Group #16845734 — subgroups

Subgroups — #16845734

Order 3968.

336 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂³×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758396	C₂	1	2	yes	maximal, complemented
3	#14520909	C₂	1	1	yes	maximal, characteristic
4	#14520811	C₂	1	2	yes	maximal, complemented
5	C₃₁×#380	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
8	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
9	C₂×C₂²⋊Dic₃₁	—	2	1	no	—
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	C₂²×C₃₁⋊C₈	—	2	4	no	—
12	C₂²×C₃₁⋊C₈	C₄	1	4	yes	complemented
13	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₃₁×#95	—	2	1	no	—
16	C₃₁⋊#95	—	2	1	no	—
17	#11758396	—	2	4	no	—
18	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	—	2	2	no	—
20	C₂²×C₄×C₃₁	—	2	2	no	—
21	C₂²×C₄×C₃₁	—	2	2	no	—
22	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₁	—	2	2	no	—
27	C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₂²⋊C₄×C₃₁	—	4	2	no	—
30	C₂²⋊Dic₃₁	—	4	2	no	—
31	C₄⋊Dic₃₁	—	2	2	no	—
32	C₄⋊Dic₃₁	—	2	2	no	—
33	C₂×C₃₁⋊C₈	—	2	2	no	—
34	C₂×C₃₁⋊C₈	—	2	8	no	—
35	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
36	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
37	C₂×C₃₁⋊C₈	D₄	1	4	yes	—
38	C₂³×C₃₁	—	2	2	no	—
39	C₂³×C₃₁	—	2	2	no	—
40	C₂³×C₃₁	C₄⋊C₄	1	2	yes	—
41	C₂³×C₃₁	SD₁₆	1	2	yes	—
42	C₂³×C₃₁	SD₁₆	1	2	yes	—
43	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₃₁	—	2	2	no	—
45	C₂×C₄×C₃₁	—	2	2	no	—
46	C₂×C₄×C₃₁	—	2	2	no	—
47	C₂×C₄×C₃₁	—	2	4	no	—
48	C₂×C₄×C₃₁	—	4	2	no	—
49	C₂×C₄×C₃₁	—	4	2	no	—
50	C₂×C₄×C₃₁	—	4	2	no	—
51	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
52	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₃₁	#41	1	2	yes	—
56	C₂×Dic₃₁	—	4	2	no	—
57	C₂×Dic₃₁	—	4	2	no	—
58	C₂×Dic₃₁	—	4	2	no	—
59	C₃₁⋊C₈	—	2	2	no	—
60	C₃₁⋊C₈	C₂²⋊C₄	1	4	yes	complemented
61	#1807	—	31	1	no	maximal, Sylow
62	C₂²×C₃₁	—	2	2	no	—
63	C₂²×C₃₁	—	2	2	no	—
64	C₂²×C₃₁	—	2	4	no	—
65	C₂²×C₃₁	—	2	4	no	—
66	C₂²×C₃₁	#106	1	4	yes	complemented
67	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₃₁	#121	1	2	yes	—
69	C₂²×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
70	C₂²×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₂²×C₃₁	#135	1	2	yes	—
72	C₄×C₃₁	—	2	2	no	—
73	C₄×C₃₁	—	8	2	no	—
74	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₄×C₃₁	#117	1	1	yes	characteristic
76	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
77	Dic₃₁	—	8	2	no	—
78	C₂³×C₈	—	31	1	no	—
79	C₂×#106	—	31	2	no	—
80	#380	—	31	1	no	—
81	#380	D₃₁	1	1	yes	characteristic
82	#340	—	31	2	no	—
83	C₂×C₃₁	—	2	4	no	—
84	C₂×C₃₁	#380	1	1	yes	characteristic
85	C₂×C₃₁	C₂×#106	1	1	yes	characteristic
86	C₂×C₃₁	#427	1	1	yes	characteristic
87	C₂×C₃₁	#465	1	2	yes	—
88	C₂×C₃₁	#465	1	2	yes	—
89	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
90	C₂²×C₈	—	31	1	no	—
91	C₂²×C₈	—	31	4	no	—
92	C₂²×C₈	—	31	1	no	—
93	C₂²×C₈	—	62	4	no	—
94	C₂×C₄⋊C₄	—	31	2	no	—
95	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
96	C₂×C₂²⋊C₄	—	2	1	no	—
97	C₂×C₂²⋊C₄	—	62	1	no	—
98	#106	—	62	4	no	—
99	#95	—	2	1	no	—
100	#95	—	62	1	no	—
101	C₃₁	#1807	1	1	yes	characteristic, complemented, Sylow
102	C₂⁴	C₂²⋊D₃₁	1	1	yes	characteristic
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	62	2	no	—
107	C₂²×C₄	Dic₆₂	1	2	yes	—
108	C₂²×C₄	C₄×D₃₁	1	2	yes	—
109	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
111	C₂×C₈	—	31	2	no	—
112	C₂×C₈	—	31	4	no	—
113	C₂×C₈	—	31	2	no	—
114	C₂×C₈	—	62	8	no	—
115	C₂×C₈	—	62	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	62	2	no	—
119	C₄⋊C₄	—	62	2	no	—
120	C₂²⋊C₄	—	4	2	no	—
121	C₂²⋊C₄	—	124	2	no	—
122	C₂³	—	2	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	Dic₃₁⋊C₄	1	2	yes	—
125	C₂³	#92480	1	2	yes	—
126	C₂³	#92481	1	2	yes	—
127	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	124	2	no	—
136	C₂×C₄	—	124	2	no	—
137	C₂×C₄	—	124	2	no	—
138	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
139	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
141	C₂×C₄	D₄×D₃₁	1	2	yes	—
142	C₂×C₄	#92497	1	2	yes	—
143	C₈	—	31	4	no	—
144	C₈	—	62	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	2	4	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	2	2	no	—
149	C₂²	#11758396	1	4	yes	complemented
150	C₂²	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
151	C₂²	C₂×#92480	1	1	yes	characteristic
152	C₂²	#11758516	1	2	yes	—
153	C₂²	C₂×#92481	1	1	yes	characteristic
154	C₂²	#11758527	1	2	yes	—
155	C₄	—	2	2	no	—
156	C₄	—	8	2	no	—
157	C₄	—	248	2	no	—
158	C₄	#11758454	1	1	yes	characteristic
159	C₄	#11758455	1	2	yes	—
160	C₄	C₂²⋊C₄×D₃₁	1	1	yes	characteristic
161	C₂	—	2	4	no	—
162	C₂	#14520907	1	1	yes	characteristic
163	C₂	C₂×#11758396	1	1	yes	characteristic
164	C₂	#14521085	1	1	yes	characteristic
165	C₂	#14521276	1	2	yes	—
166	C₂	#14521280	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₃₁⋊C₈

C₂

yes

maximal, characteristic

C₂×#11758396

C₂

yes

maximal, complemented

#14520909

C₂

yes

maximal, characteristic

#14520811

C₂

yes

maximal, complemented

C₃₁×#380

C₂

yes

maximal, characteristic

C₂³×C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₃₁

—

C₂×C₂²⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

—

C₂²×C₃₁⋊C₈

C₄

yes

complemented

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₃₁×#95

—

C₃₁⋊#95

—

#11758396

—

C₂⁴×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

Q₈

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₄⋊C₄

yes

—

C₂³×C₃₁

SD₁₆

yes

—

C₂³×C₃₁

SD₁₆

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

C₂²⋊C₄

yes

complemented

#1807

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#106

yes

complemented

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#121

yes

—

C₂²×C₃₁

C₂×SD₁₆

yes

characteristic

C₂²×C₃₁

C₂×SD₁₆

yes

characteristic

C₂²×C₃₁

#135

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₁

#117

yes

characteristic

C₄×C₃₁

C₂²⋊Q₈

yes

—

Dic₃₁

—

C₂³×C₈

—

C₂×#106

—

#380

—

#380

D₃₁

yes

characteristic

#340

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#465

yes

—

C₂³×C₄

C₂×D₃₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#106

—

#95

—

100

#95

—

101

C₃₁

#1807

yes

characteristic, complemented, Sylow

102

C₂⁴

C₂²⋊D₃₁

yes

characteristic

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

Dic₆₂

yes

—

108

C₂²×C₄

C₄×D₃₁

yes

—

109

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

110

C₂²×C₄

C₂²×D₃₁

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

124

—

122

C₂³

—

123

C₂³

—

124

C₂³

Dic₃₁⋊C₄

yes

—

125

C₂³

#92480

yes

—

126

C₂³

#92481

yes

—

127

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

124

—

136

C₂×C₄

—

124

—

137

C₂×C₄

—

124

—

138

C₂×C₄

Dic₃₁⋊C₄

yes

—

139

C₂×C₄

C₂×Dic₆₂

yes

characteristic

140

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

141

C₂×C₄

D₄×D₃₁

yes

—

142

C₂×C₄

#92497

yes

—

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#11758396

yes

complemented

150

C₂²

C₂×Dic₃₁⋊C₄

yes

characteristic

151

C₂²

C₂×#92480

yes

characteristic

152

C₂²

#11758516

yes

—

153

C₂²

C₂×#92481

yes

characteristic

154

C₂²

#11758527

yes

—

155

C₄

—

156

C₄

—

157

C₄

—

248

—

158

C₄

#11758454

yes

characteristic

159

C₄

#11758455

yes

—

160

C₄

C₂²⋊C₄×D₃₁

yes

characteristic

161

C₂

—

162

C₂

#14520907

yes

characteristic

163

C₂

C₂×#11758396

yes

characteristic

164

C₂

#14521085

yes

characteristic

165

C₂

#14521276

yes

—

166

C₂

#14521280

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#380	C₂
socle	C₂³×C₃₁	C₂×D₄
prime radical	C₂⁴×C₃₁	D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#380	D₃₁
hypocenter	C₃₁	#1807
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#380

C₂

socle

C₂³×C₃₁

C₂×D₄

prime radical

C₂⁴×C₃₁

D₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

N	H ≅ G/N	type
2. C₂×#11758396	C₂	semidirect
4. #14520811	C₂	semidirect
12. C₂²×C₃₁⋊C₈	C₄	semidirect
60. C₃₁⋊C₈	C₂²⋊C₄	semidirect
66. C₂²×C₃₁	#106	semidirect
101. C₃₁	#1807	semidirect
149. C₂²	#11758396	semidirect

H ≅ G/N

type

2. C₂×#11758396

C₂

semidirect

4. #14520811

C₂

semidirect

12. C₂²×C₃₁⋊C₈

C₄

semidirect

60. C₃₁⋊C₈

C₂²⋊C₄

semidirect

66. C₂²×C₃₁

#106

semidirect

101. C₃₁

#1807

semidirect

149. C₂²

#11758396

semidirect

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

subgroup	index
C₂³×C₃₁⋊C₈	2
C₂×#11758396	2
#14520909	2
#14520811	2
C₃₁×#380	2
#1807	31

subgroup

index

C₂³×C₃₁⋊C₈

C₃₁×#380

Subgroups — #16845734

Conjugacy classes of subgroups

Complements

Maximal subgroups