Group #14580908 — subgroups

Subgroups — #14580908

Order 2624.

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

normal subgroups only

This group has 270 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	1	yes	maximal, characteristic
2	C₂²×Dic₈₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#13050163	C₂	1	2	yes	maximal, complemented
4	#13050150	C₂	1	2	yes	maximal, complemented
5	#13050148	C₂	1	1	yes	maximal, characteristic, complemented
6	#13050147	C₂	1	1	yes	maximal, characteristic, complemented
7	#13050145	C₂	1	1	yes	maximal, characteristic, complemented
8	#13050142	C₂	1	2	yes	maximal, complemented
9	#13050141	C₂	1	2	yes	maximal, complemented
10	#13050120	C₂	1	1	yes	maximal, characteristic, complemented
11	#123×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×C₄₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₄₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₄₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₄₁	C₂²	1	1	yes	characteristic
17	C₂²×Dic₄₁	C₂²	1	1	yes	characteristic
18	C₂²×Dic₄₁	C₂²	1	1	yes	characteristic
19	#10620308	—	2	4	no	—
20	C₂×C₄×D₄₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₈₂	—	2	4	no	—
22	C₂×Dic₈₂	C₂²	1	2	yes	—
23	C₂×Dic₈₂	C₂²	1	2	yes	—
24	C₂×Dic₈₂	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
26	C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
28	C₂²⋊Dic₄₁	C₂²	1	1	yes	characteristic
29	C₂²⋊Dic₄₁	C₂²	1	2	yes	—
30	#10620281	C₂²	1	2	yes	—
31	#10620281	C₂²	1	2	yes	—
32	C₄⋊Dic₄₁	C₂²	1	2	yes	—
33	#10620279	C₂²	1	2	yes	—
34	#10620279	C₂²	1	2	yes	—
35	#10620279	C₂²	1	2	yes	—
36	C₄×Dic₄₁	C₂²	1	2	yes	—
37	C₂³×C₄₁	C₂³	1	1	yes	characteristic
38	C₂³×C₄₁	C₂³	1	1	yes	characteristic
39	C₂²×D₄₁	C₂³	1	1	yes	characteristic
40	D₄×C₄₁	—	2	2	no	—
41	C₂×C₄×C₄₁	—	2	2	no	—
42	C₂×C₄×C₄₁	C₂³	1	2	yes	—
43	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₄₁	C₂³	1	2	yes	—
45	C₂²⋊D₄₁	—	2	2	no	—
46	C₂²⋊D₄₁	—	2	2	no	—
47	C₂×Dic₄₁	—	2	2	no	—
48	C₂×Dic₄₁	—	2	2	no	—
49	C₂×Dic₄₁	D₄	1	4	yes	—
50	C₂×Dic₄₁	C₂³	1	2	yes	—
51	C₂×Dic₄₁	C₂³	1	1	yes	characteristic
52	C₂×Dic₄₁	C₂³	1	2	yes	—
53	C₂×Dic₄₁	C₂³	1	1	yes	characteristic
54	C₂×Dic₄₁	C₂³	1	1	yes	characteristic
55	C₄×D₄₁	—	2	2	no	—
56	Dic₈₂	—	2	4	no	—
57	Dic₈₂	—	2	4	no	—
58	Dic₈₂	—	2	2	no	—
59	C₂²×C₄₁	—	2	2	no	—
60	C₂²×C₄₁	—	2	1	no	—
61	C₂²×C₄₁	—	2	2	no	—
62	C₂²×C₄₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
64	C₂×D₄₁	—	2	1	no	—
65	C₂×D₄₁	—	2	2	no	—
66	C₄×C₄₁	—	2	2	no	—
67	C₄×C₄₁	—	2	1	no	—
68	C₄×C₄₁	—	2	2	no	—
69	Dic₄₁	—	2	2	no	—
70	Dic₄₁	—	2	1	no	—
71	Dic₄₁	—	2	2	no	—
72	Dic₄₁	C₂×D₄	1	2	yes	—
73	Dic₄₁	C₂×D₄	1	2	yes	—
74	C₂×C₄₁	—	2	2	no	—
75	C₂×C₄₁	—	4	1	no	—
76	C₂×C₄₁	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄₁	2₋¹⁺⁴	1	2	yes	—
78	D₄₁	—	4	1	no	—
79	#536	—	41	1	no	maximal, Sylow
80	C₄₁	#536	1	1	yes	characteristic, complemented, Sylow
81	C₂×#41	—	41	1	no	—
82	C₂²×Q₈	—	41	1	no	—
83	C₄⋊Q₈	—	41	2	no	—
84	#124	—	41	2	no	—
85	#123	—	41	1	no	—
86	#123	—	41	1	no	—
87	#123	—	41	1	no	—
88	#123	D₄₁	1	1	yes	characteristic, complemented
89	C₂²⋊Q₈	—	41	2	no	—
90	C₂²⋊Q₈	—	41	2	no	—
91	#117	—	41	1	no	—
92	#41	—	82	4	no	—
93	C₂×Q₈	—	41	2	no	—
94	C₂×Q₈	—	41	2	no	—
95	C₂×Q₈	—	41	1	no	—
96	C₂×Q₈	—	82	4	no	—
97	C₂×D₄	—	41	1	no	—
98	C₂×D₄	—	41	1	no	—
99	C₂×D₄	C₂×D₄₁	1	1	yes	characteristic
100	C₂²×C₄	—	41	1	no	—
101	C₂²×C₄	—	41	1	no	—
102	C₂²×C₄	—	41	1	no	—
103	C₂²×C₄	—	41	1	no	—
104	C₂²×C₄	C₂×D₄₁	1	1	yes	characteristic
105	C₄⋊C₄	—	41	2	no	—
106	C₄⋊C₄	—	41	2	no	—
107	C₄⋊C₄	—	41	2	no	—
108	C₄⋊C₄	—	41	2	no	—
109	C₄⋊C₄	C₂×D₄₁	1	2	yes	—
110	C₂²⋊C₄	—	41	1	no	—
111	C₂²⋊C₄	—	41	2	no	—
112	C₂²⋊C₄	—	41	2	no	—
113	C₂²⋊C₄	—	41	2	no	—
114	C₂²⋊C₄	C₂×D₄₁	1	1	yes	characteristic
115	C₂²⋊C₄	C₂×D₄₁	1	2	yes	—
116	C₄²	—	41	2	no	—
117	C₂³	—	41	1	no	—
118	C₂³	C₂²×D₄₁	1	1	yes	characteristic
119	C₂³	C₂²×D₄₁	1	1	yes	characteristic
120	Q₈	—	82	4	no	—
121	Q₈	—	82	4	no	—
122	Q₈	—	82	2	no	—
123	D₄	—	2	2	no	—
124	D₄	—	82	2	no	—
125	D₄	—	82	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	41	4	no	—
128	C₂×C₄	—	41	1	no	—
129	C₂×C₄	—	41	1	no	—
130	C₂×C₄	—	41	2	no	—
131	C₂×C₄	—	41	2	no	—
132	C₂×C₄	—	41	1	no	—
133	C₂×C₄	—	82	2	no	—
134	C₂×C₄	—	82	2	no	—
135	C₂×C₄	—	82	2	no	—
136	C₂×C₄	C₂²×D₄₁	1	2	yes	—
137	C₂×C₄	C₂²×D₄₁	1	2	yes	—
138	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	82	1	no	—
143	C₂²	—	82	2	no	—
144	C₂²	D₄×D₄₁	1	2	yes	—
145	C₂²	C₂³×D₄₁	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	2	no	—
149	C₄	—	41	2	no	—
150	C₄	—	41	2	no	—
151	C₄	—	82	1	no	—
152	C₄	—	82	2	no	—
153	C₄	—	82	2	no	—
154	C₂	—	2	2	no	—
155	C₂	—	4	1	no	—
156	C₂	—	164	1	no	—
157	C₂	C₂×D₄×D₄₁	1	1	yes	characteristic
158	C₂	#13050211	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10620308

C₂

yes

maximal, characteristic

C₂²×Dic₈₂

C₂

yes

maximal, characteristic, complemented

#13050163

C₂

yes

maximal, complemented

#13050150

C₂

yes

maximal, complemented

#13050148

C₂

yes

maximal, characteristic, complemented

#13050147

C₂

yes

maximal, characteristic, complemented

#13050145

C₂

yes

maximal, characteristic, complemented

#13050142

C₂

yes

maximal, complemented

#13050141

C₂

yes

maximal, complemented

#13050120

C₂

yes

maximal, characteristic, complemented

#123×C₄₁

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₄₁

C₂²

yes

characteristic

C₂²×C₄×C₄₁

C₂²

yes

characteristic

C₂×C₂²⋊D₄₁

C₂²

yes

characteristic

C₂×C₂²⋊D₄₁

C₂²

yes

characteristic

C₂²×Dic₄₁

C₂²

yes

characteristic

C₂²×Dic₄₁

C₂²

yes

characteristic

C₂²×Dic₄₁

C₂²

yes

characteristic

#10620308

—

C₂×C₄×D₄₁

C₂²

yes

characteristic

C₂×Dic₈₂

—

C₂×Dic₈₂

C₂²

yes

—

C₂×Dic₈₂

C₂²

yes

—

C₂×Dic₈₂

C₂²

yes

characteristic

C₄⋊C₄×C₄₁

C₂²

yes

—

C₂²⋊C₄×C₄₁

C₂²

yes

characteristic

C₂²⋊C₄×C₄₁

C₂²

yes

—

C₂²⋊Dic₄₁

C₂²

yes

characteristic

C₂²⋊Dic₄₁

C₂²

yes

—

#10620281

C₂²

yes

—

#10620281

C₂²

yes

—

C₄⋊Dic₄₁

C₂²

yes

—

#10620279

C₂²

yes

—

#10620279

C₂²

yes

—

#10620279

C₂²

yes

—

C₄×Dic₄₁

C₂²

yes

—

C₂³×C₄₁

C₂³

yes

characteristic

C₂³×C₄₁

C₂³

yes

characteristic

C₂²×D₄₁

C₂³

yes

characteristic

D₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

C₂³

yes

—

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂×C₄×C₄₁

C₂³

yes

—

C₂²⋊D₄₁

—

C₂²⋊D₄₁

—

C₂×Dic₄₁

—

C₂×Dic₄₁

—

C₂×Dic₄₁

D₄

yes

—

C₂×Dic₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

characteristic

C₂×Dic₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

characteristic

C₂×Dic₄₁

C₂³

yes

characteristic

C₄×D₄₁

—

Dic₈₂

—

Dic₈₂

—

Dic₈₂

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

C₂×D₄

yes

—

C₂²×C₄₁

C₂⁴

yes

characteristic

C₂×D₄₁

—

C₂×D₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

Dic₄₁

—

Dic₄₁

—

Dic₄₁

—

Dic₄₁

C₂×D₄

yes

—

Dic₄₁

C₂×D₄

yes

—

C₂×C₄₁

—

C₂×C₄₁

—

C₂×C₄₁

C₂²×D₄

yes

characteristic

C₂×C₄₁

2₋¹⁺⁴

yes

—

D₄₁

—

#536

—

maximal, Sylow

C₄₁

#536

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

—

C₄⋊Q₈

—

—

—

—

—

D₄₁

yes

characteristic, complemented

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#117

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₁

yes

characteristic

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×D₄₁

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

C₂×D₄₁

yes

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

C₂×D₄₁

yes

characteristic

115

C₂²⋊C₄

C₂×D₄₁

yes

—

116

C₄²

—

117

C₂³

—

118

C₂³

C₂²×D₄₁

yes

characteristic

119

C₂³

C₂²×D₄₁

yes

characteristic

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₂²×D₄₁

yes

—

137

C₂×C₄

C₂²×D₄₁

yes

—

138

C₂×C₄

C₂²×D₄₁

yes

characteristic

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

D₄×D₄₁

yes

—

145

C₂²

C₂³×D₄₁

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₂

—

155

C₂

—

156

C₂

—

164

—

157

C₂

C₂×D₄×D₄₁

yes

characteristic

158

C₂

#13050211

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator, socle, cosocle	C₂²×C₄₁	C₂⁴
Fitting	#123×C₄₁	C₂
prime radical	C₂×#10620308	C₂
hypercenter	#123	D₄₁
hypocenter	C₄₁	#536
LCS	G ▷ C₂²×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₁

commutator, socle, cosocle

C₂²×C₄₁

C₂⁴

Fitting

#123×C₄₁

C₂

C₂

D₄₁

C₄₁

G ▷ C₂²×C₄₁ ▷ C₄₁

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
2. C₂²×Dic₈₂	C₂	semidirect
3. #13050163	C₂	semidirect
4. #13050150	C₂	semidirect
5. #13050148	C₂	semidirect
6. #13050147	C₂	semidirect
7. #13050145	C₂	semidirect
8. #13050142	C₂	semidirect
9. #13050141	C₂	semidirect
10. #13050120	C₂	semidirect
11. #123×C₄₁	C₂	semidirect
80. C₄₁	#536	semidirect
88. #123	D₄₁	semidirect

H ≅ G/N

type

2. C₂²×Dic₈₂

C₂

semidirect

3. #13050163

C₂

semidirect

4. #13050150

C₂

semidirect

5. #13050148

C₂

semidirect

6. #13050147

C₂

semidirect

7. #13050145

C₂

semidirect

8. #13050142

C₂

semidirect

9. #13050141

C₂

semidirect

10. #13050120

C₂

semidirect

11. #123×C₄₁

C₂

semidirect

80. C₄₁

#536

semidirect

88. #123

D₄₁

semidirect

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

subgroup	index
C₂×#10620308	2
C₂²×Dic₈₂	2
#13050163	2
#13050150	2
#13050148	2
#13050147	2
#13050145	2
#13050142	2
#13050141	2
#13050120	2
#123×C₄₁	2
#536	41

subgroup

index

C₂²×Dic₈₂

#123×C₄₁

Subgroups — #14580908

Conjugacy classes of subgroups

Complements

Maximal subgroups