Subgroups — #31885335

Order 18176.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590649	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇₁⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#1425	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986431	—	2	1	no	—
5	Dic₁₁₃₆	C₄	1	1	yes	characteristic
6	#17985838	—	2	1	no	—
7	D₂₂₇₂	C₄	1	1	yes	characteristic
8	C₇₁⋊C₆₄	—	2	1	no	—
9	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
10	#368×C₇₁	—	2	1	no	—
11	#14556150	—	4	1	no	—
12	Dic₅₆₈	—	2	1	no	—
13	#14556073	—	4	1	no	—
14	D₁₁₃₆	—	2	1	no	—
15	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
16	#108×C₇₁	—	4	1	no	—
17	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₇₁	D₄	1	1	yes	characteristic
19	#11765865	—	8	1	no	—
20	Dic₂₈₄	—	4	1	no	—
21	#11765854	—	8	1	no	—
22	D₅₆₈	—	4	1	no	—
23	M₁₆×C₇₁	—	8	1	no	—
24	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
25	C₁₆×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
27	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
28	C₂²⋊D₇₁	—	16	1	no	—
29	D₂₈₄	—	8	1	no	—
30	C₄×D₇₁	—	16	1	no	—
31	Dic₁₄₂	—	8	1	no	—
32	C₈×C₇₁	—	16	1	no	—
33	C₈×C₇₁	#102	1	1	yes	characteristic
34	C₈×C₇₁	SD₃₂	1	1	yes	characteristic
35	C₂²×C₇₁	SD₆₄	1	1	yes	characteristic
36	C₂×D₇₁	—	16	1	no	—
37	C₄×C₇₁	#357	1	1	yes	characteristic
38	C₄×C₇₁	D₃₂	1	1	yes	characteristic
39	Dic₇₁	—	16	1	no	—
40	#7539	—	71	1	no	maximal, Sylow
41	C₂×C₇₁	—	2	1	no	—
42	C₂×C₇₁	#1415	1	1	yes	characteristic
43	D₇₁	—	32	1	no	—
44	#2262	—	71	1	no	—
45	C₂×C₆₄	—	71	1	no	—
46	#1425	D₇₁	1	1	yes	characteristic, complemented
47	C₇₁	#7539	1	1	yes	characteristic, complemented, Sylow
48	#508	—	142	1	no	—
49	Q₆₄	—	71	1	no	—
50	SD₆₄	—	142	1	no	—
51	D₃₂	—	71	1	no	—
52	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
53	#368	—	2	1	no	—
54	C₆₄	—	142	1	no	—
55	#135	—	284	1	no	—
56	Q₃₂	—	142	1	no	—
57	SD₃₂	—	284	1	no	—
58	D₁₆	—	142	1	no	—
59	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
60	#108	—	4	1	no	—
61	C₃₂	C₄×D₇₁	1	1	yes	characteristic
62	C₃₂	D₂₈₄	1	1	yes	characteristic
63	#41	—	568	1	no	—
64	Q₁₆	—	284	1	no	—
65	SD₁₆	—	568	1	no	—
66	D₈	—	284	1	no	—
67	M₁₆	—	8	1	no	—
68	C₂×C₈	#11765850	1	1	yes	characteristic
69	C₁₆	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
70	C₁₆	#11765852	1	1	yes	characteristic
71	Q₈	—	568	1	no	—
72	D₄	—	568	1	no	—
73	D₄	—	1136	1	no	—
74	C₂×C₄	—	1136	1	no	—
75	C₂×C₄	#14556069	1	1	yes	characteristic
76	C₈	—	16	1	no	—
77	C₈	#14556048	1	1	yes	characteristic
78	C₈	#14556071	1	1	yes	characteristic
79	C₂²	—	1136	1	no	—
80	C₂²	#17985836	1	1	yes	characteristic
81	C₄	—	1136	1	no	—
82	C₄	#17985799	1	1	yes	characteristic
83	C₄	#17985834	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2272	1	no	—
86	C₂	#22586517	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17985834
commutator	C₃₂×C₇₁	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₁
Fitting	C₇₁×#1425	C₂
socle	C₂×C₇₁	#1415
prime radical	#22590649	C₂
cosocle	C₂×C₃₂×C₇₁	C₂²
hypercenter	#1425	D₇₁
hypocenter	C₇₁	#7539
LCS	G ▷ C₃₂×C₇₁ ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #1425

Complements

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

N	H ≅ G/N	type
2. C₂×C₇₁⋊C₆₄	C₂	semidirect
3. C₇₁×#1425	C₂	semidirect
46. #1425	D₇₁	semidirect
47. C₇₁	#7539	semidirect

Maximal subgroups

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

subgroup	index
#22590649	2
C₂×C₇₁⋊C₆₄	2
C₇₁×#1425	2
#7539	71