Subgroups — #46830537

Order 34624.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31772616	C₂	1	1	yes	maximal, characteristic
2	#31772551	C₂	1	2	yes	maximal, complemented
3	#31772548	C₂	1	2	yes	maximal
4	#31772539	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₅₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₁₀₈₂	C₂²	1	1	yes	characteristic
7	C₂×D₂₁₆₄	C₂²	1	1	yes	characteristic
8	C₂×#16864513	C₂²	1	2	yes	—
9	#22575813	—	2	1	no	—
10	#22575813	—	2	1	no	—
11	C₂×Q₈×C₅₄₁	—	2	1	no	—
12	C₄⋊C₄×C₅₄₁	—	2	1	no	—
13	C₄⋊C₄×C₅₄₁	C₂²	1	2	yes	—
14	C₄²×C₅₄₁	C₂²	1	1	yes	characteristic
15	C₂²×D₅₄₁	—	2	1	no	—
16	Dic₁₀₈₂	—	2	2	no	—
17	D₂₁₆₄	—	2	2	no	—
18	C₂×Dic₅₄₁	—	2	1	no	—
19	#16864513	—	2	2	no	—
20	Q₈×C₅₄₁	—	4	2	no	—
21	C₂×C₄×C₅₄₁	—	2	2	no	—
22	C₂×C₄×C₅₄₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₅₄₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₅₄₁	C₂³	1	1	yes	characteristic
25	C₂²×C₅₄₁	C₂×D₄	1	1	yes	characteristic
26	C₂×D₅₄₁	—	4	1	no	—
27	C₂×D₅₄₁	—	4	1	no	—
28	C₂×D₅₄₁	—	4	1	no	—
29	C₄×C₅₄₁	—	2	1	no	—
30	C₄×C₅₄₁	—	2	1	no	—
31	C₄×C₅₄₁	—	4	2	no	—
32	C₄×C₅₄₁	#41	1	2	yes	—
33	Dic₅₄₁	—	4	1	no	—
34	C₂×C₅₄₁	#124	1	1	yes	characteristic
35	C₂×C₅₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂×C₅₄₁	#137	1	1	yes	characteristic
37	D₅₄₁	—	8	1	no	—
38	C₅₄₁	#489	1	1	yes	characteristic, complemented, Sylow
39	#489	—	541	1	no	maximal, Sylow
40	C₄⋊Q₈	D₅₄₁	1	1	yes	characteristic, complemented
41	#124	—	541	1	no	—
42	#103	—	541	2	no	—
43	#102	—	541	2	no	—
44	#97	—	541	1	no	—
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	541	1	no	—
47	C₂×D₄	—	541	1	no	—
48	C₂×C₈	—	541	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	C₂×D₅₄₁	1	2	yes	—
51	C₂²⋊C₄	—	1082	1	no	—
52	C₂²⋊C₄	—	1082	1	no	—
53	C₄²	C₂×D₅₄₁	1	1	yes	characteristic
54	C₂³	—	1082	1	no	—
55	Q₈	—	4	2	no	—
56	Q₈	—	1082	2	no	—
57	D₄	—	1082	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	1082	1	no	—
60	C₂×C₄	C₂²⋊D₅₄₁	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₅₄₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₅₄₁	1	1	yes	characteristic
63	C₈	—	1082	2	no	—
64	C₂²	—	2164	1	no	—
65	C₂²	—	2164	1	no	—
66	C₂²	—	2164	1	no	—
67	C₂²	C₂×C₂²⋊D₅₄₁	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	2	no	—
71	C₄	—	2164	1	no	—
72	C₄	#22575837	1	2	yes	—
73	C₂	—	4328	1	no	—
74	C₂	#31772617	1	1	yes	characteristic
75	C₂	#31772654	1	1	yes	characteristic
76	C₂	#31772659	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₄₁
commutator, cosocle	C₂×C₄×C₅₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₅₄₁
Fitting	C₄⋊Q₈×C₅₄₁	C₂
socle	C₂²×C₅₄₁	C₂×D₄
prime radical	C₂×D₂₁₆₄	C₂²
hypercenter	C₄⋊Q₈	D₅₄₁
hypocenter	C₅₄₁	#489
LCS	G ▷ C₂×C₄×C₅₄₁ ▷ C₂×C₅₄₁ ▷ C₅₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #31772551	C₂	semidirect
4. #31772539	C₂	semidirect
5. C₄⋊Q₈×C₅₄₁	C₂	semidirect
38. C₅₄₁	#489	semidirect
40. C₄⋊Q₈	D₅₄₁	semidirect

Maximal subgroups

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

subgroup	index
#31772616	2
#31772551	2
#31772548	2
#31772539	2
C₄⋊Q₈×C₅₄₁	2
#489	541