Subgroups — #22590245

Order 9088.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
2	Dic₇₁⋊C₁₆	C₂	1	2	yes	maximal
3	#17985774	C₂	1	2	yes	maximal
4	C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₄₂	C₄	1	1	yes	characteristic
7	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₄⋊(C₇₁⋊C₈)	C₄	1	1	yes	characteristic
10	Dic₇₁⋊C₈	C₄	1	2	yes	—
11	C₈×Dic₇₁	C₂²	1	2	yes	—
12	C₂×C₁₆×C₇₁	C₂²	1	2	yes	—
13	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₄₂	C₈	1	1	yes	characteristic
15	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₇₁	C₈	1	1	yes	characteristic
17	Dic₇₁⋊C₄	C₈	1	2	yes	—
18	C₄×Dic₇₁	C₂×C₄	1	2	yes	—
19	C₇₁⋊C₁₆	—	2	1	no	—
20	C₇₁⋊C₁₆	Q₈	1	2	yes	complemented
21	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
23	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₁₆×C₇₁	—	2	2	no	—
25	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₇₁	C₂×C₈	1	2	yes	—
29	C₂×Dic₇₁	C₂×C₈	1	2	yes	—
30	Dic₁₄₂	—	2	2	no	—
31	C₈×C₇₁	—	2	1	no	—
32	C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
33	C₈×C₇₁	#41	1	1	yes	characteristic
34	C₇₁⋊C₈	—	2	2	no	—
35	C₂²×C₇₁	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₇₁	—	2	1	no	—
37	C₄×C₇₁	M₃₂	1	2	yes	—
38	C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₇₁	#131	1	1	yes	characteristic
40	Dic₇₁	—	2	2	no	—
41	Dic₇₁	—	2	2	no	—
42	C₂×C₇₁	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₇₁	C₂×M₃₂	1	1	yes	characteristic
44	C₂×C₇₁	#504	1	1	yes	characteristic
45	#2183	—	71	1	no	maximal, Sylow
46	C₇₁	#2183	1	1	yes	characteristic, complemented, Sylow
47	C₈×Q₈	—	71	1	no	—
48	C₄⋊C₁₆	—	71	2	no	—
49	C₄⋊C₁₆	D₇₁	1	1	yes	characteristic
50	#346	—	71	2	no	—
51	C₄×C₁₆	—	71	1	no	—
52	C₄×Q₈	—	71	1	no	—
53	C₂×C₁₆	—	71	1	no	—
54	C₂×C₁₆	—	71	1	no	—
55	C₂×C₁₆	C₂×D₇₁	1	2	yes	—
56	C₄⋊C₈	—	71	2	no	—
57	C₄⋊C₈	—	71	1	no	—
58	C₄×C₈	—	71	2	no	—
59	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
60	C₂×Q₈	—	71	1	no	—
61	C₂×C₈	—	71	2	no	—
62	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
63	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
64	C₄⋊C₄	—	71	2	no	—
65	C₄⋊C₄	—	71	1	no	—
66	C₄²	—	71	2	no	—
67	C₄²	C₄×D₇₁	1	1	yes	characteristic
68	C₁₆	—	2	2	no	—
69	C₁₆	—	71	2	no	—
70	C₁₆	—	142	1	no	—
71	Q₈	—	142	2	no	—
72	C₂×C₄	—	71	2	no	—
73	C₂×C₄	—	71	2	no	—
74	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
75	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	142	2	no	—
79	C₈	Q₈×D₇₁	1	1	yes	characteristic
80	C₈	#11765864	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₇₁	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	142	2	no	—
84	C₄	—	142	2	no	—
85	C₄	#14556067	1	2	yes	—
86	C₄	#14556093	1	1	yes	characteristic
87	C₄	#14556136	1	1	yes	characteristic
88	C₂	#17986123	1	1	yes	characteristic
89	C₂	C₂×#14556067	1	1	yes	characteristic
90	C₂	#17986417	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂²×C₈
Fitting	C₄⋊C₁₆×C₇₁	C₂
hypercenter	C₄⋊C₁₆	D₇₁
hypocenter	C₇₁	#2183
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
20. C₇₁⋊C₁₆	Q₈	semidirect
46. C₇₁	#2183	semidirect