Group #31918359 — subgroups

Subgroups — #31918359

Order 18176.

158 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22590617	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22586542	C₂	1	1	yes	maximal, characteristic
4	#22586539	C₂	1	1	yes	maximal, characteristic
5	Dic₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	#22586521	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1416	C₂	1	1	yes	maximal, characteristic
8	#17986400	—	2	1	no	—
9	Q₁₆×Dic₇₁	—	2	1	no	—
10	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
11	#17985826	—	2	1	no	—
12	#17985825	C₂²	1	1	yes	characteristic
13	#17985825	C₂²	1	1	yes	characteristic
14	#17985803	—	2	1	no	—
15	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
18	#366×C₇₁	C₂²	1	1	yes	characteristic
19	#14556119	—	4	1	no	—
20	Q₈×Dic₇₁	—	4	1	no	—
21	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
22	#14556062	—	2	1	no	—
23	#14556062	—	2	1	no	—
24	#14556059	—	4	1	no	—
25	#14556052	—	4	1	no	—
26	C₈×Dic₇₁	D₄	1	1	yes	characteristic
27	C₇₁⋊C₃₂	—	2	1	no	—
28	C₂×Q₁₆×C₇₁	—	2	1	no	—
29	Q₃₂×C₇₁	—	2	1	no	—
30	Q₃₂×C₇₁	Q₈	1	2	yes	—
31	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
32	#107×C₇₁	—	2	1	no	—
33	C₃₂×C₇₁	—	2	1	no	—
34	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
35	C₄⋊Dic₇₁	—	4	1	no	—
36	C₄⋊Dic₇₁	—	4	1	no	—
37	C₄⋊Dic₇₁	—	8	1	no	—
38	C₄⋊Dic₇₁	—	8	1	no	—
39	Dic₇₁⋊C₄	—	8	1	no	—
40	C₄×Dic₇₁	—	8	1	no	—
41	C₄×Dic₇₁	D₈	1	1	yes	characteristic
42	C₇₁⋊C₁₆	—	2	1	no	—
43	C₂×Q₈×C₇₁	—	4	1	no	—
44	Q₁₆×C₇₁	—	2	2	no	—
45	Q₁₆×C₇₁	—	4	1	no	—
46	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₇₁	—	4	1	no	—
48	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
49	C₁₆×C₇₁	#41	1	1	yes	characteristic
50	Q₈×C₇₁	—	4	2	no	—
51	Q₈×C₇₁	—	8	1	no	—
52	C₂×C₄×C₇₁	—	8	1	no	—
53	C₂×C₄×C₇₁	—	8	1	no	—
54	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₇₁	—	8	1	no	—
56	C₂×Dic₇₁	—	8	1	no	—
57	C₂×Dic₇₁	D₁₆	1	1	yes	characteristic
58	C₂×Dic₇₁	D₁₆	1	1	yes	characteristic
59	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
60	C₈×C₇₁	#137	1	1	yes	characteristic
61	C₇₁⋊C₈	—	2	1	no	—
62	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
63	C₄×C₇₁	—	8	2	no	—
64	C₄×C₇₁	—	16	1	no	—
65	C₄×C₇₁	#474	1	1	yes	characteristic
66	C₄×C₇₁	#510	1	1	yes	characteristic
67	Dic₇₁	—	2	1	no	—
68	Dic₇₁	—	2	1	no	—
69	Dic₇₁	—	16	1	no	—
70	Dic₇₁	—	16	1	no	—
71	#13708	—	71	1	no	maximal, Sylow
72	C₂×C₇₁	#2224	1	1	yes	characteristic
73	C₂×C₇₁	#2262	1	1	yes	characteristic
74	C₂×C₇₁	#2264	1	1	yes	characteristic
75	#2253	—	71	1	no	—
76	C₄×Q₃₂	—	71	1	no	—
77	#1424	—	71	1	no	—
78	#1423	—	71	1	no	—
79	C₄⋊C₃₂	—	71	1	no	—
80	#1416	—	71	1	no	—
81	#1416	D₇₁	1	1	yes	characteristic
82	C₇₁	#13708	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₃₂	C₂×D₇₁	1	1	yes	characteristic
84	#499	—	142	1	no	—
85	C₄×Q₁₆	—	142	1	no	—
86	C₂×C₃₂	—	71	1	no	—
87	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
88	#367	—	142	1	no	—
89	#366	—	71	1	no	—
90	#366	—	71	1	no	—
91	#366	C₂×D₇₁	1	1	yes	characteristic
92	#358	—	142	1	no	—
93	C₄×C₁₆	—	71	1	no	—
94	C₂×Q₁₆	—	2	1	no	—
95	#125	—	284	1	no	—
96	C₄×Q₈	—	284	1	no	—
97	Q₃₂	—	2	1	no	—
98	Q₃₂	Dic₁₄₂	1	2	yes	—
99	C₂×C₁₆	—	71	1	no	—
100	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
101	#107	—	2	1	no	—
102	#107	—	142	1	no	—
103	#107	—	142	1	no	—
104	#106	—	284	1	no	—
105	#103	—	284	1	no	—
106	C₄×C₈	—	71	1	no	—
107	C₃₂	—	2	1	no	—
108	C₃₂	—	142	1	no	—
109	C₂×Q₈	—	4	1	no	—
110	Q₁₆	—	2	2	no	—
111	Q₁₆	—	4	1	no	—
112	C₂×C₈	—	71	1	no	—
113	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	284	1	no	—
116	C₄⋊C₄	—	284	1	no	—
117	C₄⋊C₄	—	568	1	no	—
118	C₄⋊C₄	—	568	1	no	—
119	C₄⋊C₄	—	568	1	no	—
120	C₄²	—	568	1	no	—
121	C₄²	—	71	1	no	—
122	C₁₆	—	142	1	no	—
123	C₁₆	C₂×Dic₁₄₂	1	1	yes	characteristic
124	C₁₆	#11765864	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	Q₈	—	8	1	no	—
127	C₂×C₄	—	8	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	568	1	no	—
130	C₂×C₄	—	568	1	no	—
131	C₂×C₄	—	71	1	no	—
132	C₂×C₄	—	71	1	no	—
133	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
134	C₈	—	142	1	no	—
135	C₈	#14556105	1	1	yes	characteristic
136	C₈	#14556157	1	1	yes	characteristic
137	C₂²	D₁₆×D₇₁	1	1	yes	characteristic
138	C₄	—	8	2	no	—
139	C₄	—	16	1	no	—
140	C₄	—	1136	1	no	—
141	C₄	—	1136	1	no	—
142	C₄	—	142	1	no	—
143	C₄	—	142	1	no	—
144	C₄	#17986277	1	1	yes	characteristic
145	C₄	#17986438	1	1	yes	characteristic
146	C₂	#22590475	1	1	yes	characteristic
147	C₂	#22590648	1	1	yes	characteristic
148	C₂	#22590659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590617

C₂

yes

maximal, characteristic

Q₃₂×Dic₇₁

C₂

yes

maximal, characteristic

#22586542

C₂

yes

maximal, characteristic

#22586539

C₂

yes

maximal, characteristic

Dic₇₁⋊C₃₂

C₂

yes

maximal, characteristic

#22586521

C₂

yes

maximal, characteristic

C₇₁×#1416

C₂

yes

maximal, characteristic

#17986400

—

Q₁₆×Dic₇₁

—

C₂×C₇₁⋊C₃₂

C₂²

yes

characteristic

#17985826

—

#17985825

C₂²

yes

characteristic

#17985825

C₂²

yes

characteristic

#17985803

—

C₁₆×Dic₇₁

C₂²

yes

characteristic

C₂×Q₃₂×C₇₁

C₂²

yes

characteristic

C₂×C₃₂×C₇₁

C₂²

yes

characteristic

#366×C₇₁

C₂²

yes

characteristic

#14556119

—

Q₈×Dic₇₁

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

—

—

—

C₈×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₃₂

—

C₂×Q₁₆×C₇₁

—

Q₃₂×C₇₁

—

Q₃₂×C₇₁

Q₈

yes

—

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

#107×C₇₁

—

C₃₂×C₇₁

—

C₂×C₇₁⋊C₈

D₈

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

D₈

yes

characteristic

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

C₂×Q₈

yes

characteristic

C₁₆×C₇₁

#41

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

D₁₆

yes

characteristic

C₂×Dic₇₁

D₁₆

yes

characteristic

C₈×C₇₁

C₂²⋊Q₈

yes

characteristic

C₈×C₇₁

#137

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×D₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₄×Q₃₂

—

#1424

—

#1423

—

C₄⋊C₃₂

—

#1416

—

#1416

D₇₁

yes

characteristic

C₇₁

#13708

yes

characteristic, complemented, Sylow

C₂×Q₃₂

C₂×D₇₁

yes

characteristic

#499

—

142

—

C₄×Q₁₆

—

142

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₁

yes

characteristic

#367

—

142

—

#366

—

#366

—

#366

C₂×D₇₁

yes

characteristic

#358

—

142

—

C₄×C₁₆

—

C₂×Q₁₆

—

#125

—

284

—

C₄×Q₈

—

284

—

Q₃₂

—

Q₃₂

Dic₁₄₂

yes

—

C₂×C₁₆

—

100

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

101

#107

—

102

#107

—

142

—

103

#107

—

142

—

104

#106

—

284

—

105

#103

—

284

—

106

C₄×C₈

—

107

C₃₂

—

108

C₃₂

—

142

—

109

C₂×Q₈

—

110

Q₁₆

—

111

Q₁₆

—

112

C₂×C₈

—

113

C₂×C₈

D₄×D₇₁

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

284

—

116

C₄⋊C₄

—

284

—

117

C₄⋊C₄

—

568

—

118

C₄⋊C₄

—

568

—

119

C₄⋊C₄

—

568

—

120

C₄²

—

568

—

121

C₄²

—

122

C₁₆

—

142

—

123

C₁₆

C₂×Dic₁₄₂

yes

characteristic

124

C₁₆

#11765864

yes

characteristic

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

568

—

130

C₂×C₄

—

568

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

D₈×D₇₁

yes

characteristic

134

C₈

—

142

—

135

C₈

#14556105

yes

characteristic

136

C₈

#14556157

yes

characteristic

137

C₂²

D₁₆×D₇₁

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

1136

—

141

C₄

—

1136

—

142

C₄

—

142

—

143

C₄

—

142

—

144

C₄

#17986277

yes

characteristic

145

C₄

#17986438

yes

characteristic

146

C₂

#22590475

yes

characteristic

147

C₂

#22590648

yes

characteristic

148

C₂

#22590659

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₇₁
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#1416	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₁₆
hypercenter	#1416	D₇₁
hypocenter	C₇₁	#13708
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₆×D₇₁

commutator, cosocle

C₂×C₁₆×C₇₁

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₇₁×#1416

C₂

socle, prime radical

C₂²×C₇₁

C₂×D₁₆

D₇₁

C₇₁

G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
82. C₇₁	#13708	semidirect

H ≅ G/N

type

82. C₇₁

#13708

semidirect

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

subgroup	index
#22590617	2
Q₃₂×Dic₇₁	2
#22586542	2
#22586539	2
Dic₇₁⋊C₃₂	2
#22586521	2
C₇₁×#1416	2
#13708	71

subgroup

index

Q₃₂×Dic₇₁

Dic₇₁⋊C₃₂

C₇₁×#1416

Subgroups — #31918359

Conjugacy classes of subgroups

Complements

Maximal subgroups