Group #19137736 — subgroups

Subgroups — #19137736

Order 5504.

688 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 213 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	#14582802	C₂	1	3	yes	maximal, complemented
2	#14582792	C₂	1	6	yes	maximal, complemented
3	#14582779	C₂	1	3	yes	maximal, complemented
4	#14582755	C₂	1	2	yes	maximal, complemented
5	C₂×#120×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
7	C₂×C₄⋊Dic₄₃	C₂²	1	3	yes	complemented
8	C₂×C₂²⋊Dic₄₃	C₂²	1	6	yes	—
9	C₂×C₂²⋊Dic₄₃	C₂²	1	6	yes	—
10	C₂×C₂²⋊Dic₄₃	C₂²	1	3	yes	complemented
11	C₂×C₄×Dic₄₃	C₂²	1	3	yes	complemented
12	C₄₃⋊#95	C₂²	1	6	yes	complemented
13	C₂⁵×C₄₃	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₄₃	C₂²	1	3	yes	—
15	#120×C₄₃	C₄	1	8	yes	complemented
16	C₂×C₂²⋊C₄×C₄₃	C₂²	1	3	yes	—
17	C₂⁴×C₄₃	—	4	3	no	—
18	C₂⁴×C₄₃	—	4	3	no	—
19	C₂⁴×C₄₃	C₂×C₄	1	4	yes	complemented
20	C₂⁴×C₄₃	C₂³	1	3	yes	—
21	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
22	C₂×D₄×C₄₃	—	4	3	no	—
23	C₂×D₄×C₄₃	—	4	3	no	—
24	C₂×D₄×C₄₃	C₂×C₄	1	12	yes	—
25	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
26	C₂²×Dic₄₃	—	4	3	no	—
27	C₂²×Dic₄₃	C₂³	1	2	yes	—
28	C₂²×Dic₄₃	C₂³	1	6	yes	—
29	C₂²⋊C₄×C₄₃	C₂×C₄	1	12	yes	complemented
30	C₂²⋊Dic₄₃	—	4	6	no	—
31	C₂²⋊Dic₄₃	—	4	6	no	—
32	C₂²⋊Dic₄₃	—	4	3	no	—
33	C₄⋊Dic₄₃	—	4	3	no	—
34	C₄×Dic₄₃	—	4	3	no	—
35	C₂³×C₄₃	—	2	3	no	—
36	C₂³×C₄₃	—	2	6	no	—
37	C₂³×C₄₃	—	2	6	no	—
38	C₂³×C₄₃	—	2	3	no	—
39	C₂³×C₄₃	—	2	3	no	—
40	C₂³×C₄₃	—	2	3	no	—
41	C₂³×C₄₃	—	4	12	no	—
42	C₂³×C₄₃	—	4	12	no	—
43	C₂³×C₄₃	—	4	4	no	—
44	C₂³×C₄₃	C₂²×C₄	1	6	yes	—
45	C₂³×C₄₃	C₂²×C₄	1	2	yes	—
46	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
47	D₄×C₄₃	—	4	12	no	—
48	C₂×C₄×C₄₃	—	2	3	no	—
49	C₂×C₄×C₄₃	—	2	3	no	—
50	C₂×C₄×C₄₃	C₂²×C₄	1	6	yes	—
51	C₂×Dic₄₃	—	2	6	no	—
52	C₂×Dic₄₃	—	2	6	no	—
53	C₂×Dic₄₃	—	2	6	no	—
54	C₂×Dic₄₃	—	2	6	no	—
55	C₂×Dic₄₃	—	4	4	no	—
56	C₂²×C₄₃	—	2	12	no	—
57	C₂²×C₄₃	—	2	6	no	—
58	C₂²×C₄₃	—	2	6	no	—
59	C₂²×C₄₃	—	4	16	no	—
60	C₂²×C₄₃	—	4	3	no	—
61	C₂²×C₄₃	—	4	3	no	—
62	C₂²×C₄₃	—	4	6	no	—
63	C₂²×C₄₃	—	4	3	no	—
64	C₂²×C₄₃	—	4	1	no	—
65	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
66	C₂²×C₄₃	H(5,𝔽₂)	1	3	yes	—
67	C₂²×C₄₃	H(5,𝔽₂)	1	3	yes	—
68	C₄×C₄₃	—	2	6	no	—
69	Dic₄₃	—	4	6	no	—
70	Dic₄₃	—	4	2	no	—
71	#2375	—	43	1	no	maximal, Sylow
72	C₂×C₄₃	—	4	2	no	—
73	C₂×C₄₃	—	4	6	no	—
74	C₂×C₄₃	#518	1	3	yes	—
75	C₂×C₄₃	#560	1	3	yes	—
76	C₂×C₄₃	H(3,𝔽₄)	1	1	yes	characteristic
77	C₂×#120	D₄₃	1	1	yes	characteristic
78	#390	—	43	3	no	—
79	#388	—	43	6	no	—
80	#386	—	43	3	no	—
81	#379	—	43	2	no	—
82	C₄₃	#2375	1	1	yes	characteristic, complemented, Sylow
83	C₂⁵	C₂×D₄₃	1	1	yes	characteristic
84	C₂²×D₄	C₂×D₄₃	1	3	yes	—
85	C₂³×C₄	—	43	1	no	—
86	#120	Dic₄₃	1	8	yes	complemented
87	C₂×C₄⋊C₄	—	43	3	no	—
88	C₂×C₂²⋊C₄	—	43	6	no	—
89	C₂×C₂²⋊C₄	—	43	6	no	—
90	C₂×C₂²⋊C₄	—	43	3	no	—
91	C₂×C₂²⋊C₄	C₂×D₄₃	1	3	yes	—
92	C₂×C₄²	—	43	3	no	—
93	#95	—	43	6	no	—
94	C₂⁴	—	4	3	no	—
95	C₂⁴	—	4	3	no	—
96	C₂⁴	C₂×Dic₄₃	1	4	yes	complemented
97	C₂⁴	C₂²×D₄₃	1	3	yes	—
98	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
99	C₂×D₄	—	4	3	no	—
100	C₂×D₄	—	4	3	no	—
101	C₂×D₄	C₂×Dic₄₃	1	12	yes	—
102	C₂²×C₄	—	43	2	no	—
103	C₂²×C₄	—	43	6	no	—
104	C₂²×C₄	—	172	3	no	—
105	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
106	C₄⋊C₄	—	172	3	no	—
107	C₂²⋊C₄	—	172	6	no	—
108	C₂²⋊C₄	—	172	3	no	—
109	C₂²⋊C₄	—	172	6	no	—
110	C₂²⋊C₄	C₂×Dic₄₃	1	12	yes	complemented
111	C₄²	—	172	3	no	—
112	C₂³	—	2	3	no	—
113	C₂³	—	2	6	no	—
114	C₂³	—	2	6	no	—
115	C₂³	—	2	3	no	—
116	C₂³	—	2	3	no	—
117	C₂³	—	2	3	no	—
118	C₂³	—	4	12	no	—
119	C₂³	—	4	12	no	—
120	C₂³	—	4	4	no	—
121	C₂³	C₂²×Dic₄₃	1	6	yes	—
122	C₂³	C₂²×Dic₄₃	1	2	yes	—
123	C₂³	C₂³×D₄₃	1	1	yes	characteristic
124	D₄	—	4	12	no	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	2	3	no	—
127	C₂×C₄	—	86	6	no	—
128	C₂×C₄	—	86	6	no	—
129	C₂×C₄	—	86	6	no	—
130	C₂×C₄	—	86	6	no	—
131	C₂×C₄	—	172	4	no	—
132	C₂×C₄	C₂²×Dic₄₃	1	6	yes	—
133	C₂²	—	2	6	no	—
134	C₂²	—	2	6	no	—
135	C₂²	—	2	12	no	—
136	C₂²	—	4	16	no	—
137	C₂²	—	4	3	no	—
138	C₂²	—	4	6	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	3	no	—
142	C₂²	C₂³×Dic₄₃	1	1	yes	characteristic
143	C₂²	#13063370	1	3	yes	—
144	C₂²	#13063370	1	3	yes	—
145	C₄	—	2	6	no	—
146	C₄	—	172	6	no	—
147	C₄	—	172	2	no	—
148	C₂	—	4	6	no	—
149	C₂	—	4	2	no	—
150	C₂	#14583374	1	3	yes	—
151	C₂	#14583638	1	3	yes	—
152	C₂	#14583642	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14582802

C₂

yes

maximal, complemented

#14582792

C₂

yes

maximal, complemented

#14582779

C₂

yes

maximal, complemented

#14582755

C₂

yes

maximal, complemented

C₂×#120×C₄₃

C₂

yes

maximal, characteristic

C₂³×Dic₄₃

C₂²

yes

characteristic, complemented

C₂×C₄⋊Dic₄₃

C₂²

yes

complemented

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

complemented

C₂×C₄×Dic₄₃

C₂²

yes

complemented

C₄₃⋊#95

C₂²

yes

complemented

C₂⁵×C₄₃

C₂²

yes

characteristic

C₂²×D₄×C₄₃

C₂²

yes

—

#120×C₄₃

C₄

yes

complemented

C₂×C₂²⋊C₄×C₄₃

C₂²

yes

—

C₂⁴×C₄₃

—

C₂⁴×C₄₃

—

C₂⁴×C₄₃

C₂×C₄

yes

complemented

C₂⁴×C₄₃

C₂³

yes

—

C₂⁴×C₄₃

C₂³

yes

characteristic

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²⋊C₄×C₄₃

C₂×C₄

yes

complemented

C₂²⋊Dic₄₃

—

C₂²⋊Dic₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄×Dic₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂²×C₄

yes

—

C₂³×C₄₃

C₂²×C₄

yes

—

C₂³×C₄₃

C₂⁴

yes

characteristic

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂³×C₄

yes

characteristic

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₄×C₄₃

—

Dic₄₃

—

Dic₄₃

—

#2375

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#518

yes

—

C₂×C₄₃

#560

yes

—

C₂×C₄₃

H(3,𝔽₄)

yes

characteristic

C₂×#120

D₄₃

yes

—

—

—

—

C₄₃

#2375

yes

characteristic, complemented, Sylow

C₂⁵

C₂×D₄₃

yes

characteristic

C₂²×D₄

C₂×D₄₃

yes

—

C₂³×C₄

—

#120

Dic₄₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₄₃

yes

—

C₂×C₄²

—

#95

—

C₂⁴

—

C₂⁴

—

C₂⁴

C₂×Dic₄₃

yes

complemented

C₂⁴

C₂²×D₄₃

yes

—

C₂⁴

C₂²×D₄₃

yes

characteristic

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

C₂×Dic₄₃

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

172

—

105

C₂²×C₄

C₂²×D₄₃

yes

—

106

C₄⋊C₄

—

172

—

107

C₂²⋊C₄

—

172

—

108

C₂²⋊C₄

—

172

—

109

C₂²⋊C₄

—

172

—

110

C₂²⋊C₄

C₂×Dic₄₃

yes

complemented

111

C₄²

—

172

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

C₂²×Dic₄₃

yes

—

122

C₂³

C₂²×Dic₄₃

yes

—

123

C₂³

C₂³×D₄₃

yes

characteristic

124

D₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

172

—

132

C₂×C₄

C₂²×Dic₄₃

yes

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂³×Dic₄₃

yes

characteristic

143

C₂²

#13063370

yes

—

144

C₂²

#13063370

yes

—

145

C₄

—

146

C₄

—

172

—

147

C₄

—

172

—

148

C₂

—

149

C₂

—

150

C₂

#14583374

yes

—

151

C₂

#14583638

yes

—

152

C₂

#14583642

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₂×#120×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂×#120	D₄₃
hypocenter	C₄₃	#2375
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×#120

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator

C₂²×C₄₃

C₂³×C₄

Fitting

C₂×#120×C₄₃

C₂

socle

C₂³×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₂×#120

N	H ≅ G/N	type
1. #14582802	C₂	semidirect
2. #14582792	C₂	semidirect
3. #14582779	C₂	semidirect
4. #14582755	C₂	semidirect
6. C₂³×Dic₄₃	C₂²	semidirect
7. C₂×C₄⋊Dic₄₃	C₂²	semidirect
10. C₂×C₂²⋊Dic₄₃	C₂²	semidirect
11. C₂×C₄×Dic₄₃	C₂²	semidirect
12. C₄₃⋊#95	C₂²	semidirect
15. #120×C₄₃	C₄	semidirect
19. C₂⁴×C₄₃	C₂×C₄	semidirect
29. C₂²⋊C₄×C₄₃	C₂×C₄	semidirect
82. C₄₃	#2375	semidirect
86. #120	Dic₄₃	semidirect
96. C₂⁴	C₂×Dic₄₃	semidirect
110. C₂²⋊C₄	C₂×Dic₄₃	semidirect

H ≅ G/N

type

1. #14582802

C₂

semidirect

2. #14582792

C₂

semidirect

3. #14582779

C₂

semidirect

4. #14582755

C₂

semidirect

6. C₂³×Dic₄₃

C₂²

semidirect

7. C₂×C₄⋊Dic₄₃

C₂²

semidirect

10. C₂×C₂²⋊Dic₄₃

C₂²

semidirect

11. C₂×C₄×Dic₄₃

C₂²

semidirect

12. C₄₃⋊#95

C₂²

semidirect

15. #120×C₄₃

C₄

semidirect

19. C₂⁴×C₄₃

C₂×C₄

semidirect

29. C₂²⋊C₄×C₄₃

C₂×C₄

semidirect

82. C₄₃

#2375

semidirect

86. #120

Dic₄₃

semidirect

96. C₂⁴

C₂×Dic₄₃

semidirect

110. C₂²⋊C₄

C₂×Dic₄₃

semidirect

Subgroups — #19137736

Conjugacy classes of subgroups

Complements