Subgroups — #31885119

Order 18176.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22586538	C₂	1	1	yes	maximal, characteristic
2	#22586256	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1422	C₂	1	1	yes	maximal, characteristic
4	#17985833	—	2	1	no	—
5	C₄×C₇₁⋊C₁₆	C₄	1	1	yes	characteristic
6	C₈×C₇₁⋊C₈	C₄	1	1	yes	characteristic
7	M₆₄×C₇₁	—	2	1	no	—
8	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×C₇₁⋊C₁₆	C₈	1	1	yes	characteristic
10	C₂×C₇₁⋊C₁₆	C₈	1	1	yes	characteristic
11	C₄×C₇₁⋊C₈	—	2	1	no	—
12	C₇₁⋊C₃₂	—	4	1	no	—
13	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₇₁	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₇₁	—	4	1	no	—
17	C₂×C₇₁⋊C₈	—	2	2	no	—
18	C₇₁⋊C₁₆	—	2	1	no	—
19	C₇₁⋊C₁₆	C₁₆	1	2	yes	—
20	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
22	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
23	C₁₆×C₇₁	—	2	1	no	—
24	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
25	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
26	C₂×C₄×C₇₁	—	2	1	no	—
27	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
28	C₈×C₇₁	#106	1	1	yes	characteristic
29	C₈×C₇₁	#108	1	1	yes	characteristic
30	C₈×C₇₁	C₂×C₁₆	1	1	yes	characteristic
31	C₈×C₇₁	M₃₂	1	1	yes	characteristic
32	C₇₁⋊C₈	—	4	2	no	—
33	C₂²×C₇₁	#364	1	1	yes	characteristic
34	C₄×C₇₁	—	2	2	no	—
35	C₄×C₇₁	#334	1	1	yes	characteristic
36	C₄×C₇₁	C₄⋊C₁₆	1	1	yes	characteristic
37	#7456	—	71	1	no	maximal, Sylow
38	C₂×C₇₁	—	2	1	no	—
39	C₂×C₇₁	#1367	1	1	yes	characteristic
40	#1422	—	71	1	no	—
41	#1422	D₇₁	1	1	yes	characteristic
42	#1311	—	71	1	no	—
43	C₇₁	#7456	1	1	yes	characteristic, complemented, Sylow
44	M₆₄	—	2	1	no	—
45	M₆₄	—	142	1	no	—
46	C₄×C₁₆	—	71	1	no	—
47	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
48	C₈²	—	71	1	no	—
49	C₂×C₁₆	—	71	1	no	—
50	C₂×C₁₆	—	71	1	no	—
51	C₂×C₁₆	Dic₁₄₂	1	1	yes	characteristic
52	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
53	C₄×C₈	—	142	1	no	—
54	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
55	C₃₂	—	4	1	no	—
56	C₃₂	—	284	1	no	—
57	C₂×C₈	—	142	2	no	—
58	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
59	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
60	C₄²	#11765848	1	1	yes	characteristic
61	C₁₆	—	2	1	no	—
62	C₁₆	—	71	2	no	—
63	C₁₆	—	142	1	no	—
64	C₁₆	#11765852	1	1	yes	characteristic
65	C₁₆	#11765853	1	1	yes	characteristic
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	Dic₇₁⋊C₈	1	1	yes	characteristic
68	C₈	—	284	2	no	—
69	C₈	#14556059	1	1	yes	characteristic
70	C₈	#14556063	1	1	yes	characteristic
71	C₈	C₁₆×D₇₁	1	1	yes	characteristic
72	C₈	#14556067	1	1	yes	characteristic
73	C₂²	#17985819	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	#17985749	1	1	yes	characteristic
76	C₄	Dic₇₁⋊C₁₆	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	#22586403	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14556059
commutator	C₈×C₇₁	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₇₁
Fitting	C₇₁×#1422	C₂
socle	C₂×C₇₁	#1367
prime radical	C₂²×C₇₁	#364
cosocle	C₄×C₁₆×C₇₁	C₂²
hypercenter	#1422	D₇₁
hypocenter	C₇₁	#7456
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ #1422

Complements

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

N	H ≅ G/N	type
43. C₇₁	#7456	semidirect

Maximal subgroups

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

subgroup	index
#22586538	2
#22586256	2
C₇₁×#1422	2
#7456	71