Group #31911851 — subgroups

Subgroups — #31911851

Order 18176.

172 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 59 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	#22590619	C₂	1	1	yes	maximal, characteristic, complemented
2	#22590517	C₂	1	2	yes	maximal
3	#22590178	C₂	1	1	yes	maximal, characteristic
4	#22586400	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986415	—	2	1	no	—
7	#17986401	C₂²	1	2	yes	—
8	#17986117	C₂²	1	1	yes	characteristic
9	#17985827	—	2	1	no	—
10	#17985825	—	2	1	no	—
11	#17985822	—	2	2	no	—
12	#17985820	C₂²	1	1	yes	characteristic
13	#17985810	—	2	2	no	—
14	#17985782	—	2	1	no	—
15	#364×C₇₁	C₂²	1	2	yes	—
16	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	#14556150	D₄	1	1	yes	characteristic
18	#14556134	—	2	1	no	—
19	#14556120	—	2	2	no	—
20	#14556068	—	2	1	no	—
21	#14556067	—	4	1	no	—
22	C₁₆×D₇₁	—	4	1	no	—
23	#14556064	D₄	1	1	yes	characteristic
24	#14556062	—	2	2	no	—
25	#14556060	—	2	2	no	—
26	#14556056	—	2	1	no	—
27	M₃₂×C₇₁	—	2	2	no	—
28	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	#11765865	—	2	1	no	—
32	Dic₂₈₄	Q₁₆	1	1	yes	characteristic
33	#11765854	—	2	1	no	—
34	D₅₆₈	Q₁₆	1	1	yes	characteristic
35	#11765849	—	2	1	no	—
36	#11765848	—	4	1	no	—
37	C₈×D₇₁	—	4	1	no	—
38	C₄⋊Dic₇₁	—	4	2	no	—
39	Dic₇₁⋊C₄	—	8	2	no	—
40	C₇₁⋊C₁₆	—	4	1	no	—
41	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₇₁	#41	1	1	yes	characteristic
44	C₁₆×C₇₁	—	2	1	no	—
45	C₁₆×C₇₁	—	4	2	no	—
46	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
47	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
48	C₂×C₄×C₇₁	—	2	1	no	—
49	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
50	C₂²⋊D₇₁	—	4	1	no	—
51	C₂×Dic₇₁	—	8	2	no	—
52	D₂₈₄	—	2	1	no	—
53	C₄×D₇₁	—	4	1	no	—
54	Dic₁₄₂	—	2	1	no	—
55	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
56	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
57	C₈×C₇₁	#135	1	1	yes	characteristic
58	C₈×C₇₁	#137	1	1	yes	characteristic
59	C₇₁⋊C₈	—	4	1	no	—
60	C₂²×C₇₁	#472	1	1	yes	characteristic
61	C₂×D₇₁	—	4	1	no	—
62	C₄×C₇₁	—	4	1	no	—
63	C₄×C₇₁	#461	1	1	yes	characteristic
64	C₄×C₇₁	#467	1	1	yes	characteristic
65	Dic₇₁	—	4	1	no	—
66	Dic₇₁	—	16	2	no	—
67	#12464	—	71	1	no	maximal, Sylow
68	C₂×C₇₁	—	2	1	no	—
69	C₂×C₇₁	#1675	1	1	yes	characteristic
70	D₇₁	—	8	1	no	—
71	#2253	—	71	1	no	—
72	#2231	—	71	2	no	—
73	#2170	—	71	1	no	—
74	#1373	D₇₁	1	1	yes	characteristic, complemented
75	#1366	—	71	2	no	—
76	C₇₁	#12464	1	1	yes	characteristic, complemented, Sylow
77	#504	—	142	1	no	—
78	#499	—	71	2	no	—
79	#443	—	71	1	no	—
80	#367	—	142	1	no	—
81	#366	—	142	1	no	—
82	#365	—	142	2	no	—
83	#364	—	71	1	no	—
84	#364	C₂×D₇₁	1	2	yes	—
85	#360	—	142	2	no	—
86	#350	—	142	1	no	—
87	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
88	#135	—	71	1	no	—
89	#131	—	142	1	no	—
90	#125	—	142	2	no	—
91	M₃₂	—	2	2	no	—
92	M₃₂	—	284	1	no	—
93	M₃₂	—	142	1	no	—
94	C₂×C₁₆	—	284	1	no	—
95	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
96	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
97	#108	—	71	1	no	—
98	#107	—	142	2	no	—
99	#106	—	142	2	no	—
100	#104	—	142	1	no	—
101	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
102	#41	—	142	1	no	—
103	Q₁₆	—	71	1	no	—
104	SD₁₆	—	142	1	no	—
105	D₈	—	71	1	no	—
106	M₁₆	—	284	1	no	—
107	M₁₆	—	142	1	no	—
108	C₂×C₈	—	284	1	no	—
109	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
110	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
111	C₄⋊C₄	—	284	2	no	—
112	C₄⋊C₄	—	568	2	no	—
113	C₄²	#11765866	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	4	2	no	—
116	C₁₆	—	284	1	no	—
117	C₁₆	#11765854	1	1	yes	characteristic
118	C₁₆	#11765854	1	1	yes	characteristic
119	Q₈	—	142	1	no	—
120	D₄	—	284	1	no	—
121	D₄	—	142	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	284	1	no	—
124	C₂×C₄	—	568	2	no	—
125	C₂×C₄	#14556103	1	1	yes	characteristic
126	C₈	—	284	1	no	—
127	C₈	C₂×#11765854	1	1	yes	characteristic
128	C₈	Q₁₆×D₇₁	1	1	yes	characteristic
129	C₈	#14556147	1	1	yes	characteristic
130	C₈	#14556161	1	1	yes	characteristic
131	C₂²	—	284	1	no	—
132	C₂²	#17986270	1	1	yes	characteristic
133	C₄	—	4	1	no	—
134	C₄	—	284	1	no	—
135	C₄	—	1136	2	no	—
136	C₄	#17986205	1	1	yes	characteristic
137	C₄	#17986241	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	568	1	no	—
140	C₂	#22587740	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590619

C₂

yes

maximal, characteristic, complemented

#22590517

C₂

yes

maximal

#22590178

C₂

yes

maximal, characteristic

#22586400

C₂

yes

maximal, complemented

C₇₁×#1373

C₂

yes

maximal, characteristic, complemented

#17986415

—

#17986401

C₂²

yes

—

#17986117

C₂²

yes

—

—

—

C₂²

yes

characteristic

#17985810

—

#17985782

—

#364×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

#14556150

D₄

yes

—

—

—

—

C₁₆×D₇₁

—

#14556064

D₄

yes

—

—

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

#11765865

—

Dic₂₈₄

Q₁₆

yes

characteristic

#11765854

—

D₅₆₈

Q₁₆

yes

characteristic

#11765849

—

#11765848

—

C₈×D₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

#41

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₁₆×C₇₁

SD₁₆

yes

characteristic

C₁₆×C₇₁

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#121

yes

characteristic

C₂²⋊D₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

C₄×D₇₁

—

Dic₁₄₂

—

C₈×C₇₁

C₂×SD₁₆

yes

characteristic

C₈×C₇₁

C₂×Q₁₆

yes

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

—

C₂²×C₇₁

#472

yes

characteristic

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

#12464

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1675

yes

characteristic

D₇₁

—

—

—

—

D₇₁

yes

characteristic, complemented

#1366

—

C₇₁

#12464

yes

characteristic, complemented, Sylow

#504

—

142

—

#499

—

#443

—

#367

—

142

—

#366

—

142

—

#365

—

142

—

#364

—

#364

C₂×D₇₁

yes

—

#360

—

142

—

#350

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#135

—

#131

—

142

—

#125

—

142

—

M₃₂

—

M₃₂

—

284

—

M₃₂

—

142

—

C₂×C₁₆

—

284

—

C₂×C₁₆

D₂₈₄

yes

characteristic

C₂×C₁₆

D₂₈₄

yes

characteristic

#108

—

#107

—

142

—

#106

—

142

—

100

#104

—

142

—

101

C₄×C₈

C₂²×D₇₁

yes

characteristic

102

#41

—

142

—

103

Q₁₆

—

104

SD₁₆

—

142

—

105

D₈

—

106

M₁₆

—

284

—

107

M₁₆

—

142

—

108

C₂×C₈

—

284

—

109

C₂×C₈

D₄×D₇₁

yes

characteristic

110

C₂×C₈

C₂×D₂₈₄

yes

characteristic

111

C₄⋊C₄

—

284

—

112

C₄⋊C₄

—

568

—

113

C₄²

#11765866

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

284

—

117

C₁₆

#11765854

yes

characteristic

118

C₁₆

#11765854

yes

characteristic

119

Q₈

—

142

—

120

D₄

—

284

—

121

D₄

—

142

—

122

C₂×C₄

—

123

C₂×C₄

—

284

—

124

C₂×C₄

—

568

—

125

C₂×C₄

#14556103

yes

characteristic

126

C₈

—

284

—

127

C₈

C₂×#11765854

yes

characteristic

128

C₈

Q₁₆×D₇₁

yes

characteristic

129

C₈

#14556147

yes

characteristic

130

C₈

#14556161

yes

characteristic

131

C₂²

—

284

—

132

C₂²

#17986270

yes

characteristic

133

C₄

—

134

C₄

—

284

—

135

C₄

—

1136

—

136

C₄

#17986205

yes

characteristic

137

C₄

#17986241

yes

characteristic

138

C₂

—

139

C₂

—

568

—

140

C₂

#22587740

yes

characteristic

	subgroup	G/H
center	C₂	#22587740
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1373	C₂
socle	C₂×C₇₁	#1675
prime radical	#14556150	D₄
hypercenter	#1373	D₇₁
hypocenter	C₇₁	#12464
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₂

#22587740

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1373

C₂

C₂×C₇₁

D₄

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
1. #22590619	C₂	semidirect
4. #22586400	C₂	semidirect
5. C₇₁×#1373	C₂	semidirect
74. #1373	D₇₁	semidirect
76. C₇₁	#12464	semidirect

H ≅ G/N

type

1. #22590619

C₂

semidirect

4. #22586400

C₂

semidirect

5. C₇₁×#1373

C₂

semidirect

74. #1373

D₇₁

semidirect

76. C₇₁

#12464

semidirect

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

subgroup	index
#22590619	2
#22590517	2
#22590178	2
#22586400	2
C₇₁×#1373	2
#12464	71

subgroup

index

C₇₁×#1373

Subgroups — #31911851

Conjugacy classes of subgroups

Complements

Maximal subgroups