Group #14581075 — subgroups

Subgroups — #14581075

Order 2624.

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	#13050160	C₂	1	1	yes	maximal, characteristic, complemented
2	#13050157	C₂	1	2	yes	maximal, complemented
3	#13050146	C₂	1	2	yes	maximal, complemented
4	#13050145	C₂	1	1	yes	maximal, characteristic, complemented
5	#13050137	C₂	1	2	yes	maximal, complemented
6	#13050136	C₂	1	2	yes	maximal, complemented
7	#13050135	C₂	1	2	yes	maximal, complemented
8	#13050133	C₂	1	2	yes	maximal, complemented
9	#120×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₄₁	C₂²	1	2	yes	—
12	C₂×D₄×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₄₁	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₄₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₄₁	C₂²	1	2	yes	—
16	C₂×C₂²⋊D₄₁	C₂²	1	2	yes	complemented
17	C₂²×Dic₄₁	C₂²	1	1	yes	characteristic, complemented
18	C₂×D₁₆₄	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₄×D₄₁	C₂²	1	2	yes	complemented
20	C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
21	C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
22	C₂²⋊Dic₄₁	C₂²	1	2	yes	—
23	C₂²⋊Dic₄₁	C₂²	1	2	yes	—
24	C₂²⋊Dic₄₁	C₂²	1	2	yes	—
25	C₂²⋊Dic₄₁	C₂²	1	1	yes	characteristic, complemented
26	#10620281	C₂²	1	2	yes	complemented
27	C₄⋊Dic₄₁	C₂²	1	2	yes	complemented
28	#10620279	C₂²	1	2	yes	complemented
29	#10620279	C₂²	1	2	yes	complemented
30	C₄×Dic₄₁	C₂²	1	1	yes	characteristic, complemented
31	C₂³×C₄₁	—	4	2	no	—
32	C₂³×C₄₁	—	4	1	no	—
33	C₂³×C₄₁	C₂³	1	2	yes	—
34	C₂³×C₄₁	C₂³	1	1	yes	characteristic
35	C₂³×C₄₁	C₂³	1	1	yes	characteristic
36	C₂²×D₄₁	C₂³	1	2	yes	—
37	D₄×C₄₁	—	4	2	no	—
38	D₄×C₄₁	—	4	1	no	—
39	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₄₁	C₂³	1	2	yes	—
41	C₂²⋊D₄₁	—	4	2	no	—
42	C₂²⋊D₄₁	—	4	2	no	—
43	C₂²⋊D₄₁	—	4	2	no	—
44	C₂²⋊D₄₁	—	4	2	no	—
45	C₂×Dic₄₁	—	4	1	no	—
46	C₂×Dic₄₁	C₂³	1	2	yes	—
47	C₂×Dic₄₁	C₂³	1	2	yes	—
48	C₂×Dic₄₁	C₂³	1	2	yes	—
49	D₁₆₄	—	4	1	no	—
50	C₄×D₄₁	—	4	2	no	—
51	C₂²×C₄₁	—	2	1	no	—
52	C₂²×C₄₁	—	2	2	no	—
53	C₂²×C₄₁	—	2	2	no	—
54	C₂²×C₄₁	—	2	2	no	—
55	C₂²×C₄₁	—	2	2	no	—
56	C₂²×C₄₁	—	2	2	no	—
57	C₂²×C₄₁	—	2	1	no	—
58	C₂²×C₄₁	—	4	4	no	—
59	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
60	C₂×D₄₁	—	2	2	no	—
61	C₂×D₄₁	—	2	2	no	—
62	C₂×D₄₁	—	2	2	no	—
63	C₄×C₄₁	—	2	2	no	—
64	C₄×C₄₁	—	2	1	no	—
65	Dic₄₁	—	2	2	no	—
66	Dic₄₁	—	2	2	no	—
67	Dic₄₁	—	2	2	no	—
68	C₂×C₄₁	—	4	1	no	—
69	C₂×C₄₁	—	4	2	no	—
70	C₂×C₄₁	—	4	1	no	—
71	C₂×C₄₁	H(5,𝔽₂)	1	2	yes	—
72	C₂×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
73	D₄₁	—	4	2	no	—
74	#560	—	41	1	no	maximal, Sylow
75	C₄₁	#560	1	1	yes	characteristic, complemented, Sylow
76	#127	—	41	1	no	—
77	#126	—	41	2	no	—
78	#123	—	41	1	no	—
79	#123	—	41	2	no	—
80	#121	—	41	2	no	—
81	#121	—	41	2	no	—
82	#121	—	41	2	no	—
83	#120	—	41	2	no	—
84	#120	D₄₁	1	1	yes	characteristic, complemented
85	C₂⁴	C₂×D₄₁	1	1	yes	characteristic
86	C₂×D₄	—	41	2	no	—
87	C₂×D₄	—	41	2	no	—
88	C₂×D₄	—	41	2	no	—
89	C₂×D₄	—	41	2	no	—
90	C₂×D₄	—	41	1	no	—
91	C₂×D₄	C₂×D₄₁	1	2	yes	—
92	C₂×D₄	C₂×D₄₁	1	1	yes	characteristic
93	C₂²×C₄	—	41	1	no	—
94	C₂²×C₄	—	41	2	no	—
95	C₄⋊C₄	—	41	2	no	—
96	C₄⋊C₄	—	41	2	no	—
97	C₄⋊C₄	—	41	2	no	—
98	C₂²⋊C₄	—	41	2	no	—
99	C₂²⋊C₄	—	41	2	no	—
100	C₂²⋊C₄	—	41	2	no	—
101	C₂²⋊C₄	—	41	1	no	—
102	C₂²⋊C₄	—	41	2	no	—
103	C₂²⋊C₄	C₂×D₄₁	1	2	yes	—
104	C₂²⋊C₄	C₂×D₄₁	1	1	yes	characteristic
105	C₄²	—	41	1	no	—
106	C₂³	—	4	2	no	—
107	C₂³	—	4	1	no	—
108	C₂³	—	41	2	no	—
109	C₂³	C₂²×D₄₁	1	2	yes	—
110	C₂³	C₂²×D₄₁	1	1	yes	characteristic
111	C₂³	C₂²×D₄₁	1	1	yes	characteristic
112	D₄	—	4	2	no	—
113	D₄	—	4	1	no	—
114	D₄	—	164	1	no	—
115	D₄	—	164	2	no	—
116	D₄	—	164	2	no	—
117	D₄	—	164	2	no	—
118	D₄	—	164	2	no	—
119	C₂×C₄	—	41	2	no	—
120	C₂×C₄	—	41	2	no	—
121	C₂×C₄	—	41	2	no	—
122	C₂×C₄	—	164	2	no	—
123	C₂×C₄	—	164	1	no	—
124	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₄₁	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	—	82	2	no	—
135	C₂²	—	82	2	no	—
136	C₂²	—	82	2	no	—
137	C₂²	C₂³×D₄₁	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	82	2	no	—
141	C₄	—	82	2	no	—
142	C₄	—	82	2	no	—
143	C₂	—	4	2	no	—
144	C₂	—	4	1	no	—
145	C₂	—	4	1	no	—
146	C₂	—	164	2	no	—
147	C₂	#13050209	1	2	yes	—
148	C₂	#13050209	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13050160

C₂

yes

maximal, characteristic, complemented

#13050157

C₂

yes

maximal, complemented

#13050146

C₂

yes

maximal, complemented

#13050145

C₂

yes

maximal, characteristic, complemented

#13050137

C₂

yes

maximal, complemented

#13050136

C₂

yes

maximal, complemented

#13050135

C₂

yes

maximal, complemented

#13050133

C₂

yes

maximal, complemented

#120×C₄₁

C₂

yes

maximal, characteristic, complemented

C₂⁴×C₄₁

C₂²

yes

characteristic

C₂×D₄×C₄₁

C₂²

yes

—

C₂×D₄×C₄₁

C₂²

yes

characteristic

C₂×C₂²⋊D₄₁

C₂²

yes

—

C₂×C₂²⋊D₄₁

C₂²

yes

—

C₂×C₂²⋊D₄₁

C₂²

yes

—

C₂×C₂²⋊D₄₁

C₂²

yes

complemented

C₂²×Dic₄₁

C₂²

yes

characteristic, complemented

C₂×D₁₆₄

C₂²

yes

characteristic, complemented

C₂×C₄×D₄₁

C₂²

yes

complemented

C₂²⋊C₄×C₄₁

C₂²

yes

—

C₂²⋊C₄×C₄₁

C₂²

yes

characteristic

C₂²⋊Dic₄₁

C₂²

yes

—

C₂²⋊Dic₄₁

C₂²

yes

—

C₂²⋊Dic₄₁

C₂²

yes

—

C₂²⋊Dic₄₁

C₂²

yes

characteristic, complemented

#10620281

C₂²

yes

complemented

C₄⋊Dic₄₁

C₂²

yes

complemented

#10620279

C₂²

yes

complemented

#10620279

C₂²

yes

complemented

C₄×Dic₄₁

C₂²

yes

characteristic, complemented

C₂³×C₄₁

—

C₂³×C₄₁

—

C₂³×C₄₁

C₂³

yes

—

C₂³×C₄₁

C₂³

yes

characteristic

C₂³×C₄₁

C₂³

yes

characteristic

C₂²×D₄₁

C₂³

yes

—

D₄×C₄₁

—

D₄×C₄₁

—

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂×C₄×C₄₁

C₂³

yes

—

C₂²⋊D₄₁

—

C₂²⋊D₄₁

—

C₂²⋊D₄₁

—

C₂²⋊D₄₁

—

C₂×Dic₄₁

—

C₂×Dic₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

—

D₁₆₄

—

C₄×D₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

C₂⁴

yes

characteristic

C₂×D₄₁

—

C₂×D₄₁

—

C₂×D₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

Dic₄₁

—

Dic₄₁

—

Dic₄₁

—

C₂×C₄₁

—

C₂×C₄₁

—

C₂×C₄₁

—

C₂×C₄₁

H(5,𝔽₂)

yes

—

C₂×C₄₁

H(5,𝔽₂)

yes

characteristic

D₄₁

—

#560

—

maximal, Sylow

C₄₁

#560

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

—

—

D₄₁

yes

characteristic, complemented

C₂⁴

C₂×D₄₁

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₁

yes

—

C₂×D₄

C₂×D₄₁

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×D₄₁

yes

—

104

C₂²⋊C₄

C₂×D₄₁

yes

characteristic

105

C₄²

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

C₂²×D₄₁

yes

—

110

C₂³

C₂²×D₄₁

yes

characteristic

111

C₂³

C₂²×D₄₁

yes

characteristic

112

D₄

—

113

D₄

—

114

D₄

—

164

—

115

D₄

—

164

—

116

D₄

—

164

—

117

D₄

—

164

—

118

D₄

—

164

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

164

—

123

C₂×C₄

—

164

—

124

C₂×C₄

C₂²×D₄₁

yes

characteristic

125

C₂×C₄

C₂²×D₄₁

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×D₄₁

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

—

164

—

147

C₂

#13050209

yes

—

148

C₂

#13050209

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator, socle, cosocle	C₂²×C₄₁	C₂⁴
Fitting	#120×C₄₁	C₂
hypercenter	#120	D₄₁
hypocenter	C₄₁	#560
LCS	G ▷ C₂²×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ #120

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₁

commutator, socle, cosocle

C₂²×C₄₁

C₂⁴

Fitting

#120×C₄₁

C₂

D₄₁

C₄₁

G ▷ C₂²×C₄₁ ▷ C₄₁

C₁ ◁ C₂² ◁ #120

N	H ≅ G/N	type
1. #13050160	C₂	semidirect
2. #13050157	C₂	semidirect
3. #13050146	C₂	semidirect
4. #13050145	C₂	semidirect
5. #13050137	C₂	semidirect
6. #13050136	C₂	semidirect
7. #13050135	C₂	semidirect
8. #13050133	C₂	semidirect
9. #120×C₄₁	C₂	semidirect
16. C₂×C₂²⋊D₄₁	C₂²	semidirect
17. C₂²×Dic₄₁	C₂²	semidirect
18. C₂×D₁₆₄	C₂²	semidirect
19. C₂×C₄×D₄₁	C₂²	semidirect
25. C₂²⋊Dic₄₁	C₂²	semidirect
26. #10620281	C₂²	semidirect
27. C₄⋊Dic₄₁	C₂²	semidirect
28. #10620279	C₂²	semidirect
29. #10620279	C₂²	semidirect
30. C₄×Dic₄₁	C₂²	semidirect
75. C₄₁	#560	semidirect
84. #120	D₄₁	semidirect

H ≅ G/N

type

1. #13050160

C₂

semidirect

2. #13050157

C₂

semidirect

3. #13050146

C₂

semidirect

4. #13050145

C₂

semidirect

5. #13050137

C₂

semidirect

6. #13050136

C₂

semidirect

7. #13050135

C₂

semidirect

8. #13050133

C₂

semidirect

9. #120×C₄₁

C₂

semidirect

16. C₂×C₂²⋊D₄₁

C₂²

semidirect

17. C₂²×Dic₄₁

C₂²

semidirect

18. C₂×D₁₆₄

C₂²

semidirect

19. C₂×C₄×D₄₁

C₂²

semidirect

25. C₂²⋊Dic₄₁

C₂²

semidirect

26. #10620281

C₂²

semidirect

27. C₄⋊Dic₄₁

C₂²

semidirect

28. #10620279

C₂²

semidirect

29. #10620279

C₂²

semidirect

30. C₄×Dic₄₁

C₂²

semidirect

75. C₄₁

#560

semidirect

84. #120

D₄₁

semidirect

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

subgroup	index
#13050160	2
#13050157	2
#13050146	2
#13050145	2
#13050137	2
#13050136	2
#13050135	2
#13050133	2
#120×C₄₁	2
#560	41

subgroup

index

#120×C₄₁

Subgroups — #14581075

Conjugacy classes of subgroups

Complements

Maximal subgroups