Subgroups — #31885134

Order 18176.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22586474	C₂	1	1	yes	maximal, characteristic
2	#22586414	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1372	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₁⋊C₃₂	C₄	1	1	yes	characteristic
5	C₂×C₇₁⋊C₃₂	C₄	1	1	yes	characteristic
6	#17985752	—	2	1	no	—
7	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
8	#335×C₇₁	—	2	1	no	—
9	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
10	C₇₁⋊C₃₂	—	2	1	no	—
11	C₇₁⋊C₃₂	C₈	1	2	yes	complemented
12	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₇₁	Q₈	1	1	yes	characteristic
14	C₄⋊C₈×C₇₁	—	4	1	no	—
15	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
16	C₂×C₇₁⋊C₈	—	8	1	no	—
17	C₂×C₈×C₇₁	—	8	1	no	—
18	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
19	C₂×C₈×C₇₁	Q₁₆	1	1	yes	characteristic
20	C₄²×C₇₁	D₈	1	1	yes	characteristic
21	C₁₆×C₇₁	—	2	1	no	—
22	C₁₆×C₇₁	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₇₁	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
25	C₂×C₄×C₇₁	SD₃₂	1	1	yes	characteristic
26	C₂×C₄×C₇₁	SD₃₂	1	1	yes	characteristic
27	C₈×C₇₁	—	2	1	no	—
28	C₈×C₇₁	—	16	1	no	—
29	C₈×C₇₁	C₄⋊C₈	1	1	yes	characteristic
30	C₈×C₇₁	#108	1	1	yes	characteristic
31	C₇₁⋊C₈	—	16	1	no	—
32	C₂²×C₇₁	#367	1	1	yes	characteristic
33	C₄×C₇₁	—	2	1	no	—
34	C₄×C₇₁	—	2	1	no	—
35	C₄×C₇₁	#335	1	1	yes	characteristic
36	C₄×C₇₁	#368	1	1	yes	characteristic
37	#7463	—	71	1	no	maximal, Sylow
38	C₂×C₇₁	#1372	1	1	yes	characteristic
39	C₂×C₇₁	#1426	1	1	yes	characteristic
40	C₂×C₇₁	#1426	1	1	yes	characteristic
41	#1397	—	71	1	no	—
42	#1372	—	71	1	no	—
43	#1372	D₇₁	1	1	yes	characteristic
44	C₇₁	#7463	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₃₂	—	71	1	no	—
46	C₂×C₃₂	—	71	1	no	—
47	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
48	#335	—	2	1	no	—
49	#335	—	142	1	no	—
50	C₂×C₁₆	Dic₁₄₂	1	1	yes	characteristic
51	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
52	C₄⋊C₈	—	4	1	no	—
53	C₄⋊C₈	—	284	1	no	—
54	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
55	C₃₂	—	71	2	no	—
56	C₃₂	—	142	1	no	—
57	C₂×C₈	—	8	1	no	—
58	C₂×C₈	—	568	1	no	—
59	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
60	C₂×C₈	#11765855	1	1	yes	characteristic
61	C₄²	#11765850	1	1	yes	characteristic
62	C₁₆	—	2	1	no	—
63	C₁₆	C₈×D₇₁	1	1	yes	characteristic
64	C₁₆	#11765848	1	1	yes	characteristic
65	C₂×C₄	#14556061	1	1	yes	characteristic
66	C₂×C₄	#14556071	1	1	yes	characteristic
67	C₂×C₄	#14556072	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	—	16	1	no	—
70	C₈	—	1136	1	no	—
71	C₈	Dic₇₁⋊C₈	1	1	yes	characteristic
72	C₈	#14556063	1	1	yes	characteristic
73	C₂²	#17985826	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	#17985751	1	1	yes	characteristic
77	C₄	#17985828	1	1	yes	characteristic
78	C₂	#22586413	1	1	yes	characteristic
79	C₂	#22586545	1	1	yes	characteristic
80	C₂	#22586547	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985826
commutator	C₁₆×C₇₁	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₇₁
Fitting	C₇₁×#1372	C₂
socle, prime radical	C₂²×C₇₁	#367
cosocle	C₄×C₁₆×C₇₁	C₂²
hypercenter	#1372	D₇₁
hypocenter	C₇₁	#7463
LCS	G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1372

Complements

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

N	H ≅ G/N	type
11. C₇₁⋊C₃₂	C₈	semidirect
44. C₇₁	#7463	semidirect

Maximal subgroups

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

subgroup	index
#22586474	2
#22586414	2
C₇₁×#1372	2
#7463	71