Group #26109800 — subgroups

Subgroups — #26109800

Order 12032.

566 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 107 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	#20243312	C₂	1	2	yes	maximal
2	#20241296	C₂	1	4	yes	maximal
3	C₄₇×#2310	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093943	—	2	4	no	—
5	#15689087	C₂²	1	4	yes	—
6	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
7	C₂×#117×C₄₇	C₄	1	2	yes	—
8	C₄₇×#391	—	2	4	no	—
9	C₄₇×#381	C₄	1	2	yes	—
10	C₄₇×#381	C₂²	1	2	yes	—
11	C₂×#10625329	—	2	4	no	—
12	#13093943	—	4	8	no	—
13	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
14	C₂²×Q₈×C₄₇	—	2	4	no	—
15	C₂²×Q₈×C₄₇	—	4	4	no	—
16	C₂²×Q₈×C₄₇	D₄	1	4	yes	—
17	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
20	#117×C₄₇	—	4	4	no	—
21	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
22	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₄₇	—	2	2	no	—
24	#95×C₄₇	—	2	4	no	—
25	#95×C₄₇	—	2	4	no	—
26	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×Q₈×C₄₇	—	2	8	no	—
28	C₂×Q₈×C₄₇	—	2	8	no	—
29	C₂×Q₈×C₄₇	—	4	4	no	—
30	C₂×Q₈×C₄₇	—	4	16	no	—
31	C₂²×C₄×C₄₇	—	2	2	no	—
32	C₂²×C₄×C₄₇	—	2	4	no	—
33	C₂²×C₄×C₄₇	—	2	2	no	—
34	C₂²×C₄×C₄₇	—	2	4	no	—
35	C₂²×C₄×C₄₇	—	4	2	no	—
36	C₂²×C₄×C₄₇	—	4	2	no	—
37	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₄₇	#41	1	4	yes	—
41	C₄⋊C₄×C₄₇	—	4	4	no	—
42	C₂²⋊C₄×C₄₇	—	4	4	no	—
43	C₄²×C₄₇	—	4	4	no	—
44	#10625329	—	4	8	no	—
45	C₂×C₄₇⋊C₈	—	4	4	no	—
46	C₂³×C₄₇	—	2	2	no	—
47	C₂³×C₄₇	—	4	2	no	—
48	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂³×C₄₇	#117	1	2	yes	—
50	Q₈×C₄₇	—	4	16	no	—
51	C₂×C₄×C₄₇	—	2	8	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	2	4	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	—	2	2	no	—
56	C₂×C₄×C₄₇	—	2	4	no	—
57	C₂×C₄×C₄₇	—	4	4	no	—
58	C₂×C₄×C₄₇	—	4	4	no	—
59	C₂×C₄×C₄₇	—	4	8	no	—
60	C₂×C₄×C₄₇	—	4	4	no	—
61	C₂×C₄×C₄₇	#99	1	4	yes	—
62	C₂×C₄×C₄₇	#123	1	4	yes	—
63	C₄₇⋊C₈	—	8	4	no	—
64	#8753	—	47	1	no	maximal, Sylow
65	C₂²×C₄₇	—	2	2	no	—
66	C₂²×C₄₇	—	2	2	no	—
67	C₂²×C₄₇	—	2	4	no	—
68	C₂²×C₄₇	—	4	4	no	—
69	C₂²×C₄₇	#381	1	1	yes	characteristic
70	C₂²×C₄₇	C₂×#99	1	2	yes	—
71	C₄×C₄₇	—	2	4	no	—
72	C₄×C₄₇	—	2	4	no	—
73	C₄×C₄₇	—	4	2	no	—
74	C₄×C₄₇	—	8	4	no	—
75	#2310	D₄₇	1	1	yes	characteristic
76	#1784	—	47	2	no	—
77	#1306	—	47	4	no	—
78	C₂×C₄₇	—	2	2	no	—
79	C₂×C₄₇	—	4	2	no	—
80	C₂×C₄₇	#1819	1	1	yes	characteristic
81	C₂×C₄₇	#1830	1	2	yes	—
82	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
83	C₂×#117	Dic₄₇	1	2	yes	—
84	C₂×#101	—	94	4	no	—
85	#407	—	47	4	no	—
86	#391	—	2	4	no	—
87	#381	Dic₄₇	1	2	yes	—
88	#381	C₂×D₄₇	1	2	yes	—
89	C₄₇	#8753	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	—	2	4	no	—
91	C₂²×Q₈	—	4	4	no	—
92	C₂²×Q₈	C₂²⋊D₄₇	1	4	yes	—
93	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
94	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
95	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
96	C₂×M₁₆	—	94	4	no	—
97	#117	—	4	4	no	—
98	C₂×C₄⋊C₄	—	2	2	no	—
99	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
100	C₂×C₄²	—	2	2	no	—
101	#101	—	188	8	no	—
102	C₂²⋊C₈	—	94	8	no	—
103	#95	—	2	4	no	—
104	#95	—	2	4	no	—
105	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
106	C₂×Q₈	—	2	8	no	—
107	C₂×Q₈	—	2	8	no	—
108	C₂×Q₈	—	4	4	no	—
109	C₂×Q₈	—	4	16	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	2	4	no	—
114	C₂²×C₄	—	4	2	no	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
117	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
118	C₂²×C₄	#10625350	1	4	yes	—
119	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
120	M₁₆	—	188	8	no	—
121	C₂×C₈	—	188	4	no	—
122	C₄⋊C₄	—	4	4	no	—
123	C₂²⋊C₄	—	4	4	no	—
124	C₄²	—	4	4	no	—
125	C₂³	—	2	2	no	—
126	C₂³	—	4	2	no	—
127	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
128	C₂³	#13093983	1	2	yes	—
129	Q₈	—	4	16	no	—
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	4	8	no	—
139	C₂×C₄	—	4	4	no	—
140	C₂×C₄	#13093939	1	4	yes	—
141	C₂×C₄	#13094007	1	4	yes	—
142	C₈	—	376	4	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	4	no	—
146	C₂²	—	4	4	no	—
147	C₂²	#15688986	1	1	yes	characteristic
148	C₂²	C₂×#13093939	1	2	yes	—
149	C₄	—	2	4	no	—
150	C₄	—	2	4	no	—
151	C₄	—	4	2	no	—
152	C₄	—	8	4	no	—
153	C₂	—	2	2	no	—
154	C₂	—	4	2	no	—
155	C₂	#20243462	1	1	yes	characteristic
156	C₂	#20243502	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243312

C₂

yes

maximal

#20241296

C₂

yes

maximal

C₄₇×#2310

C₂

yes

maximal, characteristic

C₂×#13093943

—

#15689087

C₂²

yes

—

C₂³×Q₈×C₄₇

C₂²

yes

characteristic

C₂×#117×C₄₇

C₄

yes

—

C₄₇×#391

—

C₄₇×#381

C₄

yes

—

C₄₇×#381

C₂²

yes

—

C₂×#10625329

—

#13093943

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

D₄

yes

—

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

C₂³

yes

characteristic

#117×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

—

#95×C₄₇

—

#95×C₄₇

—

C₂⁴×C₄₇

C₂²×C₄

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

#117

yes

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#99

yes

—

C₂×C₄×C₄₇

#123

yes

—

C₄₇⋊C₈

—

#8753

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#381

yes

characteristic

C₂²×C₄₇

C₂×#99

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

#2310

D₄₇

yes

characteristic

#1784

—

#1306

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1819

yes

characteristic

C₂×C₄₇

#1830

yes

—

C₂³×Q₈

C₂×D₄₇

yes

characteristic

C₂×#117

Dic₄₇

yes

—

—

—

—

Dic₄₇

yes

—

#381

C₂×D₄₇

yes

—

C₄₇

#8753

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂²⋊D₄₇

yes

—

C₂³×C₄

C₂×Dic₄₇

yes

characteristic

C₂³×C₄

C₂×Dic₄₇

yes

characteristic

C₂³×C₄

C₂²×D₄₇

yes

characteristic

C₂×M₁₆

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

C₂×Dic₄₇

yes

—

100

C₂×C₄²

—

101

#101

—

188

—

102

C₂²⋊C₈

—

103

#95

—

104

#95

—

105

C₂⁴

C₂²×Dic₄₇

yes

characteristic

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

117

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

118

C₂²×C₄

#10625350

yes

—

119

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

120

M₁₆

—

188

—

121

C₂×C₈

—

188

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₂×C₂²⋊Dic₄₇

yes

characteristic

128

C₂³

#13093983

yes

—

129

Q₈

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#13093939

yes

—

141

C₂×C₄

#13094007

yes

—

142

C₈

—

376

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#15688986

yes

characteristic

148

C₂²

C₂×#13093939

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₂

—

154

C₂

—

155

C₂

#20243462

yes

characteristic

156

C₂

#20243502

yes

—

	subgroup	G/H
center	C₂²	#15688986
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2310	C₂
socle	C₂²×C₄₇	#381
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2310	D₄₇
hypocenter	C₄₇	#8753
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2310

subgroup

G/H

center

C₂²

#15688986

commutator, prime radical

C₂⁴×C₄₇

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₄₇

Fitting

C₄₇×#2310

C₂

socle

C₂²×C₄₇

#381

cosocle

C₂³×C₄×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂⁴×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2310

N	H ≅ G/N	type
89. C₄₇	#8753	semidirect

H ≅ G/N

type

89. C₄₇

#8753

semidirect

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

subgroup	index
#20243312	2
#20241296	2
C₄₇×#2310	2
#8753	47

subgroup

index

#20243312

#20241296

C₄₇×#2310

#8753

Subgroups — #26109800

Conjugacy classes of subgroups

Complements

Maximal subgroups