Group #26128083 — subgroups

Subgroups — #26128083

Order 12032.

206 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20245293	C₂	1	2	yes	maximal
2	C₈×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
3	#20241366	C₂	1	2	yes	maximal
4	C₈×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1370	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₉₄	C₂²	1	2	yes	—
7	C₄×Dic₁₈₈	C₄	1	1	yes	characteristic
8	Dic₄₇⋊C₁₆	—	2	2	no	—
9	#15688898	—	2	2	no	—
10	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#15688876	C₄	1	1	yes	characteristic
13	#15688855	C₄	1	2	yes	—
14	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₈²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₈₈	C₈	1	1	yes	characteristic
17	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
18	C₂×C₄₇⋊C₁₆	—	2	1	no	—
19	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
20	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
21	#13093958	C₈	1	1	yes	characteristic
22	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
23	Dic₄₇⋊C₈	—	2	2	no	—
24	#13093949	C₈	1	2	yes	—
25	C₈×Dic₄₇	—	2	2	no	—
26	C₂×C₁₆×C₄₇	—	2	2	no	—
27	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×Dic₉₄	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₄₇	—	2	1	no	—
32	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
36	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊Dic₄₇	C₂×C₈	1	2	yes	—
38	Dic₄₇⋊C₄	—	2	2	no	—
39	C₄×Dic₄₇	—	2	2	no	—
40	C₂×C₄₇⋊C₈	—	2	2	no	—
41	Dic₁₈₈	—	2	2	no	—
42	C₁₆×C₄₇	—	4	2	no	—
43	C₄₇⋊C₁₆	—	2	2	no	—
44	C₄₇⋊C₁₆	—	2	1	no	—
45	C₄₇⋊C₁₆	Q₁₆	1	2	yes	complemented
46	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
49	C₂×Dic₄₇	—	2	2	no	—
50	C₂×Dic₄₇	—	2	2	no	—
51	Dic₉₄	—	2	2	no	—
52	Dic₉₄	—	2	2	no	—
53	C₈×C₄₇	—	2	2	no	—
54	C₈×C₄₇	—	2	1	no	—
55	C₈×C₄₇	—	2	1	no	—
56	C₈×C₄₇	M₃₂	1	2	yes	—
57	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₄₇	#135	1	1	yes	characteristic
59	C₄₇⋊C₈	—	4	2	no	—
60	#12054	—	47	1	no	maximal, Sylow
61	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
62	C₄×C₄₇	—	2	1	no	—
63	C₄×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
64	C₄×C₄₇	#443	1	1	yes	characteristic
65	C₄×C₄₇	C₂×M₃₂	1	1	yes	characteristic
66	C₄×C₄₇	#504	1	1	yes	characteristic
67	Dic₄₇	—	4	2	no	—
68	Dic₄₇	—	4	2	no	—
69	#2183	—	47	2	no	—
70	C₈×Q₁₆	—	47	1	no	—
71	#1370	D₄₇	1	1	yes	characteristic
72	Q₈⋊C₁₆	—	47	2	no	—
73	C₈×C₁₆	—	47	1	no	—
74	C₂×C₄₇	C₈×Q₁₆	1	1	yes	characteristic
75	C₂×C₄₇	#2169	1	1	yes	characteristic
76	C₂×C₄₇	#2170	1	1	yes	characteristic
77	C₈×Q₈	—	47	2	no	—
78	C₄×Q₁₆	—	47	1	no	—
79	C₄⋊C₁₆	—	94	2	no	—
80	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	—
81	#346	—	94	2	no	—
82	C₄×C₁₆	—	47	1	no	—
83	C₄×C₁₆	—	47	1	no	—
84	#335	—	47	1	no	—
85	Q₈⋊C₈	—	47	2	no	—
86	C₈²	C₂×D₄₇	1	1	yes	characteristic
87	C₄₇	#12054	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	—	47	1	no	—
89	C₄×Q₈	—	47	2	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	47	1	no	—
92	C₂×C₁₆	—	47	1	no	—
93	C₂×C₁₆	—	94	1	no	—
94	#107	—	47	1	no	—
95	C₄⋊C₈	—	47	2	no	—
96	C₄⋊C₈	—	94	2	no	—
97	#103	—	47	2	no	—
98	C₄×C₈	—	94	2	no	—
99	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
100	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
102	C₂×Q₈	—	47	2	no	—
103	Q₁₆	—	94	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	94	2	no	—
106	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
107	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
108	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
109	C₂×C₈	#10625354	1	1	yes	characteristic
110	C₄⋊C₄	—	47	2	no	—
111	C₄⋊C₄	—	94	2	no	—
112	C₄²	—	94	2	no	—
113	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
114	C₁₆	—	4	2	no	—
115	C₁₆	—	47	2	no	—
116	C₁₆	—	94	2	no	—
117	C₁₆	—	94	1	no	—
118	Q₈	—	94	2	no	—
119	Q₈	—	94	2	no	—
120	C₂×C₄	—	94	2	no	—
121	C₂×C₄	—	94	2	no	—
122	C₂×C₄	#13093986	1	1	yes	characteristic
123	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
124	C₂×C₄	#13094032	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	188	2	no	—
129	C₈	#13093963	1	2	yes	—
130	C₈	Q₁₆×D₄₇	1	1	yes	characteristic
131	C₈	#13094043	1	1	yes	characteristic
132	C₂²	#15689188	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	188	2	no	—
135	C₄	—	188	2	no	—
136	C₄	#15689217	1	1	yes	characteristic
137	C₄	#15689239	1	1	yes	characteristic
138	C₄	C₂×#13093963	1	1	yes	characteristic
139	C₄	#15689541	1	1	yes	characteristic
140	C₂	#20242194	1	1	yes	characteristic
141	C₂	#20245219	1	1	yes	characteristic
142	C₂	#20245224	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245293

C₂

yes

maximal

C₈×Dic₁₈₈

C₂

yes

maximal, characteristic

#20241366

C₂

yes

maximal

C₈×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic

C₄₇×#1370

C₂

yes

maximal, characteristic

C₈×Dic₉₄

C₂²

yes

—

C₄×Dic₁₈₈

C₄

yes

characteristic

Dic₄₇⋊C₁₆

—

#15688898

—

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₄×C₄₇⋊C₁₆

C₂²

yes

characteristic

#15688876

C₄

yes

characteristic

#15688855

C₄

yes

—

C₄₇×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

C₈

yes

characteristic

C₄×Dic₉₄

C₂×C₄

yes

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

D₄

yes

characteristic

C₂×C₄₇⋊C₁₆

D₄

yes

characteristic

#13093958

C₈

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂×C₄

yes

—

Dic₄₇⋊C₈

—

#13093949

C₈

yes

—

C₈×Dic₄₇

—

C₂×C₁₆×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

C₂×C₈

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₈

yes

characteristic

C₂×C₈×C₄₇

C₂×C₈

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄⋊Dic₄₇

C₂×C₈

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

Q₁₆

yes

complemented

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

#131

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

M₃₂

yes

—

C₈×C₄₇

C₂×Q₁₆

yes

characteristic

C₈×C₄₇

#135

yes

characteristic

C₄₇⋊C₈

—

#12054

—

maximal, Sylow

C₂²×C₄₇

C₈×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₄×Q₁₆

yes

characteristic

C₄×C₄₇

#443

yes

characteristic

C₄×C₄₇

C₂×M₃₂

yes

characteristic

C₄×C₄₇

#504

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#2183

—

C₈×Q₁₆

—

#1370

D₄₇

yes

characteristic

Q₈⋊C₁₆

—

C₈×C₁₆

—

C₂×C₄₇

C₈×Q₁₆

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₄₇

yes

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12054

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₄×D₄₇

yes

characteristic

100

C₄×C₈

C₄×D₄₇

yes

characteristic

101

C₄×C₈

C₂²×D₄₇

yes

characteristic

102

C₂×Q₈

—

103

Q₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₈×D₄₇

yes

characteristic

107

C₂×C₈

C₈×D₄₇

yes

characteristic

108

C₂×C₈

D₄×D₄₇

yes

characteristic

109

C₂×C₈

#10625354

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

C₂×C₄×D₄₇

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#13093986

yes

characteristic

123

C₂×C₄

C₂×C₈×D₄₇

yes

characteristic

124

C₂×C₄

#13094032

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

188

—

129

C₈

#13093963

yes

—

130

C₈

Q₁₆×D₄₇

yes

characteristic

131

C₈

#13094043

yes

characteristic

132

C₂²

#15689188

yes

characteristic

133

C₄

—

134

C₄

—

188

—

135

C₄

—

188

—

136

C₄

#15689217

yes

characteristic

137

C₄

#15689239

yes

characteristic

138

C₄

C₂×#13093963

yes

characteristic

139

C₄

#15689541

yes

characteristic

140

C₂

#20242194

yes

characteristic

141

C₂

#20245219

yes

characteristic

142

C₂

#20245224

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1370	C₂
socle, prime radical	C₂²×C₄₇	C₈×D₄
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1370	D₄₇
hypocenter	C₄₇	#12054
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

subgroup

G/H

center

C₂×C₈

D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1370

C₂

socle, prime radical

C₂²×C₄₇

C₈×D₄

cosocle

C₄×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

N	H ≅ G/N	type
45. C₄₇⋊C₁₆	Q₁₆	semidirect
87. C₄₇	#12054	semidirect

H ≅ G/N

type

45. C₄₇⋊C₁₆

Q₁₆

semidirect

87. C₄₇

#12054

semidirect

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

subgroup	index
#20245293	2
C₈×Dic₁₈₈	2
#20241366	2
C₈×C₄₇⋊C₁₆	2
C₄₇×#1370	2
#12054	47

subgroup

index

#20245293

C₈×Dic₁₈₈

#20241366

C₈×C₄₇⋊C₁₆

C₄₇×#1370

#12054

Subgroups — #26128083

Conjugacy classes of subgroups

Complements

Maximal subgroups