Subgroups — #46838674

Order 34688.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774330	C₂	1	2	yes	maximal, complemented
2	#31773734	C₂	1	2	yes	maximal, complemented
3	D₈₆₇₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₆₄×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22576183	—	2	2	no	—
7	#22576103	—	2	2	no	—
8	D₄₃₃₆	C₂²	1	2	yes	—
9	C₁₆×D₂₇₁	C₂²	1	1	yes	characteristic
10	C₂₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
11	Q₃₂×C₂₇₁	C₂²	1	2	yes	—
12	C₃₂×C₂₇₁	C₂²	1	1	yes	characteristic
13	#16864655	—	4	2	no	—
14	#16864641	—	4	2	no	—
15	D₂₁₆₈	—	2	2	no	—
16	C₈×D₂₇₁	D₄	1	1	yes	characteristic
17	C₂₇₁⋊C₁₆	D₄	1	1	yes	characteristic
18	Q₁₆×C₂₇₁	—	2	2	no	—
19	C₁₆×C₂₇₁	C₂³	1	1	yes	characteristic
20	D₁₀₈₄	—	4	2	no	—
21	D₁₀₈₄	—	8	2	no	—
22	C₄×D₂₇₁	—	8	2	no	—
23	C₄×D₂₇₁	D₈	1	1	yes	characteristic
24	C₂₇₁⋊C₈	D₈	1	1	yes	characteristic
25	Q₈×C₂₇₁	—	4	2	no	—
26	C₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
27	C₂×D₂₇₁	—	8	2	no	—
28	C₂×D₂₇₁	D₁₆	1	1	yes	characteristic
29	C₄×C₂₇₁	—	8	2	no	—
30	C₄×C₂₇₁	C₂×D₈	1	1	yes	characteristic
31	Dic₂₇₁	D₁₆	1	1	yes	characteristic
32	C₂×C₂₇₁	C₂×D₁₆	1	1	yes	characteristic
33	D₂₇₁	—	2	1	no	—
34	D₂₇₁	—	16	2	no	—
35	C₂₇₁	#2262	1	1	yes	characteristic, complemented, Sylow
36	#2262	—	271	1	no	maximal, Sylow
37	#508	—	271	2	no	—
38	Q₆₄	D₂₇₁	1	1	yes	characteristic, complemented
39	SD₆₄	—	271	2	no	—
40	D₃₂	—	271	1	no	—
41	C₂×C₃₂	—	271	1	no	—
42	#135	—	542	2	no	—
43	Q₃₂	C₂×D₂₇₁	1	2	yes	—
44	SD₃₂	—	542	2	no	—
45	D₁₆	—	271	2	no	—
46	C₂×C₁₆	—	271	1	no	—
47	C₃₂	—	271	1	no	—
48	C₃₂	C₂×D₂₇₁	1	1	yes	characteristic
49	#41	—	1084	2	no	—
50	Q₁₆	—	2	2	no	—
51	SD₁₆	—	1084	2	no	—
52	D₈	—	542	2	no	—
53	C₂×C₈	—	271	1	no	—
54	C₁₆	—	271	1	no	—
55	C₁₆	C₂²×D₂₇₁	1	1	yes	characteristic
56	Q₈	—	4	2	no	—
57	D₄	—	1084	2	no	—
58	D₄	—	2168	2	no	—
59	C₂×C₄	—	271	1	no	—
60	C₂×C₄	—	2168	2	no	—
61	C₈	—	271	1	no	—
62	C₈	D₄×D₂₇₁	1	1	yes	characteristic
63	C₂²	—	271	1	no	—
64	C₂²	—	2168	2	no	—
65	C₄	—	8	2	no	—
66	C₄	—	271	1	no	—
67	C₄	D₈×D₂₇₁	1	1	yes	characteristic
68	C₂	—	542	1	no	—
69	C₂	—	4336	2	no	—
70	C₂	D₁₆×D₂₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₂₇₁
commutator, cosocle	C₁₆×C₂₇₁	C₂³
Frattini	C₁₆	C₂²×D₂₇₁
Fitting	Q₆₄×C₂₇₁	C₂
socle	C₂×C₂₇₁	C₂×D₁₆
hypercenter	Q₆₄	D₂₇₁
hypocenter	C₂₇₁	#2262
LCS	G ▷ C₁₆×C₂₇₁ ▷ C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
1. #31774330	C₂	semidirect
2. #31773734	C₂	semidirect
3. D₈₆₇₂	C₂	semidirect
4. C₃₂×D₂₇₁	C₂	semidirect
5. Q₆₄×C₂₇₁	C₂	semidirect
35. C₂₇₁	#2262	semidirect
38. Q₆₄	D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#31774330	2
#31773734	2
D₈₆₇₂	2
C₃₂×D₂₇₁	2
Q₆₄×C₂₇₁	2
#2262	271