Subgroups — #34067284

Order 19328.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752566	C₂	1	1	yes	maximal, characteristic
2	#23752009	C₂	1	1	yes	maximal, characteristic
3	#354×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#18012731	C₄	1	1	yes	characteristic, complemented
5	#18012663	—	2	1	no	—
6	#18012660	—	2	1	no	—
7	#18012658	—	2	1	no	—
8	C₄×C₁₅₁⋊C₈	C₄	1	1	yes	characteristic, complemented
9	#124×C₁₅₁	C₂²	1	1	yes	characteristic
10	#99×C₁₅₁	—	2	1	no	—
11	#14558988	—	4	1	no	—
12	C₂×C₁₅₁⋊C₈	—	2	1	no	—
13	C₄⋊Dic₁₅₁	—	2	1	no	—
14	Dic₁₅₁⋊C₄	—	4	1	no	—
15	C₂×Q₈×C₁₅₁	D₄	1	1	yes	characteristic
16	C₂×D₄×C₁₅₁	D₄	1	1	yes	characteristic
17	C₂²⋊C₄×C₁₅₁	—	4	1	no	—
18	C₂²⋊C₄×C₁₅₁	—	4	1	no	—
19	C₄²×C₁₅₁	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₅₁	—	4	1	no	—
21	C₁₅₁⋊C₈	—	4	1	no	—
22	C₁₅₁⋊C₈	—	8	1	no	—
23	C₂³×C₁₅₁	—	2	1	no	—
24	Q₈×C₁₅₁	—	4	1	no	—
25	D₄×C₁₅₁	—	4	1	no	—
26	C₂×C₄×C₁₅₁	—	2	1	no	—
27	C₂×C₄×C₁₅₁	—	2	1	no	—
28	C₂×C₄×C₁₅₁	—	8	1	no	—
29	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₁₅₁	—	4	1	no	—
31	C₂²×C₁₅₁	—	8	1	no	—
32	C₂²×C₁₅₁	#99	1	1	yes	characteristic
33	C₄×C₁₅₁	—	2	1	no	—
34	C₄×C₁₅₁	—	4	1	no	—
35	C₄×C₁₅₁	—	4	1	no	—
36	C₄×C₁₅₁	—	16	1	no	—
37	Dic₁₅₁	—	8	1	no	—
38	C₂×C₁₅₁	—	2	1	no	—
39	C₂×C₁₅₁	—	8	1	no	—
40	C₂×C₁₅₁	#354	1	1	yes	characteristic
41	C₁₅₁	#1414	1	1	yes	characteristic, complemented, Sylow
42	#1414	—	151	1	no	maximal, Sylow
43	#488	—	151	1	no	—
44	#355	—	151	1	no	—
45	#354	D₁₅₁	1	1	yes	characteristic
46	#125	—	151	1	no	—
47	#124	C₂×D₁₅₁	1	1	yes	characteristic
48	#103	—	302	1	no	—
49	#102	—	302	1	no	—
50	#101	—	302	1	no	—
51	#99	—	2	1	no	—
52	C₄×C₈	—	151	1	no	—
53	C₂×Q₈	D₆₀₄	1	1	yes	characteristic
54	C₂×D₄	C₂²⋊D₁₅₁	1	1	yes	characteristic
55	M₁₆	—	604	1	no	—
56	C₂×C₈	—	302	1	no	—
57	C₄⋊C₄	—	302	1	no	—
58	C₄⋊C₄	—	604	1	no	—
59	C₂²⋊C₄	—	4	1	no	—
60	C₂²⋊C₄	—	4	1	no	—
61	C₄²	C₄×D₁₅₁	1	1	yes	characteristic
62	C₂³	—	2	1	no	—
63	Q₈	—	4	1	no	—
64	D₄	—	4	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	8	1	no	—
68	C₂×C₄	—	604	1	no	—
69	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
70	C₈	—	604	1	no	—
71	C₈	—	1208	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	8	1	no	—
74	C₂²	#18012653	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	16	1	no	—
79	C₄	—	1208	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#23752006	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23752006
commutator	C₄²×C₁₅₁	C₂×C₄
Frattini	#124	C₂×D₁₅₁
Fitting	#354×C₁₅₁	C₂
socle	C₂×C₁₅₁	#354
prime radical	C₂×D₄×C₁₅₁	D₄
cosocle	#124×C₁₅₁	C₂²
hypercenter	#354	D₁₅₁
hypocenter	C₁₅₁	#1414
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂×C₄×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #354

Complements

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

N	H ≅ G/N	type
4. #18012731	C₄	semidirect
8. C₄×C₁₅₁⋊C₈	C₄	semidirect
41. C₁₅₁	#1414	semidirect

Maximal subgroups

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

subgroup	index
#23752566	2
#23752009	2
#354×C₁₅₁	2
#1414	151