Group #19132918 — subgroups

Subgroups — #19132918

Order 5504.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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×C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	#14582952	C₂	1	1	yes	maximal, characteristic
3	#14582624	C₂	1	4	yes	maximal
4	C₄₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₈₆	C₄	1	1	yes	characteristic
6	C₄×Dic₈₆	—	2	1	no	—
7	C₄×Dic₈₆	—	2	1	no	—
8	C₄×Dic₈₆	—	2	1	no	—
9	C₄×Dic₈₆	—	2	1	no	—
10	C₂×C₄⋊Dic₄₃	C₄	1	1	yes	characteristic
11	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄×Dic₄₃	C₄	1	2	yes	—
13	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
14	C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
15	C₄×C₄₃⋊C₈	—	2	1	no	—
16	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₄₃	—	2	1	no	—
18	C₂²⋊C₈×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	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₄₃	C₂×C₄	1	2	yes	—
24	C₂²×Dic₄₃	C₂×C₄	1	2	yes	—
25	C₂×Dic₈₆	—	2	1	no	—
26	C₂×Dic₈₆	—	2	1	no	—
27	C₂×Dic₈₆	—	2	2	no	—
28	C₂×Dic₈₆	C₈	1	4	yes	complemented
29	C₂×C₈×C₄₃	—	2	2	no	—
30	C₄²×C₄₃	D₄	1	1	yes	characteristic
31	C₄²×C₄₃	D₄	1	1	yes	characteristic
32	C₄²×C₄₃	D₄	1	1	yes	characteristic
33	C₄²×C₄₃	D₄	1	1	yes	characteristic
34	C₄⋊Dic₄₃	—	2	1	no	—
35	C₄⋊Dic₄₃	—	2	1	no	—
36	Dic₄₃⋊C₄	—	2	2	no	—
37	Dic₄₃⋊C₄	—	2	2	no	—
38	C₄×Dic₄₃	—	2	2	no	—
39	C₄×Dic₄₃	—	2	2	no	—
40	C₂×C₄₃⋊C₈	—	2	2	no	—
41	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₄₃	—	2	2	no	—
43	C₂×C₄×C₄₃	—	4	1	no	—
44	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×C₈	1	2	yes	—
49	C₂×C₄×C₄₃	M₁₆	1	2	yes	—
50	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₄₃	—	2	2	no	—
53	C₂×Dic₄₃	—	2	2	no	—
54	C₂×Dic₄₃	—	2	2	no	—
55	C₂×Dic₄₃	—	4	2	no	—
56	C₂×Dic₄₃	C₂×C₈	1	4	yes	—
57	Dic₈₆	—	2	4	no	—
58	Dic₈₆	—	2	4	no	—
59	C₈×C₄₃	—	4	2	no	—
60	C₄₃⋊C₈	—	4	2	no	—
61	C₂²×C₄₃	—	2	1	no	—
62	C₂²×C₄₃	—	2	1	no	—
63	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄₃	C₂²×C₈	1	1	yes	characteristic
65	C₂²×C₄₃	C₂×M₁₆	1	1	yes	characteristic
66	C₄×C₄₃	—	2	1	no	—
67	C₄×C₄₃	—	2	1	no	—
68	C₄×C₄₃	—	2	1	no	—
69	C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
70	C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
71	C₄×C₄₃	#101	1	2	yes	—
72	Dic₄₃	—	2	4	no	—
73	Dic₄₃	—	4	2	no	—
74	#1461	—	43	1	no	maximal, Sylow
75	C₂×C₄₃	—	2	1	no	—
76	C₂×C₄₃	—	2	1	no	—
77	C₂×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄₃	#410	1	1	yes	characteristic
79	C₂×C₄₃	C₂×#101	1	1	yes	characteristic
80	C₂×C₄×Q₈	—	43	1	no	—
81	#431	—	43	1	no	—
82	#431	D₄₃	1	1	yes	characteristic
83	#324	—	43	4	no	—
84	C₄₃	#1461	1	1	yes	characteristic, complemented, Sylow
85	C₂²×Q₈	—	43	1	no	—
86	C₄×Q₈	—	86	1	no	—
87	C₄×Q₈	—	86	1	no	—
88	C₄×Q₈	—	86	1	no	—
89	C₄×Q₈	—	86	1	no	—
90	C₂×C₄⋊C₄	—	43	2	no	—
91	C₂×C₄⋊C₄	—	43	1	no	—
92	C₂×C₄²	—	43	2	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	—	43	2	no	—
97	C₂²⋊C₈	C₂×D₄₃	1	2	yes	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	86	1	no	—
100	C₂×Q₈	—	43	4	no	—
101	C₂×Q₈	—	86	2	no	—
102	C₂×Q₈	—	86	1	no	—
103	C₂×Q₈	—	86	1	no	—
104	C₂²×C₄	—	43	2	no	—
105	C₂²×C₄	—	43	2	no	—
106	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
107	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
108	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	86	2	no	—
111	C₄⋊C₄	—	86	1	no	—
112	C₄⋊C₄	—	86	1	no	—
113	C₄⋊C₄	—	86	2	no	—
114	C₄⋊C₄	—	86	2	no	—
115	C₄²	—	86	2	no	—
116	C₄²	—	86	2	no	—
117	C₄²	D₁₇₂	1	1	yes	characteristic
118	C₄²	D₁₇₂	1	1	yes	characteristic
119	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
120	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
121	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
122	Q₈	—	86	4	no	—
123	Q₈	—	86	4	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	43	4	no	—
127	C₂×C₄	—	86	2	no	—
128	C₂×C₄	—	86	2	no	—
129	C₂×C₄	—	86	2	no	—
130	C₂×C₄	—	172	2	no	—
131	C₂×C₄	C₈×D₄₃	1	2	yes	—
132	C₂×C₄	#10621872	1	2	yes	—
133	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
134	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
135	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
139	C₈	—	4	2	no	—
140	C₈	—	172	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
144	C₂²	C₂×C₈×D₄₃	1	1	yes	characteristic
145	C₂²	C₂×#10621872	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	86	4	no	—
150	C₄	—	172	2	no	—
151	C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
152	C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
153	C₄	#13063243	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	C₂×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
157	C₂	#14582872	1	1	yes	characteristic
158	C₂	C₂×#13063243	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₈₆

C₂

yes

maximal, characteristic

#14582952

C₂

yes

maximal, characteristic

#14582624

C₂

yes

maximal

C₄₃×#431

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₄

yes

characteristic

C₄×Dic₈₆

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

C₂×C₄⋊Dic₄₃

C₄

yes

characteristic

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₃

C₄

yes

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

C₂²

yes

—

C₄×C₄₃⋊C₈

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

C₂²

yes

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂×C₄

yes

—

C₂²×Dic₄₃

C₂×C₄

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₈

yes

complemented

C₂×C₈×C₄₃

—

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²×C₄

yes

characteristic

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×C₈

yes

—

C₂×C₄×C₄₃

M₁₆

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×C₈

yes

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

C₂²×C₈

yes

characteristic

C₂²×C₄₃

C₂×M₁₆

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊C₈

yes

—

C₄×C₄₃

C₂²⋊C₈

yes

—

C₄×C₄₃

#101

yes

—

Dic₄₃

—

Dic₄₃

—

#1461

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

C₂×C₂²⋊C₈

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄×Q₈

—

#431

—

#431

D₄₃

yes

characteristic

#324

—

C₄₃

#1461

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₄₃

yes

—

C₄×C₈

—

C₄×C₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₄×D₄₃

yes

characteristic

107

C₂²×C₄

C₄×D₄₃

yes

characteristic

108

C₂²×C₄

C₂²×D₄₃

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

D₁₇₂

yes

characteristic

118

C₄²

D₁₇₂

yes

characteristic

119

C₄²

C₂²⋊D₄₃

yes

characteristic

120

C₄²

C₂²⋊D₄₃

yes

characteristic

121

C₂³

C₂×C₄×D₄₃

yes

characteristic

122

Q₈

—

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

172

—

131

C₂×C₄

C₈×D₄₃

yes

—

132

C₂×C₄

#10621872

yes

—

133

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

134

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

135

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

136

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

137

C₂×C₄

C₂×D₁₇₂

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

139

C₈

—

140

C₈

—

172

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

144

C₂²

C₂×C₈×D₄₃

yes

characteristic

145

C₂²

C₂×#10621872

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

172

—

151

C₄

(C₂×D₄₃)⋊C₈

yes

—

152

C₄

(C₂×D₄₃)⋊C₈

yes

—

153

C₄

#13063243

yes

—

154

C₂

—

155

C₂

—

156

C₂

C₂×(C₂×D₄₃)⋊C₈

yes

characteristic

157

C₂

#14582872

yes

characteristic

158

C₂

C₂×#13063243

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₄₃)⋊C₄
commutator	C₂²×C₄₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#431	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	#431	D₄₃
hypocenter	C₄₃	#1461
LCS	G ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

(C₂×D₄₃)⋊C₄

commutator

C₂²×C₄₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#431

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
28. C₂×Dic₈₆	C₈	semidirect
84. C₄₃	#1461	semidirect

H ≅ G/N

type

28. C₂×Dic₈₆

C₈

semidirect

84. C₄₃

#1461

semidirect

Subgroups — #19132918

Conjugacy classes of subgroups

Complements