	—	2	2	no	—
22	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
26	C₂×D₂₁₁	—	4	1	no	—
27	C₂×D₂₁₁	—	4	1	no	—
28	C₂×D₂₁₁	—	4	1	no	—
29	C₄×C₂₁₁	—	2	1	no	—
30	C₄×C₂₁₁	—	2	1	no	—
31	C₄×C₂₁₁	—	4	2	no	—
32	C₄×C₂₁₁	#41	1	2	yes	—
33	Dic₂₁₁	—	4	1	no	—
34	C₂×C₂₁₁	#124	1	1	yes	characteristic
35	C₂×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂×C₂₁₁	#137	1	1	yes	characteristic
37	D₂₁₁	—	8	1	no	—
38	C₂₁₁	#489	1	1	yes	characteristic, complemented, Sylow
39	#489	—	211	1	no	maximal, Sylow
40	C₄⋊Q₈	D₂₁₁	1	1	yes	characteristic, complemented
41	#124	—	211	1	no	—
42	#103	—	211	2	no	—
43	#102	—	211	2	no	—
44	#97	—	211	1	no	—
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	211	1	no	—
47	C₂×D₄	—	211	1	no	—
48	C₂×C₈	—	211	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
51	C₂²⋊C₄	—	422	1	no	—
52	C₂²⋊C₄	—	422	1	no	—
53	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
54	C₂³	—	422	1	no	—
55	Q₈	—	4	2	no	—
56	Q₈	—	422	2	no	—
57	D₄	—	422	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	422	1	no	—
60	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
63	C₈	—	422	2	no	—
64	C₂²	—	844	1	no	—
65	C₂²	—	844	1	no	—
66	C₂²	—	844	1	no	—
67	C₂²	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	2	no	—
71	C₄	—	844	1	no	—
72	C₄	#16809963	1	2	yes	—
73	C₂	—	1688	1	no	—
74	C₂	#20269966	1	1	yes	characteristic
75	C₂	#20270003	1	1	yes	characteristic
76	C₂	#20270008	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₁₁
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	C₄⋊Q₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
prime radical	C₂×D₈₄₄	C₂²
hypercenter	C₄⋊Q₈	D₂₁₁
hypocenter	C₂₁₁	#489
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #20269900	C₂	semidirect
4. #20269888	C₂	semidirect
5. C₄⋊Q₈×C₂₁₁	C₂	semidirect
38. C₂₁₁	#489	semidirect
40. C₄⋊Q₈	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20269965	2
#20269900	2
#20269897	2
#20269888	2
C₄⋊Q₈×C₂₁₁	2
#489	211