Group #31912453 — subgroups

Subgroups — #31912453

Order 18176.

390 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 203 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	C₂×#17985775	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22586417	C₂	1	4	yes	maximal
4	C₂×#346×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
8	C₂×C₈×Dic₇₁	C₂²	1	2	yes	—
9	C₄²×Dic₇₁	C₄	1	1	yes	characteristic
10	#17985775	C₄	1	4	yes	complemented
11	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	#346×C₇₁	C₄	1	4	yes	complemented
14	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₇₁	C₈	1	4	yes	—
16	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
17	C₂×C₇₁⋊C₁₆	—	2	4	no	—
18	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
19	C₄×C₇₁⋊C₈	C₈	1	4	yes	—
20	C₈×Dic₇₁	—	2	4	no	—
21	C₈×Dic₇₁	—	2	4	no	—
22	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₁	—	2	4	no	—
26	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
29	C₂²×Dic₇₁	C₂×C₈	1	4	yes	—
30	C₂×C₇₁⋊C₈	—	2	2	no	—
31	C₂×C₇₁⋊C₈	—	2	2	no	—
32	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
33	C₄×Dic₇₁	—	2	2	no	—
34	C₄×Dic₇₁	—	2	2	no	—
35	C₄×Dic₇₁	—	2	8	no	—
36	C₇₁⋊C₁₆	—	2	4	no	—
37	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₁	C₄²	1	2	yes	—
40	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
41	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	#41	1	4	yes	—
44	C₄²×C₇₁	C₄²	1	2	yes	—
45	C₄²×C₇₁	C₂×C₈	1	2	yes	—
46	C₁₆×C₇₁	—	2	4	no	—
47	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₁	—	2	4	no	—
49	C₂×C₄×C₇₁	C₄×C₈	1	4	yes	—
50	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₇₁	#117	1	2	yes	—
52	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
54	C₂×Dic₇₁	—	2	4	no	—
55	C₂×Dic₇₁	—	2	4	no	—
56	C₂×Dic₇₁	M₃₂	1	8	yes	—
57	C₈×C₇₁	#97	1	4	yes	—
58	C₈×C₇₁	#117	1	2	yes	—
59	C₈×C₇₁	#117	1	2	yes	—
60	C₇₁⋊C₈	—	2	4	no	—
61	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
62	C₂²×C₇₁	#431	1	2	yes	—
63	C₂²×C₇₁	C₂×M₃₂	1	4	yes	—
64	C₄×C₇₁	—	2	4	no	—
65	C₄×C₇₁	#376	1	1	yes	characteristic
66	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
67	C₄×C₇₁	#431	1	2	yes	—
68	Dic₇₁	—	2	8	no	—
69	#12591	—	71	1	no	maximal, Sylow
70	C₂×C₇₁	#1744	1	1	yes	characteristic
71	C₂×C₇₁	C₄×M₃₂	1	2	yes	—
72	C₂×C₇₁	#2163	1	4	yes	—
73	C₂×#346	—	71	1	no	—
74	C₂×#346	D₇₁	1	1	yes	characteristic
75	C₄²×C₈	—	71	1	no	—
76	#1374	—	71	4	no	—
77	C₇₁	#12591	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₁₆	—	71	2	no	—
79	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
80	C₂×C₄×C₈	—	71	1	no	—
81	C₂×C₄×C₈	—	71	2	no	—
82	C₂×C₄×C₈	—	71	2	no	—
83	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
84	C₄³	—	71	1	no	—
85	#346	—	71	4	no	—
86	#346	Dic₇₁	1	4	yes	complemented
87	C₂²×C₈	—	71	2	no	—
88	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
89	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
90	C₂×C₄²	—	71	4	no	—
91	C₂×C₄²	—	71	2	no	—
92	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
93	C₂×C₁₆	—	2	4	no	—
94	C₂×C₁₆	—	71	4	no	—
95	C₂×C₁₆	—	142	4	no	—
96	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	—
97	C₄×C₈	—	71	4	no	—
98	C₄×C₈	—	142	4	no	—
99	C₄×C₈	—	142	4	no	—
100	C₄×C₈	C₄×D₇₁	1	2	yes	—
101	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
102	C₂²×C₄	—	71	4	no	—
103	C₂²×C₄	C₈×D₇₁	1	2	yes	—
104	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
105	C₂×C₈	—	71	4	no	—
106	C₂×C₈	—	142	2	no	—
107	C₂×C₈	—	142	2	no	—
108	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
109	C₂×C₈	#11765848	1	4	yes	—
110	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
111	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
112	C₂×C₈	#11765865	1	4	yes	—
113	C₄²	—	142	8	no	—
114	C₄²	—	142	2	no	—
115	C₄²	—	142	2	no	—
116	C₄²	C₄×Dic₇₁	1	2	yes	—
117	C₄²	C₈×D₇₁	1	2	yes	—
118	C₁₆	—	2	4	no	—
119	C₁₆	—	142	4	no	—
120	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	71	8	no	—
123	C₂×C₄	—	142	4	no	—
124	C₂×C₄	—	142	4	no	—
125	C₂×C₄	C₈×Dic₇₁	1	4	yes	—
126	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
127	C₂×C₄	#14556085	1	2	yes	—
128	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
129	C₂×C₄	C₂×#11765848	1	2	yes	—
130	C₈	—	142	4	no	—
131	C₈	#14556038	1	4	yes	—
132	C₈	#14556085	1	2	yes	—
133	C₈	#14556087	1	2	yes	—
134	C₂²	C₂×C₈×Dic₇₁	1	1	yes	characteristic
135	C₂²	#17986050	1	2	yes	—
136	C₂²	M₃₂×D₇₁	1	4	yes	—
137	C₄	—	2	4	no	—
138	C₄	—	142	8	no	—
139	C₄	#17985845	1	1	yes	characteristic
140	C₄	C₂×#14556038	1	1	yes	characteristic
141	C₄	#17986050	1	2	yes	—
142	C₂	#22588085	1	1	yes	characteristic
143	C₂	M₃₂×Dic₇₁	1	2	yes	—
144	C₂	#22590141	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985775

C₂

yes

maximal, characteristic

C₄×C₈×Dic₇₁

C₂

yes

maximal, characteristic

#22586417

C₂

yes

maximal

C₂×#346×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

—

C₂×C₄×C₇₁⋊C₈

C₄

yes

characteristic

C₂×C₈×Dic₇₁

C₄

yes

—

C₂×C₈×Dic₇₁

C₂²

yes

—

C₄²×Dic₇₁

C₄

yes

characteristic

#17985775

C₄

yes

complemented

C₂²×C₁₆×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#346×C₇₁

C₄

yes

complemented

C₂²×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₇₁

C₈

yes

—

C₂×C₄×Dic₇₁

C₂×C₄

yes

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

C₄×C₇₁⋊C₈

C₈

yes

—

C₈×Dic₇₁

—

C₈×Dic₇₁

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₂²×Dic₇₁

C₂×C₈

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×C₈

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₄²

yes

—

C₂×C₈×C₇₁

M₁₆

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₄²×C₇₁

C₄²

yes

—

C₄²×C₇₁

C₂×C₈

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄×C₈

yes

—

C₂×C₄×C₇₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

M₃₂

yes

—

C₈×C₇₁

#97

yes

—

C₈×C₇₁

#117

yes

—

C₈×C₇₁

#117

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₄×C₈

yes

characteristic

C₂²×C₇₁

#431

yes

—

C₂²×C₇₁

C₂×M₃₂

yes

—

C₄×C₇₁

—

C₄×C₇₁

#376

yes

characteristic

C₄×C₇₁

C₂×#97

yes

characteristic

C₄×C₇₁

#431

yes

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

C₄×M₃₂

yes

—

C₂×C₇₁

#2163

yes

—

C₂×#346

—

C₂×#346

D₇₁

yes

characteristic

C₄²×C₈

—

#1374

—

C₇₁

#12591

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₁

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄³

—

#346

—

#346

Dic₇₁

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₄×D₇₁

yes

characteristic

C₂²×C₈

C₂²×D₇₁

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₄×D₇₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

142

—

C₂×C₁₆

C₂×Dic₇₁

yes

—

C₄×C₈

—

C₄×C₈

—

142

—

C₄×C₈

—

142

—

100

C₄×C₈

C₄×D₇₁

yes

—

101

C₄×C₈

C₂×Dic₇₁

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

C₈×D₇₁

yes

—

104

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

142

—

107

C₂×C₈

—

142

—

108

C₂×C₈

C₄×Dic₇₁

yes

—

109

C₂×C₈

#11765848

yes

—

110

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

111

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

112

C₂×C₈

#11765865

yes

—

113

C₄²

—

142

—

114

C₄²

—

142

—

115

C₄²

—

142

—

116

C₄²

C₄×Dic₇₁

yes

—

117

C₄²

C₈×D₇₁

yes

—

118

C₁₆

—

119

C₁₆

—

142

—

120

C₂³

C₂×C₈×D₇₁

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

142

—

124

C₂×C₄

—

142

—

125

C₂×C₄

C₈×Dic₇₁

yes

—

126

C₂×C₄

C₂×C₄×Dic₇₁

yes

characteristic

127

C₂×C₄

#14556085

yes

—

128

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

129

C₂×C₄

C₂×#11765848

yes

—

130

C₈

—

142

—

131

C₈

#14556038

yes

—

132

C₈

#14556085

yes

—

133

C₈

#14556087

yes

—

134

C₂²

C₂×C₈×Dic₇₁

yes

characteristic

135

C₂²

#17986050

yes

—

136

C₂²

M₃₂×D₇₁

yes

—

137

C₄

—

138

C₄

—

142

—

139

C₄

#17985845

yes

characteristic

140

C₄

C₂×#14556038

yes

characteristic

141

C₄

#17986050

yes

—

142

C₂

#22588085

yes

characteristic

143

C₂

M₃₂×Dic₇₁

yes

—

144

C₂

#22590141

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄×C₈
Fitting	C₂×#346×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#346	D₇₁
hypocenter	C₇₁	#12591
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₇₁

commutator

C₂²×C₇₁

C₂×C₄×C₈

Fitting

C₂×#346×C₇₁

C₂

socle, prime radical

C₂³×C₇₁

C₂²×C₈

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₈ ◁ C₂×#346

N	H ≅ G/N	type
10. #17985775	C₄	semidirect
13. #346×C₇₁	C₄	semidirect
77. C₇₁	#12591	semidirect
86. #346	Dic₇₁	semidirect

H ≅ G/N

type

10. #17985775

C₄

semidirect

13. #346×C₇₁

C₄

semidirect

77. C₇₁

#12591

semidirect

86. #346

Dic₇₁

semidirect

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

subgroup	index
C₂×#17985775	2
C₄×C₈×Dic₇₁	2
#22586417	2
C₂×#346×C₇₁	2
#12591	71

subgroup

index

C₂×#17985775

C₄×C₈×Dic₇₁

#22586417

C₂×#346×C₇₁

#12591

Subgroups — #31912453

Conjugacy classes of subgroups

Complements

Maximal subgroups