Subgroups — #34067165

Order 19328.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₅₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#23751965	C₂	1	1	yes	maximal, characteristic
3	#500×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₅₁⋊C₁₆	—	2	1	no	—
5	C₄⋊(C₁₅₁⋊C₈)	—	2	1	no	—
6	C₄⋊Q₈×C₁₅₁	—	2	1	no	—
7	#107×C₁₅₁	—	2	1	no	—
8	#107×C₁₅₁	C₄	1	2	yes	—
9	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
10	C₂×C₁₅₁⋊C₈	—	4	1	no	—
11	C₁₅₁⋊C₁₆	—	4	1	no	—
12	C₂×Q₈×C₁₅₁	—	4	1	no	—
13	C₂×C₈×C₁₅₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₅₁	D₄	1	1	yes	characteristic
15	C₄⋊C₄×C₁₅₁	—	2	2	no	—
16	C₄⋊C₄×C₁₅₁	—	4	1	no	—
17	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
18	C₁₅₁⋊C₈	—	8	1	no	—
19	Q₈×C₁₅₁	—	8	1	no	—
20	Q₈×C₁₅₁	—	8	1	no	—
21	C₂×C₄×C₁₅₁	—	4	2	no	—
22	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₅₁	D₈	1	1	yes	characteristic
24	C₂×C₄×C₁₅₁	SD₁₆	1	1	yes	characteristic
25	C₈×C₁₅₁	—	2	1	no	—
26	C₈×C₁₅₁	SD₁₆	1	1	yes	characteristic
27	C₈×C₁₅₁	Q₁₆	1	1	yes	characteristic
28	C₂²×C₁₅₁	#102	1	1	yes	characteristic
29	C₄×C₁₅₁	—	2	1	no	—
30	C₄×C₁₅₁	—	8	2	no	—
31	C₄×C₁₅₁	#101	1	1	yes	characteristic
32	C₄×C₁₅₁	#103	1	1	yes	characteristic
33	C₄×C₁₅₁	D₁₆	1	1	yes	characteristic
34	C₄×C₁₅₁	SD₃₂	1	1	yes	characteristic
35	C₂×C₁₅₁	#332	1	1	yes	characteristic
36	C₂×C₁₅₁	#357	1	1	yes	characteristic
37	C₂×C₁₅₁	#362	1	1	yes	characteristic
38	C₁₅₁	#1364	1	1	yes	characteristic, complemented, Sylow
39	#1364	—	151	1	no	maximal, Sylow
40	#500	D₁₅₁	1	1	yes	characteristic
41	C₄⋊C₁₆	—	151	1	no	—
42	#335	—	151	1	no	—
43	C₄⋊Q₈	—	2	1	no	—
44	C₂×C₁₆	—	302	1	no	—
45	#107	—	2	1	no	—
46	#107	Dic₁₅₁	1	2	yes	—
47	C₄⋊C₈	—	302	1	no	—
48	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
49	C₂×Q₈	—	4	1	no	—
50	C₂×C₈	—	604	1	no	—
51	C₂×C₈	C₂×Dic₁₅₁	1	1	yes	characteristic
52	C₂×C₈	C₂²⋊D₁₅₁	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	4	1	no	—
55	C₄²	C₂²⋊D₁₅₁	1	1	yes	characteristic
56	C₁₆	—	604	1	no	—
57	Q₈	—	8	1	no	—
58	Q₈	—	8	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	C₂²⋊Dic₁₅₁	1	1	yes	characteristic
61	C₂×C₄	#14558989	1	1	yes	characteristic
62	C₂×C₄	#14558992	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	1208	1	no	—
65	C₈	#14558991	1	1	yes	characteristic
66	C₈	#14558994	1	1	yes	characteristic
67	C₂²	#18012660	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	8	2	no	—
70	C₄	#18012658	1	1	yes	characteristic
71	C₄	#18012663	1	1	yes	characteristic
72	C₄	#18012680	1	1	yes	characteristic
73	C₄	#18012683	1	1	yes	characteristic
74	C₂	#23751958	1	1	yes	characteristic
75	C₂	#23752013	1	1	yes	characteristic
76	C₂	#23752028	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#18012660
commutator	C₂×C₈×C₁₅₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₅₁
Fitting	#500×C₁₅₁	C₂
socle, prime radical	C₂²×C₁₅₁	#102
cosocle	C₄×C₈×C₁₅₁	C₂²
hypercenter	#500	D₁₅₁
hypocenter	C₁₅₁	#1364
LCS	G ▷ C₂×C₈×C₁₅₁ ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
38. C₁₅₁	#1364	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₁₅₁⋊C₁₆)	2
#23751965	2
#500×C₁₅₁	2
#1364	151