Group #31909321 — subgroups

Subgroups — #31909321

Order 18176.

236 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22590173	C₂	1	2	yes	maximal, complemented
2	C₈×D₅₆₈	C₂	1	1	yes	maximal, characteristic
3	#22586404	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586296	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556067	—	2	2	no	—
7	C₄×D₅₆₈	C₄	1	1	yes	characteristic
8	C₈×D₂₈₄	C₂²	1	2	yes	—
9	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	—
10	(C₂×D₇₁)⋊C₁₆	—	2	2	no	—
11	#17985752	C₄	1	1	yes	characteristic
12	#17985728	C₄	1	2	yes	—
13	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₈²×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×D₅₆₈	C₈	1	1	yes	characteristic
17	C₂×C₈×D₇₁	—	2	2	no	—
18	C₄×D₂₈₄	C₂×C₄	1	2	yes	—
19	#14556067	—	4	4	no	—
20	C₂×C₇₁⋊C₁₆	—	2	2	no	—
21	#14556062	C₈	1	1	yes	characteristic
22	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
23	#14556050	C₈	1	2	yes	—
24	(C₂×D₇₁)⋊C₈	—	2	2	no	—
25	C₂×C₁₆×C₇₁	—	2	1	no	—
26	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
27	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
30	C₂×D₂₈₄	C₂×C₈	1	2	yes	—
31	C₂×C₄×D₇₁	—	2	2	no	—
32	D₅₆₈	—	2	2	no	—
33	C₂×C₇₁⋊C₈	—	2	2	no	—
34	C₈×D₇₁	—	4	2	no	—
35	C₈×D₇₁	—	4	2	no	—
36	C₄⋊Dic₇₁	C₂×C₈	1	2	yes	—
37	(C₂×D₇₁)⋊C₄	—	2	2	no	—
38	C₇₁⋊C₁₆	—	4	2	no	—
39	C₂×C₈×C₇₁	—	2	1	no	—
40	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
44	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₁₆×C₇₁	—	2	2	no	—
46	C₁₆×C₇₁	—	4	1	no	—
47	C₂²×D₇₁	—	2	2	no	—
48	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
51	C₂×Dic₇₁	—	2	2	no	—
52	D₂₈₄	—	2	2	no	—
53	D₂₈₄	—	2	2	no	—
54	C₄×D₇₁	—	4	2	no	—
55	C₄×D₇₁	—	4	2	no	—
56	C₈×C₇₁	—	2	1	no	—
57	C₈×C₇₁	—	2	1	no	—
58	C₈×C₇₁	—	4	1	no	—
59	C₈×C₇₁	M₃₂	1	2	yes	complemented
60	C₈×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
61	C₇₁⋊C₈	—	4	2	no	—
62	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
63	C₂×D₇₁	—	4	2	no	—
64	C₂×D₇₁	—	4	2	no	—
65	C₂×D₇₁	—	4	2	no	—
66	C₄×C₇₁	—	2	1	no	—
67	C₄×C₇₁	#442	1	1	yes	characteristic
68	C₄×C₇₁	#444	1	1	yes	characteristic
69	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
70	C₄×C₇₁	#504	1	1	yes	characteristic
71	Dic₇₁	—	4	2	no	—
72	#12064	—	71	1	no	maximal, Sylow
73	C₂×C₇₁	#1580	1	1	yes	characteristic
74	C₂×C₇₁	#2169	1	1	yes	characteristic
75	C₂×C₇₁	#2171	1	1	yes	characteristic
76	D₇₁	—	8	2	no	—
77	#2169	—	71	2	no	—
78	C₈×D₈	—	71	1	no	—
79	#1367	—	71	1	no	—
80	D₄⋊C₁₆	—	71	2	no	—
81	#1312	D₇₁	1	1	yes	characteristic, complemented
82	C₇₁	#12064	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₃₂	—	142	2	no	—
84	C₄×D₈	—	71	1	no	—
85	C₈×D₄	—	71	2	no	—
86	C₄⋊C₁₆	—	71	2	no	—
87	C₂²⋊C₁₆	—	142	2	no	—
88	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
89	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
90	#335	—	71	1	no	—
91	D₄⋊C₈	—	71	2	no	—
92	C₈²	C₂×D₇₁	1	1	yes	characteristic
93	C₂×D₈	—	71	1	no	—
94	C₂²×C₈	—	142	2	no	—
95	C₄×D₄	—	71	2	no	—
96	M₃₂	—	284	4	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	142	2	no	—
99	C₂×C₁₆	D₂₈₄	1	2	yes	—
100	#107	—	71	1	no	—
101	C₄⋊C₈	—	71	2	no	—
102	#102	—	71	2	no	—
103	C₂²⋊C₈	—	142	2	no	—
104	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
105	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
107	C₂×D₄	—	71	2	no	—
108	C₂²×C₄	—	142	2	no	—
109	D₈	—	142	2	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	284	2	no	—
112	C₂×C₈	—	284	2	no	—
113	C₂×C₈	—	142	2	no	—
114	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
115	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
116	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
117	C₂×C₈	#11765865	1	1	yes	characteristic
118	C₄⋊C₄	—	71	2	no	—
119	C₂²⋊C₄	—	142	2	no	—
120	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
121	C₁₆	—	2	2	no	—
122	C₁₆	—	4	1	no	—
123	C₁₆	—	284	2	no	—
124	C₂³	—	142	2	no	—
125	D₄	—	142	2	no	—
126	D₄	—	142	2	no	—
127	C₂×C₄	—	284	2	no	—
128	C₂×C₄	—	284	2	no	—
129	C₂×C₄	—	142	2	no	—
130	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
131	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
132	C₂×C₄	#14556134	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	284	2	no	—
137	C₈	#14556067	1	2	yes	complemented
138	C₈	#14556156	1	2	yes	—
139	C₂²	—	284	2	no	—
140	C₂²	—	284	2	no	—
141	C₂²	—	284	2	no	—
142	C₂²	C₈×D₂₈₄	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	284	2	no	—
145	C₄	#17986112	1	1	yes	characteristic
146	C₄	#17986122	1	1	yes	characteristic
147	C₄	C₂×#14556067	1	1	yes	characteristic
148	C₄	#17986415	1	1	yes	characteristic
149	C₂	—	568	2	no	—
150	C₂	#22587165	1	1	yes	characteristic
151	C₂	#22590173	1	1	yes	characteristic
152	C₂	#22590183	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590173

C₂

yes

maximal, complemented

C₈×D₅₆₈

C₂

yes

maximal, characteristic

#22586404

C₂

yes

maximal, characteristic, complemented

#22586296

C₂

yes

maximal, complemented

C₇₁×#1312

C₂

yes

maximal, characteristic, complemented

C₂×#14556067

—

C₄×D₅₆₈

C₄

yes

characteristic

C₈×D₂₈₄

C₂²

yes

—

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

—

(C₂×D₇₁)⋊C₁₆

—

#17985752

C₄

yes

characteristic

#17985728

C₄

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₈²×C₇₁

C₂²

yes

characteristic

C₂×D₅₆₈

C₈

yes

characteristic

C₂×C₈×D₇₁

—

C₄×D₂₈₄

C₂×C₄

yes

—

#14556067

—

C₂×C₇₁⋊C₁₆

—

#14556062

C₈

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂×C₄

yes

—

#14556050

C₈

yes

—

(C₂×D₇₁)⋊C₈

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×D₂₈₄

C₂×C₈

yes

—

C₂×C₄×D₇₁

—

D₅₆₈

—

C₂×C₇₁⋊C₈

—

C₈×D₇₁

—

C₈×D₇₁

—

C₄⋊Dic₇₁

C₂×C₈

yes

—

(C₂×D₇₁)⋊C₄

—

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₂²×D₇₁

—

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

#131

yes

characteristic

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

M₃₂

yes

complemented

C₈×C₇₁

AGL(1,ℤ/8)

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₈×D₄

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×M₃₂

yes

characteristic

C₄×C₇₁

#504

yes

characteristic

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#2169

—

C₈×D₈

—

#1367

—

D₄⋊C₁₆

—

#1312

D₇₁

yes

characteristic, complemented

C₇₁

#12064

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

142

—

C₄×D₈

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

—

D₄⋊C₈

—

C₈²

C₂×D₇₁

yes

characteristic

C₂×D₈

—

C₂²×C₈

—

142

—

C₄×D₄

—

M₃₂

—

284

—

C₂×C₁₆

—

C₂×C₁₆

—

142

—

C₂×C₁₆

D₂₈₄

yes

—

100

#107

—

101

C₄⋊C₈

—

102

#102

—

103

C₂²⋊C₈

—

142

—

104

C₄×C₈

C₄×D₇₁

yes

characteristic

105

C₄×C₈

C₄×D₇₁

yes

characteristic

106

C₄×C₈

C₂²×D₇₁

yes

characteristic

107

C₂×D₄

—

108

C₂²×C₄

—

142

—

109

D₈

—

142

—

110

C₂×C₈

—

111

C₂×C₈

—

284

—

112

C₂×C₈

—

284

—

113

C₂×C₈

—

142

—

114

C₂×C₈

C₈×D₇₁

yes

characteristic

115

C₂×C₈

C₈×D₇₁

yes

characteristic

116

C₂×C₈

C₂×D₂₈₄

yes

characteristic

117

C₂×C₈

#11765865

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

142

—

120

C₄²

C₂×C₄×D₇₁

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

284

—

124

C₂³

—

142

—

125

D₄

—

142

—

126

D₄

—

142

—

127

C₂×C₄

—

284

—

128

C₂×C₄

—

284

—

129

C₂×C₄

—

142

—

130

C₂×C₄

C₄×D₂₈₄

yes

characteristic

131

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

132

C₂×C₄

#14556134

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

284

—

137

C₈

#14556067

yes

complemented

138

C₈

#14556156

yes

—

139

C₂²

—

284

—

140

C₂²

—

284

—

141

C₂²

—

284

—

142

C₂²

C₈×D₂₈₄

yes

characteristic

143

C₄

—

144

C₄

—

284

—

145

C₄

#17986112

yes

characteristic

146

C₄

#17986122

yes

characteristic

147

C₄

C₂×#14556067

yes

characteristic

148

C₄

#17986415

yes

characteristic

149

C₂

—

568

—

150

C₂

#22587165

yes

characteristic

151

C₂

#22590173

yes

characteristic

152

C₂

#22590183

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1312	C₂
socle	C₂²×C₇₁	C₈×D₄
prime radical	C₂×D₅₆₈	C₈
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1312	D₇₁
hypocenter	C₇₁	#12064
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

subgroup

G/H

center

C₂×C₈

C₂×D₂₈₄

commutator

C₂×C₄×C₇₁

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1312

C₂

socle

C₂²×C₇₁

C₈×D₄

prime radical

C₂×D₅₆₈

C₈

cosocle

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

N	H ≅ G/N	type
1. #22590173	C₂	semidirect
3. #22586404	C₂	semidirect
4. #22586296	C₂	semidirect
5. C₇₁×#1312	C₂	semidirect
59. C₈×C₇₁	M₃₂	semidirect
81. #1312	D₇₁	semidirect
82. C₇₁	#12064	semidirect
137. C₈	#14556067	semidirect

H ≅ G/N

type

1. #22590173

C₂

semidirect

3. #22586404

C₂

semidirect

4. #22586296

C₂

semidirect

5. C₇₁×#1312

C₂

semidirect

59. C₈×C₇₁

M₃₂

semidirect

81. #1312

D₇₁

semidirect

82. C₇₁

#12064

semidirect

137. C₈

#14556067

semidirect

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

subgroup	index
#22590173	2
C₈×D₅₆₈	2
#22586404	2
#22586296	2
C₇₁×#1312	2
#12064	71

subgroup

index

C₈×D₅₆₈

C₇₁×#1312

Subgroups — #31909321

Conjugacy classes of subgroups

Complements

Maximal subgroups