Group #19133229 — subgroups

Subgroups — #19133229

Order 5504.

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

normal subgroups only

This group has 238 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	#14583449	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	C₂×#10621898	C₄	1	2	yes	—
6	#13063325	C₄	1	1	yes	characteristic
7	#13063322	C₄	1	1	yes	characteristic
8	#13063313	C₂²	1	2	yes	—
9	#13063299	—	2	1	no	—
10	#13063299	—	2	1	no	—
11	C₄×D₁₇₂	—	2	1	no	—
12	C₄×D₁₇₂	—	2	1	no	—
13	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
14	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
15	#13063236	C₂²	1	2	yes	—
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	—
20	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₄₃	—	2	2	no	—
24	#10621898	—	4	4	no	—
25	C₂×D₁₇₂	—	2	1	no	—
26	C₂×D₁₇₂	—	2	1	no	—
27	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₄₃	—	2	2	no	—
29	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₄₃	—	2	2	no	—
31	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₄₃	D₄	1	1	yes	characteristic
33	C₄²×C₄₃	D₄	1	1	yes	characteristic
34	C₄²×C₄₃	C₂³	1	1	yes	characteristic
35	(C₂×D₄₃)⋊C₄	—	2	2	no	—
36	(C₂×D₄₃)⋊C₄	—	2	2	no	—
37	C₄⋊Dic₄₃	—	2	1	no	—
38	C₄⋊Dic₄₃	—	2	1	no	—
39	C₂×C₄₃⋊C₈	—	2	2	no	—
40	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×D₄₃	—	2	2	no	—
42	C₂×C₄×C₄₃	—	2	2	no	—
43	C₂×C₄×C₄₃	—	4	1	no	—
44	C₂×C₄×C₄₃	—	4	1	no	—
45	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂²⋊D₄₃	—	4	4	no	—
52	C₂×Dic₄₃	—	2	2	no	—
53	D₁₇₂	—	4	2	no	—
54	D₁₇₂	—	4	2	no	—
55	D₁₇₂	—	4	2	no	—
56	C₄×D₄₃	—	4	4	no	—
57	Dic₈₆	—	4	2	no	—
58	C₈×C₄₃	—	4	2	no	—
59	C₄₃⋊C₈	—	4	2	no	—
60	C₂²×C₄₃	—	2	1	no	—
61	C₂²×C₄₃	—	2	1	no	—
62	C₂²×C₄₃	—	2	1	no	—
63	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×D₄₃	—	4	2	no	—
65	C₂×D₄₃	—	4	2	no	—
66	C₂×D₄₃	—	4	2	no	—
67	C₄×C₄₃	—	2	1	no	—
68	C₄×C₄₃	—	2	1	no	—
69	C₄×C₄₃	—	2	1	no	—
70	C₄×C₄₃	—	2	1	no	—
71	C₄×C₄₃	—	2	1	no	—
72	C₄×C₄₃	#100	1	2	yes	—
73	Dic₄₃	—	4	2	no	—
74	#1536	—	43	1	no	maximal, Sylow
75	C₂×C₄₃	—	4	1	no	—
76	C₂×C₄₃	C₂×#100	1	1	yes	characteristic
77	C₂×C₄₃	#421	1	1	yes	characteristic
78	C₂×C₄₃	#421	1	1	yes	characteristic
79	D₄₃	—	8	2	no	—
80	#532	—	43	1	no	—
81	#432	—	43	1	no	—
82	#432	D₄₃	1	1	yes	characteristic, complemented
83	#329	—	43	4	no	—
84	C₄₃	#1536	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	—	43	2	no	—
86	C₄⋊Q₈	—	43	1	no	—
87	#127	—	43	1	no	—
88	#124	—	43	2	no	—
89	#121	—	86	1	no	—
90	#121	—	86	1	no	—
91	C₄×D₄	—	86	1	no	—
92	C₄×D₄	—	86	1	no	—
93	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
94	C₄⋊C₈	—	2	1	no	—
95	C₄⋊C₈	—	86	1	no	—
96	C₂²⋊C₈	—	2	1	no	—
97	C₂²⋊C₈	—	86	1	no	—
98	#97	—	43	2	no	—
99	#97	C₂×D₄₃	1	2	yes	—
100	#41	—	172	4	no	—
101	C₂×Q₈	—	43	2	no	—
102	C₂×D₄	—	43	2	no	—
103	C₂×D₄	—	86	2	no	—
104	C₂×D₄	—	86	1	no	—
105	C₂×D₄	—	86	1	no	—
106	C₂²×C₄	—	86	2	no	—
107	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
108	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	86	2	no	—
112	C₄⋊C₄	—	86	1	no	—
113	C₄⋊C₄	—	86	1	no	—
114	C₂²⋊C₄	—	86	2	no	—
115	C₂²⋊C₄	—	86	2	no	—
116	C₄²	C₄×D₄₃	1	1	yes	characteristic
117	C₄²	D₁₇₂	1	1	yes	characteristic
118	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
119	C₄²	C₂²×D₄₃	1	1	yes	characteristic
120	C₂³	—	86	2	no	—
121	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
122	Q₈	—	172	2	no	—
123	D₄	—	172	2	no	—
124	D₄	—	172	2	no	—
125	D₄	—	172	2	no	—
126	D₄	—	172	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	86	2	no	—
131	C₂×C₄	—	172	4	no	—
132	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
133	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
134	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
136	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	172	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	172	2	no	—
144	C₂²	—	172	2	no	—
145	C₂²	—	172	2	no	—
146	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	172	2	no	—
153	C₄	#13063241	1	2	yes	—
154	C₂	—	4	1	no	—
155	C₂	—	344	2	no	—
156	C₂	C₂×#13063241	1	1	yes	characteristic
157	C₂	#14582915	1	1	yes	characteristic
158	C₂	#14582917	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583449

C₂

yes

maximal, characteristic

#14582956

C₂

yes

maximal, characteristic, complemented

#14582638

C₂

yes

maximal, complemented

C₄₃×#432

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

C₄

yes

—

#13063325

C₄

yes

characteristic

#13063322

C₄

yes

characteristic

#13063313

C₂²

yes

—

#13063299

—

#13063299

—

C₄×D₁₇₂

—

C₄×D₁₇₂

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

#97×C₄₃

C₂²

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂×C₂²⋊D₄₃

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×C₄×D₄₃

—

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

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

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

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#100

yes

—

Dic₄₃

—

#1536

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

#532

—

#432

—

#432

D₄₃

yes

characteristic, complemented

#329

—

C₄₃

#1536

yes

characteristic, complemented, Sylow

C₂×#41

—

C₄⋊Q₈

—

—

—

—

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

C₂×D₄₃

yes

—

100

#41

—

172

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₄×D₄₃

yes

characteristic

108

C₂²×C₄

D₁₇₂

yes

characteristic

109

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

C₄×D₄₃

yes

characteristic

117

C₄²

D₁₇₂

yes

characteristic

118

C₄²

C₂²⋊D₄₃

yes

characteristic

119

C₄²

C₂²×D₄₃

yes

characteristic

120

C₂³

—

121

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

122

Q₈

—

172

—

123

D₄

—

172

—

124

D₄

—

172

—

125

D₄

—

172

—

126

D₄

—

172

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

172

—

132

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

133

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

134

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

135

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

136

C₂×C₄

C₂×D₁₇₂

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

138

C₈

—

139

C₈

—

172

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

172

—

144

C₂²

—

172

—

145

C₂²

—

172

—

146

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

172

—

153

C₄

#13063241

yes

—

154

C₂

—

155

C₂

—

344

—

156

C₂

C₂×#13063241

yes

characteristic

157

C₂

#14582915

yes

characteristic

158

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	#1536
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
82. #432	D₄₃	semidirect
84. C₄₃	#1536	semidirect

H ≅ G/N

type

2. #14582956

C₂

semidirect

3. #14582638

C₂

semidirect

4. C₄₃×#432

C₂

semidirect

82. #432

D₄₃

semidirect

84. C₄₃

#1536

semidirect

Subgroups — #19133229

Conjugacy classes of subgroups

Complements