Group #14580913 — subgroups

Subgroups — #14580913

Order 2624.

294 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 103 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×#10620308	C₂	1	2	yes	maximal, complemented
2	#13050142	C₂	1	2	yes	maximal, complemented
3	#13050141	C₂	1	2	yes	maximal, complemented
4	#13050138	C₂	1	1	yes	maximal, characteristic, complemented
5	#13050137	C₂	1	2	yes	maximal, complemented
6	#13050136	C₂	1	2	yes	maximal, complemented
7	#13050136	C₂	1	2	yes	maximal, complemented
8	C₂×#10620279	C₂	1	1	yes	maximal, characteristic, complemented
9	#121×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₄×C₄₁	C₂²	1	2	yes	—
11	C₂×D₄×C₄₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄×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	C₂²×Dic₄₁	C₂²	1	2	yes	—
17	C₂²×Dic₄₁	C₂²	1	1	yes	characteristic
18	C₂²×Dic₄₁	C₂²	1	1	yes	characteristic
19	#10620308	—	2	8	no	—
20	C₂×D₁₆₄	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄₁	C₂²	1	2	yes	—
22	C₂×Dic₈₂	C₂²	1	2	yes	—
23	C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
24	C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
25	C₂²⋊Dic₄₁	C₂²	1	2	yes	—
26	#10620281	C₂²	1	2	yes	—
27	#10620281	C₂²	1	2	yes	—
28	C₄⋊Dic₄₁	C₂²	1	1	yes	characteristic
29	#10620279	C₂²	1	2	yes	—
30	#10620279	C₂²	1	2	yes	—
31	#10620279	C₂²	1	2	yes	—
32	C₂³×C₄₁	C₂³	1	2	yes	—
33	C₂³×C₄₁	C₂³	1	1	yes	characteristic
34	C₂²×D₄₁	C₂³	1	2	yes	—
35	D₄×C₄₁	—	2	4	no	—
36	D₄×C₄₁	—	4	1	no	—
37	C₂×C₄×C₄₁	—	4	1	no	—
38	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₄₁	C₂³	1	2	yes	—
41	C₂²⋊D₄₁	—	2	4	no	—
42	C₂²⋊D₄₁	—	2	4	no	—
43	C₂²⋊D₄₁	—	4	2	no	—
44	C₂×Dic₄₁	—	2	4	no	—
45	C₂×Dic₄₁	—	4	1	no	—
46	C₂×Dic₄₁	D₄	1	4	yes	complemented
47	C₂×Dic₄₁	C₂³	1	2	yes	—
48	C₂×Dic₄₁	C₂³	1	2	yes	—
49	C₂×Dic₄₁	C₂³	1	2	yes	—
50	D₁₆₄	—	4	1	no	—
51	C₄×D₄₁	—	2	4	no	—
52	Dic₈₂	—	2	4	no	—
53	C₂²×C₄₁	—	2	1	no	—
54	C₂²×C₄₁	—	2	1	no	—
55	C₂²×C₄₁	—	2	2	no	—
56	C₂²×C₄₁	—	2	2	no	—
57	C₂²×C₄₁	—	2	2	no	—
58	C₂²×C₄₁	C₂×D₄	1	2	yes	—
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	C₄×C₄₁	—	2	1	no	—
66	Dic₄₁	—	2	2	no	—
67	Dic₄₁	—	2	2	no	—
68	Dic₄₁	C₂×D₄	1	4	yes	—
69	C₂×C₄₁	—	2	2	no	—
70	C₂×C₄₁	—	4	2	no	—
71	C₂×C₄₁	C₂²×D₄	1	1	yes	characteristic
72	C₂×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
73	C₂×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
74	D₄₁	—	4	2	no	—
75	#537	—	41	1	no	maximal, Sylow
76	C₄₁	#537	1	1	yes	characteristic, complemented, Sylow
77	C₂×#41	—	41	2	no	—
78	C₂²⋊Q₈	—	41	2	no	—
79	C₂²⋊Q₈	—	41	2	no	—
80	#121	—	41	2	no	—
81	#121	—	41	2	no	—
82	#121	—	41	1	no	—
83	#121	—	41	2	no	—
84	#121	D₄₁	1	1	yes	characteristic, complemented
85	C₂×C₄⋊C₄	—	41	1	no	—
86	#41	—	82	8	no	—
87	C₂×Q₈	—	41	2	no	—
88	C₂×D₄	—	41	2	no	—
89	C₂×D₄	—	41	2	no	—
90	C₂×D₄	—	41	2	no	—
91	C₂×D₄	—	41	1	no	—
92	C₂×D₄	C₂×D₄₁	1	2	yes	—
93	C₂×D₄	C₂×D₄₁	1	1	yes	characteristic
94	C₂²×C₄	—	41	1	no	—
95	C₂²×C₄	—	41	2	no	—
96	C₂²×C₄	—	41	1	no	—
97	C₂²×C₄	—	41	2	no	—
98	C₂²×C₄	C₂×D₄₁	1	1	yes	characteristic
99	C₄⋊C₄	—	41	2	no	—
100	C₄⋊C₄	—	41	2	no	—
101	C₄⋊C₄	—	41	2	no	—
102	C₄⋊C₄	—	41	1	no	—
103	C₄⋊C₄	C₂×D₄₁	1	1	yes	characteristic
104	C₂²⋊C₄	—	41	2	no	—
105	C₂²⋊C₄	—	41	2	no	—
106	C₂²⋊C₄	—	41	2	no	—
107	C₂²⋊C₄	C₂×D₄₁	1	2	yes	—
108	C₂³	—	41	2	no	—
109	C₂³	C₂²×D₄₁	1	2	yes	—
110	C₂³	C₂²×D₄₁	1	1	yes	characteristic
111	Q₈	—	82	4	no	—
112	D₄	—	2	4	no	—
113	D₄	—	4	1	no	—
114	D₄	—	82	4	no	—
115	D₄	—	82	4	no	—
116	D₄	—	164	1	no	—
117	D₄	—	164	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	41	4	no	—
120	C₂×C₄	—	41	2	no	—
121	C₂×C₄	—	41	2	no	—
122	C₂×C₄	—	41	2	no	—
123	C₂×C₄	—	82	4	no	—
124	C₂×C₄	—	82	4	no	—
125	C₂×C₄	—	164	1	no	—
126	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
127	C₂×C₄	C₂²×D₄₁	1	2	yes	—
128	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	2	2	no	—
134	C₂²	—	82	2	no	—
135	C₂²	—	82	2	no	—
136	C₂²	—	82	2	no	—
137	C₂²	D₄×D₄₁	1	2	yes	—
138	C₂²	C₂³×D₄₁	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	41	4	no	—
143	C₄	—	82	2	no	—
144	C₄	—	82	2	no	—
145	C₂	—	2	2	no	—
146	C₂	—	4	2	no	—
147	C₂	—	164	2	no	—
148	C₂	C₂×D₄×D₄₁	1	1	yes	characteristic
149	C₂	#13050209	1	1	yes	characteristic
150	C₂	#13050211	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10620308

C₂

yes

maximal, complemented

#13050142

C₂

yes

maximal, complemented

#13050141

C₂

yes

maximal, complemented

#13050138

C₂

yes

maximal, characteristic, complemented

#13050137

C₂

yes

maximal, complemented

#13050136

C₂

yes

maximal, complemented

#13050136

C₂

yes

maximal, complemented

C₂×#10620279

C₂

yes

maximal, characteristic, complemented

#121×C₄₁

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₄₁

C₂²

yes

—

C₂×D₄×C₄₁

C₂²

yes

characteristic

C₂²×C₄×C₄₁

C₂²

yes

characteristic

C₂×C₂²⋊D₄₁

C₂²

yes

—

C₂×C₂²⋊D₄₁

C₂²

yes

—

C₂×C₂²⋊D₄₁

C₂²

yes

—

C₂²×Dic₄₁

C₂²

yes

—

C₂²×Dic₄₁

C₂²

yes

characteristic

C₂²×Dic₄₁

C₂²

yes

characteristic

#10620308

—

C₂×D₁₆₄

C₂²

yes

characteristic

C₂×C₄×D₄₁

C₂²

yes

—

C₂×Dic₈₂

C₂²

yes

—

C₄⋊C₄×C₄₁

C₂²

yes

characteristic

C₂²⋊C₄×C₄₁

C₂²

yes

—

C₂²⋊Dic₄₁

C₂²

yes

—

#10620281

C₂²

yes

—

#10620281

C₂²

yes

—

C₄⋊Dic₄₁

C₂²

yes

characteristic

#10620279

C₂²

yes

—

#10620279

C₂²

yes

—

#10620279

C₂²

yes

—

C₂³×C₄₁

C₂³

yes

—

C₂³×C₄₁

C₂³

yes

characteristic

C₂²×D₄₁

C₂³

yes

—

D₄×C₄₁

—

D₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂×C₄×C₄₁

C₂³

yes

—

C₂²⋊D₄₁

—

C₂²⋊D₄₁

—

C₂²⋊D₄₁

—

C₂×Dic₄₁

—

C₂×Dic₄₁

—

C₂×Dic₄₁

D₄

yes

complemented

C₂×Dic₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

—

D₁₆₄

—

C₄×D₄₁

—

Dic₈₂

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

C₂×D₄

yes

—

C₂²×C₄₁

C₂⁴

yes

characteristic

C₂×D₄₁

—

C₂×D₄₁

—

C₂×D₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

Dic₄₁

—

Dic₄₁

—

Dic₄₁

C₂×D₄

yes

—

C₂×C₄₁

—

C₂×C₄₁

—

C₂×C₄₁

C₂²×D₄

yes

characteristic

C₂×C₄₁

H(5,𝔽₂)

yes

characteristic

C₂×C₄₁

2₋¹⁺⁴

yes

characteristic

D₄₁

—

#537

—

maximal, Sylow

C₄₁

#537

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

—

—

—

—

D₄₁

yes

characteristic, complemented

C₂×C₄⋊C₄

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₁

yes

—

C₂×D₄

C₂×D₄₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄₁

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

C₂×D₄₁

yes

characteristic

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

C₂×D₄₁

yes

—

108

C₂³

—

109

C₂³

C₂²×D₄₁

yes

—

110

C₂³

C₂²×D₄₁

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

D₄

—

116

D₄

—

164

—

117

D₄

—

164

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

164

—

126

C₂×C₄

C₂²×D₄₁

yes

characteristic

127

C₂×C₄

C₂²×D₄₁

yes

—

128

C₂×C₄

C₂²×D₄₁

yes

characteristic

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

D₄×D₄₁

yes

—

138

C₂²

C₂³×D₄₁

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₂

—

146

C₂

—

147

C₂

—

164

—

148

C₂

C₂×D₄×D₄₁

yes

characteristic

149

C₂

#13050209

yes

characteristic

150

C₂

#13050211

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator, socle, cosocle	C₂²×C₄₁	C₂⁴
Fitting	#121×C₄₁	C₂
hypercenter	#121	D₄₁
hypocenter	C₄₁	#537
LCS	G ▷ C₂²×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₁

commutator, socle, cosocle

C₂²×C₄₁

C₂⁴

Fitting

#121×C₄₁

C₂

D₄₁

C₄₁

G ▷ C₂²×C₄₁ ▷ C₄₁

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. C₂×#10620308	C₂	semidirect
2. #13050142	C₂	semidirect
3. #13050141	C₂	semidirect
4. #13050138	C₂	semidirect
5. #13050137	C₂	semidirect
6. #13050136	C₂	semidirect
7. #13050136	C₂	semidirect
8. C₂×#10620279	C₂	semidirect
9. #121×C₄₁	C₂	semidirect
46. C₂×Dic₄₁	D₄	semidirect
76. C₄₁	#537	semidirect
84. #121	D₄₁	semidirect

H ≅ G/N

type

1. C₂×#10620308

C₂

semidirect

2. #13050142

C₂

semidirect

3. #13050141

C₂

semidirect

4. #13050138

C₂

semidirect

5. #13050137

C₂

semidirect

6. #13050136

C₂

semidirect

7. #13050136

C₂

semidirect

8. C₂×#10620279

C₂

semidirect

9. #121×C₄₁

C₂

semidirect

46. C₂×Dic₄₁

D₄

semidirect

76. C₄₁

#537

semidirect

84. #121

D₄₁

semidirect

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

subgroup	index
C₂×#10620308	2
#13050142	2
#13050141	2
#13050138	2
#13050137	2
#13050136	2
C₂×#10620279	2
#121×C₄₁	2
#537	41

subgroup

index

#121×C₄₁

Subgroups — #14580913

Conjugacy classes of subgroups

Complements

Maximal subgroups