Group #26103094 — subgroups

Subgroups — #26103094

Order 12032.

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

normal subgroups only

This group has 206 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	#20241468	C₂	1	2	yes	maximal
2	C₄₇×#1992	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093964	—	2	2	no	—
4	C₂×#125×C₄₇	C₄	1	1	yes	characteristic
5	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
6	C₄₇×#384	C₄	1	1	yes	characteristic
7	C₄₇×#340	—	2	2	no	—
8	#13093964	—	4	2	no	—
9	#13093964	—	4	2	no	—
10	C₂×C₄₇⋊C₁₆	—	2	4	no	—
11	C₂²×C₈×C₄₇	D₄	1	2	yes	—
12	#125×C₄₇	—	2	1	no	—
13	#125×C₄₇	—	4	1	no	—
14	#125×C₄₇	C₈	1	2	yes	—
15	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
16	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
17	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
18	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
19	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
20	#97×C₄₇	—	4	1	no	—
21	C₂²×C₄×C₄₇	—	2	1	no	—
22	C₂²×C₄×C₄₇	—	4	1	no	—
23	C₂²×C₄×C₄₇	—	4	1	no	—
24	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₄₇	—	2	2	no	—
26	C₂×C₈×C₄₇	—	2	2	no	—
27	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
28	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
29	C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₄⋊C₄×C₄₇	—	4	2	no	—
31	C₄⋊C₄×C₄₇	—	4	2	no	—
32	C₄⋊C₄×C₄₇	—	4	1	no	—
33	C₄⋊C₄×C₄₇	—	4	2	no	—
34	C₄⋊C₄×C₄₇	—	8	2	no	—
35	C₄²×C₄₇	—	2	1	no	—
36	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
38	C₄₇⋊C₁₆	—	4	4	no	—
39	C₂³×C₄₇	#100	1	1	yes	characteristic
40	C₂×C₄×C₄₇	—	2	1	no	—
41	C₂×C₄×C₄₇	—	2	1	no	—
42	C₂×C₄×C₄₇	—	4	1	no	—
43	C₂×C₄×C₄₇	—	4	1	no	—
44	C₂×C₄×C₄₇	—	4	2	no	—
45	C₂×C₄×C₄₇	—	8	1	no	—
46	C₂×C₄×C₄₇	—	8	1	no	—
47	C₂×C₄×C₄₇	—	8	2	no	—
48	C₂×C₄×C₄₇	—	8	1	no	—
49	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	#99	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#103	1	2	yes	—
52	C₂×C₄×C₄₇	#104	1	2	yes	—
53	C₈×C₄₇	—	2	2	no	—
54	C₈×C₄₇	—	2	2	no	—
55	#7130	—	47	1	no	maximal, Sylow
56	C₂²×C₄₇	—	2	1	no	—
57	C₂²×C₄₇	—	2	1	no	—
58	C₂²×C₄₇	#323	1	1	yes	characteristic
59	C₂²×C₄₇	#329	1	1	yes	characteristic
60	C₂²×C₄₇	#333	1	1	yes	characteristic
61	C₄×C₄₇	—	4	1	no	—
62	C₄×C₄₇	—	4	1	no	—
63	C₄×C₄₇	—	8	2	no	—
64	C₄×C₄₇	—	16	1	no	—
65	C₄×C₄₇	Q₈⋊C₈	1	2	yes	—
66	C₄×C₄₇	#328	1	2	yes	—
67	#1992	D₄₇	1	1	yes	characteristic
68	#1375	—	47	2	no	—
69	C₂×C₄₇	—	2	1	no	—
70	C₂×C₄₇	—	2	1	no	—
71	C₂×C₄₇	#1272	1	1	yes	characteristic
72	C₂×C₄₇	#1355	1	1	yes	characteristic
73	C₂×C₄₇	#1356	1	1	yes	characteristic
74	C₂×#125	Dic₄₇	1	1	yes	characteristic
75	C₂×M₃₂	—	94	2	no	—
76	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
77	#384	Dic₄₇	1	1	yes	characteristic
78	#340	—	2	2	no	—
79	C₄₇	#7130	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
81	#125	—	2	1	no	—
82	#125	—	4	1	no	—
83	#125	C₄₇⋊C₈	1	2	yes	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	4	1	no	—
87	C₂×C₄⋊C₄	—	4	1	no	—
88	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
89	M₃₂	—	188	2	no	—
90	M₃₂	—	188	2	no	—
91	C₂×C₁₆	—	94	4	no	—
92	#97	—	4	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	—	4	1	no	—
96	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	#10625336	1	2	yes	—
100	C₂×C₈	#10625337	1	2	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	8	2	no	—
107	C₄²	—	2	1	no	—
108	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
109	C₄²	#10625329	1	1	yes	characteristic
110	C₁₆	—	188	4	no	—
111	C₂³	#13093941	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	8	1	no	—
118	C₂×C₄	—	8	1	no	—
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	—	8	1	no	—
121	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
122	C₂×C₄	#13093939	1	1	yes	characteristic
123	C₂×C₄	#13093948	1	2	yes	—
124	C₂×C₄	#13093951	1	2	yes	—
125	C₈	—	2	2	no	—
126	C₈	—	2	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	#15688847	1	1	yes	characteristic
130	C₂²	#15688863	1	1	yes	characteristic
131	C₂²	#15688872	1	1	yes	characteristic
132	C₄	—	4	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	8	2	no	—
135	C₄	—	16	1	no	—
136	C₄	Q₈⋊(C₄₇⋊C₈)	1	2	yes	—
137	C₄	#15688860	1	2	yes	—
138	C₂	—	2	1	no	—
139	C₂	—	2	1	no	—
140	C₂	#20241214	1	1	yes	characteristic
141	C₂	#20241421	1	1	yes	characteristic
142	C₂	#20241423	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20241468

C₂

yes

maximal

C₄₇×#1992

C₂

yes

maximal, characteristic

C₂×#13093964

—

C₂×#125×C₄₇

C₄

yes

characteristic

C₂×#97×C₄₇

C₂²

yes

characteristic

C₄₇×#384

C₄

yes

characteristic

C₄₇×#340

—

#13093964

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₂²×C₈×C₄₇

D₄

yes

—

#125×C₄₇

—

#125×C₄₇

—

#125×C₄₇

C₈

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

#97×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₄²×C₄₇

M₁₆

yes

characteristic

C₄₇⋊C₁₆

—

C₂³×C₄₇

#100

yes

characteristic

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

C₂²⋊C₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#104

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

#7130

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

Q₈⋊C₈

yes

—

C₄×C₄₇

#328

yes

—

#1992

D₄₇

yes

characteristic

#1375

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

Dic₄₇

yes

characteristic

C₂×M₃₂

—

C₂×#97

C₂×D₄₇

yes

characteristic

#384

Dic₄₇

yes

characteristic

#340

—

C₄₇

#7130

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₄₇

yes

—

#125

—

#125

—

#125

C₄₇⋊C₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

M₃₂

—

188

—

M₃₂

—

188

—

C₂×C₁₆

—

#97

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#10625336

yes

—

100

C₂×C₈

#10625337

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

C₂×C₄₇⋊C₈

yes

characteristic

109

C₄²

#10625329

yes

characteristic

110

C₁₆

—

188

—

111

C₂³

#13093941

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

122

C₂×C₄

#13093939

yes

characteristic

123

C₂×C₄

#13093948

yes

—

124

C₂×C₄

#13093951

yes

—

125

C₈

—

126

C₈

—

127

C₂²

—

128

C₂²

—

129

C₂²

#15688847

yes

characteristic

130

C₂²

#15688863

yes

characteristic

131

C₂²

#15688872

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

Q₈⋊(C₄₇⋊C₈)

yes

—

137

C₄

#15688860

yes

—

138

C₂

—

139

C₂

—

140

C₂

#20241214

yes

characteristic

141

C₂

#20241421

yes

characteristic

142

C₂

#20241423

yes

characteristic

	subgroup	G/H
center	C₂²	#15688863
commutator	C₄²×C₄₇	C₂×C₈
Frattini	C₂×#97	C₂×D₄₇
Fitting	C₄₇×#1992	C₂
socle	C₂²×C₄₇	#329
prime radical	C₂³×C₄₇	#100
cosocle	C₂×#97×C₄₇	C₂²
hypercenter	#1992	D₄₇
hypocenter	C₄₇	#7130
LCS	G ▷ C₄²×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1992

subgroup

G/H

center

C₂²

#15688863

commutator

C₄²×C₄₇

C₂×C₈

Frattini

C₂×#97

C₂×D₄₇

Fitting

C₄₇×#1992

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×#97×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1992

N	H ≅ G/N	type
79. C₄₇	#7130	semidirect

H ≅ G/N

type

79. C₄₇

#7130

semidirect

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

subgroup	index
#20241468	2
C₄₇×#1992	2
#7130	47

subgroup

index

#20241468

C₄₇×#1992

#7130

Subgroups — #26103094

Conjugacy classes of subgroups

Complements

Maximal subgroups