Group #19135738 — subgroups

Subgroups — #19135738

Order 5504.

294 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 87 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×#13063245	C₂	1	2	yes	maximal
2	#14582801	C₂	1	1	yes	maximal, characteristic
3	#14582661	C₂	1	2	yes	maximal, complemented
4	#14582654	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
7	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
9	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
10	#13063245	—	2	4	no	—
11	#125×C₄₃	—	2	4	no	—
12	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
13	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	C₂³×D₄₃	—	2	1	no	—
16	C₂²×C₄×C₄₃	—	2	2	no	—
17	C₂²×C₄×C₄₃	D₄	1	2	yes	—
18	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₄₃	—	2	1	no	—
20	C₂×D₁₇₂	—	2	1	no	—
21	C₂×D₁₇₂	—	2	2	no	—
22	C₂×D₁₇₂	—	2	2	no	—
23	C₂×D₁₇₂	Q₈	1	2	yes	—
24	C₄⋊C₄×C₄₃	—	2	8	no	—
25	C₄⋊C₄×C₄₃	—	2	2	no	—
26	C₄⋊C₄×C₄₃	D₄	1	4	yes	—
27	C₄²×C₄₃	—	2	2	no	—
28	(C₂×D₄₃)⋊C₄	—	4	4	no	—
29	C₄⋊Dic₄₃	—	2	2	no	—
30	C₂×C₄₃⋊C₈	—	2	2	no	—
31	C₂×C₄₃⋊C₈	—	2	2	no	—
32	C₂×C₄₃⋊C₈	—	2	2	no	—
33	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×D₄₃	—	2	2	no	—
35	C₂²×D₄₃	—	4	1	no	—
36	C₂²×D₄₃	—	4	1	no	—
37	C₂²×D₄₃	—	4	2	no	—
38	C₂²×D₄₃	—	4	2	no	—
39	C₂×C₄×C₄₃	—	2	2	no	—
40	C₂×C₄×C₄₃	—	2	4	no	—
41	C₂×C₄×C₄₃	—	2	2	no	—
42	C₂×C₄×C₄₃	—	2	2	no	—
43	C₂×C₄×C₄₃	—	4	2	no	—
44	C₂×C₄×C₄₃	—	4	2	no	—
45	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₄₃	#41	1	2	yes	—
48	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
49	C₂×Dic₄₃	—	4	1	no	—
50	C₂×Dic₄₃	—	4	2	no	—
51	D₁₇₂	—	2	4	no	—
52	D₁₇₂	—	4	2	no	—
53	C₄₃⋊C₈	—	4	2	no	—
54	C₂²×C₄₃	#120	1	1	yes	characteristic
55	C₂²×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₄₃	#124	1	1	yes	characteristic
57	C₂²×C₄₃	#135	1	2	yes	—
58	C₂²×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
59	C₂×D₄₃	—	4	2	no	—
60	C₂×D₄₃	—	4	4	no	—
61	C₂×D₄₃	—	8	2	no	—
62	C₂×D₄₃	—	8	1	no	—
63	C₂×D₄₃	—	8	1	no	—
64	C₄×C₄₃	—	4	2	no	—
65	C₄×C₄₃	—	4	4	no	—
66	C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₄₃	#124	1	2	yes	—
68	Dic₄₃	—	8	1	no	—
69	#2025	—	43	1	no	maximal, Sylow
70	C₂×C₄₃	#393	1	1	yes	characteristic
71	C₂×C₄₃	#449	1	2	yes	—
72	C₂×C₄₃	#478	1	2	yes	—
73	C₂×C₄₃	#488	1	1	yes	characteristic
74	C₂×C₄₃	#490	1	1	yes	characteristic
75	D₄₃	—	8	2	no	—
76	C₂×#125	D₄₃	1	1	yes	characteristic, complemented
77	C₂×#102	—	43	2	no	—
78	#390	—	43	1	no	—
79	#340	—	43	2	no	—
80	#336	—	43	1	no	—
81	C₄₃	#2025	1	1	yes	characteristic, complemented, Sylow
82	C₂²×D₄	—	43	1	no	—
83	C₂²×C₈	—	43	2	no	—
84	#125	—	2	4	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄⋊C₄	—	43	1	no	—
87	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
88	C₂×C₂²⋊C₄	—	86	2	no	—
89	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
90	#102	—	86	4	no	—
91	C₂⁴	—	86	1	no	—
92	C₂×D₄	—	43	2	no	—
93	C₂×D₄	—	86	2	no	—
94	C₂×D₄	—	86	1	no	—
95	C₂×D₄	—	86	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	86	1	no	—
98	C₂²×C₄	C₂²⋊D₄₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
100	C₂×C₈	—	86	2	no	—
101	C₂×C₈	—	86	2	no	—
102	C₂×C₈	—	86	2	no	—
103	C₄⋊C₄	—	2	8	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	86	2	no	—
106	C₄⋊C₄	C₂²⋊D₄₃	1	4	yes	—
107	C₂²⋊C₄	—	172	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	86	2	no	—
110	C₂³	—	172	1	no	—
111	C₂³	—	172	1	no	—
112	C₂³	—	172	2	no	—
113	C₂³	—	172	2	no	—
114	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
115	D₄	—	86	4	no	—
116	D₄	—	172	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	172	1	no	—
124	C₂×C₄	—	172	2	no	—
125	C₂×C₄	Q₈×D₄₃	1	1	yes	characteristic
126	C₂×C₄	#10621902	1	2	yes	—
127	C₂×C₄	#10621902	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
129	C₈	—	172	2	no	—
130	C₂²	—	344	2	no	—
131	C₂²	—	344	1	no	—
132	C₂²	—	344	1	no	—
133	C₂²	—	172	2	no	—
134	C₂²	—	172	4	no	—
135	C₂²	#13063294	1	1	yes	characteristic
136	C₂²	#13063301	1	1	yes	characteristic
137	C₂²	#13063314	1	1	yes	characteristic
138	C₂²	#13063345	1	2	yes	—
139	C₂²	#13063350	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	4	4	no	—
142	C₄	—	344	1	no	—
143	C₄	#13063301	1	2	yes	—
144	C₄	#13063314	1	2	yes	—
145	C₂	—	344	2	no	—
146	C₂	#14582810	1	1	yes	characteristic
147	C₂	#14583044	1	2	yes	—
148	C₂	#14583196	1	2	yes	—
149	C₂	#14583256	1	1	yes	characteristic
150	C₂	#14583265	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063245

C₂

yes

maximal

#14582801

C₂

yes

maximal, characteristic

#14582661

C₂

yes

maximal, complemented

#14582654

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×(C₂×D₄₃)⋊C₄

—

#13063245

—

#125×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂³×D₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

Q₈

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

D₄

yes

—

C₄²×C₄₃

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₃

#41

yes

—

C₂×C₄×C₄₃

#41

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₄₃⋊C₈

—

C₂²×C₄₃

#120

yes

characteristic

C₂²×C₄₃

C₂²⋊Q₈

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂²⋊Q₈

yes

—

C₄×C₄₃

#124

yes

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#478

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

C₂×#125

D₄₃

yes

characteristic, complemented

—

—

—

—

C₄₃

#2025

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

#102

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₄₃

yes

—

C₂²×C₄

C₂²×D₄₃

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂²⋊D₄₃

yes

—

107

C₂²⋊C₄

—

172

—

108

C₄²

—

109

C₂³

—

110

C₂³

—

172

—

111

C₂³

—

172

—

112

C₂³

—

172

—

113

C₂³

—

172

—

114

C₂³

C₂×C₂²⋊D₄₃

yes

characteristic

115

D₄

—

116

D₄

—

172

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

172

—

124

C₂×C₄

—

172

—

125

C₂×C₄

Q₈×D₄₃

yes

characteristic

126

C₂×C₄

#10621902

yes

—

127

C₂×C₄

#10621902

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

129

C₈

—

172

—

130

C₂²

—

344

—

131

C₂²

—

344

—

132

C₂²

—

344

—

133

C₂²

—

172

—

134

C₂²

—

172

—

135

C₂²

#13063294

yes

characteristic

136

C₂²

#13063301

yes

characteristic

137

C₂²

#13063314

yes

characteristic

138

C₂²

#13063345

yes

—

139

C₂²

#13063350

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

344

—

143

C₄

#13063301

yes

—

144

C₄

#13063314

yes

—

145

C₂

—

344

—

146

C₂

#14582810

yes

characteristic

147

C₂

#14583044

yes

—

148

C₂

#14583196

yes

—

149

C₂

#14583256

yes

characteristic

150

C₂

#14583265

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₃
commutator	C₂²×C₄×C₄₃	C₂³
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×#125×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×#125	D₄₃
hypocenter	C₄₃	#2025
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₃

commutator

C₂²×C₄×C₄₃

C₂³

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×#125×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
3. #14582661	C₂	semidirect
4. #14582654	C₂	semidirect
5. C₂×#125×C₄₃	C₂	semidirect
76. C₂×#125	D₄₃	semidirect
81. C₄₃	#2025	semidirect

H ≅ G/N

type

3. #14582661

C₂

semidirect

4. #14582654

C₂

semidirect

5. C₂×#125×C₄₃

C₂

semidirect

76. C₂×#125

D₄₃

semidirect

81. C₄₃

#2025

semidirect

Subgroups — #19135738

Conjugacy classes of subgroups

Complements