Group #26214822 — subgroups

Subgroups — #26214822

Order 12032.

1054 conjugacy classes of subgroups; only the 403 normal subgroups are shown.

normal subgroups only

This group has 1054 conjugacy classes of subgroups — too many to draw. Only its 403 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×C₄×Dic₁₈₈	C₂	1	3	yes	maximal
2	#20245751	C₂	1	2	yes	maximal
3	C₄×#13093958	C₂	1	3	yes	maximal
4	#20243219	C₂	1	6	yes	maximal
5	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₉₄	C₂²	1	6	yes	—
8	C₄×Dic₁₈₈	C₄	1	24	yes	complemented
9	C₂×#13093958	C₂²	1	3	yes	—
10	C₂×#13093949	C₂²	1	6	yes	—
11	C₄×C₄⋊Dic₄₇	C₂²	1	6	yes	—
12	#15688974	—	2	6	no	—
13	#15688968	—	2	2	no	—
14	#15688886	C₂²	1	6	yes	—
15	C₂×#97×C₄₇	C₂²	1	3	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
17	C₄₇×#376	C₂²	1	1	yes	characteristic
18	C₂²×Dic₉₄	C₂³	1	2	yes	—
19	C₂×Dic₁₈₈	C₂×C₄	1	12	yes	—
20	C₄×Dic₉₄	—	2	12	no	—
21	C₄×Dic₉₄	C₂×C₄	1	24	yes	—
22	C₂×C₄⋊Dic₄₇	C₂³	1	6	yes	—
23	C₂×Dic₄₇⋊C₄	—	2	6	no	—
24	C₂×C₄×Dic₄₇	—	2	6	no	—
25	C₂×C₄×Dic₄₇	—	2	6	no	—
26	#13093958	C₂×C₄	1	12	yes	—
27	#13093949	C₂×C₄	1	24	yes	—
28	C₄₇⋊#95	—	2	6	no	—
29	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
30	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
32	#97×C₄₇	—	2	6	no	—
33	C₄×C₈×C₄₇	C₂×C₄	1	12	yes	—
34	#95×C₄₇	D₄	1	1	yes	characteristic
35	#95×C₄₇	D₄	1	3	yes	—
36	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
38	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
39	C₂²×Dic₄₇	—	2	2	no	—
40	C₂²×Dic₄₇	—	2	6	no	—
41	C₂×Dic₉₄	—	2	6	no	—
42	C₂×Dic₉₄	C₂²×C₄	1	12	yes	—
43	C₂×C₈×C₄₇	—	2	6	no	—
44	C₂×C₈×C₄₇	C₂²×C₄	1	6	yes	—
45	C₂×C₈×C₄₇	C₂²×C₄	1	6	yes	—
46	C₄²×C₄₇	C₂²×C₄	1	6	yes	—
47	C₄²×C₄₇	#41	1	6	yes	—
48	C₄⋊Dic₄₇	—	2	6	no	—
49	C₄⋊Dic₄₇	C₂²×C₄	1	12	yes	—
50	Dic₄₇⋊C₄	—	2	24	no	—
51	C₄×Dic₄₇	—	2	24	no	—
52	C₄×Dic₄₇	—	8	6	no	—
53	Dic₁₈₈	C₄²	1	16	yes	complemented
54	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	—	2	3	no	—
56	C₂×C₄×C₄₇	—	2	6	no	—
57	C₂×C₄×C₄₇	—	2	3	no	—
58	C₂×C₄×C₄₇	C₄×D₄	1	6	yes	—
59	C₂×C₄×C₄₇	C₄×D₄	1	6	yes	—
60	C₂×C₄×C₄₇	C₂³×C₄	1	3	yes	—
61	C₂×C₄×C₄₇	C₂×#41	1	3	yes	—
62	C₂×Dic₄₇	—	2	12	no	—
63	C₂×Dic₄₇	—	2	12	no	—
64	C₂×Dic₄₇	—	4	6	no	—
65	C₂×Dic₄₇	—	4	6	no	—
66	Dic₉₄	—	2	12	no	—
67	Dic₉₄	C₂×C₄²	1	8	yes	—
68	C₈×C₄₇	—	2	6	no	—
69	C₈×C₄₇	C₂×C₄²	1	4	yes	—
70	#20507	—	47	1	no	maximal, Sylow
71	C₂²×C₄₇	C₂×C₄×D₄	1	3	yes	—
72	C₂²×C₄₇	#576	1	1	yes	characteristic
73	C₂²×C₄₇	#578	1	3	yes	—
74	C₄×C₄₇	—	2	6	no	—
75	C₄×C₄₇	—	2	6	no	—
76	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
77	C₄×C₄₇	C₄×#41	1	3	yes	—
78	Dic₄₇	—	2	8	no	—
79	Dic₄₇	—	4	12	no	—
80	C₂×C₄×Q₁₆	—	47	3	no	—
81	#2276	—	47	2	no	—
82	C₄×#107	—	47	3	no	—
83	#1763	—	47	6	no	—
84	#1726	D₄₇	1	1	yes	characteristic
85	C₂×C₄₇	C₄²×D₄	1	1	yes	characteristic
86	C₂×C₄₇	#2950	1	3	yes	—
87	C₂×C₄₇	#2951	1	3	yes	—
88	C₂²×Q₁₆	—	47	1	no	—
89	C₂×C₄×Q₈	—	47	6	no	—
90	C₄×Q₁₆	—	47	24	no	—
91	C₂×#107	—	47	3	no	—
92	C₂×#103	—	47	6	no	—
93	C₂×#97	C₂×D₄₇	1	3	yes	—
94	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
95	C₄×C₄⋊C₄	—	47	6	no	—
96	C₄×C₄⋊C₄	—	94	6	no	—
97	#376	—	94	2	no	—
98	#376	C₂×D₄₇	1	1	yes	characteristic
99	#340	—	47	6	no	—
100	C₄₇	#20507	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	—	47	2	no	—
102	C₂×Q₁₆	—	47	12	no	—
103	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
104	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
105	C₄×Q₈	—	47	24	no	—
106	C₄×Q₈	—	94	12	no	—
107	C₂×C₄⋊C₄	—	47	6	no	—
108	C₂×C₄⋊C₄	—	94	6	no	—
109	C₂×C₄²	—	94	6	no	—
110	C₂×C₄²	—	94	6	no	—
111	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
112	#107	—	47	12	no	—
113	#103	—	47	24	no	—
114	#97	—	2	6	no	—
115	C₄×C₈	C₄×D₄₇	1	12	yes	—
116	#95	—	94	6	no	—
117	#95	D₁₈₈	1	1	yes	characteristic
118	#95	D₁₈₈	1	3	yes	—
119	C₂×Q₈	—	47	12	no	—
120	C₂×Q₈	—	94	6	no	—
121	C₂²×C₄	—	94	2	no	—
122	C₂²×C₄	—	94	6	no	—
123	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
124	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
125	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
126	Q₁₆	—	47	16	no	—
127	C₂×C₈	—	2	6	no	—
128	C₂×C₈	C₂×C₄×D₄₇	1	6	yes	—
129	C₂×C₈	C₂×C₄×D₄₇	1	6	yes	—
130	C₄⋊C₄	—	47	12	no	—
131	C₄⋊C₄	—	94	24	no	—
132	C₄⋊C₄	—	94	6	no	—
133	C₄²	—	94	24	no	—
134	C₄²	—	376	6	no	—
135	C₄²	C₂×C₄×D₄₇	1	6	yes	—
136	C₄²	#10625350	1	6	yes	—
137	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
138	Q₈	—	47	8	no	—
139	Q₈	—	94	12	no	—
140	C₂×C₄	—	2	3	no	—
141	C₂×C₄	—	2	6	no	—
142	C₂×C₄	—	2	3	no	—
143	C₂×C₄	—	94	12	no	—
144	C₂×C₄	—	94	12	no	—
145	C₂×C₄	—	188	6	no	—
146	C₂×C₄	—	188	6	no	—
147	C₂×C₄	C₄×D₁₈₈	1	6	yes	—
148	C₂×C₄	C₄×D₁₈₈	1	6	yes	—
149	C₂×C₄	C₂²×C₄×D₄₇	1	3	yes	—
150	C₂×C₄	C₂×#10625350	1	3	yes	—
151	C₈	—	2	6	no	—
152	C₈	C₄²×D₄₇	1	4	yes	—
153	C₂²	C₂×C₄×D₁₈₈	1	3	yes	—
154	C₂²	#15689928	1	1	yes	characteristic
155	C₂²	#15689940	1	3	yes	—
156	C₄	—	2	6	no	—
157	C₄	—	2	6	no	—
158	C₄	—	94	8	no	—
159	C₄	—	188	12	no	—
160	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
161	C₄	C₄×#10625350	1	3	yes	—
162	C₂	C₄²×D₁₈₈	1	1	yes	characteristic
163	C₂	#20252257	1	3	yes	—
164	C₂	#20252265	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₈₈

C₂

yes

maximal

#20245751

C₂

yes

maximal

C₄×#13093958

C₂

yes

maximal

#20243219

C₂

yes

maximal

C₄₇×#1726

C₂

yes

maximal, characteristic

C₂²×Dic₁₈₈

C₂²

yes

characteristic

C₂×C₄×Dic₉₄

C₂²

yes

—

C₄×Dic₁₈₈

C₄

yes

complemented

C₂×#13093958

C₂²

yes

—

C₂×#13093949

C₂²

yes

—

C₄×C₄⋊Dic₄₇

C₂²

yes

—

#15688974

—

#15688968

—

#15688886

C₂²

yes

—

C₂×#97×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₄₇×#376

C₂²

yes

characteristic

C₂²×Dic₉₄

C₂³

yes

—

C₂×Dic₁₈₈

C₂×C₄

yes

—

C₄×Dic₉₄

—

C₄×Dic₉₄

C₂×C₄

yes

—

C₂×C₄⋊Dic₄₇

C₂³

yes

—

C₂×Dic₄₇⋊C₄

—

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

—

#13093958

C₂×C₄

yes

—

#13093949

C₂×C₄

yes

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

C₂³

yes

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

—

#97×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

#95×C₄₇

D₄

yes

characteristic

#95×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂²×C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂²×C₄

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

Dic₁₈₈

C₄²

yes

complemented

C₂³×C₄₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₂³×C₄

yes

—

C₂×C₄×C₄₇

C₂×#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

C₂×C₄²

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₄²

yes

—

#20507

—

maximal, Sylow

C₂²×C₄₇

C₂×C₄×D₄

yes

—

C₂²×C₄₇

#576

yes

characteristic

C₂²×C₄₇

#578

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²×C₄²

yes

characteristic

C₄×C₄₇

C₄×#41

yes

—

Dic₄₇

—

Dic₄₇

—

C₂×C₄×Q₁₆

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

C₄²×D₄

yes

characteristic

C₂×C₄₇

#2950

yes

—

C₂×C₄₇

#2951

yes

—

C₂²×Q₁₆

—

C₂×C₄×Q₈

—

C₄×Q₁₆

—

C₂×#107

—

C₂×#103

—

C₂×#97

C₂×D₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

#376

—

#376

C₂×D₄₇

yes

characteristic

#340

—

100

C₄₇

#20507

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

—

102

C₂×Q₁₆

—

103

C₂²×C₈

C₂²×D₄₇

yes

—

104

C₂²×C₈

C₂²×D₄₇

yes

characteristic

105

C₄×Q₈

—

106

C₄×Q₈

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄²

—

110

C₂×C₄²

—

111

C₂×C₄²

C₂²×D₄₇

yes

—

112

#107

—

113

#103

—

114

#97

—

115

C₄×C₈

C₄×D₄₇

yes

—

116

#95

—

117

#95

D₁₈₈

yes

characteristic

118

#95

D₁₈₈

yes

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂×D₁₈₈

yes

—

124

C₂²×C₄

C₂×D₁₈₈

yes

—

125

C₂²×C₄

C₂³×D₄₇

yes

characteristic

126

Q₁₆

—

127

C₂×C₈

—

128

C₂×C₈

C₂×C₄×D₄₇

yes

—

129

C₂×C₈

C₂×C₄×D₄₇

yes

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄²

—

134

C₄²

—

376

—

135

C₄²

C₂×C₄×D₄₇

yes

—

136

C₄²

#10625350

yes

—

137

C₂³

C₂²×D₁₈₈

yes

characteristic

138

Q₈

—

139

Q₈

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

188

—

146

C₂×C₄

—

188

—

147

C₂×C₄

C₄×D₁₈₈

yes

—

148

C₂×C₄

C₄×D₁₈₈

yes

—

149

C₂×C₄

C₂²×C₄×D₄₇

yes

—

150

C₂×C₄

C₂×#10625350

yes

—

151

C₈

—

152

C₈

C₄²×D₄₇

yes

—

153

C₂²

C₂×C₄×D₁₈₈

yes

—

154

C₂²

#15689928

yes

characteristic

155

C₂²

#15689940

yes

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

188

—

160

C₄

C₂×C₄²×D₄₇

yes

characteristic

161

C₄

C₄×#10625350

yes

—

162

C₂

C₄²×D₁₈₈

yes

characteristic

163

C₂

#20252257

yes

—

164

C₂

#20252265

yes

—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1726	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#20507
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

subgroup

G/H

center

C₂³

C₂²×D₁₈₈

commutator

C₄×C₄₇

C₂²×C₄²

Frattini

C₂²×C₄

C₂³×D₄₇

Fitting

C₄₇×#1726

C₂

socle, prime radical

C₂³×C₄₇

C₂²×D₄

cosocle

C₂²×C₄×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ #376 ◁ #1726

N	H ≅ G/N	type
8. C₄×Dic₁₈₈	C₄	semidirect
53. Dic₁₈₈	C₄²	semidirect
100. C₄₇	#20507	semidirect

H ≅ G/N

type

8. C₄×Dic₁₈₈

C₄

semidirect

53. Dic₁₈₈

C₄²

semidirect

100. C₄₇

#20507

semidirect

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

subgroup	index
C₂×C₄×Dic₁₈₈	2
#20245751	2
C₄×#13093958	2
#20243219	2
C₄₇×#1726	2
#20507	47

subgroup

index

C₂×C₄×Dic₁₈₈

C₄₇×#1726

Subgroups — #26214822

Conjugacy classes of subgroups

Complements

Maximal subgroups