Group #19135918 — subgroups

Subgroups — #19135918

Order 5504.

288 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 89 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	1	yes	maximal, characteristic
2	C₂×#13063250	C₂	1	2	yes	maximal
3	#14582805	C₂	1	2	yes	maximal
4	#14582662	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₈₆	C₂²	1	2	yes	—
7	C₂×Dic₁₇₂	—	2	4	no	—
8	C₂×Dic₁₇₂	—	2	2	no	—
9	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
10	#13063250	—	2	4	no	—
11	C₄₃⋊#95	—	2	2	no	—
12	C₄₃⋊#95	—	2	2	no	—
13	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₄₃	—	2	2	no	—
17	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₄₃	—	2	2	no	—
21	C₂²×Dic₄₃	—	2	2	no	—
22	C₂×Dic₈₆	—	2	2	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	D₄	1	4	yes	—
28	C₂×C₈×C₄₃	—	2	2	no	—
29	C₂×C₈×C₄₃	—	2	1	no	—
30	C₂×C₈×C₄₃	—	2	2	no	—
31	C₂×C₈×C₄₃	D₄	1	2	yes	—
32	C₄²×C₄₃	—	2	1	no	—
33	C₄²×C₄₃	—	2	1	no	—
34	C₄⋊Dic₄₃	—	2	2	no	—
35	C₄⋊Dic₄₃	—	2	2	no	—
36	Dic₁₇₂	—	2	8	no	—
37	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₄₃	—	2	1	no	—
39	C₂×C₄×C₄₃	—	2	1	no	—
40	C₂×C₄×C₄₃	—	2	1	no	—
41	C₂×C₄×C₄₃	—	2	2	no	—
42	C₂×C₄×C₄₃	Q₁₆	1	2	yes	complemented
43	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₄₃	#41	1	2	yes	—
47	C₂×Dic₄₃	—	2	4	no	—
48	C₂×Dic₄₃	—	4	2	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	Dic₈₆	—	2	4	no	—
54	Dic₈₆	—	2	4	no	—
55	Dic₈₆	—	4	2	no	—
56	Dic₈₆	—	4	2	no	—
57	C₈×C₄₃	—	2	2	no	—
58	C₈×C₄₃	—	4	1	no	—
59	C₂²×C₄₃	#121	1	1	yes	characteristic
60	C₂²×C₄₃	#121	1	2	yes	—
61	C₂²×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₄₃	#135	1	1	yes	characteristic
63	C₂²×C₄₃	#137	1	2	yes	—
64	C₄×C₄₃	—	2	2	no	—
65	C₄×C₄₃	—	4	1	no	—
66	C₄×C₄₃	#120	1	1	yes	characteristic
67	C₄×C₄₃	#123	1	2	yes	—
68	C₄×C₄₃	#124	1	1	yes	characteristic
69	Dic₄₃	—	4	4	no	—
70	Dic₄₃	—	8	2	no	—
71	#2056	—	43	1	no	maximal, Sylow
72	C₂×C₄₃	#394	1	1	yes	characteristic
73	C₂×C₄₃	#462	1	2	yes	—
74	C₂×C₄₃	#464	1	2	yes	—
75	C₂×C₄₃	#467	1	1	yes	characteristic
76	C₂×C₄₃	#470	1	1	yes	characteristic
77	C₂²×Q₁₆	—	43	1	no	—
78	C₂×C₄⋊C₈	D₄₃	1	1	yes	characteristic
79	C₂×#103	—	43	2	no	—
80	#391	—	43	2	no	—
81	#340	—	43	1	no	—
82	C₄₃	#2056	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	43	2	no	—
84	C₂×Q₁₆	—	86	4	no	—
85	C₂×Q₁₆	—	86	2	no	—
86	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	43	2	no	—
89	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	#103	—	86	4	no	—
92	#95	—	86	2	no	—
93	#95	—	86	2	no	—
94	C₂×Q₈	—	43	4	no	—
95	C₂×Q₈	—	86	2	no	—
96	C₂×Q₈	—	86	2	no	—
97	C₂×Q₈	—	86	2	no	—
98	C₂×Q₈	—	86	2	no	—
99	C₂×Q₈	—	86	2	no	—
100	C₂²×C₄	—	86	2	no	—
101	C₂²×C₄	—	86	2	no	—
102	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
103	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
105	Q₁₆	—	86	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₂²⋊D₄₃	1	2	yes	—
110	C₄⋊C₄	—	86	2	no	—
111	C₄⋊C₄	—	86	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
115	Q₈	—	86	4	no	—
116	Q₈	—	86	4	no	—
117	Q₈	—	172	2	no	—
118	Q₈	—	172	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	86	4	no	—
124	C₂×C₄	—	172	2	no	—
125	C₂×C₄	—	172	2	no	—
126	C₂×C₄	—	172	2	no	—
127	C₂×C₄	—	172	2	no	—
128	C₂×C₄	—	172	2	no	—
129	C₂×C₄	Dic₁₇₂	1	2	yes	complemented
130	C₂×C₄	#10621898	1	1	yes	characteristic
131	C₂×C₄	D₄×D₄₃	1	2	yes	—
132	C₂×C₄	#10621902	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	4	1	no	—
136	C₂²	#13063299	1	1	yes	characteristic
137	C₂²	#13063300	1	2	yes	—
138	C₂²	C₂×Dic₁₇₂	1	1	yes	characteristic
139	C₂²	#13063347	1	1	yes	characteristic
140	C₂²	#13063358	1	2	yes	—
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	344	2	no	—
144	C₄	—	172	4	no	—
145	C₄	#13063293	1	1	yes	characteristic
146	C₄	#13063310	1	2	yes	—
147	C₄	#13063314	1	1	yes	characteristic
148	C₂	#14582816	1	1	yes	characteristic
149	C₂	#14583110	1	2	yes	—
150	C₂	#14583122	1	2	yes	—
151	C₂	#14583140	1	1	yes	characteristic
152	C₂	#14583157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₇₂

C₂

yes

maximal, characteristic

C₂×#13063250

C₂

yes

maximal

#14582805

C₂

yes

maximal

#14582662

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

#13063250

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

D₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₄

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

Q₁₆

yes

complemented

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₂²×C₄₃

#121

yes

characteristic

C₂²×C₄₃

#121

yes

—

C₂²×C₄₃

C₂×Q₁₆

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#120

yes

characteristic

C₄×C₄₃

#123

yes

—

C₄×C₄₃

#124

yes

characteristic

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#464

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂²×Q₁₆

—

C₂×C₄⋊C₈

D₄₃

yes

—

—

—

C₄₃

#2056

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₄₃

yes

characteristic

C₂²×C₈

C₂×D₄₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

#103

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

D₁₇₂

yes

characteristic

103

C₂²×C₄

D₁₇₂

yes

characteristic

104

C₂²×C₄

C₂²×D₄₃

yes

characteristic

105

Q₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂²⋊D₄₃

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₂³

C₂×D₁₇₂

yes

characteristic

115

Q₈

—

116

Q₈

—

117

Q₈

—

172

—

118

Q₈

—

172

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

172

—

125

C₂×C₄

—

172

—

126

C₂×C₄

—

172

—

127

C₂×C₄

—

172

—

128

C₂×C₄

—

172

—

129

C₂×C₄

Dic₁₇₂

yes

complemented

130

C₂×C₄

#10621898

yes

characteristic

131

C₂×C₄

D₄×D₄₃

yes

—

132

C₂×C₄

#10621902

yes

—

133

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

134

C₈

—

135

C₈

—

136

C₂²

#13063299

yes

characteristic

137

C₂²

#13063300

yes

—

138

C₂²

C₂×Dic₁₇₂

yes

characteristic

139

C₂²

#13063347

yes

characteristic

140

C₂²

#13063358

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

344

—

144

C₄

—

172

—

145

C₄

#13063293

yes

characteristic

146

C₄

#13063310

yes

—

147

C₄

#13063314

yes

characteristic

148

C₂

#14582816

yes

characteristic

149

C₂

#14583110

yes

—

150

C₂

#14583122

yes

—

151

C₂

#14583140

yes

characteristic

152

C₂

#14583157

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₇₂
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×C₄⋊C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#2056
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₇₂

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×C₄⋊C₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₄₃

hypocenter

C₄₃

#2056

LCS

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
42. C₂×C₄×C₄₃	Q₁₆	semidirect
82. C₄₃	#2056	semidirect
129. C₂×C₄	Dic₁₇₂	semidirect

H ≅ G/N

type

42. C₂×C₄×C₄₃

Q₁₆

semidirect

82. C₄₃

#2056

semidirect

129. C₂×C₄

Dic₁₇₂

semidirect

Subgroups — #19135918

Conjugacy classes of subgroups

Complements