Group #19136658 — subgroups

Subgroups — #19136658

Order 5504.

158 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 51 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×Dic₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13063270	C₂	1	1	yes	maximal, characteristic
3	#14583283	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582704	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582701	C₂	1	1	yes	maximal, characteristic
7	C₄₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
10	C₂×#10621873	—	2	1	no	—
11	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
12	#13063313	—	2	1	no	—
13	C₄×Dic₈₆	—	2	1	no	—
14	Dic₃₄₄	—	2	2	no	—
15	#13063270	—	2	2	no	—
16	#13063259	C₂²	1	1	yes	characteristic
17	#13063250	—	2	1	no	—
18	C₂×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×D₁₇₂	—	2	1	no	—
22	C₂×Dic₈₆	—	2	1	no	—
23	C₂×Dic₈₆	—	2	1	no	—
24	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
26	C₄²×C₄₃	D₄	1	1	yes	characteristic
27	(C₂×D₄₃)⋊C₄	—	4	1	no	—
28	(C₂×D₄₃)⋊C₄	—	4	1	no	—
29	C₄⋊Dic₄₃	—	2	1	no	—
30	Dic₄₃⋊C₄	—	4	1	no	—
31	C₄×Dic₄₃	—	4	1	no	—
32	Dic₁₇₂	—	2	1	no	—
33	Dic₁₇₂	—	2	1	no	—
34	Dic₁₇₂	—	2	1	no	—
35	Dic₁₇₂	D₄	1	2	yes	—
36	D₃₄₄	—	2	1	no	—
37	D₃₄₄	—	2	1	no	—
38	#10621873	—	4	2	no	—
39	C₁₆×C₄₃	—	2	1	no	—
40	C₁₆×C₄₃	—	2	1	no	—
41	C₂²×D₄₃	—	4	1	no	—
42	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
43	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
44	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₄₃	—	4	1	no	—
46	C₂×Dic₄₃	—	4	1	no	—
47	C₂×Dic₄₃	—	4	1	no	—
48	D₁₇₂	—	4	1	no	—
49	D₁₇₂	—	4	1	no	—
50	Dic₈₆	—	2	2	no	—
51	Dic₈₆	—	4	1	no	—
52	Dic₈₆	—	4	1	no	—
53	Dic₈₆	—	4	1	no	—
54	C₈×C₄₃	—	2	1	no	—
55	C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₈×C₄₃	#41	1	1	yes	characteristic
57	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
58	C₂×D₄₃	—	8	1	no	—
59	C₂×D₄₃	—	8	1	no	—
60	C₂×D₄₃	—	8	1	no	—
61	C₄×C₄₃	—	2	1	no	—
62	C₄×C₄₃	—	2	1	no	—
63	C₄×C₄₃	#121	1	1	yes	characteristic
64	C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	Dic₄₃	—	4	2	no	—
66	Dic₄₃	—	8	1	no	—
67	Dic₄₃	—	8	1	no	—
68	#2211	—	43	1	no	maximal, Sylow
69	C₂×C₄₃	#459	1	1	yes	characteristic
70	C₂×C₄₃	#508	1	1	yes	characteristic
71	C₂×C₄₃	#510	1	1	yes	characteristic
72	D₄₃	—	16	1	no	—
73	C₂×Q₃₂	—	43	1	no	—
74	C₂×SD₃₂	—	43	1	no	—
75	#495	—	43	1	no	—
76	C₄×Q₁₆	—	43	1	no	—
77	C₄⋊C₁₆	D₄₃	1	1	yes	characteristic, complemented
78	#358	—	43	1	no	—
79	#357	—	43	1	no	—
80	C₄₃	#2211	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₁₆	—	43	1	no	—
82	C₂×Q₁₆	—	43	1	no	—
83	C₂×SD₁₆	—	86	1	no	—
84	C₂×D₈	—	43	1	no	—
85	#124	—	86	1	no	—
86	C₄×Q₈	—	86	1	no	—
87	Q₃₂	—	86	2	no	—
88	SD₃₂	—	86	2	no	—
89	C₂×C₁₆	C₂×D₄₃	1	1	yes	characteristic
90	C₂×C₁₆	C₂×D₄₃	1	1	yes	characteristic
91	#107	—	43	1	no	—
92	#103	—	86	1	no	—
93	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
94	C₂×Q₈	—	86	1	no	—
95	C₂×Q₈	—	86	1	no	—
96	C₂×D₄	—	86	1	no	—
97	Q₁₆	—	43	2	no	—
98	Q₁₆	—	86	1	no	—
99	Q₁₆	—	86	1	no	—
100	Q₁₆	—	86	1	no	—
101	SD₁₆	—	172	2	no	—
102	D₈	—	86	1	no	—
103	D₈	—	86	1	no	—
104	C₂×C₈	D₁₇₂	1	1	yes	characteristic
105	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
106	C₄⋊C₄	—	86	1	no	—
107	C₄⋊C₄	—	172	1	no	—
108	C₂²⋊C₄	—	172	1	no	—
109	C₂²⋊C₄	—	172	1	no	—
110	C₄²	—	172	1	no	—
111	C₄²	D₁₇₂	1	1	yes	characteristic
112	C₁₆	—	2	1	no	—
113	C₁₆	—	2	1	no	—
114	C₂³	—	172	1	no	—
115	Q₈	—	86	2	no	—
116	Q₈	—	172	1	no	—
117	Q₈	—	172	1	no	—
118	Q₈	—	172	1	no	—
119	D₄	—	172	1	no	—
120	D₄	—	172	1	no	—
121	C₂×C₄	—	172	1	no	—
122	C₂×C₄	—	172	1	no	—
123	C₂×C₄	—	172	1	no	—
124	C₂×C₄	D₃₄₄	1	1	yes	characteristic
125	C₂×C₄	D₃₄₄	1	1	yes	characteristic
126	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
127	C₈	—	2	1	no	—
128	C₈	D₄×D₄₃	1	1	yes	characteristic
129	C₈	#10621902	1	1	yes	characteristic
130	C₂²	—	344	1	no	—
131	C₂²	—	344	1	no	—
132	C₂²	—	344	1	no	—
133	C₂²	C₂×D₃₄₄	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	344	1	no	—
137	C₄	—	344	1	no	—
138	C₄	—	172	2	no	—
139	C₄	#13063300	1	1	yes	characteristic
140	C₄	#13063353	1	1	yes	characteristic
141	C₂	—	688	1	no	—
142	C₂	#14583095	1	1	yes	characteristic
143	C₂	#14583331	1	1	yes	characteristic
144	C₂	#14583342	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₃₄₄

C₂

yes

maximal, characteristic, complemented

C₂×#13063270

C₂

yes

maximal, characteristic

#14583283

C₂

yes

maximal, characteristic

C₄×Dic₁₇₂

C₂

yes

maximal, characteristic, complemented

#14582704

C₂

yes

maximal, characteristic, complemented

#14582701

C₂

yes

maximal, characteristic

C₄₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₇₂

C₂²

yes

characteristic

C₂×Dic₁₇₂

C₂²

yes

characteristic

C₂×#10621873

—

C₂×D₃₄₄

C₂²

yes

characteristic

#13063313

—

C₄×Dic₈₆

—

Dic₃₄₄

—

#13063270

—

#13063259

C₂²

yes

characteristic

#13063250

—

C₂×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×C₁₆×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×D₁₇₂

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

D₄

yes

characteristic

C₂×C₈×C₄₃

C₂³

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

D₄

yes

—

D₃₄₄

—

D₃₄₄

—

#10621873

—

C₁₆×C₄₃

—

C₁₆×C₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

D₈

yes

characteristic

C₂×C₄×C₄₃

D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂×D₄

yes

characteristic

C₈×C₄₃

#41

yes

characteristic

C₂²×C₄₃

C₂×D₈

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

C₂×Q₃₂

—

C₂×SD₃₂

—

#495

—

C₄×Q₁₆

—

C₄⋊C₁₆

D₄₃

yes

characteristic, complemented

#358

—

#357

—

C₄₃

#2211

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

#124

—

C₄×Q₈

—

Q₃₂

—

SD₃₂

—

C₂×C₁₆

C₂×D₄₃

yes

characteristic

C₂×C₁₆

C₂×D₄₃

yes

characteristic

#107

—

#103

—

C₄×C₈

C₂×D₄₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

100

Q₁₆

—

101

SD₁₆

—

172

—

102

D₈

—

103

D₈

—

104

C₂×C₈

D₁₇₂

yes

characteristic

105

C₂×C₈

C₂²×D₄₃

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

172

—

108

C₂²⋊C₄

—

172

—

109

C₂²⋊C₄

—

172

—

110

C₄²

—

172

—

111

C₄²

D₁₇₂

yes

characteristic

112

C₁₆

—

113

C₁₆

—

114

C₂³

—

172

—

115

Q₈

—

116

Q₈

—

172

—

117

Q₈

—

172

—

118

Q₈

—

172

—

119

D₄

—

172

—

120

D₄

—

172

—

121

C₂×C₄

—

172

—

122

C₂×C₄

—

172

—

123

C₂×C₄

—

172

—

124

C₂×C₄

D₃₄₄

yes

characteristic

125

C₂×C₄

D₃₄₄

yes

characteristic

126

C₂×C₄

C₂×D₁₇₂

yes

characteristic

127

C₈

—

128

C₈

D₄×D₄₃

yes

characteristic

129

C₈

#10621902

yes

characteristic

130

C₂²

—

344

—

131

C₂²

—

344

—

132

C₂²

—

344

—

133

C₂²

C₂×D₃₄₄

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

344

—

137

C₄

—

344

—

138

C₄

—

172

—

139

C₄

#13063300

yes

characteristic

140

C₄

#13063353

yes

characteristic

141

C₂

—

688

—

142

C₂

#14583095

yes

characteristic

143

C₂

#14583331

yes

characteristic

144

C₂

#14583342

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₃₄₄
commutator	C₂×C₈×C₄₃	C₂³
Frattini	C₂×C₈	C₂²×D₄₃
Fitting	C₄₃×C₄⋊C₁₆	C₂
socle	C₂²×C₄₃	C₂×D₈
hypercenter	C₄⋊C₁₆	D₄₃
hypocenter	C₄₃	#2211
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₃₄₄

commutator

C₂×C₈×C₄₃

C₂³

Frattini

C₂×C₈

C₂²×D₄₃

Fitting

C₄₃×C₄⋊C₁₆

C₂

socle

C₂²×C₄₃

C₂×D₈

hypercenter

C₄⋊C₁₆

D₄₃

hypocenter

C₄₃

#2211

LCS

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
1. C₂×Dic₃₄₄	C₂	semidirect
4. C₄×Dic₁₇₂	C₂	semidirect
5. #14582704	C₂	semidirect
7. C₄₃×C₄⋊C₁₆	C₂	semidirect
77. C₄⋊C₁₆	D₄₃	semidirect
80. C₄₃	#2211	semidirect

H ≅ G/N

type

1. C₂×Dic₃₄₄

C₂

semidirect

4. C₄×Dic₁₇₂

C₂

semidirect

5. #14582704

C₂

semidirect

7. C₄₃×C₄⋊C₁₆

C₂

semidirect

77. C₄⋊C₁₆

D₄₃

semidirect

80. C₄₃

#2211

semidirect

Subgroups — #19136658

Conjugacy classes of subgroups

Complements