Group #31917943 — subgroups

Subgroups — #31917943

Order 18176.

198 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 99 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	M₆₄×D₇₁	C₂	1	2	yes	maximal
2	#22589935	C₂	1	1	yes	maximal, characteristic
3	#22586538	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586537	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1422	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556067	C₄	1	1	yes	characteristic
7	C₂×C₁₆×D₇₁	C₂²	1	1	yes	characteristic
8	#17985953	C₄	1	1	yes	characteristic
9	#17985833	C₂²	1	2	yes	—
10	#17985832	—	2	2	no	—
11	C₃₂×D₇₁	—	2	2	no	—
12	#17985775	C₄	1	1	yes	characteristic
13	#17985774	C₄	1	1	yes	characteristic
14	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
15	M₆₄×C₇₁	C₂²	1	2	yes	—
16	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×#11765848	C₈	1	1	yes	characteristic
18	C₂×C₈×D₇₁	C₂×C₄	1	1	yes	characteristic
19	#14556085	C₈	1	1	yes	characteristic
20	#14556067	—	2	2	no	—
21	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₁₆×D₇₁	D₄	1	2	yes	—
24	C₁₆×D₇₁	Q₈	1	2	yes	—
25	#14556039	C₈	1	1	yes	characteristic
26	#14556038	C₈	1	1	yes	characteristic
27	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
28	C₇₁⋊C₃₂	—	2	2	no	—
29	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₃₂×C₇₁	—	2	2	no	—
33	C₂×C₄×D₇₁	C₂×C₈	1	1	yes	characteristic
34	#11765848	C₁₆	1	4	yes	—
35	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
36	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
37	C₈×D₇₁	C₄⋊C₄	1	2	yes	—
38	C₈×D₇₁	M₁₆	1	2	yes	—
39	Dic₇₁⋊C₄	—	2	1	no	—
40	(C₂×D₇₁)⋊C₄	—	2	1	no	—
41	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
42	C₇₁⋊C₁₆	—	2	1	no	—
43	C₇₁⋊C₁₆	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
47	C₁₆×C₇₁	—	2	1	no	—
48	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
50	C₂²×D₇₁	M₃₂	1	1	yes	characteristic
51	C₂×C₄×C₇₁	—	2	1	no	—
52	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
53	C₂×Dic₇₁	—	2	1	no	—
54	C₂×Dic₇₁	M₃₂	1	1	yes	characteristic
55	C₄×D₇₁	C₄⋊C₈	1	2	yes	—
56	C₄×D₇₁	C₂×C₁₆	1	2	yes	—
57	C₈×C₇₁	C₂×C₁₆	1	2	yes	—
58	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
60	C₇₁⋊C₈	C₄⋊C₈	1	2	yes	—
61	C₇₁⋊C₈	C₂×C₁₆	1	2	yes	—
62	C₂²×C₇₁	C₂×M₃₂	1	1	yes	characteristic
63	C₂×D₇₁	—	4	1	no	—
64	C₂×D₇₁	C₄⋊C₁₆	1	2	yes	—
65	C₄×C₇₁	—	4	1	no	—
66	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
67	C₄×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
68	Dic₇₁	—	4	1	no	—
69	Dic₇₁	C₄⋊C₁₆	1	2	yes	—
70	#13630	—	71	1	no	maximal, Sylow
71	C₂×C₇₁	—	2	1	no	—
72	C₂×C₇₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
73	D₇₁	—	2	2	no	—
74	C₂×M₆₄	—	71	2	no	—
75	#2108	—	71	1	no	—
76	#1422	—	71	2	no	—
77	#1422	—	71	1	no	—
78	#1422	D₇₁	1	1	yes	characteristic, complemented
79	C₇₁	#13630	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	—	71	1	no	—
81	C₂²×C₁₆	—	71	1	no	—
82	#405	—	71	1	no	—
83	M₆₄	—	71	2	no	—
84	M₆₄	—	142	2	no	—
85	M₆₄	C₂×D₇₁	1	2	yes	—
86	C₂×C₃₂	—	142	2	no	—
87	#346	—	71	1	no	—
88	#346	—	71	1	no	—
89	C₄×C₁₆	—	71	1	no	—
90	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
91	C₂×M₁₆	—	71	1	no	—
92	C₂²×C₈	—	71	1	no	—
93	#117	—	71	1	no	—
94	M₃₂	—	142	2	no	—
95	C₂×C₁₆	—	71	1	no	—
96	C₂×C₁₆	—	71	2	no	—
97	C₂×C₁₆	—	71	2	no	—
98	C₂×C₁₆	—	71	1	no	—
99	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
100	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
101	#97	—	71	1	no	—
102	#97	—	71	1	no	—
103	C₄×C₈	—	71	1	no	—
104	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
105	C₃₂	—	2	2	no	—
106	C₃₂	—	142	2	no	—
107	C₂²×C₄	—	71	1	no	—
108	M₁₆	—	71	4	no	—
109	C₂×C₈	—	71	1	no	—
110	C₂×C₈	—	71	1	no	—
111	C₂×C₈	—	71	2	no	—
112	C₂×C₈	—	71	2	no	—
113	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
114	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
115	C₄⋊C₄	—	142	1	no	—
116	C₂²⋊C₄	—	142	1	no	—
117	C₄²	—	71	1	no	—
118	C₄²	C₈×D₇₁	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	71	2	no	—
121	C₁₆	—	142	1	no	—
122	C₁₆	D₄×D₇₁	1	1	yes	characteristic
123	C₁₆	Q₈×D₇₁	1	1	yes	characteristic
124	C₂³	—	71	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	71	1	no	—
127	C₂×C₄	—	71	2	no	—
128	C₂×C₄	—	71	2	no	—
129	C₂×C₄	—	142	1	no	—
130	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
131	C₈	—	71	2	no	—
132	C₈	—	71	2	no	—
133	C₈	C₁₆×D₇₁	1	2	yes	—
134	C₈	C₄⋊C₄×D₇₁	1	1	yes	characteristic
135	C₈	M₁₆×D₇₁	1	1	yes	characteristic
136	C₂²	—	284	1	no	—
137	C₂²	—	71	2	no	—
138	C₂²	M₃₂×D₇₁	1	1	yes	characteristic
139	C₄	—	4	1	no	—
140	C₄	—	284	1	no	—
141	C₄	—	71	2	no	—
142	C₄	C₄⋊C₈×D₇₁	1	1	yes	characteristic
143	C₄	C₂×C₁₆×D₇₁	1	1	yes	characteristic
144	C₂	—	2	1	no	—
145	C₂	—	142	2	no	—
146	C₂	C₄⋊C₁₆×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₆₄×D₇₁

C₂

yes

maximal

#22589935

C₂

yes

maximal, characteristic

#22586538

C₂

yes

maximal, characteristic, complemented

#22586537

C₂

yes

maximal, complemented

C₇₁×#1422

C₂

yes

maximal, characteristic, complemented

C₂×#14556067

C₄

yes

characteristic

C₂×C₁₆×D₇₁

C₂²

yes

characteristic

#17985953

C₄

yes

characteristic

#17985833

C₂²

yes

—

#17985832

—

C₃₂×D₇₁

—

#17985775

C₄

yes

characteristic

#17985774

C₄

yes

characteristic

C₁₆×Dic₇₁

C₂²

yes

characteristic

M₆₄×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#11765848

C₈

yes

characteristic

C₂×C₈×D₇₁

C₂×C₄

yes

characteristic

#14556085

C₈

yes

characteristic

#14556067

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₁₆×D₇₁

D₄

yes

—

C₁₆×D₇₁

Q₈

yes

—

#14556039

C₈

yes

characteristic

#14556038

C₈

yes

characteristic

C₈×Dic₇₁

C₂×C₄

yes

characteristic

C₇₁⋊C₃₂

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₃₂×C₇₁

—

C₂×C₄×D₇₁

C₂×C₈

yes

characteristic

#11765848

C₁₆

yes

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₈

yes

characteristic

C₈×D₇₁

C₄⋊C₄

yes

—

C₈×D₇₁

M₁₆

yes

—

Dic₇₁⋊C₄

—

(C₂×D₇₁)⋊C₄

—

C₄×Dic₇₁

C₂×C₈

yes

characteristic

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

C₂×Q₈

yes

characteristic

C₂²×D₇₁

M₃₂

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

M₃₂

yes

characteristic

C₄×D₇₁

C₄⋊C₈

yes

—

C₄×D₇₁

C₂×C₁₆

yes

—

C₈×C₇₁

C₂×C₁₆

yes

—

C₈×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₇₁⋊C₈

C₄⋊C₈

yes

—

C₇₁⋊C₈

C₂×C₁₆

yes

—

C₂²×C₇₁

C₂×M₃₂

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

C₄⋊C₁₆

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₁

C₂²×C₁₆

yes

characteristic

Dic₇₁

—

Dic₇₁

C₄⋊C₁₆

yes

—

#13630

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

C₂×C₄⋊C₁₆

yes

characteristic

D₇₁

—

C₂×M₆₄

—

—

—

—

D₇₁

yes

characteristic, complemented

C₇₁

#13630

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

M₆₄

—

M₆₄

—

142

—

M₆₄

C₂×D₇₁

yes

—

C₂×C₃₂

—

142

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₁

yes

characteristic

100

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

101

#97

—

102

#97

—

103

C₄×C₈

—

104

C₄×C₈

C₄×D₇₁

yes

characteristic

105

C₃₂

—

106

C₃₂

—

142

—

107

C₂²×C₄

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₈×D₇₁

yes

characteristic

114

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

115

C₄⋊C₄

—

142

—

116

C₂²⋊C₄

—

142

—

117

C₄²

—

118

C₄²

C₈×D₇₁

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

—

142

—

122

C₁₆

D₄×D₇₁

yes

characteristic

123

C₁₆

Q₈×D₇₁

yes

characteristic

124

C₂³

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

142

—

130

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

C₁₆×D₇₁

yes

—

134

C₈

C₄⋊C₄×D₇₁

yes

characteristic

135

C₈

M₁₆×D₇₁

yes

characteristic

136

C₂²

—

284

—

137

C₂²

—

138

C₂²

M₃₂×D₇₁

yes

characteristic

139

C₄

—

140

C₄

—

284

—

141

C₄

—

142

C₄

C₄⋊C₈×D₇₁

yes

characteristic

143

C₄

C₂×C₁₆×D₇₁

yes

characteristic

144

C₂

—

145

C₂

—

142

—

146

C₂

C₄⋊C₁₆×D₇₁

yes

characteristic

	subgroup	G/H
center	C₁₆	D₄×D₇₁
commutator	C₄×C₇₁	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#1422	C₂
socle	C₂×C₇₁	C₂×C₄⋊C₁₆
prime radical	C₂²×D₇₁	M₃₂
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	#1422	D₇₁
hypocenter	C₇₁	#13630
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #1422

subgroup

G/H

center

C₁₆

D₄×D₇₁

commutator

C₄×C₇₁

C₂²×C₁₆

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₇₁×#1422

C₂

socle

C₂×C₇₁

C₂×C₄⋊C₁₆

prime radical

C₂²×D₇₁

M₃₂

cosocle

C₂×C₁₆×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #1422

N	H ≅ G/N	type
3. #22586538	C₂	semidirect
4. #22586537	C₂	semidirect
5. C₇₁×#1422	C₂	semidirect
78. #1422	D₇₁	semidirect
79. C₇₁	#13630	semidirect

H ≅ G/N

type

3. #22586538

C₂

semidirect

4. #22586537

C₂

semidirect

5. C₇₁×#1422

C₂

semidirect

78. #1422

D₇₁

semidirect

79. C₇₁

#13630

semidirect

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

subgroup	index
M₆₄×D₇₁	2
#22589935	2
#22586538	2
#22586537	2
C₇₁×#1422	2
#13630	71

subgroup

index

M₆₄×D₇₁

C₇₁×#1422

Subgroups — #31917943

Conjugacy classes of subgroups

Complements

Maximal subgroups