Group #26103120 — subgroups

Subgroups — #26103120

Order 12032.

246 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 119 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	#20245187	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	C₄×C₄₇⋊C₁₆	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	2	no	—
32	C₄²×C₄₇	—	4	2	no	—
33	C₄²×C₄₇	—	4	2	no	—
34	C₄²×C₄₇	C₁₆	1	4	yes	complemented
35	C₄²×C₄₇	C₄²	1	1	yes	characteristic
36	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
39	C₄₇⋊C₁₆	—	4	4	no	—
40	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
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	C₂×C₄×C₄₇	C₂×C₁₆	1	2	yes	—
53	C₂×C₄×C₄₇	M₃₂	1	2	yes	—
54	C₈×C₄₇	—	2	2	no	—
55	C₈×C₄₇	—	2	2	no	—
56	C₈×C₄₇	—	4	2	no	—
57	#7142	—	47	1	no	maximal, Sylow
58	C₂²×C₄₇	—	2	1	no	—
59	C₂²×C₄₇	—	2	1	no	—
60	C₂²×C₄₇	#336	1	1	yes	characteristic
61	C₂²×C₄₇	C₂²⋊C₁₆	1	2	yes	—
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	C₄×C₁₆	1	1	yes	characteristic
67	C₄×C₄₇	#346	1	1	yes	characteristic
68	C₄×C₄₇	#347	1	2	yes	—
69	C₄×C₄₇	C₄⋊C₁₆	1	2	yes	—
70	#2162	—	47	2	no	—
71	#1744	D₄₇	1	1	yes	characteristic
72	C₂×C₄₇	—	2	2	no	—
73	C₂×C₄₇	#1274	1	1	yes	characteristic
74	C₂×C₄₇	#1374	1	1	yes	characteristic
75	C₂×C₄₇	#1375	1	1	yes	characteristic
76	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
78	C₄³	Dic₄₇	1	1	yes	characteristic
79	C₄⋊C₁₆	—	94	2	no	—
80	C₂²⋊C₁₆	—	94	2	no	—
81	C₄×C₁₆	—	47	4	no	—
82	#336	—	2	2	no	—
83	C₄₇	#7142	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	2	1	no	—
85	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
86	C₂×C₄²	—	2	1	no	—
87	C₂×C₄²	—	2	1	no	—
88	C₂×C₄²	C₄₇⋊C₈	1	2	yes	—
89	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
90	C₂×C₁₆	—	94	4	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	C₄₇⋊C₈	1	2	yes	—
93	C₄×C₈	Dic₉₄	1	1	yes	characteristic
94	C₄×C₈	C₄×D₄₇	1	2	yes	—
95	C₄×C₈	D₁₈₈	1	1	yes	characteristic
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
98	C₂²×C₄	#10625329	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
104	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
105	C₄²	—	2	2	no	—
106	C₄²	—	4	2	no	—
107	C₄²	—	4	2	no	—
108	C₄²	C₄₇⋊C₁₆	1	4	yes	complemented
109	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
110	C₄²	#10625329	1	1	yes	characteristic
111	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
112	C₄²	C₄⋊Dic₄₇	1	1	yes	characteristic
113	C₁₆	—	188	4	no	—
114	C₂³	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
121	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
122	C₂×C₄	#13093935	1	1	yes	characteristic
123	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
124	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
125	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄₇⋊C₁₆	1	2	yes	—
127	C₂×C₄	#13093964	1	2	yes	—
128	C₈	—	2	2	no	—
129	C₈	—	2	2	no	—
130	C₈	—	4	2	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	#15688878	1	1	yes	characteristic
134	C₂²	C₂²⋊(C₄₇⋊C₁₆)	1	2	yes	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	4	4	no	—
138	C₄	#15688879	1	2	yes	—
139	C₄	C₄×C₄₇⋊C₁₆	1	1	yes	characteristic
140	C₄	#15688899	1	1	yes	characteristic
141	C₄	#15688901	1	2	yes	—
142	C₄	C₄⋊(C₄₇⋊C₁₆)	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	#20241217	1	1	yes	characteristic
145	C₂	#20241466	1	1	yes	characteristic
146	C₂	#20241468	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245187

C₂

yes

maximal

C₄₇×#1744

C₂

yes

maximal, characteristic

C₄⋊(C₄₇⋊C₁₆)

—

C₂²⋊(C₄₇⋊C₁₆)

—

C₄×C₄₇⋊C₁₆

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

C₁₆

yes

complemented

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

#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

C₂×C₄×C₄₇

C₂×C₁₆

yes

—

C₂×C₄×C₄₇

M₃₂

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

#7142

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#336

yes

characteristic

C₂²×C₄₇

C₂²⋊C₁₆

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

H(3,ℤ/4)

yes

—

C₄×C₄₇

C₄×C₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₄×C₄₇

C₄⋊C₁₆

yes

—

#2162

—

#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

—

C₄×C₁₆

—

#336

—

C₄₇

#7142

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

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

Dic₄₇⋊C₄

yes

—

104

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

C₄₇⋊C₁₆

yes

complemented

109

C₄²

C₂×C₄₇⋊C₈

yes

characteristic

110

C₄²

#10625329

yes

characteristic

111

C₄²

C₄×Dic₄₇

yes

characteristic

112

C₄²

C₄⋊Dic₄₇

yes

characteristic

113

C₁₆

—

188

—

114

C₂³

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₄₇⋊#95

yes

characteristic

121

C₂×C₄

C₄×C₄₇⋊C₈

yes

characteristic

122

C₂×C₄

#13093935

yes

characteristic

123

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

124

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

characteristic

125

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

characteristic

126

C₂×C₄

C₂×C₄₇⋊C₁₆

yes

—

127

C₂×C₄

#13093964

yes

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₂²

—

132

C₂²

—

133

C₂²

#15688878

yes

characteristic

134

C₂²

C₂²⋊(C₄₇⋊C₁₆)

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#15688879

yes

—

139

C₄

C₄×C₄₇⋊C₁₆

yes

characteristic

140

C₄

#15688899

yes

characteristic

141

C₄

#15688901

yes

—

142

C₄

C₄⋊(C₄₇⋊C₁₆)

yes

—

143

C₂

—

144

C₂

#20241217

yes

characteristic

145

C₂

#20241466

yes

characteristic

146

C₂

#20241468

yes

characteristic

	subgroup	G/H
center	C₄²	C₄×Dic₄₇
commutator	C₄×C₄₇	C₄×C₁₆
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	#7142
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ #1744

subgroup

G/H

center

C₄²

C₄×Dic₄₇

commutator

C₄×C₄₇

C₄×C₁₆

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 ▷ C₄×C₄₇ ▷ C₄₇

C₁ ◁ C₄² ◁ #1744

N	H ≅ G/N	type
5. C₄×C₄₇⋊C₁₆	C₄	semidirect
34. C₄²×C₄₇	C₁₆	semidirect
83. C₄₇	#7142	semidirect
108. C₄²	C₄₇⋊C₁₆	semidirect

H ≅ G/N

type

5. C₄×C₄₇⋊C₁₆

C₄

semidirect

34. C₄²×C₄₇

C₁₆

semidirect

83. C₄₇

#7142

semidirect

108. C₄²

C₄₇⋊C₁₆

semidirect

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

subgroup	index
#20245187	2
C₄₇×#1744	2
#7142	47

subgroup

index

#20245187

C₄₇×#1744

#7142

Subgroups — #26103120

Conjugacy classes of subgroups

Complements

Maximal subgroups