Subgroups — #44548537

Order 32128.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632055	C₂	1	1	yes	maximal, characteristic
2	#30631486	C₂	1	1	yes	maximal, characteristic
3	#356×C₂₅₁	C₂	1	1	yes	maximal, characteristic
4	#22548707	C₄	1	1	yes	characteristic
5	#22548639	—	2	1	no	—
6	#22548639	—	2	1	no	—
7	#22548634	—	2	1	no	—
8	#22548626	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
10	#101×C₂₅₁	—	2	1	no	—
11	#16860794	—	4	1	no	—
12	C₂×C₂₅₁⋊C₈	—	2	1	no	—
13	C₄⋊Dic₂₅₁	—	2	1	no	—
14	Dic₂₅₁⋊C₄	—	4	1	no	—
15	C₂×Q₈×C₂₅₁	D₄	1	1	yes	characteristic
16	C₂×Q₈×C₂₅₁	D₄	1	1	yes	characteristic
17	M₁₆×C₂₅₁	—	4	1	no	—
18	C₄⋊C₄×C₂₅₁	—	4	1	no	—
19	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₅₁	—	4	1	no	—
21	C₂₅₁⋊C₈	—	4	1	no	—
22	C₂₅₁⋊C₈	—	8	1	no	—
23	Q₈×C₂₅₁	—	4	1	no	—
24	Q₈×C₂₅₁	—	4	1	no	—
25	C₂×C₄×C₂₅₁	—	2	1	no	—
26	C₂×C₄×C₂₅₁	—	2	1	no	—
27	C₂×C₄×C₂₅₁	—	2	1	no	—
28	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₂₅₁	—	8	1	no	—
30	C₂²×C₂₅₁	#99	1	1	yes	characteristic
31	C₄×C₂₅₁	—	2	1	no	—
32	C₄×C₂₅₁	—	4	1	no	—
33	C₄×C₂₅₁	—	4	1	no	—
34	C₄×C₂₅₁	—	4	1	no	—
35	Dic₂₅₁	—	8	1	no	—
36	C₂×C₂₅₁	—	2	1	no	—
37	C₂×C₂₅₁	#354	1	1	yes	characteristic
38	C₂₅₁	#1413	1	1	yes	characteristic, complemented, Sylow
39	#1413	—	251	1	no	maximal, Sylow
40	#491	—	251	1	no	—
41	#356	—	251	1	no	—
42	#356	D₂₅₁	1	1	yes	characteristic
43	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
44	#125	—	251	1	no	—
45	#103	—	502	1	no	—
46	#103	—	502	1	no	—
47	#101	—	2	1	no	—
48	#101	—	502	1	no	—
49	#97	—	251	1	no	—
50	C₂×Q₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
51	C₂×Q₈	D₁₀₀₄	1	1	yes	characteristic
52	M₁₆	—	4	1	no	—
53	M₁₆	—	1004	1	no	—
54	C₂×C₈	—	502	1	no	—
55	C₄⋊C₄	—	4	1	no	—
56	C₄⋊C₄	—	1004	1	no	—
57	C₄⋊C₄	—	502	1	no	—
58	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
59	Q₈	—	4	1	no	—
60	Q₈	—	4	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	1004	1	no	—
65	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
66	C₈	—	8	1	no	—
67	C₈	—	2008	1	no	—
68	C₈	—	1004	1	no	—
69	C₂²	#22548629	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	2008	1	no	—
75	C₂	—	2	1	no	—
76	C₂	#30631481	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30631481
commutator	C₄²×C₂₅₁	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₂₅₁
Fitting	#356×C₂₅₁	C₂
socle	C₂×C₂₅₁	#354
prime radical	C₂²×C₂₅₁	#99
cosocle	C₄⋊Q₈×C₂₅₁	C₂²
hypercenter	#356	D₂₅₁
hypocenter	C₂₅₁	#1413
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂×C₄×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
38. C₂₅₁	#1413	semidirect

Maximal subgroups

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

subgroup	index
#30632055	2
#30631486	2
#356×C₂₅₁	2
#1413	251