Group #14583638 — subgroups

Subgroups — #14583638

Order 2752.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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	#13063321	C₂	1	1	yes	maximal, characteristic, complemented
2	#13063318	C₂	1	2	yes	maximal, complemented
3	#13063307	C₂	1	2	yes	maximal, complemented
4	#13063306	C₂	1	1	yes	maximal, characteristic, complemented
5	#13063298	C₂	1	2	yes	maximal, complemented
6	#13063297	C₂	1	2	yes	maximal, complemented
7	#13063296	C₂	1	2	yes	maximal, complemented
8	#13063294	C₂	1	2	yes	maximal, complemented
9	#120×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×C₄₃	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₄₃	C₂²	1	2	yes	—
12	C₂×D₄×C₄₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₄₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₄₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₄₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊D₄₃	C₂²	1	2	yes	complemented
17	C₂²×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
18	C₂×D₁₇₂	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₄×D₄₃	C₂²	1	2	yes	complemented
20	C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
21	C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
22	C₂²⋊Dic₄₃	C₂²	1	2	yes	—
23	C₂²⋊Dic₄₃	C₂²	1	2	yes	—
24	C₂²⋊Dic₄₃	C₂²	1	2	yes	—
25	C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic, complemented
26	(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	complemented
27	C₄⋊Dic₄₃	C₂²	1	2	yes	complemented
28	Dic₄₃⋊C₄	C₂²	1	2	yes	complemented
29	Dic₄₃⋊C₄	C₂²	1	2	yes	complemented
30	C₄×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
31	C₂³×C₄₃	—	4	2	no	—
32	C₂³×C₄₃	—	4	1	no	—
33	C₂³×C₄₃	C₂³	1	2	yes	—
34	C₂³×C₄₃	C₂³	1	1	yes	characteristic
35	C₂³×C₄₃	C₂³	1	1	yes	characteristic
36	C₂²×D₄₃	C₂³	1	2	yes	—
37	D₄×C₄₃	—	4	2	no	—
38	D₄×C₄₃	—	4	1	no	—
39	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₄₃	C₂³	1	2	yes	—
41	C₂²⋊D₄₃	—	4	2	no	—
42	C₂²⋊D₄₃	—	4	2	no	—
43	C₂²⋊D₄₃	—	4	2	no	—
44	C₂²⋊D₄₃	—	4	2	no	—
45	C₂×Dic₄₃	—	4	1	no	—
46	C₂×Dic₄₃	C₂³	1	2	yes	—
47	C₂×Dic₄₃	C₂³	1	2	yes	—
48	C₂×Dic₄₃	C₂³	1	2	yes	—
49	D₁₇₂	—	4	1	no	—
50	C₄×D₄₃	—	4	2	no	—
51	C₂²×C₄₃	—	2	2	no	—
52	C₂²×C₄₃	—	2	2	no	—
53	C₂²×C₄₃	—	2	2	no	—
54	C₂²×C₄₃	—	2	2	no	—
55	C₂²×C₄₃	—	2	1	no	—
56	C₂²×C₄₃	—	2	1	no	—
57	C₂²×C₄₃	—	2	2	no	—
58	C₂²×C₄₃	—	4	4	no	—
59	C₂²×C₄₃	C₂⁴	1	1	yes	characteristic
60	C₂×D₄₃	—	2	2	no	—
61	C₂×D₄₃	—	2	2	no	—
62	C₂×D₄₃	—	2	2	no	—
63	C₄×C₄₃	—	2	2	no	—
64	C₄×C₄₃	—	2	1	no	—
65	Dic₄₃	—	2	2	no	—
66	Dic₄₃	—	2	2	no	—
67	Dic₄₃	—	2	2	no	—
68	C₂×C₄₃	—	4	1	no	—
69	C₂×C₄₃	—	4	2	no	—
70	C₂×C₄₃	—	4	1	no	—
71	C₂×C₄₃	H(5,𝔽₂)	1	2	yes	—
72	C₂×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
73	D₄₃	—	4	2	no	—
74	#560	—	43	1	no	maximal, Sylow
75	C₄₃	#560	1	1	yes	characteristic, complemented, Sylow
76	#127	—	43	1	no	—
77	#126	—	43	2	no	—
78	#123	—	43	1	no	—
79	#123	—	43	2	no	—
80	#121	—	43	2	no	—
81	#121	—	43	2	no	—
82	#121	—	43	2	no	—
83	#120	—	43	2	no	—
84	#120	D₄₃	1	1	yes	characteristic, complemented
85	C₂⁴	C₂×D₄₃	1	1	yes	characteristic
86	C₂×D₄	—	43	2	no	—
87	C₂×D₄	—	43	2	no	—
88	C₂×D₄	—	43	2	no	—
89	C₂×D₄	—	43	2	no	—
90	C₂×D₄	—	43	1	no	—
91	C₂×D₄	C₂×D₄₃	1	2	yes	—
92	C₂×D₄	C₂×D₄₃	1	1	yes	characteristic
93	C₂²×C₄	—	43	1	no	—
94	C₂²×C₄	—	43	2	no	—
95	C₄⋊C₄	—	43	2	no	—
96	C₄⋊C₄	—	43	2	no	—
97	C₄⋊C₄	—	43	2	no	—
98	C₂²⋊C₄	—	43	2	no	—
99	C₂²⋊C₄	—	43	2	no	—
100	C₂²⋊C₄	—	43	2	no	—
101	C₂²⋊C₄	—	43	1	no	—
102	C₂²⋊C₄	—	43	2	no	—
103	C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
104	C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
105	C₄²	—	43	1	no	—
106	C₂³	—	4	2	no	—
107	C₂³	—	4	1	no	—
108	C₂³	—	43	2	no	—
109	C₂³	C₂²×D₄₃	1	2	yes	—
110	C₂³	C₂²×D₄₃	1	1	yes	characteristic
111	C₂³	C₂²×D₄₃	1	1	yes	characteristic
112	D₄	—	4	2	no	—
113	D₄	—	4	1	no	—
114	D₄	—	172	1	no	—
115	D₄	—	172	2	no	—
116	D₄	—	172	2	no	—
117	D₄	—	172	2	no	—
118	D₄	—	172	2	no	—
119	C₂×C₄	—	43	2	no	—
120	C₂×C₄	—	43	2	no	—
121	C₂×C₄	—	43	2	no	—
122	C₂×C₄	—	172	2	no	—
123	C₂×C₄	—	172	1	no	—
124	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₄₃	1	2	yes	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	4	4	no	—
134	C₂²	—	86	2	no	—
135	C₂²	—	86	2	no	—
136	C₂²	—	86	2	no	—
137	C₂²	C₂³×D₄₃	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	86	2	no	—
141	C₄	—	86	2	no	—
142	C₄	—	86	2	no	—
143	C₂	—	4	2	no	—
144	C₂	—	4	1	no	—
145	C₂	—	4	1	no	—
146	C₂	—	172	2	no	—
147	C₂	#13063370	1	2	yes	—
148	C₂	#13063370	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13063321

C₂

yes

maximal, characteristic, complemented

#13063318

C₂

yes

maximal, complemented

#13063307

C₂

yes

maximal, complemented

#13063306

C₂

yes

maximal, characteristic, complemented

#13063298

C₂

yes

maximal, complemented

#13063297

C₂

yes

maximal, complemented

#13063296

C₂

yes

maximal, complemented

#13063294

C₂

yes

maximal, complemented

#120×C₄₃

C₂

yes

maximal, characteristic, complemented

C₂⁴×C₄₃

C₂²

yes

characteristic

C₂×D₄×C₄₃

C₂²

yes

—

C₂×D₄×C₄₃

C₂²

yes

characteristic

C₂×C₂²⋊D₄₃

C₂²

yes

—

C₂×C₂²⋊D₄₃

C₂²

yes

—

C₂×C₂²⋊D₄₃

C₂²

yes

—

C₂×C₂²⋊D₄₃

C₂²

yes

complemented

C₂²×Dic₄₃

C₂²

yes

characteristic, complemented

C₂×D₁₇₂

C₂²

yes

characteristic, complemented

C₂×C₄×D₄₃

C₂²

yes

complemented

C₂²⋊C₄×C₄₃

C₂²

yes

—

C₂²⋊C₄×C₄₃

C₂²

yes

characteristic

C₂²⋊Dic₄₃

C₂²

yes

—

C₂²⋊Dic₄₃

C₂²

yes

—

C₂²⋊Dic₄₃

C₂²

yes

—

C₂²⋊Dic₄₃

C₂²

yes

characteristic, complemented

(C₂×D₄₃)⋊C₄

C₂²

yes

complemented

C₄⋊Dic₄₃

C₂²

yes

complemented

Dic₄₃⋊C₄

C₂²

yes

complemented

Dic₄₃⋊C₄

C₂²

yes

complemented

C₄×Dic₄₃

C₂²

yes

characteristic, complemented

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂³

yes

—

C₂³×C₄₃

C₂³

yes

characteristic

C₂³×C₄₃

C₂³

yes

characteristic

C₂²×D₄₃

C₂³

yes

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

C₂³

yes

characteristic

C₂×C₄×C₄₃

C₂³

yes

—

C₂²⋊D₄₃

—

C₂²⋊D₄₃

—

C₂²⋊D₄₃

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

—

C₂×Dic₄₃

C₂³

yes

—

D₁₇₂

—

C₄×D₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂⁴

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

H(5,𝔽₂)

yes

—

C₂×C₄₃

H(5,𝔽₂)

yes

characteristic

D₄₃

—

#560

—

maximal, Sylow

C₄₃

#560

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

—

—

D₄₃

yes

characteristic, complemented

C₂⁴

C₂×D₄₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₃

yes

—

C₂×D₄

C₂×D₄₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

C₂×D₄₃

yes

—

104

C₂²⋊C₄

C₂×D₄₃

yes

characteristic

105

C₄²

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

C₂²×D₄₃

yes

—

110

C₂³

C₂²×D₄₃

yes

characteristic

111

C₂³

C₂²×D₄₃

yes

characteristic

112

D₄

—

113

D₄

—

114

D₄

—

172

—

115

D₄

—

172

—

116

D₄

—

172

—

117

D₄

—

172

—

118

D₄

—

172

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

172

—

123

C₂×C₄

—

172

—

124

C₂×C₄

C₂²×D₄₃

yes

characteristic

125

C₂×C₄

C₂²×D₄₃

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂³×D₄₃

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

—

172

—

147

C₂

#13063370

yes

—

148

C₂

#13063370

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃
commutator, socle, cosocle	C₂²×C₄₃	C₂⁴
Fitting	#120×C₄₃	C₂
hypercenter	#120	D₄₃
hypocenter	C₄₃	#560
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #120

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₃

commutator, socle, cosocle

C₂²×C₄₃

C₂⁴

Fitting

#120×C₄₃

C₂

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂² ◁ #120

N	H ≅ G/N	type
1. #13063321	C₂	semidirect
2. #13063318	C₂	semidirect
3. #13063307	C₂	semidirect
4. #13063306	C₂	semidirect
5. #13063298	C₂	semidirect
6. #13063297	C₂	semidirect
7. #13063296	C₂	semidirect
8. #13063294	C₂	semidirect
9. #120×C₄₃	C₂	semidirect
16. C₂×C₂²⋊D₄₃	C₂²	semidirect
17. C₂²×Dic₄₃	C₂²	semidirect
18. C₂×D₁₇₂	C₂²	semidirect
19. C₂×C₄×D₄₃	C₂²	semidirect
25. C₂²⋊Dic₄₃	C₂²	semidirect
26. (C₂×D₄₃)⋊C₄	C₂²	semidirect
27. C₄⋊Dic₄₃	C₂²	semidirect
28. Dic₄₃⋊C₄	C₂²	semidirect
29. Dic₄₃⋊C₄	C₂²	semidirect
30. C₄×Dic₄₃	C₂²	semidirect
75. C₄₃	#560	semidirect
84. #120	D₄₃	semidirect

H ≅ G/N

type

1. #13063321

C₂

semidirect

2. #13063318

C₂

semidirect

3. #13063307

C₂

semidirect

4. #13063306

C₂

semidirect

5. #13063298

C₂

semidirect

6. #13063297

C₂

semidirect

7. #13063296

C₂

semidirect

8. #13063294

C₂

semidirect

9. #120×C₄₃

C₂

semidirect

16. C₂×C₂²⋊D₄₃

C₂²

semidirect

17. C₂²×Dic₄₃

C₂²

semidirect

18. C₂×D₁₇₂

C₂²

semidirect

19. C₂×C₄×D₄₃

C₂²

semidirect

25. C₂²⋊Dic₄₃

C₂²

semidirect

26. (C₂×D₄₃)⋊C₄

C₂²

semidirect

27. C₄⋊Dic₄₃

C₂²

semidirect

28. Dic₄₃⋊C₄

C₂²

semidirect

29. Dic₄₃⋊C₄

C₂²

semidirect

30. C₄×Dic₄₃

C₂²

semidirect

75. C₄₃

#560

semidirect

84. #120

D₄₃

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13063321	2
#13063318	2
#13063307	2
#13063306	2
#13063298	2
#13063297	2
#13063296	2
#13063294	2
#120×C₄₃	2
#560	43

subgroup

index

#120×C₄₃

Subgroups — #14583638

Conjugacy classes of subgroups

Complements

Maximal subgroups