Group #16845802 — subgroups

Subgroups — #16845802

Order 3968.

344 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×C₄⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic
2	#14520909	C₂	1	1	yes	maximal, characteristic
3	#14520812	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₂³×Dic₃₁	—	2	1	no	—
7	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₃₁	—	2	1	no	—
9	C₂×C₄⋊Dic₃₁	—	2	1	no	—
10	C₂×C₄⋊Dic₃₁	—	2	2	no	—
11	C₂×C₄⋊Dic₃₁	C₄	1	4	yes	complemented
12	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
14	C₃₁×C₂²⋊C₈	—	2	2	no	—
15	C₃₁⋊#95	—	2	1	no	—
16	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₁	—	2	4	no	—
18	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₃₁	D₄	1	2	yes	—
21	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃₁	—	2	4	no	—
23	C₂²×Dic₃₁	—	2	2	no	—
24	C₂²×Dic₃₁	—	2	2	no	—
25	C₂²×Dic₃₁	—	4	4	no	—
26	C₂×C₈×C₃₁	—	2	2	no	—
27	C₂×C₈×C₃₁	—	2	4	no	—
28	C₂²⋊Dic₃₁	—	4	2	no	—
29	C₄⋊Dic₃₁	—	2	4	no	—
30	C₄⋊Dic₃₁	—	2	2	no	—
31	C₄⋊Dic₃₁	—	2	4	no	—
32	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
33	C₂³×C₃₁	—	2	2	no	—
34	C₂³×C₃₁	—	2	2	no	—
35	C₂³×C₃₁	C₂²⋊C₄	1	2	yes	—
36	C₂³×C₃₁	SD₁₆	1	2	yes	—
37	C₂³×C₃₁	Q₁₆	1	2	yes	—
38	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	—	2	2	no	—
40	C₂×C₄×C₃₁	—	2	4	no	—
41	C₂×C₄×C₃₁	—	2	4	no	—
42	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	#41	1	4	yes	—
46	C₂×Dic₃₁	—	2	4	no	—
47	C₂×Dic₃₁	—	4	8	no	—
48	C₂×Dic₃₁	—	4	4	no	—
49	C₂×Dic₃₁	—	4	2	no	—
50	C₂×Dic₃₁	—	4	2	no	—
51	C₂×Dic₃₁	—	4	2	no	—
52	C₈×C₃₁	—	4	2	no	—
53	#1824	—	31	1	no	maximal, Sylow
54	C₂²×C₃₁	—	2	2	no	—
55	C₂²×C₃₁	—	2	2	no	—
56	C₂²×C₃₁	—	2	4	no	—
57	C₂²×C₃₁	—	2	4	no	—
58	C₂²×C₃₁	#103	1	4	yes	—
59	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₃₁	#123	1	2	yes	—
61	C₂²×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
64	C₄×C₃₁	—	2	2	no	—
65	C₄×C₃₁	#117	1	2	yes	—
66	C₄×C₃₁	#123	1	2	yes	—
67	Dic₃₁	—	4	4	no	—
68	Dic₃₁	—	8	2	no	—
69	C₂²×C₄⋊C₄	—	31	1	no	—
70	C₂×C₂²⋊C₈	D₃₁	1	1	yes	characteristic
71	#380	—	31	1	no	—
72	#340	—	31	4	no	—
73	C₂×C₃₁	—	2	4	no	—
74	C₂×C₃₁	#381	1	1	yes	characteristic
75	C₂×C₃₁	C₂×#103	1	1	yes	characteristic
76	C₂×C₃₁	#418	1	1	yes	characteristic
77	C₂×C₃₁	#481	1	2	yes	—
78	C₂×C₃₁	#484	1	2	yes	—
79	C₂³×C₄	—	62	1	no	—
80	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₃₁	1	2	yes	—
82	C₂×C₄⋊C₄	—	31	4	no	—
83	C₂×C₄⋊C₄	—	31	2	no	—
84	C₂×C₄⋊C₄	—	31	2	no	—
85	C₂×C₄⋊C₄	—	62	1	no	—
86	C₂×C₄⋊C₄	—	62	2	no	—
87	C₂×C₂²⋊C₄	—	62	1	no	—
88	C₂²⋊C₈	—	2	2	no	—
89	#95	—	62	1	no	—
90	C₃₁	#1824	1	1	yes	characteristic, complemented, Sylow
91	C₂⁴	D₁₂₄	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	62	4	no	—
94	C₂²×C₄	—	62	2	no	—
95	C₂²×C₄	—	62	2	no	—
96	C₂²×C₄	—	124	4	no	—
97	C₂²×C₄	C₄×D₃₁	1	2	yes	—
98	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
100	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	31	4	no	—
104	C₄⋊C₄	—	62	4	no	—
105	C₄⋊C₄	—	62	2	no	—
106	C₄⋊C₄	—	62	4	no	—
107	C₂²⋊C₄	—	124	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	#92470	1	2	yes	—
111	C₂³	Dic₁₂₄	1	2	yes	—
112	C₂³	(C₂×D₃₁)⋊C₄	1	2	yes	—
113	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	62	4	no	—
118	C₂×C₄	—	124	8	no	—
119	C₂×C₄	—	124	4	no	—
120	C₂×C₄	—	124	2	no	—
121	C₂×C₄	—	124	2	no	—
122	C₂×C₄	—	124	2	no	—
123	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
125	C₂×C₄	#92497	1	4	yes	—
126	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₂²	—	2	4	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	#11758404	1	4	yes	—
133	C₂²	#11758463	1	2	yes	—
134	C₂²	C₂×#92470	1	1	yes	characteristic
135	C₂²	C₂×Dic₁₂₄	1	1	yes	characteristic
136	C₂²	#11758485	1	2	yes	—
137	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	124	4	no	—
140	C₄	—	248	2	no	—
141	C₄	#11758454	1	2	yes	—
142	C₄	#11758512	1	2	yes	—
143	C₂	—	2	4	no	—
144	C₂	#14520910	1	1	yes	characteristic
145	C₂	C₂×#11758404	1	1	yes	characteristic
146	C₂	#14521055	1	1	yes	characteristic
147	C₂	#14521368	1	2	yes	—
148	C₂	#14521385	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₃₁

C₂

yes

maximal, characteristic

#14520909

C₂

yes

maximal, characteristic

#14520812

C₂

yes

maximal, complemented

C₂×C₃₁×C₂²⋊C₈

C₂

yes

maximal, characteristic

C₂³×C₄×C₃₁

C₂²

yes

characteristic

C₂³×Dic₃₁

—

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₂²⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₄

yes

complemented

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₃₁×C₂²⋊C₈

—

C₃₁⋊#95

—

C₂⁴×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂²⋊C₄

yes

—

C₂³×C₃₁

SD₁₆

yes

—

C₂³×C₃₁

Q₁₆

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₈×C₃₁

—

#1824

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#103

yes

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

#123

yes

—

C₂²×C₃₁

C₂×SD₁₆

yes

characteristic

C₂²×C₃₁

C₂×Q₁₆

yes

characteristic

C₂²×C₃₁

AGL(1,ℤ/8)

yes

—

C₄×C₃₁

—

C₄×C₃₁

#117

yes

—

C₄×C₃₁

#123

yes

—

Dic₃₁

—

Dic₃₁

—

C₂²×C₄⋊C₄

—

C₂×C₂²⋊C₈

D₃₁

yes

characteristic

#380

—

#340

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#484

yes

—

C₂³×C₄

—

C₂³×C₄

C₂×D₃₁

yes

characteristic

C₂²×C₈

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

#95

—

C₃₁

#1824

yes

characteristic, complemented, Sylow

C₂⁴

D₁₂₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

124

—

C₂²×C₄

C₄×D₃₁

yes

—

C₂²×C₄

D₁₂₄

yes

characteristic

C₂²×C₄

C₂²⋊D₃₁

yes

—

100

C₂²×C₄

C₂²×D₃₁

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

124

—

108

C₂³

—

109

C₂³

—

110

C₂³

#92470

yes

—

111

C₂³

Dic₁₂₄

yes

—

112

C₂³

(C₂×D₃₁)⋊C₄

yes

—

113

C₂³

C₂×D₁₂₄

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

124

—

119

C₂×C₄

—

124

—

120

C₂×C₄

—

124

—

121

C₂×C₄

—

124

—

122

C₂×C₄

—

124

—

123

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

124

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

125

C₂×C₄

#92497

yes

—

126

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#11758404

yes

—

133

C₂²

#11758463

yes

—

134

C₂²

C₂×#92470

yes

characteristic

135

C₂²

C₂×Dic₁₂₄

yes

characteristic

136

C₂²

#11758485

yes

—

137

C₂²

C₂×(C₂×D₃₁)⋊C₄

yes

characteristic

138

C₄

—

139

C₄

—

124

—

140

C₄

—

248

—

141

C₄

#11758454

yes

—

142

C₄

#11758512

yes

—

143

C₂

—

144

C₂

#14520910

yes

characteristic

145

C₂

C₂×#11758404

yes

characteristic

146

C₂

#14521055

yes

characteristic

147

C₂

#14521368

yes

—

148

C₂

#14521385

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₂₄
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×C₂²⋊C₈	C₂
socle	C₂³×C₃₁	C₂×D₄
prime radical	C₂⁴×C₃₁	D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×C₂²⋊C₈	D₃₁
hypocenter	C₃₁	#1824
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₂₄

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×C₂²⋊C₈

C₂

socle

C₂³×C₃₁

C₂×D₄

prime radical

C₂⁴×C₃₁

D₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×C₂²⋊C₈

D₃₁

hypocenter

C₃₁

#1824

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. #14520812	C₂	semidirect
11. C₂×C₄⋊Dic₃₁	C₄	semidirect
90. C₃₁	#1824	semidirect

H ≅ G/N

type

3. #14520812

C₂

semidirect

11. C₂×C₄⋊Dic₃₁

C₄

semidirect

90. C₃₁

#1824

semidirect

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

subgroup	index
C₂²×C₄⋊Dic₃₁	2
#14520909	2
#14520812	2
C₂×C₃₁×C₂²⋊C₈	2
#1824	31

subgroup

index

C₂²×C₄⋊Dic₃₁

#14520909

#14520812

C₂×C₃₁×C₂²⋊C₈

#1824

Subgroups — #16845802

Conjugacy classes of subgroups

Complements

Maximal subgroups