Group #26103709 — subgroups

Subgroups — #26103709

Order 12032.

170 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 39 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	#20243607	C₂	1	1	yes	maximal, characteristic
2	#20241314	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1321	C₂	1	1	yes	maximal, characteristic
4	#15689890	C₄	1	1	yes	characteristic
5	#15689171	—	2	1	no	—
6	#15689107	C₄	1	1	yes	characteristic
7	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	#15688888	—	2	1	no	—
9	C₄₇×#408	C₂²	1	1	yes	characteristic
10	C₄₇×#349	—	2	1	no	—
11	#13094031	—	4	2	no	—
12	C₂×#10625329	—	2	1	no	—
13	C₂×#10625329	—	2	1	no	—
14	C₂×#10625329	—	4	1	no	—
15	C₂×C₄₇⋊C₁₆	—	4	1	no	—
16	#13093959	—	4	2	no	—
17	#13093943	—	2	1	no	—
18	#13093941	C₈	1	2	yes	—
19	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
22	M₃₂×C₄₇	—	4	1	no	—
23	C₂²⋊C₈×C₄₇	—	4	1	no	—
24	#41×C₄₇	—	4	1	no	—
25	#41×C₄₇	—	4	1	no	—
26	C₂×Q₈×C₄₇	M₁₆	1	1	yes	characteristic
27	C₂×D₄×C₄₇	—	2	1	no	—
28	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	—	2	1	no	—
30	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
31	M₁₆×C₄₇	—	4	1	no	—
32	C₂×C₈×C₄₇	—	2	1	no	—
33	C₂×C₈×C₄₇	—	2	1	no	—
34	C₂×C₈×C₄₇	—	4	1	no	—
35	#10625329	—	2	2	no	—
36	#10625329	—	2	2	no	—
37	#10625329	—	4	1	no	—
38	#10625329	—	4	2	no	—
39	#10625329	—	4	1	no	—
40	C₂×C₄₇⋊C₈	—	4	1	no	—
41	C₂×C₄₇⋊C₈	—	4	1	no	—
42	C₂×C₄₇⋊C₈	—	4	2	no	—
43	C₁₆×C₄₇	—	8	1	no	—
44	C₄₇⋊C₁₆	—	8	1	no	—
45	C₂³×C₄₇	—	2	1	no	—
46	C₂³×C₄₇	#101	1	1	yes	characteristic
47	Q₈×C₄₇	—	4	1	no	—
48	D₄×C₄₇	—	4	1	no	—
49	D₄×C₄₇	—	4	2	no	—
50	C₂×C₄×C₄₇	—	2	1	no	—
51	C₂×C₄×C₄₇	—	2	1	no	—
52	C₂×C₄×C₄₇	—	2	1	no	—
53	C₂×C₄×C₄₇	—	4	1	no	—
54	C₂×C₄×C₄₇	—	4	1	no	—
55	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#99	1	1	yes	characteristic
57	C₈×C₄₇	—	4	1	no	—
58	C₈×C₄₇	—	4	1	no	—
59	C₄₇⋊C₈	—	4	2	no	—
60	C₄₇⋊C₈	—	4	2	no	—
61	#7377	—	47	1	no	maximal, Sylow
62	C₂²×C₄₇	—	2	1	no	—
63	C₂²×C₄₇	—	4	1	no	—
64	C₂²×C₄₇	—	4	1	no	—
65	C₂²×C₄₇	—	8	1	no	—
66	C₂²×C₄₇	#324	1	1	yes	characteristic
67	C₄×C₄₇	—	2	1	no	—
68	C₄×C₄₇	—	4	1	no	—
69	C₄×C₄₇	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
70	C₄×C₄₇	#352	1	1	yes	characteristic
71	#1858	—	47	1	no	—
72	#1321	D₄₇	1	1	yes	characteristic
73	#1315	—	47	1	no	—
74	C₂×C₄₇	—	2	1	no	—
75	C₂×C₄₇	—	4	1	no	—
76	C₂×C₄₇	—	8	1	no	—
77	C₂×C₄₇	#1326	1	1	yes	characteristic
78	#568	—	47	1	no	—
79	#430	—	94	1	no	—
80	#413	—	47	1	no	—
81	#408	C₂×D₄₇	1	1	yes	characteristic
82	#349	—	2	1	no	—
83	C₂²⋊C₁₆	—	94	1	no	—
84	#341	—	94	1	no	—
85	C₄₇	#7377	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	C₄×D₄₇	1	1	yes	characteristic
87	#131	—	188	2	no	—
88	C₂×M₁₆	—	94	1	no	—
89	C₂×M₁₆	—	94	1	no	—
90	C₂×M₁₆	—	188	1	no	—
91	C₂×M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
92	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
93	M₃₂	—	4	1	no	—
94	C₂×C₁₆	—	188	1	no	—
95	#108	—	188	2	no	—
96	#101	—	94	1	no	—
97	#100	—	47	2	no	—
98	C₂²⋊C₈	—	4	1	no	—
99	#41	—	4	1	no	—
100	#41	—	4	1	no	—
101	C₂×Q₈	#10625324	1	1	yes	characteristic
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	C₈×D₄₇	1	1	yes	characteristic
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
106	M₁₆	—	4	1	no	—
107	M₁₆	—	94	2	no	—
108	M₁₆	—	94	2	no	—
109	M₁₆	—	188	1	no	—
110	M₁₆	—	188	2	no	—
111	M₁₆	—	188	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	188	1	no	—
116	C₂×C₈	—	188	1	no	—
117	C₂×C₈	—	188	2	no	—
118	C₁₆	—	8	1	no	—
119	C₁₆	—	376	1	no	—
120	C₂³	—	2	1	no	—
121	C₂³	#13093942	1	1	yes	characteristic
122	Q₈	—	4	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
131	C₂×C₄	#13093938	1	1	yes	characteristic
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	188	2	no	—
135	C₈	—	188	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	#15688848	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	1	no	—
143	C₄	#15688909	1	1	yes	characteristic
144	C₄	#15688914	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	4	1	no	—
147	C₂	—	8	1	no	—
148	C₂	#20241335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243607

C₂

yes

maximal, characteristic

#20241314

C₂

yes

maximal, characteristic

C₄₇×#1321

C₂

yes

maximal, characteristic

#15689890

C₄

yes

characteristic

#15689171

—

#15689107

C₄

yes

characteristic

C₂²⋊(C₄₇⋊C₁₆)

—

#15688888

—

C₄₇×#408

C₂²

yes

characteristic

C₄₇×#349

—

—

—

—

—

C₂×C₄₇⋊C₁₆

—

#13093959

—

#13093943

—

#13093941

C₈

yes

—

C₂×#41×C₄₇

C₂×C₄

yes

characteristic

C₂×M₁₆×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

D₄

yes

characteristic

M₃₂×C₄₇

—

C₂²⋊C₈×C₄₇

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

M₁₆

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

—

—

—

—

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

C₂³×C₄₇

#101

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₄₇

#99

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#7377

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#324

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

AΣL(1,𝔽₁₆)

yes

C₄×C₄₇

yes

—

D₄₇

yes

characteristic

#1315

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

—

—

—

C₂×D₄₇

yes

characteristic

#349

—

C₂²⋊C₁₆

—

#341

—

C₄₇

#7377

yes

characteristic, complemented, Sylow

C₂×#41

C₄×D₄₇

yes

characteristic

#131

—

188

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

188

—

C₂×M₁₆

C₂²⋊D₄₇

yes

characteristic

C₂²×C₈

D₁₈₈

yes

characteristic

M₃₂

—

C₂×C₁₆

—

188

—

#108

—

188

—

#101

—

#100

—

C₂²⋊C₈

—

#41

—

100

#41

—

101

C₂×Q₈

#10625324

yes

characteristic

102

C₂×D₄

—

103

C₂×D₄

C₈×D₄₇

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

188

—

110

M₁₆

—

188

—

111

M₁₆

—

188

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

188

—

116

C₂×C₈

—

188

—

117

C₂×C₈

—

188

—

118

C₁₆

—

119

C₁₆

—

376

—

120

C₂³

—

121

C₂³

#13093942

yes

characteristic

122

Q₈

—

123

D₄

—

124

D₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

(C₂×D₄₇)⋊C₈

yes

characteristic

131

C₂×C₄

#13093938

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

188

—

135

C₈

—

188

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#15688848

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

#15688909

yes

characteristic

144

C₄

#15688914

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#20241335

yes

characteristic

	subgroup	G/H
center	C₄	#15688909
commutator	C₂×D₄×C₄₇	C₂×C₈
Frattini	#408	C₂×D₄₇
Fitting	C₄₇×#1321	C₂
socle	C₂×C₄₇	#1326
prime radical	C₂×#41×C₄₇	C₂×C₄
cosocle	C₄₇×#408	C₂²
hypercenter	#1321	D₄₇
hypocenter	C₄₇	#7377
LCS	G ▷ C₂×D₄×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ #1321

subgroup

G/H

center

C₄

#15688909

commutator

C₂×D₄×C₄₇

C₂×C₈

Frattini

#408

C₂×D₄₇

Fitting

C₄₇×#1321

C₂

socle

C₂×C₄₇

#1326

prime radical

C₂×#41×C₄₇

C₂×C₄

cosocle

C₄₇×#408

C₂²

D₄₇

C₄₇

G ▷ C₂×D₄×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ #1321

N	H ≅ G/N	type
85. C₄₇	#7377	semidirect

H ≅ G/N

type

85. C₄₇

#7377

semidirect

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

subgroup	index
#20243607	2
#20241314	2
C₄₇×#1321	2
#7377	47

subgroup

index

#20243607

#20241314

C₄₇×#1321

#7377

Subgroups — #26103709

Conjugacy classes of subgroups

Complements

Maximal subgroups