Subgroups — #22586481

Order 9088.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986415	C₂	1	1	yes	maximal, characteristic
2	#17985833	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#14556134	C₄	1	1	yes	characteristic
5	#14556068	C₄	1	1	yes	characteristic
6	#14556067	—	2	1	no	—
7	C₁₆×D₇₁	—	2	1	no	—
8	C₇₁⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
10	C₃₂×C₇₁	—	2	1	no	—
11	#11765865	C₈	1	1	yes	characteristic
12	#11765849	C₈	1	1	yes	characteristic
13	#11765848	—	2	1	no	—
14	C₈×D₇₁	—	2	1	no	—
15	C₇₁⋊C₁₆	—	2	1	no	—
16	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₇₁	D₄	1	1	yes	characteristic
18	C₁₆×C₇₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
20	C₂²⋊D₇₁	—	2	1	no	—
21	D₂₈₄	C₁₆	1	1	yes	characteristic
22	C₄×D₇₁	—	2	1	no	—
23	Dic₁₄₂	C₁₆	1	1	yes	characteristic
24	C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₇₁	M₁₆	1	1	yes	characteristic
26	C₇₁⋊C₈	—	2	1	no	—
27	C₂²×C₇₁	M₃₂	1	1	yes	characteristic
28	C₂×D₇₁	—	2	1	no	—
29	C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
31	Dic₇₁	—	2	1	no	—
32	C₂×C₇₁	—	2	1	no	—
33	C₂×C₇₁	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₇₁	—	4	1	no	—
35	#1401	—	71	1	no	maximal, Sylow
36	C₇₁	#1401	1	1	yes	characteristic, complemented, Sylow
37	#504	—	71	1	no	—
38	M₆₄	—	71	1	no	—
39	C₂×C₃₂	D₇₁	1	1	yes	characteristic, complemented
40	#131	—	71	1	no	—
41	M₃₂	—	71	1	no	—
42	M₃₂	—	142	1	no	—
43	C₂×C₁₆	—	142	1	no	—
44	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	142	1	no	—
47	#41	—	71	1	no	—
48	M₁₆	—	71	1	no	—
49	M₁₆	—	142	1	no	—
50	C₂×C₈	—	142	1	no	—
51	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
52	C₁₆	—	142	1	no	—
53	C₁₆	D₂₈₄	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
55	Q₈	—	71	1	no	—
56	D₄	—	71	1	no	—
57	D₄	—	142	1	no	—
58	C₂×C₄	—	142	1	no	—
59	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
60	C₈	—	142	1	no	—
61	C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
62	C₈	#11765848	1	1	yes	characteristic
63	C₂²	—	142	1	no	—
64	C₂²	#14556067	1	1	yes	characteristic
65	C₄	—	142	1	no	—
66	C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
67	C₄	C₁₆×D₇₁	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	284	1	no	—
70	C₂	(C₂×D₇₁)⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₂₈₄
commutator	C₄×C₇₁	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₇₁
Fitting	C₂×C₃₂×C₇₁	C₂
socle	C₂×C₇₁	C₂²⋊C₁₆
hypercenter	C₂×C₃₂	D₇₁
hypocenter	C₇₁	#1401
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #17985833	C₂	semidirect
3. C₂×C₃₂×C₇₁	C₂	semidirect
36. C₇₁	#1401	semidirect
39. C₂×C₃₂	D₇₁	semidirect