Group #26103100 — subgroups

Subgroups — #26103100

Order 12032.

216 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20241472	C₂	1	2	yes	maximal
2	C₄₇×#1985	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093964	—	2	2	no	—
4	C₂×C₄×Q₈×C₄₇	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₈×C₄₇	C₄	1	1	yes	characteristic
6	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
7	C₄₇×#336	—	2	2	no	—
8	#13093964	—	2	4	no	—
9	#13093964	—	4	2	no	—
10	C₂×C₄₇⋊C₁₆	—	4	2	no	—
11	C₂²×Q₈×C₄₇	C₈	1	1	yes	characteristic
12	C₂²×C₈×C₄₇	—	2	1	no	—
13	C₂²×C₈×C₄₇	D₄	1	2	yes	—
14	C₄×Q₈×C₄₇	—	4	1	no	—
15	C₄×Q₈×C₄₇	—	4	1	no	—
16	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
17	C₂×C₄⋊C₄×C₄₇	C₈	1	1	yes	characteristic
18	C₂×C₄²×C₄₇	—	2	1	no	—
19	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₄⋊C₈×C₄₇	—	4	1	no	—
21	#97×C₄₇	—	4	1	no	—
22	C₂×Q₈×C₄₇	—	4	2	no	—
23	C₂×Q₈×C₄₇	—	4	1	no	—
24	C₂²×C₄×C₄₇	—	2	1	no	—
25	C₂²×C₄×C₄₇	—	2	1	no	—
26	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
29	C₂×C₈×C₄₇	—	2	2	no	—
30	C₂×C₈×C₄₇	—	2	2	no	—
31	C₂×C₈×C₄₇	—	4	1	no	—
32	C₂×C₈×C₄₇	—	4	2	no	—
33	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
34	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
35	C₄⋊C₄×C₄₇	—	4	2	no	—
36	C₄⋊C₄×C₄₇	—	4	1	no	—
37	C₄²×C₄₇	—	2	1	no	—
38	C₄²×C₄₇	—	2	1	no	—
39	C₄²×C₄₇	—	4	1	no	—
40	C₄²×C₄₇	—	4	1	no	—
41	C₄₇⋊C₁₆	—	4	4	no	—
42	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
43	Q₈×C₄₇	—	4	2	no	—
44	Q₈×C₄₇	—	4	2	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	—	4	1	no	—
49	C₂×C₄×C₄₇	—	4	1	no	—
50	C₂×C₄×C₄₇	—	4	1	no	—
51	C₂×C₄×C₄₇	—	4	1	no	—
52	C₂×C₄×C₄₇	—	4	2	no	—
53	C₂×C₄×C₄₇	—	4	1	no	—
54	C₂×C₄×C₄₇	#99	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#103	1	2	yes	—
57	C₂×C₄×C₄₇	#104	1	2	yes	—
58	C₈×C₄₇	—	2	2	no	—
59	C₈×C₄₇	—	2	2	no	—
60	C₈×C₄₇	—	8	1	no	—
61	#7133	—	47	1	no	maximal, Sylow
62	C₂²×C₄₇	—	2	1	no	—
63	C₂²×C₄₇	—	2	1	no	—
64	C₂²×C₄₇	#323	1	1	yes	characteristic
65	C₂²×C₄₇	Q₈⋊C₈	1	2	yes	—
66	C₄×C₄₇	—	4	2	no	—
67	C₄×C₄₇	—	4	1	no	—
68	C₄×C₄₇	—	4	1	no	—
69	C₄×C₄₇	—	8	1	no	—
70	C₄×C₄₇	#328	1	2	yes	—
71	C₄×C₄₇	#329	1	1	yes	characteristic
72	C₄×C₄₇	#333	1	1	yes	characteristic
73	#1985	D₄₇	1	1	yes	characteristic
74	#1377	—	47	2	no	—
75	C₂×C₄₇	—	2	2	no	—
76	C₂×C₄₇	#1272	1	1	yes	characteristic
77	C₂×C₄₇	#1320	1	1	yes	characteristic
78	C₂×C₄₇	#1321	1	1	yes	characteristic
79	C₂×C₄×Q₈	Dic₄₇	1	1	yes	characteristic
80	C₂×M₃₂	—	94	2	no	—
81	C₂×C₄⋊C₈	Dic₄₇	1	1	yes	characteristic
82	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
83	#336	—	2	2	no	—
84	C₄₇	#7133	1	1	yes	characteristic, complemented, Sylow
85	C₂²×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
86	C₂²×C₈	—	2	1	no	—
87	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
88	C₄×Q₈	—	4	1	no	—
89	C₄×Q₈	—	4	1	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	C₄₇⋊C₈	1	1	yes	characteristic
92	C₂×C₄²	—	2	1	no	—
93	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
94	M₃₂	—	94	4	no	—
95	M₃₂	—	188	2	no	—
96	C₂×C₁₆	—	188	2	no	—
97	C₄⋊C₈	—	4	1	no	—
98	#97	—	4	1	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
104	C₂²×C₄	#10625329	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	4	1	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	#10625336	1	2	yes	—
111	C₂×C₈	#10625337	1	2	yes	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	4	1	no	—
117	C₄²	—	4	1	no	—
118	C₁₆	—	188	4	no	—
119	C₂³	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
120	Q₈	—	4	2	no	—
121	Q₈	—	4	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	#13093939	1	1	yes	characteristic
132	C₂×C₄	#13093941	1	1	yes	characteristic
133	C₂×C₄	#13093948	1	2	yes	—
134	C₂×C₄	#13093951	1	2	yes	—
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	—	8	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	#15688847	1	1	yes	characteristic
141	C₂²	Q₈⋊(C₄₇⋊C₈)	1	2	yes	—
142	C₄	—	4	2	no	—
143	C₄	—	4	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	8	1	no	—
146	C₄	#15688860	1	2	yes	—
147	C₄	#15688863	1	1	yes	characteristic
148	C₄	#15688872	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	#20241214	1	1	yes	characteristic
151	C₂	#20241324	1	1	yes	characteristic
152	C₂	#20241326	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20241472

C₂

yes

maximal

C₄₇×#1985

C₂

yes

maximal, characteristic

C₂×#13093964

—

C₂×C₄×Q₈×C₄₇

C₄

yes

characteristic

C₂×C₄⋊C₈×C₄₇

C₄

yes

characteristic

C₂×#97×C₄₇

C₂²

yes

characteristic

C₄₇×#336

—

#13093964

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₂²×Q₈×C₄₇

C₈

yes

characteristic

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₈

yes

characteristic

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₄₇

—

#97×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

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

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂²⋊C₈

yes

characteristic

Q₈×C₄₇

—

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

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#104

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

#7133

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#323

yes

characteristic

C₂²×C₄₇

Q₈⋊C₈

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#328

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

D₄₇

yes

characteristic

#1377

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄×Q₈

Dic₄₇

yes

characteristic

C₂×M₃₂

—

C₂×C₄⋊C₈

Dic₄₇

yes

characteristic

C₂×#97

C₂×D₄₇

yes

characteristic

#336

—

C₄₇

#7133

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₄₇⋊C₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₄₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₄₇⋊C₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

M₃₂

—

M₃₂

—

188

—

C₂×C₁₆

—

188

—

C₄⋊C₈

—

#97

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂×C₄₇⋊C₈

yes

characteristic

104

C₂²×C₄

#10625329

yes

characteristic

105

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

#10625336

yes

—

111

C₂×C₈

#10625337

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₁₆

—

188

—

119

C₂³

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#13093939

yes

characteristic

132

C₂×C₄

#13093941

yes

characteristic

133

C₂×C₄

#13093948

yes

—

134

C₂×C₄

#13093951

yes

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

#15688847

yes

characteristic

141

C₂²

Q₈⋊(C₄₇⋊C₈)

yes

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#15688860

yes

—

147

C₄

#15688863

yes

characteristic

148

C₄

#15688872

yes

characteristic

149

C₂

—

150

C₂

#20241214

yes

characteristic

151

C₂

#20241324

yes

characteristic

152

C₂

#20241326

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13093939
commutator	C₂²×C₄×C₄₇	C₂×C₈
Frattini	C₂×#97	C₂×D₄₇
Fitting	C₄₇×#1985	C₂
socle	C₂²×C₄₇	#323
prime radical	C₂³×C₄₇	C₂²⋊C₈
cosocle	C₂×#97×C₄₇	C₂²
hypercenter	#1985	D₄₇
hypocenter	C₄₇	#7133
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1985

subgroup

G/H

center

C₂×C₄

#13093939

commutator

C₂²×C₄×C₄₇

C₂×C₈

Frattini

C₂×#97

C₂×D₄₇

Fitting

C₄₇×#1985

C₂

socle

C₂²×C₄₇

#323

prime radical

C₂³×C₄₇

C₂²⋊C₈

cosocle

C₂×#97×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1985

N	H ≅ G/N	type
84. C₄₇	#7133	semidirect

H ≅ G/N

type

84. C₄₇

#7133

semidirect

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

subgroup	index
#20241472	2
C₄₇×#1985	2
#7133	47

subgroup

index

#20241472

C₄₇×#1985

#7133

Subgroups — #26103100

Conjugacy classes of subgroups

Complements

Maximal subgroups