Group #31891150 — subgroups

Subgroups — #31891150

Order 18176.

270 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 115 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	#22588512	C₂	1	1	yes	maximal, characteristic
2	#22588471	C₂	1	1	yes	maximal, characteristic
3	#22586199	C₂	1	4	yes	maximal
4	C₇₁×#1452	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556064	—	2	2	no	—
6	#17986023	C₂²	1	2	yes	—
7	#17985865	C₂²	1	2	yes	—
8	#17985762	—	2	2	no	—
9	#431×C₇₁	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
11	#322×C₇₁	C₄	1	4	yes	complemented
12	C₂×#11765849	—	2	2	no	—
13	C₂×C₄⋊Dic₇₁	—	2	2	no	—
14	#14556064	—	4	4	no	—
15	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
16	C₇₁⋊#95	—	2	4	no	—
17	C₂²×C₈×C₇₁	D₄	1	2	yes	—
18	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
19	C₄⋊C₈×C₇₁	—	2	2	no	—
20	C₂²⋊C₈×C₇₁	—	2	2	no	—
21	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
22	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₇₁	Q₈	1	2	yes	—
24	C₂²×Dic₇₁	—	4	2	no	—
25	#11765849	—	4	4	no	—
26	C₂×C₇₁⋊C₈	—	4	2	no	—
27	C₄⋊Dic₇₁	—	4	2	no	—
28	C₄⋊Dic₇₁	—	4	2	no	—
29	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₁	#41	1	2	yes	—
31	C₂×C₈×C₇₁	—	2	2	no	—
32	C₂×C₈×C₇₁	—	2	2	no	—
33	C₂×C₈×C₇₁	—	2	4	no	—
34	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
35	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₇₁	#41	1	2	yes	—
38	C₂³×C₇₁	#124	1	1	yes	characteristic
39	C₂×C₄×C₇₁	—	4	2	no	—
40	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	#117	1	2	yes	—
42	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
43	C₂×C₄×C₇₁	#125	1	2	yes	—
44	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
45	C₂×C₄×C₇₁	#137	1	2	yes	—
46	C₂×Dic₇₁	—	8	4	no	—
47	C₂×Dic₇₁	—	8	2	no	—
48	C₈×C₇₁	—	2	4	no	—
49	C₈×C₇₁	—	4	4	no	—
50	C₇₁⋊C₈	—	8	2	no	—
51	C₂²×C₇₁	—	2	2	no	—
52	C₂²×C₇₁	#382	1	1	yes	characteristic
53	C₂²×C₇₁	#489	1	2	yes	—
54	C₄×C₇₁	—	2	2	no	—
55	C₄×C₇₁	—	2	2	no	—
56	C₄×C₇₁	#428	1	2	yes	—
57	C₄×C₇₁	#490	1	1	yes	characteristic
58	C₄×C₇₁	#491	1	1	yes	characteristic
59	Dic₇₁	—	16	2	no	—
60	#8794	—	71	1	no	maximal, Sylow
61	C₂×C₇₁	—	2	2	no	—
62	C₂×C₇₁	#1833	1	2	yes	—
63	C₂×C₇₁	#1836	1	1	yes	characteristic
64	#1848	—	71	1	no	—
65	#1835	—	71	1	no	—
66	#1452	D₇₁	1	1	yes	characteristic
67	#1286	—	71	4	no	—
68	C₇₁	#8794	1	1	yes	characteristic, complemented, Sylow
69	#431	C₂×D₇₁	1	2	yes	—
70	C₂×#108	—	142	2	no	—
71	#423	—	71	2	no	—
72	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
73	#382	—	71	2	no	—
74	#340	—	142	2	no	—
75	#322	Dic₇₁	1	4	yes	complemented
76	C₂×M₁₆	—	142	2	no	—
77	C₂²×C₈	D₂₈₄	1	2	yes	—
78	C₂×C₄⋊C₄	—	142	2	no	—
79	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
80	#108	—	284	4	no	—
81	C₄⋊C₈	—	2	2	no	—
82	C₄⋊C₈	—	142	4	no	—
83	C₂²⋊C₈	—	2	2	no	—
84	C₄×C₈	Dic₁₄₂	1	2	yes	—
85	C₄×C₈	C₂×Dic₇₁	1	4	yes	—
86	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
87	#95	—	142	4	no	—
88	C₂²×C₄	—	284	2	no	—
89	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
90	C₂²×C₄	#11765865	1	2	yes	—
91	M₁₆	—	284	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	284	2	no	—
96	C₂×C₈	C₄⋊Dic₇₁	1	4	yes	—
97	C₄⋊C₄	—	284	2	no	—
98	C₄⋊C₄	—	284	2	no	—
99	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
100	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
101	C₄²	#11765865	1	2	yes	—
102	C₂³	#14556116	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	568	4	no	—
105	C₂×C₄	—	568	2	no	—
106	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
107	C₂×C₄	#14556087	1	2	yes	—
108	C₂×C₄	#14556116	1	1	yes	characteristic
109	C₂×C₄	#14556120	1	2	yes	—
110	C₂×C₄	#14556156	1	2	yes	—
111	C₂×C₄	#14556161	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	4	4	no	—
114	C₈	—	568	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#17985865	1	1	yes	characteristic
117	C₂²	#17986359	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	1136	2	no	—
121	C₄	#17986041	1	2	yes	—
122	C₄	#17986364	1	1	yes	characteristic
123	C₄	#17986368	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#22588464	1	2	yes	—
126	C₂	#22588474	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22588512

C₂

yes

maximal, characteristic

#22588471

C₂

yes

maximal, characteristic

#22586199

C₂

yes

maximal

C₇₁×#1452

C₂

yes

maximal, characteristic

C₂×#14556064

—

#17986023

C₂²

yes

—

#17985865

C₂²

yes

—

#17985762

—

#431×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#322×C₇₁

C₄

yes

complemented

C₂×#11765849

—

C₂×C₄⋊Dic₇₁

—

#14556064

—

C₄⋊(C₇₁⋊C₈)

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

D₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

C₂²×Dic₇₁

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

#124

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

#124

yes

characteristic

C₂×C₄×C₇₁

#125

yes

—

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₁

#137

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

#382

yes

characteristic

C₂²×C₇₁

#489

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#428

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

#8794

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1833

yes

—

C₂×C₇₁

yes

—

—

D₇₁

yes

characteristic

#1286

—

C₇₁

#8794

yes

characteristic, complemented, Sylow

#431

C₂×D₇₁

yes

—

C₂×#108

—

142

—

#423

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#382

—

#340

—

142

—

#322

Dic₇₁

yes

complemented

C₂×M₁₆

—

142

—

C₂²×C₈

D₂₈₄

yes

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

#108

—

284

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₂²⋊C₈

—

C₄×C₈

Dic₁₄₂

yes

—

C₄×C₈

C₂×Dic₇₁

yes

—

C₄×C₈

C₂×Dic₇₁

yes

—

#95

—

142

—

C₂²×C₄

—

284

—

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

C₂²×C₄

#11765865

yes

—

M₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

284

—

C₂×C₈

C₄⋊Dic₇₁

yes

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄²

C₂²×Dic₇₁

yes

characteristic

100

C₄²

C₂×Dic₁₄₂

yes

characteristic

101

C₄²

#11765865

yes

—

102

C₂³

#14556116

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

568

—

105

C₂×C₄

—

568

—

106

C₂×C₄

C₂×C₄⋊Dic₇₁

yes

characteristic

107

C₂×C₄

#14556087

yes

—

108

C₂×C₄

#14556116

yes

characteristic

109

C₂×C₄

#14556120

yes

—

110

C₂×C₄

#14556156

yes

—

111

C₂×C₄

#14556161

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

568

—

115

C₂²

—

116

C₂²

#17985865

yes

characteristic

117

C₂²

#17986359

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

1136

—

121

C₄

#17986041

yes

—

122

C₄

#17986364

yes

characteristic

123

C₄

#17986368

yes

characteristic

124

C₂

—

125

C₂

#22588464

yes

—

126

C₂

#22588474

yes

characteristic

	subgroup	G/H
center	C₂²	#17985865
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1452	C₂
socle	C₂²×C₇₁	#382
prime radical	C₂³×C₇₁	#124
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1452	D₇₁
hypocenter	C₇₁	#8794
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#17985865

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1452

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
11. #322×C₇₁	C₄	semidirect
68. C₇₁	#8794	semidirect
75. #322	Dic₇₁	semidirect

H ≅ G/N

type

11. #322×C₇₁

C₄

semidirect

68. C₇₁

#8794

semidirect

75. #322

Dic₇₁

semidirect

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

subgroup	index
#22588512	2
#22588471	2
#22586199	2
C₇₁×#1452	2
#8794	71

subgroup

index

C₇₁×#1452

Subgroups — #31891150

Conjugacy classes of subgroups

Complements

Maximal subgroups