Group #14583471 — subgroups

Subgroups — #14583471

Order 2752.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×#10621900	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#13063324	C₂	1	2	yes	maximal, complemented
4	#13063311	C₂	1	2	yes	maximal, complemented
5	#13063309	C₂	1	1	yes	maximal, characteristic, complemented
6	#13063308	C₂	1	1	yes	maximal, characteristic, complemented
7	#13063306	C₂	1	1	yes	maximal, characteristic, complemented
8	#13063303	C₂	1	2	yes	maximal, complemented
9	#13063302	C₂	1	2	yes	maximal, complemented
10	#13063281	C₂	1	1	yes	maximal, characteristic, complemented
11	#123×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×C₄₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₄₃	C₂²	1	1	yes	characteristic
16	C₂²×Dic₄₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₄₃	C₂²	1	1	yes	characteristic
18	C₂²×Dic₄₃	C₂²	1	1	yes	characteristic
19	#10621900	—	2	4	no	—
20	C₂×C₄×D₄₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₈₆	—	2	4	no	—
22	C₂×Dic₈₆	C₂²	1	2	yes	—
23	C₂×Dic₈₆	C₂²	1	2	yes	—
24	C₂×Dic₈₆	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
26	C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
28	C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
29	C₂²⋊Dic₄₃	C₂²	1	2	yes	—
30	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
31	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
32	C₄⋊Dic₄₃	C₂²	1	2	yes	—
33	Dic₄₃⋊C₄	C₂²	1	2	yes	—
34	Dic₄₃⋊C₄	C₂²	1	2	yes	—
35	Dic₄₃⋊C₄	C₂²	1	2	yes	—
36	C₄×Dic₄₃	C₂²	1	2	yes	—
37	C₂³×C₄₃	C₂³	1	1	yes	characteristic
38	C₂³×C₄₃	C₂³	1	1	yes	characteristic
39	C₂²×D₄₃	C₂³	1	1	yes	characteristic
40	D₄×C₄₃	—	2	2	no	—
41	C₂×C₄×C₄₃	—	2	2	no	—
42	C₂×C₄×C₄₃	C₂³	1	2	yes	—
43	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₄₃	C₂³	1	2	yes	—
45	C₂²⋊D₄₃	—	2	2	no	—
46	C₂²⋊D₄₃	—	2	2	no	—
47	C₂×Dic₄₃	—	2	2	no	—
48	C₂×Dic₄₃	—	2	2	no	—
49	C₂×Dic₄₃	D₄	1	4	yes	—
50	C₂×Dic₄₃	C₂³	1	2	yes	—
51	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
52	C₂×Dic₄₃	C₂³	1	2	yes	—
53	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
54	C₂×Dic₄₃	C₂³	1	1	yes	characteristic
55	C₄×D₄₃	—	2	2	no	—
56	Dic₈₆	—	2	4	no	—
57	Dic₈₆	—	2	4	no	—
58	Dic₈₆	—	2	2	no	—
59	C₂²×C₄₃	—	2	2	no	—
60	C₂²×C₄₃	—	2	1	no	—
61	C₂²×C₄₃	—	2	2	no	—
62	C₂²×C₄₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄₃	C₂⁴	1	1	yes	characteristic
64	C₂×D₄₃	—	2	1	no	—
65	C₂×D₄₃	—	2	2	no	—
66	C₄×C₄₃	—	2	2	no	—
67	C₄×C₄₃	—	2	1	no	—
68	C₄×C₄₃	—	2	2	no	—
69	Dic₄₃	—	2	2	no	—
70	Dic₄₃	—	2	1	no	—
71	Dic₄₃	—	2	2	no	—
72	Dic₄₃	C₂×D₄	1	2	yes	—
73	Dic₄₃	C₂×D₄	1	2	yes	—
74	C₂×C₄₃	—	2	2	no	—
75	C₂×C₄₃	—	4	1	no	—
76	C₂×C₄₃	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄₃	2₋¹⁺⁴	1	2	yes	—
78	D₄₃	—	4	1	no	—
79	#536	—	43	1	no	maximal, Sylow
80	C₄₃	#536	1	1	yes	characteristic, complemented, Sylow
81	C₂×#41	—	43	1	no	—
82	C₂²×Q₈	—	43	1	no	—
83	C₄⋊Q₈	—	43	2	no	—
84	#124	—	43	2	no	—
85	#123	—	43	1	no	—
86	#123	—	43	1	no	—
87	#123	—	43	1	no	—
88	#123	D₄₃	1	1	yes	characteristic, complemented
89	C₂²⋊Q₈	—	43	2	no	—
90	C₂²⋊Q₈	—	43	2	no	—
91	#117	—	43	1	no	—
92	#41	—	86	4	no	—
93	C₂×Q₈	—	43	2	no	—
94	C₂×Q₈	—	43	2	no	—
95	C₂×Q₈	—	43	1	no	—
96	C₂×Q₈	—	86	4	no	—
97	C₂×D₄	—	43	1	no	—
98	C₂×D₄	—	43	1	no	—
99	C₂×D₄	C₂×D₄₃	1	1	yes	characteristic
100	C₂²×C₄	—	43	1	no	—
101	C₂²×C₄	—	43	1	no	—
102	C₂²×C₄	—	43	1	no	—
103	C₂²×C₄	—	43	1	no	—
104	C₂²×C₄	C₂×D₄₃	1	1	yes	characteristic
105	C₄⋊C₄	—	43	2	no	—
106	C₄⋊C₄	—	43	2	no	—
107	C₄⋊C₄	—	43	2	no	—
108	C₄⋊C₄	—	43	2	no	—
109	C₄⋊C₄	C₂×D₄₃	1	2	yes	—
110	C₂²⋊C₄	—	43	1	no	—
111	C₂²⋊C₄	—	43	2	no	—
112	C₂²⋊C₄	—	43	2	no	—
113	C₂²⋊C₄	—	43	2	no	—
114	C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
115	C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
116	C₄²	—	43	2	no	—
117	C₂³	—	43	1	no	—
118	C₂³	C₂²×D₄₃	1	1	yes	characteristic
119	C₂³	C₂²×D₄₃	1	1	yes	characteristic
120	Q₈	—	86	4	no	—
121	Q₈	—	86	4	no	—
122	Q₈	—	86	2	no	—
123	D₄	—	2	2	no	—
124	D₄	—	86	2	no	—
125	D₄	—	86	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	43	4	no	—
128	C₂×C₄	—	43	1	no	—
129	C₂×C₄	—	43	1	no	—
130	C₂×C₄	—	43	2	no	—
131	C₂×C₄	—	43	2	no	—
132	C₂×C₄	—	43	1	no	—
133	C₂×C₄	—	86	2	no	—
134	C₂×C₄	—	86	2	no	—
135	C₂×C₄	—	86	2	no	—
136	C₂×C₄	C₂²×D₄₃	1	2	yes	—
137	C₂×C₄	C₂²×D₄₃	1	2	yes	—
138	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	86	1	no	—
143	C₂²	—	86	2	no	—
144	C₂²	D₄×D₄₃	1	2	yes	—
145	C₂²	C₂³×D₄₃	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	2	no	—
149	C₄	—	43	2	no	—
150	C₄	—	43	2	no	—
151	C₄	—	86	1	no	—
152	C₄	—	86	2	no	—
153	C₄	—	86	2	no	—
154	C₂	—	2	2	no	—
155	C₂	—	4	1	no	—
156	C₂	—	172	1	no	—
157	C₂	C₂×D₄×D₄₃	1	1	yes	characteristic
158	C₂	#13063372	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10621900

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂

yes

maximal, characteristic, complemented

#13063324

C₂

yes

maximal, complemented

#13063311

C₂

yes

maximal, complemented

#13063309

C₂

yes

maximal, characteristic, complemented

#13063308

C₂

yes

maximal, characteristic, complemented

#13063306

C₂

yes

maximal, characteristic, complemented

#13063303

C₂

yes

maximal, complemented

#13063302

C₂

yes

maximal, complemented

#13063281

C₂

yes

maximal, characteristic, complemented

#123×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

C₂²

yes

characteristic

C₂×C₂²⋊D₄₃

C₂²

yes

characteristic

C₂×C₂²⋊D₄₃

C₂²

yes

characteristic

C₂²×Dic₄₃

C₂²

yes

characteristic

C₂²×Dic₄₃

C₂²

yes

characteristic

C₂²×Dic₄₃

C₂²

yes

characteristic

#10621900

—

C₂×C₄×D₄₃

C₂²

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂²

yes

—

C₂×Dic₈₆

C₂²

yes

—

C₂×Dic₈₆

C₂²

yes

characteristic

C₄⋊C₄×C₄₃

C₂²

yes

—

C₂²⋊C₄×C₄₃

C₂²

yes

characteristic

C₂²⋊C₄×C₄₃

C₂²

yes

—

C₂²⋊Dic₄₃

C₂²

yes

characteristic

C₂²⋊Dic₄₃

C₂²

yes

—

(C₂×D₄₃)⋊C₄

C₂²

yes

—

(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₄₃

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

Dic₄₃⋊C₄

C₂²

yes

—

C₄×Dic₄₃

C₂²

yes

—

C₂³×C₄₃

C₂³

yes

characteristic

C₂³×C₄₃

C₂³

yes

characteristic

C₂²×D₄₃

C₂³

yes

characteristic

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂³

yes

—

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₄×C₄₃

C₂³

yes

—

C₂²⋊D₄₃

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

D₄

yes

—

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

characteristic

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

characteristic

C₂×Dic₄₃

C₂³

yes

characteristic

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×D₄

yes

—

C₂²×C₄₃

C₂⁴

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

C₂×D₄

yes

—

Dic₄₃

C₂×D₄

yes

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

C₂²×D₄

yes

characteristic

C₂×C₄₃

2₋¹⁺⁴

yes

—

D₄₃

—

#536

—

maximal, Sylow

C₄₃

#536

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

—

C₄⋊Q₈

—

—

—

—

—

D₄₃

yes

characteristic, complemented

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#117

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₃

yes

characteristic

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×D₄₃

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

C₂×D₄₃

yes

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

C₂×D₄₃

yes

characteristic

115

C₂²⋊C₄

C₂×D₄₃

yes

—

116

C₄²

—

117

C₂³

—

118

C₂³

C₂²×D₄₃

yes

characteristic

119

C₂³

C₂²×D₄₃

yes

characteristic

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₂²×D₄₃

yes

—

137

C₂×C₄

C₂²×D₄₃

yes

—

138

C₂×C₄

C₂²×D₄₃

yes

characteristic

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

D₄×D₄₃

yes

—

145

C₂²

C₂³×D₄₃

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₂

—

155

C₂

—

156

C₂

—

172

—

157

C₂

C₂×D₄×D₄₃

yes

characteristic

158

C₂

#13063372

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃
commutator, socle, cosocle	C₂²×C₄₃	C₂⁴
Fitting	#123×C₄₃	C₂
prime radical	C₂×#10621900	C₂
hypercenter	#123	D₄₃
hypocenter	C₄₃	#536
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₃

commutator, socle, cosocle

C₂²×C₄₃

C₂⁴

Fitting

#123×C₄₃

C₂

C₂

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
2. C₂²×Dic₈₆	C₂	semidirect
3. #13063324	C₂	semidirect
4. #13063311	C₂	semidirect
5. #13063309	C₂	semidirect
6. #13063308	C₂	semidirect
7. #13063306	C₂	semidirect
8. #13063303	C₂	semidirect
9. #13063302	C₂	semidirect
10. #13063281	C₂	semidirect
11. #123×C₄₃	C₂	semidirect
80. C₄₃	#536	semidirect
88. #123	D₄₃	semidirect

H ≅ G/N

type

2. C₂²×Dic₈₆

C₂

semidirect

3. #13063324

C₂

semidirect

4. #13063311

C₂

semidirect

5. #13063309

C₂

semidirect

6. #13063308

C₂

semidirect

7. #13063306

C₂

semidirect

8. #13063303

C₂

semidirect

9. #13063302

C₂

semidirect

10. #13063281

C₂

semidirect

11. #123×C₄₃

C₂

semidirect

80. C₄₃

#536

semidirect

88. #123

D₄₃

semidirect

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

subgroup	index
C₂×#10621900	2
C₂²×Dic₈₆	2
#13063324	2
#13063311	2
#13063309	2
#13063308	2
#13063306	2
#13063303	2
#13063302	2
#13063281	2
#123×C₄₃	2
#536	43

subgroup

index

C₂²×Dic₈₆

#123×C₄₃

Subgroups — #14583471

Conjugacy classes of subgroups

Complements

Maximal subgroups