Group #16846500 — subgroups

Subgroups — #16846500

Order 3968.

184 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 79 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	#14521098	C₂	1	1	yes	maximal, characteristic
2	#14521097	C₂	1	1	yes	maximal, characteristic
3	C₂×#11758410	C₂	1	1	yes	maximal, characteristic
4	C₂×#11758410	C₂	1	1	yes	maximal, characteristic
5	#14521005	C₂	1	1	yes	maximal, characteristic
6	#14520819	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#344	C₂	1	1	yes	maximal, characteristic
8	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#117	C₂²	1	1	yes	characteristic
11	C₂×#92474	C₂²	1	1	yes	characteristic
12	C₂×#92474	C₂²	1	1	yes	characteristic
13	C₂×#92474	C₂²	1	1	yes	characteristic
14	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	#11758410	—	2	2	no	—
16	#11758410	—	2	1	no	—
17	#11758410	—	2	1	no	—
18	#11758410	C₄	1	4	yes	complemented
19	#11758407	—	2	2	no	—
20	#11758391	—	2	1	no	—
21	C₄×C₃₁⋊C₈	—	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	M₁₆×C₃₁	—	2	1	no	—
26	M₁₆×C₃₁	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₃₁	—	2	1	no	—
28	C₂×C₈×C₃₁	—	2	1	no	—
29	C₄⋊C₄×C₃₁	—	2	1	no	—
30	C₂²⋊C₄×C₃₁	Q₈	1	2	yes	—
31	C₄²×C₃₁	—	2	1	no	—
32	#92474	—	2	1	no	—
33	#92474	—	2	1	no	—
34	#92474	—	2	1	no	—
35	#92474	—	2	1	no	—
36	#92474	C₂×C₄	1	2	yes	—
37	#92474	D₄	1	2	yes	—
38	C₂×C₃₁⋊C₈	—	2	1	no	—
39	C₂×C₃₁⋊C₈	—	2	1	no	—
40	C₂×C₃₁⋊C₈	—	2	1	no	—
41	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
43	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
44	C₂×C₃₁⋊C₈	Q₈	1	2	yes	—
45	C₂³×C₃₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₃₁	—	2	2	no	—
47	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
52	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
53	C₈×C₃₁	—	2	2	no	—
54	C₈×C₃₁	—	4	1	no	—
55	C₃₁⋊C₈	—	2	2	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	#1953	—	31	1	no	maximal, Sylow
61	C₂²×C₃₁	—	4	1	no	—
62	C₂²×C₃₁	C₄×Q₈	1	1	yes	characteristic
63	C₂²×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
65	C₄×C₃₁	—	4	2	no	—
66	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
68	C₄×C₃₁	#121	1	1	yes	characteristic
69	C₄×C₃₁	#125	1	1	yes	characteristic
70	#430	—	31	1	no	—
71	#430	—	31	1	no	—
72	C₂×#108	—	31	1	no	—
73	C₂×#108	—	31	1	no	—
74	#405	—	31	1	no	—
75	#344	—	31	1	no	—
76	#344	D₃₁	1	1	yes	characteristic
77	C₂×C₃₁	—	2	1	no	—
78	C₂×C₃₁	—	2	1	no	—
79	C₂×C₃₁	—	2	1	no	—
80	C₂×C₃₁	#389	1	1	yes	characteristic
81	C₂×M₁₆	—	31	1	no	—
82	C₂×M₁₆	—	31	1	no	—
83	C₂×M₁₆	—	31	1	no	—
84	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
85	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
86	C₂²×C₈	—	31	1	no	—
87	#117	C₂×D₃₁	1	1	yes	characteristic
88	#108	—	31	4	no	—
89	#108	—	62	2	no	—
90	#108	—	62	1	no	—
91	#108	—	62	1	no	—
92	#108	—	62	2	no	—
93	#97	—	62	1	no	—
94	C₄×C₈	—	62	1	no	—
95	C₃₁	#1953	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
97	M₁₆	—	2	1	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	2	1	no	—
100	M₁₆	—	31	2	no	—
101	M₁₆	—	31	2	no	—
102	M₁₆	—	62	1	no	—
103	M₁₆	—	62	1	no	—
104	M₁₆	—	62	1	no	—
105	M₁₆	—	62	1	no	—
106	M₁₆	C₄×D₃₁	1	2	yes	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	31	2	no	—
110	C₂×C₈	—	31	1	no	—
111	C₂×C₈	—	31	1	no	—
112	C₂×C₈	—	31	2	no	—
113	C₂×C₈	—	62	1	no	—
114	C₂×C₈	—	62	1	no	—
115	C₂×C₈	—	62	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	Dic₆₂	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₂³	C₂×Dic₆₂	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
122	C₂×C₄	#92495	1	1	yes	characteristic
123	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
124	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
125	C₂×C₄	#92497	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	4	1	no	—
129	C₈	—	31	2	no	—
130	C₈	—	31	2	no	—
131	C₈	—	62	2	no	—
132	C₈	—	62	2	no	—
133	C₈	—	124	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	C₄×Dic₆₂	1	1	yes	characteristic
136	C₂²	#11758455	1	1	yes	characteristic
137	C₂²	#11758465	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	(C₂²⋊D₃₁)⋊C₄	1	1	yes	characteristic
140	C₄	#11758461	1	1	yes	characteristic
141	C₄	#11758466	1	1	yes	characteristic
142	C₄	C₄⋊C₄×D₃₁	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#14520946	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521098

C₂

yes

maximal, characteristic

#14521097

C₂

yes

maximal, characteristic

C₂×#11758410

C₂

yes

maximal, characteristic

C₂×#11758410

C₂

yes

maximal, characteristic

#14521005

C₂

yes

maximal, characteristic

#14520819

C₂

yes

maximal, characteristic

C₃₁×#344

C₂

yes

maximal, characteristic

C₂×M₁₆×C₃₁

C₂²

yes

characteristic

C₂×M₁₆×C₃₁

C₂²

yes

characteristic

C₃₁×#117

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

—

—

—

C₄

yes

complemented

#11758407

—

#11758391

—

C₄×C₃₁⋊C₈

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

M₁₆×C₃₁

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

M₁₆×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

Q₈

yes

—

C₄²×C₃₁

—

—

—

—

—

C₂×C₄

yes

—

#92474

D₄

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

D₄

yes

characteristic

C₂×C₃₁⋊C₈

D₄

yes

characteristic

C₂×C₃₁⋊C₈

Q₈

yes

—

C₂³×C₃₁

C₂×Q₈

yes

characteristic

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

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₈×C₃₁

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

C₃₁⋊C₈

C₄⋊C₄

yes

—

C₃₁⋊C₈

C₄⋊C₄

yes

—

#1953

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₄×Q₈

yes

characteristic

C₂²×C₃₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₃₁

C₄⋊Q₈

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₁

C₄×D₄

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

—

—

—

—

—

D₃₁

yes

characteristic

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#389

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

characteristic

C₂×M₁₆

C₂×D₃₁

yes

characteristic

C₂²×C₈

—

#117

C₂×D₃₁

yes

—

—

—

—

—

—

C₄×C₈

—

C₃₁

#1953

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

C₄×D₃₁

yes

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

Dic₆₂

yes

—

118

C₄²

—

119

C₂³

C₂×Dic₆₂

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

122

C₂×C₄

#92495

yes

characteristic

123

C₂×C₄

D₄×D₃₁

yes

characteristic

124

C₂×C₄

D₄×D₃₁

yes

characteristic

125

C₂×C₄

#92497

yes

characteristic

126

C₂×C₄

Q₈×D₃₁

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

124

—

134

C₂²

—

135

C₂²

C₄×Dic₆₂

yes

characteristic

136

C₂²

#11758455

yes

characteristic

137

C₂²

#11758465

yes

characteristic

138

C₄

—

139

C₄

(C₂²⋊D₃₁)⋊C₄

yes

characteristic

140

C₄

#11758461

yes

characteristic

141

C₄

#11758466

yes

characteristic

142

C₄

C₄⋊C₄×D₃₁

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#14520946

yes

characteristic

	subgroup	G/H
center	C₄	#11758461
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#344	C₂
socle	C₂×C₃₁	#389
prime radical	C₂³×C₃₁	C₂×Q₈
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#344	D₃₁
hypocenter	C₃₁	#1953
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#11758461

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#344

C₂

socle

C₂×C₃₁

#389

prime radical

C₂³×C₃₁

C₂×Q₈

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
18. #11758410	C₄	semidirect
95. C₃₁	#1953	semidirect

H ≅ G/N

type

18. #11758410

C₄

semidirect

95. C₃₁

#1953

semidirect

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

subgroup	index
#14521098	2
#14521097	2
C₂×#11758410	2
#14521005	2
#14520819	2
C₃₁×#344	2
#1953	31

subgroup

index

C₃₁×#344

Subgroups — #16846500

Conjugacy classes of subgroups

Complements

Maximal subgroups