Group #19133239 — subgroups

Subgroups — C₂×#14582645 #19133239

Order 5504.

256 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 125 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⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14582645	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₃⋊C₈	—	2	1	no	—
6	C₂²×C₄₃⋊C₈	—	2	1	no	—
7	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
8	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
9	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
10	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
11	C₂²×Q₈×C₄₃	—	2	1	no	—
12	C₄⋊Q₈×C₄₃	—	2	1	no	—
13	C₄⋊Q₈×C₄₃	—	2	1	no	—
14	C₄⋊Q₈×C₄₃	C₄	1	2	yes	—
15	C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
17	C₂×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
18	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₄₃	—	2	2	no	—
20	C₂×Q₈×C₄₃	—	2	2	no	—
21	C₂×Q₈×C₄₃	—	4	2	no	—
22	C₂×Q₈×C₄₃	—	4	2	no	—
23	C₂²×C₄×C₄₃	—	2	2	no	—
24	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₄₃	—	2	2	no	—
28	C₄⋊C₄×C₄₃	—	2	2	no	—
29	C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
31	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₄₃	D₄	1	1	yes	characteristic
33	C₄²×C₄₃	D₄	1	1	yes	characteristic
34	C₄²×C₄₃	C₂³	1	1	yes	characteristic
35	C₂×C₄₃⋊C₈	—	2	2	no	—
36	C₂×C₄₃⋊C₈	—	2	2	no	—
37	C₂×C₄₃⋊C₈	—	4	2	no	—
38	C₂×C₄₃⋊C₈	—	4	2	no	—
39	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
40	Q₈×C₄₃	—	4	4	no	—
41	Q₈×C₄₃	—	4	2	no	—
42	Q₈×C₄₃	—	4	2	no	—
43	C₂×C₄×C₄₃	—	2	2	no	—
44	C₂×C₄×C₄₃	—	2	4	no	—
45	C₂×C₄×C₄₃	—	4	4	no	—
46	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₄₃	D₈	1	2	yes	—
50	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
51	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
52	C₂×C₄×C₄₃	Q₁₆	1	2	yes	—
53	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₄₃⋊C₈	—	4	2	no	—
57	C₄₃⋊C₈	—	4	2	no	—
58	C₂²×C₄₃	#101	1	2	yes	—
59	C₂²×C₄₃	#102	1	2	yes	—
60	C₂²×C₄₃	#103	1	2	yes	—
61	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₄₃	—	2	1	no	—
63	C₄×C₄₃	—	2	1	no	—
64	C₄×C₄₃	—	4	4	no	—
65	C₄×C₄₃	#102	1	2	yes	—
66	C₄×C₄₃	#103	1	2	yes	—
67	C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
68	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×#332	—	43	1	no	maximal, Sylow
72	C₂×C₄₃	#332	1	4	yes	complemented
73	C₂×C₄₃	C₂×#101	1	1	yes	characteristic
74	C₂×C₄₃	C₂×#102	1	1	yes	characteristic
75	C₂×C₄₃	C₂×#103	1	1	yes	characteristic
76	C₂×C₄⋊Q₈	D₄₃	1	1	yes	characteristic
77	C₂×C₄⋊C₈	—	43	1	no	—
78	C₂×C₄⋊C₈	—	43	1	no	—
79	#332	—	43	4	no	—
80	C₄₃	C₂×#332	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	—	2	1	no	—
82	C₂²×C₈	—	86	1	no	—
83	C₂²×C₈	—	86	1	no	—
84	C₄⋊Q₈	—	2	1	no	—
85	C₄⋊Q₈	—	2	1	no	—
86	C₄⋊Q₈	Dic₄₃	1	2	yes	—
87	C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
88	C₂×C₄⋊C₄	—	2	1	no	—
89	C₂×C₄⋊C₄	Dic₄₃	1	2	yes	—
90	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
91	C₄⋊C₈	—	43	2	no	—
92	C₄⋊C₈	—	43	2	no	—
93	C₄⋊C₈	—	86	1	no	—
94	C₄⋊C₈	—	86	1	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	4	2	no	—
98	C₂×Q₈	—	4	2	no	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
101	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
103	C₂×C₈	—	86	2	no	—
104	C₂×C₈	—	86	2	no	—
105	C₂×C₈	—	172	2	no	—
106	C₂×C₈	—	172	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
111	C₄²	C₂×Dic₄₃	1	1	yes	characteristic
112	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
113	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
114	C₄²	C₂²×D₄₃	1	1	yes	characteristic
115	C₂³	C₂²⋊Dic₄₃	1	1	yes	characteristic
116	Q₈	—	4	4	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	4	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	4	4	no	—
122	C₂×C₄	#10621882	1	2	yes	—
123	C₂×C₄	#10621883	1	2	yes	—
124	C₂×C₄	#10621884	1	2	yes	—
125	C₂×C₄	#10621885	1	2	yes	—
126	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
127	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
128	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
129	C₂×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
132	C₈	—	172	2	no	—
133	C₈	—	172	2	no	—
134	C₂²	#13063244	1	2	yes	—
135	C₂²	#13063246	1	2	yes	—
136	C₂²	#13063249	1	2	yes	—
137	C₂²	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	4	no	—
141	C₄	#13063246	1	2	yes	—
142	C₄	#13063249	1	2	yes	—
143	C₄	C₂×#10621882	1	1	yes	characteristic
144	C₄	C₂×#10621884	1	1	yes	characteristic
145	C₄	C₂×#10621883	1	1	yes	characteristic
146	C₄	C₂×#10621885	1	1	yes	characteristic
147	C₂	#14582645	1	4	yes	direct product factor
148	C₂	C₂×#13063244	1	1	yes	characteristic
149	C₂	C₂×#13063246	1	1	yes	characteristic
150	C₂	C₂×#13063249	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic

#14582645

C₂

yes

maximal, direct product factor

C₂×C₄⋊Q₈×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

—

C₂²×C₄₃⋊C₈

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₂²×Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

C₄

yes

—

C₄⋊Q₈×C₄₃

C₂²

yes

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₄

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

C₂³

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

D₈

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

Q₁₆

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

#101

yes

—

C₂²×C₄₃

#102

yes

—

C₂²×C₄₃

#103

yes

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#102

yes

—

C₄×C₄₃

#103

yes

—

C₄×C₄₃

C₂×D₈

yes

characteristic

C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₄×C₄₃

C₂×Q₁₆

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄⋊Q₈

D₄₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

#332

—

C₄₃

C₂×#332

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₄₃

yes

—

C₄⋊Q₈

C₂×D₄₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₄₃

yes

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

100

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

101

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

102

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

172

—

106

C₂×C₈

—

172

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

C₂×Dic₄₃

yes

—

111

C₄²

C₂×Dic₄₃

yes

characteristic

112

C₄²

C₂²⋊D₄₃

yes

characteristic

113

C₄²

C₂²⋊D₄₃

yes

characteristic

114

C₄²

C₂²×D₄₃

yes

characteristic

115

C₂³

C₂²⋊Dic₄₃

yes

characteristic

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#10621882

yes

—

123

C₂×C₄

#10621883

yes

—

124

C₂×C₄

#10621884

yes

—

125

C₂×C₄

#10621885

yes

—

126

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

127

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

128

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

129

C₂×C₄

C₂²×Dic₄₃

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

132

C₈

—

172

—

133

C₈

—

172

—

134

C₂²

#13063244

yes

—

135

C₂²

#13063246

yes

—

136

C₂²

#13063249

yes

—

137

C₂²

C₂×C₂²⋊Dic₄₃

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#13063246

yes

—

142

C₄

#13063249

yes

—

143

C₄

C₂×#10621882

yes

characteristic

144

C₄

C₂×#10621884

yes

characteristic

145

C₄

C₂×#10621883

yes

characteristic

146

C₄

C₂×#10621885

yes

characteristic

147

C₂

#14582645

yes

direct product factor

148

C₂

C₂×#13063244

yes

characteristic

149

C₂

C₂×#13063246

yes

characteristic

150

C₂

C₂×#13063249

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₄
Frattini	C₄²	C₂²×D₄₃
Fitting	C₂×C₄⋊Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²⋊C₄
hypercenter	C₂×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	C₂×#332
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂²⋊Dic₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₄

Frattini

C₄²

C₂²×D₄₃

Fitting

C₂×C₄⋊Q₈×C₄₃

C₂

socle

C₂³×C₄₃

C₂²⋊C₄

hypercenter

C₂×C₄⋊Q₈

D₄₃

hypocenter

C₄₃

C₂×#332

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #14582645	C₂	direct (N × H)
72. C₂×C₄₃	#332	semidirect
80. C₄₃	C₂×#332	semidirect
147. C₂	#14582645	direct (N × H)

H ≅ G/N

type

3. #14582645

C₂

direct (N × H)

72. C₂×C₄₃

#332

semidirect

80. C₄₃

C₂×#332

semidirect

147. C₂

#14582645

direct (N × H)

Subgroups — C₂×#14582645 #19133239

Conjugacy classes of subgroups

Complements

Subgroups — C2×#14582645 #19133239

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14582645 #19133239