Group #31917987 — subgroups

Subgroups — #31917987

Order 18176.

180 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22589935	C₂	1	1	yes	maximal, characteristic
2	Dic₇₁⋊C₃₂	C₂	1	2	yes	maximal, complemented
3	(C₂×D₇₁)⋊C₃₂	C₂	1	2	yes	maximal
4	C₄×C₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556067	C₄	1	1	yes	characteristic
7	C₂×C₁₆×D₇₁	C₂²	1	1	yes	characteristic
8	#17985953	C₄	1	1	yes	characteristic
9	C₂×C₇₁⋊C₃₂	C₂²	1	2	yes	—
10	#17985775	C₄	1	1	yes	characteristic
11	#17985774	C₄	1	1	yes	characteristic
12	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
14	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#11765848	C₈	1	1	yes	characteristic
16	C₂×C₈×D₇₁	C₂×C₄	1	1	yes	characteristic
17	#14556085	C₈	1	1	yes	characteristic
18	#14556067	—	2	2	no	—
19	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₁₆×D₇₁	—	2	2	no	—
22	#14556039	C₈	1	1	yes	characteristic
23	#14556038	C₈	1	1	yes	characteristic
24	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
25	C₇₁⋊C₃₂	—	2	2	no	—
26	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
28	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₃₂×C₇₁	—	2	2	no	—
30	C₂×C₄×D₇₁	C₂×C₈	1	1	yes	characteristic
31	#11765848	—	2	2	no	—
32	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
33	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
34	C₈×D₇₁	—	2	1	no	—
35	C₈×D₇₁	—	2	1	no	—
36	Dic₇₁⋊C₄	C₁₆	1	2	yes	—
37	(C₂×D₇₁)⋊C₄	C₁₆	1	2	yes	—
38	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
39	C₇₁⋊C₁₆	—	2	1	no	—
40	C₇₁⋊C₁₆	—	2	1	no	—
41	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
44	C₁₆×C₇₁	M₁₆	1	2	yes	—
45	C₁₆×C₇₁	#41	1	2	yes	—
46	C₂²×D₇₁	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
49	C₂×Dic₇₁	C₂×C₁₆	1	2	yes	—
50	C₂×Dic₇₁	C₂×C₁₆	1	1	yes	characteristic
51	C₄×D₇₁	—	2	1	no	—
52	C₄×D₇₁	—	2	1	no	—
53	C₈×C₇₁	M₃₂	1	2	yes	—
54	C₈×C₇₁	#117	1	1	yes	characteristic
55	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
56	C₇₁⋊C₈	—	2	1	no	—
57	C₇₁⋊C₈	—	2	1	no	—
58	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
59	C₂×D₇₁	—	2	1	no	—
60	C₂×D₇₁	—	2	2	no	—
61	C₄×C₇₁	—	2	2	no	—
62	C₄×C₇₁	#431	1	1	yes	characteristic
63	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
64	Dic₇₁	—	2	1	no	—
65	Dic₇₁	—	2	2	no	—
66	#13641	—	71	1	no	maximal, Sylow
67	C₂×C₇₁	#2162	1	1	yes	characteristic
68	C₂×C₇₁	#2258	1	2	yes	—
69	D₇₁	—	4	1	no	—
70	#2108	—	71	1	no	—
71	C₄⋊C₃₂	—	71	2	no	—
72	C₂²⋊C₃₂	—	71	2	no	—
73	#1397	D₇₁	1	1	yes	characteristic, complemented
74	C₄×C₃₂	—	71	1	no	—
75	C₇₁	#13641	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₃₂	—	71	1	no	—
77	C₂²×C₁₆	—	71	1	no	—
78	#405	—	71	1	no	—
79	C₂×C₃₂	—	71	2	no	—
80	C₂×C₃₂	C₂×D₇₁	1	2	yes	—
81	#346	—	71	1	no	—
82	#346	—	71	1	no	—
83	C₄×C₁₆	—	71	1	no	—
84	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
85	C₂×M₁₆	—	71	1	no	—
86	C₂²×C₈	—	71	1	no	—
87	#117	—	71	1	no	—
88	M₃₂	—	142	2	no	—
89	C₂×C₁₆	—	71	1	no	—
90	C₂×C₁₆	—	71	1	no	—
91	C₂×C₁₆	—	142	2	no	—
92	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
93	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
94	#97	—	71	1	no	—
95	#97	—	71	1	no	—
96	C₄×C₈	—	71	1	no	—
97	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
98	C₃₂	—	2	2	no	—
99	C₃₂	—	142	2	no	—
100	C₂²×C₄	—	71	1	no	—
101	M₁₆	—	142	2	no	—
102	C₂×C₈	—	71	1	no	—
103	C₂×C₈	—	71	1	no	—
104	C₂×C₈	—	142	1	no	—
105	C₂×C₈	—	142	1	no	—
106	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
107	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
108	C₄⋊C₄	—	71	2	no	—
109	C₂²⋊C₄	—	71	2	no	—
110	C₄²	—	71	1	no	—
111	C₄²	C₈×D₇₁	1	1	yes	characteristic
112	C₁₆	—	142	1	no	—
113	C₁₆	—	142	1	no	—
114	C₁₆	#11765848	1	2	yes	—
115	C₁₆	#11765865	1	2	yes	—
116	C₂³	—	71	1	no	—
117	C₂×C₄	—	71	2	no	—
118	C₂×C₄	—	71	1	no	—
119	C₂×C₄	—	142	1	no	—
120	C₂×C₄	—	142	1	no	—
121	C₂×C₄	C₁₆×D₇₁	1	2	yes	—
122	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
123	C₈	—	142	1	no	—
124	C₈	—	142	1	no	—
125	C₈	#14556067	1	2	yes	—
126	C₈	#14556085	1	1	yes	characteristic
127	C₈	C₂×#11765848	1	1	yes	characteristic
128	C₂²	—	142	2	no	—
129	C₂²	—	142	1	no	—
130	C₂²	C₂×C₁₆×D₇₁	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	142	2	no	—
133	C₄	—	142	1	no	—
134	C₄	#17986050	1	1	yes	characteristic
135	C₄	C₂×#14556067	1	1	yes	characteristic
136	C₂	—	284	1	no	—
137	C₂	#22590137	1	1	yes	characteristic
138	C₂	#22590635	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22589935

C₂

yes

maximal, characteristic

Dic₇₁⋊C₃₂

C₂

yes

maximal, complemented

(C₂×D₇₁)⋊C₃₂

C₂

yes

maximal

C₄×C₇₁⋊C₃₂

C₂

yes

maximal, characteristic, complemented

C₇₁×#1397

C₂

yes

maximal, characteristic, complemented

C₂×#14556067

C₄

yes

characteristic

C₂×C₁₆×D₇₁

C₂²

yes

characteristic

#17985953

C₄

yes

characteristic

C₂×C₇₁⋊C₃₂

C₂²

yes

—

#17985775

C₄

yes

characteristic

#17985774

C₄

yes

characteristic

C₁₆×Dic₇₁

C₂²

yes

characteristic

C₂×C₃₂×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#11765848

C₈

yes

characteristic

C₂×C₈×D₇₁

C₂×C₄

yes

characteristic

#14556085

C₈

yes

characteristic

#14556067

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₁₆×D₇₁

—

#14556039

C₈

yes

characteristic

#14556038

C₈

yes

characteristic

C₈×Dic₇₁

C₂×C₄

yes

characteristic

C₇₁⋊C₃₂

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₃₂×C₇₁

—

C₂×C₄×D₇₁

C₂×C₈

yes

characteristic

#11765848

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₈

yes

characteristic

C₈×D₇₁

—

C₈×D₇₁

—

Dic₇₁⋊C₄

C₁₆

yes

—

(C₂×D₇₁)⋊C₄

C₁₆

yes

—

C₄×Dic₇₁

C₂×C₈

yes

characteristic

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₁₆×C₇₁

M₁₆

yes

—

C₁₆×C₇₁

#41

yes

—

C₂²×D₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×C₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×Dic₇₁

C₂×C₁₆

yes

—

C₂×Dic₇₁

C₂×C₁₆

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

M₃₂

yes

—

C₈×C₇₁

#117

yes

characteristic

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂²×C₁₆

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#431

yes

characteristic

C₄×C₇₁

C₂×M₃₂

yes

characteristic

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

—

#2108

—

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

#1397

D₇₁

yes

characteristic, complemented

C₄×C₃₂

—

C₇₁

#13641

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₁

yes

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

142

—

C₂×C₁₆

C₄×D₇₁

yes

characteristic

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₄×D₇₁

yes

characteristic

C₃₂

—

C₃₂

—

142

—

100

C₂²×C₄

—

101

M₁₆

—

142

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

142

—

105

C₂×C₈

—

142

—

106

C₂×C₈

C₈×D₇₁

yes

characteristic

107

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

C₈×D₇₁

yes

characteristic

112

C₁₆

—

142

—

113

C₁₆

—

142

—

114

C₁₆

#11765848

yes

—

115

C₁₆

#11765865

yes

—

116

C₂³

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

142

—

120

C₂×C₄

—

142

—

121

C₂×C₄

C₁₆×D₇₁

yes

—

122

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

123

C₈

—

142

—

124

C₈

—

142

—

125

C₈

#14556067

yes

—

126

C₈

#14556085

yes

characteristic

127

C₈

C₂×#11765848

yes

characteristic

128

C₂²

—

142

—

129

C₂²

—

142

—

130

C₂²

C₂×C₁₆×D₇₁

yes

characteristic

131

C₄

—

132

C₄

—

142

—

133

C₄

—

142

—

134

C₄

#17986050

yes

characteristic

135

C₄

C₂×#14556067

yes

characteristic

136

C₂

—

284

—

137

C₂

#22590137

yes

characteristic

138

C₂

#22590635

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂²×C₁₆
Fitting	C₇₁×#1397	C₂
prime radical	C₂²×D₇₁	C₂×C₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	#1397	D₇₁
hypocenter	C₇₁	#13641
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ #1397

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇₁

commutator, socle

C₂²×C₇₁

C₂²×C₁₆

Fitting

C₇₁×#1397

C₂

prime radical

C₂²×D₇₁

C₂×C₁₆

cosocle

C₂×C₁₆×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₁₆ ◁ #1397

N	H ≅ G/N	type
2. Dic₇₁⋊C₃₂	C₂	semidirect
4. C₄×C₇₁⋊C₃₂	C₂	semidirect
5. C₇₁×#1397	C₂	semidirect
73. #1397	D₇₁	semidirect
75. C₇₁	#13641	semidirect

H ≅ G/N

type

2. Dic₇₁⋊C₃₂

C₂

semidirect

4. C₄×C₇₁⋊C₃₂

C₂

semidirect

5. C₇₁×#1397

C₂

semidirect

73. #1397

D₇₁

semidirect

75. C₇₁

#13641

semidirect

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

subgroup	index
#22589935	2
Dic₇₁⋊C₃₂	2
(C₂×D₇₁)⋊C₃₂	2
C₄×C₇₁⋊C₃₂	2
C₇₁×#1397	2
#13641	71

subgroup

index

#22589935

Dic₇₁⋊C₃₂

(C₂×D₇₁)⋊C₃₂

C₄×C₇₁⋊C₃₂

C₇₁×#1397

#13641

Subgroups — #31917987

Conjugacy classes of subgroups

Complements

Maximal subgroups