Subgroups — #31885333

Order 18176.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇₁⋊C₆₄	C₂	1	1	yes	maximal, characteristic
2	#22586540	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic
4	#17985825	—	2	1	no	—
5	C₇₁⋊C₆₄	—	2	1	no	—
6	C₂×Q₃₂×C₇₁	—	2	1	no	—
7	Q₆₄×C₇₁	—	2	1	no	—
8	Q₆₄×C₇₁	C₄	1	2	yes	complemented
9	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
10	#14556062	—	4	1	no	—
11	C₂×Q₁₆×C₇₁	—	4	1	no	—
12	Q₃₂×C₇₁	—	2	2	no	—
13	Q₃₂×C₇₁	—	4	1	no	—
14	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
15	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₇₁	D₄	1	1	yes	characteristic
17	C₄⋊Dic₇₁	—	8	1	no	—
18	C₂×Q₈×C₇₁	—	8	1	no	—
19	Q₁₆×C₇₁	—	4	2	no	—
20	Q₁₆×C₇₁	—	8	1	no	—
21	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
22	C₁₆×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
24	Q₈×C₇₁	—	8	2	no	—
25	Q₈×C₇₁	—	16	1	no	—
26	C₂×C₄×C₇₁	—	16	1	no	—
27	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
28	C₂×Dic₇₁	—	16	1	no	—
29	C₈×C₇₁	#102	1	1	yes	characteristic
30	C₈×C₇₁	SD₃₂	1	1	yes	characteristic
31	C₂²×C₇₁	D₃₂	1	1	yes	characteristic
32	C₄×C₇₁	—	16	2	no	—
33	C₄×C₇₁	#357	1	1	yes	characteristic
34	C₄×C₇₁	SD₆₄	1	1	yes	characteristic
35	Dic₇₁	—	32	1	no	—
36	#7538	—	71	1	no	maximal, Sylow
37	C₂×C₇₁	#1415	1	1	yes	characteristic
38	C₂×C₇₁	SD₁₂₈	1	1	yes	characteristic
39	C₂×C₇₁	Q₁₂₈	1	1	yes	characteristic
40	C₂×Q₆₄	D₇₁	1	1	yes	characteristic
41	C₂×C₆₄	—	71	1	no	—
42	#1423	—	71	1	no	—
43	C₇₁	#7538	1	1	yes	characteristic, complemented, Sylow
44	C₂×Q₃₂	—	2	1	no	—
45	Q₆₄	—	2	1	no	—
46	Q₆₄	Dic₇₁	1	2	yes	complemented
47	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
48	#366	—	142	1	no	—
49	C₆₄	—	142	1	no	—
50	C₂×Q₁₆	—	4	1	no	—
51	Q₃₂	—	2	2	no	—
52	Q₃₂	—	4	1	no	—
53	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
54	#107	—	284	1	no	—
55	C₃₂	C₂×Dic₇₁	1	1	yes	characteristic
56	C₃₂	C₂²⋊D₇₁	1	1	yes	characteristic
57	C₂×Q₈	—	8	1	no	—
58	Q₁₆	—	4	2	no	—
59	Q₁₆	—	8	1	no	—
60	C₂×C₈	#11765850	1	1	yes	characteristic
61	C₄⋊C₄	—	568	1	no	—
62	C₁₆	C₂²⋊Dic₇₁	1	1	yes	characteristic
63	C₁₆	#11765853	1	1	yes	characteristic
64	Q₈	—	8	2	no	—
65	Q₈	—	16	1	no	—
66	C₂×C₄	—	16	1	no	—
67	C₂×C₄	—	1136	1	no	—
68	C₂×C₄	#14556069	1	1	yes	characteristic
69	C₈	#14556049	1	1	yes	characteristic
70	C₈	#14556072	1	1	yes	characteristic
71	C₂²	#17985834	1	1	yes	characteristic
72	C₄	—	16	2	no	—
73	C₄	—	2272	1	no	—
74	C₄	#17985800	1	1	yes	characteristic
75	C₄	#17985837	1	1	yes	characteristic
76	C₂	#22586518	1	1	yes	characteristic
77	C₂	#22586554	1	1	yes	characteristic
78	C₂	#22586557	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985834
commutator	C₃₂×C₇₁	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₁
Fitting	C₂×Q₆₄×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	D₃₂
cosocle	C₂×C₃₂×C₇₁	C₂²
hypercenter	C₂×Q₆₄	D₇₁
hypocenter	C₇₁	#7538
LCS	G ▷ C₃₂×C₇₁ ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×Q₆₄

Complements

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

N	H ≅ G/N	type
8. Q₆₄×C₇₁	C₄	semidirect
43. C₇₁	#7538	semidirect
46. Q₆₄	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇₁⋊C₆₄	2
#22586540	2
C₂×Q₆₄×C₇₁	2
#7538	71