Subgroups — #34067289

Order 19328.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₄₁₆	C₂	1	1	yes	maximal, characteristic
2	#23752038	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₂₀₈	—	2	1	no	—
5	D₂₄₁₆	—	2	1	no	—
6	D₂₄₁₆	C₄	1	2	yes	complemented
7	#18012673	—	2	1	no	—
8	C₂×C₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
9	C₃₂×C₁₅₁	—	2	1	no	—
10	C₂×D₆₀₄	—	4	1	no	—
11	D₁₂₀₈	—	2	2	no	—
12	D₁₂₀₈	—	4	1	no	—
13	C₄⋊Dic₁₅₁	—	4	1	no	—
14	C₂×C₈×C₁₅₁	D₄	1	1	yes	characteristic
15	C₁₆×C₁₅₁	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₅₁	D₄	1	1	yes	characteristic
17	C₂²×D₁₅₁	—	8	1	no	—
18	D₆₀₄	—	4	2	no	—
19	D₆₀₄	—	8	1	no	—
20	C₂×Dic₁₅₁	—	8	1	no	—
21	C₂×C₄×C₁₅₁	D₈	1	1	yes	characteristic
22	C₈×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₅₁	SD₁₆	1	1	yes	characteristic
24	C₂²×C₁₅₁	D₁₆	1	1	yes	characteristic
25	C₂×D₁₅₁	—	8	2	no	—
26	C₂×D₁₅₁	—	16	1	no	—
27	C₂×D₁₅₁	—	16	1	no	—
28	C₄×C₁₅₁	#102	1	1	yes	characteristic
29	C₄×C₁₅₁	SD₃₂	1	1	yes	characteristic
30	Dic₁₅₁	—	16	1	no	—
31	C₂×C₁₅₁	#357	1	1	yes	characteristic
32	C₂×C₁₅₁	D₃₂	1	1	yes	characteristic
33	C₂×C₁₅₁	SD₆₄	1	1	yes	characteristic
34	D₁₅₁	—	16	2	no	—
35	C₁₅₁	#1415	1	1	yes	characteristic, complemented, Sylow
36	#1415	—	151	1	no	maximal, Sylow
37	C₂×D₁₆	—	151	1	no	—
38	C₂×C₃₂	D₁₅₁	1	1	yes	characteristic, complemented
39	#366	—	151	1	no	—
40	C₂×D₈	—	302	1	no	—
41	D₁₆	—	302	1	no	—
42	D₁₆	—	151	2	no	—
43	C₂×C₁₆	C₂×D₁₅₁	1	1	yes	characteristic
44	#107	—	302	1	no	—
45	C₃₂	—	2	1	no	—
46	C₂×D₄	—	604	1	no	—
47	D₈	—	302	2	no	—
48	D₈	—	604	1	no	—
49	C₂×C₈	D₆₀₄	1	1	yes	characteristic
50	C₄⋊C₄	—	604	1	no	—
51	C₁₆	C₄×D₁₅₁	1	1	yes	characteristic
52	C₁₆	C₂²⋊D₁₅₁	1	1	yes	characteristic
53	C₂³	—	1208	1	no	—
54	D₄	—	604	2	no	—
55	D₄	—	1208	1	no	—
56	C₂×C₄	—	1208	1	no	—
57	C₂×C₄	D₁₂₀₈	1	1	yes	characteristic
58	C₈	(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
59	C₈	#14558993	1	1	yes	characteristic
60	C₂²	—	2416	1	no	—
61	C₂²	—	2416	1	no	—
62	C₂²	—	1208	2	no	—
63	C₂²	D₂₄₁₆	1	1	yes	characteristic
64	C₄	—	2416	1	no	—
65	C₄	#18012661	1	1	yes	characteristic
66	C₄	#18012684	1	1	yes	characteristic
67	C₂	—	2416	2	no	—
68	C₂	#23752014	1	1	yes	characteristic
69	C₂	D₄₈₃₂	1	1	yes	characteristic
70	C₂	#23752051	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₂₄₁₆
commutator	C₁₆×C₁₅₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₅₁
Fitting	C₂×C₃₂×C₁₅₁	C₂
socle	C₂²×C₁₅₁	D₁₆
prime radical	C₂×D₂₄₁₆	C₂
cosocle	C₂×C₁₆×C₁₅₁	C₂²
hypercenter	C₂×C₃₂	D₁₅₁
hypocenter	C₁₅₁	#1415
LCS	G ▷ C₁₆×C₁₅₁ ▷ C₈×C₁₅₁ ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #23752038	C₂	semidirect
3. C₂×C₃₂×C₁₅₁	C₂	semidirect
6. D₂₄₁₆	C₄	semidirect
35. C₁₅₁	#1415	semidirect
38. C₂×C₃₂	D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₄₁₆	2
#23752038	2
C₂×C₃₂×C₁₅₁	2
#1415	151