Group #31884794 — subgroups

Subgroups — #31884794

Order 18176.

170 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 45 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	#22589950	C₂	1	1	yes	maximal, characteristic
2	#22586480	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#1400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₈×D₇₁	C₄	1	1	yes	characteristic
5	C₂×#14556068	C₄	1	1	yes	characteristic
6	#17985833	—	2	1	no	—
7	(C₂×D₇₁)⋊C₁₆	—	2	1	no	—
8	(C₂×D₇₁)⋊C₁₆	—	2	1	no	—
9	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
10	M₆₄×C₇₁	—	2	1	no	—
11	C₂²×C₄×D₇₁	C₈	1	1	yes	characteristic
12	C₂²×C₇₁⋊C₈	C₈	1	1	yes	characteristic
13	C₂×C₈×D₇₁	—	2	1	no	—
14	C₂×C₈×D₇₁	—	2	1	no	—
15	C₂×C₈×D₇₁	—	4	1	no	—
16	C₂×C₈×D₇₁	—	4	1	no	—
17	#14556068	—	4	1	no	—
18	C₂×C₇₁⋊C₁₆	—	2	1	no	—
19	C₇₁⋊C₃₂	—	4	1	no	—
20	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
21	M₃₂×C₇₁	—	4	1	no	—
22	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
24	C₃₂×C₇₁	—	4	1	no	—
25	C₂³×D₇₁	C₁₆	1	1	yes	characteristic
26	C₂²×Dic₇₁	C₁₆	1	1	yes	characteristic
27	C₂×C₄×D₇₁	—	2	1	no	—
28	C₂×C₄×D₇₁	—	2	1	no	—
29	C₂×C₄×D₇₁	—	4	1	no	—
30	C₂×C₄×D₇₁	—	4	1	no	—
31	C₂×C₇₁⋊C₈	—	2	1	no	—
32	C₂×C₇₁⋊C₈	—	4	1	no	—
33	C₈×D₇₁	—	4	2	no	—
34	C₈×D₇₁	—	4	1	no	—
35	C₈×D₇₁	—	4	1	no	—
36	C₇₁⋊C₁₆	—	4	1	no	—
37	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₇₁	—	2	1	no	—
39	C₂×C₈×C₇₁	—	2	1	no	—
40	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	M₁₆	1	1	yes	characteristic
42	C₁₆×C₇₁	—	2	1	no	—
43	C₁₆×C₇₁	—	2	1	no	—
44	C₂³×C₇₁	C₂×C₁₆	1	1	yes	characteristic
45	C₂²×D₇₁	—	2	1	no	—
46	C₂²×D₇₁	—	4	2	no	—
47	C₂²×D₇₁	—	4	1	no	—
48	C₂×C₄×C₇₁	—	2	1	no	—
49	C₂×C₄×C₇₁	—	2	1	no	—
50	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₁	M₃₂	1	1	yes	characteristic
52	C₂×Dic₇₁	—	2	1	no	—
53	C₂×Dic₇₁	—	4	1	no	—
54	C₄×D₇₁	—	4	2	no	—
55	C₄×D₇₁	—	4	1	no	—
56	C₄×D₇₁	—	4	1	no	—
57	C₈×C₇₁	—	2	1	no	—
58	C₈×C₇₁	#99	1	1	yes	characteristic
59	C₈×C₇₁	#100	1	1	yes	characteristic
60	C₇₁⋊C₈	—	4	1	no	—
61	C₂²×C₇₁	—	2	1	no	—
62	C₂²×C₇₁	—	4	1	no	—
63	C₂²×C₇₁	C₂²⋊C₁₆	1	1	yes	characteristic
64	C₂×D₇₁	—	2	2	no	—
65	C₂×D₇₁	—	4	1	no	—
66	C₂×D₇₁	—	4	2	no	—
67	C₂×D₇₁	—	4	1	no	—
68	C₂×D₇₁	—	4	2	no	—
69	C₄×C₇₁	—	2	1	no	—
70	C₄×C₇₁	#323	1	1	yes	characteristic
71	C₄×C₇₁	#349	1	1	yes	characteristic
72	Dic₇₁	—	4	1	no	—
73	#7336	—	71	1	no	maximal, Sylow
74	C₂×C₇₁	—	2	1	no	—
75	C₂×C₇₁	—	4	1	no	—
76	C₂×C₇₁	#1314	1	1	yes	characteristic
77	D₇₁	—	4	2	no	—
78	#2112	—	71	1	no	—
79	#1400	—	71	1	no	—
80	#1400	D₇₁	1	1	yes	characteristic, complemented
81	C₇₁	#7336	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₈	—	71	1	no	—
83	C₂×M₃₂	—	71	1	no	—
84	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
85	M₆₄	—	2	1	no	—
86	M₆₄	—	142	1	no	—
87	C₂²⋊C₁₆	—	142	1	no	—
88	C₂²⋊C₁₆	—	142	1	no	—
89	C₂³×C₄	—	71	1	no	—
90	C₂²×C₈	—	284	1	no	—
91	C₂²×C₈	—	284	1	no	—
92	C₂²×C₈	—	71	1	no	—
93	C₂²×C₈	—	142	1	no	—
94	C₂²×C₈	—	142	1	no	—
95	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
96	M₃₂	—	4	1	no	—
97	M₃₂	—	284	1	no	—
98	C₂×C₁₆	—	142	1	no	—
99	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
100	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
101	C₃₂	—	4	1	no	—
102	C₃₂	—	284	1	no	—
103	C₂⁴	—	71	1	no	—
104	C₂²×C₄	—	284	1	no	—
105	C₂²×C₄	—	284	1	no	—
106	C₂²×C₄	—	71	1	no	—
107	C₂²×C₄	—	142	1	no	—
108	C₂²×C₄	—	142	1	no	—
109	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	284	2	no	—
113	C₂×C₈	—	284	1	no	—
114	C₂×C₈	—	284	1	no	—
115	C₂×C₈	—	284	1	no	—
116	C₂×C₈	—	142	1	no	—
117	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
118	C₂×C₈	#11765848	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	2	1	no	—
121	C₁₆	—	284	1	no	—
122	C₂³	—	284	1	no	—
123	C₂³	—	284	2	no	—
124	C₂³	—	142	1	no	—
125	C₂³	C₁₆×D₇₁	1	1	yes	characteristic
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	284	2	no	—
129	C₂×C₄	—	284	1	no	—
130	C₂×C₄	—	284	1	no	—
131	C₂×C₄	—	284	1	no	—
132	C₂×C₄	—	142	1	no	—
133	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
134	C₂×C₄	#14556067	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	—	284	1	no	—
137	C₈	#14556042	1	1	yes	characteristic
138	C₈	#14556044	1	1	yes	characteristic
139	C₂²	—	2	1	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	284	1	no	—
142	C₂²	—	284	2	no	—
143	C₂²	—	284	2	no	—
144	C₂²	—	284	1	no	—
145	C₂²	—	142	2	no	—
146	C₂²	(C₂×D₇₁)⋊C₁₆	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	284	1	no	—
149	C₄	#17985722	1	1	yes	characteristic
150	C₄	#17985780	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	4	1	no	—
153	C₂	—	284	2	no	—
154	C₂	#22586263	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22589950

C₂

yes

maximal, characteristic

#22586480

C₂

yes

maximal, characteristic, complemented

C₇₁×#1400

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₇₁

C₄

yes

characteristic

C₂×#14556068

C₄

yes

characteristic

#17985833

—

(C₂×D₇₁)⋊C₁₆

—

(C₂×D₇₁)⋊C₁₆

—

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

M₆₄×C₇₁

—

C₂²×C₄×D₇₁

C₈

yes

characteristic

C₂²×C₇₁⋊C₈

C₈

yes

characteristic

C₂×C₈×D₇₁

—

C₂×C₈×D₇₁

—

C₂×C₈×D₇₁

—

C₂×C₈×D₇₁

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₇₁⋊C₃₂

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₃₂×C₇₁

—

C₂³×D₇₁

C₁₆

yes

characteristic

C₂²×Dic₇₁

C₁₆

yes

characteristic

C₂×C₄×D₇₁

—

C₂×C₄×D₇₁

—

C₂×C₄×D₇₁

—

C₂×C₄×D₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₈×D₇₁

—

C₈×D₇₁

—

C₈×D₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

M₁₆

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×C₁₆

yes

characteristic

C₂²×D₇₁

—

C₂²×D₇₁

—

C₂²×D₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₁

M₃₂

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²⋊C₁₆

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

#7336

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1314

yes

characteristic

D₇₁

—

#2112

—

#1400

—

#1400

D₇₁

yes

characteristic, complemented

C₇₁

#7336

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₇₁

yes

characteristic

M₆₄

—

M₆₄

—

142

—

C₂²⋊C₁₆

—

142

—

C₂²⋊C₁₆

—

142

—

C₂³×C₄

—

C₂²×C₈

—

284

—

C₂²×C₈

—

284

—

C₂²×C₈

—

C₂²×C₈

—

142

—

C₂²×C₈

—

142

—

C₂²×C₈

C₄×D₇₁

yes

characteristic

M₃₂

—

M₃₂

—

284

—

C₂×C₁₆

—

142

—

C₂×C₁₆

D₂₈₄

yes

characteristic

100

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

101

C₃₂

—

102

C₃₂

—

284

—

103

C₂⁴

—

104

C₂²×C₄

—

284

—

105

C₂²×C₄

—

284

—

106

C₂²×C₄

—

107

C₂²×C₄

—

142

—

108

C₂²×C₄

—

142

—

109

C₂²×C₄

C₈×D₇₁

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

284

—

113

C₂×C₈

—

284

—

114

C₂×C₈

—

284

—

115

C₂×C₈

—

284

—

116

C₂×C₈

—

142

—

117

C₂×C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

118

C₂×C₈

#11765848

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

—

284

—

122

C₂³

—

284

—

123

C₂³

—

284

—

124

C₂³

—

142

—

125

C₂³

C₁₆×D₇₁

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

284

—

129

C₂×C₄

—

284

—

130

C₂×C₄

—

284

—

131

C₂×C₄

—

284

—

132

C₂×C₄

—

142

—

133

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

characteristic

134

C₂×C₄

#14556067

yes

characteristic

135

C₈

—

136

C₈

—

284

—

137

C₈

#14556042

yes

characteristic

138

C₈

#14556044

yes

characteristic

139

C₂²

—

140

C₂²

—

141

C₂²

—

284

—

142

C₂²

—

284

—

143

C₂²

—

284

—

144

C₂²

—

284

—

145

C₂²

—

142

—

146

C₂²

(C₂×D₇₁)⋊C₁₆

yes

characteristic

147

C₄

—

148

C₄

—

284

—

149

C₄

#17985722

yes

characteristic

150

C₄

#17985780

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

—

284

—

154

C₂

#22586263

yes

characteristic

	subgroup	G/H
center	C₈	#14556042
commutator	C₂³×C₇₁	C₂×C₁₆
Frattini	C₂×M₃₂	C₂×D₇₁
Fitting	C₇₁×#1400	C₂
socle	C₂×C₇₁	#1314
prime radical	C₂³×D₇₁	C₁₆
cosocle	C₂×M₃₂×C₇₁	C₂²
hypercenter	#1400	D₇₁
hypocenter	C₇₁	#7336
LCS	G ▷ C₂³×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×M₃₂ ◁ #1400

subgroup

G/H

center

C₈

#14556042

commutator

C₂³×C₇₁

C₂×C₁₆

Frattini

C₂×M₃₂

C₂×D₇₁

Fitting

C₇₁×#1400

C₂

socle

C₂×C₇₁

#1314

prime radical

C₂³×D₇₁

C₁₆

cosocle

C₂×M₃₂×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₂³×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×M₃₂ ◁ #1400

N	H ≅ G/N	type
2. #22586480	C₂	semidirect
3. C₇₁×#1400	C₂	semidirect
80. #1400	D₇₁	semidirect
81. C₇₁	#7336	semidirect

H ≅ G/N

type

2. #22586480

C₂

semidirect

3. C₇₁×#1400

C₂

semidirect

80. #1400

D₇₁

semidirect

81. C₇₁

#7336

semidirect

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

subgroup	index
#22589950	2
#22586480	2
C₇₁×#1400	2
#7336	71

subgroup

index

#22589950

#22586480

C₇₁×#1400

#7336

Subgroups — #31884794

Conjugacy classes of subgroups

Complements

Maximal subgroups