Group #19134416 — subgroups

Subgroups — C₄×#13063250 #19134416

Order 5504.

332 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 145 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×#13063250	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
4	#14582662	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
7	C₄×Dic₈₆	—	2	2	no	—
8	C₄×Dic₈₆	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	C₂×Dic₄₃⋊C₄	—	2	1	no	—
12	C₂×C₄×Dic₄₃	—	2	1	no	—
13	C₂×C₄×Dic₄₃	—	2	1	no	—
14	#13063250	C₄	1	8	yes	direct product factor
15	C₄₃⋊#95	—	2	1	no	—
16	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₄₃	—	2	2	no	—
19	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₄₃	—	2	1	no	—
23	C₂²×Dic₄₃	—	2	2	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	C₂×C₈×C₄₃	—	2	2	no	—
30	C₂×C₈×C₄₃	—	2	2	no	—
31	C₂×C₈×C₄₃	C₂×C₄	1	4	yes	—
32	C₄²×C₄₃	C₂×C₄	1	2	yes	—
33	C₄²×C₄₃	D₄	1	2	yes	—
34	C₄⋊Dic₄₃	—	2	1	no	—
35	C₄⋊Dic₄₃	—	2	2	no	—
36	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
37	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
38	Dic₄₃⋊C₄	—	2	4	no	—
39	C₄×Dic₄₃	—	2	4	no	—
40	C₄×Dic₄₃	—	4	2	no	—
41	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₄₃	—	2	2	no	—
43	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
46	C₂×C₄×C₄₃	Q₁₆	1	2	yes	—
47	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	#41	1	2	yes	—
51	C₂×Dic₄₃	—	2	2	no	—
52	C₂×Dic₄₃	—	2	4	no	—
53	C₂×Dic₄₃	—	2	2	no	—
54	C₂×Dic₄₃	—	2	4	no	—
55	C₂×Dic₄₃	—	4	2	no	—
56	C₂×Dic₄₃	—	4	2	no	—
57	C₂×Dic₄₃	—	4	2	no	—
58	Dic₈₆	—	2	2	no	—
59	Dic₈₆	—	2	4	no	—
60	Dic₈₆	C₄²	1	4	yes	complemented
61	C₈×C₄₃	—	2	4	no	—
62	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄₃	C₄×D₄	1	2	yes	—
64	C₂²×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₂²×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂²×C₄₃	#135	1	2	yes	—
67	C₄×C₄₃	—	2	2	no	—
68	C₄×C₄₃	#103	1	4	yes	complemented
69	C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
70	C₄×C₄₃	#117	1	1	yes	characteristic
71	C₄×C₄₃	C₄×D₄	1	2	yes	—
72	Dic₄₃	—	2	4	no	—
73	Dic₄₃	—	4	2	no	—
74	Dic₄₃	—	4	4	no	—
75	C₄×#103	—	43	1	no	maximal, Sylow
76	C₂×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄₃	C₂×#103	1	1	yes	characteristic
78	C₂×C₄₃	#416	1	1	yes	characteristic
79	C₂×C₄₃	C₄×SD₁₆	1	2	yes	—
80	C₂×C₄₃	C₄×Q₁₆	1	2	yes	—
81	C₂×C₄×Q₈	—	43	1	no	—
82	C₂×#103	—	43	2	no	—
83	C₂×C₄×C₈	D₄₃	1	1	yes	characteristic
84	C₄×C₄⋊C₄	—	43	1	no	—
85	#340	—	43	2	no	—
86	C₄₃	C₄×#103	1	1	yes	characteristic, complemented, Sylow
87	C₂²×Q₈	—	43	1	no	—
88	C₂²×C₈	C₂×D₄₃	1	2	yes	—
89	C₄×Q₈	—	43	4	no	—
90	C₄×Q₈	—	86	2	no	—
91	C₂×C₄⋊C₄	—	43	1	no	—
92	C₂×C₄⋊C₄	—	43	2	no	—
93	C₂×C₄⋊C₄	—	86	1	no	—
94	C₂×C₄²	—	86	1	no	—
95	C₂×C₄²	—	86	1	no	—
96	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
97	#103	—	43	8	no	—
98	C₄×C₈	—	2	2	no	—
99	#95	—	86	1	no	—
100	C₂×Q₈	—	43	4	no	—
101	C₂×Q₈	—	43	2	no	—
102	C₂×Q₈	—	86	2	no	—
103	C₂×Q₈	—	86	1	no	—
104	C₂²×C₄	—	86	1	no	—
105	C₂²×C₄	—	86	2	no	—
106	C₂²×C₄	—	86	1	no	—
107	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
108	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	C₄×D₄₃	1	4	yes	—
113	C₄⋊C₄	—	43	4	no	—
114	C₄⋊C₄	—	43	2	no	—
115	C₄⋊C₄	—	86	4	no	—
116	C₄⋊C₄	—	86	2	no	—
117	C₄⋊C₄	—	86	1	no	—
118	C₄²	—	86	4	no	—
119	C₄²	—	172	2	no	—
120	C₄²	C₄×D₄₃	1	2	yes	—
121	C₄²	C₂²⋊D₄₃	1	2	yes	—
122	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
123	Q₈	—	43	4	no	—
124	Q₈	—	86	2	no	—
125	Q₈	—	86	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	86	2	no	—
128	C₂×C₄	—	86	4	no	—
129	C₂×C₄	—	86	2	no	—
130	C₂×C₄	—	86	4	no	—
131	C₂×C₄	—	172	2	no	—
132	C₂×C₄	—	172	2	no	—
133	C₂×C₄	—	172	2	no	—
134	C₂×C₄	#10621873	1	2	yes	—
135	C₂×C₄	Dic₁₇₂	1	2	yes	—
136	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
137	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
138	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
140	C₂×C₄	#10621898	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
144	C₂²	C₄×D₁₇₂	1	2	yes	—
145	C₂²	C₂×#10621873	1	1	yes	characteristic
146	C₂²	C₂×Dic₁₇₂	1	1	yes	characteristic
147	C₂²	#13063347	1	2	yes	—
148	C₄	—	2	2	no	—
149	C₄	—	86	4	no	—
150	C₄	—	172	2	no	—
151	C₄	—	172	4	no	—
152	C₄	#13063250	1	4	yes	direct product factor
153	C₄	C₄²×D₄₃	1	1	yes	characteristic
154	C₄	#13063282	1	1	yes	characteristic
155	C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
156	C₂	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
157	C₂	C₂×#13063250	1	1	yes	characteristic
158	C₂	#14582896	1	1	yes	characteristic
159	C₂	C₄×#10621873	1	2	yes	—
160	C₂	C₄×Dic₁₇₂	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₈₆

C₂

yes

maximal, characteristic

C₂×#13063250

C₂

yes

maximal

C₄×C₄⋊Dic₄₃

C₂

yes

maximal, characteristic

#14582662

C₂

yes

maximal

C₂×C₄×C₈×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

C₄×Dic₈₆

—

C₄×Dic₈₆

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

—

C₂×C₄×Dic₄₃

—

#13063250

C₄

yes

direct product factor

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×Dic₈₆

C₂×C₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

D₄

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

Q₁₆

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

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

C₄²

yes

complemented

C₈×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

C₄×D₄

yes

—

C₂²×C₄₃

C₂×SD₁₆

yes

characteristic

C₂²×C₄₃

C₂×Q₁₆

yes

characteristic

C₂²×C₄₃

#135

yes

—

C₄×C₄₃

—

C₄×C₄₃

#103

yes

complemented

C₄×C₄₃

C₂×C₄²

yes

characteristic

C₄×C₄₃

#117

yes

characteristic

C₄×C₄₃

C₄×D₄

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

C₄×#103

—

maximal, Sylow

C₂×C₄₃

C₄×C₂²⋊C₄

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

C₄×SD₁₆

yes

—

C₂×C₄₃

C₄×Q₁₆

yes

—

C₂×C₄×Q₈

—

C₂×#103

—

C₂×C₄×C₈

D₄₃

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₄₃

C₄×#103

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂×D₄₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

#103

—

C₄×C₈

—

#95

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

D₁₇₂

yes

characteristic

108

C₂²×C₄

D₁₇₂

yes

characteristic

109

C₂²×C₄

C₂²×D₄₃

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₄×D₄₃

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

—

119

C₄²

—

172

—

120

C₄²

C₄×D₄₃

yes

—

121

C₄²

C₂²⋊D₄₃

yes

—

122

C₂³

C₂×D₁₇₂

yes

characteristic

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

172

—

132

C₂×C₄

—

172

—

133

C₂×C₄

—

172

—

134

C₂×C₄

#10621873

yes

—

135

C₂×C₄

Dic₁₇₂

yes

—

136

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

137

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

138

C₂×C₄

C₂×C₄×D₄₃

yes

characteristic

139

C₂×C₄

C₂×C₄×D₄₃

yes

—

140

C₂×C₄

#10621898

yes

—

141

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

142

C₈

—

143

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

144

C₂²

C₄×D₁₇₂

yes

—

145

C₂²

C₂×#10621873

yes

characteristic

146

C₂²

C₂×Dic₁₇₂

yes

characteristic

147

C₂²

#13063347

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

172

—

151

C₄

—

172

—

152

C₄

#13063250

yes

direct product factor

153

C₄

C₄²×D₄₃

yes

characteristic

154

C₄

#13063282

yes

characteristic

155

C₄

C₄×C₂²⋊D₄₃

yes

—

156

C₂

C₄×(C₂×D₄₃)⋊C₄

yes

characteristic

157

C₂

C₂×#13063250

yes

characteristic

158

C₂

#14582896

yes

characteristic

159

C₂

C₄×#10621873

yes

—

160

C₂

C₄×Dic₁₇₂

yes

—

	subgroup	G/H
center	C₂²×C₄	D₁₇₂
commutator	C₄×C₄₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×C₄×C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₄₃
hypocenter	C₄₃	C₄×#103
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₁₇₂

commutator

C₄×C₄₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×C₄×C₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₄₃

hypocenter

C₄₃

C₄×#103

LCS

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
8. C₄×Dic₈₆	C₄	semidirect
14. #13063250	C₄	direct (N × H)
60. Dic₈₆	C₄²	semidirect
68. C₄×C₄₃	#103	semidirect
86. C₄₃	C₄×#103	semidirect
152. C₄	#13063250	direct (N × H)

H ≅ G/N

type

8. C₄×Dic₈₆

C₄

semidirect

14. #13063250

C₄

direct (N × H)

60. Dic₈₆

C₄²

semidirect

68. C₄×C₄₃

#103

semidirect

86. C₄₃

C₄×#103

semidirect

152. C₄

#13063250

direct (N × H)

Subgroups — C₄×#13063250 #19134416

Conjugacy classes of subgroups

Complements

Subgroups — C4×#13063250 #19134416

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#13063250 #19134416