Group #19145114 — subgroups

Subgroups — C₂×#14583308 #19145114

Order 5504.

384 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 181 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×#13063325	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063317	C₂	1	1	yes	maximal, characteristic
3	#14583308	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13063259	C₂	1	2	yes	maximal
5	C₂×#13063257	C₂	1	2	yes	maximal
6	C₂×#97×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₈₆	—	2	1	no	—
8	#13063325	—	2	2	no	—
9	#13063325	C₂²	1	4	yes	—
10	#13063317	—	2	2	no	—
11	#13063317	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₄₃	—	2	1	no	—
13	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
16	C₂×Dic₄₃⋊C₄	—	2	1	no	—
17	C₂×Dic₄₃⋊C₄	—	2	1	no	—
18	#13063259	C₂²	1	8	yes	—
19	#13063257	C₂²	1	8	yes	—
20	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
22	#97×C₄₃	C₂²	1	4	yes	—
23	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₃	—	2	1	no	—
27	C₂²×Dic₄₃	—	2	1	no	—
28	C₂²×Dic₄₃	—	2	2	no	—
29	C₂×Dic₈₆	—	2	4	no	—
30	C₂×Dic₈₆	—	4	4	no	—
31	C₂×C₈×C₄₃	Q₈	1	8	yes	complemented
32	C₂×C₈×C₄₃	C₂³	1	4	yes	—
33	C₄²×C₄₃	D₄	1	2	yes	—
34	C₄²×C₄₃	C₂³	1	2	yes	—
35	C₄⋊Dic₄₃	—	2	4	no	—
36	C₄⋊Dic₄₃	—	2	1	no	—
37	C₄⋊Dic₄₃	—	2	1	no	—
38	C₄⋊Dic₄₃	—	2	2	no	—
39	C₄⋊Dic₄₃	C₂³	1	2	yes	—
40	C₄⋊Dic₄₃	C₂³	1	4	yes	—
41	C₄⋊Dic₄₃	C₂³	1	2	yes	—
42	Dic₄₃⋊C₄	—	2	4	no	—
43	Dic₄₃⋊C₄	—	2	4	no	—
44	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₄₃	—	2	2	no	—
46	C₂×C₄×C₄₃	—	2	2	no	—
47	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×Q₈	1	4	yes	—
51	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₄₃	—	2	2	no	—
53	C₂×Dic₄₃	—	2	2	no	—
54	C₂×Dic₄₃	—	2	4	no	—
55	C₂×Dic₄₃	—	4	4	no	—
56	C₂×Dic₄₃	—	4	2	no	—
57	C₂×Dic₄₃	—	4	2	no	—
58	Dic₈₆	—	4	8	no	—
59	C₈×C₄₃	C₂×Q₈	1	8	yes	complemented
60	C₂²×C₄₃	C₄⋊Q₈	1	2	yes	—
61	C₂²×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
62	C₂²×C₄₃	#137	1	2	yes	—
63	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₄₃	—	2	2	no	—
65	C₄×C₄₃	—	2	2	no	—
66	C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
67	C₄×C₄₃	C₂²×Q₈	1	2	yes	—
68	Dic₄₃	—	4	4	no	—
69	Dic₄₃	—	4	2	no	—
70	Dic₄₃	—	4	2	no	—
71	C₂×#501	—	43	1	no	maximal, Sylow
72	C₂×C₄₃	#501	1	4	yes	complemented
73	C₂×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
76	C₂×C₄⋊Q₈	—	43	1	no	—
77	C₂×#125	—	43	1	no	—
78	#501	—	43	8	no	—
79	C₂×#107	—	43	2	no	—
80	C₂×#106	—	43	2	no	—
81	C₂×#97	D₄₃	1	1	yes	characteristic
82	C₄₃	C₂×#501	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	86	1	no	—
84	C₂²×C₈	C₂×D₄₃	1	2	yes	—
85	C₄⋊Q₈	—	43	4	no	—
86	C₄⋊Q₈	—	86	2	no	—
87	#125	—	43	4	no	—
88	#125	—	86	2	no	—
89	C₂×C₄⋊C₄	—	43	1	no	—
90	C₂×C₄⋊C₄	—	43	1	no	—
91	C₂×C₄⋊C₄	—	43	2	no	—
92	C₂×C₄⋊C₄	—	86	1	no	—
93	C₂×C₄⋊C₄	—	86	1	no	—
94	C₂×C₄⋊C₄	—	86	1	no	—
95	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
96	#107	—	43	8	no	—
97	#106	—	43	8	no	—
98	#97	C₂×D₄₃	1	4	yes	—
99	C₂×Q₈	—	86	4	no	—
100	C₂×Q₈	—	172	4	no	—
101	C₂²×C₄	—	86	1	no	—
102	C₂²×C₄	—	86	1	no	—
103	C₂²×C₄	—	86	2	no	—
104	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
105	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
107	C₂×C₈	Dic₈₆	1	8	yes	complemented
108	C₂×C₈	C₂²×D₄₃	1	4	yes	—
109	C₄⋊C₄	—	43	2	no	—
110	C₄⋊C₄	—	43	4	no	—
111	C₄⋊C₄	—	43	2	no	—
112	C₄⋊C₄	—	86	1	no	—
113	C₄⋊C₄	—	86	1	no	—
114	C₄⋊C₄	—	86	2	no	—
115	C₄⋊C₄	—	86	4	no	—
116	C₄⋊C₄	—	86	4	no	—
117	C₄⋊C₄	—	86	4	no	—
118	C₄²	D₁₇₂	1	2	yes	—
119	C₄²	C₂²×D₄₃	1	2	yes	—
120	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
121	Q₈	—	172	8	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	86	2	no	—
125	C₂×C₄	—	86	2	no	—
126	C₂×C₄	—	86	4	no	—
127	C₂×C₄	—	172	4	no	—
128	C₂×C₄	—	172	2	no	—
129	C₂×C₄	—	172	2	no	—
130	C₂×C₄	C₂×Dic₈₆	1	4	yes	—
131	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
132	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
133	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
135	C₈	C₂×Dic₈₆	1	8	yes	complemented
136	C₂²	#13063325	1	2	yes	—
137	C₂²	#13063353	1	2	yes	—
138	C₂²	#13063358	1	2	yes	—
139	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	172	4	no	—
143	C₄	—	172	2	no	—
144	C₄	—	172	2	no	—
145	C₄	#13063325	1	2	yes	—
146	C₄	C₂²×Dic₈₆	1	2	yes	—
147	C₂	#14583308	1	4	yes	direct product factor
148	C₂	C₂×#13063325	1	1	yes	characteristic
149	C₂	C₂×#13063353	1	1	yes	characteristic
150	C₂	C₂×#13063358	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063325

C₂

yes

maximal, characteristic

C₂×#13063317

C₂

yes

maximal, characteristic

#14583308

C₂

yes

maximal, direct product factor

C₂×#13063259

C₂

yes

maximal

C₂×#13063257

C₂

yes

maximal

C₂×#97×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

—

#13063325

—

#13063325

C₂²

yes

—

#13063317

—

#13063317

C₂²

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

—

C₂×Dic₄₃⋊C₄

—

#13063259

C₂²

yes

—

#13063257

C₂²

yes

—

C₂²×C₈×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#97×C₄₃

C₂²

yes

—

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×C₈×C₄₃

Q₈

yes

complemented

C₂×C₈×C₄₃

C₂³

yes

—

C₄²×C₄₃

D₄

yes

—

C₄²×C₄₃

C₂³

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂³

yes

—

C₄⋊Dic₄₃

C₂³

yes

—

C₄⋊Dic₄₃

C₂³

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₂³×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

C₂×Q₈

yes

complemented

C₂²×C₄₃

C₄⋊Q₈

yes

—

C₂²×C₄₃

AGL(1,ℤ/8)

yes

—

C₂²×C₄₃

#137

yes

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄⋊Q₈

yes

—

C₄×C₄₃

C₂²×Q₈

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

C₂×C₄⋊Q₈

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄⋊Q₈

—

—

—

—

—

C₂×#97

D₄₃

yes

characteristic

C₄₃

C₂×#501

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂×D₄₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₃

yes

—

—

C₂×D₄₃

yes

—

C₂×Q₈

—

100

C₂×Q₈

—

172

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

D₁₇₂

yes

characteristic

105

C₂²×C₄

D₁₇₂

yes

characteristic

106

C₂²×C₄

C₂²×D₄₃

yes

characteristic

107

C₂×C₈

Dic₈₆

yes

complemented

108

C₂×C₈

C₂²×D₄₃

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

D₁₇₂

yes

—

119

C₄²

C₂²×D₄₃

yes

—

120

C₂³

C₂×D₁₇₂

yes

characteristic

121

Q₈

—

172

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

172

—

128

C₂×C₄

—

172

—

129

C₂×C₄

—

172

—

130

C₂×C₄

C₂×Dic₈₆

yes

—

131

C₂×C₄

C₂×D₁₇₂

yes

—

132

C₂×C₄

C₂×D₁₇₂

yes

—

133

C₂×C₄

C₂×D₁₇₂

yes

characteristic

134

C₂×C₄

C₂³×D₄₃

yes

characteristic

135

C₈

C₂×Dic₈₆

yes

complemented

136

C₂²

#13063325

yes

—

137

C₂²

#13063353

yes

—

138

C₂²

#13063358

yes

—

139

C₂²

C₂²×D₁₇₂

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

172

—

143

C₄

—

172

—

144

C₄

—

172

—

145

C₄

#13063325

yes

—

146

C₄

C₂²×Dic₈₆

yes

—

147

C₂

#14583308

yes

direct product factor

148

C₂

C₂×#13063325

yes

characteristic

149

C₂

C₂×#13063353

yes

characteristic

150

C₂

C₂×#13063358

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₂×#97×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₄
hypercenter	C₂×#97	D₄₃
hypocenter	C₄₃	C₂×#501
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₇₂

commutator

C₂×C₄×C₄₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₂×#97×C₄₃

C₂

socle

C₂³×C₄₃

C₂×D₄

hypercenter

C₂×#97

D₄₃

hypocenter

C₄₃

C₂×#501

LCS

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
3. #14583308	C₂	direct (N × H)
31. C₂×C₈×C₄₃	Q₈	semidirect
59. C₈×C₄₃	C₂×Q₈	semidirect
72. C₂×C₄₃	#501	semidirect
82. C₄₃	C₂×#501	semidirect
107. C₂×C₈	Dic₈₆	semidirect
135. C₈	C₂×Dic₈₆	semidirect
147. C₂	#14583308	direct (N × H)

H ≅ G/N

type

3. #14583308

C₂

direct (N × H)

31. C₂×C₈×C₄₃

Q₈

semidirect

59. C₈×C₄₃

C₂×Q₈

semidirect

72. C₂×C₄₃

#501

semidirect

82. C₄₃

C₂×#501

semidirect

107. C₂×C₈

Dic₈₆

semidirect

135. C₈

C₂×Dic₈₆

semidirect

147. C₂

#14583308

direct (N × H)

Subgroups — C₂×#14583308 #19145114

Conjugacy classes of subgroups

Complements

Subgroups — C2×#14583308 #19145114

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583308 #19145114