Group #31911698 — subgroups

Subgroups — #31911698

Order 18176.

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

normal subgroups only

This group has 214 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	#22590593	C₂	1	1	yes	maximal, characteristic, complemented
2	#22590422	C₂	1	2	yes	maximal
3	#22590178	C₂	1	1	yes	maximal, characteristic
4	#22586378	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986442	—	2	2	no	—
7	C₂×Dic₅₆₈	—	2	1	no	—
8	C₂×Dic₅₆₈	—	2	1	no	—
9	#17986415	—	2	1	no	—
10	#17986379	C₂²	1	2	yes	—
11	#17986117	C₂²	1	1	yes	characteristic
12	#17985820	C₂²	1	1	yes	characteristic
13	#17985816	—	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	C₂×Dic₂₈₄	—	2	2	no	—
19	C₂×Dic₂₈₄	—	2	2	no	—
20	#14556134	—	2	1	no	—
21	#14556128	—	2	2	no	—
22	Dic₅₆₈	—	4	2	no	—
23	Dic₅₆₈	—	4	2	no	—
24	Dic₅₆₈	—	4	1	no	—
25	Dic₅₆₈	—	4	1	no	—
26	#14556073	—	4	2	no	—
27	#14556068	—	2	1	no	—
28	#14556067	—	4	1	no	—
29	C₁₆×D₇₁	—	4	1	no	—
30	#14556064	D₄	1	1	yes	characteristic
31	#14556056	—	2	1	no	—
32	M₃₂×C₇₁	—	2	2	no	—
33	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
35	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
36	#11765865	—	2	1	no	—
37	C₂×Dic₁₄₂	—	4	2	no	—
38	Dic₂₈₄	—	4	2	no	—
39	Dic₂₈₄	—	4	2	no	—
40	Dic₂₈₄	—	4	2	no	—
41	Dic₂₈₄	—	4	2	no	—
42	Dic₂₈₄	D₈	1	1	yes	characteristic
43	#11765854	—	2	1	no	—
44	D₅₆₈	D₈	1	1	yes	characteristic
45	#11765849	—	2	1	no	—
46	#11765848	—	4	1	no	—
47	C₈×D₇₁	—	4	1	no	—
48	C₄⋊Dic₇₁	—	8	2	no	—
49	C₇₁⋊C₁₆	—	4	1	no	—
50	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₇₁	#41	1	1	yes	characteristic
53	C₁₆×C₇₁	—	2	1	no	—
54	C₁₆×C₇₁	—	4	2	no	—
55	C₁₆×C₇₁	D₈	1	1	yes	characteristic
56	C₁₆×C₇₁	D₈	1	1	yes	characteristic
57	C₂×C₄×C₇₁	—	2	1	no	—
58	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
59	C₂²⋊D₇₁	—	4	1	no	—
60	C₂×Dic₇₁	—	8	2	no	—
61	D₂₈₄	—	2	1	no	—
62	C₄×D₇₁	—	4	1	no	—
63	Dic₁₄₂	—	2	1	no	—
64	Dic₁₄₂	—	8	2	no	—
65	Dic₁₄₂	—	8	2	no	—
66	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
67	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
68	C₈×C₇₁	#135	1	1	yes	characteristic
69	C₈×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₇₁⋊C₈	—	4	1	no	—
71	C₂²×C₇₁	#472	1	1	yes	characteristic
72	C₂×D₇₁	—	4	1	no	—
73	C₄×C₇₁	—	2	2	no	—
74	C₄×C₇₁	#459	1	1	yes	characteristic
75	C₄×C₇₁	#466	1	1	yes	characteristic
76	Dic₇₁	—	4	1	no	—
77	Dic₇₁	—	16	2	no	—
78	#12437	—	71	1	no	maximal, Sylow
79	C₂×C₇₁	—	2	1	no	—
80	C₂×C₇₁	#1667	1	1	yes	characteristic
81	D₇₁	—	8	1	no	—
82	#2247	—	71	1	no	—
83	#2214	—	71	2	no	—
84	#2170	—	71	1	no	—
85	#1373	D₇₁	1	1	yes	characteristic, complemented
86	#1358	—	71	2	no	—
87	C₇₁	#12437	1	1	yes	characteristic, complemented, Sylow
88	#510	—	142	2	no	—
89	C₂×Q₃₂	—	142	1	no	—
90	C₂×Q₃₂	—	142	1	no	—
91	#504	—	142	1	no	—
92	#494	—	71	2	no	—
93	#443	—	71	1	no	—
94	#364	—	71	1	no	—
95	#364	C₂×D₇₁	1	2	yes	—
96	#362	—	142	2	no	—
97	#350	—	142	1	no	—
98	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
99	#135	—	71	1	no	—
100	C₂×Q₁₆	—	142	2	no	—
101	C₂×Q₁₆	—	142	2	no	—
102	#131	—	142	1	no	—
103	C₄⋊Q₈	—	142	2	no	—
104	Q₃₂	—	284	2	no	—
105	Q₃₂	—	284	2	no	—
106	Q₃₂	—	284	1	no	—
107	Q₃₂	—	284	1	no	—
108	SD₃₂	—	284	2	no	—
109	M₃₂	—	2	2	no	—
110	M₃₂	—	284	1	no	—
111	M₃₂	—	142	1	no	—
112	C₂×C₁₆	—	284	1	no	—
113	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
114	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
115	#108	—	71	1	no	—
116	#104	—	142	1	no	—
117	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
118	#41	—	142	1	no	—
119	C₂×Q₈	—	284	2	no	—
120	Q₁₆	—	284	2	no	—
121	Q₁₆	—	284	2	no	—
122	Q₁₆	—	284	2	no	—
123	Q₁₆	—	284	2	no	—
124	Q₁₆	—	71	1	no	—
125	SD₁₆	—	142	1	no	—
126	D₈	—	71	1	no	—
127	M₁₆	—	284	1	no	—
128	M₁₆	—	142	1	no	—
129	C₂×C₈	—	284	1	no	—
130	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
131	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
132	C₄⋊C₄	—	568	2	no	—
133	C₄²	#11765866	1	1	yes	characteristic
134	C₁₆	—	2	1	no	—
135	C₁₆	—	4	2	no	—
136	C₁₆	—	284	1	no	—
137	C₁₆	D₅₆₈	1	1	yes	characteristic
138	C₁₆	D₅₆₈	1	1	yes	characteristic
139	Q₈	—	568	2	no	—
140	Q₈	—	568	2	no	—
141	Q₈	—	142	1	no	—
142	D₄	—	284	1	no	—
143	D₄	—	142	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	284	1	no	—
146	C₂×C₄	—	568	2	no	—
147	C₂×C₄	#14556103	1	1	yes	characteristic
148	C₈	—	284	1	no	—
149	C₈	D₈×D₇₁	1	1	yes	characteristic
150	C₈	C₂×D₅₆₈	1	1	yes	characteristic
151	C₈	#14556151	1	1	yes	characteristic
152	C₈	#14556156	1	1	yes	characteristic
153	C₂²	—	284	1	no	—
154	C₂²	#17986270	1	1	yes	characteristic
155	C₄	—	2	2	no	—
156	C₄	—	284	1	no	—
157	C₄	—	1136	2	no	—
158	C₄	#17986195	1	1	yes	characteristic
159	C₄	#17986236	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	568	1	no	—
162	C₂	#22587696	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590593

C₂

yes

maximal, characteristic, complemented

#22590422

C₂

yes

maximal

#22590178

C₂

yes

maximal, characteristic

#22586378

C₂

yes

maximal, complemented

C₇₁×#1373

C₂

yes

maximal, characteristic, complemented

#17986442

—

C₂×Dic₅₆₈

—

C₂×Dic₅₆₈

—

#17986415

—

#17986379

C₂²

yes

—

#17986117

C₂²

yes

characteristic

#17985820

C₂²

yes

characteristic

#17985816

—

#17985782

—

#364×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

#14556150

D₄

yes

characteristic

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

#14556134

—

#14556128

—

Dic₅₆₈

—

Dic₅₆₈

—

Dic₅₆₈

—

Dic₅₆₈

—

#14556073

—

#14556068

—

#14556067

—

C₁₆×D₇₁

—

#14556064

D₄

yes

characteristic

#14556056

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

#11765865

—

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

D₈

yes

characteristic

#11765854

—

D₅₆₈

D₈

yes

characteristic

#11765849

—

#11765848

—

C₈×D₇₁

—

C₄⋊Dic₇₁

—

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

D₈

yes

characteristic

C₁₆×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#121

yes

characteristic

C₂²⋊D₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

C₄×D₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

C₂×D₈

yes

characteristic

C₈×C₇₁

C₂×D₈

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

—

#12437

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1667

yes

characteristic

D₇₁

—

—

—

—

D₇₁

yes

characteristic, complemented

#1358

—

C₇₁

#12437

yes

characteristic, complemented, Sylow

#510

—

142

—

C₂×Q₃₂

—

142

—

C₂×Q₃₂

—

142

—

#504

—

142

—

—

—

—

C₂×D₇₁

yes

—

#362

—

142

—

#350

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#135

—

100

C₂×Q₁₆

—

142

—

101

C₂×Q₁₆

—

142

—

102

#131

—

142

—

103

C₄⋊Q₈

—

142

—

104

Q₃₂

—

284

—

105

Q₃₂

—

284

—

106

Q₃₂

—

284

—

107

Q₃₂

—

284

—

108

SD₃₂

—

284

—

109

M₃₂

—

110

M₃₂

—

284

—

111

M₃₂

—

142

—

112

C₂×C₁₆

—

284

—

113

C₂×C₁₆

D₂₈₄

yes

characteristic

114

C₂×C₁₆

D₂₈₄

yes

characteristic

115

#108

—

116

#104

—

142

—

117

C₄×C₈

C₂²×D₇₁

yes

characteristic

118

#41

—

142

—

119

C₂×Q₈

—

284

—

120

Q₁₆

—

284

—

121

Q₁₆

—

284

—

122

Q₁₆

—

284

—

123

Q₁₆

—

284

—

124

Q₁₆

—

125

SD₁₆

—

142

—

126

D₈

—

127

M₁₆

—

284

—

128

M₁₆

—

142

—

129

C₂×C₈

—

284

—

130

C₂×C₈

D₄×D₇₁

yes

characteristic

131

C₂×C₈

C₂×D₂₈₄

yes

characteristic

132

C₄⋊C₄

—

568

—

133

C₄²

#11765866

yes

characteristic

134

C₁₆

—

135

C₁₆

—

136

C₁₆

—

284

—

137

C₁₆

D₅₆₈

yes

characteristic

138

C₁₆

D₅₆₈

yes

characteristic

139

Q₈

—

568

—

140

Q₈

—

568

—

141

Q₈

—

142

—

142

D₄

—

284

—

143

D₄

—

142

—

144

C₂×C₄

—

145

C₂×C₄

—

284

—

146

C₂×C₄

—

568

—

147

C₂×C₄

#14556103

yes

characteristic

148

C₈

—

284

—

149

C₈

D₈×D₇₁

yes

characteristic

150

C₈

C₂×D₅₆₈

yes

characteristic

151

C₈

#14556151

yes

characteristic

152

C₈

#14556156

yes

characteristic

153

C₂²

—

284

—

154

C₂²

#17986270

yes

characteristic

155

C₄

—

156

C₄

—

284

—

157

C₄

—

1136

—

158

C₄

#17986195

yes

characteristic

159

C₄

#17986236

yes

characteristic

160

C₂

—

161

C₂

—

568

—

162

C₂

#22587696

yes

characteristic

	subgroup	G/H
center	C₂	#22587696
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1373	C₂
socle	C₂×C₇₁	#1667
prime radical	#14556150	D₄
hypercenter	#1373	D₇₁
hypocenter	C₇₁	#12437
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₂

#22587696

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. #22590593	C₂	semidirect
4. #22586378	C₂	semidirect
5. C₇₁×#1373	C₂	semidirect
85. #1373	D₇₁	semidirect
87. C₇₁	#12437	semidirect

H ≅ G/N

type

1. #22590593

C₂

semidirect

4. #22586378

C₂

semidirect

5. C₇₁×#1373

C₂

semidirect

85. #1373

D₇₁

semidirect

87. C₇₁

#12437

semidirect

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

subgroup	index
#22590593	2
#22590422	2
#22590178	2
#22586378	2
C₇₁×#1373	2
#12437	71

subgroup

index

C₇₁×#1373

Subgroups — #31911698

Conjugacy classes of subgroups

Complements

Maximal subgroups