Subgroups — #31918476

Order 18176.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590619	C₂	1	2	yes	maximal
2	#22586542	C₂	1	2	yes	maximal
3	#22586540	C₂	1	2	yes	maximal
4	C₄×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17986401	—	2	2	no	—
6	#17985827	—	2	2	no	—
7	#17985825	C₂²	1	4	yes	—
8	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
11	#14556120	—	4	2	no	—
12	#14556062	—	2	4	no	—
13	#14556060	—	4	2	no	—
14	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
17	C₃₂×C₇₁	Q₈	1	2	yes	—
18	C₃₂×C₇₁	Q₈	1	2	yes	—
19	C₄⋊Dic₇₁	—	4	4	no	—
20	Dic₇₁⋊C₄	—	8	4	no	—
21	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
22	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
23	C₄²×C₇₁	D₈	1	1	yes	characteristic
24	C₁₆×C₇₁	Q₁₆	1	2	yes	—
25	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
26	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₄×C₇₁	D₁₆	1	2	yes	—
28	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
29	C₂×Dic₇₁	—	8	4	no	—
30	C₈×C₇₁	Q₃₂	1	2	yes	—
31	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
32	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
33	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
34	C₄×C₇₁	—	2	2	no	—
35	C₄×C₇₁	#500	1	1	yes	characteristic
36	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
37	Dic₇₁	—	16	4	no	—
38	#13732	—	71	1	no	maximal, Sylow
39	C₂×C₇₁	#2252	1	1	yes	characteristic
40	C₂×C₇₁	#2262	1	2	yes	—
41	#2253	—	71	2	no	—
42	#1424	—	71	2	no	—
43	#1423	—	71	2	no	—
44	C₄×C₃₂	D₇₁	1	1	yes	characteristic
45	C₇₁	#13732	1	1	yes	characteristic, complemented, Sylow
46	#499	—	142	2	no	—
47	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
48	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
49	#367	—	142	2	no	—
50	#366	—	71	4	no	—
51	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
52	#125	—	284	2	no	—
53	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
54	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
55	#107	—	142	4	no	—
56	#106	—	284	2	no	—
57	C₄×C₈	D₂₈₄	1	1	yes	characteristic
58	C₃₂	Dic₁₄₂	1	2	yes	—
59	C₃₂	Dic₁₄₂	1	2	yes	—
60	C₂×C₈	D₅₆₈	1	1	yes	characteristic
61	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
62	C₄⋊C₄	—	284	4	no	—
63	C₄⋊C₄	—	568	4	no	—
64	C₄²	D₅₆₈	1	1	yes	characteristic
65	C₁₆	Dic₂₈₄	1	2	yes	—
66	C₁₆	C₂×Dic₁₄₂	1	1	yes	characteristic
67	C₁₆	C₂×Dic₁₄₂	1	1	yes	characteristic
68	C₂×C₄	—	568	4	no	—
69	C₂×C₄	D₁₁₃₆	1	2	yes	—
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	—	2	2	no	—
76	C₄	—	1136	4	no	—
77	C₄	#17986404	1	1	yes	characteristic
78	C₄	C₂×Dic₅₆₈	1	1	yes	characteristic
79	C₂	#22590615	1	1	yes	characteristic
80	C₂	#22590649	1	2	yes	—

	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	#13732
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
45. C₇₁	#13732	semidirect

Maximal subgroups

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

subgroup	index
#22590619	2
#22586542	2
#22586540	2
C₄×C₃₂×C₇₁	2
#13732	71