Group #31909046 — subgroups

Subgroups — #31909046

Order 18176.

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

normal subgroups only

This group has 220 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	#364×D₇₁	C₂	1	2	yes	maximal
2	C₄×#14556067	C₂	1	1	yes	maximal, characteristic
3	#22586416	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586415	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556067	C₄	1	1	yes	characteristic
7	C₂×#14556067	C₄	1	1	yes	characteristic
8	M₃₂×D₇₁	—	2	2	no	—
9	C₄×C₈×D₇₁	C₂²	1	1	yes	characteristic
10	#17985820	C₂²	1	2	yes	—
11	#17985819	—	2	2	no	—
12	#17985774	C₄	1	1	yes	characteristic
13	#17985774	C₄	1	1	yes	characteristic
14	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
15	#364×C₇₁	C₂²	1	2	yes	—
16	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₇₁	D₄	1	1	yes	characteristic
19	C₄²×D₇₁	Q₈	1	1	yes	characteristic
20	#14556068	—	2	2	no	—
21	#14556067	—	2	2	no	—
22	#14556067	—	4	2	no	—
23	#14556067	C₈	1	4	yes	—
24	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₁₆×D₇₁	—	4	2	no	—
27	C₄×C₇₁⋊C₈	Q₈	1	1	yes	characteristic
28	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₇₁	D₄	1	1	yes	characteristic
30	M₃₂×C₇₁	—	2	2	no	—
31	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₇₁	—	2	1	no	—
35	C₂×C₄×D₇₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₈×D₇₁	C₂×C₈	1	2	yes	—
39	C₈×D₇₁	M₁₆	1	2	yes	—
40	C₈×D₇₁	D₈	1	2	yes	—
41	C₈×D₇₁	Q₁₆	1	2	yes	—
42	C₄×Dic₇₁	—	2	1	no	—
43	C₄×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₇₁⋊C₁₆	—	2	1	no	—
45	C₇₁⋊C₁₆	—	4	2	no	—
46	C₇₁⋊C₁₆	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
50	C₁₆×C₇₁	—	2	1	no	—
51	C₁₆×C₇₁	—	4	2	no	—
52	C₁₆×C₇₁	C₂×C₈	1	2	yes	—
53	C₂²×D₇₁	#108	1	1	yes	characteristic
54	C₂×C₄×C₇₁	—	2	1	no	—
55	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×Dic₇₁	—	2	1	no	—
57	C₂×Dic₇₁	#108	1	1	yes	characteristic
58	C₄×D₇₁	—	4	2	no	—
59	C₄×D₇₁	C₄⋊C₈	1	2	yes	—
60	C₄×D₇₁	#107	1	2	yes	—
61	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
62	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
63	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
64	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
65	C₇₁⋊C₈	C₄⋊C₈	1	2	yes	—
66	C₇₁⋊C₈	#107	1	2	yes	—
67	C₂²×C₇₁	C₂×#108	1	1	yes	characteristic
68	C₂×D₇₁	—	4	1	no	—
69	C₂×D₇₁	#335	1	2	yes	—
70	C₄×C₇₁	—	2	2	no	—
71	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
72	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
73	Dic₇₁	—	4	1	no	—
74	Dic₇₁	#335	1	2	yes	—
75	#12011	—	71	1	no	maximal, Sylow
76	C₂×C₇₁	—	2	1	no	—
77	C₂×C₇₁	C₂×#335	1	1	yes	characteristic
78	D₇₁	—	2	2	no	—
79	C₂×#364	—	71	2	no	—
80	C₄×M₃₂	—	71	1	no	—
81	#1373	—	71	2	no	—
82	#1373	—	71	1	no	—
83	#1373	D₇₁	1	1	yes	characteristic, complemented
84	C₇₁	#12011	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₃₂	—	71	1	no	—
86	C₂×M₃₂	—	71	1	no	—
87	C₂×M₃₂	—	142	2	no	—
88	C₂×C₄×C₈	—	71	1	no	—
89	#364	—	71	2	no	—
90	#364	—	142	2	no	—
91	#364	C₂×D₇₁	1	2	yes	—
92	#346	—	71	1	no	—
93	#346	—	71	1	no	—
94	C₄×C₁₆	—	71	1	no	—
95	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
96	C₂²×C₈	—	71	1	no	—
97	C₂²×C₈	—	71	1	no	—
98	C₂×C₄²	—	71	1	no	—
99	M₃₂	—	2	2	no	—
100	M₃₂	—	284	2	no	—
101	M₃₂	—	71	4	no	—
102	M₃₂	—	142	2	no	—
103	M₃₂	—	142	2	no	—
104	C₂×C₁₆	—	284	2	no	—
105	C₂×C₁₆	—	71	1	no	—
106	C₂×C₁₆	—	71	1	no	—
107	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
108	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
109	C₄×C₈	—	71	1	no	—
110	C₄×C₈	—	71	1	no	—
111	C₄×C₈	—	71	1	no	—
112	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
113	C₂²×C₄	—	71	1	no	—
114	C₂²×C₄	—	142	1	no	—
115	C₂×C₈	—	71	1	no	—
116	C₂×C₈	—	71	1	no	—
117	C₂×C₈	—	71	2	no	—
118	C₂×C₈	—	71	2	no	—
119	C₂×C₈	—	71	2	no	—
120	C₂×C₈	—	71	2	no	—
121	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
122	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
123	C₄²	—	71	1	no	—
124	C₄²	—	142	1	no	—
125	C₄²	Q₈×D₇₁	1	1	yes	characteristic
126	C₁₆	—	2	1	no	—
127	C₁₆	—	4	2	no	—
128	C₁₆	—	284	2	no	—
129	C₁₆	—	71	2	no	—
130	C₁₆	—	142	1	no	—
131	C₁₆	C₈×D₇₁	1	2	yes	—
132	C₂³	—	71	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	284	2	no	—
135	C₂×C₄	—	71	1	no	—
136	C₂×C₄	—	71	2	no	—
137	C₂×C₄	—	71	2	no	—
138	C₂×C₄	—	142	1	no	—
139	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
140	C₈	—	71	2	no	—
141	C₈	—	71	2	no	—
142	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
143	C₈	M₁₆×D₇₁	1	1	yes	characteristic
144	C₈	D₈×D₇₁	1	1	yes	characteristic
145	C₈	Q₁₆×D₇₁	1	1	yes	characteristic
146	C₂²	—	284	1	no	—
147	C₂²	—	71	2	no	—
148	C₂²	#108×D₇₁	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	284	1	no	—
151	C₄	—	71	2	no	—
152	C₄	C₄⋊C₈×D₇₁	1	1	yes	characteristic
153	C₄	#107×D₇₁	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	142	2	no	—
156	C₂	#335×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#364×D₇₁

C₂

yes

maximal

C₄×#14556067

C₂

yes

maximal, characteristic

#22586416

C₂

yes

maximal, characteristic, complemented

#22586415

C₂

yes

maximal, complemented

C₇₁×#1373

C₂

yes

maximal, characteristic, complemented

C₂×#14556067

C₄

yes

characteristic

C₂×#14556067

C₄

yes

characteristic

M₃₂×D₇₁

—

C₄×C₈×D₇₁

C₂²

yes

characteristic

#17985820

C₂²

yes

—

#17985819

—

#17985774

C₄

yes

characteristic

#17985774

C₄

yes

characteristic

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

#364×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₈×D₇₁

C₂×C₄

yes

characteristic

C₂×C₈×D₇₁

D₄

yes

characteristic

C₄²×D₇₁

Q₈

yes

—

—

—

C₈

yes

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₁₆×D₇₁

—

C₄×C₇₁⋊C₈

Q₈

yes

characteristic

C₈×Dic₇₁

C₂×C₄

yes

characteristic

C₈×Dic₇₁

D₄

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄×D₇₁

—

C₂×C₄×D₇₁

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₈×D₇₁

C₂×C₈

yes

—

C₈×D₇₁

M₁₆

yes

—

C₈×D₇₁

D₈

yes

—

C₈×D₇₁

Q₁₆

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₄⋊C₄

yes

characteristic

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

C₂×C₈

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₁₆×C₇₁

C₂×C₈

yes

—

C₂²×D₇₁

#108

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

#108

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

C₄⋊C₈

yes

—

C₄×D₇₁

#107

yes

—

C₈×C₇₁

C₂²×C₈

yes

characteristic

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₈×C₇₁

C₂×D₈

yes

characteristic

C₈×C₇₁

C₂×Q₁₆

yes

characteristic

C₇₁⋊C₈

C₄⋊C₈

yes

—

C₇₁⋊C₈

#107

yes

—

C₂²×C₇₁

C₂×#108

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

#335

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₁

C₂×#107

yes

characteristic

Dic₇₁

—

Dic₇₁

#335

yes

—

#12011

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

C₂×#335

yes

characteristic

D₇₁

—

C₂×#364

—

C₄×M₃₂

—

#1373

—

#1373

—

#1373

D₇₁

yes

characteristic, complemented

C₇₁

#12011

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

142

—

C₂×C₄×C₈

—

#364

—

#364

—

142

—

#364

C₂×D₇₁

yes

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

100

M₃₂

—

284

—

101

M₃₂

—

102

M₃₂

—

142

—

103

M₃₂

—

142

—

104

C₂×C₁₆

—

284

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

C₂×C₁₆

C₄×D₇₁

yes

characteristic

108

C₂×C₁₆

C₄×D₇₁

yes

characteristic

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

C₂²×D₇₁

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

142

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

122

C₂×C₈

D₄×D₇₁

yes

characteristic

123

C₄²

—

124

C₄²

—

142

—

125

C₄²

Q₈×D₇₁

yes

characteristic

126

C₁₆

—

127

C₁₆

—

128

C₁₆

—

284

—

129

C₁₆

—

130

C₁₆

—

142

—

131

C₁₆

C₈×D₇₁

yes

—

132

C₂³

—

133

C₂×C₄

—

134

C₂×C₄

—

284

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

142

—

139

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

C₂×C₈×D₇₁

yes

characteristic

143

C₈

M₁₆×D₇₁

yes

characteristic

144

C₈

D₈×D₇₁

yes

characteristic

145

C₈

Q₁₆×D₇₁

yes

characteristic

146

C₂²

—

284

—

147

C₂²

—

148

C₂²

#108×D₇₁

yes

characteristic

149

C₄

—

150

C₄

—

284

—

151

C₄

—

152

C₄

C₄⋊C₈×D₇₁

yes

characteristic

153

C₄

#107×D₇₁

yes

characteristic

154

C₂

—

155

C₂

—

142

—

156

C₂

#335×D₇₁

yes

characteristic

	subgroup	G/H
center	C₈	D₈×D₇₁
commutator	C₈×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1373	C₂
socle	C₂×C₇₁	C₂×#335
prime radical	C₂²×D₇₁	#108
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1373	D₇₁
hypocenter	C₇₁	#12011
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₈

D₈×D₇₁

commutator

C₈×C₇₁

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1373

C₂

C₂×C₇₁

C₂²×D₇₁

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
3. #22586416	C₂	semidirect
4. #22586415	C₂	semidirect
5. C₇₁×#1373	C₂	semidirect
83. #1373	D₇₁	semidirect
84. C₇₁	#12011	semidirect

H ≅ G/N

type

3. #22586416

C₂

semidirect

4. #22586415

C₂

semidirect

5. C₇₁×#1373

C₂

semidirect

83. #1373

D₇₁

semidirect

84. C₇₁

#12011

semidirect

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

subgroup	index
#364×D₇₁	2
C₄×#14556067	2
#22586416	2
#22586415	2
C₇₁×#1373	2
#12011	71

subgroup

index

#364×D₇₁

C₇₁×#1373

Subgroups — #31909046

Conjugacy classes of subgroups

Complements

Maximal subgroups