Subgroups — #46838659

Order 34688.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774313	C₂	1	1	yes	maximal, characteristic
2	#31773730	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₂₇₁	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22576166	C₄	1	1	yes	characteristic
7	#22576100	C₄	1	1	yes	characteristic
8	#22576099	C₄	1	2	yes	—
9	C₁₆×D₂₇₁	C₂²	1	2	yes	—
10	C₂₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
11	C₂₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
13	C₃₂×C₂₇₁	C₂²	1	2	yes	—
14	#16864654	C₈	1	1	yes	characteristic
15	#16864638	C₈	1	1	yes	characteristic
16	#16864637	C₈	1	2	yes	—
17	C₈×D₂₇₁	C₂×C₄	1	2	yes	—
18	C₂₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₂₇₁	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₂₇₁	C₂×C₄	1	1	yes	characteristic
21	C₁₆×C₂₇₁	C₂³	1	1	yes	characteristic
22	Dic₅₄₂	C₁₆	1	1	yes	characteristic
23	D₁₀₈₄	C₁₆	1	1	yes	characteristic
24	C₂²⋊D₂₇₁	C₁₆	1	2	yes	—
25	C₄×D₂₇₁	C₂×C₈	1	2	yes	—
26	C₂₇₁⋊C₈	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₂₇₁	C₂×C₈	1	1	yes	characteristic
28	C₈×C₂₇₁	C₂×C₈	1	1	yes	characteristic
29	C₈×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₂₇₁	C₂×C₁₆	1	1	yes	characteristic
31	C₂×D₂₇₁	C₂×C₁₆	1	2	yes	—
32	C₄×C₂₇₁	C₂×C₁₆	1	1	yes	characteristic
33	C₄×C₂₇₁	C₂²×C₈	1	1	yes	characteristic
34	Dic₂₇₁	C₂×C₁₆	1	2	yes	—
35	C₂×C₂₇₁	—	2	1	no	—
36	C₂×C₂₇₁	C₂²×C₁₆	1	1	yes	characteristic
37	D₂₇₁	—	2	2	no	—
38	C₂₇₁	#2258	1	1	yes	characteristic, complemented, Sylow
39	#2258	—	271	1	no	maximal, Sylow
40	#504	—	271	1	no	—
41	M₆₄	—	271	2	no	—
42	M₆₄	D₂₇₁	1	1	yes	characteristic, complemented
43	C₂×C₃₂	—	271	1	no	—
44	C₂×C₃₂	—	271	2	no	—
45	#131	—	271	1	no	—
46	M₃₂	—	271	1	no	—
47	M₃₂	—	271	2	no	—
48	C₂×C₁₆	—	271	2	no	—
49	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
50	C₃₂	—	271	1	no	—
51	C₃₂	—	271	1	no	—
52	C₃₂	C₂×D₂₇₁	1	2	yes	—
53	#41	—	271	1	no	—
54	M₁₆	—	271	1	no	—
55	M₁₆	—	271	2	no	—
56	C₂×C₈	—	271	2	no	—
57	C₂×C₈	C₄×D₂₇₁	1	1	yes	characteristic
58	C₁₆	—	271	2	no	—
59	C₁₆	C₄×D₂₇₁	1	1	yes	characteristic
60	C₁₆	C₂²×D₂₇₁	1	1	yes	characteristic
61	Q₈	—	271	1	no	—
62	D₄	—	271	2	no	—
63	D₄	—	271	1	no	—
64	C₂×C₄	—	271	2	no	—
65	C₂×C₄	C₈×D₂₇₁	1	1	yes	characteristic
66	C₈	—	271	2	no	—
67	C₈	C₈×D₂₇₁	1	1	yes	characteristic
68	C₈	C₂×C₄×D₂₇₁	1	1	yes	characteristic
69	C₂²	—	271	2	no	—
70	C₂²	C₁₆×D₂₇₁	1	1	yes	characteristic
71	C₄	—	271	2	no	—
72	C₄	C₁₆×D₂₇₁	1	1	yes	characteristic
73	C₄	C₂×C₈×D₂₇₁	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	542	2	no	—
76	C₂	C₂×C₁₆×D₂₇₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂²×D₂₇₁
commutator, socle	C₂×C₂₇₁	C₂²×C₁₆
Fitting	M₆₄×C₂₇₁	C₂
prime radical	#16864654	C₈
cosocle	C₁₆×C₂₇₁	C₂³
hypercenter	M₆₄	D₂₇₁
hypocenter	C₂₇₁	#2258
LCS	G ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #31773730	C₂	semidirect
3. C₂×C₂₇₁⋊C₃₂	C₂	semidirect
4. C₃₂×D₂₇₁	C₂	semidirect
5. M₆₄×C₂₇₁	C₂	semidirect
38. C₂₇₁	#2258	semidirect
42. M₆₄	D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#31774313	2
#31773730	2
C₂×C₂₇₁⋊C₃₂	2
C₃₂×D₂₇₁	2
M₆₄×C₂₇₁	2
#2258	271