Subgroups — #31918472

Order 18176.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590615	C₂	1	2	yes	maximal
2	#22586540	C₂	1	4	yes	maximal
3	C₄×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
4	#17986404	—	2	2	no	—
5	#17985825	—	2	2	no	—
6	#17985825	C₂²	1	4	yes	—
7	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
8	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
9	#14556128	—	4	2	no	—
10	#14556062	—	2	4	no	—
11	#14556062	—	4	2	no	—
12	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
14	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
15	C₃₂×C₇₁	Q₈	1	4	yes	complemented
16	C₂×Dic₁₄₂	—	8	2	no	—
17	C₄⋊Dic₇₁	—	4	4	no	—
18	C₄⋊Dic₇₁	—	8	2	no	—
19	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
20	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
21	C₄²×C₇₁	D₈	1	1	yes	characteristic
22	C₁₆×C₇₁	Q₁₆	1	2	yes	—
23	C₁₆×C₇₁	C₂×Q₈	1	2	yes	—
24	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
25	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
26	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
27	C₂×Dic₇₁	—	8	4	no	—
28	Dic₁₄₂	—	16	4	no	—
29	C₈×C₇₁	Q₃₂	1	2	yes	—
30	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
31	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
32	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
33	C₄×C₇₁	D₃₂	1	2	yes	—
34	C₄×C₇₁	Q₆₄	1	2	yes	—
35	C₄×C₇₁	#500	1	1	yes	characteristic
36	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
37	Dic₇₁	—	16	4	no	—
38	#13731	—	71	1	no	maximal, Sylow
39	C₂×C₇₁	#2252	1	1	yes	characteristic
40	C₂×C₇₁	C₂×D₃₂	1	1	yes	characteristic
41	C₂×C₇₁	C₂×Q₆₄	1	1	yes	characteristic
42	#2252	—	71	2	no	—
43	#1423	—	71	4	no	—
44	C₄×C₃₂	D₇₁	1	1	yes	characteristic
45	C₇₁	#13731	1	1	yes	characteristic, complemented, Sylow
46	#500	—	142	2	no	—
47	C₂×C₃₂	C₂×D₇₁	1	2	yes	—
48	#366	—	71	4	no	—
49	#366	—	142	2	no	—
50	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
51	C₄⋊Q₈	—	284	2	no	—
52	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
53	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
54	#107	—	284	2	no	—
55	#107	—	142	4	no	—
56	C₄×C₈	D₂₈₄	1	1	yes	characteristic
57	C₃₂	Dic₁₄₂	1	4	yes	complemented
58	C₂×Q₈	—	568	2	no	—
59	C₂×C₈	D₅₆₈	1	1	yes	characteristic
60	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
61	C₄⋊C₄	—	284	4	no	—
62	C₄⋊C₄	—	568	2	no	—
63	C₄²	D₅₆₈	1	1	yes	characteristic
64	C₁₆	Dic₂₈₄	1	2	yes	—
65	C₁₆	C₂×Dic₁₄₂	1	2	yes	—
66	Q₈	—	1136	4	no	—
67	C₂×C₄	—	568	4	no	—
68	C₂×C₄	D₁₁₃₆	1	1	yes	characteristic
69	C₂×C₄	D₁₁₃₆	1	1	yes	characteristic
70	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
71	C₈	Dic₅₆₈	1	2	yes	—
72	C₈	#14556128	1	1	yes	characteristic
73	C₈	C₂×Dic₂₈₄	1	1	yes	characteristic
74	C₂²	C₂×D₁₁₃₆	1	1	yes	characteristic
75	C₄	—	1136	4	no	—
76	C₄	D₂₂₇₂	1	2	yes	—
77	C₄	Dic₁₁₃₆	1	2	yes	—
78	C₄	#17986404	1	1	yes	characteristic
79	C₄	C₂×Dic₅₆₈	1	1	yes	characteristic
80	C₂	#22590615	1	1	yes	characteristic
81	C₂	C₂×D₂₂₇₂	1	1	yes	characteristic
82	C₂	C₂×Dic₁₁₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₁₃₆
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₄×C₃₂×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₁₆
hypercenter	C₄×C₃₂	D₇₁
hypocenter	C₇₁	#13731
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₇₁	Q₈	semidirect
45. C₇₁	#13731	semidirect
57. C₃₂	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
#22590615	2
#22586540	2
C₄×C₃₂×C₇₁	2
#13731	71