Group #26124961 — subgroups

Subgroups — #26124961

Order 12032.

686 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 235 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	#20251978	C₂	1	3	yes	maximal
2	#20243086	C₂	1	3	yes	maximal
3	C₂×C₄₇×#338	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
5	C₂×#13093983	C₂²	1	3	yes	—
6	#15689165	—	2	12	no	—
7	#15689161	—	2	6	no	—
8	#15688877	—	2	6	no	—
9	C₂²×M₁₆×C₄₇	C₂²	1	3	yes	—
10	C₄₇×#338	C₄	1	4	yes	complemented
11	C₂²×C₄₇⋊C₈	—	2	3	no	—
12	C₂²×C₄₇⋊C₈	—	2	3	no	—
13	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
14	#13093983	—	2	6	no	—
15	#13093983	—	2	6	no	—
16	C₂×C₄⋊Dic₄₇	D₄	1	6	yes	—
17	C₂×C₂²⋊Dic₄₇	—	2	3	no	—
18	C₂×C₄×Dic₄₇	—	2	3	no	—
19	#13093958	—	2	12	no	—
20	#13093956	—	2	12	no	—
21	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₄₇	—	2	6	no	—
23	C₂×M₁₆×C₄₇	—	2	6	no	—
24	C₂×M₁₆×C₄₇	C₂×C₄	1	6	yes	—
25	C₂×M₁₆×C₄₇	Q₈	1	6	yes	—
26	C₂²×C₈×C₄₇	D₄	1	6	yes	—
27	C₂⁴×C₄₇	#41	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
30	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
31	C₂²×C₄×C₄₇	C₂×Q₈	1	3	yes	—
32	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	#41	1	3	yes	—
34	C₂²×Dic₄₇	—	2	6	no	—
35	M₁₆×C₄₇	—	2	12	no	—
36	M₁₆×C₄₇	—	2	12	no	—
37	C₂×C₈×C₄₇	—	2	6	no	—
38	C₂×C₈×C₄₇	—	2	6	no	—
39	C₂×C₈×C₄₇	C₄⋊C₄	1	12	yes	—
40	C₂²⋊Dic₄₇	—	4	6	no	—
41	C₄⋊Dic₄₇	—	2	12	no	—
42	C₄×Dic₄₇	—	4	3	no	—
43	C₄×Dic₄₇	—	4	3	no	—
44	C₂×C₄₇⋊C₈	—	2	6	no	—
45	C₂×C₄₇⋊C₈	—	4	3	no	—
46	C₂×C₄₇⋊C₈	—	4	1	no	—
47	C₂³×C₄₇	—	4	2	no	—
48	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
49	C₂³×C₄₇	#121	1	3	yes	—
50	C₂³×C₄₇	#125	1	3	yes	—
51	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
52	C₂×C₄×C₄₇	C₄×D₄	1	3	yes	—
53	C₂×C₄×C₄₇	#120	1	3	yes	—
54	C₂×C₄×C₄₇	C₂²⋊Q₈	1	6	yes	—
55	C₂×C₄×C₄₇	C₂²⋊Q₈	1	3	yes	—
56	C₂×C₄×C₄₇	#123	1	3	yes	—
57	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₄⋊Q₈	1	3	yes	—
59	C₂×C₄×C₄₇	C₄⋊Q₈	1	3	yes	—
60	C₂×Dic₄₇	—	4	12	no	—
61	C₂×Dic₄₇	—	4	6	no	—
62	C₈×C₄₇	—	2	12	no	—
63	C₄₇⋊C₈	—	4	2	no	—
64	#11337	—	47	1	no	maximal, Sylow
65	C₂²×C₄₇	—	2	6	no	—
66	C₂²×C₄₇	—	4	1	no	—
67	C₂²×C₄₇	—	4	3	no	—
68	C₂²×C₄₇	#389	1	3	yes	—
69	C₂²×C₄₇	#392	1	1	yes	characteristic
70	C₂²×C₄₇	#398	1	3	yes	—
71	C₄×C₄₇	#384	1	1	yes	characteristic
72	C₄×C₄₇	#385	1	3	yes	—
73	C₄×C₄₇	#395	1	3	yes	—
74	C₄×C₄₇	#399	1	1	yes	characteristic
75	Dic₄₇	—	8	6	no	—
76	#2911	—	47	3	no	—
77	C₂×#338	D₄₇	1	1	yes	characteristic
78	#1729	—	47	3	no	—
79	C₂×C₄₇	—	2	3	no	—
80	C₂×C₄₇	—	2	3	no	—
81	C₂×C₄₇	#1444	1	1	yes	characteristic
82	C₂×C₄₇	#2017	1	2	yes	—
83	C₂²×M₁₆	C₂×D₄₇	1	3	yes	—
84	C₂³×C₈	—	47	1	no	—
85	C₂×#117	—	47	3	no	—
86	#428	—	94	12	no	—
87	#427	—	94	6	no	—
88	#338	Dic₄₇	1	4	yes	complemented
89	#336	—	94	6	no	—
90	C₄₇	#11337	1	1	yes	characteristic, complemented, Sylow
91	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
92	C₂×M₁₆	—	2	6	no	—
93	C₂×M₁₆	—	2	6	no	—
94	C₂×M₁₆	Dic₉₄	1	6	yes	—
95	C₂×M₁₆	C₂×Dic₄₇	1	6	yes	—
96	C₂²×C₈	—	47	2	no	—
97	C₂²×C₈	—	94	3	no	—
98	C₂²×C₈	—	94	3	no	—
99	C₂²×C₈	D₁₈₈	1	6	yes	—
100	#117	—	94	6	no	—
101	#117	—	94	6	no	—
102	C₂×C₄⋊C₄	—	47	6	no	—
103	C₂×C₂²⋊C₄	—	94	3	no	—
104	C₂×C₄²	—	94	3	no	—
105	#107	—	94	12	no	—
106	#106	—	94	12	no	—
107	C₂⁴	#10625354	1	1	yes	characteristic
108	C₂²×C₄	—	94	6	no	—
109	C₂²×C₄	C₂×Dic₉₄	1	3	yes	—
110	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
111	C₂²×C₄	D₄×D₄₇	1	3	yes	—
112	C₂²×C₄	#10625352	1	3	yes	—
113	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
114	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
115	M₁₆	—	2	12	no	—
116	M₁₆	—	2	12	no	—
117	C₂×C₈	—	2	6	no	—
118	C₂×C₈	—	2	6	no	—
119	C₂×C₈	—	94	6	no	—
120	C₂×C₈	—	188	3	no	—
121	C₂×C₈	—	188	1	no	—
122	C₂×C₈	C₄⋊Dic₄₇	1	12	yes	—
123	C₄⋊C₄	—	94	12	no	—
124	C₂²⋊C₄	—	188	6	no	—
125	C₄²	—	188	3	no	—
126	C₄²	—	188	3	no	—
127	C₂³	—	4	2	no	—
128	C₂³	Q₈×Dic₄₇	1	1	yes	characteristic
129	C₂³	#13093999	1	3	yes	—
130	C₂³	#13094015	1	3	yes	—
131	C₂×C₄	—	188	12	no	—
132	C₂×C₄	—	188	6	no	—
133	C₂×C₄	C₂×C₄⋊Dic₄₇	1	3	yes	—
134	C₂×C₄	D₄×Dic₄₇	1	3	yes	—
135	C₂×C₄	#13093994	1	3	yes	—
136	C₂×C₄	#13094001	1	6	yes	—
137	C₂×C₄	#13094004	1	3	yes	—
138	C₂×C₄	#13094008	1	3	yes	—
139	C₂×C₄	#13094021	1	1	yes	characteristic
140	C₂×C₄	#13094023	1	3	yes	—
141	C₂×C₄	#13094024	1	3	yes	—
142	C₈	—	2	12	no	—
143	C₈	—	188	2	no	—
144	C₂²	—	2	6	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	4	3	no	—
147	C₂²	#15689022	1	3	yes	—
148	C₂²	#15689033	1	1	yes	characteristic
149	C₂²	#15689055	1	3	yes	—
150	C₄	—	376	6	no	—
151	C₄	#15688993	1	1	yes	characteristic
152	C₄	#15688998	1	3	yes	—
153	C₄	#15689044	1	3	yes	—
154	C₄	#15689060	1	1	yes	characteristic
155	C₂	—	2	3	no	—
156	C₂	—	2	3	no	—
157	C₂	#20241653	1	1	yes	characteristic
158	C₂	#20244493	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20251978

C₂

yes

maximal

#20243086

C₂

yes

maximal

C₂×C₄₇×#338

C₂

yes

maximal, characteristic

C₂³×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×#13093983

C₂²

yes

—

#15689165

—

#15689161

—

#15688877

—

C₂²×M₁₆×C₄₇

C₂²

yes

—

C₄₇×#338

C₄

yes

complemented

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

Q₈

yes

—

#13093983

—

#13093983

—

C₂×C₄⋊Dic₄₇

D₄

yes

—

C₂×C₂²⋊Dic₄₇

—

C₂×C₄×Dic₄₇

—

#13093958

—

#13093956

—

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

C₂×C₄

yes

—

C₂×M₁₆×C₄₇

Q₈

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂⁴×C₄₇

#41

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×Q₈

yes

—

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂²×Dic₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

C₄×Q₈

yes

characteristic

C₂³×C₄₇

#121

yes

—

C₂³×C₄₇

#125

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#120

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#11337

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#389

yes

—

C₂²×C₄₇

#392

yes

characteristic

C₂²×C₄₇

#398

yes

—

C₄×C₄₇

#384

yes

characteristic

C₄×C₄₇

#385

yes

—

C₄×C₄₇

#395

yes

—

C₄×C₄₇

#399

yes

characteristic

Dic₄₇

—

#2911

—

C₂×#338

D₄₇

yes

characteristic

#1729

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1444

yes

characteristic

C₂×C₄₇

#2017

yes

—

C₂²×M₁₆

C₂×D₄₇

yes

—

C₂³×C₈

—

—

—

—

Dic₄₇

yes

complemented

#336

—

C₄₇

#11337

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂²×D₄₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

Dic₉₄

yes

—

C₂×M₁₆

C₂×Dic₄₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₁₈₈

yes

—

100

#117

—

101

#117

—

102

C₂×C₄⋊C₄

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₄²

—

105

#107

—

106

#106

—

107

C₂⁴

#10625354

yes

characteristic

108

C₂²×C₄

—

109

C₂²×C₄

C₂×Dic₉₄

yes

—

110

C₂²×C₄

C₂×D₁₈₈

yes

—

111

C₂²×C₄

D₄×D₄₇

yes

—

112

C₂²×C₄

#10625352

yes

—

113

C₂²×C₄

Q₈×D₄₇

yes

characteristic

114

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

188

—

121

C₂×C₈

—

188

—

122

C₂×C₈

C₄⋊Dic₄₇

yes

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

188

—

125

C₄²

—

188

—

126

C₄²

—

188

—

127

C₂³

—

128

C₂³

Q₈×Dic₄₇

yes

characteristic

129

C₂³

#13093999

yes

—

130

C₂³

#13094015

yes

—

131

C₂×C₄

—

188

—

132

C₂×C₄

—

188

—

133

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

—

134

C₂×C₄

D₄×Dic₄₇

yes

—

135

C₂×C₄

#13093994

yes

—

136

C₂×C₄

#13094001

yes

—

137

C₂×C₄

#13094004

yes

—

138

C₂×C₄

#13094008

yes

—

139

C₂×C₄

#13094021

yes

characteristic

140

C₂×C₄

#13094023

yes

—

141

C₂×C₄

#13094024

yes

—

142

C₈

—

143

C₈

—

188

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#15689022

yes

—

148

C₂²

#15689033

yes

characteristic

149

C₂²

#15689055

yes

—

150

C₄

—

376

—

151

C₄

#15688993

yes

characteristic

152

C₄

#15688998

yes

—

153

C₄

#15689044

yes

—

154

C₄

#15689060

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#20241653

yes

characteristic

158

C₂

#20244493

yes

—

	subgroup	G/H
center	C₂×C₄	#13094021
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#338	C₂
socle	C₂²×C₄₇	#392
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#338	D₄₇
hypocenter	C₄₇	#11337
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

subgroup

G/H

center

C₂×C₄

#13094021

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₄₇

Fitting

C₂×C₄₇×#338

C₂

C₂²×C₄₇

C₂⁴×C₄₇

C₂³×C₄×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

N	H ≅ G/N	type
10. C₄₇×#338	C₄	semidirect
88. #338	Dic₄₇	semidirect
90. C₄₇	#11337	semidirect

H ≅ G/N

type

10. C₄₇×#338

C₄

semidirect

88. #338

Dic₄₇

semidirect

90. C₄₇

#11337

semidirect

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

subgroup	index
#20251978	2
#20243086	2
C₂×C₄₇×#338	2
#11337	47

subgroup

index

#20251978

#20243086

C₂×C₄₇×#338

#11337

Subgroups — #26124961

Conjugacy classes of subgroups

Complements

Maximal subgroups