Group #31909443 — subgroups

Subgroups — #31909443

Order 18176.

210 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 85 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	#22587237	C₂	1	2	yes	maximal
2	#22586412	C₂	1	1	yes	maximal, characteristic
3	#22586304	C₂	1	2	yes	maximal
4	C₁₆×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#17986130	—	2	2	no	—
7	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
8	#17985752	C₂²	1	2	yes	—
9	Q₈⋊(C₇₁⋊C₈)	—	2	2	no	—
10	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₇₁	C₄	1	1	yes	characteristic
12	C₄×Q₁₆×C₇₁	C₂²	1	2	yes	—
13	#366×C₇₁	C₄	1	1	yes	characteristic
14	#358×C₇₁	C₄	1	2	yes	—
15	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₇₁⋊C₁₆	—	2	1	no	—
17	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
18	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
19	#14556039	—	4	2	no	—
20	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
21	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
22	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	—
23	C₄×Q₈×C₇₁	—	2	2	no	—
24	Q₃₂×C₇₁	—	2	2	no	—
25	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
27	#107×C₇₁	C₂×C₄	1	2	yes	—
28	#103×C₇₁	—	2	2	no	—
29	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
30	C₂×C₇₁⋊C₈	—	2	1	no	—
31	C₂×C₇₁⋊C₈	—	4	2	no	—
32	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
33	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
34	C₇₁⋊C₁₆	—	2	2	no	—
35	C₂×Q₈×C₇₁	—	2	2	no	—
36	Q₁₆×C₇₁	—	2	2	no	—
37	Q₁₆×C₇₁	—	2	2	no	—
38	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₇₁	—	2	2	no	—
41	C₄⋊C₄×C₇₁	—	4	2	no	—
42	C₄²×C₇₁	—	4	2	no	—
43	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₁	—	2	1	no	—
45	C₁₆×C₇₁	M₁₆	1	2	yes	—
46	Q₈×C₇₁	—	4	2	no	—
47	Q₈×C₇₁	—	4	2	no	—
48	C₂×C₄×C₇₁	—	4	2	no	—
49	C₂×C₄×C₇₁	—	4	2	no	—
50	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
53	C₈×C₇₁	—	2	1	no	—
54	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
55	C₈×C₇₁	#131	1	1	yes	characteristic
56	C₇₁⋊C₈	—	2	2	no	—
57	C₇₁⋊C₈	—	2	1	no	—
58	C₇₁⋊C₈	—	8	2	no	—
59	C₇₁⋊C₈	Q₃₂	1	2	yes	complemented
60	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
61	C₄×C₇₁	—	2	1	no	—
62	C₄×C₇₁	—	8	2	no	—
63	C₄×C₇₁	—	8	2	no	—
64	C₄×C₇₁	#436	1	1	yes	characteristic
65	C₄×C₇₁	#443	1	1	yes	characteristic
66	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
67	C₄×C₇₁	#508	1	1	yes	characteristic
68	#12083	—	71	1	no	maximal, Sylow
69	C₂×C₇₁	#1589	1	1	yes	characteristic
70	C₂×C₇₁	C₄×Q₃₂	1	1	yes	characteristic
71	C₂×C₇₁	#2178	1	1	yes	characteristic
72	C₄×Q₃₂	D₇₁	1	1	yes	characteristic
73	#1591	—	71	2	no	—
74	#1371	—	71	1	no	—
75	#1332	—	71	2	no	—
76	C₈×C₁₆	—	71	1	no	—
77	C₇₁	#12083	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₃₂	Dic₇₁	1	1	yes	characteristic
79	#446	—	142	2	no	—
80	C₄×Q₁₆	C₂×D₇₁	1	2	yes	—
81	#366	Dic₇₁	1	1	yes	characteristic
82	#358	Dic₇₁	1	2	yes	—
83	C₄×C₁₆	—	71	1	no	—
84	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
85	#335	—	71	2	no	—
86	Q₈⋊C₈	—	142	2	no	—
87	C₈²	—	71	1	no	—
88	C₂×Q₁₆	C₂×Dic₇₁	1	2	yes	—
89	C₄×Q₈	—	2	2	no	—
90	Q₃₂	—	2	2	no	—
91	C₂×C₁₆	—	142	1	no	—
92	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
93	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
94	#107	C₂×Dic₇₁	1	2	yes	—
95	C₄⋊C₈	—	284	2	no	—
96	C₄⋊C₈	—	142	2	no	—
97	#103	—	2	2	no	—
98	#97	—	284	2	no	—
99	C₄×C₈	—	71	1	no	—
100	C₄×C₈	—	71	1	no	—
101	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
102	C₂×Q₈	—	2	2	no	—
103	Q₁₆	—	2	2	no	—
104	Q₁₆	—	2	2	no	—
105	C₂×C₈	—	284	2	no	—
106	C₂×C₈	—	71	1	no	—
107	C₂×C₈	—	71	1	no	—
108	C₂×C₈	—	142	1	no	—
109	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
110	C₂×C₈	#11765864	1	1	yes	characteristic
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄²	—	4	2	no	—
114	C₄²	D₄×D₇₁	1	1	yes	characteristic
115	C₁₆	—	2	1	no	—
116	C₁₆	—	142	2	no	—
117	C₁₆	#11765849	1	2	yes	—
118	Q₈	—	4	2	no	—
119	Q₈	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
123	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
124	C₂×C₄	#14556147	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	568	2	no	—
127	C₈	—	71	2	no	—
128	C₈	—	142	2	no	—
129	C₈	—	142	1	no	—
130	C₈	C₂×#11765849	1	1	yes	characteristic
131	C₈	#14556135	1	1	yes	characteristic
132	C₂²	D₈×Dic₇₁	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	2	no	—
135	C₄	—	8	2	no	—
136	C₄	#17986077	1	1	yes	characteristic
137	C₄	#17986116	1	1	yes	characteristic
138	C₄	Q₃₂×D₇₁	1	1	yes	characteristic
139	C₄	#17986432	1	1	yes	characteristic
140	C₂	#22587225	1	1	yes	characteristic
141	C₂	Q₃₂×Dic₇₁	1	1	yes	characteristic
142	C₂	#22590221	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22587237

C₂

yes

maximal

#22586412

C₂

yes

maximal, characteristic

#22586304

C₂

yes

maximal

C₁₆×C₇₁⋊C₈

C₂

yes

maximal, characteristic

C₄×Q₃₂×C₇₁

C₂

yes

maximal, characteristic

#17986130

—

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

#17985752

C₂²

yes

—

Q₈⋊(C₇₁⋊C₈)

—

C₈×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×Q₃₂×C₇₁

C₄

yes

characteristic

C₄×Q₁₆×C₇₁

C₂²

yes

—

#366×C₇₁

C₄

yes

characteristic

#358×C₇₁

C₄

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

#14556039

—

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₂×Q₁₆×C₇₁

C₂×C₄

yes

—

C₄×Q₈×C₇₁

—

Q₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

#107×C₇₁

C₂×C₄

yes

—

#103×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₈

yes

characteristic

C₂×C₇₁⋊C₈

D₈

yes

characteristic

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

M₁₆

yes

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₈×C₇₁

#131

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

Q₃₂

yes

complemented

C₂²×C₇₁

C₄×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×Q₃₂

yes

C₄×C₇₁

yes

—

C₂×C₇₁

yes

C₂×C₇₁

C₄×Q₃₂

yes

characteristic

C₂×C₇₁

#2178

yes

characteristic

C₄×Q₃₂

D₇₁

yes

—

—

—

C₈×C₁₆

—

C₇₁

#12083

yes

characteristic, complemented, Sylow

C₂×Q₃₂

Dic₇₁

yes

characteristic

#446

—

142

—

C₄×Q₁₆

C₂×D₇₁

yes

—

#366

Dic₇₁

yes

characteristic

#358

Dic₇₁

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

—

Q₈⋊C₈

—

142

—

C₈²

—

C₂×Q₁₆

C₂×Dic₇₁

yes

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

142

—

C₂×C₁₆

C₂×Dic₇₁

yes

characteristic

C₂×C₁₆

C₂×Dic₇₁

yes

characteristic

#107

C₂×Dic₇₁

yes

—

C₄⋊C₈

—

284

—

C₄⋊C₈

—

142

—

#103

—

#97

—

284

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₂²×D₇₁

yes

characteristic

102

C₂×Q₈

—

103

Q₁₆

—

104

Q₁₆

—

105

C₂×C₈

—

284

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

142

—

109

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

110

C₂×C₈

#11765864

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

—

114

C₄²

D₄×D₇₁

yes

characteristic

115

C₁₆

—

116

C₁₆

—

142

—

117

C₁₆

#11765849

yes

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

D₄×Dic₇₁

yes

characteristic

123

C₂×C₄

D₈×D₇₁

yes

characteristic

124

C₂×C₄

#14556147

yes

characteristic

125

C₈

—

126

C₈

—

568

—

127

C₈

—

128

C₈

—

142

—

129

C₈

—

142

—

130

C₈

C₂×#11765849

yes

characteristic

131

C₈

#14556135

yes

characteristic

132

C₂²

D₈×Dic₇₁

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#17986077

yes

characteristic

137

C₄

#17986116

yes

characteristic

138

C₄

Q₃₂×D₇₁

yes

characteristic

139

C₄

#17986432

yes

characteristic

140

C₂

#22587225

yes

characteristic

141

C₂

Q₃₂×Dic₇₁

yes

characteristic

142

C₂

#22590221

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₄×Q₃₂×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₄×D₈
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₄×Q₃₂	D₇₁
hypocenter	C₇₁	#12083
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂×C₄

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₄×Q₃₂×C₇₁

C₂

socle, prime radical

C₂²×C₇₁

C₄×D₈

cosocle

C₄×C₈×C₇₁

C₂³

hypercenter

C₄×Q₃₂

D₇₁

hypocenter

C₇₁

#12083

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

N	H ≅ G/N	type
59. C₇₁⋊C₈	Q₃₂	semidirect
77. C₇₁	#12083	semidirect

H ≅ G/N

type

59. C₇₁⋊C₈

Q₃₂

semidirect

77. C₇₁

#12083

semidirect

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

subgroup	index
#22587237	2
#22586412	2
#22586304	2
C₁₆×C₇₁⋊C₈	2
C₄×Q₃₂×C₇₁	2
#12083	71

subgroup

index

#22587237

#22586412

#22586304

C₁₆×C₇₁⋊C₈

C₄×Q₃₂×C₇₁

#12083

Subgroups — #31909443

Conjugacy classes of subgroups

Complements

Maximal subgroups