Group #16848852 — subgroups

Subgroups — #16848852

Order 3968.

688 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 213 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	#14520952	C₂	1	3	yes	maximal, complemented
2	#14520942	C₂	1	6	yes	maximal, complemented
3	#14520929	C₂	1	3	yes	maximal, complemented
4	#14520905	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#120	C₂	1	1	yes	maximal, characteristic
6	C₂⁵×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₃₁	C₂²	1	3	yes	—
8	C₂³×Dic₃₁	C₂²	1	1	yes	characteristic, complemented
9	C₃₁×#120	C₄	1	8	yes	complemented
10	C₂×C₂²⋊C₄×C₃₁	C₂²	1	3	yes	—
11	C₂×C₂²⋊Dic₃₁	C₂²	1	6	yes	—
12	C₂×C₂²⋊Dic₃₁	C₂²	1	6	yes	—
13	C₂×C₂²⋊Dic₃₁	C₂²	1	3	yes	complemented
14	C₂×C₄⋊Dic₃₁	C₂²	1	3	yes	complemented
15	C₂×C₄×Dic₃₁	C₂²	1	3	yes	complemented
16	C₃₁⋊#95	C₂²	1	6	yes	complemented
17	C₂⁴×C₃₁	—	4	3	no	—
18	C₂⁴×C₃₁	—	4	3	no	—
19	C₂⁴×C₃₁	C₂×C₄	1	4	yes	complemented
20	C₂⁴×C₃₁	C₂³	1	3	yes	—
21	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
22	C₂×D₄×C₃₁	—	4	3	no	—
23	C₂×D₄×C₃₁	—	4	3	no	—
24	C₂×D₄×C₃₁	C₂×C₄	1	12	yes	—
25	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
26	C₂²×Dic₃₁	—	4	3	no	—
27	C₂²×Dic₃₁	C₂³	1	2	yes	—
28	C₂²×Dic₃₁	C₂³	1	6	yes	—
29	C₂²⋊C₄×C₃₁	C₂×C₄	1	12	yes	complemented
30	C₂²⋊Dic₃₁	—	4	6	no	—
31	C₂²⋊Dic₃₁	—	4	6	no	—
32	C₂²⋊Dic₃₁	—	4	3	no	—
33	C₄⋊Dic₃₁	—	4	3	no	—
34	C₄×Dic₃₁	—	4	3	no	—
35	C₂³×C₃₁	—	2	3	no	—
36	C₂³×C₃₁	—	2	6	no	—
37	C₂³×C₃₁	—	2	6	no	—
38	C₂³×C₃₁	—	2	3	no	—
39	C₂³×C₃₁	—	2	3	no	—
40	C₂³×C₃₁	—	2	3	no	—
41	C₂³×C₃₁	—	4	12	no	—
42	C₂³×C₃₁	—	4	12	no	—
43	C₂³×C₃₁	—	4	4	no	—
44	C₂³×C₃₁	C₂²×C₄	1	6	yes	—
45	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
46	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
47	D₄×C₃₁	—	4	12	no	—
48	C₂×C₄×C₃₁	—	2	3	no	—
49	C₂×C₄×C₃₁	—	2	3	no	—
50	C₂×C₄×C₃₁	C₂²×C₄	1	6	yes	—
51	C₂×Dic₃₁	—	2	6	no	—
52	C₂×Dic₃₁	—	2	6	no	—
53	C₂×Dic₃₁	—	2	6	no	—
54	C₂×Dic₃₁	—	2	6	no	—
55	C₂×Dic₃₁	—	4	4	no	—
56	#2375	—	31	1	no	maximal, Sylow
57	C₂²×C₃₁	—	2	6	no	—
58	C₂²×C₃₁	—	2	6	no	—
59	C₂²×C₃₁	—	2	12	no	—
60	C₂²×C₃₁	—	4	16	no	—
61	C₂²×C₃₁	—	4	3	no	—
62	C₂²×C₃₁	—	4	3	no	—
63	C₂²×C₃₁	—	4	6	no	—
64	C₂²×C₃₁	—	4	1	no	—
65	C₂²×C₃₁	—	4	3	no	—
66	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
67	C₂²×C₃₁	H(5,𝔽₂)	1	3	yes	—
68	C₂²×C₃₁	H(5,𝔽₂)	1	3	yes	—
69	C₄×C₃₁	—	2	6	no	—
70	Dic₃₁	—	4	6	no	—
71	Dic₃₁	—	4	2	no	—
72	C₂×#120	D₃₁	1	1	yes	characteristic
73	#390	—	31	3	no	—
74	#388	—	31	6	no	—
75	#386	—	31	3	no	—
76	#379	—	31	2	no	—
77	C₂×C₃₁	—	4	2	no	—
78	C₂×C₃₁	—	4	6	no	—
79	C₂×C₃₁	#518	1	3	yes	—
80	C₂×C₃₁	#560	1	3	yes	—
81	C₂×C₃₁	H(3,𝔽₄)	1	1	yes	characteristic
82	C₂⁵	C₂×D₃₁	1	1	yes	characteristic
83	C₂²×D₄	C₂×D₃₁	1	3	yes	—
84	C₂³×C₄	—	31	1	no	—
85	#120	Dic₃₁	1	8	yes	complemented
86	C₂×C₄⋊C₄	—	31	3	no	—
87	C₂×C₂²⋊C₄	—	31	6	no	—
88	C₂×C₂²⋊C₄	—	31	6	no	—
89	C₂×C₂²⋊C₄	—	31	3	no	—
90	C₂×C₂²⋊C₄	C₂×D₃₁	1	3	yes	—
91	C₂×C₄²	—	31	3	no	—
92	#95	—	31	6	no	—
93	C₃₁	#2375	1	1	yes	characteristic, complemented, Sylow
94	C₂⁴	—	4	3	no	—
95	C₂⁴	—	4	3	no	—
96	C₂⁴	C₂×Dic₃₁	1	4	yes	complemented
97	C₂⁴	C₂²×D₃₁	1	3	yes	—
98	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
99	C₂×D₄	—	4	3	no	—
100	C₂×D₄	—	4	3	no	—
101	C₂×D₄	C₂×Dic₃₁	1	12	yes	—
102	C₂²×C₄	—	31	2	no	—
103	C₂²×C₄	—	31	6	no	—
104	C₂²×C₄	—	124	3	no	—
105	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
106	C₄⋊C₄	—	124	3	no	—
107	C₂²⋊C₄	—	124	6	no	—
108	C₂²⋊C₄	—	124	3	no	—
109	C₂²⋊C₄	—	124	6	no	—
110	C₂²⋊C₄	C₂×Dic₃₁	1	12	yes	complemented
111	C₄²	—	124	3	no	—
112	C₂³	—	2	3	no	—
113	C₂³	—	2	6	no	—
114	C₂³	—	2	6	no	—
115	C₂³	—	2	3	no	—
116	C₂³	—	2	3	no	—
117	C₂³	—	2	3	no	—
118	C₂³	—	4	12	no	—
119	C₂³	—	4	12	no	—
120	C₂³	—	4	4	no	—
121	C₂³	C₂²×Dic₃₁	1	6	yes	—
122	C₂³	C₂²×Dic₃₁	1	2	yes	—
123	C₂³	C₂³×D₃₁	1	1	yes	characteristic
124	D₄	—	4	12	no	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	2	3	no	—
127	C₂×C₄	—	62	6	no	—
128	C₂×C₄	—	62	6	no	—
129	C₂×C₄	—	62	6	no	—
130	C₂×C₄	—	62	6	no	—
131	C₂×C₄	—	124	4	no	—
132	C₂×C₄	C₂²×Dic₃₁	1	6	yes	—
133	C₂²	—	2	6	no	—
134	C₂²	—	2	6	no	—
135	C₂²	—	2	12	no	—
136	C₂²	—	4	16	no	—
137	C₂²	—	4	3	no	—
138	C₂²	—	4	6	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	3	no	—
142	C₂²	#11758561	1	3	yes	—
143	C₂²	#11758561	1	3	yes	—
144	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
145	C₄	—	2	6	no	—
146	C₄	—	124	6	no	—
147	C₄	—	124	2	no	—
148	C₂	—	4	6	no	—
149	C₂	—	4	2	no	—
150	C₂	#14521524	1	3	yes	—
151	C₂	#14521788	1	3	yes	—
152	C₂	#14521792	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14520952

C₂

yes

maximal, complemented

#14520942

C₂

yes

maximal, complemented

#14520929

C₂

yes

maximal, complemented

#14520905

C₂

yes

maximal, complemented

C₂×C₃₁×#120

C₂

yes

maximal, characteristic

C₂⁵×C₃₁

C₂²

yes

characteristic

C₂²×D₄×C₃₁

C₂²

yes

—

C₂³×Dic₃₁

C₂²

yes

characteristic, complemented

C₃₁×#120

C₄

yes

complemented

C₂×C₂²⋊C₄×C₃₁

C₂²

yes

—

C₂×C₂²⋊Dic₃₁

C₂²

yes

—

C₂×C₂²⋊Dic₃₁

C₂²

yes

—

C₂×C₂²⋊Dic₃₁

C₂²

yes

complemented

C₂×C₄⋊Dic₃₁

C₂²

yes

complemented

C₂×C₄×Dic₃₁

C₂²

yes

complemented

C₃₁⋊#95

C₂²

yes

complemented

C₂⁴×C₃₁

—

C₂⁴×C₃₁

—

C₂⁴×C₃₁

C₂×C₄

yes

complemented

C₂⁴×C₃₁

C₂³

yes

—

C₂⁴×C₃₁

C₂³

yes

characteristic

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×Dic₃₁

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²⋊C₄×C₃₁

C₂×C₄

yes

complemented

C₂²⋊Dic₃₁

—

C₂²⋊Dic₃₁

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂²×C₄

yes

—

C₂³×C₃₁

C₂²×C₄

yes

—

C₂³×C₃₁

C₂⁴

yes

characteristic

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

#2375

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₂²×C₃₁

H(5,𝔽₂)

yes

—

C₂²×C₃₁

H(5,𝔽₂)

yes

—

C₄×C₃₁

—

Dic₃₁

—

Dic₃₁

—

C₂×#120

D₃₁

yes

—

—

—

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#518

yes

—

C₂×C₃₁

#560

yes

—

C₂×C₃₁

H(3,𝔽₄)

yes

characteristic

C₂⁵

C₂×D₃₁

yes

characteristic

C₂²×D₄

C₂×D₃₁

yes

—

C₂³×C₄

—

#120

Dic₃₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

—

#95

—

C₃₁

#2375

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂⁴

—

C₂⁴

C₂×Dic₃₁

yes

complemented

C₂⁴

C₂²×D₃₁

yes

—

C₂⁴

C₂²×D₃₁

yes

characteristic

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

C₂×Dic₃₁

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

124

—

105

C₂²×C₄

C₂²×D₃₁

yes

—

106

C₄⋊C₄

—

124

—

107

C₂²⋊C₄

—

124

—

108

C₂²⋊C₄

—

124

—

109

C₂²⋊C₄

—

124

—

110

C₂²⋊C₄

C₂×Dic₃₁

yes

complemented

111

C₄²

—

124

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

C₂²×Dic₃₁

yes

—

122

C₂³

C₂²×Dic₃₁

yes

—

123

C₂³

C₂³×D₃₁

yes

characteristic

124

D₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

124

—

132

C₂×C₄

C₂²×Dic₃₁

yes

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

#11758561

yes

—

143

C₂²

#11758561

yes

—

144

C₂²

C₂³×Dic₃₁

yes

characteristic

145

C₄

—

146

C₄

—

124

—

147

C₄

—

124

—

148

C₂

—

149

C₂

—

150

C₂

#14521524

yes

—

151

C₂

#14521788

yes

—

152

C₂

#14521792

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting, prime radical	C₂×C₃₁×#120	C₂
socle, cosocle	C₂³×C₃₁	C₂⁴
hypercenter	C₂×#120	D₃₁
hypocenter	C₃₁	#2375
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×#120

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator

C₂²×C₃₁

C₂³×C₄

Fitting, prime radical

C₂×C₃₁×#120

C₂

socle, cosocle

C₂³×C₃₁

C₂⁴

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ C₂×#120

N	H ≅ G/N	type
1. #14520952	C₂	semidirect
2. #14520942	C₂	semidirect
3. #14520929	C₂	semidirect
4. #14520905	C₂	semidirect
8. C₂³×Dic₃₁	C₂²	semidirect
9. C₃₁×#120	C₄	semidirect
13. C₂×C₂²⋊Dic₃₁	C₂²	semidirect
14. C₂×C₄⋊Dic₃₁	C₂²	semidirect
15. C₂×C₄×Dic₃₁	C₂²	semidirect
16. C₃₁⋊#95	C₂²	semidirect
19. C₂⁴×C₃₁	C₂×C₄	semidirect
29. C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
85. #120	Dic₃₁	semidirect
93. C₃₁	#2375	semidirect
96. C₂⁴	C₂×Dic₃₁	semidirect
110. C₂²⋊C₄	C₂×Dic₃₁	semidirect

H ≅ G/N

type

1. #14520952

C₂

semidirect

2. #14520942

C₂

semidirect

3. #14520929

C₂

semidirect

4. #14520905

C₂

semidirect

8. C₂³×Dic₃₁

C₂²

semidirect

9. C₃₁×#120

C₄

semidirect

13. C₂×C₂²⋊Dic₃₁

C₂²

semidirect

14. C₂×C₄⋊Dic₃₁

C₂²

semidirect

15. C₂×C₄×Dic₃₁

C₂²

semidirect

16. C₃₁⋊#95

C₂²

semidirect

19. C₂⁴×C₃₁

C₂×C₄

semidirect

29. C₂²⋊C₄×C₃₁

C₂×C₄

semidirect

85. #120

Dic₃₁

semidirect

93. C₃₁

#2375

semidirect

96. C₂⁴

C₂×Dic₃₁

semidirect

110. C₂²⋊C₄

C₂×Dic₃₁

semidirect

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

subgroup	index
#14520952	2
#14520942	2
#14520929	2
#14520905	2
C₂×C₃₁×#120	2
#2375	31

subgroup

index

C₂×C₃₁×#120

Subgroups — #16848852

Conjugacy classes of subgroups

Complements

Maximal subgroups