Group #19135697 — subgroups

Subgroups — #19135697

Order 5504.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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×Dic₁₇₂	C₂	1	2	yes	maximal
2	C₂×#13063325	C₂	1	1	yes	maximal, characteristic
3	C₂×#13063250	C₂	1	2	yes	maximal
4	#14582805	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
7	C₂²×Dic₈₆	C₂²	1	2	yes	—
8	C₂×Dic₁₇₂	—	2	4	no	—
9	C₂×Dic₁₇₂	—	2	4	no	—
10	C₂×Dic₁₇₂	—	2	4	no	—
11	#13063325	—	2	4	no	—
12	C₂×C₄⋊Dic₄₃	—	2	2	no	—
13	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
14	#13063250	—	2	4	no	—
15	C₄₃⋊#95	—	2	2	no	—
16	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₄₃	D₄	1	2	yes	—
19	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₄₃	—	2	1	no	—
21	C₂²×Dic₄₃	—	2	2	no	—
22	C₂²×Dic₄₃	—	2	1	no	—
23	C₂×Dic₈₆	—	2	2	no	—
24	C₂×Dic₈₆	—	2	2	no	—
25	C₂×Dic₈₆	—	2	2	no	—
26	C₂×Dic₈₆	—	2	2	no	—
27	C₂×Dic₈₆	—	2	2	no	—
28	C₂×Dic₈₆	—	2	2	no	—
29	C₂×Dic₈₆	—	2	2	no	—
30	C₂×Dic₈₆	—	2	1	no	—
31	C₂×Dic₈₆	D₄	1	4	yes	—
32	C₂×Dic₈₆	D₄	1	2	yes	—
33	C₂×C₈×C₄₃	—	2	2	no	—
34	C₂×C₈×C₄₃	—	2	2	no	—
35	C₂×C₈×C₄₃	D₄	1	4	yes	—
36	C₄²×C₄₃	—	2	2	no	—
37	C₄⋊Dic₄₃	—	2	8	no	—
38	C₄⋊Dic₄₃	—	2	2	no	—
39	Dic₁₇₂	—	2	16	no	—
40	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₄₃	—	2	2	no	—
42	C₂×C₄×C₄₃	—	2	2	no	—
43	C₂×C₄×C₄₃	Q₁₆	1	4	yes	complemented
44	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
48	C₂×Dic₄₃	—	2	4	no	—
49	C₂×Dic₄₃	—	2	2	no	—
50	C₂×Dic₄₃	—	4	1	no	—
51	C₂×Dic₄₃	—	4	2	no	—
52	C₂×Dic₄₃	—	4	2	no	—
53	C₂×Dic₄₃	—	4	1	no	—
54	C₂×Dic₄₃	—	4	1	no	—
55	C₂×Dic₄₃	—	4	2	no	—
56	Dic₈₆	—	2	4	no	—
57	Dic₈₆	—	2	4	no	—
58	Dic₈₆	—	2	4	no	—
59	Dic₈₆	—	4	2	no	—
60	Dic₈₆	—	4	2	no	—
61	Dic₈₆	—	4	2	no	—
62	C₈×C₄₃	—	2	4	no	—
63	C₂²×C₄₃	#120	1	1	yes	characteristic
64	C₂²×C₄₃	#121	1	1	yes	characteristic
65	C₂²×C₄₃	#127	1	1	yes	characteristic
66	C₂²×C₄₃	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₄₃	#137	1	2	yes	—
68	C₄×C₄₃	—	2	4	no	—
69	C₄×C₄₃	#120	1	2	yes	—
70	C₄×C₄₃	#121	1	2	yes	—
71	Dic₄₃	—	4	2	no	—
72	Dic₄₃	—	4	4	no	—
73	Dic₄₃	—	8	1	no	—
74	#2016	—	43	1	no	maximal, Sylow
75	C₂×C₄₃	#392	1	1	yes	characteristic
76	C₂×C₄₃	#451	1	2	yes	—
77	C₂×C₄₃	#462	1	2	yes	—
78	C₂×C₄₃	#494	1	1	yes	characteristic
79	C₂×C₄₃	#497	1	1	yes	characteristic
80	C₂²×Q₁₆	—	43	2	no	—
81	C₂×C₄⋊Q₈	—	43	1	no	—
82	C₂×#103	—	43	2	no	—
83	#391	—	43	1	no	—
84	#336	D₄₃	1	1	yes	characteristic
85	C₄₃	#2016	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	43	2	no	—
87	C₂²×Q₈	—	43	1	no	—
88	C₂×Q₁₆	—	86	4	no	—
89	C₂×Q₁₆	—	86	4	no	—
90	C₂×Q₁₆	—	86	4	no	—
91	C₂²×C₈	C₂×D₄₃	1	2	yes	—
92	C₄⋊Q₈	—	86	4	no	—
93	C₂×C₄⋊C₄	—	43	1	no	—
94	C₂×C₄⋊C₄	—	86	2	no	—
95	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
96	#103	—	86	4	no	—
97	#95	—	86	2	no	—
98	C₂×Q₈	—	43	4	no	—
99	C₂×Q₈	—	43	2	no	—
100	C₂×Q₈	—	86	2	no	—
101	C₂×Q₈	—	86	1	no	—
102	C₂×Q₈	—	86	2	no	—
103	C₂×Q₈	—	86	2	no	—
104	C₂×Q₈	—	86	2	no	—
105	C₂×Q₈	—	86	2	no	—
106	C₂×Q₈	—	86	2	no	—
107	C₂×Q₈	—	86	2	no	—
108	C₂²×C₄	—	86	1	no	—
109	C₂²×C₄	—	86	2	no	—
110	C₂²×C₄	—	86	1	no	—
111	C₂²×C₄	D₁₇₂	1	2	yes	—
112	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
113	Q₁₆	—	86	16	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	D₁₇₂	1	4	yes	—
117	C₄⋊C₄	—	86	8	no	—
118	C₄⋊C₄	—	86	2	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
121	Q₈	—	86	4	no	—
122	Q₈	—	86	4	no	—
123	Q₈	—	86	4	no	—
124	Q₈	—	172	2	no	—
125	Q₈	—	172	2	no	—
126	Q₈	—	172	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	86	4	no	—
130	C₂×C₄	—	86	2	no	—
131	C₂×C₄	—	172	1	no	—
132	C₂×C₄	—	172	2	no	—
133	C₂×C₄	—	172	1	no	—
134	C₂×C₄	—	172	2	no	—
135	C₂×C₄	—	172	1	no	—
136	C₂×C₄	—	172	2	no	—
137	C₂×C₄	Dic₁₇₂	1	4	yes	complemented
138	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
139	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
140	C₂×C₄	D₄×D₄₃	1	2	yes	—
141	C₂×C₄	#10621902	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₂²	#13063295	1	1	yes	characteristic
144	C₂²	#13063300	1	1	yes	characteristic
145	C₂²	#13063322	1	1	yes	characteristic
146	C₂²	C₂×Dic₁₇₂	1	2	yes	—
147	C₂²	#13063358	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	344	1	no	—
150	C₄	—	172	4	no	—
151	C₄	—	172	2	no	—
152	C₄	#13063295	1	2	yes	—
153	C₄	#13063300	1	2	yes	—
154	C₂	#14582809	1	1	yes	characteristic
155	C₂	#14583057	1	2	yes	—
156	C₂	#14583110	1	2	yes	—
157	C₂	#14583279	1	1	yes	characteristic
158	C₂	#14583293	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₇₂

C₂

yes

maximal

C₂×#13063325

C₂

yes

maximal, characteristic

C₂×#13063250

C₂

yes

maximal

#14582805

C₂

yes

maximal, characteristic

C₄₃×#336

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

C₂²×Dic₈₆

C₂²

yes

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

#13063325

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

#13063250

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

D₄

yes

—

C₂×Dic₈₆

D₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₄

yes

—

C₄²×C₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

Q₁₆

yes

complemented

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

#41

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

C₂×Q₁₆

yes

—

C₂²×C₄₃

#137

yes

—

C₄×C₄₃

—

C₄×C₄₃

#120

yes

—

C₄×C₄₃

#121

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#462

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×#103

—

#391

—

#336

D₄₃

yes

characteristic

C₄₃

#2016

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₄₃

yes

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

#103

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

D₁₇₂

yes

—

112

C₂²×C₄

C₂²×D₄₃

yes

characteristic

113

Q₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

D₁₇₂

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

—

120

C₂³

C₂×D₁₇₂

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

172

—

125

Q₈

—

172

—

126

Q₈

—

172

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

172

—

132

C₂×C₄

—

172

—

133

C₂×C₄

—

172

—

134

C₂×C₄

—

172

—

135

C₂×C₄

—

172

—

136

C₂×C₄

—

172

—

137

C₂×C₄

Dic₁₇₂

yes

complemented

138

C₂×C₄

C₂×D₁₇₂

yes

—

139

C₂×C₄

D₄×D₄₃

yes

characteristic

140

C₂×C₄

D₄×D₄₃

yes

—

141

C₂×C₄

#10621902

yes

characteristic

142

C₈

—

143

C₂²

#13063295

yes

characteristic

144

C₂²

#13063300

yes

characteristic

145

C₂²

#13063322

yes

characteristic

146

C₂²

C₂×Dic₁₇₂

yes

—

147

C₂²

#13063358

yes

—

148

C₄

—

149

C₄

—

344

—

150

C₄

—

172

—

151

C₄

—

172

—

152

C₄

#13063295

yes

—

153

C₄

#13063300

yes

—

154

C₂

#14582809

yes

characteristic

155

C₂

#14583057

yes

—

156

C₂

#14583110

yes

—

157

C₂

#14583279

yes

characteristic

158

C₂

#14583293

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₇₂
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#336	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	#336	D₄₃
hypocenter	C₄₃	#2016
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₁₇₂

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#336

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
43. C₂×C₄×C₄₃	Q₁₆	semidirect
85. C₄₃	#2016	semidirect
137. C₂×C₄	Dic₁₇₂	semidirect

H ≅ G/N

type

43. C₂×C₄×C₄₃

Q₁₆

semidirect

85. C₄₃

#2016

semidirect

137. C₂×C₄

Dic₁₇₂

semidirect

Subgroups — #19135697

Conjugacy classes of subgroups

Complements