Group #19133569 — subgroups

Subgroups — #19133569

Order 5504.

238 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 57 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	#14583054	C₂	1	2	yes	maximal
2	#14582819	C₂	1	1	yes	maximal, characteristic
3	#14582636	C₂	1	2	yes	maximal
4	#14582625	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#544	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
7	C₂×#10621885	—	2	2	no	—
8	C₂×Dic₄₃⋊C₄	—	2	1	no	—
9	#13063248	—	2	2	no	—
10	C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
11	C₄₃⋊#95	—	2	1	no	—
12	C₄₃⋊#95	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₄₃	—	2	2	no	—
14	#125×C₄₃	—	2	2	no	—
15	C₂²⋊Q₈×C₄₃	—	2	1	no	—
16	C₂²⋊Q₈×C₄₃	C₂²	1	2	yes	—
17	#117×C₄₃	—	2	1	no	—
18	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₄₃	—	2	1	no	—
20	C₂×Q₈×C₄₃	D₄	1	2	yes	—
21	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₄₃	—	2	1	no	—
25	C₂²×Dic₄₃	—	2	1	no	—
26	C₂×Dic₈₆	—	2	2	no	—
27	C₂×Dic₈₆	—	4	2	no	—
28	C₄⋊C₄×C₄₃	—	2	1	no	—
29	C₄⋊C₄×C₄₃	—	2	1	no	—
30	C₄⋊C₄×C₄₃	—	2	1	no	—
31	C₄⋊C₄×C₄₃	—	2	2	no	—
32	C₄⋊C₄×C₄₃	—	2	2	no	—
33	C₄⋊C₄×C₄₃	—	2	2	no	—
34	C₄⋊C₄×C₄₃	D₄	1	2	yes	—
35	C₂²⋊C₄×C₄₃	—	2	2	no	—
36	C₄²×C₄₃	—	2	2	no	—
37	#10621885	—	4	4	no	—
38	Dic₄₃⋊C₄	—	4	2	no	—
39	C₂×C₄₃⋊C₈	—	2	2	no	—
40	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₄₃	—	4	2	no	—
42	Q₈×C₄₃	—	8	1	no	—
43	C₂×C₄×C₄₃	—	2	1	no	—
44	C₂×C₄×C₄₃	—	2	1	no	—
45	C₂×C₄×C₄₃	—	2	2	no	—
46	C₂×C₄×C₄₃	—	2	2	no	—
47	C₂×C₄×C₄₃	—	2	1	no	—
48	C₂×C₄×C₄₃	—	4	1	no	—
49	C₂×C₄×C₄₃	—	4	1	no	—
50	C₂×C₄×C₄₃	Q₁₆	1	2	yes	complemented
51	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
52	C₂×Dic₄₃	—	2	2	no	—
53	C₂×Dic₄₃	—	2	2	no	—
54	C₂×Dic₄₃	—	4	1	no	—
55	C₂×Dic₄₃	—	4	1	no	—
56	C₂×Dic₄₃	—	4	1	no	—
57	C₂×Dic₄₃	—	4	1	no	—
58	Dic₈₆	—	2	4	no	—
59	Dic₈₆	—	4	2	no	—
60	C₄₃⋊C₈	—	4	2	no	—
61	C₂²×C₄₃	—	2	1	no	—
62	C₂²×C₄₃	—	2	1	no	—
63	C₂²×C₄₃	#120	1	1	yes	characteristic
64	C₂²×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₂²×C₄₃	#137	1	1	yes	characteristic
66	C₄×C₄₃	—	2	2	no	—
67	C₄×C₄₃	—	2	2	no	—
68	C₄×C₄₃	—	4	2	no	—
69	C₄×C₄₃	—	4	1	no	—
70	C₄×C₄₃	—	4	2	no	—
71	Dic₄₃	—	4	2	no	—
72	Dic₄₃	—	4	2	no	—
73	#1605	—	43	1	no	maximal, Sylow
74	C₂×C₄₃	—	2	1	no	—
75	C₂×C₄₃	—	2	1	no	—
76	C₂×C₄₃	#451	1	1	yes	characteristic
77	C₂×C₄₃	#456	1	1	yes	characteristic
78	C₂×C₄₃	B(4,𝔽₂)	1	1	yes	characteristic
79	#544	D₄₃	1	1	yes	characteristic
80	#451	—	43	2	no	—
81	#395	—	43	1	no	—
82	#328	—	43	2	no	—
83	#324	—	43	1	no	—
84	C₄₃	#1605	1	1	yes	characteristic, complemented, Sylow
85	C₂²×Q₈	—	43	1	no	—
86	C₂×Q₁₆	—	86	2	no	—
87	C₄⋊Q₈	—	2	2	no	—
88	#125	—	2	2	no	—
89	C₂²⋊Q₈	—	2	1	no	—
90	C₂²⋊Q₈	C₂×D₄₃	1	2	yes	—
91	#117	—	2	1	no	—
92	C₂×C₄⋊C₄	—	86	1	no	—
93	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
94	#103	—	86	2	no	—
95	C₂²⋊C₈	—	43	2	no	—
96	#95	—	43	1	no	—
97	#95	—	86	1	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	86	2	no	—
100	C₂×Q₈	—	172	2	no	—
101	C₂×Q₈	C₂²⋊D₄₃	1	2	yes	—
102	C₂²×C₄	—	86	1	no	—
103	C₂²×C₄	—	86	1	no	—
104	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
107	Q₁₆	—	172	4	no	—
108	C₂×C₈	—	86	2	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	172	2	no	—
116	C₄⋊C₄	C₂²⋊D₄₃	1	2	yes	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₄²	—	2	2	no	—
119	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
120	Q₈	—	4	2	no	—
121	Q₈	—	8	1	no	—
122	Q₈	—	86	4	no	—
123	Q₈	—	172	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	86	2	no	—
132	C₂×C₄	—	86	2	no	—
133	C₂×C₄	—	172	1	no	—
134	C₂×C₄	—	172	1	no	—
135	C₂×C₄	—	172	1	no	—
136	C₂×C₄	—	172	1	no	—
137	C₂×C₄	#10621885	1	2	yes	complemented
138	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
139	C₈	—	172	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	#13063294	1	1	yes	characteristic
143	C₂²	C₂×#10621885	1	1	yes	characteristic
144	C₂²	#13063356	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	2	no	—
148	C₄	—	4	1	no	—
149	C₄	—	4	2	no	—
150	C₄	—	172	2	no	—
151	C₄	—	172	2	no	—
152	C₂	—	2	1	no	—
153	C₂	—	2	1	no	—
154	C₂	#14583055	1	1	yes	characteristic
155	C₂	#14583082	1	1	yes	characteristic
156	C₂	#14583086	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583054

C₂

yes

maximal

#14582819

C₂

yes

maximal, characteristic

#14582636

C₂

yes

maximal

#14582625

C₂

yes

maximal, characteristic

C₄₃×#544

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

C₂×#10621885

—

C₂×Dic₄₃⋊C₄

—

#13063248

—

C₂²⋊(C₄₃⋊C₈)

C₂²

yes

—

C₄₃⋊#95

—

C₄₃⋊#95

C₂²

yes

characteristic

C₄⋊Q₈×C₄₃

—

#125×C₄₃

—

C₂²⋊Q₈×C₄₃

—

C₂²⋊Q₈×C₄₃

C₂²

yes

—

#117×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

#10621885

—

Dic₄₃⋊C₄

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

Q₁₆

yes

complemented

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#120

yes

characteristic

C₂²×C₄₃

C₂×Q₁₆

yes

characteristic

C₂²×C₄₃

#137

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

Dic₄₃

—

Dic₄₃

—

#1605

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

yes

—

—

—

—

C₄₃

#1605

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂×D₄₃

yes

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

#103

—

C₂²⋊C₈

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

172

—

101

C₂×Q₈

C₂²⋊D₄₃

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

105

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

106

C₂²×C₄

C₂²×D₄₃

yes

characteristic

107

Q₁₆

—

172

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

172

—

116

C₄⋊C₄

C₂²⋊D₄₃

yes

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

172

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

172

—

134

C₂×C₄

—

172

—

135

C₂×C₄

—

172

—

136

C₂×C₄

—

172

—

137

C₂×C₄

#10621885

yes

complemented

138

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

139

C₈

—

172

—

140

C₂²

—

141

C₂²

—

142

C₂²

#13063294

yes

characteristic

143

C₂²

C₂×#10621885

yes

characteristic

144

C₂²

#13063356

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

172

—

151

C₄

—

172

—

152

C₂

—

153

C₂

—

154

C₂

#14583055

yes

characteristic

155

C₂

#14583082

yes

characteristic

156

C₂

#14583086

yes

characteristic

	subgroup	G/H
center	C₂²	#13063294
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#544	C₂
socle	C₂²×C₄₃	#120
hypercenter	#544	D₄₃
hypocenter	C₄₃	#1605
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

subgroup

G/H

center

C₂²

#13063294

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#544

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

N	H ≅ G/N	type
50. C₂×C₄×C₄₃	Q₁₆	semidirect
84. C₄₃	#1605	semidirect
137. C₂×C₄	#10621885	semidirect

H ≅ G/N

type

50. C₂×C₄×C₄₃

Q₁₆

semidirect

84. C₄₃

#1605

semidirect

137. C₂×C₄

#10621885

semidirect

Subgroups — #19133569

Conjugacy classes of subgroups

Complements