Group #19148709 — subgroups

Subgroups — C₂×#14583636 #19148709

Order 5504.

1344 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 469 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	#14583636	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13063319	C₂	1	2	yes	maximal, complemented
3	C₂×#13063306	C₂	1	3	yes	maximal, complemented
4	C₂×#13063297	C₂	1	6	yes	maximal, complemented
5	C₂×#13063293	C₂	1	3	yes	maximal, complemented
6	C₂×#127×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×D₄₃	C₂²	1	1	yes	characteristic
8	C₂²×C₂²⋊D₄₃	C₂²	1	6	yes	—
9	C₂³×Dic₄₃	C₂²	1	3	yes	complemented
10	#13063319	C₂²	1	16	yes	complemented
11	#13063306	C₂²	1	24	yes	complemented
12	#13063297	C₂²	1	48	yes	complemented
13	#13063293	C₂²	1	24	yes	complemented
14	C₂×Dic₄₃⋊C₄	C₂²	1	6	yes	complemented
15	C₂×C₂²⋊Dic₄₃	C₂²	1	3	yes	—
16	C₂×C₂²⋊Dic₄₃	C₂²	1	3	yes	complemented
17	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	6	yes	—
18	C₂²×D₄×C₄₃	C₂²	1	3	yes	—
19	C₂²×D₄×C₄₃	C₂²	1	3	yes	complemented
20	#127×C₄₃	C₂²	1	8	yes	complemented
21	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic, complemented
22	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
23	C₂⁴×C₄₃	C₂³	1	3	yes	—
24	C₂³×D₄₃	—	4	6	no	—
25	C₂³×D₄₃	C₂³	1	2	yes	—
26	C₂³×D₄₃	C₂³	1	4	yes	—
27	C₂×D₄×C₄₃	—	4	6	no	—
28	C₂×D₄×C₄₃	—	4	6	no	—
29	C₂×D₄×C₄₃	C₂³	1	12	yes	—
30	C₂×D₄×C₄₃	C₂³	1	12	yes	complemented
31	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
32	C₂×C₂²⋊D₄₃	—	4	12	no	—
33	C₂×C₂²⋊D₄₃	C₂³	1	24	yes	—
34	C₂²×Dic₄₃	—	4	6	no	—
35	C₂²×Dic₄₃	C₂³	1	6	yes	—
36	C₂²×Dic₄₃	C₂³	1	12	yes	complemented
37	C₄²×C₄₃	C₂³	1	4	yes	complemented
38	C₂²⋊Dic₄₃	C₂³	1	12	yes	—
39	C₂²⋊Dic₄₃	C₂³	1	12	yes	complemented
40	(C₂×D₄₃)⋊C₄	C₂³	1	24	yes	—
41	Dic₄₃⋊C₄	C₂³	1	24	yes	complemented
42	C₂³×C₄₃	—	2	6	no	—
43	C₂³×C₄₃	—	2	12	no	—
44	C₂³×C₄₃	—	2	6	no	—
45	C₂³×C₄₃	C₂⁴	1	2	yes	—
46	C₂³×C₄₃	C₂⁴	1	6	yes	—
47	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
48	C₂²×D₄₃	—	2	12	no	—
49	C₂²×D₄₃	—	4	12	no	—
50	C₂²×D₄₃	—	4	16	no	—
51	C₂²×D₄₃	C₂⁴	1	4	yes	—
52	D₄×C₄₃	—	4	12	no	—
53	D₄×C₄₃	—	4	12	no	—
54	C₂×C₄×C₄₃	—	2	6	no	—
55	C₂×C₄×C₄₃	C₂⁴	1	6	yes	—
56	C₂²⋊D₄₃	—	4	24	no	—
57	C₂×Dic₄₃	—	2	12	no	—
58	C₂×Dic₄₃	—	4	12	no	—
59	C₂×Dic₄₃	C₂⁴	1	12	yes	—
60	C₂²×C₄₃	—	2	6	no	—
61	C₂²×C₄₃	—	2	6	no	—
62	C₂²×C₄₃	—	2	12	no	—
63	C₂²×C₄₃	—	4	4	no	—
64	C₂²×C₄₃	—	4	12	no	—
65	C₂²×C₄₃	H(5,𝔽₂)	1	6	yes	—
66	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
67	C₂×D₄₃	—	2	12	no	—
68	C₂×D₄₃	—	4	16	no	—
69	C₂×D₄₃	—	4	8	no	—
70	C₄×C₄₃	—	2	6	no	—
71	Dic₄₃	—	2	12	no	—
72	C₂×#560	—	43	1	no	maximal, Sylow
73	C₂×C₄₃	—	4	2	no	—
74	C₂×C₄₃	—	4	6	no	—
75	C₂×C₄₃	#560	1	4	yes	complemented
76	C₂×C₄₃	C₂×H(5,𝔽₂)	1	3	yes	—
77	D₄₃	—	4	4	no	—
78	#560	—	43	16	no	—
79	C₂×#127	D₄₃	1	1	yes	characteristic, complemented
80	C₂×#126	—	43	2	no	—
81	C₂×#123	—	43	3	no	—
82	C₂×#121	—	43	6	no	—
83	C₂×#120	—	43	3	no	—
84	C₄₃	C₂×#560	1	1	yes	characteristic, complemented, Sylow
85	C₂⁵	—	43	1	no	—
86	C₂²×D₄	—	43	6	no	—
87	C₂²×D₄	C₂×D₄₃	1	3	yes	—
88	C₂²×D₄	C₂×D₄₃	1	3	yes	complemented
89	C₂³×C₄	—	43	3	no	—
90	#127	C₂×D₄₃	1	8	yes	complemented
91	#126	—	43	16	no	—
92	#123	—	43	24	no	—
93	#121	—	43	48	no	—
94	#120	—	43	24	no	—
95	C₂×C₄⋊C₄	—	43	6	no	—
96	C₂×C₂²⋊C₄	—	43	3	no	—
97	C₂×C₂²⋊C₄	—	43	3	no	—
98	C₂×C₂²⋊C₄	—	43	6	no	—
99	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic, complemented
100	C₂⁴	—	43	2	no	—
101	C₂⁴	—	43	4	no	—
102	C₂⁴	—	172	6	no	—
103	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
104	C₂⁴	C₂²×D₄₃	1	3	yes	—
105	C₂×D₄	—	4	6	no	—
106	C₂×D₄	—	4	6	no	—
107	C₂×D₄	—	43	24	no	—
108	C₂×D₄	—	172	12	no	—
109	C₂×D₄	C₂²×D₄₃	1	12	yes	—
110	C₂×D₄	C₂²×D₄₃	1	12	yes	complemented
111	C₂²×C₄	—	43	6	no	—
112	C₂²×C₄	—	43	12	no	—
113	C₂²×C₄	—	172	6	no	—
114	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
115	C₄⋊C₄	—	43	24	no	—
116	C₂²⋊C₄	—	43	12	no	—
117	C₂²⋊C₄	—	43	12	no	—
118	C₂²⋊C₄	—	43	24	no	—
119	C₄²	C₂²×D₄₃	1	4	yes	complemented
120	C₂³	—	2	6	no	—
121	C₂³	—	2	12	no	—
122	C₂³	—	2	6	no	—
123	C₂³	—	43	4	no	—
124	C₂³	—	86	12	no	—
125	C₂³	—	172	12	no	—
126	C₂³	—	172	16	no	—
127	C₂³	C₂³×D₄₃	1	2	yes	—
128	C₂³	C₂³×D₄₃	1	6	yes	—
129	C₂³	C₂³×D₄₃	1	1	yes	characteristic
130	D₄	—	4	12	no	—
131	D₄	—	4	12	no	—
132	D₄	—	172	24	no	—
133	C₂×C₄	—	2	6	no	—
134	C₂×C₄	—	43	12	no	—
135	C₂×C₄	—	86	12	no	—
136	C₂×C₄	—	172	12	no	—
137	C₂×C₄	C₂³×D₄₃	1	6	yes	—
138	C₂²	—	2	6	no	—
139	C₂²	—	2	12	no	—
140	C₂²	—	2	6	no	—
141	C₂²	—	4	4	no	—
142	C₂²	—	4	12	no	—
143	C₂²	—	86	12	no	—
144	C₂²	—	172	16	no	—
145	C₂²	—	172	8	no	—
146	C₂²	#13063370	1	6	yes	—
147	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
148	C₄	—	2	6	no	—
149	C₄	—	86	12	no	—
150	C₂	—	4	2	no	—
151	C₂	—	4	6	no	—
152	C₂	—	172	4	no	—
153	C₂	#14583636	1	4	yes	direct product factor
154	C₂	C₂×#13063370	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14583636

C₂

yes

maximal, direct product factor

C₂×#13063319

C₂

yes

maximal, complemented

C₂×#13063306

C₂

yes

maximal, complemented

C₂×#13063297

C₂

yes

maximal, complemented

C₂×#13063293

C₂

yes

maximal, complemented

C₂×#127×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₄₃

C₂²

yes

characteristic

C₂²×C₂²⋊D₄₃

C₂²

yes

—

C₂³×Dic₄₃

C₂²

yes

complemented

#13063319

C₂²

yes

complemented

#13063306

C₂²

yes

complemented

#13063297

C₂²

yes

complemented

#13063293

C₂²

yes

complemented

C₂×Dic₄₃⋊C₄

C₂²

yes

complemented

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

complemented

C₂×(C₂×D₄₃)⋊C₄

C₂²

yes

—

C₂²×D₄×C₄₃

C₂²

yes

—

C₂²×D₄×C₄₃

C₂²

yes

complemented

#127×C₄₃

C₂²

yes

complemented

C₂×C₄²×C₄₃

C₂²

yes

characteristic, complemented

C₂⁴×C₄₃

C₂³

yes

characteristic

C₂⁴×C₄₃

C₂³

yes

—

C₂³×D₄₃

—

C₂³×D₄₃

C₂³

yes

—

C₂³×D₄₃

C₂³

yes

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

C₂³

yes

—

C₂×D₄×C₄₃

C₂³

yes

complemented

C₂²×C₄×C₄₃

C₂³

yes

—

C₂×C₂²⋊D₄₃

—

C₂×C₂²⋊D₄₃

C₂³

yes

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

complemented

C₄²×C₄₃

C₂³

yes

complemented

C₂²⋊Dic₄₃

C₂³

yes

—

C₂²⋊Dic₄₃

C₂³

yes

complemented

(C₂×D₄₃)⋊C₄

C₂³

yes

—

Dic₄₃⋊C₄

C₂³

yes

complemented

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂⁴

yes

—

C₂³×C₄₃

C₂⁴

yes

—

C₂³×C₄₃

C₂⁴

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

C₂⁴

yes

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂⁴

yes

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂⁴

yes

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₂²×C₄₃

C₂⁵

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

Dic₄₃

—

C₂×#560

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#560

yes

complemented

C₂×C₄₃

C₂×H(5,𝔽₂)

yes

—

D₄₃

—

#560

—

C₂×#127

D₄₃

yes

characteristic, complemented

—

—

—

—

C₄₃

C₂×#560

yes

characteristic, complemented, Sylow

C₂⁵

—

C₂²×D₄

—

C₂²×D₄

C₂×D₄₃

yes

—

C₂²×D₄

C₂×D₄₃

yes

complemented

C₂³×C₄

—

#127

C₂×D₄₃

yes

—

—

—

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₄₃

yes

characteristic, complemented

100

C₂⁴

—

101

C₂⁴

—

102

C₂⁴

—

172

—

103

C₂⁴

C₂²×D₄₃

yes

characteristic

104

C₂⁴

C₂²×D₄₃

yes

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

172

—

109

C₂×D₄

C₂²×D₄₃

yes

—

110

C₂×D₄

C₂²×D₄₃

yes

complemented

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

172

—

114

C₂²×C₄

C₂²×D₄₃

yes

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

C₂²×D₄₃

yes

complemented

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

172

—

126

C₂³

—

172

—

127

C₂³

C₂³×D₄₃

yes

—

128

C₂³

C₂³×D₄₃

yes

—

129

C₂³

C₂³×D₄₃

yes

characteristic

130

D₄

—

131

D₄

—

132

D₄

—

172

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

172

—

137

C₂×C₄

C₂³×D₄₃

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

172

—

145

C₂²

—

172

—

146

C₂²

#13063370

yes

—

147

C₂²

C₂⁴×D₄₃

yes

characteristic

148

C₄

—

149

C₄

—

150

C₂

—

151

C₂

—

152

C₂

—

172

—

153

C₂

#14583636

yes

direct product factor

154

C₂

C₂×#13063370

yes

—

	subgroup	G/H
center	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂⁵
Frattini	C₂²	C₂⁴×D₄₃
Fitting	C₂×#127×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂×#127	D₄₃
hypocenter	C₄₃	C₂×#560
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂³×D₄₃

commutator

C₂²×C₄₃

C₂⁵

Frattini

C₂²

C₂⁴×D₄₃

Fitting

C₂×#127×C₄₃

C₂

socle

C₂³×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. #14583636	C₂	direct (N × H)
2. C₂×#13063319	C₂	semidirect
3. C₂×#13063306	C₂	semidirect
4. C₂×#13063297	C₂	semidirect
5. C₂×#13063293	C₂	semidirect
6. C₂×#127×C₄₃	C₂	semidirect
9. C₂³×Dic₄₃	C₂²	semidirect
10. #13063319	C₂²	semidirect
11. #13063306	C₂²	semidirect
12. #13063297	C₂²	semidirect
13. #13063293	C₂²	semidirect
14. C₂×Dic₄₃⋊C₄	C₂²	semidirect
16. C₂×C₂²⋊Dic₄₃	C₂²	semidirect
19. C₂²×D₄×C₄₃	C₂²	semidirect
20. #127×C₄₃	C₂²	semidirect
21. C₂×C₄²×C₄₃	C₂²	semidirect
30. C₂×D₄×C₄₃	C₂³	semidirect
36. C₂²×Dic₄₃	C₂³	semidirect
37. C₄²×C₄₃	C₂³	semidirect
39. C₂²⋊Dic₄₃	C₂³	semidirect
41. Dic₄₃⋊C₄	C₂³	semidirect
75. C₂×C₄₃	#560	semidirect
79. C₂×#127	D₄₃	semidirect
84. C₄₃	C₂×#560	semidirect
88. C₂²×D₄	C₂×D₄₃	semidirect
90. #127	C₂×D₄₃	semidirect
99. C₂×C₄²	C₂×D₄₃	semidirect
110. C₂×D₄	C₂²×D₄₃	semidirect
119. C₄²	C₂²×D₄₃	semidirect
153. C₂	#14583636	direct (N × H)

H ≅ G/N

type

1. #14583636

C₂

direct (N × H)

2. C₂×#13063319

C₂

semidirect

3. C₂×#13063306

C₂

semidirect

4. C₂×#13063297

C₂

semidirect

5. C₂×#13063293

C₂

semidirect

6. C₂×#127×C₄₃

C₂

semidirect

9. C₂³×Dic₄₃

C₂²

semidirect

10. #13063319

C₂²

semidirect

11. #13063306

C₂²

semidirect

12. #13063297

C₂²

semidirect

13. #13063293

C₂²

semidirect

14. C₂×Dic₄₃⋊C₄

C₂²

semidirect

16. C₂×C₂²⋊Dic₄₃

C₂²

semidirect

19. C₂²×D₄×C₄₃

C₂²

semidirect

20. #127×C₄₃

C₂²

semidirect

21. C₂×C₄²×C₄₃

C₂²

semidirect

30. C₂×D₄×C₄₃

C₂³

semidirect

36. C₂²×Dic₄₃

C₂³

semidirect

37. C₄²×C₄₃

C₂³

semidirect

39. C₂²⋊Dic₄₃

C₂³

semidirect

41. Dic₄₃⋊C₄

C₂³

semidirect

75. C₂×C₄₃

#560

semidirect

79. C₂×#127

D₄₃

semidirect

84. C₄₃

C₂×#560

semidirect

88. C₂²×D₄

C₂×D₄₃

semidirect

90. #127

C₂×D₄₃

semidirect

99. C₂×C₄²

C₂×D₄₃

semidirect

110. C₂×D₄

C₂²×D₄₃

semidirect

119. C₄²

C₂²×D₄₃

semidirect

153. C₂

#14583636

direct (N × H)

Subgroups — C₂×#14583636 #19148709

Conjugacy classes of subgroups

Complements

Subgroups — C2×#14583636 #19148709

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583636 #19148709