Group #31909207 — subgroups

Subgroups — #31909207

Order 18176.

216 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 75 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	#22590222	C₂	1	1	yes	maximal, characteristic
2	#22590176	C₂	1	2	yes	maximal, complemented
3	#22586312	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986431	C₄	1	1	yes	characteristic
7	#17986417	—	2	2	no	—
8	#17986117	C₂²	1	2	yes	—
9	#17985829	C₄	1	1	yes	characteristic
10	#17985807	C₄	1	2	yes	—
11	#17985784	—	2	2	no	—
12	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
14	#364×C₇₁	C₂²	1	2	yes	—
15	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
16	#14556150	C₂×C₄	1	2	yes	—
17	#14556134	—	2	2	no	—
18	Dic₅₆₈	C₈	1	1	yes	characteristic, complemented
19	#14556073	C₈	1	2	yes	complemented
20	D₁₁₃₆	C₈	1	1	yes	characteristic, complemented
21	#14556067	—	4	2	no	—
22	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
23	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
24	C₁₆×D₇₁	—	4	2	no	—
25	#14556064	C₂×C₄	1	2	yes	—
26	#14556056	—	2	2	no	—
27	C₄×C₇₁⋊C₈	—	2	1	no	—
28	M₃₂×C₇₁	—	2	2	no	—
29	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	#11765865	—	2	2	no	—
33	Dic₂₈₄	C₂×C₈	1	2	yes	—
34	#11765854	—	2	2	no	—
35	D₅₆₈	C₂×C₈	1	2	yes	—
36	#11765849	—	2	2	no	—
37	#11765848	—	4	2	no	—
38	C₂×C₇₁⋊C₈	—	2	2	no	—
39	C₈×D₇₁	—	4	2	no	—
40	C₇₁⋊C₁₆	—	2	1	no	—
41	C₇₁⋊C₁₆	D₈	1	1	yes	characteristic
42	C₇₁⋊C₁₆	D₈	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₇₁	#41	1	1	yes	characteristic
46	C₁₆×C₇₁	—	2	1	no	—
47	C₁₆×C₇₁	—	4	2	no	—
48	C₁₆×C₇₁	C₂×C₈	1	1	yes	characteristic
49	C₁₆×C₇₁	C₂×C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₁	—	2	1	no	—
51	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
52	C₂²⋊D₇₁	—	4	2	no	—
53	D₂₈₄	—	2	2	no	—
54	C₄×D₇₁	—	4	2	no	—
55	Dic₁₄₂	—	2	2	no	—
56	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₇₁	#131	1	1	yes	characteristic
58	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
59	C₈×C₇₁	#135	1	1	yes	characteristic
60	C₇₁⋊C₈	—	2	4	no	—
61	C₇₁⋊C₈	—	4	2	no	—
62	C₂²×C₇₁	#443	1	1	yes	characteristic
63	C₂×D₇₁	—	4	2	no	—
64	C₄×C₇₁	—	2	2	no	—
65	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
66	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
67	Dic₇₁	—	4	2	no	—
68	#12046	—	71	1	no	maximal, Sylow
69	C₂×C₇₁	—	2	1	no	—
70	C₂×C₇₁	C₈×D₈	1	1	yes	characteristic
71	D₇₁	—	8	2	no	—
72	#2178	—	71	1	no	—
73	#2170	—	71	2	no	—
74	#1373	D₇₁	1	1	yes	characteristic, complemented
75	#1335	—	71	2	no	—
76	C₈×C₁₆	—	71	1	no	—
77	C₇₁	#12046	1	1	yes	characteristic, complemented, Sylow
78	#508	—	71	1	no	—
79	#504	—	142	2	no	—
80	#443	—	71	2	no	—
81	#368	—	71	1	no	—
82	#364	C₂×D₇₁	1	2	yes	—
83	#359	—	71	2	no	—
84	#350	—	142	2	no	—
85	C₄×C₁₆	—	71	1	no	—
86	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
87	C₈²	—	71	1	no	—
88	#135	—	71	2	no	—
89	#131	—	142	2	no	—
90	Q₃₂	—	71	1	no	—
91	SD₃₂	—	71	2	no	—
92	D₁₆	—	71	1	no	—
93	M₃₂	—	2	2	no	—
94	M₃₂	—	284	2	no	—
95	C₂×C₁₆	—	284	2	no	—
96	C₂×C₁₆	—	71	1	no	—
97	C₂×C₁₆	—	71	1	no	—
98	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
99	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
100	#108	—	71	2	no	—
101	#104	—	142	2	no	—
102	C₄×C₈	—	142	1	no	—
103	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
104	#41	—	142	2	no	—
105	Q₁₆	—	71	2	no	—
106	SD₁₆	—	142	2	no	—
107	D₈	—	71	2	no	—
108	M₁₆	—	284	2	no	—
109	M₁₆	—	142	2	no	—
110	C₂×C₈	—	284	2	no	—
111	C₂×C₈	—	142	2	no	—
112	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
113	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
114	C₄²	#11765866	1	1	yes	characteristic
115	C₁₆	—	2	1	no	—
116	C₁₆	—	4	2	no	—
117	C₁₆	—	71	1	no	—
118	C₁₆	—	71	1	no	—
119	C₁₆	—	142	1	no	—
120	C₁₆	C₈×D₇₁	1	1	yes	characteristic
121	C₁₆	C₈×D₇₁	1	1	yes	characteristic
122	Q₈	—	142	2	no	—
123	D₄	—	284	2	no	—
124	D₄	—	142	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	284	2	no	—
127	C₂×C₄	#14556090	1	1	yes	characteristic
128	C₈	—	284	2	no	—
129	C₈	—	142	4	no	—
130	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
131	C₈	#14556136	1	1	yes	characteristic
132	C₈	D₈×D₇₁	1	1	yes	characteristic
133	C₈	#14556151	1	1	yes	characteristic
134	C₂²	—	284	2	no	—
135	C₂²	#17986115	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	284	2	no	—
138	C₄	#17986064	1	1	yes	characteristic
139	C₄	#17986081	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	568	2	no	—
142	C₂	#22587134	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590222

C₂

yes

maximal, characteristic

#22590176

C₂

yes

maximal, complemented

#22586312

C₂

yes

maximal, complemented

C₁₆×C₇₁⋊C₈

C₂

yes

maximal, characteristic, complemented

C₇₁×#1373

C₂

yes

maximal, characteristic, complemented

#17986431

C₄

yes

characteristic

#17986417

—

#17986117

C₂²

yes

—

#17985829

C₄

yes

characteristic

#17985807

C₄

yes

—

#17985784

—

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₈×C₇₁⋊C₈

C₂²

yes

characteristic

#364×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

#14556150

C₂×C₄

yes

—

#14556134

—

Dic₅₆₈

C₈

yes

characteristic, complemented

#14556073

C₈

yes

complemented

D₁₁₃₆

C₈

yes

characteristic, complemented

#14556067

—

C₂×C₇₁⋊C₁₆

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

D₄

yes

characteristic

C₁₆×D₇₁

—

#14556064

C₂×C₄

yes

—

#14556056

—

C₄×C₇₁⋊C₈

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

#11765865

—

Dic₂₈₄

C₂×C₈

yes

—

#11765854

—

D₅₆₈

C₂×C₈

yes

—

#11765849

—

#11765848

—

C₂×C₇₁⋊C₈

—

C₈×D₇₁

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

D₈

yes

characteristic

C₇₁⋊C₁₆

D₈

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

#41

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₁₆×C₇₁

C₂×C₈

yes

characteristic

C₁₆×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄×D₄

yes

characteristic

C₂²⋊D₇₁

—

D₂₈₄

—

C₄×D₇₁

—

Dic₁₄₂

—

C₈×C₇₁

C₂²×C₈

yes

characteristic

C₈×C₇₁

#131

yes

characteristic

C₈×C₇₁

C₂×D₈

yes

characteristic

C₈×C₇₁

#135

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

#443

yes

characteristic

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₈×D₄

yes

characteristic

C₄×C₇₁

C₄×D₈

yes

characteristic

Dic₇₁

—

#12046

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

C₈×D₈

yes

characteristic

D₇₁

—

#2178

—

#2170

—

#1373

D₇₁

yes

characteristic, complemented

#1335

—

C₈×C₁₆

—

C₇₁

#12046

yes

characteristic, complemented, Sylow

#508

—

#504

—

142

—

#443

—

#368

—

#364

C₂×D₇₁

yes

—

#359

—

#350

—

142

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₈²

—

#135

—

#131

—

142

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

284

—

C₂×C₁₆

—

284

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₁

yes

characteristic

C₂×C₁₆

C₄×D₇₁

yes

characteristic

100

#108

—

101

#104

—

142

—

102

C₄×C₈

—

142

—

103

C₄×C₈

C₂²×D₇₁

yes

characteristic

104

#41

—

142

—

105

Q₁₆

—

106

SD₁₆

—

142

—

107

D₈

—

108

M₁₆

—

284

—

109

M₁₆

—

142

—

110

C₂×C₈

—

284

—

111

C₂×C₈

—

142

—

112

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

113

C₂×C₈

D₄×D₇₁

yes

characteristic

114

C₄²

#11765866

yes

characteristic

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

C₁₆

—

119

C₁₆

—

142

—

120

C₁₆

C₈×D₇₁

yes

characteristic

121

C₁₆

C₈×D₇₁

yes

characteristic

122

Q₈

—

142

—

123

D₄

—

284

—

124

D₄

—

142

—

125

C₂×C₄

—

126

C₂×C₄

—

284

—

127

C₂×C₄

#14556090

yes

characteristic

128

C₈

—

284

—

129

C₈

—

142

—

130

C₈

C₂×C₈×D₇₁

yes

characteristic

131

C₈

#14556136

yes

characteristic

132

C₈

D₈×D₇₁

yes

characteristic

133

C₈

#14556151

yes

characteristic

134

C₂²

—

284

—

135

C₂²

#17986115

yes

characteristic

136

C₄

—

137

C₄

—

284

—

138

C₄

#17986064

yes

characteristic

139

C₄

#17986081

yes

characteristic

140

C₂

—

141

C₂

—

568

—

142

C₂

#22587134

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	C₈×D₈
prime radical	#17986431	C₄
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1373	D₇₁
hypocenter	C₇₁	#12046
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

socle

C₂×C₇₁

C₈×D₈

prime radical

#17986431

C₄

cosocle

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
2. #22590176	C₂	semidirect
3. #22586312	C₂	semidirect
4. C₁₆×C₇₁⋊C₈	C₂	semidirect
5. C₇₁×#1373	C₂	semidirect
18. Dic₅₆₈	C₈	semidirect
19. #14556073	C₈	semidirect
20. D₁₁₃₆	C₈	semidirect
74. #1373	D₇₁	semidirect
77. C₇₁	#12046	semidirect

H ≅ G/N

type

2. #22590176

C₂

semidirect

3. #22586312

C₂

semidirect

4. C₁₆×C₇₁⋊C₈

C₂

semidirect

5. C₇₁×#1373

C₂

semidirect

18. Dic₅₆₈

C₈

semidirect

19. #14556073

C₈

semidirect

20. D₁₁₃₆

C₈

semidirect

74. #1373

D₇₁

semidirect

77. C₇₁

#12046

semidirect

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

subgroup	index
#22590222	2
#22590176	2
#22586312	2
C₁₆×C₇₁⋊C₈	2
C₇₁×#1373	2
#12046	71

subgroup

index

#22590222

#22590176

#22586312

C₁₆×C₇₁⋊C₈

C₇₁×#1373

#12046

Subgroups — #31909207

Conjugacy classes of subgroups

Complements

Maximal subgroups