Group #14780328 — subgroups

Subgroups — #14780328

Order 2816.

912 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 255 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	#13068914	C₂	1	12	yes	maximal, complemented
2	#13068759	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₁×#127	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₁×#127	C₄	1	1	yes	characteristic
5	C₂×C₄×D₄×C₁₁	C₂²	1	6	yes	—
6	C₂²×#4318	C₂²	1	4	yes	—
7	C₁₁×#390	C₄	1	6	yes	—
8	C₁₁×#384	C₄	1	1	yes	characteristic
9	C₄³×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊(C₁₁⋊C₈)	C₂²	1	12	yes	—
11	#10622162	C₂²	1	12	yes	—
12	C₂²×D₄×C₁₁	C₂×C₄	1	6	yes	—
13	C₂³×C₄×C₁₁	C₂³	1	4	yes	—
14	C₁₁×#127	—	2	4	no	—
15	C₄×D₄×C₁₁	—	4	6	no	—
16	C₄×D₄×C₁₁	—	4	6	no	—
17	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	6	yes	—
18	C₂×C₁₁×C₂²⋊C₄	C₂×C₄	1	12	yes	—
19	C₂×C₄²×C₁₁	C₂×C₄	1	3	yes	—
20	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
22	C₂×#4318	—	4	12	no	—
23	C₂²×C₁₁⋊C₈	C₂³	1	8	yes	—
24	C₂²⋊(C₁₁⋊C₈)	—	4	12	no	—
25	#23357	—	11	1	no	maximal, Sylow
26	C₂⁴×C₁₁	C₂²×C₄	1	4	yes	—
27	C₂×D₄×C₁₁	—	2	12	no	—
28	C₂×D₄×C₁₁	—	4	6	no	—
29	C₂×D₄×C₁₁	—	4	6	no	—
30	C₂²×C₄×C₁₁	—	2	12	no	—
31	C₂²×C₄×C₁₁	—	2	12	no	—
32	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
33	C₂²×C₄×C₁₁	C₂²×C₄	1	4	yes	—
34	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
35	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
36	C₁₁×C₄⋊C₄	—	4	6	no	—
37	C₁₁×C₂²⋊C₄	—	4	12	no	—
38	C₄²×C₁₁	—	2	6	no	—
39	C₄²×C₁₁	—	4	3	no	—
40	C₄²×C₁₁	M₁₆	1	4	yes	complemented
41	#4318	—	4	16	no	—
42	C₂×C₁₁⋊C₈	—	2	24	no	—
43	C₄×#127	D₁₁	1	1	yes	characteristic
44	#1870	—	11	12	no	—
45	#1839	—	11	2	no	—
46	C₂³×C₁₁	—	2	12	no	—
47	C₂³×C₁₁	—	2	4	no	—
48	C₂³×C₁₁	—	2	12	no	—
49	C₂³×C₁₁	C₂³×C₄	1	1	yes	characteristic
50	D₄×C₁₁	—	4	24	no	—
51	C₂×C₄×C₁₁	—	2	3	no	—
52	C₂×C₄×C₁₁	—	2	3	no	—
53	C₂×C₄×C₁₁	—	2	3	no	—
54	C₂×C₄×C₁₁	—	2	12	no	—
55	C₂×C₄×C₁₁	—	2	6	no	—
56	C₂×C₄×C₁₁	—	4	4	no	—
57	C₂×C₄×C₁₁	—	4	12	no	—
58	C₂×C₄×C₁₁	C₂×M₁₆	1	6	yes	—
59	C₂×C₄×C₁₁	H(5,𝔽₂)	1	6	yes	—
60	C₁₁⋊C₈	—	4	8	no	—
61	C₂²×M₁₆	—	11	4	no	—
62	C₂×#127	Dic₁₁	1	1	yes	characteristic
63	C₂×C₄×D₄	C₂×D₁₁	1	6	yes	—
64	C₂×C₂²⋊C₈	—	11	12	no	—
65	#390	Dic₁₁	1	6	yes	—
66	#384	Dic₁₁	1	1	yes	characteristic
67	C₄³	C₂×D₁₁	1	1	yes	characteristic
68	#336	—	11	12	no	—
69	C₂²×C₁₁	—	4	4	no	—
70	C₂²×C₁₁	—	4	12	no	—
71	C₂²×C₁₁	—	4	12	no	—
72	C₂²×C₁₁	#518	1	3	yes	—
73	C₂²×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
74	C₂²×C₁₁	#568	1	3	yes	—
75	C₄×C₁₁	—	2	6	no	—
76	C₄×C₁₁	—	4	4	no	—
77	C₄×C₁₁	—	4	3	no	—
78	C₄×C₁₁	#560	1	4	yes	—
79	C₂²×D₄	C₂×Dic₁₁	1	6	yes	—
80	C₂³×C₄	C₂²×D₁₁	1	4	yes	—
81	C₂×M₁₆	—	44	12	no	—
82	C₂²×C₈	—	11	8	no	—
83	#127	—	2	4	no	—
84	C₄×D₄	—	4	6	no	—
85	C₄×D₄	—	4	6	no	—
86	C₂×C₄⋊C₄	C₂×Dic₁₁	1	6	yes	—
87	C₂×C₂²⋊C₄	C₂×Dic₁₁	1	12	yes	—
88	C₂×C₄²	C₂×Dic₁₁	1	3	yes	—
89	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
90	C₂×C₄²	C₂²×D₁₁	1	3	yes	—
91	C₂²⋊C₈	—	44	12	no	—
92	C₂×C₁₁	—	8	4	no	—
93	C₂×C₁₁	#2380	1	1	yes	characteristic
94	C₂×C₁₁	#2975	1	3	yes	—
95	C₂×C₁₁	#2980	1	3	yes	—
96	C₂⁴	C₂²×Dic₁₁	1	4	yes	—
97	C₂×D₄	—	2	12	no	—
98	C₂×D₄	—	4	6	no	—
99	C₂×D₄	—	4	6	no	—
100	C₂²×C₄	—	2	12	no	—
101	C₂²×C₄	—	2	12	no	—
102	C₂²×C₄	C₂²×Dic₁₁	1	3	yes	—
103	C₂²×C₄	C₂²×Dic₁₁	1	4	yes	—
104	C₂²×C₄	C₂²×Dic₁₁	1	3	yes	—
105	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
106	M₁₆	—	44	16	no	—
107	C₂×C₈	—	22	24	no	—
108	C₄⋊C₄	—	4	6	no	—
109	C₂²⋊C₄	—	4	12	no	—
110	C₄²	—	2	6	no	—
111	C₄²	—	4	3	no	—
112	C₄²	#4318	1	4	yes	complemented
113	C₁₁	#23357	1	1	yes	characteristic, complemented, Sylow
114	C₂³	—	2	4	no	—
115	C₂³	—	2	12	no	—
116	C₂³	—	2	12	no	—
117	C₂³	C₂³×Dic₁₁	1	1	yes	characteristic
118	D₄	—	4	24	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	2	3	no	—
121	C₂×C₄	—	2	3	no	—
122	C₂×C₄	—	2	12	no	—
123	C₂×C₄	—	2	6	no	—
124	C₂×C₄	—	4	4	no	—
125	C₂×C₄	—	4	12	no	—
126	C₂×C₄	C₂×#4318	1	6	yes	—
127	C₂×C₄	#67037	1	6	yes	—
128	C₈	—	44	8	no	—
129	C₂²	—	4	12	no	—
130	C₂²	—	4	4	no	—
131	C₂²	—	4	12	no	—
132	C₂²	C₂²×#4318	1	1	yes	characteristic
133	C₂²	#10623329	1	3	yes	—
134	C₂²	#10623350	1	3	yes	—
135	C₄	—	2	6	no	—
136	C₄	—	4	4	no	—
137	C₄	—	4	3	no	—
138	C₄	#10623251	1	4	yes	—
139	C₂	—	8	4	no	—
140	C₂	#13071793	1	1	yes	characteristic
141	C₂	#13077657	1	3	yes	—
142	C₂	#13077687	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13068914

C₂

yes

maximal, complemented

#13068759

C₂

yes

maximal, complemented

C₄×C₁₁×#127

C₂

yes

maximal, characteristic

C₂×C₁₁×#127

C₄

yes

characteristic

C₂×C₄×D₄×C₁₁

C₂²

yes

—

C₂²×#4318

C₂²

yes

—

C₁₁×#390

C₄

yes

—

C₁₁×#384

C₄

yes

characteristic

C₄³×C₁₁

C₂²

yes

characteristic

C₂×C₂²⋊(C₁₁⋊C₈)

C₂²

yes

—

#10622162

C₂²

yes

—

C₂²×D₄×C₁₁

C₂×C₄

yes

—

C₂³×C₄×C₁₁

C₂³

yes

—

C₁₁×#127

—

C₄×D₄×C₁₁

—

C₄×D₄×C₁₁

—

C₂×C₁₁×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₁₁×C₂²⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₁₁

C₂×C₄

yes

—

C₂×C₄²×C₁₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₁

C₂³

yes

—

C₂×#4318

—

C₂²×C₁₁⋊C₈

C₂³

yes

—

C₂²⋊(C₁₁⋊C₈)

—

#23357

—

maximal, Sylow

C₂⁴×C₁₁

C₂²×C₄

yes

—

C₂×D₄×C₁₁

—

C₂×D₄×C₁₁

—

C₂×D₄×C₁₁

—

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

—

C₂²×C₄×C₁₁

C₂²×C₄

yes

—

C₂²×C₄×C₁₁

C₂²×C₄

yes

—

C₂²×C₄×C₁₁

C₂²×C₄

yes

—

C₂²×C₄×C₁₁

C₂⁴

yes

characteristic

C₁₁×C₄⋊C₄

—

C₁₁×C₂²⋊C₄

—

C₄²×C₁₁

—

C₄²×C₁₁

—

C₄²×C₁₁

M₁₆

yes

complemented

#4318

—

C₂×C₁₁⋊C₈

—

C₄×#127

D₁₁

yes

characteristic

#1870

—

#1839

—

C₂³×C₁₁

—

C₂³×C₁₁

—

C₂³×C₁₁

—

C₂³×C₁₁

C₂³×C₄

yes

characteristic

D₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₂×M₁₆

yes

—

C₂×C₄×C₁₁

H(5,𝔽₂)

yes

—

C₁₁⋊C₈

—

C₂²×M₁₆

—

C₂×#127

Dic₁₁

yes

characteristic

C₂×C₄×D₄

C₂×D₁₁

yes

—

C₂×C₂²⋊C₈

—

#390

Dic₁₁

yes

—

#384

Dic₁₁

yes

characteristic

C₄³

C₂×D₁₁

yes

characteristic

#336

—

C₂²×C₁₁

—

C₂²×C₁₁

—

C₂²×C₁₁

—

C₂²×C₁₁

#518

yes

—

C₂²×C₁₁

C₂²×M₁₆

yes

characteristic

C₂²×C₁₁

#568

yes

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

—

C₄×C₁₁

#560

yes

—

C₂²×D₄

C₂×Dic₁₁

yes

—

C₂³×C₄

C₂²×D₁₁

yes

—

C₂×M₁₆

—

C₂²×C₈

—

#127

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

C₂×Dic₁₁

yes

—

C₂×C₂²⋊C₄

C₂×Dic₁₁

yes

—

C₂×C₄²

C₂×Dic₁₁

yes

—

C₂×C₄²

C₂×Dic₁₁

yes

characteristic

C₂×C₄²

C₂²×D₁₁

yes

—

C₂²⋊C₈

—

C₂×C₁₁

—

C₂×C₁₁

#2380

yes

characteristic

C₂×C₁₁

#2975

yes

—

C₂×C₁₁

#2980

yes

—

C₂⁴

C₂²×Dic₁₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×Dic₁₁

yes

—

103

C₂²×C₄

C₂²×Dic₁₁

yes

—

104

C₂²×C₄

C₂²×Dic₁₁

yes

—

105

C₂²×C₄

C₂³×D₁₁

yes

characteristic

106

M₁₆

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₄²

#4318

yes

complemented

113

C₁₁

#23357

yes

characteristic, complemented, Sylow

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂³×Dic₁₁

yes

characteristic

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂×#4318

yes

—

127

C₂×C₄

#67037

yes

—

128

C₈

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂²×#4318

yes

characteristic

133

C₂²

#10623329

yes

—

134

C₂²

#10623350

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#10623251

yes

—

139

C₂

—

140

C₂

#13071793

yes

characteristic

141

C₂

#13077657

yes

—

142

C₂

#13077687

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₁
commutator, socle	C₂³×C₁₁	C₂³×C₄
Fitting	C₄×C₁₁×#127	C₂
prime radical	C₂×C₁₁×#127	C₄
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₄×#127	D₁₁
hypocenter	C₁₁	#23357
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#127

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₁₁

commutator, socle

C₂³×C₁₁

C₂³×C₄

Fitting

C₄×C₁₁×#127

C₂

prime radical

C₂×C₁₁×#127

C₄

cosocle

C₂²×C₄×C₁₁

C₂⁴

D₁₁

C₁₁

G ▷ C₂³×C₁₁ ▷ C₁₁

C₁ ◁ C₂²×C₄ ◁ C₄×#127

N	H ≅ G/N	type
1. #13068914	C₂	semidirect
2. #13068759	C₂	semidirect
40. C₄²×C₁₁	M₁₆	semidirect
112. C₄²	#4318	semidirect
113. C₁₁	#23357	semidirect

H ≅ G/N

type

1. #13068914

C₂

semidirect

2. #13068759

C₂

semidirect

40. C₄²×C₁₁

M₁₆

semidirect

112. C₄²

#4318

semidirect

113. C₁₁

#23357

semidirect

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

subgroup	index
#13068914	2
#13068759	2
C₄×C₁₁×#127	2
#23357	11

subgroup

index

#13068914

#13068759

C₄×C₁₁×#127

#23357

Subgroups — #14780328

Conjugacy classes of subgroups

Complements

Maximal subgroups