Subgroups — #44537087

Order 31424.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30628325	C₂	1	1	yes	maximal, characteristic
2	#30628260	C₂	1	2	yes	maximal, complemented
3	#30628257	C₂	1	2	yes	maximal
4	#30628248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₄₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₉₈₂	C₂²	1	1	yes	characteristic
7	C₂×D₁₉₆₄	C₂²	1	1	yes	characteristic
8	C₂×C₄₉₁⋊C₈	C₂²	1	2	yes	—
9	(C₂×D₄₉₁)⋊C₄	—	2	1	no	—
10	(C₂×D₄₉₁)⋊C₄	—	2	1	no	—
11	C₂×Q₈×C₄₉₁	—	2	1	no	—
12	C₄⋊C₄×C₄₉₁	—	2	1	no	—
13	C₄⋊C₄×C₄₉₁	C₂²	1	2	yes	—
14	C₄²×C₄₉₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄₉₁	—	2	1	no	—
16	Dic₉₈₂	—	2	2	no	—
17	D₁₉₆₄	—	2	2	no	—
18	C₂×Dic₄₉₁	—	2	1	no	—
19	C₄₉₁⋊C₈	—	2	2	no	—
20	Q₈×C₄₉₁	—	4	2	no	—
21	C₂×C₄×C₄₉₁	—	2	2	no	—
22	C₂×C₄×C₄₉₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₉₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₄₉₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄₉₁	C₂×D₄	1	1	yes	characteristic
26	C₂×D₄₉₁	—	4	1	no	—
27	C₂×D₄₉₁	—	4	1	no	—
28	C₂×D₄₉₁	—	4	1	no	—
29	C₄×C₄₉₁	—	2	1	no	—
30	C₄×C₄₉₁	—	2	1	no	—
31	C₄×C₄₉₁	—	4	2	no	—
32	C₄×C₄₉₁	#41	1	2	yes	—
33	Dic₄₉₁	—	4	1	no	—
34	C₂×C₄₉₁	#124	1	1	yes	characteristic
35	C₂×C₄₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂×C₄₉₁	#137	1	1	yes	characteristic
37	D₄₉₁	—	8	1	no	—
38	C₄₉₁	#489	1	1	yes	characteristic, complemented, Sylow
39	#489	—	491	1	no	maximal, Sylow
40	C₄⋊Q₈	D₄₉₁	1	1	yes	characteristic, complemented
41	#124	—	491	1	no	—
42	#103	—	491	2	no	—
43	#102	—	491	2	no	—
44	#97	—	491	1	no	—
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	491	1	no	—
47	C₂×D₄	—	491	1	no	—
48	C₂×C₈	—	491	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	C₂×D₄₉₁	1	2	yes	—
51	C₂²⋊C₄	—	982	1	no	—
52	C₂²⋊C₄	—	982	1	no	—
53	C₄²	C₂×D₄₉₁	1	1	yes	characteristic
54	C₂³	—	982	1	no	—
55	Q₈	—	4	2	no	—
56	Q₈	—	982	2	no	—
57	D₄	—	982	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	982	1	no	—
60	C₂×C₄	C₂²⋊D₄₉₁	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₄₉₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₄₉₁	1	1	yes	characteristic
63	C₈	—	982	2	no	—
64	C₂²	—	1964	1	no	—
65	C₂²	—	1964	1	no	—
66	C₂²	—	1964	1	no	—
67	C₂²	C₂×C₂²⋊D₄₉₁	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	2	no	—
71	C₄	—	1964	1	no	—
72	C₄	#21463329	1	2	yes	—
73	C₂	—	3928	1	no	—
74	C₂	#30628326	1	1	yes	characteristic
75	C₂	#30628363	1	1	yes	characteristic
76	C₂	#30628368	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₉₁
commutator, cosocle	C₂×C₄×C₄₉₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₉₁
Fitting	C₄⋊Q₈×C₄₉₁	C₂
socle	C₂²×C₄₉₁	C₂×D₄
prime radical	C₂×D₁₉₆₄	C₂²
hypercenter	C₄⋊Q₈	D₄₉₁
hypocenter	C₄₉₁	#489
LCS	G ▷ C₂×C₄×C₄₉₁ ▷ C₂×C₄₉₁ ▷ C₄₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #30628260	C₂	semidirect
4. #30628248	C₂	semidirect
5. C₄⋊Q₈×C₄₉₁	C₂	semidirect
38. C₄₉₁	#489	semidirect
40. C₄⋊Q₈	D₄₉₁	semidirect

Maximal subgroups

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

subgroup	index
#30628325	2
#30628260	2
#30628257	2
#30628248	2
C₄⋊Q₈×C₄₉₁	2
#489	491