Group #31918474 — subgroups

Subgroups — #31918474

Order 18176.

146 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590617	C₂	1	2	yes	maximal
2	#22586541	C₂	1	2	yes	maximal
3	#22586540	C₂	1	1	yes	maximal, characteristic
4	C₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1423	C₂	1	1	yes	maximal, characteristic
6	#17986400	—	2	2	no	—
7	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
8	#17985826	—	2	2	no	—
9	#17985825	C₂²	1	2	yes	—
10	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
12	#366×C₇₁	C₂²	1	2	yes	—
13	#14556119	—	4	2	no	—
14	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
15	#14556062	—	2	2	no	—
16	#14556059	—	4	2	no	—
17	C₈×Dic₇₁	D₄	1	1	yes	characteristic
18	C₇₁⋊C₃₂	Q₈	1	2	yes	—
19	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
20	#107×C₇₁	—	2	2	no	—
21	C₃₂×C₇₁	Q₈	1	2	yes	—
22	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
23	C₄⋊Dic₇₁	—	4	2	no	—
24	C₄⋊Dic₇₁	—	8	2	no	—
25	Dic₇₁⋊C₄	—	8	2	no	—
26	C₄×Dic₇₁	D₈	1	1	yes	characteristic
27	C₇₁⋊C₁₆	Q₁₆	1	2	yes	—
28	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₇₁	—	4	2	no	—
30	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
31	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₇₁	—	8	2	no	—
33	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
34	C₂×Dic₇₁	—	8	2	no	—
35	C₂×Dic₇₁	D₁₆	1	1	yes	characteristic
36	C₂×Dic₇₁	D₁₆	1	1	yes	characteristic
37	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
38	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
39	C₇₁⋊C₈	Q₃₂	1	2	yes	—
40	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
41	C₄×C₇₁	—	16	2	no	—
42	C₄×C₇₁	#500	1	1	yes	characteristic
43	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
44	Dic₇₁	—	2	1	no	—
45	Dic₇₁	—	2	1	no	—
46	Dic₇₁	—	16	2	no	—
47	#13732	—	71	1	no	maximal, Sylow
48	C₂×C₇₁	#2252	1	1	yes	characteristic
49	C₂×C₇₁	#2262	1	1	yes	characteristic
50	C₂×C₇₁	#2262	1	1	yes	characteristic
51	#2253	—	71	2	no	—
52	#1424	—	71	2	no	—
53	#1423	—	71	1	no	—
54	#1423	D₇₁	1	1	yes	characteristic
55	C₄×C₃₂	—	71	1	no	—
56	C₇₁	#13732	1	1	yes	characteristic, complemented, Sylow
57	#499	—	142	2	no	—
58	C₂×C₃₂	—	71	1	no	—
59	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
60	#367	—	142	2	no	—
61	#366	—	71	2	no	—
62	#366	C₂×D₇₁	1	2	yes	—
63	C₄×C₁₆	—	71	1	no	—
64	#125	—	284	2	no	—
65	C₂×C₁₆	—	71	1	no	—
66	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
67	#107	—	2	2	no	—
68	#107	—	142	2	no	—
69	#106	—	284	2	no	—
70	C₄×C₈	—	71	1	no	—
71	C₃₂	—	71	2	no	—
72	C₃₂	Dic₁₄₂	1	2	yes	—
73	C₂×C₈	—	71	1	no	—
74	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
75	C₄⋊C₄	—	4	2	no	—
76	C₄⋊C₄	—	284	2	no	—
77	C₄⋊C₄	—	568	2	no	—
78	C₄⋊C₄	—	568	2	no	—
79	C₄²	—	71	1	no	—
80	C₁₆	—	71	2	no	—
81	C₁₆	Q₈×D₇₁	1	1	yes	characteristic
82	C₁₆	C₂×Dic₁₄₂	1	1	yes	characteristic
83	C₂×C₄	—	8	2	no	—
84	C₂×C₄	—	568	2	no	—
85	C₂×C₄	—	71	1	no	—
86	C₂×C₄	—	71	1	no	—
87	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
88	C₈	—	71	2	no	—
89	C₈	#14556127	1	1	yes	characteristic
90	C₈	Q₁₆×D₇₁	1	1	yes	characteristic
91	C₂²	D₁₆×D₇₁	1	1	yes	characteristic
92	C₄	—	16	2	no	—
93	C₄	—	1136	2	no	—
94	C₄	—	142	1	no	—
95	C₄	—	142	1	no	—
96	C₄	#17986403	1	1	yes	characteristic
97	C₄	Q₃₂×D₇₁	1	1	yes	characteristic
98	C₂	#22590614	1	1	yes	characteristic
99	C₂	#22590646	1	1	yes	characteristic
100	C₂	#22590650	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590617

C₂

yes

maximal

#22586541

C₂

yes

maximal

#22586540

C₂

yes

maximal, characteristic

C₃₂×Dic₇₁

C₂

yes

maximal, characteristic

C₇₁×#1423

C₂

yes

maximal, characteristic

#17986400

—

C₂×C₇₁⋊C₃₂

C₂²

yes

characteristic

#17985826

—

#17985825

C₂²

yes

—

C₁₆×Dic₇₁

C₂²

yes

characteristic

C₂×C₃₂×C₇₁

C₂²

yes

characteristic

#366×C₇₁

C₂²

yes

—

#14556119

—

C₂×C₇₁⋊C₁₆

D₄

yes

characteristic

#14556062

—

#14556059

—

C₈×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₃₂

Q₈

yes

—

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

#107×C₇₁

—

C₃₂×C₇₁

Q₈

yes

—

C₂×C₇₁⋊C₈

D₈

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

D₈

yes

characteristic

C₇₁⋊C₁₆

Q₁₆

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

C₂×Q₈

yes

characteristic

C₁₆×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

D₁₆

yes

characteristic

C₂×Dic₇₁

D₁₆

yes

characteristic

C₈×C₇₁

C₄⋊Q₈

yes

characteristic

C₈×C₇₁

C₂×Q₁₆

yes

characteristic

C₇₁⋊C₈

Q₃₂

yes

—

C₂²×C₇₁

C₂×D₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

#500

yes

characteristic

C₄×C₇₁

C₂×Q₃₂

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

—

—

D₇₁

yes

characteristic

C₄×C₃₂

—

C₇₁

#13732

yes

characteristic, complemented, Sylow

#499

—

142

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₁

yes

characteristic

#367

—

142

—

#366

—

#366

C₂×D₇₁

yes

—

C₄×C₁₆

—

#125

—

284

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

#107

—

#107

—

142

—

#106

—

284

—

C₄×C₈

—

C₃₂

—

C₃₂

Dic₁₄₂

yes

—

C₂×C₈

—

C₂×C₈

D₄×D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

568

—

C₄⋊C₄

—

568

—

C₄²

—

C₁₆

—

C₁₆

Q₈×D₇₁

yes

characteristic

C₁₆

C₂×Dic₁₄₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

568

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₈×D₇₁

yes

characteristic

C₈

—

C₈

#14556127

yes

characteristic

C₈

Q₁₆×D₇₁

yes

characteristic

C₂²

D₁₆×D₇₁

yes

characteristic

C₄

—

C₄

—

1136

—

C₄

—

142

—

C₄

—

142

—

C₄

#17986403

yes

characteristic

C₄

Q₃₂×D₇₁

yes

C₂

yes

C₂

yes

100

C₂

#22590650

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₇₁
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#1423	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₁₆
hypercenter	#1423	D₇₁
hypocenter	C₇₁	#13732
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1423

subgroup

G/H

center

C₂²

D₁₆×D₇₁

commutator, cosocle

C₂×C₁₆×C₇₁

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₇₁×#1423

C₂

socle, prime radical

C₂²×C₇₁

C₂×D₁₆

D₇₁

C₇₁

G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1423

N	H ≅ G/N	type
56. C₇₁	#13732	semidirect

H ≅ G/N

type

56. C₇₁

#13732

semidirect

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

subgroup	index
#22590617	2
#22586541	2
#22586540	2
C₃₂×Dic₇₁	2
C₇₁×#1423	2
#13732	71

subgroup

index

#22590617

#22586541

#22586540

C₃₂×Dic₇₁

C₇₁×#1423

#13732

Subgroups — #31918474

Conjugacy classes of subgroups

Complements

Maximal subgroups