Group #19133186 — subgroups

Subgroups — #19133186

Order 5504.

270 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₄×#10621877	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582638	C₂	1	4	yes	maximal, complemented
4	C₄×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10621898	C₄	1	2	yes	—
6	C₂×#10621877	—	2	1	no	—
7	#13063325	C₄	1	1	yes	characteristic
8	#13063322	C₄	1	1	yes	characteristic
9	#13063313	C₂²	1	2	yes	—
10	#13063299	—	2	1	no	—
11	#13063299	—	2	1	no	—
12	C₄×D₁₇₂	—	2	1	no	—
13	C₄×D₁₇₂	—	2	1	no	—
14	#13063236	C₂²	1	2	yes	—
15	C₄×C₄₃⋊C₈	—	2	1	no	—
16	C₂×M₁₆×C₄₃	—	2	1	no	—
17	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	#97×C₄₃	C₂²	1	2	yes	—
19	C₄×C₈×C₄₃	—	2	1	no	—
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	M₁₆×C₄₃	—	2	4	no	—
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	C₄⋊Dic₄₃	—	2	1	no	—
39	C₄⋊Dic₄₃	—	2	1	no	—
40	#10621877	—	2	4	no	—
41	C₂×C₄₃⋊C₈	—	2	2	no	—
42	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×D₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	—	2	2	no	—
45	C₂×C₄×C₄₃	—	2	2	no	—
46	C₂×C₄×C₄₃	—	2	2	no	—
47	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂²⋊D₄₃	—	4	4	no	—
54	C₂×Dic₄₃	—	2	2	no	—
55	D₁₇₂	—	4	2	no	—
56	D₁₇₂	—	4	2	no	—
57	D₁₇₂	—	4	2	no	—
58	C₄×D₄₃	—	4	4	no	—
59	Dic₈₆	—	4	2	no	—
60	C₈×C₄₃	—	2	4	no	—
61	C₄₃⋊C₈	—	2	4	no	—
62	C₂²×C₄₃	—	2	1	no	—
63	C₂²×C₄₃	—	2	1	no	—
64	C₂²×C₄₃	—	2	1	no	—
65	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×D₄₃	—	4	2	no	—
67	C₂×D₄₃	—	4	2	no	—
68	C₂×D₄₃	—	4	2	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	C₄×C₄₃	#104	1	2	yes	—
74	C₄×C₄₃	#104	1	2	yes	—
75	Dic₄₃	—	4	2	no	—
76	#1525	—	43	1	no	maximal, Sylow
77	C₂×C₄₃	—	4	1	no	—
78	C₂×C₄₃	C₂×#100	1	1	yes	characteristic
79	C₂×C₄₃	C₂×#104	1	1	yes	characteristic
80	C₂×C₄₃	C₂×#104	1	1	yes	characteristic
81	D₄₃	—	8	2	no	—
82	#532	—	43	1	no	—
83	C₄×M₁₆	—	43	1	no	—
84	C₄×M₁₆	D₄₃	1	1	yes	characteristic, complemented
85	#329	—	43	4	no	—
86	C₄₃	#1525	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	—	43	2	no	—
88	C₂×M₁₆	—	2	1	no	—
89	C₂×M₁₆	—	86	1	no	—
90	C₄⋊Q₈	—	43	1	no	—
91	#127	—	43	1	no	—
92	#124	—	43	2	no	—
93	#121	—	86	1	no	—
94	#121	—	86	1	no	—
95	C₄×D₄	—	86	1	no	—
96	C₄×D₄	—	86	1	no	—
97	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
98	#97	—	43	2	no	—
99	#97	C₂×D₄₃	1	2	yes	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	86	1	no	—
102	#41	—	172	4	no	—
103	C₂×Q₈	—	43	2	no	—
104	C₂×D₄	—	43	2	no	—
105	C₂×D₄	—	86	2	no	—
106	C₂×D₄	—	86	1	no	—
107	C₂×D₄	—	86	1	no	—
108	C₂²×C₄	—	86	2	no	—
109	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
110	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
111	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	86	4	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	86	2	no	—
116	C₄⋊C₄	—	86	1	no	—
117	C₄⋊C₄	—	86	1	no	—
118	C₂²⋊C₄	—	86	2	no	—
119	C₂²⋊C₄	—	86	2	no	—
120	C₄²	C₄×D₄₃	1	1	yes	characteristic
121	C₄²	D₁₇₂	1	1	yes	characteristic
122	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
123	C₄²	C₂²×D₄₃	1	1	yes	characteristic
124	C₂³	—	86	2	no	—
125	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
126	Q₈	—	172	2	no	—
127	D₄	—	172	2	no	—
128	D₄	—	172	4	no	—
129	D₄	—	172	2	no	—
130	D₄	—	172	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	86	2	no	—
135	C₂×C₄	—	172	4	no	—
136	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
138	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
140	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₈	—	86	4	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	172	2	no	—
148	C₂²	—	172	2	no	—
149	C₂²	—	172	2	no	—
150	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	172	2	no	—
155	C₄	#13063241	1	2	yes	—
156	C₄	#13063251	1	2	yes	—
157	C₄	#13063253	1	2	yes	—
158	C₂	—	4	1	no	—
159	C₂	—	344	2	no	—
160	C₂	C₂×#13063241	1	1	yes	characteristic
161	C₂	C₂×#13063251	1	1	yes	characteristic
162	C₂	C₂×#13063253	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583449

C₂

yes

maximal, characteristic

C₄×#10621877

C₂

yes

maximal, characteristic, complemented

#14582638

C₂

yes

maximal, complemented

C₄×M₁₆×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂×#10621898

C₄

yes

—

C₂×#10621877

—

#13063325

C₄

yes

characteristic

#13063322

C₄

yes

characteristic

#13063313

C₂²

yes

—

#13063299

—

#13063299

—

C₄×D₁₇₂

—

C₄×D₁₇₂

—

#13063236

C₂²

yes

—

C₄×C₄₃⋊C₈

—

C₂×M₁₆×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#97×C₄₃

C₂²

yes

—

C₄×C₈×C₄₃

—

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

—

M₁₆×C₄₃

—

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

—

#10621877

—

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

#100

yes

—

C₄×C₄₃

#104

yes

—

C₄×C₄₃

#104

yes

—

Dic₄₃

—

#1525

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

#532

—

C₄×M₁₆

—

C₄×M₁₆

D₄₃

yes

characteristic, complemented

#329

—

C₄₃

#1525

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

—

—

—

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

#97

—

#97

C₂×D₄₃

yes

—

100

C₄×C₈

—

101

C₄×C₈

—

102

#41

—

172

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₄×D₄₃

yes

characteristic

110

C₂²×C₄

D₁₇₂

yes

characteristic

111

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

C₄×D₄₃

yes

characteristic

121

C₄²

D₁₇₂

yes

characteristic

122

C₄²

C₂²⋊D₄₃

yes

characteristic

123

C₄²

C₂²×D₄₃

yes

characteristic

124

C₂³

—

125

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

126

Q₈

—

172

—

127

D₄

—

172

—

128

D₄

—

172

—

129

D₄

—

172

—

130

D₄

—

172

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

172

—

136

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

137

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

138

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

139

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

140

C₂×C₄

C₂×D₁₇₂

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

142

C₈

—

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

172

—

148

C₂²

—

172

—

149

C₂²

—

172

—

150

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

172

—

155

C₄

#13063241

yes

—

156

C₄

#13063251

yes

—

157

C₄

#13063253

yes

—

158

C₂

—

159

C₂

—

344

—

160

C₂

C₂×#13063241

yes

characteristic

161

C₂

C₂×#13063251

yes

characteristic

162

C₂

C₂×#13063253

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₄₃)⋊C₄
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₄²	C₂²×D₄₃
Fitting	C₄×M₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	C₄×M₁₆	D₄₃
hypocenter	C₄₃	#1525
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

(C₂×D₄₃)⋊C₄

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₄²

C₂²×D₄₃

Fitting

C₄×M₁₆×C₄₃

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

hypercenter

C₄×M₁₆

D₄₃

hypocenter

C₄₃

#1525

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. C₄×#10621877	C₂	semidirect
3. #14582638	C₂	semidirect
4. C₄×M₁₆×C₄₃	C₂	semidirect
84. C₄×M₁₆	D₄₃	semidirect
86. C₄₃	#1525	semidirect

H ≅ G/N

type

2. C₄×#10621877

C₂

semidirect

3. #14582638

C₂

semidirect

4. C₄×M₁₆×C₄₃

C₂

semidirect

84. C₄×M₁₆

D₄₃

semidirect

86. C₄₃

#1525

semidirect

Subgroups — #19133186

Conjugacy classes of subgroups

Complements