Subgroups — #34067293

Order 19328.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752644	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₅₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#368×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#18012761	—	2	1	no	—
5	Dic₁₂₀₈	C₄	1	1	yes	characteristic
6	#18012684	—	2	1	no	—
7	D₂₄₁₆	C₄	1	1	yes	characteristic
8	C₁₅₁⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
10	#108×C₁₅₁	—	2	1	no	—
11	#14559004	—	4	1	no	—
12	Dic₆₀₄	—	2	1	no	—
13	#14558993	—	4	1	no	—
14	D₁₂₀₈	—	2	1	no	—
15	M₁₆×C₁₅₁	—	4	1	no	—
16	C₂×C₈×C₁₅₁	D₄	1	1	yes	characteristic
17	C₁₆×C₁₅₁	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₅₁	D₄	1	1	yes	characteristic
19	Dic₃₀₂	—	4	1	no	—
20	D₆₀₄	—	4	1	no	—
21	C₂²⋊D₁₅₁	—	8	1	no	—
22	C₄×D₁₅₁	—	8	1	no	—
23	C₂×C₄×C₁₅₁	D₈	1	1	yes	characteristic
24	C₈×C₁₅₁	—	8	1	no	—
25	C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₅₁	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₅₁	SD₃₂	1	1	yes	characteristic
28	C₂×D₁₅₁	—	8	1	no	—
29	C₄×C₁₅₁	#102	1	1	yes	characteristic
30	C₄×C₁₅₁	D₁₆	1	1	yes	characteristic
31	Dic₁₅₁	—	8	1	no	—
32	C₂×C₁₅₁	—	2	1	no	—
33	C₂×C₁₅₁	#357	1	1	yes	characteristic
34	D₁₅₁	—	16	1	no	—
35	C₁₅₁	#1417	1	1	yes	characteristic, complemented, Sylow
36	#1417	—	151	1	no	maximal, Sylow
37	#508	—	151	1	no	—
38	C₂×C₃₂	—	151	1	no	—
39	#368	D₁₅₁	1	1	yes	characteristic, complemented
40	#135	—	302	1	no	—
41	Q₃₂	—	151	1	no	—
42	SD₃₂	—	302	1	no	—
43	D₁₆	—	151	1	no	—
44	C₂×C₁₆	C₂×D₁₅₁	1	1	yes	characteristic
45	#108	—	2	1	no	—
46	C₃₂	—	302	1	no	—
47	#41	—	604	1	no	—
48	Q₁₆	—	302	1	no	—
49	SD₁₆	—	604	1	no	—
50	D₈	—	302	1	no	—
51	M₁₆	—	4	1	no	—
52	C₂×C₈	C₂²⋊D₁₅₁	1	1	yes	characteristic
53	C₁₆	C₄×D₁₅₁	1	1	yes	characteristic
54	C₁₆	D₆₀₄	1	1	yes	characteristic
55	Q₈	—	604	1	no	—
56	D₄	—	604	1	no	—
57	D₄	—	1208	1	no	—
58	C₂×C₄	—	1208	1	no	—
59	C₂×C₄	#14558989	1	1	yes	characteristic
60	C₈	—	8	1	no	—
61	C₈	(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
62	C₈	#14558991	1	1	yes	characteristic
63	C₂²	—	1208	1	no	—
64	C₂²	#18012682	1	1	yes	characteristic
65	C₄	—	1208	1	no	—
66	C₄	#18012659	1	1	yes	characteristic
67	C₄	#18012680	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	2416	1	no	—
70	C₂	#23752012	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#18012680
commutator	C₁₆×C₁₅₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₅₁
Fitting	#368×C₁₅₁	C₂
socle	C₂×C₁₅₁	#357
prime radical	#23752644	C₂
cosocle	C₂×C₁₆×C₁₅₁	C₂²
hypercenter	#368	D₁₅₁
hypocenter	C₁₅₁	#1417
LCS	G ▷ C₁₆×C₁₅₁ ▷ C₈×C₁₅₁ ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₅₁⋊C₃₂	C₂	semidirect
3. #368×C₁₅₁	C₂	semidirect
35. C₁₅₁	#1417	semidirect
39. #368	D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23752644	2
C₂×C₁₅₁⋊C₃₂	2
#368×C₁₅₁	2
#1417	151