Group #19133160 — subgroups

Subgroups — #19133160

Order 5504.

220 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 85 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	#14583456	C₂	1	1	yes	maximal, characteristic
2	#14582765	C₂	1	1	yes	maximal, characteristic
3	#14582637	C₂	1	4	yes	maximal
4	C₄₃×#431	C₂	1	1	yes	maximal, characteristic
5	#13063325	C₄	1	1	yes	characteristic, complemented
6	#13063325	C₄	1	1	yes	characteristic, complemented
7	#13063317	—	2	1	no	—
8	#13063304	—	2	1	no	—
9	#13063304	C₂²	1	2	yes	—
10	C₄×Dic₈₆	—	2	1	no	—
11	C₄×Dic₈₆	C₄	1	2	yes	complemented
12	#13063284	—	2	1	no	—
13	C₂×C₄⋊Dic₄₃	—	2	1	no	—
14	C₄₃⋊#95	—	2	1	no	—
15	C₄₃⋊#95	C₂²	1	2	yes	—
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	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₄₃	—	2	2	no	—
24	C₂×Dic₈₆	—	2	1	no	—
25	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₄₃	—	2	2	no	—
27	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₄₃	D₄	1	1	yes	characteristic
30	C₄²×C₄₃	D₄	1	1	yes	characteristic
31	C₂²⋊Dic₄₃	—	2	2	no	—
32	C₄⋊Dic₄₃	—	2	1	no	—
33	C₄⋊Dic₄₃	—	2	1	no	—
34	C₄⋊Dic₄₃	—	2	1	no	—
35	C₄⋊Dic₄₃	—	4	2	no	—
36	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
37	Dic₄₃⋊C₄	—	2	2	no	—
38	Dic₄₃⋊C₄	—	2	2	no	—
39	C₄×Dic₄₃	—	2	2	no	—
40	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₄₃	—	4	1	no	—
42	C₂×C₄×C₄₃	—	4	1	no	—
43	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
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	SD₁₆	1	2	yes	—
48	C₂×C₄×C₄₃	Q₁₆	1	2	yes	—
49	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₄₃	—	2	2	no	—
52	C₂×Dic₄₃	—	2	2	no	—
53	C₂×Dic₄₃	—	4	2	no	—
54	C₂×Dic₄₃	—	4	2	no	—
55	C₂×Dic₄₃	—	4	2	no	—
56	Dic₈₆	—	4	2	no	—
57	Dic₈₆	—	4	2	no	—
58	C₈×C₄₃	—	4	2	no	—
59	C₂²×C₄₃	—	2	1	no	—
60	C₂²×C₄₃	—	2	1	no	—
61	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂²×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₄₃	—	2	1	no	—
65	C₄×C₄₃	—	2	1	no	—
66	C₄×C₄₃	—	2	1	no	—
67	C₄×C₄₃	—	2	1	no	—
68	C₄×C₄₃	#103	1	2	yes	—
69	C₄×C₄₃	#103	1	2	yes	—
70	Dic₄₃	—	4	2	no	—
71	Dic₄₃	—	4	2	no	—
72	Dic₄₃	—	8	2	no	—
73	#1518	—	43	1	no	maximal, Sylow
74	C₂×C₄₃	—	2	1	no	—
75	C₂×C₄₃	—	2	1	no	—
76	C₂×C₄₃	#410	1	1	yes	characteristic
77	C₂×C₄₃	C₂×#103	1	1	yes	characteristic
78	C₂×C₄₃	#421	1	1	yes	characteristic
79	#533	—	43	1	no	—
80	#431	D₄₃	1	1	yes	characteristic
81	#382	—	43	1	no	—
82	#328	—	43	4	no	—
83	C₄₃	#1518	1	1	yes	characteristic, complemented, Sylow
84	C₄⋊Q₈	—	43	1	no	—
85	C₄⋊Q₈	—	43	1	no	—
86	#125	—	86	1	no	—
87	C₂²⋊Q₈	—	43	2	no	—
88	C₂²⋊Q₈	—	86	1	no	—
89	C₄×Q₈	—	43	2	no	—
90	C₄×Q₈	—	86	1	no	—
91	#117	—	86	1	no	—
92	C₂×C₄⋊C₄	—	86	1	no	—
93	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
94	C₄⋊C₈	—	2	1	no	—
95	C₂²⋊C₈	C₂×D₄₃	1	2	yes	—
96	C₄×C₈	—	2	1	no	—
97	#95	—	43	2	no	—
98	#95	—	86	1	no	—
99	C₂×Q₈	—	43	2	no	—
100	C₂×Q₈	—	86	1	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	C₂×C₈	—	2	2	no	—
106	C₄⋊C₄	—	43	2	no	—
107	C₄⋊C₄	—	86	1	no	—
108	C₄⋊C₄	—	86	1	no	—
109	C₄⋊C₄	—	86	2	no	—
110	C₄⋊C₄	—	86	1	no	—
111	C₄⋊C₄	—	86	2	no	—
112	C₄⋊C₄	—	172	2	no	—
113	C₂²⋊C₄	—	86	2	no	—
114	C₄²	—	86	2	no	—
115	C₄²	C₄×D₄₃	1	1	yes	characteristic
116	C₄²	C₄×D₄₃	1	1	yes	characteristic
117	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
118	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
119	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
120	Q₈	—	172	2	no	—
121	Q₈	—	172	2	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	86	2	no	—
125	C₂×C₄	—	86	2	no	—
126	C₂×C₄	—	172	2	no	—
127	C₂×C₄	—	172	2	no	—
128	C₂×C₄	—	172	2	no	—
129	C₂×C₄	#10621873	1	2	yes	—
130	C₂×C₄	Dic₁₇₂	1	2	yes	—
131	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
132	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
133	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
134	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
137	C₈	—	4	2	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
141	C₂²	C₂×#10621873	1	1	yes	characteristic
142	C₂²	C₂×Dic₁₇₂	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	344	2	no	—
148	C₄	—	172	2	no	—
149	C₄	—	172	2	no	—
150	C₄	#13063250	1	2	yes	—
151	C₄	#13063250	1	2	yes	—
152	C₂	—	2	1	no	—
153	C₂	—	2	1	no	—
154	C₂	#14582872	1	1	yes	characteristic
155	C₂	C₂×#13063250	1	1	yes	characteristic
156	C₂	#14582915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583456

C₂

yes

maximal, characteristic

#14582765

C₂

yes

maximal, characteristic

#14582637

C₂

yes

maximal

C₄₃×#431

C₂

yes

maximal, characteristic

#13063325

C₄

yes

characteristic, complemented

#13063325

C₄

yes

characteristic, complemented

#13063317

—

#13063304

—

#13063304

C₂²

yes

—

C₄×Dic₈₆

—

C₄×Dic₈₆

C₄

yes

complemented

#13063284

—

C₂×C₄⋊Dic₄₃

—

C₄₃⋊#95

—

C₄₃⋊#95

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

C₂²

yes

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂³×C₄₃

C₂×D₄

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

SD₁₆

yes

—

C₂×C₄×C₄₃

Q₁₆

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

C₂×SD₁₆

yes

characteristic

C₂²×C₄₃

C₂×Q₁₆

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#103

yes

—

C₄×C₄₃

#103

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#1518

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

D₄₃

yes

characteristic

#382

—

#328

—

C₄₃

#1518

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

C₂×D₄₃

yes

—

C₄×C₈

—

#95

—

#95

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

D₁₇₂

yes

characteristic

103

C₂²×C₄

D₁₇₂

yes

characteristic

104

C₂²×C₄

C₂²×D₄₃

yes

characteristic

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

172

—

113

C₂²⋊C₄

—

114

C₄²

—

115

C₄²

C₄×D₄₃

yes

characteristic

116

C₄²

C₄×D₄₃

yes

characteristic

117

C₄²

C₂²⋊D₄₃

yes

characteristic

118

C₄²

C₂²⋊D₄₃

yes

characteristic

119

C₂³

C₂×D₁₇₂

yes

characteristic

120

Q₈

—

172

—

121

Q₈

—

172

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

172

—

127

C₂×C₄

—

172

—

128

C₂×C₄

—

172

—

129

C₂×C₄

#10621873

yes

—

130

C₂×C₄

Dic₁₇₂

yes

—

131

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

132

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

133

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

134

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

135

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

141

C₂²

C₂×#10621873

yes

characteristic

142

C₂²

C₂×Dic₁₇₂

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

344

—

148

C₄

—

172

—

149

C₄

—

172

—

150

C₄

#13063250

yes

—

151

C₄

#13063250

yes

—

152

C₂

—

153

C₂

—

154

C₂

#14582872

yes

characteristic

155

C₂

C₂×#13063250

yes

characteristic

156

C₂

#14582915

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₄₃)⋊C₄
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄₃×#431	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	#431	D₄₃
hypocenter	C₄₃	#1518
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×(C₂×D₄₃)⋊C₄

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄₃×#431

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
5. #13063325	C₄	semidirect
6. #13063325	C₄	semidirect
11. C₄×Dic₈₆	C₄	semidirect
83. C₄₃	#1518	semidirect

H ≅ G/N

type

5. #13063325

C₄

semidirect

6. #13063325

C₄

semidirect

11. C₄×Dic₈₆

C₄

semidirect

83. C₄₃

#1518

semidirect

Subgroups — #19133160

Conjugacy classes of subgroups

Complements