Group #19135406 — subgroups

Subgroups — #19135406

Order 5504.

336 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 109 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×#13063313	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063250	C₂	1	2	yes	maximal, complemented
3	C₂×#13063247	C₂	1	2	yes	maximal
4	#14582765	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
7	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
8	#13063313	—	2	2	no	—
9	#13063313	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
12	#13063250	—	2	4	no	—
13	#13063247	—	2	4	no	—
14	C₄₃⋊#95	—	2	2	no	—
15	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₄₃	—	2	2	no	—
18	C₂³×D₄₃	—	2	1	no	—
19	C₂²×C₄×C₄₃	D₄	1	2	yes	—
20	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₄₃	—	2	1	no	—
22	C₂²×Dic₄₃	—	2	2	no	—
23	C₂×D₁₇₂	—	2	1	no	—
24	C₂×D₁₇₂	—	2	2	no	—
25	C₂×D₁₇₂	—	2	2	no	—
26	C₂×D₁₇₂	C₂×C₄	1	2	yes	—
27	C₂×Dic₈₆	—	2	2	no	—
28	C₂×Dic₈₆	—	2	2	no	—
29	C₂×Dic₈₆	—	2	1	no	—
30	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₄₃	—	2	2	no	—
32	C₂×C₈×C₄₃	—	2	2	no	—
33	C₂×C₈×C₄₃	D₄	1	4	yes	—
34	C₄²×C₄₃	C₂×C₄	1	2	yes	—
35	C₄²×C₄₃	D₄	1	2	yes	—
36	(C₂×D₄₃)⋊C₄	—	2	8	no	—
37	C₄⋊Dic₄₃	—	2	2	no	—
38	C₄⋊Dic₄₃	—	2	2	no	—
39	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×D₄₃	—	2	2	no	—
41	C₂²×D₄₃	—	4	1	no	—
42	C₂²×D₄₃	—	4	1	no	—
43	C₂²×D₄₃	—	4	4	no	—
44	C₂×C₄×C₄₃	—	2	4	no	—
45	C₂×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
46	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	#41	1	2	yes	—
50	C₂×Dic₄₃	—	2	2	no	—
51	C₂×Dic₄₃	—	4	1	no	—
52	C₂×Dic₄₃	—	4	1	no	—
53	C₂×Dic₄₃	—	4	2	no	—
54	C₂×Dic₄₃	—	4	4	no	—
55	D₁₇₂	—	2	4	no	—
56	D₁₇₂	—	4	2	no	—
57	Dic₈₆	—	2	4	no	—
58	Dic₈₆	—	4	2	no	—
59	C₈×C₄₃	—	2	4	no	—
60	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄₃	#124	1	1	yes	characteristic
62	C₂²×C₄₃	#127	1	1	yes	characteristic
63	C₂²×C₄₃	#135	1	4	yes	—
64	C₂×D₄₃	—	4	2	no	—
65	C₂×D₄₃	—	4	4	no	—
66	C₂×D₄₃	—	8	2	no	—
67	C₂×D₄₃	—	8	2	no	—
68	C₄×C₄₃	—	2	4	no	—
69	C₄×C₄₃	C₄×D₄	1	2	yes	—
70	C₄×C₄₃	#121	1	2	yes	—
71	Dic₄₃	—	4	2	no	—
72	Dic₄₃	—	8	2	no	—
73	#1958	—	43	1	no	maximal, Sylow
74	C₂×C₄₃	#390	1	1	yes	characteristic
75	C₂×C₄₃	#416	1	2	yes	—
76	C₂×C₄₃	#488	1	2	yes	—
77	C₂×C₄₃	#495	1	2	yes	—
78	D₄₃	—	8	2	no	—
79	C₂×#124	—	43	1	no	—
80	C₂×#103	—	43	2	no	—
81	C₂×#102	—	43	2	no	—
82	C₂×C₄×C₈	D₄₃	1	1	yes	characteristic, complemented
83	#382	—	43	1	no	—
84	C₄₃	#1958	1	1	yes	characteristic, complemented, Sylow
85	C₂²×Q₈	—	43	1	no	—
86	C₂²×D₄	—	43	1	no	—
87	C₂²×C₈	C₂×D₄₃	1	2	yes	—
88	#124	—	43	4	no	—
89	#124	—	86	2	no	—
90	C₂×C₄⋊C₄	—	43	2	no	—
91	C₂×C₂²⋊C₄	—	86	2	no	—
92	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
93	#103	—	86	4	no	—
94	#102	—	86	4	no	—
95	C₄×C₈	—	2	2	no	—
96	#95	—	86	2	no	—
97	C₂⁴	—	86	1	no	—
98	C₂×Q₈	—	43	2	no	—
99	C₂×Q₈	—	86	1	no	—
100	C₂×Q₈	—	86	2	no	—
101	C₂×Q₈	—	86	2	no	—
102	C₂×D₄	—	43	2	no	—
103	C₂×D₄	—	86	2	no	—
104	C₂×D₄	—	86	2	no	—
105	C₂×D₄	—	86	1	no	—
106	C₂²×C₄	—	86	1	no	—
107	C₂²×C₄	—	86	2	no	—
108	C₂²×C₄	D₁₇₂	1	2	yes	—
109	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	D₁₇₂	1	4	yes	—
113	C₄⋊C₄	—	86	2	no	—
114	C₄⋊C₄	—	86	2	no	—
115	C₂²⋊C₄	—	86	8	no	—
116	C₄²	C₄×D₄₃	1	2	yes	—
117	C₄²	C₂²⋊D₄₃	1	2	yes	—
118	C₂³	—	86	2	no	—
119	C₂³	—	172	1	no	—
120	C₂³	—	172	1	no	—
121	C₂³	—	172	4	no	—
122	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
123	Q₈	—	86	4	no	—
124	Q₈	—	172	2	no	—
125	D₄	—	86	4	no	—
126	D₄	—	172	2	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	86	2	no	—
129	C₂×C₄	—	172	2	no	—
130	C₂×C₄	—	172	4	no	—
131	C₂×C₄	—	172	1	no	—
132	C₂×C₄	—	172	1	no	—
133	C₂×C₄	(C₂×D₄₃)⋊C₄	1	4	yes	—
134	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
135	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
136	C₂×C₄	#10621898	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₂²	—	344	2	no	—
140	C₂²	—	344	2	no	—
141	C₂²	—	172	2	no	—
142	C₂²	—	172	4	no	—
143	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
144	C₂²	#13063313	1	1	yes	characteristic
145	C₂²	#13063322	1	1	yes	characteristic
146	C₂²	#13063347	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	344	2	no	—
149	C₄	—	172	2	no	—
150	C₄	C₄×D₁₇₂	1	2	yes	—
151	C₄	#13063299	1	2	yes	—
152	C₂	—	344	2	no	—
153	C₂	#14582801	1	1	yes	characteristic
154	C₂	#14582896	1	2	yes	—
155	C₂	#14583254	1	2	yes	—
156	C₂	#14583283	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063313

C₂

yes

maximal, characteristic

C₂×#13063250

C₂

yes

maximal, complemented

C₂×#13063247

C₂

yes

maximal

#14582765

C₂

yes

maximal, characteristic, complemented

C₂×C₄×C₈×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₈₆

C₂²

yes

characteristic

C₂²×D₁₇₂

C₂²

yes

characteristic

#13063313

—

#13063313

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×(C₂×D₄₃)⋊C₄

—

#13063250

—

#13063247

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

—

C₂³×D₄₃

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₄

yes

—

C₄²×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

D₄

yes

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄×D₄

yes

—

C₄×C₄₃

#121

yes

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#488

yes

—

C₂×C₄₃

#495

yes

—

D₄₃

—

C₂×#124

—

C₂×#103

—

C₂×#102

—

C₂×C₄×C₈

D₄₃

yes

characteristic, complemented

#382

—

C₄₃

#1958

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂²×C₈

C₂×D₄₃

yes

—

#124

—

#124

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

#103

—

#102

—

C₄×C₈

—

#95

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

D₁₇₂

yes

—

109

C₂²×C₄

C₂²×D₄₃

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

D₁₇₂

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

C₄×D₄₃

yes

—

117

C₄²

C₂²⋊D₄₃

yes

—

118

C₂³

—

119

C₂³

—

172

—

120

C₂³

—

172

—

121

C₂³

—

172

—

122

C₂³

C₂×D₁₇₂

yes

characteristic

123

Q₈

—

124

Q₈

—

172

—

125

D₄

—

126

D₄

—

172

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

172

—

130

C₂×C₄

—

172

—

131

C₂×C₄

—

172

—

132

C₂×C₄

—

172

—

133

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

134

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

135

C₂×C₄

C₂×D₁₇₂

yes

—

136

C₂×C₄

#10621898

yes

—

137

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

138

C₈

—

139

C₂²

—

344

—

140

C₂²

—

344

—

141

C₂²

—

172

—

142

C₂²

—

172

—

143

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

144

C₂²

#13063313

yes

characteristic

145

C₂²

#13063322

yes

characteristic

146

C₂²

#13063347

yes

—

147

C₄

—

148

C₄

—

344

—

149

C₄

—

172

—

150

C₄

C₄×D₁₇₂

yes

—

151

C₄

#13063299

yes

—

152

C₂

—

344

—

153

C₂

#14582801

yes

characteristic

154

C₂

#14582896

yes

—

155

C₂

#14583254

yes

—

156

C₂

#14583283

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×C₄×C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₄₃
hypocenter	C₄₃	#1958
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×C₄×C₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₄₃

hypocenter

C₄₃

#1958

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. C₂×#13063250	C₂	semidirect
4. #14582765	C₂	semidirect
5. C₂×C₄×C₈×C₄₃	C₂	semidirect
9. #13063313	C₄	semidirect
82. C₂×C₄×C₈	D₄₃	semidirect
84. C₄₃	#1958	semidirect

H ≅ G/N

type

2. C₂×#13063250

C₂

semidirect

4. #14582765

C₂

semidirect

5. C₂×C₄×C₈×C₄₃

C₂

semidirect

9. #13063313

C₄

semidirect

82. C₂×C₄×C₈

D₄₃

semidirect

84. C₄₃

#1958

semidirect

Subgroups — #19135406

Conjugacy classes of subgroups

Complements