Group #19136524 — subgroups

Subgroups — #19136524

Order 5504.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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×#13063269	C₂	1	2	yes	maximal, complemented
2	#14583140	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582700	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic, complemented
7	#13063304	—	2	1	no	—
8	#13063269	—	2	4	no	—
9	C₂×C₄₃⋊C₁₆	C₂²	1	2	yes	complemented
10	#13063259	C₂²	1	1	yes	characteristic, complemented
11	#13063250	—	2	1	no	—
12	C₂²×D₄×C₄₃	—	2	1	no	—
13	C₂×D₈×C₄₃	—	2	4	no	—
14	C₂×D₈×C₄₃	—	2	1	no	—
15	C₂×D₈×C₄₃	C₂²	1	2	yes	—
16	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄₃	—	4	1	no	—
18	C₂×D₄×C₄₃	—	2	4	no	—
19	C₂×D₄×C₄₃	—	2	2	no	—
20	C₂×D₄×C₄₃	—	4	2	no	—
21	C₂×D₄×C₄₃	—	4	1	no	—
22	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
23	C₂×Dic₈₆	—	2	1	no	—
24	D₈×C₄₃	—	2	4	no	—
25	D₈×C₄₃	—	2	2	no	—
26	D₈×C₄₃	D₄	1	4	yes	—
27	C₂×C₈×C₄₃	—	2	2	no	—
28	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
29	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
30	C₂²⋊Dic₄₃	—	4	1	no	—
31	C₄⋊Dic₄₃	—	2	1	no	—
32	Dic₄₃⋊C₄	—	4	1	no	—
33	Dic₁₇₂	—	2	2	no	—
34	C₄₃⋊C₁₆	—	2	2	no	—
35	C₂³×C₄₃	—	4	4	no	—
36	C₂³×C₄₃	—	4	2	no	—
37	C₂³×C₄₃	—	8	2	no	—
38	C₂³×C₄₃	—	8	2	no	—
39	C₂³×C₄₃	D₈	1	1	yes	characteristic
40	D₄×C₄₃	—	2	4	no	—
41	D₄×C₄₃	—	4	4	no	—
42	D₄×C₄₃	—	4	2	no	—
43	C₂×C₄×C₄₃	—	2	1	no	—
44	C₂×C₄×C₄₃	—	2	1	no	—
45	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
46	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₄₃	—	4	1	no	—
48	C₂×Dic₄₃	—	4	1	no	—
49	Dic₈₆	—	4	1	no	—
50	Dic₈₆	—	4	1	no	—
51	C₈×C₄₃	—	2	1	no	—
52	C₈×C₄₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄₃	—	2	1	no	—
54	C₂²×C₄₃	—	2	1	no	—
55	C₂²×C₄₃	—	4	4	no	—
56	C₂²×C₄₃	—	8	8	no	—
57	C₂²×C₄₃	—	8	2	no	—
58	C₂²×C₄₃	—	8	2	no	—
59	C₂²×C₄₃	SD₃₂	1	2	yes	complemented
60	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
61	C₄×C₄₃	—	2	1	no	—
62	C₄×C₄₃	#120	1	1	yes	characteristic
63	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	Dic₄₃	—	8	1	no	—
65	Dic₄₃	—	8	1	no	—
66	#2187	—	43	1	no	maximal, Sylow
67	C₂×C₄₃	—	2	2	no	—
68	C₂×C₄₃	—	8	4	no	—
69	C₂×C₄₃	#447	1	1	yes	characteristic
70	C₂×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
71	C₂×C₄₃	#509	1	1	yes	characteristic
72	C₂²×D₈	D₄₃	1	1	yes	characteristic
73	C₂×SD₃₂	—	43	2	no	—
74	#467	—	43	1	no	—
75	#357	—	43	2	no	—
76	C₂²⋊C₁₆	—	43	1	no	—
77	C₄₃	#2187	1	1	yes	characteristic, complemented, Sylow
78	C₂²×D₄	—	2	1	no	—
79	C₂×Q₁₆	—	43	1	no	—
80	C₂×D₈	—	2	4	no	—
81	C₂×D₈	—	2	1	no	—
82	C₂×D₈	C₂×D₄₃	1	2	yes	—
83	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
84	C₂²⋊Q₈	—	86	1	no	—
85	SD₃₂	—	86	4	no	—
86	C₂×C₁₆	—	43	2	no	—
87	#107	—	43	1	no	—
88	#103	—	86	1	no	—
89	C₂⁴	—	4	1	no	—
90	C₂×Q₈	—	86	1	no	—
91	C₂×D₄	—	2	4	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	4	2	no	—
94	C₂×D₄	—	4	1	no	—
95	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
96	Q₁₆	—	86	2	no	—
97	D₈	—	2	4	no	—
98	D₈	—	2	2	no	—
99	D₈	C₂²⋊D₄₃	1	4	yes	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
103	C₄⋊C₄	—	86	1	no	—
104	C₄⋊C₄	—	172	1	no	—
105	C₂²⋊C₄	—	172	1	no	—
106	C₁₆	—	86	2	no	—
107	C₂³	—	4	4	no	—
108	C₂³	—	4	2	no	—
109	C₂³	—	8	2	no	—
110	C₂³	—	8	2	no	—
111	C₂³	#10621882	1	1	yes	characteristic
112	Q₈	—	172	1	no	—
113	Q₈	—	172	1	no	—
114	D₄	—	2	4	no	—
115	D₄	—	4	4	no	—
116	D₄	—	4	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	172	1	no	—
120	C₂×C₄	—	172	1	no	—
121	C₂×C₄	#10621882	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	4	4	no	—
128	C₂²	—	8	8	no	—
129	C₂²	—	8	2	no	—
130	C₂²	—	8	2	no	—
131	C₂²	#13063269	1	2	yes	complemented
132	C₂²	C₂×#10621882	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	344	1	no	—
135	C₄	—	344	1	no	—
136	C₄	#13063294	1	1	yes	characteristic
137	C₄	#13063351	1	1	yes	characteristic
138	C₂	—	2	2	no	—
139	C₂	—	8	4	no	—
140	C₂	#14583033	1	1	yes	characteristic
141	C₂	C₂×#13063269	1	1	yes	characteristic
142	C₂	#14583335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063269

C₂

yes

maximal, complemented

#14583140

C₂

yes

maximal, characteristic, complemented

#14582700

C₂

yes

maximal, complemented

C₂²⋊(C₄₃⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₄₃

C₂

yes

maximal, characteristic

C₂×Dic₁₇₂

C₂²

yes

characteristic, complemented

#13063304

—

#13063269

—

C₂×C₄₃⋊C₁₆

C₂²

yes

complemented

#13063259

C₂²

yes

characteristic, complemented

#13063250

—

C₂²×D₄×C₄₃

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

—

C₂×D₈×C₄₃

C₂²

yes

—

C₂²×C₈×C₄₃

C₂²

yes

characteristic

C₂⁴×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂×Dic₈₆

—

D₈×C₄₃

—

D₈×C₄₃

—

D₈×C₄₃

D₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₄

yes

characteristic

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₁₇₂

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

D₈

yes

characteristic

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂×D₄

yes

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

SD₃₂

yes

complemented

C₂²×C₄₃

C₂×D₈

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

#2187

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#447

yes

characteristic

C₂×C₄₃

C₂×SD₃₂

yes

characteristic

C₂×C₄₃

#509

yes

characteristic

C₂²×D₈

D₄₃

yes

characteristic

C₂×SD₃₂

—

#467

—

#357

—

C₂²⋊C₁₆

—

C₄₃

#2187

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

C₂×D₄₃

yes

—

C₂²×C₈

C₂×D₄₃

yes

characteristic

C₂²⋊Q₈

—

SD₃₂

—

C₂×C₁₆

—

#107

—

#103

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

Q₁₆

—

D₈

—

D₈

—

D₈

C₂²⋊D₄₃

yes

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₄₃

yes

characteristic

102

C₂×C₈

C₂²×D₄₃

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

172

—

105

C₂²⋊C₄

—

172

—

106

C₁₆

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

#10621882

yes

characteristic

112

Q₈

—

172

—

113

Q₈

—

172

—

114

D₄

—

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

172

—

120

C₂×C₄

—

172

—

121

C₂×C₄

#10621882

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

123

C₈

—

124

C₈

C₂×C₂²⋊D₄₃

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

#13063269

yes

complemented

132

C₂²

C₂×#10621882

yes

characteristic

133

C₄

—

134

C₄

—

344

—

135

C₄

—

344

—

136

C₄

#13063294

yes

characteristic

137

C₄

#13063351

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

#14583033

yes

characteristic

141

C₂

C₂×#13063269

yes

characteristic

142

C₂

#14583335

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#10621882
commutator	C₂×C₈×C₄₃	C₂³
Frattini	C₂×C₈	C₂²×D₄₃
Fitting	C₂²×D₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂×D₈
hypercenter	C₂²×D₈	D₄₃
hypocenter	C₄₃	#2187
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₈

subgroup

G/H

center

C₂²

C₂×#10621882

commutator

C₂×C₈×C₄₃

C₂³

Frattini

C₂×C₈

C₂²×D₄₃

Fitting

C₂²×D₈×C₄₃

C₂

socle

C₂²×C₄₃

C₂×D₈

hypercenter

C₂²×D₈

D₄₃

hypocenter

C₄₃

#2187

LCS

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₈

N	H ≅ G/N	type
1. C₂×#13063269	C₂	semidirect
2. #14583140	C₂	semidirect
3. #14582700	C₂	semidirect
4. C₂²⋊(C₄₃⋊C₁₆)	C₂	semidirect
6. C₂×Dic₁₇₂	C₂²	semidirect
9. C₂×C₄₃⋊C₁₆	C₂²	semidirect
10. #13063259	C₂²	semidirect
59. C₂²×C₄₃	SD₃₂	semidirect
77. C₄₃	#2187	semidirect
131. C₂²	#13063269	semidirect

H ≅ G/N

type

1. C₂×#13063269

C₂

semidirect

2. #14583140

C₂

semidirect

3. #14582700

C₂

semidirect

4. C₂²⋊(C₄₃⋊C₁₆)

C₂

semidirect

6. C₂×Dic₁₇₂

C₂²

semidirect

9. C₂×C₄₃⋊C₁₆

C₂²

semidirect

10. #13063259

C₂²

semidirect

59. C₂²×C₄₃

SD₃₂

semidirect

77. C₄₃

#2187

semidirect

131. C₂²

#13063269

semidirect

Subgroups — #19136524

Conjugacy classes of subgroups

Complements