Subgroups — #31885115

Order 18176.

96 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	#22586416	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#2253	C₂	1	1	yes	maximal, characteristic
4	#17985833	—	2	1	no	—
5	#17985820	—	2	1	no	—
6	#499×C₇₁	—	2	1	no	—
7	#367×C₇₁	C₄	1	2	yes	—
8	#366×C₇₁	—	2	1	no	—
9	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
10	#14556068	—	4	1	no	—
11	C₇₁⋊C₃₂	—	4	1	no	—
12	#125×C₇₁	—	4	1	no	—
13	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
15	#107×C₇₁	—	2	2	no	—
16	#106×C₇₁	—	4	1	no	—
17	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
18	C₇₁⋊C₁₆	—	8	1	no	—
19	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
21	C₄⋊C₄×C₇₁	—	4	2	no	—
22	C₄⋊C₄×C₇₁	—	8	2	no	—
23	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
24	C₁₆×C₇₁	—	2	1	no	—
25	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
26	C₁₆×C₇₁	Q₁₆	1	1	yes	characteristic
27	C₂×C₄×C₇₁	—	2	1	no	—
28	C₂×C₄×C₇₁	—	8	2	no	—
29	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
30	C₈×C₇₁	#101	1	1	yes	characteristic
31	C₈×C₇₁	#103	1	1	yes	characteristic
32	C₈×C₇₁	SD₃₂	1	1	yes	characteristic
33	C₈×C₇₁	Q₃₂	1	1	yes	characteristic
34	C₂²×C₇₁	#361	1	1	yes	characteristic
35	C₄×C₇₁	—	4	1	no	—
36	C₄×C₇₁	—	16	2	no	—
37	C₄×C₇₁	#332	1	1	yes	characteristic
38	C₄×C₇₁	#358	1	1	yes	characteristic
39	#7454	—	71	1	no	maximal, Sylow
40	C₂×C₇₁	—	2	1	no	—
41	C₂×C₇₁	#1365	1	1	yes	characteristic
42	#2253	D₇₁	1	1	yes	characteristic
43	#1422	—	71	1	no	—
44	#1373	—	71	1	no	—
45	C₇₁	#7454	1	1	yes	characteristic, complemented, Sylow
46	#499	—	2	1	no	—
47	M₆₄	—	142	1	no	—
48	#367	Dic₇₁	1	2	yes	—
49	#366	—	2	1	no	—
50	#364	—	142	1	no	—
51	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
52	#125	—	4	1	no	—
53	M₃₂	—	284	1	no	—
54	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
55	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
56	#107	—	2	2	no	—
57	#106	—	4	1	no	—
58	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
59	C₃₂	—	284	1	no	—
60	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
61	C₂×C₈	#11765850	1	1	yes	characteristic
62	C₄⋊C₄	—	4	2	no	—
63	C₄⋊C₄	—	8	2	no	—
64	C₄²	#11765853	1	1	yes	characteristic
65	C₁₆	—	2	1	no	—
66	C₁₆	—	568	1	no	—
67	C₁₆	#11765852	1	1	yes	characteristic
68	C₁₆	#11765855	1	1	yes	characteristic
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	8	2	no	—
71	C₂×C₄	#14556049	1	1	yes	characteristic
72	C₈	#14556047	1	1	yes	characteristic
73	C₈	#14556052	1	1	yes	characteristic
74	C₈	#14556071	1	1	yes	characteristic
75	C₈	#14556074	1	1	yes	characteristic
76	C₂²	#17985812	1	1	yes	characteristic
77	C₄	—	4	1	no	—
78	C₄	—	16	2	no	—
79	C₄	#17985745	1	1	yes	characteristic
80	C₄	#17985803	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	#22586398	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22586398
commutator	C₂×C₁₆×C₇₁	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₇₁
Fitting	C₇₁×#2253	C₂
socle	C₂×C₇₁	#1365
prime radical	C₂²×C₇₁	#361
cosocle	C₄×C₁₆×C₇₁	C₂²
hypercenter	#2253	D₇₁
hypocenter	C₇₁	#7454
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2253

Complements

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

N	H ≅ G/N	type
45. C₇₁	#7454	semidirect

Maximal subgroups

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

subgroup	index
#22586538	2
#22586416	2
C₇₁×#2253	2
#7454	71