Group #14581050 — subgroups

Subgroups — #14581050

Order 2624.

236 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 97 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	#13050163	C₂	1	1	yes	maximal, characteristic, complemented
2	#13050151	C₂	1	2	yes	maximal, complemented
3	#13050150	C₂	1	2	yes	maximal, complemented
4	#13050136	C₂	1	2	yes	maximal, complemented
5	#13050135	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×Dic₄₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊D₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#13050123	C₂	1	1	yes	maximal, characteristic, complemented
9	#13050122	C₂	1	1	yes	maximal, characteristic, complemented
10	#13050120	C₂	1	2	yes	maximal, complemented
11	#121×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₄₁	C₂²	1	2	yes	—
14	C₂²×C₄×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₄₁	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊D₄₁	C₂²	1	2	yes	—
17	C₂²×Dic₄₁	C₂²	1	2	yes	—
18	C₂×C₄×D₄₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₈₂	C₂²	1	2	yes	—
20	C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
21	C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
22	C₂²⋊Dic₄₁	C₂²	1	2	yes	—
23	C₂²⋊Dic₄₁	C₂²	1	2	yes	—
24	C₂²⋊Dic₄₁	C₂²	1	1	yes	characteristic
25	C₂²⋊Dic₄₁	C₂²	1	2	yes	—
26	#10620281	C₂²	1	2	yes	—
27	#10620281	C₂²	1	1	yes	characteristic
28	C₄⋊Dic₄₁	C₂²	1	1	yes	characteristic
29	C₄⋊Dic₄₁	C₂²	1	1	yes	characteristic
30	#10620279	C₂²	1	1	yes	characteristic
31	#10620279	C₂²	1	2	yes	—
32	C₄×Dic₄₁	C₂²	1	2	yes	—
33	C₄×Dic₄₁	C₂²	1	2	yes	—
34	C₄×Dic₄₁	C₂²	1	1	yes	characteristic
35	C₂³×C₄₁	C₂³	1	1	yes	characteristic
36	C₂³×C₄₁	C₂³	1	2	yes	—
37	C₂²×D₄₁	C₂³	1	1	yes	characteristic
38	D₄×C₄₁	—	2	2	no	—
39	D₄×C₄₁	—	4	2	no	—
40	C₂×C₄×C₄₁	—	2	2	no	—
41	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₄₁	C₂³	1	2	yes	—
43	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
44	C₂²⋊D₄₁	—	2	2	no	—
45	C₂²⋊D₄₁	—	4	2	no	—
46	C₂×Dic₄₁	—	2	4	no	—
47	C₂×Dic₄₁	C₂³	1	1	yes	characteristic
48	C₂×Dic₄₁	C₂³	1	2	yes	—
49	C₂×Dic₄₁	C₂³	1	2	yes	—
50	C₂×Dic₄₁	C₂³	1	1	yes	characteristic
51	C₂×Dic₄₁	C₂³	1	1	yes	characteristic
52	C₄×D₄₁	—	2	2	no	—
53	Dic₈₂	—	4	2	no	—
54	C₂²×C₄₁	—	2	2	no	—
55	C₂²×C₄₁	—	2	1	no	—
56	C₂²×C₄₁	—	2	1	no	—
57	C₂²×C₄₁	—	2	2	no	—
58	C₂²×C₄₁	—	2	2	no	—
59	C₂²×C₄₁	—	2	1	no	—
60	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
61	C₂×D₄₁	—	2	1	no	—
62	C₂×D₄₁	—	2	1	no	—
63	C₂×D₄₁	—	2	1	no	—
64	C₄×C₄₁	—	2	1	no	—
65	C₄×C₄₁	—	2	2	no	—
66	C₄×C₄₁	#41	1	2	yes	—
67	Dic₄₁	—	2	1	no	—
68	Dic₄₁	—	2	2	no	—
69	Dic₄₁	—	2	1	no	—
70	Dic₄₁	—	2	2	no	—
71	Dic₄₁	#41	1	2	yes	—
72	C₂×C₄₁	—	4	2	no	—
73	C₂×C₄₁	—	4	1	no	—
74	C₂×C₄₁	C₂×#41	1	1	yes	characteristic
75	C₂×C₄₁	C₂×#41	1	1	yes	characteristic
76	C₂×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
77	D₄₁	—	4	1	no	—
78	#555	—	41	1	no	maximal, Sylow
79	C₄₁	#555	1	1	yes	characteristic, complemented, Sylow
80	C₄⋊Q₈	—	41	1	no	—
81	#124	—	41	2	no	—
82	#124	—	41	2	no	—
83	#121	—	41	1	no	—
84	#121	—	41	2	no	—
85	#121	D₄₁	1	1	yes	characteristic, complemented
86	C₄×D₄	—	41	1	no	—
87	C₄×D₄	—	41	1	no	—
88	#117	—	41	1	no	—
89	#117	—	41	1	no	—
90	#117	—	41	2	no	—
91	C₂×Q₈	—	41	2	no	—
92	C₂×D₄	—	41	1	no	—
93	C₂×D₄	—	41	2	no	—
94	C₂×D₄	C₂×D₄₁	1	1	yes	characteristic
95	C₂×D₄	C₂×D₄₁	1	2	yes	—
96	C₂²×C₄	—	41	2	no	—
97	C₂²×C₄	—	41	1	no	—
98	C₂²×C₄	C₂×D₄₁	1	1	yes	characteristic
99	C₄⋊C₄	—	41	1	no	—
100	C₄⋊C₄	—	41	1	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	1	no	—
107	C₂²⋊C₄	—	41	2	no	—
108	C₂²⋊C₄	—	41	2	no	—
109	C₂²⋊C₄	—	41	1	no	—
110	C₂²⋊C₄	C₂×D₄₁	1	2	yes	—
111	C₄²	—	41	2	no	—
112	C₄²	—	41	2	no	—
113	C₄²	—	41	1	no	—
114	C₂³	—	41	1	no	—
115	C₂³	C₂²×D₄₁	1	1	yes	characteristic
116	C₂³	C₂²×D₄₁	1	2	yes	—
117	Q₈	—	164	2	no	—
118	D₄	—	2	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	82	2	no	—
121	D₄	—	164	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	41	1	no	—
124	C₂×C₄	—	41	2	no	—
125	C₂×C₄	—	41	1	no	—
126	C₂×C₄	—	41	1	no	—
127	C₂×C₄	—	41	2	no	—
128	C₂×C₄	—	82	4	no	—
129	C₂×C₄	—	82	2	no	—
130	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₄₁	1	2	yes	—
132	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
133	C₂²	—	2	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	2	no	—
139	C₂²	—	82	1	no	—
140	C₂²	—	82	1	no	—
141	C₂²	—	82	1	no	—
142	C₂²	C₂³×D₄₁	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	2	no	—
145	C₄	—	41	2	no	—
146	C₄	—	82	2	no	—
147	C₄	—	82	1	no	—
148	C₄	—	82	2	no	—
149	C₄	—	82	1	no	—
150	C₄	#10620308	1	2	yes	—
151	C₂	—	4	2	no	—
152	C₂	—	4	1	no	—
153	C₂	—	164	1	no	—
154	C₂	#41×D₄₁	1	1	yes	characteristic
155	C₂	C₂×#10620308	1	1	yes	characteristic
156	C₂	#13050209	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13050163

C₂

yes

maximal, characteristic, complemented

#13050151

C₂

yes

maximal, complemented

#13050150

C₂

yes

maximal, complemented

#13050136

C₂

yes

maximal, complemented

#13050135

C₂

yes

maximal, characteristic, complemented

D₄×Dic₄₁

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₄₁

C₂

yes

maximal, characteristic, complemented

#13050123

C₂

yes

maximal, characteristic, complemented

#13050122

C₂

yes

maximal, characteristic, complemented

#13050120

C₂

yes

maximal, complemented

#121×C₄₁

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₄₁

C₂²

yes

characteristic

C₂×D₄×C₄₁

C₂²

yes

—

C₂²×C₄×C₄₁

C₂²

yes

characteristic

C₂×C₂²⋊D₄₁

C₂²

yes

characteristic

C₂×C₂²⋊D₄₁

C₂²

yes

—

C₂²×Dic₄₁

C₂²

yes

—

C₂×C₄×D₄₁

C₂²

yes

characteristic

C₂×Dic₈₂

C₂²

yes

—

C₄⋊C₄×C₄₁

C₂²

yes

characteristic

C₂²⋊C₄×C₄₁

C₂²

yes

—

C₂²⋊Dic₄₁

C₂²

yes

—

C₂²⋊Dic₄₁

C₂²

yes

—

C₂²⋊Dic₄₁

C₂²

yes

characteristic

C₂²⋊Dic₄₁

C₂²

yes

—

#10620281

C₂²

yes

—

#10620281

C₂²

yes

characteristic

C₄⋊Dic₄₁

C₂²

yes

characteristic

C₄⋊Dic₄₁

C₂²

yes

characteristic

#10620279

C₂²

yes

characteristic

#10620279

C₂²

yes

—

C₄×Dic₄₁

C₂²

yes

—

C₄×Dic₄₁

C₂²

yes

—

C₄×Dic₄₁

C₂²

yes

characteristic

C₂³×C₄₁

C₂³

yes

characteristic

C₂³×C₄₁

C₂³

yes

—

C₂²×D₄₁

C₂³

yes

characteristic

D₄×C₄₁

—

D₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂×C₄×C₄₁

C₂³

yes

—

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂²⋊D₄₁

—

C₂²⋊D₄₁

—

C₂×Dic₄₁

—

C₂×Dic₄₁

C₂³

yes

characteristic

C₂×Dic₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

characteristic

C₂×Dic₄₁

C₂³

yes

characteristic

C₄×D₄₁

—

Dic₈₂

—

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

—

C₄×C₄₁

#41

yes

—

Dic₄₁

—

Dic₄₁

—

Dic₄₁

—

Dic₄₁

—

Dic₄₁

#41

yes

—

C₂×C₄₁

—

C₂×C₄₁

—

C₂×C₄₁

C₂×#41

yes

characteristic

C₂×C₄₁

C₂×#41

yes

characteristic

C₂×C₄₁

H(5,𝔽₂)

yes

characteristic

D₄₁

—

#555

—

maximal, Sylow

C₄₁

#555

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

—

D₄₁

yes

characteristic, complemented

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

#117

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₁

yes

characteristic

C₂×D₄

C₂×D₄₁

yes

—

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

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

C₂×D₄₁

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₂³

—

115

C₂³

C₂²×D₄₁

yes

characteristic

116

C₂³

C₂²×D₄₁

yes

—

117

Q₈

—

164

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

164

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂²×D₄₁

yes

characteristic

131

C₂×C₄

C₂²×D₄₁

yes

—

132

C₂×C₄

C₂²×D₄₁

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂³×D₄₁

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#10620308

yes

—

151

C₂

—

152

C₂

—

153

C₂

—

164

—

154

C₂

#41×D₄₁

yes

characteristic

155

C₂

C₂×#10620308

yes

characteristic

156

C₂

#13050209

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	#555
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. #13050163	C₂	semidirect
2. #13050151	C₂	semidirect
3. #13050150	C₂	semidirect
4. #13050136	C₂	semidirect
5. #13050135	C₂	semidirect
6. D₄×Dic₄₁	C₂	semidirect
7. C₄×C₂²⋊D₄₁	C₂	semidirect
8. #13050123	C₂	semidirect
9. #13050122	C₂	semidirect
10. #13050120	C₂	semidirect
11. #121×C₄₁	C₂	semidirect
79. C₄₁	#555	semidirect
85. #121	D₄₁	semidirect

H ≅ G/N

type

1. #13050163

C₂

semidirect

2. #13050151

C₂

semidirect

3. #13050150

C₂

semidirect

4. #13050136

C₂

semidirect

5. #13050135

C₂

semidirect

6. D₄×Dic₄₁

C₂

semidirect

7. C₄×C₂²⋊D₄₁

C₂

semidirect

8. #13050123

C₂

semidirect

9. #13050122

C₂

semidirect

10. #13050120

C₂

semidirect

11. #121×C₄₁

C₂

semidirect

79. C₄₁

#555

semidirect

85. #121

D₄₁

semidirect

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

subgroup	index
#13050163	2
#13050151	2
#13050150	2
#13050136	2
#13050135	2
D₄×Dic₄₁	2
C₄×C₂²⋊D₄₁	2
#13050123	2
#13050122	2
#13050120	2
#121×C₄₁	2
#555	41

subgroup

index

D₄×Dic₄₁

C₄×C₂²⋊D₄₁

#121×C₄₁

Subgroups — #14581050

Conjugacy classes of subgroups

Complements

Maximal subgroups