Group #31918468 — subgroups

Subgroups — #31918468

Order 18176.

232 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₂×Dic₁₁₃₆	C₂	1	2	yes	maximal, complemented
2	C₂×#17985838	C₂	1	2	yes	maximal
3	#22590601	C₂	1	1	yes	maximal, characteristic
4	#22590593	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₆₈	—	2	1	no	—
7	C₂×Dic₅₆₈	C₂²	1	2	yes	—
8	C₂×Dic₅₆₈	C₂²	1	1	yes	characteristic
9	C₂×#14556073	—	2	1	no	—
10	C₂×D₁₁₃₆	C₂²	1	1	yes	characteristic
11	#17986383	—	2	1	no	—
12	#17986379	—	2	1	no	—
13	Dic₁₁₃₆	—	2	4	no	—
14	#17985838	—	2	4	no	—
15	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
16	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₈₄	—	2	2	no	—
18	C₂×Dic₂₈₄	—	2	1	no	—
19	C₂×Dic₂₈₄	—	4	1	no	—
20	C₂×#11765854	—	4	1	no	—
21	C₂×D₅₆₈	—	2	1	no	—
22	#14556128	—	4	1	no	—
23	#14556116	—	4	1	no	—
24	Dic₅₆₈	—	2	2	no	—
25	Dic₅₆₈	—	2	2	no	—
26	Dic₅₆₈	—	2	2	no	—
27	Dic₅₆₈	—	4	2	no	—
28	#14556073	—	4	2	no	—
29	D₁₁₃₆	—	2	2	no	—
30	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
33	C₃₂×C₇₁	D₄	1	4	yes	—
34	C₂×Dic₁₄₂	—	4	1	no	—
35	C₂×Dic₁₄₂	—	4	2	no	—
36	C₂×D₂₈₄	—	4	1	no	—
37	Dic₂₈₄	—	4	2	no	—
38	Dic₂₈₄	—	4	1	no	—
39	Dic₂₈₄	—	4	1	no	—
40	Dic₂₈₄	—	4	2	no	—
41	Dic₂₈₄	—	8	2	no	—
42	#11765854	—	8	2	no	—
43	D₅₆₈	—	4	1	no	—
44	D₅₆₈	—	4	1	no	—
45	C₄⋊Dic₇₁	—	8	2	no	—
46	(C₂×D₇₁)⋊C₄	—	8	2	no	—
47	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₇₁	D₈	1	1	yes	characteristic
50	C₁₆×C₇₁	D₈	1	2	yes	—
51	C₁₆×C₇₁	C₂×D₄	1	2	yes	—
52	C₂²×D₇₁	—	8	1	no	—
53	C₂×C₄×C₇₁	D₁₆	1	2	yes	—
54	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₇₁	—	8	1	no	—
56	C₂×Dic₇₁	—	8	2	no	—
57	D₂₈₄	—	8	1	no	—
58	D₂₈₄	—	8	1	no	—
59	Dic₁₄₂	—	8	1	no	—
60	Dic₁₄₂	—	8	2	no	—
61	Dic₁₄₂	—	8	1	no	—
62	Dic₁₄₂	—	8	2	no	—
63	C₈×C₇₁	D₁₆	1	2	yes	—
64	C₈×C₇₁	#127	1	1	yes	characteristic
65	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
66	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
67	C₂×D₇₁	—	16	1	no	—
68	C₂×D₇₁	—	16	2	no	—
69	C₄×C₇₁	—	2	2	no	—
70	C₄×C₇₁	#493	1	1	yes	characteristic
71	C₄×C₇₁	C₂×D₁₆	1	1	yes	characteristic
72	Dic₇₁	—	16	1	no	—
73	Dic₇₁	—	16	2	no	—
74	#13730	—	71	1	no	maximal, Sylow
75	C₂×C₇₁	#2246	1	1	yes	characteristic
76	C₂×C₇₁	#2264	1	2	yes	—
77	D₇₁	—	32	1	no	—
78	C₂×Q₆₄	—	71	2	no	—
79	C₂×SD₆₄	—	71	2	no	—
80	#2249	—	71	1	no	—
81	#2247	—	71	1	no	—
82	#1397	D₇₁	1	1	yes	characteristic, complemented
83	C₇₁	#13730	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₃₂	—	71	2	no	—
85	C₂×Q₃₂	—	71	1	no	—
86	C₂×Q₃₂	—	142	1	no	—
87	C₂×SD₃₂	—	142	1	no	—
88	C₂×D₁₆	—	71	1	no	—
89	#495	—	142	1	no	—
90	#494	—	142	1	no	—
91	Q₆₄	—	142	4	no	—
92	SD₆₄	—	142	4	no	—
93	C₂×C₃₂	C₂×D₇₁	1	2	yes	—
94	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
95	C₂×Q₁₆	—	284	1	no	—
96	C₂×Q₁₆	—	142	2	no	—
97	C₂×Q₁₆	—	142	1	no	—
98	C₂×SD₁₆	—	284	1	no	—
99	C₂×D₈	—	142	1	no	—
100	C₄⋊Q₈	—	284	1	no	—
101	#124	—	284	1	no	—
102	Q₃₂	—	284	2	no	—
103	Q₃₂	—	142	2	no	—
104	Q₃₂	—	142	2	no	—
105	Q₃₂	—	142	2	no	—
106	SD₃₂	—	284	2	no	—
107	D₁₆	—	142	2	no	—
108	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
109	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
110	C₄×C₈	D₂₈₄	1	1	yes	characteristic
111	C₃₂	D₂₈₄	1	4	yes	—
112	C₂×Q₈	—	284	1	no	—
113	C₂×Q₈	—	284	2	no	—
114	C₂×D₄	—	284	1	no	—
115	Q₁₆	—	284	2	no	—
116	Q₁₆	—	284	1	no	—
117	Q₁₆	—	284	1	no	—
118	Q₁₆	—	284	2	no	—
119	Q₁₆	—	568	2	no	—
120	SD₁₆	—	568	2	no	—
121	D₈	—	284	1	no	—
122	D₈	—	284	1	no	—
123	C₂×C₈	D₅₆₈	1	1	yes	characteristic
124	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
125	C₄⋊C₄	—	568	2	no	—
126	C₂²⋊C₄	—	568	2	no	—
127	C₄²	D₅₆₈	1	1	yes	characteristic
128	C₁₆	D₅₆₈	1	2	yes	—
129	C₁₆	C₂×D₂₈₄	1	2	yes	—
130	C₂³	—	568	1	no	—
131	Q₈	—	568	1	no	—
132	Q₈	—	568	2	no	—
133	Q₈	—	568	1	no	—
134	Q₈	—	568	2	no	—
135	D₄	—	568	1	no	—
136	D₄	—	568	1	no	—
137	C₂×C₄	—	568	1	no	—
138	C₂×C₄	—	568	2	no	—
139	C₂×C₄	D₁₁₃₆	1	2	yes	—
140	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
141	C₈	D₁₁₃₆	1	2	yes	—
142	C₈	#14556125	1	1	yes	characteristic
143	C₈	C₂×D₅₆₈	1	1	yes	characteristic
144	C₂²	—	1136	1	no	—
145	C₂²	—	1136	2	no	—
146	C₂²	C₂×D₁₁₃₆	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	1136	1	no	—
149	C₄	—	1136	2	no	—
150	C₄	#17986376	1	1	yes	characteristic
151	C₄	C₂×D₁₁₃₆	1	1	yes	characteristic
152	C₂	—	2272	1	no	—
153	C₂	#22590590	1	1	yes	characteristic
154	C₂	#22590660	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₁₃₆

C₂

yes

maximal, complemented

C₂×#17985838

C₂

yes

maximal

#22590601

C₂

yes

maximal, characteristic

#22590593

C₂

yes

maximal, characteristic, complemented

C₇₁×#1397

C₂

yes

maximal, characteristic, complemented

C₂×Dic₅₆₈

—

C₂×Dic₅₆₈

C₂²

yes

—

C₂×Dic₅₆₈

C₂²

yes

characteristic

C₂×#14556073

—

C₂×D₁₁₃₆

C₂²

yes

characteristic

#17986383

—

#17986379

—

Dic₁₁₃₆

—

#17985838

—

C₂×C₃₂×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

C₂×#11765854

—

C₂×D₅₆₈

—

#14556128

—

#14556116

—

Dic₅₆₈

—

Dic₅₆₈

—

Dic₅₆₈

—

Dic₅₆₈

—

#14556073

—

D₁₁₃₆

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₃₂×C₇₁

D₄

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×D₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

#11765854

—

D₅₆₈

—

D₅₆₈

—

C₄⋊Dic₇₁

—

(C₂×D₇₁)⋊C₄

—

C₂×C₈×C₇₁

D₈

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

D₈

yes

characteristic

C₁₆×C₇₁

D₈

yes

—

C₁₆×C₇₁

C₂×D₄

yes

—

C₂²×D₇₁

—

C₂×C₄×C₇₁

D₁₆

yes

—

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

D₁₆

yes

—

C₈×C₇₁

#127

yes

characteristic

C₈×C₇₁

C₂×D₈

yes

characteristic

C₂²×C₇₁

C₂×D₁₆

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#493

yes

characteristic

C₄×C₇₁

C₂×D₁₆

yes

characteristic

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

—

C₂×Q₆₄

—

C₂×SD₆₄

—

#2249

—

#2247

—

#1397

D₇₁

yes

characteristic, complemented

C₇₁

#13730

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

142

—

C₂×SD₃₂

—

142

—

C₂×D₁₆

—

#495

—

142

—

#494

—

142

—

Q₆₄

—

142

—

SD₆₄

—

142

—

C₂×C₃₂

C₂×D₇₁

yes

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

—

284

—

C₂×Q₁₆

—

142

—

C₂×Q₁₆

—

142

—

C₂×SD₁₆

—

284

—

C₂×D₈

—

142

—

100

C₄⋊Q₈

—

284

—

101

#124

—

284

—

102

Q₃₂

—

284

—

103

Q₃₂

—

142

—

104

Q₃₂

—

142

—

105

Q₃₂

—

142

—

106

SD₃₂

—

284

—

107

D₁₆

—

142

—

108

C₂×C₁₆

D₂₈₄

yes

characteristic

109

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

110

C₄×C₈

D₂₈₄

yes

characteristic

111

C₃₂

D₂₈₄

yes

—

112

C₂×Q₈

—

284

—

113

C₂×Q₈

—

284

—

114

C₂×D₄

—

284

—

115

Q₁₆

—

284

—

116

Q₁₆

—

284

—

117

Q₁₆

—

284

—

118

Q₁₆

—

284

—

119

Q₁₆

—

568

—

120

SD₁₆

—

568

—

121

D₈

—

284

—

122

D₈

—

284

—

123

C₂×C₈

D₅₆₈

yes

characteristic

124

C₂×C₈

C₂×D₂₈₄

yes

characteristic

125

C₄⋊C₄

—

568

—

126

C₂²⋊C₄

—

568

—

127

C₄²

D₅₆₈

yes

characteristic

128

C₁₆

D₅₆₈

yes

—

129

C₁₆

C₂×D₂₈₄

yes

—

130

C₂³

—

568

—

131

Q₈

—

568

—

132

Q₈

—

568

—

133

Q₈

—

568

—

134

Q₈

—

568

—

135

D₄

—

568

—

136

D₄

—

568

—

137

C₂×C₄

—

568

—

138

C₂×C₄

—

568

—

139

C₂×C₄

D₁₁₃₆

yes

—

140

C₂×C₄

C₂×D₅₆₈

yes

characteristic

141

C₈

D₁₁₃₆

yes

—

142

C₈

#14556125

yes

characteristic

143

C₈

C₂×D₅₆₈

yes

characteristic

144

C₂²

—

1136

—

145

C₂²

—

1136

—

146

C₂²

C₂×D₁₁₃₆

yes

characteristic

147

C₄

—

148

C₄

—

1136

—

149

C₄

—

1136

—

150

C₄

#17986376

yes

characteristic

151

C₄

C₂×D₁₁₃₆

yes

characteristic

152

C₂

—

2272

—

153

C₂

#22590590

yes

characteristic

154

C₂

#22590660

yes

—

	subgroup	G/H
center	C₂²	C₂×D₁₁₃₆
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#1397	C₂
socle	C₂²×C₇₁	C₂×D₁₆
prime radical	C₂×D₁₁₃₆	C₂²
hypercenter	#1397	D₇₁
hypocenter	C₇₁	#13730
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

subgroup

G/H

center

C₂²

C₂×D₁₁₃₆

commutator, cosocle

C₂×C₁₆×C₇₁

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₇₁×#1397

C₂

socle

C₂²×C₇₁

C₂×D₁₆

prime radical

C₂×D₁₁₃₆

C₂²

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

N	H ≅ G/N	type
1. C₂×Dic₁₁₃₆	C₂	semidirect
4. #22590593	C₂	semidirect
5. C₇₁×#1397	C₂	semidirect
82. #1397	D₇₁	semidirect
83. C₇₁	#13730	semidirect

H ≅ G/N

type

1. C₂×Dic₁₁₃₆

C₂

semidirect

4. #22590593

C₂

semidirect

5. C₇₁×#1397

C₂

semidirect

82. #1397

D₇₁

semidirect

83. C₇₁

#13730

semidirect

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

subgroup	index
C₂×Dic₁₁₃₆	2
C₂×#17985838	2
#22590601	2
#22590593	2
C₇₁×#1397	2
#13730	71

subgroup

index

C₂×Dic₁₁₃₆

C₇₁×#1397

Subgroups — #31918468

Conjugacy classes of subgroups

Complements

Maximal subgroups