Group #26103122 — subgroups

Subgroups — #26103122

Order 12032.

234 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20245191	C₂	1	2	yes	maximal
2	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
4	C₂²⋊(C₄₇⋊C₁₆)	—	2	2	no	—
5	#15688899	C₄	1	4	yes	complemented
6	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₄³×C₄₇	C₄	1	1	yes	characteristic
9	C₄₇×#336	—	2	2	no	—
10	C₂×C₄₇⋊C₁₆	—	2	4	no	—
11	C₂²×C₈×C₄₇	—	2	1	no	—
12	C₂²×C₈×C₄₇	D₄	1	2	yes	—
13	C₂×C₄²×C₄₇	—	2	1	no	—
14	C₂×C₄²×C₄₇	—	2	1	no	—
15	C₂×C₄²×C₄₇	C₈	1	2	yes	—
16	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
17	C₄×C₈×C₄₇	—	2	1	no	—
18	C₄×C₈×C₄₇	C₈	1	2	yes	—
19	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
21	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	—	2	2	no	—
23	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
26	C₂×C₈×C₄₇	—	2	2	no	—
27	C₂×C₈×C₄₇	—	4	2	no	—
28	C₂×C₈×C₄₇	—	4	2	no	—
29	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
30	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
31	C₄²×C₄₇	—	2	4	no	—
32	C₄²×C₄₇	—	4	2	no	—
33	C₄²×C₄₇	—	4	2	no	—
34	C₄²×C₄₇	C₄²	1	1	yes	characteristic
35	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
36	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
38	C₄₇⋊C₁₆	—	4	4	no	—
39	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	—	2	2	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	4	no	—
44	C₂×C₄×C₄₇	—	4	2	no	—
45	C₂×C₄×C₄₇	—	4	2	no	—
46	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
52	C₈×C₄₇	—	2	2	no	—
53	C₈×C₄₇	—	2	2	no	—
54	C₈×C₄₇	—	4	2	no	—
55	#7143	—	47	1	no	maximal, Sylow
56	C₂²×C₄₇	—	2	1	no	—
57	C₂²×C₄₇	—	2	1	no	—
58	C₂²×C₄₇	#336	1	1	yes	characteristic
59	C₂²×C₄₇	#349	1	2	yes	—
60	C₄×C₄₇	—	2	1	no	—
61	C₄×C₄₇	—	2	1	no	—
62	C₄×C₄₇	—	2	1	no	—
63	C₄×C₄₇	—	2	1	no	—
64	C₄×C₄₇	—	4	4	no	—
65	C₄×C₄₇	H(3,ℤ/4)	1	2	yes	—
66	C₄×C₄₇	#364	1	2	yes	—
67	#2163	—	47	2	no	—
68	#1744	D₄₇	1	1	yes	characteristic
69	C₂×C₄₇	—	2	2	no	—
70	C₂×C₄₇	#1274	1	1	yes	characteristic
71	C₂×C₄₇	#1376	1	1	yes	characteristic
72	C₂×C₄₇	#1377	1	1	yes	characteristic
73	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
75	C₄³	Dic₄₇	1	1	yes	characteristic
76	C₄⋊C₁₆	—	94	2	no	—
77	C₂²⋊C₁₆	—	94	2	no	—
78	#346	—	47	4	no	—
79	#336	—	2	2	no	—
80	C₄₇	#7143	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	2	1	no	—
82	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	C₄₇⋊C₈	1	2	yes	—
86	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
87	C₂×C₁₆	—	94	4	no	—
88	C₄×C₈	—	2	1	no	—
89	C₄×C₈	C₄₇⋊C₈	1	2	yes	—
90	C₄×C₈	Dic₉₄	1	1	yes	characteristic
91	C₄×C₈	C₄×D₄₇	1	2	yes	—
92	C₄×C₈	D₁₈₈	1	1	yes	characteristic
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
95	C₂²×C₄	#10625329	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
101	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
102	C₄²	—	2	4	no	—
103	C₄²	—	4	2	no	—
104	C₄²	—	4	2	no	—
105	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
106	C₄²	#10625329	1	1	yes	characteristic
107	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
108	C₄²	C₄⋊Dic₄₇	1	1	yes	characteristic
109	C₁₆	—	188	4	no	—
110	C₂³	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
118	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
119	C₂×C₄	#13093935	1	1	yes	characteristic
120	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
121	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
122	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	4	2	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	#15688878	1	1	yes	characteristic
129	C₂²	#15688905	1	2	yes	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	4	4	no	—
135	C₄	#15688879	1	2	yes	—
136	C₄	#15688944	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	#20241217	1	1	yes	characteristic
139	C₂	#20241470	1	1	yes	characteristic
140	C₂	#20241472	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245191

C₂

yes

maximal

C₄₇×#1744

C₂

yes

maximal, characteristic

C₄⋊(C₄₇⋊C₁₆)

—

C₂²⋊(C₄₇⋊C₁₆)

—

#15688899

C₄

yes

complemented

C₂×C₄×C₈×C₄₇

C₄

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄³×C₄₇

C₄

yes

characteristic

C₄₇×#336

—

C₂×C₄₇⋊C₁₆

—

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₈

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₈

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

Q₈

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₇

M₁₆

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₄²

yes

characteristic

C₄²×C₄₇

C₄⋊C₄

yes

characteristic

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₄²×C₄₇

M₁₆

yes

characteristic

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

C₄×C₈

yes

characteristic

C₂×C₄×C₄₇

#97

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₈

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

#7143

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#336

yes

characteristic

C₂²×C₄₇

#349

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

H(3,ℤ/4)

yes

—

C₄×C₄₇

#364

yes

—

#2163

—

#1744

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄×C₈

Dic₄₇

yes

characteristic

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄³

Dic₄₇

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#336

—

C₄₇

#7143

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₄₇⋊C₈

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

C₄₇⋊C₈

yes

—

C₄×C₈

Dic₉₄

yes

characteristic

C₄×C₈

C₄×D₄₇

yes

—

C₄×C₈

D₁₈₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×C₄₇⋊C₈

yes

characteristic

C₂²×C₄

#10625329

yes

characteristic

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

Dic₄₇⋊C₄

yes

—

101

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂×C₄₇⋊C₈

yes

characteristic

106

C₄²

#10625329

yes

characteristic

107

C₄²

C₄×Dic₄₇

yes

characteristic

108

C₄²

C₄⋊Dic₄₇

yes

characteristic

109

C₁₆

—

188

—

110

C₂³

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₄₇⋊#95

yes

characteristic

118

C₂×C₄

C₄×C₄₇⋊C₈

yes

characteristic

119

C₂×C₄

#13093935

yes

characteristic

120

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

121

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

characteristic

122

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₂²

—

127

C₂²

—

128

C₂²

#15688878

yes

characteristic

129

C₂²

#15688905

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#15688879

yes

—

136

C₄

#15688944

yes

—

137

C₂

—

138

C₂

#20241217

yes

characteristic

139

C₂

#20241470

yes

characteristic

140

C₂

#20241472

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄₇⋊#95
commutator	C₂×C₄×C₄₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₄₇×#1744	C₂
socle	C₂²×C₄₇	#336
prime radical	C₂³×C₄₇	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#7143
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1744

subgroup

G/H

center

C₂×C₄

C₄₇⋊#95

commutator

C₂×C₄×C₄₇

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₄₇×#1744

C₂

socle

C₂²×C₄₇

#336

prime radical

C₂³×C₄₇

C₂²⋊C₈

cosocle

C₂×C₄×C₈×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1744

N	H ≅ G/N	type
5. #15688899	C₄	semidirect
80. C₄₇	#7143	semidirect

H ≅ G/N

type

5. #15688899

C₄

semidirect

80. C₄₇

#7143

semidirect

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

subgroup	index
#20245191	2
C₄₇×#1744	2
#7143	47

subgroup

index

#20245191

C₄₇×#1744

#7143

Subgroups — #26103122

Conjugacy classes of subgroups

Complements

Maximal subgroups