Group #26224683 — subgroups

Subgroups — #26224683

Order 12032.

1154 conjugacy classes of subgroups; only the 299 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 299 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×#15689202	C₂	1	6	yes	maximal, complemented
2	#20246669	C₂	1	1	yes	maximal, characteristic
3	#20244186	C₂	1	6	yes	maximal, complemented
4	#20243542	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10625324	C₂²	1	4	yes	—
7	C₂×C₄×D₁₈₈	C₂²	1	6	yes	—
8	#15689202	—	2	24	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	6	yes	—
10	C₂×(C₂×D₄₇)⋊C₈	C₂²	1	12	yes	—
11	#15689060	C₄	1	4	yes	—
12	#15689033	C₄	1	4	yes	—
13	C₂×#97×C₄₇	C₂²	1	3	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
15	C₄₇×#376	C₂²	1	1	yes	characteristic
16	C₂²×D₁₈₈	C₂×C₄	1	6	yes	—
17	C₂²×C₄×D₄₇	C₂³	1	4	yes	—
18	C₂×#10625324	—	2	24	no	—
19	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
20	C₄×D₁₈₈	—	2	12	no	—
21	C₄×D₁₈₈	—	2	12	no	—
22	C₂×C₄⋊Dic₄₇	C₂×C₄	1	6	yes	—
23	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	12	yes	—
24	C₄⋊(C₄₇⋊C₈)	—	2	12	no	—
25	(C₂×D₄₇)⋊C₈	—	2	24	no	—
26	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
27	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
29	#97×C₄₇	—	2	6	no	—
30	C₄×C₈×C₄₇	D₄	1	12	yes	complemented
31	#95×C₄₇	C₂×C₄	1	4	yes	—
32	C₂³×D₄₇	C₂²×C₄	1	4	yes	—
33	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
34	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
35	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
36	C₂×D₁₈₈	—	2	12	no	—
37	C₂×D₁₈₈	—	4	12	no	—
38	C₂×C₄×D₄₇	—	2	12	no	—
39	C₂×C₄×D₄₇	—	2	12	no	—
40	C₂×C₈×C₄₇	—	2	6	no	—
41	C₂×C₈×C₄₇	C₂×D₄	1	6	yes	—
42	C₂×C₈×C₄₇	C₂×D₄	1	6	yes	—
43	C₄²×C₄₇	C₂×D₄	1	6	yes	—
44	C₄²×C₄₇	#41	1	6	yes	—
45	(C₂×D₄₇)⋊C₄	—	2	24	no	—
46	C₄⋊Dic₄₇	—	2	12	no	—
47	C₂×C₄₇⋊C₈	—	2	12	no	—
48	#10625324	—	4	16	no	—
49	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
50	C₂²×D₄₇	—	2	12	no	—
51	C₂²×D₄₇	—	2	4	no	—
52	C₂²×D₄₇	—	2	12	no	—
53	C₂×C₄×C₄₇	—	2	12	no	—
54	C₂×C₄×C₄₇	C₄×D₄	1	12	yes	—
55	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
56	C₂×C₄×C₄₇	C₂×#41	1	3	yes	—
57	C₂×Dic₄₇	—	2	12	no	—
58	D₁₈₈	—	4	24	no	—
59	C₄×D₄₇	—	4	4	no	—
60	C₄×D₄₇	—	4	12	no	—
61	C₈×C₄₇	—	2	6	no	—
62	C₈×C₄₇	#127	1	4	yes	—
63	C₄₇⋊C₈	—	4	4	no	—
64	#21684	—	47	1	no	maximal, Sylow
65	C₂²×C₄₇	C₂×C₄×D₄	1	3	yes	—
66	C₂²×C₄₇	#568	1	4	yes	—
67	C₂×D₄₇	—	4	4	no	—
68	C₂×D₄₇	—	4	12	no	—
69	C₂×D₄₇	—	4	12	no	—
70	C₄×C₄₇	—	2	12	no	—
71	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
72	C₄×C₄₇	#532	1	3	yes	—
73	Dic₄₇	—	4	4	no	—
74	C₂×#436	—	47	6	no	—
75	#2397	—	47	1	no	—
76	#1955	—	47	6	no	—
77	#1843	—	47	1	no	—
78	#1726	D₄₇	1	1	yes	characteristic, complemented
79	C₂×C₄₇	C₄×#127	1	1	yes	characteristic
80	C₂×C₄₇	#2930	1	6	yes	—
81	D₄₇	—	8	4	no	—
82	C₂²×M₁₆	—	47	4	no	—
83	C₂×C₄×D₄	—	47	6	no	—
84	#436	—	94	24	no	—
85	C₂×C₄⋊C₈	—	47	6	no	—
86	C₂×C₂²⋊C₈	—	47	12	no	—
87	C₂×#97	C₂×D₄₇	1	3	yes	—
88	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
89	#399	—	47	4	no	—
90	#392	—	47	4	no	—
91	#376	C₂×D₄₇	1	1	yes	characteristic
92	C₄₇	#21684	1	1	yes	characteristic, complemented, Sylow
93	C₂²×D₄	—	47	6	no	—
94	C₂³×C₄	—	47	4	no	—
95	C₂×M₁₆	—	94	24	no	—
96	C₂²×C₈	—	47	4	no	—
97	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
98	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
99	C₄×D₄	—	94	12	no	—
100	C₄×D₄	—	94	12	no	—
101	C₂×C₄⋊C₄	—	47	6	no	—
102	C₂×C₂²⋊C₄	—	47	12	no	—
103	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
104	C₄⋊C₈	—	94	12	no	—
105	C₂²⋊C₈	—	94	24	no	—
106	#97	—	2	6	no	—
107	C₄×C₈	D₁₈₈	1	12	yes	complemented
108	#95	C₄×D₄₇	1	4	yes	—
109	C₂⁴	—	47	4	no	—
110	C₂×D₄	—	94	12	no	—
111	C₂×D₄	—	188	12	no	—
112	C₂²×C₄	—	47	4	no	—
113	C₂²×C₄	—	94	12	no	—
114	C₂²×C₄	—	94	12	no	—
115	C₂²×C₄	C₂×C₄×D₄₇	1	6	yes	—
116	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
117	M₁₆	—	188	16	no	—
118	C₂×C₈	—	2	6	no	—
119	C₂×C₈	—	94	12	no	—
120	C₂×C₈	C₂×D₁₈₈	1	6	yes	—
121	C₂×C₈	C₂×D₁₈₈	1	6	yes	—
122	C₄⋊C₄	—	94	12	no	—
123	C₂²⋊C₄	—	94	24	no	—
124	C₄²	C₂×D₁₈₈	1	6	yes	—
125	C₄²	#10625350	1	6	yes	—
126	C₂³	—	94	12	no	—
127	C₂³	—	94	4	no	—
128	C₂³	—	94	12	no	—
129	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
130	D₄	—	188	24	no	—
131	C₂×C₄	—	2	12	no	—
132	C₂×C₄	—	94	12	no	—
133	C₂×C₄	—	188	4	no	—
134	C₂×C₄	—	188	12	no	—
135	C₂×C₄	C₄×D₁₈₈	1	12	yes	—
136	C₂×C₄	C₂²×D₁₈₈	1	3	yes	—
137	C₂×C₄	C₂×#10625350	1	3	yes	—
138	C₈	—	2	6	no	—
139	C₈	—	188	4	no	—
140	C₈	#13094021	1	4	yes	—
141	C₂²	—	188	12	no	—
142	C₂²	—	188	12	no	—
143	C₂²	—	188	4	no	—
144	C₂²	C₂×C₄×D₁₈₈	1	3	yes	—
145	C₂²	#15689888	1	4	yes	—
146	C₄	—	2	12	no	—
147	C₄	—	188	4	no	—
148	C₄	C₂×#13094021	1	1	yes	characteristic
149	C₄	#15689673	1	3	yes	—
150	C₂	—	376	4	no	—
151	C₂	C₄×#13094021	1	1	yes	characteristic
152	C₂	#20252093	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689202

C₂

yes

maximal, complemented

#20246669

C₂

yes

maximal, characteristic

#20244186

C₂

yes

maximal, complemented

#20243542

C₂

yes

maximal, characteristic, complemented

C₄₇×#1726

C₂

yes

maximal, characteristic, complemented

C₂²×#10625324

C₂²

yes

—

C₂×C₄×D₁₈₈

C₂²

yes

—

#15689202

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₈

C₂²

yes

—

#15689060

C₄

yes

—

#15689033

C₄

yes

—

C₂×#97×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₄₇×#376

C₂²

yes

characteristic

C₂²×D₁₈₈

C₂×C₄

yes

—

C₂²×C₄×D₄₇

C₂³

yes

—

C₂×#10625324

—

C₂²×C₄₇⋊C₈

C₂³

yes

—

C₄×D₁₈₈

—

C₄×D₁₈₈

—

C₂×C₄⋊Dic₄₇

C₂×C₄

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

(C₂×D₄₇)⋊C₈

—

C₂²×C₈×C₄₇

C₂³

yes

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

—

#97×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

complemented

#95×C₄₇

C₂×C₄

yes

—

C₂³×D₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

—

C₂²×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²×Dic₄₇

C₂²×C₄

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

#41

yes

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

#10625324

—

C₂³×C₄₇

C₂³×C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₂²×D₄

yes

—

C₂×C₄×C₄₇

C₂×#41

yes

—

C₂×Dic₄₇

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#127

yes

—

C₄₇⋊C₈

—

#21684

—

maximal, Sylow

C₂²×C₄₇

C₂×C₄×D₄

yes

—

C₂²×C₄₇

#568

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#127

yes

characteristic

C₄×C₄₇

#532

yes

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₂×C₄₇

C₄×#127

yes

characteristic

C₂×C₄₇

#2930

yes

—

D₄₇

—

C₂²×M₁₆

—

C₂×C₄×D₄

—

#436

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×#97

C₂×D₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

#399

—

#392

—

#376

C₂×D₄₇

yes

characteristic

C₄₇

#21684

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₄₇

yes

—

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₄×D₄

—

100

C₄×D₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₂²⋊C₄

—

103

C₂×C₄²

C₂²×D₄₇

yes

—

104

C₄⋊C₈

—

105

C₂²⋊C₈

—

106

#97

—

107

C₄×C₈

D₁₈₈

yes

complemented

108

#95

C₄×D₄₇

yes

—

109

C₂⁴

—

110

C₂×D₄

—

111

C₂×D₄

—

188

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₂×C₄×D₄₇

yes

—

116

C₂²×C₄

C₂³×D₄₇

yes

characteristic

117

M₁₆

—

188

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₂×D₁₈₈

yes

—

121

C₂×C₈

C₂×D₁₈₈

yes

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

C₂×D₁₈₈

yes

—

125

C₄²

#10625350

yes

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂²×C₄×D₄₇

yes

characteristic

130

D₄

—

188

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

188

—

134

C₂×C₄

—

188

—

135

C₂×C₄

C₄×D₁₈₈

yes

—

136

C₂×C₄

C₂²×D₁₈₈

yes

—

137

C₂×C₄

C₂×#10625350

yes

—

138

C₈

—

139

C₈

—

188

—

140

C₈

#13094021

yes

—

141

C₂²

—

188

—

142

C₂²

—

188

—

143

C₂²

—

188

—

144

C₂²

C₂×C₄×D₁₈₈

yes

—

145

C₂²

#15689888

yes

—

146

C₄

—

147

C₄

—

188

—

148

C₄

C₂×#13094021

yes

characteristic

149

C₄

#15689673

yes

—

150

C₂

—

376

—

151

C₂

C₄×#13094021

yes

characteristic

152

C₂

#20252093

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#1726	C₂
prime radical	#20246669	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#21684
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1726

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂³×C₄

Fitting

C₄₇×#1726

C₂

prime radical

#20246669

C₂

cosocle

C₂²×C₄×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ #1726

N	H ≅ G/N	type
1. C₂×#15689202	C₂	semidirect
3. #20244186	C₂	semidirect
4. #20243542	C₂	semidirect
5. C₄₇×#1726	C₂	semidirect
30. C₄×C₈×C₄₇	D₄	semidirect
78. #1726	D₄₇	semidirect
92. C₄₇	#21684	semidirect
107. C₄×C₈	D₁₈₈	semidirect

H ≅ G/N

type

1. C₂×#15689202

C₂

semidirect

3. #20244186

C₂

semidirect

4. #20243542

C₂

semidirect

5. C₄₇×#1726

C₂

semidirect

30. C₄×C₈×C₄₇

D₄

semidirect

78. #1726

D₄₇

semidirect

92. C₄₇

#21684

semidirect

107. C₄×C₈

D₁₈₈

semidirect

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

subgroup	index
C₂×#15689202	2
#20246669	2
#20244186	2
#20243542	2
C₄₇×#1726	2
#21684	47

subgroup

index

C₄₇×#1726

Subgroups — #26224683

Conjugacy classes of subgroups

Complements

Maximal subgroups