Group #26143540 — subgroups

Subgroups — #26143540

Order 12032.

1550 conjugacy classes of subgroups; only the 651 normal subgroups are shown.

normal subgroups only

This group has 1550 conjugacy classes of subgroups — too many to draw. Only its 651 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	C₂×#15688993	C₂	1	3	yes	maximal
2	C₂×#15688989	C₂	1	3	yes	maximal
3	#20241653	C₂	1	8	yes	maximal
4	C₂×C₄₇×#376	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊Dic₄₇	C₂²	1	3	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	12	yes	—
8	#15688993	—	2	12	no	—
9	#15688989	—	2	12	no	—
10	C₂×C₄₇⋊#95	C₂²	1	12	yes	—
11	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
12	C₄₇×#376	C₄	1	8	yes	complemented
13	C₂×#95×C₄₇	C₂²	1	4	yes	—
14	C₂³×Dic₄₇	C₂³	1	2	yes	—
15	C₂³×Dic₄₇	C₂³	1	6	yes	—
16	C₂×C₄⋊Dic₄₇	—	2	6	no	—
17	C₂×C₄⋊Dic₄₇	—	2	6	no	—
18	C₂×C₄⋊Dic₄₇	—	2	24	no	—
19	C₂×C₄⋊Dic₄₇	—	2	12	no	—
20	C₂×C₄⋊Dic₄₇	—	4	24	no	—
21	C₄₇⋊#95	—	2	24	no	—
22	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
23	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	C₂×C₄	1	12	yes	—
25	C₂×C₄²×C₄₇	D₄	1	12	yes	—
26	C₂×C₄²×C₄₇	Q₈	1	12	yes	—
27	#95×C₄₇	C₂×C₄	1	16	yes	—
28	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	—	2	12	no	—
30	C₂²×C₄×C₄₇	—	2	12	no	—
31	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
32	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
35	C₂²×C₄×C₄₇	C₂×Q₈	1	12	yes	—
36	C₂²×C₄×C₄₇	C₂×Q₈	1	6	yes	—
37	C₂²×Dic₄₇	—	2	6	no	—
38	C₂²×Dic₄₇	—	2	6	no	—
39	C₂²×Dic₄₇	—	2	12	no	—
40	C₂²×Dic₄₇	—	2	24	no	—
41	C₂²×Dic₄₇	—	2	8	no	—
42	C₄²×C₄₇	C₄⋊C₄	1	48	yes	complemented
43	C₄⋊Dic₄₇	—	4	48	no	—
44	C₄⋊Dic₄₇	—	4	4	no	—
45	C₄⋊Dic₄₇	—	4	12	no	—
46	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
47	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
48	C₂³×C₄₇	C₂²×Q₈	1	3	yes	—
49	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
50	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
51	C₂×C₄×C₄₇	—	2	24	no	—
52	C₂×C₄×C₄₇	—	2	12	no	—
53	C₂×C₄×C₄₇	—	2	12	no	—
54	C₂×C₄×C₄₇	—	2	24	no	—
55	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	24	yes	—
56	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
57	C₂×C₄×C₄₇	#127	1	4	yes	—
58	C₂×C₄×C₄₇	C₄⋊Q₈	1	12	yes	—
59	C₂×Dic₄₇	—	4	6	no	—
60	C₂×Dic₄₇	—	4	2	no	—
61	C₂×Dic₄₇	—	4	24	no	—
62	C₂×Dic₄₇	—	4	12	no	—
63	C₂×Dic₄₇	—	4	12	no	—
64	#14354	—	47	1	no	maximal, Sylow
65	C₂²×C₄₇	C₂²×C₄⋊C₄	1	3	yes	—
66	C₂²×C₄₇	#520	1	4	yes	—
67	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
68	C₂²×C₄₇	C₂×C₄⋊Q₈	1	3	yes	—
69	C₂²×C₄₇	#535	1	6	yes	—
70	C₂²×C₄₇	#536	1	6	yes	—
71	C₂²×C₄₇	#544	1	12	yes	—
72	C₄×C₄₇	—	2	24	no	—
73	C₄×C₄₇	#384	1	8	yes	—
74	Dic₄₇	—	8	6	no	—
75	Dic₄₇	—	8	2	no	—
76	C₂×#384	—	47	3	no	—
77	C₂×#382	—	47	3	no	—
78	C₂×#376	D₄₇	1	1	yes	characteristic
79	#1444	—	47	8	no	—
80	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
81	C₂×C₄₇	#2317	1	6	yes	—
82	C₂×C₄₇	#2657	1	1	yes	characteristic
83	C₂×C₄₇	#2658	1	1	yes	characteristic
84	C₂×C₄₇	#2660	1	6	yes	—
85	C₂²×C₄⋊C₄	—	47	1	no	—
86	C₂²×C₄⋊C₄	—	47	3	no	—
87	C₂²×C₄⋊C₄	—	47	12	no	—
88	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
89	#384	—	94	12	no	—
90	#382	—	94	12	no	—
91	#376	Dic₄₇	1	8	yes	complemented
92	C₂×#95	—	47	12	no	—
93	C₂×#95	C₂×D₄₇	1	4	yes	—
94	C₄₇	#14354	1	1	yes	characteristic, complemented, Sylow
95	C₂³×C₄	—	47	6	no	—
96	C₂³×C₄	—	47	2	no	—
97	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
98	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
99	C₂×C₄⋊C₄	—	94	6	no	—
100	C₂×C₄⋊C₄	—	94	6	no	—
101	C₂×C₄⋊C₄	—	94	24	no	—
102	C₂×C₄⋊C₄	—	94	12	no	—
103	C₂×C₄⋊C₄	—	188	24	no	—
104	C₂×C₄²	Dic₉₄	1	12	yes	—
105	C₂×C₄²	D₁₈₈	1	12	yes	—
106	C₂×C₄²	C₂×Dic₄₇	1	12	yes	—
107	#95	—	94	24	no	—
108	#95	C₂×Dic₄₇	1	16	yes	—
109	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
110	C₂²×C₄	—	2	12	no	—
111	C₂²×C₄	—	2	12	no	—
112	C₂²×C₄	—	94	6	no	—
113	C₂²×C₄	—	94	6	no	—
114	C₂²×C₄	—	94	12	no	—
115	C₂²×C₄	—	94	24	no	—
116	C₂²×C₄	—	94	8	no	—
117	C₂²×C₄	C₂×Dic₉₄	1	12	yes	—
118	C₂²×C₄	C₂×Dic₉₄	1	6	yes	—
119	C₂²×C₄	C₂×D₁₈₈	1	12	yes	—
120	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
121	C₂²×C₄	C₂²×Dic₄₇	1	12	yes	—
122	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
123	C₄⋊C₄	—	188	48	no	—
124	C₄⋊C₄	—	188	4	no	—
125	C₄⋊C₄	—	188	12	no	—
126	C₄²	C₄⋊Dic₄₇	1	48	yes	complemented
127	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
128	C₂³	C₂²×D₁₈₈	1	3	yes	—
129	C₂³	C₂²×Dic₉₄	1	3	yes	—
130	C₂³	#13094070	1	4	yes	—
131	C₂³	#13094071	1	4	yes	—
132	C₂×C₄	—	2	24	no	—
133	C₂×C₄	—	2	24	no	—
134	C₂×C₄	—	2	12	no	—
135	C₂×C₄	—	2	12	no	—
136	C₂×C₄	—	188	6	no	—
137	C₂×C₄	—	188	2	no	—
138	C₂×C₄	—	188	24	no	—
139	C₂×C₄	—	188	12	no	—
140	C₂×C₄	—	188	12	no	—
141	C₂×C₄	C₂×C₄⋊Dic₄₇	1	24	yes	—
142	C₂×C₄	C₂×C₄⋊Dic₄₇	1	12	yes	—
143	C₂×C₄	#13094021	1	4	yes	—
144	C₂×C₄	#13094024	1	12	yes	—
145	C₂²	C₂²×C₄⋊Dic₄₇	1	3	yes	—
146	C₂²	#15689606	1	4	yes	—
147	C₂²	C₂×#13094021	1	1	yes	characteristic
148	C₂²	C₂×#13094024	1	3	yes	—
149	C₂²	#15689692	1	6	yes	—
150	C₂²	#15689699	1	6	yes	—
151	C₂²	#15689751	1	12	yes	—
152	C₄	—	2	24	no	—
153	C₄	—	376	6	no	—
154	C₄	—	376	2	no	—
155	C₄	#15688993	1	8	yes	—
156	C₂	C₂×#15688993	1	1	yes	characteristic
157	C₂	#20246011	1	6	yes	—
158	C₂	#20249494	1	1	yes	characteristic
159	C₂	#20249498	1	1	yes	characteristic
160	C₂	#20249507	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688993

C₂

yes

maximal

C₂×#15688989

C₂

yes

maximal

#20241653

C₂

yes

maximal

C₂×C₄₇×#376

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂²×C₄⋊Dic₄₇

C₂²

yes

—

C₂²×C₄⋊Dic₄₇

C₂²

yes

—

#15688993

—

#15688989

—

C₂×C₄₇⋊#95

C₂²

yes

—

C₂²×C₄²×C₄₇

C₂²

yes

—

C₄₇×#376

C₄

yes

complemented

C₂×#95×C₄₇

C₂²

yes

—

C₂³×Dic₄₇

C₂³

yes

—

C₂³×Dic₄₇

C₂³

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₄₇⋊#95

—

C₂³×C₄×C₄₇

C₂³

yes

—

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

Q₈

yes

—

#95×C₄₇

C₂×C₄

yes

—

C₂⁴×C₄₇

C₂⁴

yes

characteristic

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

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×Q₈

yes

—

C₂²×C₄×C₄₇

C₂×Q₈

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₄²×C₄₇

C₄⋊C₄

yes

complemented

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂³×C₄₇

C₂³×C₄

yes

characteristic

C₂³×C₄₇

C₂²×D₄

yes

—

C₂³×C₄₇

C₂²×Q₈

yes

—

C₂³×C₄₇

H(5,𝔽₂)

yes

—

C₂³×C₄₇

2₋¹⁺⁴

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

#127

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

#14354

—

maximal, Sylow

C₂²×C₄₇

C₂²×C₄⋊C₄

yes

—

C₂²×C₄₇

#520

yes

—

C₂²×C₄₇

C₂×#127

yes

characteristic

C₂²×C₄₇

C₂×C₄⋊Q₈

yes

—

C₂²×C₄₇

#535

yes

—

C₂²×C₄₇

#536

yes

—

C₂²×C₄₇

#544

yes

—

C₄×C₄₇

—

C₄×C₄₇

#384

yes

—

Dic₄₇

—

Dic₄₇

—

C₂×#384

—

C₂×#382

—

C₂×#376

D₄₇

yes

characteristic

#1444

—

C₂×C₄₇

C₂×#384

yes

characteristic

C₂×C₄₇

#2317

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

—

C₂²×C₄²

C₂×D₄₇

yes

—

#384

—

#382

—

#376

Dic₄₇

yes

complemented

C₂×#95

—

C₂×#95

C₂×D₄₇

yes

—

C₄₇

#14354

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₄₇

yes

—

C₂³×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

188

—

104

C₂×C₄²

Dic₉₄

yes

—

105

C₂×C₄²

D₁₈₈

yes

—

106

C₂×C₄²

C₂×Dic₄₇

yes

—

107

#95

—

108

#95

C₂×Dic₄₇

yes

—

109

C₂⁴

C₂³×D₄₇

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂×Dic₉₄

yes

—

118

C₂²×C₄

C₂×Dic₉₄

yes

—

119

C₂²×C₄

C₂×D₁₈₈

yes

—

120

C₂²×C₄

C₂×D₁₈₈

yes

—

121

C₂²×C₄

C₂²×Dic₄₇

yes

—

122

C₂²×C₄

C₂²×Dic₄₇

yes

—

123

C₄⋊C₄

—

188

—

124

C₄⋊C₄

—

188

—

125

C₄⋊C₄

—

188

—

126

C₄²

C₄⋊Dic₄₇

yes

complemented

127

C₂³

C₂³×Dic₄₇

yes

characteristic

128

C₂³

C₂²×D₁₈₈

yes

—

129

C₂³

C₂²×Dic₉₄

yes

—

130

C₂³

#13094070

yes

—

131

C₂³

#13094071

yes

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

188

—

137

C₂×C₄

—

188

—

138

C₂×C₄

—

188

—

139

C₂×C₄

—

188

—

140

C₂×C₄

—

188

—

141

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

—

142

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

—

143

C₂×C₄

#13094021

yes

—

144

C₂×C₄

#13094024

yes

—

145

C₂²

C₂²×C₄⋊Dic₄₇

yes

—

146

C₂²

#15689606

yes

—

147

C₂²

C₂×#13094021

yes

characteristic

148

C₂²

C₂×#13094024

yes

—

149

C₂²

#15689692

yes

—

150

C₂²

#15689699

yes

—

151

C₂²

#15689751

yes

—

152

C₄

—

153

C₄

—

376

—

154

C₄

—

376

—

155

C₄

#15688993

yes

—

156

C₂

C₂×#15688993

yes

characteristic

157

C₂

#20246011

yes

—

158

C₂

#20249494

yes

characteristic

159

C₂

#20249498

yes

characteristic

160

C₂

#20249507

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#376	C₂
socle, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#376	D₄₇
hypocenter	C₄₇	#14354
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₄₇

commutator

C₂³×C₄₇

C₂³×C₄

Fitting

C₂×C₄₇×#376

C₂

socle, prime radical, cosocle

C₂⁴×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂⁴ ◁ C₂×#376

N	H ≅ G/N	type
12. C₄₇×#376	C₄	semidirect
42. C₄²×C₄₇	C₄⋊C₄	semidirect
91. #376	Dic₄₇	semidirect
94. C₄₇	#14354	semidirect
126. C₄²	C₄⋊Dic₄₇	semidirect

H ≅ G/N

type

12. C₄₇×#376

C₄

semidirect

42. C₄²×C₄₇

C₄⋊C₄

semidirect

91. #376

Dic₄₇

semidirect

94. C₄₇

#14354

semidirect

126. C₄²

C₄⋊Dic₄₇

semidirect

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

subgroup	index
C₂×#15688993	2
C₂×#15688989	2
#20241653	2
C₂×C₄₇×#376	2
#14354	47

subgroup

index

C₂×C₄₇×#376

Subgroups — #26143540

Conjugacy classes of subgroups

Complements

Maximal subgroups