Group #16846458 — subgroups

Subgroups — #16846458

Order 3968.

208 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 83 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×#11758410	C₂	1	1	yes	maximal, characteristic
2	#14521085	C₂	1	1	yes	maximal, characteristic
3	#14521077	C₂	1	1	yes	maximal, characteristic
4	#14521076	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#14520809	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#339	C₂	1	1	yes	maximal, characteristic
8	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#117	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	#11758503	C₂²	1	1	yes	characteristic
12	C₂×#92474	C₂²	1	1	yes	characteristic
13	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	#11758410	—	2	2	no	—
16	Dic₃₁⋊C₈	—	2	2	no	—
17	#11758396	C₄	1	2	yes	complemented
18	#11758395	C₄	1	2	yes	complemented
19	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
20	C₄×C₃₁⋊C₈	—	2	2	no	—
21	C₂²×C₄×C₃₁	—	2	1	no	—
22	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	M₁₆×C₃₁	—	2	1	no	—
24	M₁₆×C₃₁	—	2	1	no	—
25	C₂×C₈×C₃₁	—	2	1	no	—
26	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
27	C₂²⋊C₄×C₃₁	—	2	1	no	—
28	C₄²×C₃₁	—	2	1	no	—
29	C₄²×C₃₁	—	2	1	no	—
30	C₄²×C₃₁	Q₈	1	2	yes	—
31	C₂²⋊Dic₃₁	—	2	1	no	—
32	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
33	C₄×Dic₃₁	—	2	1	no	—
34	#92474	—	2	1	no	—
35	#92474	—	2	1	no	—
36	C₂×C₃₁⋊C₈	—	2	1	no	—
37	C₂×C₃₁⋊C₈	—	2	1	no	—
38	C₂×C₃₁⋊C₈	—	2	1	no	—
39	C₂×C₃₁⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃₁⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
42	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
43	C₂×C₃₁⋊C₈	Q₈	1	2	yes	—
44	C₂³×C₃₁	#41	1	1	yes	characteristic
45	C₂×C₄×C₃₁	—	2	4	no	—
46	C₂×C₄×C₃₁	—	2	2	no	—
47	C₂×C₄×C₃₁	—	2	2	no	—
48	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
54	C₂×Dic₃₁	—	2	2	no	—
55	C₈×C₃₁	—	4	1	no	—
56	C₃₁⋊C₈	—	2	2	no	—
57	C₃₁⋊C₈	—	4	1	no	—
58	C₃₁⋊C₈	C₄⋊C₄	1	2	yes	—
59	C₃₁⋊C₈	C₄⋊C₄	1	2	yes	—
60	#1945	—	31	1	no	maximal, Sylow
61	C₂²×C₃₁	—	2	2	no	—
62	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
63	C₂²×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₃₁	#125	1	1	yes	characteristic
65	C₄×C₃₁	—	2	4	no	—
66	C₄×C₃₁	—	4	2	no	—
67	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
69	C₄×C₃₁	#121	1	1	yes	characteristic
70	C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
71	Dic₃₁	—	4	2	no	—
72	C₂×#108	—	31	1	no	—
73	#427	—	31	1	no	—
74	#424	—	31	1	no	—
75	#424	—	31	1	no	—
76	C₂×C₄×C₈	—	31	1	no	—
77	#339	—	31	1	no	—
78	#339	D₃₁	1	1	yes	characteristic
79	C₂×C₃₁	—	2	1	no	—
80	C₂×C₃₁	—	2	1	no	—
81	C₂×C₃₁	#389	1	1	yes	characteristic
82	C₂×C₃₁	#443	1	2	yes	—
83	C₂×M₁₆	—	31	1	no	—
84	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
85	C₂²×C₈	—	31	1	no	—
86	C₂²×C₈	—	31	1	no	—
87	#117	—	31	1	no	—
88	#117	C₂×D₃₁	1	1	yes	characteristic
89	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
90	#108	—	62	2	no	—
91	#107	—	31	2	no	—
92	#106	—	31	2	no	—
93	C₄⋊C₈	—	62	2	no	—
94	C₄⋊C₈	—	62	2	no	—
95	C₄×C₈	—	62	2	no	—
96	C₃₁	#1945	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	62	1	no	—
102	M₁₆	—	62	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	31	1	no	—
105	C₂×C₈	—	31	1	no	—
106	C₂×C₈	—	31	2	no	—
107	C₂×C₈	—	31	2	no	—
108	C₂×C₈	—	31	2	no	—
109	C₂×C₈	—	62	1	no	—
110	C₂×C₈	—	62	1	no	—
111	C₂×C₈	—	62	1	no	—
112	C₄⋊C₄	—	31	2	no	—
113	C₄⋊C₄	C₄×D₃₁	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	62	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	62	1	no	—
119	C₄²	Dic₆₂	1	2	yes	—
120	C₂³	#92497	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	62	2	no	—
125	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
127	C₂×C₄	#92495	1	1	yes	characteristic
128	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
129	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
130	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
131	C₈	—	4	1	no	—
132	C₈	—	31	2	no	—
133	C₈	—	31	2	no	—
134	C₈	—	62	2	no	—
135	C₈	—	124	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	#11758455	1	1	yes	characteristic
138	C₂²	(C₂²⋊D₃₁)⋊C₄	1	1	yes	characteristic
139	C₂²	#11758466	1	1	yes	characteristic
140	C₄	—	2	4	no	—
141	C₄	—	4	2	no	—
142	C₄	—	124	2	no	—
143	C₄	C₄×Dic₆₂	1	1	yes	characteristic
144	C₄	#11758461	1	1	yes	characteristic
145	C₄	#11758465	1	1	yes	characteristic
146	C₄	C₄⋊C₄×D₃₁	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#14520946	1	1	yes	characteristic
150	C₂	#14521163	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758410

C₂

yes

maximal, characteristic

#14521085

C₂

yes

maximal, characteristic

#14521077

C₂

yes

maximal, characteristic

#14521076

C₂

yes

maximal, characteristic

C₂×C₄×C₃₁⋊C₈

C₂

yes

maximal, characteristic

#14520809

C₂

yes

maximal, characteristic

C₃₁×#339

C₂

yes

maximal, characteristic

C₂×M₁₆×C₃₁

C₂²

yes

characteristic

C₃₁×#117

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

#11758410

—

Dic₃₁⋊C₈

—

#11758396

C₄

yes

complemented

#11758395

C₄

yes

complemented

C₄⋊(C₃₁⋊C₈)

—

C₄×C₃₁⋊C₈

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

Q₈

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

—

#92474

—

#92474

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₃₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₃₁⋊C₈

D₄

yes

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₂×C₃₁⋊C₈

Q₈

yes

—

C₂³×C₃₁

#41

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

C₃₁⋊C₈

C₄⋊C₄

yes

—

C₃₁⋊C₈

C₄⋊C₄

yes

—

#1945

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₄×D₄

yes

characteristic

C₂²×C₃₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₃₁

#125

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₁

C₄×Q₈

yes

characteristic

C₄×C₃₁

#121

yes

characteristic

C₄×C₃₁

C₄⋊Q₈

yes

characteristic

Dic₃₁

—

—

—

—

—

C₂×C₄×C₈

—

#339

—

#339

D₃₁

yes

characteristic

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#389

yes

characteristic

C₂×C₃₁

#443

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₃₁

yes

characteristic

C₂×C₄²

C₂×D₃₁

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₃₁

#1945

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₄×D₃₁

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

Dic₆₂

yes

—

120

C₂³

#92497

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×Dic₆₂

yes

characteristic

126

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

127

C₂×C₄

#92495

yes

characteristic

128

C₂×C₄

D₄×D₃₁

yes

characteristic

129

C₂×C₄

D₄×D₃₁

yes

characteristic

130

C₂×C₄

Q₈×D₃₁

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

124

—

136

C₂²

—

137

C₂²

#11758455

yes

characteristic

138

C₂²

(C₂²⋊D₃₁)⋊C₄

yes

characteristic

139

C₂²

#11758466

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

124

—

143

C₄

C₄×Dic₆₂

yes

characteristic

144

C₄

#11758461

yes

characteristic

145

C₄

#11758465

yes

characteristic

146

C₄

C₄⋊C₄×D₃₁

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#14520946

yes

characteristic

150

C₂

#14521163

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#339	C₂
socle	C₂²×C₃₁	C₄×D₄
prime radical	C₂³×C₃₁	#41
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#339	D₃₁
hypocenter	C₃₁	#1945
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

subgroup

G/H

center

C₂×C₄

D₄×D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#339

C₂

socle

C₂²×C₃₁

C₄×D₄

prime radical

C₂³×C₃₁

#41

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

N	H ≅ G/N	type
17. #11758396	C₄	semidirect
18. #11758395	C₄	semidirect
96. C₃₁	#1945	semidirect

H ≅ G/N

type

17. #11758396

C₄

semidirect

18. #11758395

C₄

semidirect

96. C₃₁

#1945

semidirect

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

subgroup	index
C₂×#11758410	2
#14521085	2
#14521077	2
#14521076	2
C₂×C₄×C₃₁⋊C₈	2
#14520809	2
C₃₁×#339	2
#1945	31

subgroup

index

C₂×C₄×C₃₁⋊C₈

C₃₁×#339

Subgroups — #16846458

Conjugacy classes of subgroups

Complements

Maximal subgroups