Group #19133218 — subgroups

Subgroups — #19133218

Order 5504.

248 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 69 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×#13063313	C₂	1	1	yes	maximal, characteristic
2	#14582956	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582638	C₂	1	4	yes	maximal, complemented
4	C₄₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Dic₈₆	C₄	1	1	yes	characteristic
6	C₂²×D₁₇₂	C₄	1	1	yes	characteristic
7	#13063313	—	2	1	no	—
8	#13063313	—	2	1	no	—
9	#13063313	C₄	1	2	yes	—
10	#13063313	C₂²	1	2	yes	—
11	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
12	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
13	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
14	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
15	#13063236	C₂²	1	2	yes	complemented
16	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₄₃	—	2	1	no	—
18	C₂²⋊C₈×C₄₃	—	2	1	no	—
19	#97×C₄₃	C₂²	1	2	yes	complemented
20	C₂³×D₄₃	—	2	1	no	—
21	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
24	C₂²×Dic₄₃	—	2	1	no	—
25	C₂×D₁₇₂	—	2	1	no	—
26	C₂×D₁₇₂	—	4	2	no	—
27	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
28	C₂×Dic₈₆	—	2	1	no	—
29	C₂×Dic₈₆	—	4	2	no	—
30	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₄₃	—	2	2	no	—
32	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₄₃	D₄	1	1	yes	characteristic
34	C₄²×C₄₃	D₄	1	1	yes	characteristic
35	C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	(C₂×D₄₃)⋊C₄	—	2	2	no	—
37	(C₂×D₄₃)⋊C₄	—	2	2	no	—
38	(C₂×D₄₃)⋊C₄	—	2	2	no	—
39	(C₂×D₄₃)⋊C₄	—	2	2	no	—
40	C₂×C₄₃⋊C₈	—	2	2	no	—
41	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×D₄₃	—	2	2	no	—
43	C₂²×D₄₃	—	4	2	no	—
44	C₂²×D₄₃	—	4	2	no	—
45	C₂²×D₄₃	—	4	2	no	—
46	C₂×C₄×C₄₃	—	2	2	no	—
47	C₂×C₄×C₄₃	—	4	1	no	—
48	C₂×C₄×C₄₃	—	4	1	no	—
49	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₄₃	—	2	2	no	—
56	C₂×Dic₄₃	—	4	2	no	—
57	D₁₇₂	—	4	2	no	—
58	D₁₇₂	—	4	2	no	—
59	Dic₈₆	—	4	2	no	—
60	Dic₈₆	—	4	2	no	—
61	C₈×C₄₃	—	4	2	no	—
62	C₄₃⋊C₈	—	4	2	no	—
63	C₂²×C₄₃	—	2	1	no	—
64	C₂²×C₄₃	—	2	1	no	—
65	C₂²×C₄₃	#100	1	2	yes	—
66	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×D₄₃	—	4	2	no	—
68	C₂×D₄₃	—	4	2	no	—
69	C₂×D₄₃	—	4	2	no	—
70	C₂×D₄₃	—	8	4	no	—
71	C₄×C₄₃	—	2	1	no	—
72	C₄×C₄₃	—	2	1	no	—
73	C₄×C₄₃	—	2	1	no	—
74	C₄×C₄₃	—	2	1	no	—
75	C₄×C₄₃	—	2	1	no	—
76	C₄×C₄₃	—	2	1	no	—
77	Dic₄₃	—	4	2	no	—
78	#1533	—	43	1	no	maximal, Sylow
79	C₂×C₄₃	—	2	2	no	—
80	C₂×C₄₃	C₂×#100	1	1	yes	characteristic
81	C₂×C₄₃	#421	1	1	yes	characteristic
82	C₂×C₄₃	#421	1	1	yes	characteristic
83	D₄₃	—	8	2	no	—
84	C₂×#124	—	43	1	no	—
85	#432	—	43	1	no	—
86	#432	D₄₃	1	1	yes	characteristic, complemented
87	#329	—	43	4	no	—
88	C₄₃	#1533	1	1	yes	characteristic, complemented, Sylow
89	C₂²×Q₈	—	43	1	no	—
90	C₂²×D₄	—	43	1	no	—
91	#124	—	43	2	no	—
92	#124	—	43	2	no	—
93	#124	—	86	1	no	—
94	#124	—	86	1	no	—
95	C₂×C₂²⋊C₄	—	86	1	no	—
96	C₂×C₂²⋊C₄	—	86	1	no	—
97	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
98	C₄⋊C₈	—	2	1	no	—
99	C₄⋊C₈	—	86	1	no	—
100	C₂²⋊C₈	—	2	1	no	—
101	C₂²⋊C₈	—	86	1	no	—
102	#97	—	43	2	no	—
103	#97	C₂×D₄₃	1	2	yes	complemented
104	C₂⁴	—	86	1	no	—
105	C₂×Q₈	—	43	2	no	—
106	C₂×Q₈	—	86	1	no	—
107	C₂×Q₈	—	172	2	no	—
108	C₂×D₄	—	43	2	no	—
109	C₂×D₄	—	86	1	no	—
110	C₂×D₄	—	172	2	no	—
111	C₂²×C₄	—	86	1	no	—
112	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
113	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
114	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	86	2	no	—
117	C₂²⋊C₄	—	86	2	no	—
118	C₂²⋊C₄	—	86	2	no	—
119	C₂²⋊C₄	—	86	2	no	—
120	C₂²⋊C₄	—	86	2	no	—
121	C₄²	C₄×D₄₃	1	1	yes	characteristic
122	C₄²	D₁₇₂	1	1	yes	characteristic
123	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
124	C₄²	C₂²×D₄₃	1	1	yes	characteristic
125	C₂³	—	86	2	no	—
126	C₂³	—	172	2	no	—
127	C₂³	—	172	2	no	—
128	C₂³	—	172	2	no	—
129	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
130	Q₈	—	172	2	no	—
131	Q₈	—	172	2	no	—
132	D₄	—	172	2	no	—
133	D₄	—	172	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	86	2	no	—
138	C₂×C₄	—	172	2	no	—
139	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
140	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
141	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
143	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
145	C₈	—	4	2	no	—
146	C₈	—	172	2	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	344	4	no	—
150	C₂²	—	172	2	no	—
151	C₂²	—	172	2	no	—
152	C₂²	—	172	2	no	—
153	C₂²	#13063241	1	2	yes	—
154	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	—	172	2	no	—
162	C₂	—	2	2	no	—
163	C₂	—	344	2	no	—
164	C₂	C₂×#13063241	1	1	yes	characteristic
165	C₂	#14582915	1	1	yes	characteristic
166	C₂	#14582917	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063313

C₂

yes

maximal, characteristic

#14582956

C₂

yes

maximal, characteristic, complemented

#14582638

C₂

yes

maximal, complemented

C₄₃×#432

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₈₆

C₄

yes

characteristic

C₂²×D₁₇₂

C₄

yes

—

—

C₄

yes

—

#13063313

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₄

—

C₂×(C₂×D₄₃)⋊C₄

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

C₂²

yes

complemented

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

#97×C₄₃

C₂²

yes

complemented

C₂³×D₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#100

yes

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

Dic₄₃

—

#1533

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

C₂×#124

—

#432

—

#432

D₄₃

yes

characteristic, complemented

#329

—

C₄₃

#1533

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

—

—

—

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

C₂²⋊C₈

—

101

C₂²⋊C₈

—

102

#97

—

103

#97

C₂×D₄₃

yes

complemented

104

C₂⁴

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

172

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

172

—

111

C₂²×C₄

—

112

C₂²×C₄

C₄×D₄₃

yes

characteristic

113

C₂²×C₄

D₁₇₂

yes

characteristic

114

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

C₄×D₄₃

yes

characteristic

122

C₄²

D₁₇₂

yes

characteristic

123

C₄²

C₂²⋊D₄₃

yes

characteristic

124

C₄²

C₂²×D₄₃

yes

characteristic

125

C₂³

—

126

C₂³

—

172

—

127

C₂³

—

172

—

128

C₂³

—

172

—

129

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

130

Q₈

—

172

—

131

Q₈

—

172

—

132

D₄

—

172

—

133

D₄

—

172

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

172

—

139

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

140

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

141

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

142

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

143

C₂×C₄

C₂×D₁₇₂

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

145

C₈

—

146

C₈

—

172

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

344

—

150

C₂²

—

172

—

151

C₂²

—

172

—

152

C₂²

—

172

—

153

C₂²

#13063241

yes

—

154

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

172

—

162

C₂

—

163

C₂

—

344

—

164

C₂

C₂×#13063241

yes

characteristic

165

C₂

#14582915

yes

characteristic

166

C₂

#14582917

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₄₃)⋊C₄
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄₃×#432	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	#432	D₄₃
hypocenter	C₄₃	#1533
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×(C₂×D₄₃)⋊C₄

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄₃×#432

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
2. #14582956	C₂	semidirect
3. #14582638	C₂	semidirect
4. C₄₃×#432	C₂	semidirect
15. #13063236	C₂²	semidirect
19. #97×C₄₃	C₂²	semidirect
86. #432	D₄₃	semidirect
88. C₄₃	#1533	semidirect
103. #97	C₂×D₄₃	semidirect

H ≅ G/N

type

2. #14582956

C₂

semidirect

3. #14582638

C₂

semidirect

4. C₄₃×#432

C₂

semidirect

15. #13063236

C₂²

semidirect

19. #97×C₄₃

C₂²

semidirect

86. #432

D₄₃

semidirect

88. C₄₃

#1533

semidirect

103. #97

C₂×D₄₃

semidirect

Subgroups — #19133218

Conjugacy classes of subgroups

Complements