Group #19134682 — subgroups

Subgroups — #19134682

Order 5504.

344 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
2	#14582759	C₂	1	1	yes	maximal, characteristic
3	#14582662	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₄₃	—	2	1	no	—
6	C₂×C₄⋊Dic₄₃	—	2	1	no	—
7	C₂×C₄⋊Dic₄₃	—	2	2	no	—
8	C₂×C₄⋊Dic₄₃	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
12	C₄₃⋊#95	—	2	1	no	—
13	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₄₃	—	2	2	no	—
16	C₂⁴×C₄₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₃	—	2	4	no	—
18	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₄₃	D₄	1	2	yes	—
20	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₄₃	—	2	4	no	—
23	C₂²×Dic₄₃	—	2	2	no	—
24	C₂²×Dic₄₃	—	2	2	no	—
25	C₂²×Dic₄₃	—	4	2	no	—
26	C₂²×Dic₄₃	—	4	2	no	—
27	C₂×C₈×C₄₃	—	2	2	no	—
28	C₂×C₈×C₄₃	—	2	4	no	—
29	C₂²⋊Dic₄₃	—	4	2	no	—
30	C₄⋊Dic₄₃	—	2	4	no	—
31	C₄⋊Dic₄₃	—	2	2	no	—
32	C₄⋊Dic₄₃	—	2	4	no	—
33	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
34	C₂³×C₄₃	—	2	2	no	—
35	C₂³×C₄₃	—	2	2	no	—
36	C₂³×C₄₃	C₂²⋊C₄	1	2	yes	—
37	C₂³×C₄₃	D₈	1	2	yes	—
38	C₂³×C₄₃	SD₁₆	1	2	yes	—
39	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₄₃	—	2	2	no	—
41	C₂×C₄×C₄₃	—	2	4	no	—
42	C₂×C₄×C₄₃	—	2	4	no	—
43	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₃	#41	1	4	yes	—
47	C₂×Dic₄₃	—	2	4	no	—
48	C₂×Dic₄₃	—	4	8	no	—
49	C₂×Dic₄₃	—	4	2	no	—
50	C₂×Dic₄₃	—	4	2	no	—
51	C₂×Dic₄₃	—	4	2	no	—
52	C₂×Dic₄₃	—	4	2	no	—
53	C₂×Dic₄₃	—	4	2	no	—
54	C₈×C₄₃	—	4	2	no	—
55	C₂²×C₄₃	—	2	2	no	—
56	C₂²×C₄₃	—	2	2	no	—
57	C₂²×C₄₃	—	2	4	no	—
58	C₂²×C₄₃	—	2	4	no	—
59	C₂²×C₄₃	#102	1	4	yes	—
60	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄₃	#123	1	2	yes	—
62	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₄₃	#137	1	2	yes	—
65	C₄×C₄₃	—	2	2	no	—
66	C₄×C₄₃	#117	1	2	yes	—
67	C₄×C₄₃	#123	1	2	yes	—
68	Dic₄₃	—	4	4	no	—
69	Dic₄₃	—	8	2	no	—
70	#1823	—	43	1	no	maximal, Sylow
71	C₂×C₄₃	—	2	4	no	—
72	C₂×C₄₃	#381	1	1	yes	characteristic
73	C₂×C₄₃	C₂×#102	1	1	yes	characteristic
74	C₂×C₄₃	#419	1	1	yes	characteristic
75	C₂×C₄₃	#480	1	2	yes	—
76	C₂×C₄₃	#483	1	2	yes	—
77	C₂²×C₄⋊C₄	—	43	1	no	—
78	C₂×C₂²⋊C₈	D₄₃	1	1	yes	characteristic
79	#380	—	43	1	no	—
80	#340	—	43	4	no	—
81	C₄₃	#1823	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₄	—	86	1	no	—
83	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
84	C₂²×C₈	C₂×D₄₃	1	2	yes	—
85	C₂×C₄⋊C₄	—	43	2	no	—
86	C₂×C₄⋊C₄	—	43	4	no	—
87	C₂×C₄⋊C₄	—	43	2	no	—
88	C₂×C₄⋊C₄	—	86	1	no	—
89	C₂×C₄⋊C₄	—	86	2	no	—
90	C₂×C₂²⋊C₄	—	86	1	no	—
91	C₂²⋊C₈	—	2	2	no	—
92	#95	—	86	1	no	—
93	C₂⁴	D₁₇₂	1	1	yes	characteristic
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	—	86	4	no	—
96	C₂²×C₄	—	86	2	no	—
97	C₂²×C₄	—	86	2	no	—
98	C₂²×C₄	—	172	2	no	—
99	C₂²×C₄	—	172	2	no	—
100	C₂²×C₄	C₄×D₄₃	1	2	yes	—
101	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊D₄₃	1	2	yes	—
103	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	2	2	no	—
106	C₄⋊C₄	—	43	4	no	—
107	C₄⋊C₄	—	86	4	no	—
108	C₄⋊C₄	—	86	2	no	—
109	C₄⋊C₄	—	86	4	no	—
110	C₂²⋊C₄	—	172	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	#10621873	1	2	yes	—
114	C₂³	D₃₄₄	1	2	yes	—
115	C₂³	(C₂×D₄₃)⋊C₄	1	2	yes	—
116	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	86	4	no	—
121	C₂×C₄	—	172	2	no	—
122	C₂×C₄	—	172	2	no	—
123	C₂×C₄	—	172	2	no	—
124	C₂×C₄	—	172	8	no	—
125	C₂×C₄	—	172	2	no	—
126	C₂×C₄	—	172	2	no	—
127	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
128	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
129	C₂×C₄	#10621900	1	4	yes	—
130	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
131	C₈	—	4	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	2	4	no	—
134	C₂²	—	2	4	no	—
135	C₂²	—	2	2	no	—
136	C₂²	#13063247	1	4	yes	—
137	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
138	C₂²	#13063309	1	2	yes	—
139	C₂²	C₂×D₃₄₄	1	1	yes	characteristic
140	C₂²	C₂×#10621873	1	1	yes	characteristic
141	C₂²	#13063358	1	2	yes	—
142	C₄	—	2	2	no	—
143	C₄	—	344	2	no	—
144	C₄	—	172	4	no	—
145	C₄	#13063281	1	2	yes	—
146	C₄	#13063306	1	2	yes	—
147	C₂	—	2	4	no	—
148	C₂	#14582760	1	1	yes	characteristic
149	C₂	C₂×#13063247	1	1	yes	characteristic
150	C₂	#14582909	1	1	yes	characteristic
151	C₂	#14583213	1	2	yes	—
152	C₂	#14583230	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₄₃

C₂

yes

maximal, characteristic

#14582759

C₂

yes

maximal, characteristic

#14582662

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₄₃

C₂

yes

maximal, characteristic

C₂³×Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

—

C₄₃⋊#95

—

C₂³×C₄×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

C₂²

yes

—

C₂²⋊C₈×C₄₃

—

C₂⁴×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂²⋊C₄

yes

—

C₂³×C₄₃

D₈

yes

—

C₂³×C₄₃

SD₁₆

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#102

yes

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

#123

yes

—

C₂²×C₄₃

C₂×D₈

yes

characteristic

C₂²×C₄₃

C₂×SD₁₆

yes

characteristic

C₂²×C₄₃

#137

yes

—

C₄×C₄₃

—

C₄×C₄₃

#117

yes

—

C₄×C₄₃

#123

yes

—

Dic₄₃

—

Dic₄₃

—

#1823

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#483

yes

—

C₂²×C₄⋊C₄

—

C₂×C₂²⋊C₈

D₄₃

yes

characteristic

#380

—

#340

—

C₄₃

#1823

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

C₂×D₄₃

yes

characteristic

C₂²×C₈

C₂×D₄₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

#95

—

C₂⁴

D₁₇₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

172

—

C₂²×C₄

—

172

—

100

C₂²×C₄

C₄×D₄₃

yes

—

101

C₂²×C₄

D₁₇₂

yes

characteristic

102

C₂²×C₄

C₂²⋊D₄₃

yes

—

103

C₂²×C₄

C₂²×D₄₃

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

172

—

111

C₂³

—

112

C₂³

—

113

C₂³

#10621873

yes

—

114

C₂³

D₃₄₄

yes

—

115

C₂³

(C₂×D₄₃)⋊C₄

yes

—

116

C₂³

C₂×D₁₇₂

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

172

—

122

C₂×C₄

—

172

—

123

C₂×C₄

—

172

—

124

C₂×C₄

—

172

—

125

C₂×C₄

—

172

—

126

C₂×C₄

—

172

—

127

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

128

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

129

C₂×C₄

#10621900

yes

—

130

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#13063247

yes

—

137

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

138

C₂²

#13063309

yes

—

139

C₂²

C₂×D₃₄₄

yes

characteristic

140

C₂²

C₂×#10621873

yes

characteristic

141

C₂²

#13063358

yes

—

142

C₄

—

143

C₄

—

344

—

144

C₄

—

172

—

145

C₄

#13063281

yes

—

146

C₄

#13063306

yes

—

147

C₂

—

148

C₂

#14582760

yes

characteristic

149

C₂

C₂×#13063247

yes

characteristic

150

C₂

#14582909

yes

characteristic

151

C₂

#14583213

yes

—

152

C₂

#14583230

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×C₂²⋊C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×C₂²⋊C₈	D₄₃
hypocenter	C₄₃	#1823
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×C₂²⋊C₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₂×C₂²⋊C₈

D₄₃

hypocenter

C₄₃

#1823

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. #14582662	C₂	semidirect
8. C₂×C₄⋊Dic₄₃	C₄	semidirect
81. C₄₃	#1823	semidirect

H ≅ G/N

type

3. #14582662

C₂

semidirect

8. C₂×C₄⋊Dic₄₃

C₄

semidirect

81. C₄₃

#1823

semidirect

Subgroups — #19134682

Conjugacy classes of subgroups

Complements