Subgroups — #34071383

Order 19328.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752617	C₂	1	1	yes	maximal, characteristic
2	#23752037	C₂	1	4	yes	maximal
3	C₄×C₁₅₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#500×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	#18012739	—	2	1	no	—
6	C₂×C₁₅₁⋊C₁₆	C₂²	1	2	yes	—
7	#18012673	—	2	1	no	—
8	#18012673	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₁₅₁	—	2	1	no	—
10	#107×C₁₅₁	—	2	1	no	—
11	#107×C₁₅₁	C₂²	1	2	yes	—
12	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₀₂	—	4	1	no	—
14	C₄⋊Dic₁₅₁	—	2	2	no	—
15	C₄⋊Dic₁₅₁	—	4	1	no	—
16	C₁₅₁⋊C₁₆	Q₈	1	4	yes	complemented
17	C₂×Q₈×C₁₅₁	—	4	1	no	—
18	C₂×C₈×C₁₅₁	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₅₁	—	2	2	no	—
21	C₄⋊C₄×C₁₅₁	—	4	1	no	—
22	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
23	Dic₃₀₂	—	8	2	no	—
24	C₂×Dic₁₅₁	—	4	2	no	—
25	Q₈×C₁₅₁	—	8	2	no	—
26	C₂×C₄×C₁₅₁	—	4	2	no	—
27	C₂×C₄×C₁₅₁	D₈	1	1	yes	characteristic
28	C₂×C₄×C₁₅₁	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
30	C₈×C₁₅₁	Q₁₆	1	2	yes	—
31	C₈×C₁₅₁	C₂×Q₈	1	2	yes	—
32	C₂²×C₁₅₁	C₂×D₈	1	1	yes	characteristic
33	C₄×C₁₅₁	—	8	2	no	—
34	C₄×C₁₅₁	D₁₆	1	2	yes	—
35	C₄×C₁₅₁	Q₃₂	1	2	yes	—
36	C₄×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
37	C₄×C₁₅₁	C₂×Q₁₆	1	1	yes	characteristic
38	Dic₁₅₁	—	8	2	no	—
39	C₂×C₁₅₁	#500	1	1	yes	characteristic
40	C₂×C₁₅₁	C₂×D₁₆	1	1	yes	characteristic
41	C₂×C₁₅₁	C₂×Q₃₂	1	1	yes	characteristic
42	C₁₅₁	#2252	1	1	yes	characteristic, complemented, Sylow
43	#2252	—	151	1	no	maximal, Sylow
44	#500	—	151	1	no	—
45	#500	D₁₅₁	1	1	yes	characteristic
46	#366	—	151	4	no	—
47	C₄×C₁₆	—	151	1	no	—
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	302	1	no	—
50	C₂×C₁₆	—	151	2	no	—
51	#107	—	2	1	no	—
52	#107	—	302	1	no	—
53	#107	—	151	2	no	—
54	#107	C₂×D₁₅₁	1	2	yes	—
55	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	604	1	no	—
58	C₂×C₈	C₂²⋊D₁₅₁	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	302	2	no	—
63	C₄⋊C₄	—	604	1	no	—
64	C₄²	C₂²⋊D₁₅₁	1	1	yes	characteristic
65	C₁₆	—	151	4	no	—
66	Q₈	—	8	2	no	—
67	Q₈	—	1208	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	604	2	no	—
70	C₂×C₄	#14558989	1	1	yes	characteristic
71	C₂×C₄	#14558989	1	1	yes	characteristic
72	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
73	C₈	#14558994	1	2	yes	—
74	C₈	Q₈×D₁₅₁	1	2	yes	—
75	C₂²	C₂×#14558989	1	1	yes	characteristic
76	C₄	—	8	2	no	—
77	C₄	—	1208	2	no	—
78	C₄	#18012680	1	2	yes	—
79	C₄	#18012685	1	2	yes	—
80	C₄	#18012737	1	1	yes	characteristic
81	C₄	C₂×#14558994	1	1	yes	characteristic
82	C₂	#23752615	1	1	yes	characteristic
83	C₂	C₂×#18012680	1	1	yes	characteristic
84	C₂	C₂×#18012685	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14558989
commutator, cosocle	C₂×C₈×C₁₅₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₅₁
Fitting	#500×C₁₅₁	C₂
socle, prime radical	C₂²×C₁₅₁	C₂×D₈
hypercenter	#500	D₁₅₁
hypocenter	C₁₅₁	#2252
LCS	G ▷ C₂×C₈×C₁₅₁ ▷ 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
16. C₁₅₁⋊C₁₆	Q₈	semidirect
42. C₁₅₁	#2252	semidirect

Maximal subgroups

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

subgroup	index
#23752617	2
#23752037	2
C₄×C₁₅₁⋊C₁₆	2
#500×C₁₅₁	2
#2252	151