Group #26107448 — subgroups

Subgroups — #26107448

Order 12032.

366 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 179 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	#20242136	C₂	1	2	yes	maximal
2	#20241680	C₂	1	2	yes	maximal
3	#20241221	C₂	1	2	yes	maximal
4	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#15689176	C₂²	1	4	yes	—
6	#15688876	C₄	1	4	yes	complemented
7	#15688874	C₄	1	4	yes	complemented
8	#15688845	—	2	4	no	—
9	C₂×#97×C₄₇	C₄	1	2	yes	—
10	C₂×#97×C₄₇	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄³×C₄₇	C₂²	1	1	yes	characteristic
14	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
15	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
16	C₄×C₄₇⋊C₈	—	2	4	no	—
17	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₄₇	—	2	2	no	—
21	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
23	#97×C₄₇	—	2	4	no	—
24	#97×C₄₇	—	2	4	no	—
25	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₄₇	D₄	1	2	yes	—
28	C₄×C₈×C₄₇	Q₈	1	2	yes	—
29	C₂²×C₄×C₄₇	—	2	2	no	—
30	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	#41	1	2	yes	—
32	C₂×C₈×C₄₇	—	2	2	no	—
33	C₂×C₈×C₄₇	—	2	2	no	—
34	C₂×C₈×C₄₇	—	2	2	no	—
35	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₄₇	C₄²	1	2	yes	—
38	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
39	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
40	C₄²×C₄₇	—	2	2	no	—
41	C₄²×C₄₇	—	4	4	no	—
42	C₄²×C₄₇	M₁₆	1	4	yes	—
43	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
44	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄₇⋊C₈	—	2	8	no	—
47	C₂³×C₄₇	#117	1	1	yes	characteristic
48	C₂×C₄×C₄₇	—	2	4	no	—
49	C₂×C₄×C₄₇	—	2	4	no	—
50	C₂×C₄×C₄₇	—	4	4	no	—
51	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
54	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
55	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
57	C₈×C₄₇	—	2	1	no	—
58	C₈×C₄₇	—	2	1	no	—
59	C₈×C₄₇	—	2	4	no	—
60	C₈×C₄₇	#97	1	4	yes	—
61	C₄₇⋊C₈	—	4	8	no	—
62	#8269	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	—	2	2	no	—
64	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄₇	#432	1	2	yes	—
66	C₄×C₄₇	—	2	2	no	—
67	C₄×C₄₇	—	4	4	no	—
68	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
69	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
70	C₄×C₄₇	#423	1	2	yes	—
71	C₄×C₄₇	#444	1	4	yes	—
72	C₄×#97	D₄₇	1	1	yes	characteristic
73	#1564	—	47	2	no	—
74	#1452	—	47	2	no	—
75	#1276	—	47	2	no	—
76	C₂×C₄₇	—	2	2	no	—
77	C₂×C₄₇	#1767	1	1	yes	characteristic
78	C₂×C₄₇	#1773	1	2	yes	—
79	#431	—	47	4	no	—
80	C₂×#97	Dic₄₇	1	2	yes	—
81	C₂×#97	Dic₄₇	1	2	yes	—
82	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
84	C₄³	C₂×D₄₇	1	1	yes	characteristic
85	#335	—	47	4	no	—
86	#334	—	47	4	no	—
87	#322	—	94	4	no	—
88	C₄₇	#8269	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
90	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
91	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
92	C₂×C₄²	—	2	2	no	—
93	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
94	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
95	C₄⋊C₈	—	47	8	no	—
96	C₂²⋊C₈	—	94	4	no	—
97	#97	—	2	4	no	—
98	#97	—	2	4	no	—
99	C₄×C₈	—	94	4	no	—
100	C₄×C₈	Dic₉₄	1	2	yes	—
101	C₄×C₈	C₄×D₄₇	1	2	yes	—
102	C₄×C₈	C₄×D₄₇	1	2	yes	—
103	C₄×C₈	D₁₈₈	1	2	yes	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	#10625350	1	2	yes	—
106	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	94	8	no	—
111	C₂×C₈	#10625329	1	4	yes	—
112	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
113	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
114	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
115	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
116	C₄²	—	2	2	no	—
117	C₄²	—	4	4	no	—
118	C₄²	#10625329	1	4	yes	—
119	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
120	C₄²	C₂×C₄×D₄₇	1	2	yes	—
121	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
122	C₂³	#13093983	1	1	yes	characteristic
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	4	no	—
126	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
128	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
129	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
130	C₂×C₄	C₂×#10625329	1	2	yes	—
131	C₂×C₄	C₂×#10625329	1	2	yes	—
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	4	no	—
135	C₈	—	188	8	no	—
136	C₈	#13093935	1	4	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
139	C₂²	#15689180	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	C₂×#13093935	1	1	yes	characteristic
143	C₄	C₄×#10625329	1	1	yes	characteristic
144	C₄	#15689147	1	2	yes	—
145	C₄	#15689246	1	4	yes	—
146	C₂	—	2	2	no	—
147	C₂	#20243241	1	1	yes	characteristic
148	C₂	#20243272	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20242136

C₂

yes

maximal

#20241680

C₂

yes

maximal

#20241221

C₂

yes

maximal

C₄×#97×C₄₇

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

#15688876

C₄

yes

complemented

#15688874

C₄

yes

complemented

#15688845

—

C₂×#97×C₄₇

C₄

yes

—

C₂×#97×C₄₇

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄³×C₄₇

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#97×C₄₇

—

#97×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄²

yes

characteristic

C₂×C₈×C₄₇

C₄²

yes

characteristic

C₂×C₈×C₄₇

C₄²

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

M₁₆

yes

—

C₄²×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₄×Q₈

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#97

yes

—

C₄₇⋊C₈

—

#8269

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

#432

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#97

yes

characteristic

C₄×C₄₇

C₄×M₁₆

yes

characteristic

C₄×C₄₇

#423

yes

—

C₄×C₄₇

#444

yes

—

C₄×#97

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

#1767

yes

characteristic

C₂×C₄₇

#1773

yes

—

#431

—

C₂×#97

Dic₄₇

yes

—

C₂×#97

Dic₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄³

C₂×D₄₇

yes

—

—

—

C₄₇

#8269

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂²×C₈

C₂×Dic₄₇

yes

—

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

100

C₄×C₈

Dic₉₄

yes

—

101

C₄×C₈

C₄×D₄₇

yes

—

102

C₄×C₈

C₄×D₄₇

yes

—

103

C₄×C₈

D₁₈₈

yes

—

104

C₂²×C₄

—

105

C₂²×C₄

#10625350

yes

—

106

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

#10625329

yes

—

112

C₂×C₈

C₄×Dic₄₇

yes

characteristic

113

C₂×C₈

C₄×Dic₄₇

yes

characteristic

114

C₂×C₈

C₄×Dic₄₇

yes

—

115

C₂×C₈

C₄⋊Dic₄₇

yes

—

116

C₄²

—

117

C₄²

—

118

C₄²

#10625329

yes

—

119

C₄²

C₂×Dic₉₄

yes

characteristic

120

C₄²

C₂×C₄×D₄₇

yes

—

121

C₄²

C₂×D₁₈₈

yes

characteristic

122

C₂³

#13093983

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂×C₄×Dic₄₇

yes

characteristic

127

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

128

C₂×C₄

C₄×D₁₈₈

yes

—

129

C₂×C₄

C₄×Dic₉₄

yes

—

130

C₂×C₄

C₂×#10625329

yes

—

131

C₂×C₄

C₂×#10625329

yes

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

188

—

136

C₈

#13093935

yes

—

137

C₂²

—

138

C₂²

C₄×C₄⋊Dic₄₇

yes

characteristic

139

C₂²

#15689180

yes

—

140

C₄

—

141

C₄

—

142

C₄

C₂×#13093935

yes

characteristic

143

C₄

C₄×#10625329

yes

characteristic

144

C₄

#15689147

yes

—

145

C₄

#15689246

yes

—

146

C₂

—

147

C₂

#20243241

yes

characteristic

148

C₂

#20243272

yes

—

	subgroup	G/H
center	C₄²	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#97×C₄₇	C₂
socle	C₂²×C₄₇	C₄×C₄⋊C₄
prime radical	C₂³×C₄₇	#117
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	#8269
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×#97

subgroup

G/H

center

C₄²

C₂×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄×#97×C₄₇

C₂

socle

C₂²×C₄₇

C₄×C₄⋊C₄

prime radical

C₂³×C₄₇

#117

cosocle

C₂×C₄²×C₄₇

C₂³

hypercenter

C₄×#97

D₄₇

hypocenter

C₄₇

#8269

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×#97

N	H ≅ G/N	type
6. #15688876	C₄	semidirect
7. #15688874	C₄	semidirect
88. C₄₇	#8269	semidirect

H ≅ G/N

type

6. #15688876

C₄

semidirect

7. #15688874

C₄

semidirect

88. C₄₇

#8269

semidirect

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

subgroup	index
#20242136	2
#20241680	2
#20241221	2
C₄×#97×C₄₇	2
#8269	47

subgroup

index

C₄×#97×C₄₇

Subgroups — #26107448

Conjugacy classes of subgroups

Complements

Maximal subgroups