Group #19134428 — subgroups

Subgroups — #19134428

Order 5504.

324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂×#13063248	C₂	1	2	yes	maximal
3	C₂×#13063236	C₂	1	1	yes	maximal, characteristic
4	#14582661	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
8	C₄×Dic₈₆	—	2	2	no	—
9	C₄×Dic₈₆	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₃⋊C₄	—	2	1	no	—
12	C₂×C₄×Dic₄₃	—	2	1	no	—
13	#13063248	C₄	1	8	yes	complemented
14	#13063236	—	2	2	no	—
15	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₄₃	—	2	1	no	—
17	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	#95×C₄₃	—	2	1	no	—
19	C₂²×C₄×C₄₃	—	2	2	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	1	no	—
24	C₂²×Dic₄₃	—	2	1	no	—
25	C₂×Dic₈₆	—	2	1	no	—
26	C₂×Dic₈₆	—	2	2	no	—
27	C₂×Dic₈₆	C₂×C₄	1	4	yes	—
28	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
29	C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
31	C₄²×C₄₃	—	4	2	no	—
32	C₄²×C₄₃	C₂×C₄	1	2	yes	—
33	C₄²×C₄₃	D₄	1	2	yes	—
34	C₄⋊Dic₄₃	—	2	1	no	—
35	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
36	Dic₄₃⋊C₄	—	2	4	no	—
37	C₄×Dic₄₃	—	2	4	no	—
38	C₂×C₄₃⋊C₈	—	2	2	no	—
39	C₂×C₄₃⋊C₈	—	2	2	no	—
40	C₂×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
41	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₄₃	—	2	1	no	—
43	C₂×C₄×C₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	—	2	1	no	—
45	C₂×C₄×C₄₃	—	2	4	no	—
46	C₂×C₄×C₄₃	—	4	2	no	—
47	C₂×C₄×C₄₃	—	4	2	no	—
48	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₃	#41	1	2	yes	—
54	C₂×Dic₄₃	—	2	2	no	—
55	C₂×Dic₄₃	—	2	4	no	—
56	C₂×Dic₄₃	—	2	2	no	—
57	C₂×Dic₄₃	—	4	1	no	—
58	C₂×Dic₄₃	—	4	1	no	—
59	Dic₈₆	—	2	2	no	—
60	Dic₈₆	—	2	4	no	—
61	Dic₈₆	C₄²	1	4	yes	complemented
62	C₄₃⋊C₈	—	2	4	no	—
63	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄₃	C₄×D₄	1	2	yes	—
65	C₂²×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₄₃	#137	1	2	yes	—
67	C₂²×C₄₃	#137	1	1	yes	characteristic
68	C₄×C₄₃	—	2	2	no	—
69	C₄×C₄₃	—	2	2	no	—
70	C₄×C₄₃	—	4	4	no	—
71	C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
72	C₄×C₄₃	#117	1	1	yes	characteristic
73	C₄×C₄₃	C₄×D₄	1	2	yes	—
74	Dic₄₃	—	2	4	no	—
75	Dic₄₃	—	4	2	no	—
76	#1763	—	43	1	no	maximal, Sylow
77	C₂×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄₃	#417	1	1	yes	characteristic
79	C₂×C₄₃	#419	1	1	yes	characteristic
80	C₂×C₄₃	#440	1	2	yes	—
81	C₂×C₄₃	#441	1	2	yes	—
82	C₂×C₄×Q₈	—	43	1	no	—
83	C₂×#103	—	43	2	no	—
84	C₂×#97	—	43	1	no	—
85	C₄×C₄⋊C₄	D₄₃	1	1	yes	characteristic
86	#340	—	43	2	no	—
87	C₄₃	#1763	1	1	yes	characteristic, complemented, Sylow
88	C₂²×Q₈	—	43	1	no	—
89	C₂²×C₈	—	43	2	no	—
90	C₄×Q₈	—	43	4	no	—
91	C₄×Q₈	—	86	2	no	—
92	C₂×C₄⋊C₄	—	43	1	no	—
93	C₂×C₄⋊C₄	—	86	1	no	—
94	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
95	C₂×C₄²	—	2	1	no	—
96	C₂×C₄²	—	86	1	no	—
97	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
98	#103	—	43	8	no	—
99	#97	—	86	2	no	—
100	#95	—	2	1	no	—
101	C₂×Q₈	—	43	4	no	—
102	C₂×Q₈	—	43	2	no	—
103	C₂×Q₈	—	86	1	no	—
104	C₂×Q₈	—	86	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	86	1	no	—
107	C₂²×C₄	—	86	1	no	—
108	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
111	C₂×C₈	—	43	4	no	—
112	C₂×C₈	—	86	2	no	—
113	C₂×C₈	—	86	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	43	2	no	—
116	C₄⋊C₄	—	86	4	no	—
117	C₄⋊C₄	—	86	1	no	—
118	C₄⋊C₄	C₄×D₄₃	1	4	yes	—
119	C₄²	—	4	2	no	—
120	C₄²	—	86	4	no	—
121	C₄²	C₄×D₄₃	1	2	yes	—
122	C₄²	D₁₇₂	1	2	yes	—
123	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
124	Q₈	—	43	4	no	—
125	Q₈	—	86	2	no	—
126	Q₈	—	86	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	86	2	no	—
134	C₂×C₄	—	86	4	no	—
135	C₂×C₄	—	86	2	no	—
136	C₂×C₄	—	172	1	no	—
137	C₂×C₄	—	172	1	no	—
138	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
139	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
142	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
143	C₂×C₄	#10621898	1	2	yes	—
144	C₈	—	86	4	no	—
145	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
146	C₂²	C₄×C₂²⋊D₄₃	1	2	yes	—
147	C₂²	#13063351	1	1	yes	characteristic
148	C₂²	#13063355	1	2	yes	—
149	C₂²	#13063356	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	—	86	4	no	—
154	C₄	—	172	2	no	—
155	C₄	C₄²×D₄₃	1	1	yes	characteristic
156	C₄	#13063282	1	1	yes	characteristic
157	C₄	C₄×D₁₇₂	1	2	yes	—
158	C₂	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
159	C₂	#14582898	1	1	yes	characteristic
160	C₂	#14582907	1	1	yes	characteristic
161	C₂	#14582999	1	2	yes	—
162	C₂	#14583004	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₈₆

C₂

yes

maximal, characteristic

C₂×#13063248

C₂

yes

maximal

C₂×#13063236

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal

C₄×C₄⋊C₄×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

—

C₄×Dic₈₆

—

C₄×Dic₈₆

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

—

#13063248

C₄

yes

complemented

#13063236

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#95×C₄₃

—

C₂²×C₄×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×C₄

yes

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×C₄

yes

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

C₄²

yes

complemented

C₄₃⋊C₈

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

C₄×D₄

yes

—

C₂²×C₄₃

AGL(1,ℤ/8)

yes

characteristic

C₂²×C₄₃

#137

yes

—

C₂²×C₄₃

#137

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×C₄²

yes

characteristic

C₄×C₄₃

#117

yes

characteristic

C₄×C₄₃

C₄×D₄

yes

—

Dic₄₃

—

Dic₄₃

—

#1763

—

maximal, Sylow

C₂×C₄₃

C₄×C₂²⋊C₄

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#441

yes

—

C₂×C₄×Q₈

—

C₂×#103

—

C₂×#97

—

C₄×C₄⋊C₄

D₄₃

yes

characteristic

#340

—

C₄₃

#1763

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

#103

—

#97

—

100

#95

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

109

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

110

C₂²×C₄

C₂²×D₄₃

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₄×D₄₃

yes

—

119

C₄²

—

120

C₄²

—

121

C₄²

C₄×D₄₃

yes

—

122

C₄²

D₁₇₂

yes

—

123

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

172

—

137

C₂×C₄

—

172

—

138

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

139

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

140

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

141

C₂×C₄

C₂×C₄×D₄₃

yes

—

142

C₂×C₄

C₂×D₁₇₂

yes

characteristic

143

C₂×C₄

#10621898

yes

—

144

C₈

—

145

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

146

C₂²

C₄×C₂²⋊D₄₃

yes

—

147

C₂²

#13063351

yes

characteristic

148

C₂²

#13063355

yes

—

149

C₂²

#13063356

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

172

—

155

C₄

C₄²×D₄₃

yes

characteristic

156

C₄

#13063282

yes

characteristic

157

C₄

C₄×D₁₇₂

yes

—

158

C₂

C₄×(C₂×D₄₃)⋊C₄

yes

characteristic

159

C₂

#14582898

yes

characteristic

160

C₂

#14582907

yes

characteristic

161

C₂

#14582999

yes

—

162

C₂

#14583004

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₃
commutator	C₄×C₄₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₄×C₄⋊C₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₄×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#1763
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₃

commutator

C₄×C₄₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₄×C₄⋊C₄×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₄×C₄⋊C₄

D₄₃

hypocenter

C₄₃

#1763

LCS

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
9. C₄×Dic₈₆	C₄	semidirect
13. #13063248	C₄	semidirect
61. Dic₈₆	C₄²	semidirect
87. C₄₃	#1763	semidirect

H ≅ G/N

type

9. C₄×Dic₈₆

C₄

semidirect

13. #13063248

C₄

semidirect

61. Dic₈₆

C₄²

semidirect

87. C₄₃

#1763

semidirect

Subgroups — #19134428

Conjugacy classes of subgroups

Complements