Group #31912192 — subgroups

Subgroups — #31912192

Order 18176.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22590622	C₂	1	1	yes	maximal, characteristic
2	#22590613	C₂	1	2	yes	maximal
3	#22587949	C₂	1	2	yes	maximal
4	C₁₆×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#2254	C₂	1	1	yes	maximal, characteristic
6	#17986404	C₂²	1	2	yes	—
7	#17986351	—	2	2	no	—
8	#17986349	—	2	2	no	—
9	#17985827	—	2	2	no	—
10	#17985824	—	2	4	no	—
11	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
13	#500×C₇₁	C₂²	1	2	yes	—
14	#367×C₇₁	—	2	2	no	—
15	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
16	#14556128	—	2	2	no	—
17	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
18	#14556062	—	2	2	no	—
19	#14556062	—	2	2	no	—
20	#14556052	—	4	4	no	—
21	#14556051	—	4	4	no	—
22	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
23	C₄⋊Q₈×C₇₁	—	2	2	no	—
24	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
25	#107×C₇₁	—	2	2	no	—
26	#107×C₇₁	—	2	2	no	—
27	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₄₂	—	4	2	no	—
29	C₂×C₇₁⋊C₈	D₈	1	4	yes	—
30	C₄⋊Dic₇₁	—	4	2	no	—
31	C₄⋊Dic₇₁	—	4	2	no	—
32	C₇₁⋊C₁₆	Q₁₆	1	4	yes	—
33	C₂×Q₈×C₇₁	—	4	2	no	—
34	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₇₁	—	4	2	no	—
37	C₄⋊C₄×C₇₁	—	4	2	no	—
38	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₁₆×C₇₁	Q₁₆	1	4	yes	—
40	Q₈×C₇₁	—	8	4	no	—
41	C₂×C₄×C₇₁	—	8	2	no	—
42	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
44	C₂×Dic₇₁	—	8	2	no	—
45	Dic₁₄₂	—	8	4	no	—
46	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
47	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
48	C₇₁⋊C₈	—	2	4	no	—
49	C₂²×C₇₁	#493	1	1	yes	characteristic
50	C₄×C₇₁	—	16	2	no	—
51	C₄×C₇₁	#494	1	1	yes	characteristic
52	C₄×C₇₁	#494	1	1	yes	characteristic
53	C₄×C₇₁	#508	1	4	yes	—
54	Dic₇₁	—	16	2	no	—
55	#12522	—	71	1	no	maximal, Sylow
56	C₂×C₇₁	#1713	1	1	yes	characteristic
57	C₂×C₇₁	#2249	1	2	yes	—
58	#2254	—	71	1	no	—
59	#2254	D₇₁	1	1	yes	characteristic
60	#2252	—	71	2	no	—
61	#1710	—	71	2	no	—
62	C₈×C₁₆	—	71	1	no	—
63	C₇₁	#12522	1	1	yes	characteristic, complemented, Sylow
64	#500	—	71	2	no	—
65	#500	C₂×D₇₁	1	2	yes	—
66	#487	—	142	2	no	—
67	#487	—	142	2	no	—
68	#367	—	2	2	no	—
69	#367	—	142	2	no	—
70	#366	—	142	4	no	—
71	C₄×C₁₆	—	71	1	no	—
72	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
73	C₈²	—	71	1	no	—
74	C₄⋊Q₈	—	2	2	no	—
75	C₄⋊Q₈	—	142	2	no	—
76	C₂×C₁₆	—	71	2	no	—
77	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
78	#107	—	2	2	no	—
79	#107	—	2	2	no	—
80	#107	—	142	2	no	—
81	#107	—	142	2	no	—
82	#103	—	284	4	no	—
83	#103	—	284	4	no	—
84	C₄×C₈	—	71	2	no	—
85	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
86	C₂×Q₈	—	4	2	no	—
87	C₂×Q₈	—	284	2	no	—
88	C₂×C₈	—	71	4	no	—
89	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
90	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
91	C₄⋊C₄	—	4	2	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	284	2	no	—
94	C₄⋊C₄	—	284	2	no	—
95	C₄²	D₄×D₇₁	1	1	yes	characteristic
96	C₁₆	—	71	4	no	—
97	C₁₆	#11765855	1	4	yes	—
98	Q₈	—	8	4	no	—
99	Q₈	—	568	4	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	—	568	2	no	—
102	C₂×C₄	#14556124	1	1	yes	characteristic
103	C₂×C₄	D₈×D₇₁	1	2	yes	—
104	C₈	—	142	4	no	—
105	C₈	Q₁₆×D₇₁	1	2	yes	—
106	C₈	C₂×#11765855	1	2	yes	—
107	C₂²	#17986374	1	1	yes	characteristic
108	C₄	—	16	2	no	—
109	C₄	—	1136	2	no	—
110	C₄	#17986377	1	1	yes	characteristic
111	C₄	#17986378	1	1	yes	characteristic
112	C₄	#17986430	1	4	yes	—
113	C₂	#22587957	1	1	yes	characteristic
114	C₂	#22590600	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590622

C₂

yes

maximal, characteristic

#22590613

C₂

yes

maximal

#22587949

C₂

yes

maximal

C₁₆×C₇₁⋊C₈

C₂

yes

maximal, characteristic

C₇₁×#2254

C₂

yes

maximal, characteristic

#17986404

C₂²

yes

—

—

—

—

—

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₈×C₇₁⋊C₈

C₂²

yes

characteristic

#500×C₇₁

C₂²

yes

—

#367×C₇₁

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

#14556128

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

—

—

—

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₄⋊Q₈×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

—

#107×C₇₁

—

#107×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

C₂×C₇₁⋊C₈

D₈

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

Q₁₆

yes

—

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

Q₁₆

yes

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

C₂×Q₁₆

yes

—

C₈×C₇₁

C₂×Q₁₆

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

#493

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

#2254

—

#2254

D₇₁

yes

characteristic

#2252

—

#1710

—

C₈×C₁₆

—

C₇₁

#12522

yes

characteristic, complemented, Sylow

#500

—

#500

C₂×D₇₁

yes

—

#487

—

142

—

#487

—

142

—

#367

—

#367

—

142

—

#366

—

142

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₈²

—

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₁

yes

—

#107

—

#107

—

#107

—

142

—

#107

—

142

—

#103

—

284

—

#103

—

284

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

284

—

C₂×C₈

—

C₂×C₈

D₄×D₇₁

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄²

D₄×D₇₁

yes

characteristic

C₁₆

—

C₁₆

#11765855

yes

—

Q₈

—

Q₈

—

568

—

100

C₂×C₄

—

101

C₂×C₄

—

568

—

102

C₂×C₄

#14556124

yes

characteristic

103

C₂×C₄

D₈×D₇₁

yes

—

104

C₈

—

142

—

105

C₈

Q₁₆×D₇₁

yes

—

106

C₈

C₂×#11765855

yes

—

107

C₂²

#17986374

yes

characteristic

108

C₄

—

109

C₄

—

1136

—

110

C₄

#17986377

yes

characteristic

111

C₄

#17986378

yes

characteristic

112

C₄

#17986430

yes

—

113

C₂

#22587957

yes

characteristic

114

C₂

#22590600

yes

—

	subgroup	G/H
center	C₂²	#17986374
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#2254	C₂
socle, prime radical	C₂²×C₇₁	#493
hypercenter	#2254	D₇₁
hypocenter	C₇₁	#12522
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#17986374

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#2254

C₂

C₂²×C₇₁

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
63. C₇₁	#12522	semidirect

H ≅ G/N

type

63. C₇₁

#12522

semidirect

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

subgroup	index
#22590622	2
#22590613	2
#22587949	2
C₁₆×C₇₁⋊C₈	2
C₇₁×#2254	2
#12522	71

subgroup

index

#22590622

#22590613

#22587949

C₁₆×C₇₁⋊C₈

C₇₁×#2254

#12522

Subgroups — #31912192

Conjugacy classes of subgroups

Complements

Maximal subgroups