Subgroups — #31885111

Order 18176.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590622	C₂	1	1	yes	maximal, characteristic
2	#22586416	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1422	C₂	1	1	yes	maximal, characteristic
4	#17986404	—	2	1	no	—
5	#17985827	—	2	1	no	—
6	#17985827	C₄	1	2	yes	—
7	#17985820	—	2	1	no	—
8	M₆₄×C₇₁	—	2	1	no	—
9	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
10	#14556128	—	4	1	no	—
11	#14556068	—	4	1	no	—
12	#14556062	—	2	2	no	—
13	#14556062	—	4	1	no	—
14	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
15	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
16	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
17	C₃₂×C₇₁	—	4	1	no	—
18	C₂×Dic₁₄₂	—	8	1	no	—
19	C₄⋊Dic₇₁	—	4	2	no	—
20	C₄⋊Dic₇₁	—	8	1	no	—
21	C₇₁⋊C₁₆	—	8	1	no	—
22	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
24	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
25	C₁₆×C₇₁	—	2	1	no	—
26	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
27	C₁₆×C₇₁	Q₁₆	1	1	yes	characteristic
28	C₂×C₄×C₇₁	—	2	1	no	—
29	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
30	C₂×Dic₇₁	—	8	2	no	—
31	Dic₁₄₂	—	16	2	no	—
32	C₈×C₇₁	#101	1	1	yes	characteristic
33	C₈×C₇₁	#103	1	1	yes	characteristic
34	C₈×C₇₁	SD₃₂	1	1	yes	characteristic
35	C₈×C₇₁	Q₃₂	1	1	yes	characteristic
36	C₂²×C₇₁	#361	1	1	yes	characteristic
37	C₄×C₇₁	—	2	2	no	—
38	C₄×C₇₁	#332	1	1	yes	characteristic
39	C₄×C₇₁	#358	1	1	yes	characteristic
40	Dic₇₁	—	16	2	no	—
41	#7453	—	71	1	no	maximal, Sylow
42	C₂×C₇₁	—	2	1	no	—
43	C₂×C₇₁	#1365	1	1	yes	characteristic
44	#2254	—	71	1	no	—
45	#1422	D₇₁	1	1	yes	characteristic
46	#1373	—	71	1	no	—
47	C₇₁	#7453	1	1	yes	characteristic, complemented, Sylow
48	#500	—	142	1	no	—
49	M₆₄	—	2	1	no	—
50	#367	—	71	2	no	—
51	#367	—	142	1	no	—
52	#364	—	142	1	no	—
53	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
54	C₄⋊Q₈	—	284	1	no	—
55	M₃₂	—	284	1	no	—
56	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
57	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
58	#107	—	284	1	no	—
59	#107	—	142	2	no	—
60	C₄×C₈	D₂₈₄	1	1	yes	characteristic
61	C₃₂	—	4	1	no	—
62	C₂×Q₈	—	568	1	no	—
63	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
64	C₂×C₈	D₅₆₈	1	1	yes	characteristic
65	C₄⋊C₄	—	284	2	no	—
66	C₄⋊C₄	—	568	1	no	—
67	C₄²	#11765854	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	—	568	1	no	—
70	C₁₆	#11765853	1	1	yes	characteristic
71	C₁₆	#11765855	1	1	yes	characteristic
72	Q₈	—	1136	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	568	2	no	—
75	C₂×C₄	#14556050	1	1	yes	characteristic
76	C₈	#14556046	1	1	yes	characteristic
77	C₈	#14556051	1	1	yes	characteristic
78	C₈	#14556073	1	1	yes	characteristic
79	C₈	Dic₅₆₈	1	1	yes	characteristic
80	C₂²	#17985813	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	1136	2	no	—
83	C₄	#17985744	1	1	yes	characteristic
84	C₄	#17985804	1	1	yes	characteristic
85	C₂	—	2	1	no	—
86	C₂	#22586397	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22586397
commutator	C₂×C₁₆×C₇₁	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₇₁
Fitting	C₇₁×#1422	C₂
socle	C₂×C₇₁	#1365
prime radical	C₂²×C₇₁	#361
cosocle	C₄×C₁₆×C₇₁	C₂²
hypercenter	#1422	D₇₁
hypocenter	C₇₁	#7453
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1422

Complements

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

N	H ≅ G/N	type
47. C₇₁	#7453	semidirect

Maximal subgroups

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

subgroup	index
#22590622	2
#22586416	2
C₇₁×#1422	2
#7453	71