Subgroups — #34068717

Order 19328.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752589	C₂	1	1	yes	maximal, characteristic
2	#23752560	C₂	1	4	yes	maximal, complemented
3	C₈×C₁₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₂₀₈	—	2	2	no	—
6	#18012736	C₂²	1	2	yes	—
7	#18012659	—	2	8	no	—
8	C₄×C₁₅₁⋊C₈	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₁₅₁	—	2	2	no	—
10	C₄⋊Q₈×C₁₅₁	C₂²	1	2	yes	—
11	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
12	C₂×D₆₀₄	—	2	2	no	—
13	C₂×D₆₀₄	—	2	4	no	—
14	D₁₂₀₈	—	4	4	no	—
15	C₂×C₁₅₁⋊C₈	D₄	1	4	yes	—
16	C₂×Q₈×C₁₅₁	—	2	2	no	—
17	Q₁₆×C₁₅₁	—	4	4	no	—
18	C₂×C₈×C₁₅₁	D₄	1	2	yes	—
19	C₄⋊C₄×C₁₅₁	—	2	4	no	—
20	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
21	C₂²×D₁₅₁	—	4	2	no	—
22	D₆₀₄	—	4	4	no	—
23	D₆₀₄	—	4	8	no	—
24	C₁₅₁⋊C₈	—	2	4	no	—
25	Q₈×C₁₅₁	—	4	4	no	—
26	C₂×C₄×C₁₅₁	—	4	2	no	—
27	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
29	C₈×C₁₅₁	D₈	1	4	yes	—
30	C₂²×C₁₅₁	#127	1	1	yes	characteristic
31	C₂×D₁₅₁	—	8	2	no	—
32	C₂×D₁₅₁	—	8	4	no	—
33	C₄×C₁₅₁	—	8	2	no	—
34	C₄×C₁₅₁	C₂×D₈	1	2	yes	—
35	C₄×C₁₅₁	#135	1	4	yes	—
36	C₂×C₁₅₁	#493	1	1	yes	characteristic
37	C₂×C₁₅₁	#495	1	2	yes	—
38	D₁₅₁	—	16	2	no	—
39	C₁₅₁	#1709	1	1	yes	characteristic, complemented, Sylow
40	#1709	—	151	1	no	maximal, Sylow
41	#494	D₁₅₁	1	1	yes	characteristic, complemented
42	#493	—	151	1	no	—
43	#486	—	151	4	no	—
44	C₈²	—	151	1	no	—
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×D₈	—	302	2	no	—
47	C₄⋊Q₈	C₂×D₁₅₁	1	2	yes	—
48	#127	—	151	2	no	—
49	#102	—	302	8	no	—
50	C₄×C₈	—	151	2	no	—
51	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
52	C₂×Q₈	—	2	2	no	—
53	C₂×D₄	—	302	4	no	—
54	C₂×D₄	—	302	2	no	—
55	Q₁₆	—	4	4	no	—
56	D₈	—	604	4	no	—
57	C₂×C₈	—	151	4	no	—
58	C₂×C₈	C₂²⋊D₁₅₁	1	2	yes	—
59	C₄⋊C₄	—	2	4	no	—
60	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
61	C₂³	—	604	2	no	—
62	Q₈	—	4	4	no	—
63	D₄	—	604	4	no	—
64	D₄	—	604	8	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
68	C₈	—	302	4	no	—
69	C₈	#14558989	1	4	yes	—
70	C₂²	—	1208	2	no	—
71	C₂²	—	1208	4	no	—
72	C₂²	#18012735	1	1	yes	characteristic
73	C₄	—	8	2	no	—
74	C₄	C₂×#14558989	1	2	yes	—
75	C₄	#18012762	1	4	yes	—
76	C₂	—	2416	2	no	—
77	C₂	#23752588	1	1	yes	characteristic
78	C₂	#23752597	1	2	yes	—

	subgroup	G/H
center	C₂²	#18012735
commutator, cosocle	C₄²×C₁₅₁	C₂³
Frattini	C₄²	C₂²×D₁₅₁
Fitting	#494×C₁₅₁	C₂
socle	C₂²×C₁₅₁	#127
prime radical	#23752589	C₂
hypercenter	#494	D₁₅₁
hypocenter	C₁₅₁	#1709
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #23752560	C₂	semidirect
3. C₈×C₁₅₁⋊C₈	C₂	semidirect
4. #494×C₁₅₁	C₂	semidirect
39. C₁₅₁	#1709	semidirect
41. #494	D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23752589	2
#23752560	2
C₈×C₁₅₁⋊C₈	2
#494×C₁₅₁	2
#1709	151