Group #26132644 — subgroups

Subgroups — #26132644

Order 12032.

234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 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	#20245154	C₂	1	1	yes	maximal, characteristic
2	#20243143	C₂	1	1	yes	maximal, characteristic
3	#20241476	C₂	1	2	yes	maximal
4	#20241475	C₂	1	2	yes	maximal
5	C₄₇×#2151	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	2	yes	—
7	#15689078	C₂²	1	2	yes	—
8	#15688944	—	2	2	no	—
9	C₄₇⋊#338	C₄	1	1	yes	characteristic
10	#15688880	C₄	1	1	yes	characteristic
11	#15688879	C₄	1	2	yes	—
12	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#405	C₂²	1	1	yes	characteristic
15	C₄₇×C₄⋊C₁₆	—	2	2	no	—
16	C₂×#10625329	C₂×C₄	1	2	yes	—
17	#13093983	C₂×C₄	1	2	yes	—
18	#13093964	—	2	2	no	—
19	#13093964	—	2	2	no	—
20	C₂×C₄₇⋊C₁₆	—	2	2	no	—
21	#13093934	—	2	2	no	—
22	C₈×Dic₄₇	—	2	2	no	—
23	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
25	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₄₇	—	2	2	no	—
27	C₂×C₁₆×C₄₇	—	2	1	no	—
28	C₂×C₁₆×C₄₇	—	2	1	no	—
29	C₂×C₁₆×C₄₇	—	2	2	no	—
30	#97×C₄₇	Q₈	1	2	yes	—
31	C₄×C₈×C₄₇	D₄	1	2	yes	—
32	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
33	M₁₆×C₄₇	—	2	2	no	—
34	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₄₇	#41	1	2	yes	—
38	C₂×C₈×C₄₇	#41	1	2	yes	—
39	C₄⋊C₄×C₄₇	—	2	1	no	—
40	C₂²⋊C₄×C₄₇	—	2	1	no	—
41	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
42	C₂²⋊Dic₄₇	—	2	2	no	—
43	C₄⋊Dic₄₇	—	2	2	no	—
44	C₄×Dic₄₇	—	2	2	no	—
45	#10625329	—	2	2	no	—
46	#10625329	—	2	2	no	—
47	C₂×C₄₇⋊C₈	—	2	2	no	—
48	C₁₆×C₄₇	—	4	1	no	—
49	C₁₆×C₄₇	—	4	1	no	—
50	C₄₇⋊C₁₆	—	4	2	no	—
51	C₂³×C₄₇	#131	1	1	yes	characteristic
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#117	1	2	yes	—
55	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#131	1	2	yes	—
57	C₂×Dic₄₇	—	4	2	no	—
58	C₈×C₄₇	—	2	2	no	—
59	C₈×C₄₇	#124	1	1	yes	characteristic
60	C₈×C₄₇	#124	1	1	yes	characteristic
61	C₈×C₄₇	#125	1	2	yes	—
62	C₄₇⋊C₈	—	4	2	no	—
63	#12885	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	—	4	1	no	—
65	C₂²×C₄₇	#424	1	1	yes	characteristic
66	C₂²×C₄₇	#433	1	2	yes	—
67	C₄×C₄₇	—	4	2	no	—
68	C₄×C₄₇	#382	1	1	yes	characteristic
69	C₄×C₄₇	#424	1	1	yes	characteristic
70	C₄×C₄₇	#433	1	2	yes	—
71	Dic₄₇	—	8	2	no	—
72	#2153	—	47	1	no	—
73	#2151	D₄₇	1	1	yes	characteristic
74	#1747	—	47	1	no	—
75	#1379	—	47	2	no	—
76	#1378	—	47	2	no	—
77	C₂×C₄₇	—	2	2	no	—
78	C₂×C₄₇	—	2	1	no	—
79	C₂×C₄₇	#1832	1	1	yes	characteristic
80	C₂×M₃₂	—	47	2	no	—
81	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
82	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
83	#405	—	47	2	no	—
84	#405	C₂×D₄₇	1	1	yes	characteristic
85	#364	—	94	2	no	—
86	C₄⋊C₁₆	—	2	2	no	—
87	#338	—	47	1	no	—
88	H(3,ℤ/4)	—	47	1	no	—
89	H(3,ℤ/4)	—	47	2	no	—
90	C₄₇	#12885	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₁₆	—	47	2	no	—
92	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
94	#117	—	47	2	no	—
95	#117	C₄×D₄₇	1	1	yes	characteristic
96	M₃₂	—	2	2	no	—
97	M₃₂	—	94	2	no	—
98	M₃₂	—	94	2	no	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	—	94	2	no	—
103	#97	—	94	2	no	—
104	#97	Dic₉₄	1	2	yes	—
105	C₄×C₈	—	94	2	no	—
106	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	94	2	no	—
110	M₁₆	—	94	2	no	—
111	C₂×C₈	—	94	2	no	—
112	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
113	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
114	C₂×C₈	#10625352	1	2	yes	—
115	C₂×C₈	#10625354	1	2	yes	—
116	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	94	2	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	94	2	no	—
121	C₄²	—	94	2	no	—
122	C₄²	Dic₄₇⋊C₄	1	2	yes	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	1	no	—
125	C₁₆	—	188	2	no	—
126	C₂³	#13094030	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	188	2	no	—
129	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
130	C₂×C₄	#13093982	1	2	yes	—
131	C₂×C₄	#13094030	1	1	yes	characteristic
132	C₂×C₄	#13094032	1	2	yes	—
133	C₈	—	2	2	no	—
134	C₈	—	188	2	no	—
135	C₈	#13094011	1	1	yes	characteristic
136	C₈	#13094013	1	1	yes	characteristic
137	C₈	#13094015	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	#15689149	1	1	yes	characteristic
140	C₂²	#15689183	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	376	2	no	—
143	C₄	#15688988	1	1	yes	characteristic
144	C₄	#15689151	1	1	yes	characteristic
145	C₄	#15689183	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	#20243507	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245154

C₂

yes

maximal, characteristic

#20243143

C₂

yes

maximal, characteristic

#20241476

C₂

yes

maximal

#20241475

C₂

yes

maximal

C₄₇×#2151

C₂

yes

maximal, characteristic

C₂×#13093964

C₂²

yes

—

#15689078

C₂²

yes

—

#15688944

—

C₄₇⋊#338

C₄

yes

characteristic

#15688880

C₄

yes

characteristic

#15688879

C₄

yes

—

C₂×M₃₂×C₄₇

C₂²

yes

characteristic

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₄₇×#405

C₂²

yes

characteristic

C₄₇×C₄⋊C₁₆

—

C₂×#10625329

C₂×C₄

yes

—

#13093983

C₂×C₄

yes

—

#13093964

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

#13093934

—

C₈×Dic₄₇

—

C₂×M₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

#117×C₄₇

C₂×C₄

yes

characteristic

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

#97×C₄₇

Q₈

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#131

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#131

yes

characteristic

C₂×C₄×C₄₇

#131

yes

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₈×C₄₇

yes

—

C₄₇⋊C₈

—

#12885

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#424

yes

characteristic

C₂²×C₄₇

#433

yes

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

#2153

—

#2151

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1832

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

C₂×D₄₇

yes

characteristic

C₂²×C₁₆

C₂×D₄₇

yes

characteristic

#405

—

#405

C₂×D₄₇

yes

characteristic

#364

—

C₄⋊C₁₆

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₄₇

#12885

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₄×D₄₇

yes

characteristic

C₂²×C₈

C₂²×D₄₇

yes

characteristic

#117

—

#117

C₄×D₄₇

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

#97

—

104

#97

Dic₉₄

yes

—

105

C₄×C₈

—

106

C₄×C₈

C₂²⋊D₄₇

yes

—

107

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

Dic₄₇⋊C₄

yes

—

113

C₂×C₈

C₂×Dic₉₄

yes

characteristic

114

C₂×C₈

#10625352

yes

—

115

C₂×C₈

#10625354

yes

—

116

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

Dic₄₇⋊C₄

yes

—

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

188

—

126

C₂³

#13094030

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

188

—

129

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

130

C₂×C₄

#13093982

yes

—

131

C₂×C₄

#13094030

yes

characteristic

132

C₂×C₄

#13094032

yes

—

133

C₈

—

134

C₈

—

188

—

135

C₈

#13094011

yes

characteristic

136

C₈

#13094013

yes

characteristic

137

C₈

#13094015

yes

—

138

C₂²

—

139

C₂²

#15689149

yes

characteristic

140

C₂²

#15689183

yes

—

141

C₄

—

142

C₄

—

376

—

143

C₄

#15688988

yes

characteristic

144

C₄

#15689151

yes

characteristic

145

C₄

#15689183

yes

—

146

C₂

—

147

C₂

—

148

C₂

#20243507

yes

characteristic

	subgroup	G/H
center	C₈	#13094013
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#2151	C₂
socle	C₂×C₄₇	#1832
prime radical	C₂³×C₄₇	#131
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#2151	D₄₇
hypocenter	C₄₇	#12885
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

subgroup

G/H

center

C₈

#13094013

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#2151

C₂

C₂×C₄₇

C₂³×C₄₇

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2151

N	H ≅ G/N	type
90. C₄₇	#12885	semidirect

H ≅ G/N

type

90. C₄₇

#12885

semidirect

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

subgroup	index
#20245154	2
#20243143	2
#20241476	2
#20241475	2
C₄₇×#2151	2
#12885	47

subgroup

index

C₄₇×#2151

Subgroups — #26132644

Conjugacy classes of subgroups

Complements

Maximal subgroups