Subgroups — #22590613

Order 9088.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986404	C₂	1	1	yes	maximal, characteristic
2	#17985824	C₂	1	4	yes	maximal
3	C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#500×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#14556128	—	2	1	no	—
6	C₂×C₇₁⋊C₁₆	C₂²	1	2	yes	—
7	#14556062	—	2	1	no	—
8	#14556062	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₇₁	—	2	1	no	—
10	#107×C₇₁	—	2	1	no	—
11	#107×C₇₁	C₂²	1	2	yes	—
12	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₄₂	—	4	1	no	—
14	C₄⋊Dic₇₁	—	2	2	no	—
15	C₄⋊Dic₇₁	—	4	1	no	—
16	C₇₁⋊C₁₆	Q₈	1	4	yes	complemented
17	C₂×Q₈×C₇₁	—	4	1	no	—
18	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
19	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₇₁	—	2	2	no	—
21	C₄⋊C₄×C₇₁	—	4	1	no	—
22	C₄²×C₇₁	D₄	1	1	yes	characteristic
23	Q₈×C₇₁	—	8	2	no	—
24	C₂×C₄×C₇₁	—	4	2	no	—
25	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
26	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
27	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₇₁	—	4	2	no	—
29	Dic₁₄₂	—	8	2	no	—
30	C₈×C₇₁	Q₁₆	1	2	yes	—
31	C₈×C₇₁	C₂×Q₈	1	2	yes	—
32	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
33	C₄×C₇₁	—	8	2	no	—
34	C₄×C₇₁	D₁₆	1	2	yes	—
35	C₄×C₇₁	Q₃₂	1	2	yes	—
36	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
37	C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
38	Dic₇₁	—	8	2	no	—
39	C₂×C₇₁	#500	1	1	yes	characteristic
40	C₂×C₇₁	C₂×D₁₆	1	1	yes	characteristic
41	C₂×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
42	#2252	—	71	1	no	maximal, Sylow
43	C₇₁	#2252	1	1	yes	characteristic, complemented, Sylow
44	#500	—	71	1	no	—
45	#500	D₇₁	1	1	yes	characteristic
46	#366	—	71	4	no	—
47	C₄×C₁₆	—	71	1	no	—
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	142	1	no	—
50	C₂×C₁₆	—	71	2	no	—
51	#107	—	2	1	no	—
52	#107	—	71	2	no	—
53	#107	—	142	1	no	—
54	#107	C₂×D₇₁	1	2	yes	—
55	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	284	1	no	—
58	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	284	1	no	—
63	C₄⋊C₄	—	142	2	no	—
64	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
65	C₁₆	—	71	4	no	—
66	Q₈	—	8	2	no	—
67	Q₈	—	568	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	284	2	no	—
70	C₂×C₄	#11765850	1	1	yes	characteristic
71	C₂×C₄	#11765850	1	1	yes	characteristic
72	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
73	C₈	#11765855	1	2	yes	—
74	C₈	Q₈×D₇₁	1	2	yes	—
75	C₂²	C₂×#11765850	1	1	yes	characteristic
76	C₄	—	8	2	no	—
77	C₄	—	568	2	no	—
78	C₄	#14556069	1	2	yes	—
79	C₄	#14556074	1	2	yes	—
80	C₄	#14556126	1	1	yes	characteristic
81	C₄	C₂×#11765855	1	1	yes	characteristic
82	C₂	#17986402	1	1	yes	characteristic
83	C₂	C₂×#14556069	1	1	yes	characteristic
84	C₂	C₂×#14556074	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#500×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	#500	D₇₁
hypocenter	C₇₁	#2252
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
16. C₇₁⋊C₁₆	Q₈	semidirect
43. C₇₁	#2252	semidirect