Group #31887899 — subgroups

Subgroups — #31887899

Order 18176.

330 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 139 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	#22588026	C₂	1	1	yes	maximal, characteristic
2	#22586632	C₂	1	1	yes	maximal, characteristic
3	#22586170	C₂	1	2	yes	maximal
4	#22586168	C₂	1	2	yes	maximal
5	C₇₁×#1452	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	2	yes	—
7	C₂×#14556039	C₂²	1	1	yes	characteristic
8	C₂×#14556038	C₄	1	2	yes	—
9	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
10	#17985845	C₂²	1	1	yes	characteristic
11	#17985721	C₄	1	4	yes	complemented
12	#431×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	#322×C₇₁	C₄	1	4	yes	complemented
15	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
17	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
19	#14556039	—	2	2	no	—
20	#14556038	—	2	4	no	—
21	C₄×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
22	C₈×Dic₇₁	—	2	4	no	—
23	C₇₁⋊#95	—	2	2	no	—
24	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₇₁	—	2	2	no	—
28	C₂²⋊C₈×C₇₁	—	2	2	no	—
29	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
32	C₂²×Dic₇₁	—	2	2	no	—
33	C₂×C₇₁⋊C₈	—	2	2	no	—
34	C₂×C₇₁⋊C₈	—	2	2	no	—
35	C₂×C₇₁⋊C₈	—	2	4	no	—
36	C₂×C₇₁⋊C₈	M₁₆	1	4	yes	—
37	C₄×Dic₇₁	—	2	2	no	—
38	C₄×Dic₇₁	M₁₆	1	4	yes	—
39	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	#41	1	2	yes	—
41	C₂×C₈×C₇₁	—	2	2	no	—
42	C₂×C₈×C₇₁	—	2	2	no	—
43	C₂×C₈×C₇₁	—	2	4	no	—
44	C₂×C₈×C₇₁	C₄²	1	2	yes	—
45	C₂×C₈×C₇₁	C₄²	1	2	yes	—
46	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₇₁	#41	1	2	yes	—
49	C₂³×C₇₁	#117	1	1	yes	characteristic
50	C₂×C₄×C₇₁	—	4	2	no	—
51	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₇₁	#117	1	2	yes	—
53	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
54	C₂×C₄×C₇₁	#117	1	2	yes	—
55	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
57	C₂×Dic₇₁	—	2	4	no	—
58	C₂×Dic₇₁	—	4	4	no	—
59	C₈×C₇₁	—	2	2	no	—
60	C₈×C₇₁	—	2	2	no	—
61	C₈×C₇₁	—	4	4	no	—
62	C₇₁⋊C₈	—	2	4	no	—
63	C₇₁⋊C₈	—	4	4	no	—
64	C₂²×C₇₁	—	2	2	no	—
65	C₂²×C₇₁	#376	1	1	yes	characteristic
66	C₂²×C₇₁	#432	1	2	yes	—
67	C₄×C₇₁	—	2	2	no	—
68	C₄×C₇₁	—	2	2	no	—
69	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
70	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
71	C₄×C₇₁	#432	1	2	yes	—
72	Dic₇₁	—	4	4	no	—
73	#8139	—	71	1	no	maximal, Sylow
74	C₂×C₇₁	—	2	2	no	—
75	C₂×C₇₁	#1745	1	1	yes	characteristic
76	C₂×C₇₁	#1747	1	2	yes	—
77	#1726	—	71	1	no	—
78	#1452	—	71	1	no	—
79	#1452	D₇₁	1	1	yes	characteristic
80	#1275	—	71	2	no	—
81	#1274	—	71	2	no	—
82	C₇₁	#8139	1	1	yes	characteristic, complemented, Sylow
83	#431	—	71	2	no	—
84	#431	C₂×D₇₁	1	2	yes	—
85	C₂×#97	—	71	2	no	—
86	C₂×#97	—	71	1	no	—
87	C₂×C₄×C₈	—	71	2	no	—
88	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
89	#376	—	71	1	no	—
90	#322	—	71	4	no	—
91	#322	Dic₇₁	1	4	yes	complemented
92	C₂²×C₈	—	71	2	no	—
93	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
94	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
95	C₂×C₄²	—	71	2	no	—
96	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	142	2	no	—
99	C₂²⋊C₈	—	2	2	no	—
100	C₂²⋊C₈	—	142	2	no	—
101	#97	—	142	4	no	—
102	#97	—	142	2	no	—
103	C₄×C₈	—	71	4	no	—
104	C₄×C₈	—	142	4	no	—
105	C₄×C₈	C₄×D₇₁	1	2	yes	—
106	C₄×C₈	C₂×Dic₇₁	1	4	yes	—
107	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
108	#95	—	142	2	no	—
109	C₂²×C₄	—	142	2	no	—
110	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
111	C₂²×C₄	#11765865	1	2	yes	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	71	4	no	—
116	C₂×C₈	—	142	4	no	—
117	C₂×C₈	—	142	2	no	—
118	C₂×C₈	—	142	2	no	—
119	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
120	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
121	C₄²	—	71	4	no	—
122	C₄²	—	142	2	no	—
123	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
124	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
125	C₄²	#11765865	1	2	yes	—
126	C₂³	#14556085	1	1	yes	characteristic
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	284	4	no	—
129	C₂×C₄	—	142	4	no	—
130	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
131	C₂×C₄	#14556085	1	2	yes	—
132	C₂×C₄	#14556085	1	1	yes	characteristic
133	C₂×C₄	#14556087	1	2	yes	—
134	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
135	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
136	C₈	—	2	2	no	—
137	C₈	—	2	2	no	—
138	C₈	—	4	4	no	—
139	C₈	—	284	4	no	—
140	C₈	—	142	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	#17985845	1	1	yes	characteristic
143	C₂²	#17986054	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	284	4	no	—
147	C₄	M₁₆×Dic₇₁	1	1	yes	characteristic
148	C₄	M₁₆×Dic₇₁	1	1	yes	characteristic
149	C₄	#17986054	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	#22588088	1	1	yes	characteristic
152	C₂	#22588095	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588026

C₂

yes

maximal, characteristic

#22586632

C₂

yes

maximal, characteristic

#22586170

C₂

yes

maximal

#22586168

C₂

yes

maximal

C₇₁×#1452

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

C₂×#14556039

C₂²

yes

characteristic

C₂×#14556038

C₄

yes

—

C₂×C₈×Dic₇₁

C₄

yes

—

#17985845

C₂²

yes

characteristic

#17985721

C₄

yes

complemented

#431×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#322×C₇₁

C₄

yes

complemented

C₂²×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₇₁

C₂×C₄

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

#14556039

—

#14556038

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₈×Dic₇₁

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

M₁₆

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

M₁₆

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄²

yes

—

C₂×C₈×C₇₁

C₄²

yes

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

#117

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

#117

yes

characteristic

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

#376

yes

characteristic

C₂²×C₇₁

#432

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×M₁₆

yes

characteristic

C₄×C₇₁

C₄×M₁₆

yes

characteristic

C₄×C₇₁

#432

yes

—

Dic₇₁

—

#8139

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1745

yes

characteristic

C₂×C₇₁

#1747

yes

—

#1726

—

#1452

—

#1452

D₇₁

yes

characteristic

#1275

—

#1274

—

C₇₁

#8139

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₇₁

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₁

yes

—

—

Dic₇₁

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₄×D₇₁

yes

characteristic

C₂²×C₈

C₄×D₇₁

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂²×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₂²⋊C₈

—

100

C₂²⋊C₈

—

142

—

101

#97

—

142

—

102

#97

—

142

—

103

C₄×C₈

—

104

C₄×C₈

—

142

—

105

C₄×C₈

C₄×D₇₁

yes

—

106

C₄×C₈

C₂×Dic₇₁

yes

—

107

C₄×C₈

C₂×Dic₇₁

yes

—

108

#95

—

142

—

109

C₂²×C₄

—

142

—

110

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

111

C₂²×C₄

#11765865

yes

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

142

—

117

C₂×C₈

—

142

—

118

C₂×C₈

—

142

—

119

C₂×C₈

C₄×Dic₇₁

yes

—

120

C₂×C₈

C₄×Dic₇₁

yes

—

121

C₄²

—

122

C₄²

—

142

—

123

C₄²

C₂×C₄×D₇₁

yes

characteristic

124

C₄²

C₂²×Dic₇₁

yes

characteristic

125

C₄²

#11765865

yes

—

126

C₂³

#14556085

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

284

—

129

C₂×C₄

—

142

—

130

C₂×C₄

C₂×C₄×Dic₇₁

yes

characteristic

131

C₂×C₄

#14556085

yes

—

132

C₂×C₄

#14556085

yes

characteristic

133

C₂×C₄

#14556087

yes

—

134

C₂×C₄

M₁₆×D₇₁

yes

—

135

C₂×C₄

M₁₆×D₇₁

yes

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

284

—

140

C₈

—

142

—

141

C₂²

—

142

C₂²

#17985845

yes

characteristic

143

C₂²

#17986054

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

284

—

147

C₄

M₁₆×Dic₇₁

yes

characteristic

148

C₄

M₁₆×Dic₇₁

yes

characteristic

149

C₄

#17986054

yes

—

150

C₂

—

151

C₂

#22588088

yes

characteristic

152

C₂

#22588095

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇₁
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1452	C₂
socle	C₂²×C₇₁	#376
prime radical	C₂³×C₇₁	#117
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1452	D₇₁
hypocenter	C₇₁	#8139
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #1452

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₇₁

commutator

C₂×C₄×C₇₁

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1452

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₄ ◁ #1452

N	H ≅ G/N	type
11. #17985721	C₄	semidirect
14. #322×C₇₁	C₄	semidirect
82. C₇₁	#8139	semidirect
91. #322	Dic₇₁	semidirect

H ≅ G/N

type

11. #17985721

C₄

semidirect

14. #322×C₇₁

C₄

semidirect

82. C₇₁

#8139

semidirect

91. #322

Dic₇₁

semidirect

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

subgroup	index
#22588026	2
#22586632	2
#22586170	2
#22586168	2
C₇₁×#1452	2
#8139	71

subgroup

index

C₇₁×#1452

Subgroups — #31887899

Conjugacy classes of subgroups

Complements

Maximal subgroups