Group #26125334 — subgroups

Subgroups — #26125334

Order 12032.

228 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 123 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	1	yes	maximal, characteristic
2	#20241671	C₂	1	1	yes	maximal, characteristic
3	#20241310	C₂	1	2	yes	maximal, complemented
4	#20241308	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2163	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₄	1	2	yes	—
7	C₄⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	—
8	C₂²⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	—
9	#15688899	C₂²	1	2	yes	—
10	#15688844	C₂²	1	1	yes	characteristic
11	#15688844	C₂²	1	1	yes	characteristic
12	C₈×C₄₇⋊C₈	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×C₄⋊C₁₆	C₄	1	2	yes	—
15	C₄₇×C₂²⋊C₁₆	C₄	1	2	yes	—
16	C₄₇×#346	C₂²	1	2	yes	—
17	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
18	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
19	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
20	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
26	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
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	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
32	C₂×C₈×C₄₇	—	2	2	no	—
33	C₂×C₈×C₄₇	—	2	2	no	—
34	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₄₇	C₄²	1	1	yes	characteristic
39	C₄²×C₄₇	M₁₆	1	2	yes	—
40	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄₇⋊C₈	—	2	2	no	—
42	C₂×C₄₇⋊C₈	—	2	2	no	—
43	C₁₆×C₄₇	—	2	4	no	—
44	C₄₇⋊C₁₆	—	2	4	no	—
45	C₂³×C₄₇	#97	1	1	yes	characteristic
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	—	4	2	no	—
49	C₂×C₄×C₄₇	#97	1	2	yes	—
50	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
53	C₈×C₄₇	—	2	1	no	—
54	C₈×C₄₇	—	2	1	no	—
55	C₈×C₄₇	#131	1	2	yes	—
56	C₈×C₄₇	#131	1	2	yes	—
57	C₄₇⋊C₈	—	4	2	no	—
58	C₄₇⋊C₈	—	4	2	no	—
59	#11416	—	47	1	no	maximal, Sylow
60	C₂²×C₄₇	—	2	2	no	—
61	C₂²×C₄₇	—	2	1	no	—
62	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
63	C₄×C₄₇	—	2	2	no	—
64	C₄×C₄₇	—	2	2	no	—
65	C₄×C₄₇	—	2	1	no	—
66	C₄×C₄₇	#405	1	1	yes	characteristic
67	C₄×C₄₇	#405	1	1	yes	characteristic
68	#2163	—	47	1	no	—
69	#2163	D₄₇	1	1	yes	characteristic
70	#1450	—	47	1	no	—
71	#1313	—	47	2	no	—
72	#1313	—	47	2	no	—
73	C₂×C₄₇	—	4	1	no	—
74	C₂×C₄₇	#1452	1	1	yes	characteristic
75	C₂×C₄₇	#2110	1	2	yes	—
76	#431	—	47	2	no	—
77	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
78	C₄⋊C₁₆	—	47	2	no	—
79	C₄⋊C₁₆	Dic₄₇	1	2	yes	—
80	C₂²⋊C₁₆	—	47	2	no	—
81	C₂²⋊C₁₆	Dic₄₇	1	2	yes	—
82	#346	—	47	2	no	—
83	#346	C₂×D₄₇	1	2	yes	—
84	#322	—	47	1	no	—
85	#322	—	47	1	no	—
86	C₈²	—	47	2	no	—
87	C₄₇	#11416	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
89	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
90	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
91	C₂×C₁₆	—	47	4	no	—
92	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	—
93	C₄⋊C₈	—	94	2	no	—
94	C₂²⋊C₈	—	94	2	no	—
95	C₄×C₈	—	47	2	no	—
96	C₄×C₈	—	47	2	no	—
97	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
98	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
99	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
100	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
101	C₂²×C₄	#10625324	1	2	yes	—
102	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	94	2	no	—
106	C₂×C₈	—	94	2	no	—
107	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
108	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
109	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
110	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
111	C₄²	#10625324	1	2	yes	—
112	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
113	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
114	C₁₆	—	2	4	no	—
115	C₁₆	—	94	4	no	—
116	C₂³	#13093934	1	1	yes	characteristic
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	#13093934	1	1	yes	characteristic
121	C₂×C₄	#13093934	1	2	yes	—
122	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
123	C₂×C₄	C₂×#10625324	1	2	yes	—
124	C₈	—	2	1	no	—
125	C₈	—	2	1	no	—
126	C₈	—	188	2	no	—
127	C₈	—	188	2	no	—
128	C₈	#13094031	1	2	yes	—
129	C₈	#13094032	1	2	yes	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×#13093934	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	2	no	—
135	C₄	—	2	1	no	—
136	C₄	#15689079	1	1	yes	characteristic
137	C₄	#15689081	1	1	yes	characteristic
138	C₂	—	4	1	no	—
139	C₂	#20241681	1	1	yes	characteristic
140	C₂	#20244992	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245191

C₂

yes

maximal, characteristic

#20241671

C₂

yes

maximal, characteristic

#20241310

C₂

yes

maximal, complemented

#20241308

C₂

yes

maximal, complemented

C₄₇×#2163

C₂

yes

maximal, characteristic

#15689176

C₄

yes

—

C₄⋊(C₄₇⋊C₁₆)

C₄

yes

—

C₂²⋊(C₄₇⋊C₁₆)

C₄

yes

—

#15688899

C₂²

yes

—

#15688844

C₂²

yes

characteristic

#15688844

C₂²

yes

characteristic

C₈×C₄₇⋊C₈

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×C₄⋊C₁₆

C₄

yes

—

C₄₇×C₂²⋊C₁₆

C₄

yes

—

C₄₇×#346

C₂²

yes

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

M₁₆

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄²

yes

characteristic

C₂×C₈×C₄₇

C₄²

yes

characteristic

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₄²

yes

characteristic

C₄²×C₄₇

M₁₆

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#97

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#97

yes

—

C₂×C₄×C₄₇

#97

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#131

yes

—

C₈×C₄₇

#131

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#11416

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#97

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

#1452

yes

characteristic

C₂×C₄₇

#2110

yes

—

#431

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

Dic₄₇

yes

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

Dic₄₇

yes

—

#346

—

#346

C₂×D₄₇

yes

—

#322

—

#322

—

C₈²

—

C₄₇

#11416

yes

characteristic, complemented, Sylow

C₂²×C₈

C₄×D₄₇

yes

characteristic

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

C₄×D₄₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₄₇

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₄₇

yes

characteristic

C₄×C₈

C₄×D₄₇

yes

characteristic

C₄×C₈

C₂×Dic₄₇

yes

characteristic

100

C₄×C₈

C₂²×D₄₇

yes

characteristic

101

C₂²×C₄

#10625324

yes

—

102

C₂²×C₄

C₄×Dic₄₇

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₄×Dic₄₇

yes

characteristic

108

C₂×C₈

C₄×Dic₄₇

yes

characteristic

109

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

110

C₂×C₈

C₂²×Dic₄₇

yes

characteristic

111

C₄²

#10625324

yes

—

112

C₄²

C₄×Dic₄₇

yes

characteristic

113

C₄²

C₂×C₄×D₄₇

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₂³

#13093934

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#13093934

yes

characteristic

121

C₂×C₄

#13093934

yes

—

122

C₂×C₄

C₂×C₄×Dic₄₇

yes

characteristic

123

C₂×C₄

C₂×#10625324

yes

—

124

C₈

—

125

C₈

—

126

C₈

—

188

—

127

C₈

—

188

—

128

C₈

#13094031

yes

—

129

C₈

#13094032

yes

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×#13093934

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#15689079

yes

characteristic

137

C₄

#15689081

yes

characteristic

138

C₂

—

139

C₂

#20241681

yes

characteristic

140

C₂

#20244992

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#2163	C₂
socle	C₂²×C₄₇	C₂×#97
prime radical	C₂³×C₄₇	#97
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#2163	D₄₇
hypocenter	C₄₇	#11416
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #2163

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₄₇

commutator

C₂×C₄×C₄₇

C₂×C₄²

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#2163

C₂

socle

C₂²×C₄₇

C₂×#97

prime radical

C₂³×C₄₇

#97

cosocle

C₄×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ #2163

N	H ≅ G/N	type
3. #20241310	C₂	semidirect
4. #20241308	C₂	semidirect
87. C₄₇	#11416	semidirect

H ≅ G/N

type

3. #20241310

C₂

semidirect

4. #20241308

C₂

semidirect

87. C₄₇

#11416

semidirect

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

subgroup	index
#20245191	2
#20241671	2
#20241310	2
#20241308	2
C₄₇×#2163	2
#11416	47

subgroup

index

C₄₇×#2163

Subgroups — #26125334

Conjugacy classes of subgroups

Complements

Maximal subgroups