Group #15889474 — subgroups

Subgroups — #15889474

Order 3328.

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	#13115134	C₂	1	12	yes	maximal, complemented
2	#13114979	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₃×#127	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#127	C₄	1	1	yes	characteristic
5	C₂×C₄×D₄×C₁₃	C₂²	1	6	yes	—
6	C₂²×#6099	C₂²	1	4	yes	—
7	C₁₃×#390	C₄	1	6	yes	—
8	C₁₃×#384	C₄	1	1	yes	characteristic
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊#1055	C₂²	1	12	yes	—
11	#11718116	C₂²	1	12	yes	—
12	C₂²×D₄×C₁₃	C₂×C₄	1	6	yes	—
13	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
14	C₁₃×#127	—	2	4	no	—
15	C₄×D₄×C₁₃	—	4	6	no	—
16	C₄×D₄×C₁₃	—	4	6	no	—
17	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	6	yes	—
18	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	12	yes	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
22	C₂×#6099	—	4	12	no	—
23	C₂²×#1055	C₂³	1	8	yes	—
24	C₂²⋊#1055	—	4	12	no	—
25	#23357	—	13	1	no	maximal, Sylow
26	C₂⁴×C₁₃	C₂²×C₄	1	4	yes	—
27	C₂×D₄×C₁₃	—	2	12	no	—
28	C₂×D₄×C₁₃	—	4	6	no	—
29	C₂×D₄×C₁₃	—	4	6	no	—
30	C₂²×C₄×C₁₃	—	2	12	no	—
31	C₂²×C₄×C₁₃	—	2	12	no	—
32	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
33	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
34	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
35	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
36	C₁₃×C₄⋊C₄	—	4	6	no	—
37	C₁₃×C₂²⋊C₄	—	4	12	no	—
38	C₄²×C₁₃	—	2	6	no	—
39	C₄²×C₁₃	—	4	3	no	—
40	C₄²×C₁₃	M₁₆	1	4	yes	complemented
41	#6099	—	4	16	no	—
42	C₂×#1055	—	2	24	no	—
43	C₄×#127	D₁₃	1	1	yes	characteristic
44	#1870	—	13	12	no	—
45	#1839	—	13	2	no	—
46	C₂³×C₁₃	—	2	12	no	—
47	C₂³×C₁₃	—	2	4	no	—
48	C₂³×C₁₃	—	2	12	no	—
49	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
50	D₄×C₁₃	—	4	24	no	—
51	C₂×C₄×C₁₃	—	2	3	no	—
52	C₂×C₄×C₁₃	—	2	3	no	—
53	C₂×C₄×C₁₃	—	2	3	no	—
54	C₂×C₄×C₁₃	—	2	12	no	—
55	C₂×C₄×C₁₃	—	2	6	no	—
56	C₂×C₄×C₁₃	—	4	4	no	—
57	C₂×C₄×C₁₃	—	4	12	no	—
58	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
59	C₂×C₄×C₁₃	H(5,𝔽₂)	1	6	yes	—
60	#1055	—	4	8	no	—
61	C₂²×M₁₆	—	13	4	no	—
62	C₂×#127	Dic₁₃	1	1	yes	characteristic
63	C₂×C₄×D₄	C₂×D₁₃	1	6	yes	—
64	C₂×C₂²⋊C₈	—	13	12	no	—
65	#390	Dic₁₃	1	6	yes	—
66	#384	Dic₁₃	1	1	yes	characteristic
67	C₄³	C₂×D₁₃	1	1	yes	characteristic
68	#336	—	13	12	no	—
69	C₂²×C₁₃	—	4	4	no	—
70	C₂²×C₁₃	—	4	12	no	—
71	C₂²×C₁₃	—	4	12	no	—
72	C₂²×C₁₃	#518	1	3	yes	—
73	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
74	C₂²×C₁₃	#568	1	3	yes	—
75	C₄×C₁₃	—	2	6	no	—
76	C₄×C₁₃	—	4	4	no	—
77	C₄×C₁₃	—	4	3	no	—
78	C₄×C₁₃	#560	1	4	yes	—
79	C₂²×D₄	C₂×Dic₁₃	1	6	yes	—
80	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
81	C₂×M₁₆	—	52	12	no	—
82	C₂²×C₈	—	13	8	no	—
83	#127	—	2	4	no	—
84	C₄×D₄	—	4	6	no	—
85	C₄×D₄	—	4	6	no	—
86	C₂×C₄⋊C₄	C₂×Dic₁₃	1	6	yes	—
87	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	12	yes	—
88	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
89	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
90	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
91	C₂²⋊C₈	—	52	12	no	—
92	C₂×C₁₃	—	8	4	no	—
93	C₂×C₁₃	#2380	1	1	yes	characteristic
94	C₂×C₁₃	#2975	1	3	yes	—
95	C₂×C₁₃	#2980	1	3	yes	—
96	C₂⁴	C₂²×Dic₁₃	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×Dic₁₃	1	3	yes	—
103	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
104	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
105	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
106	M₁₆	—	52	16	no	—
107	C₂×C₈	—	26	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	#6099	1	4	yes	complemented
113	C₁₃	#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×Dic₁₃	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×#6099	1	6	yes	—
127	C₂×C₄	#88251	1	6	yes	—
128	C₈	—	52	8	no	—
129	C₂²	—	4	12	no	—
130	C₂²	—	4	4	no	—
131	C₂²	—	4	12	no	—
132	C₂²	C₂²×#6099	1	1	yes	characteristic
133	C₂²	#11719503	1	3	yes	—
134	C₂²	#11719524	1	3	yes	—
135	C₄	—	2	6	no	—
136	C₄	—	4	4	no	—
137	C₄	—	4	3	no	—
138	C₄	#11719425	1	4	yes	—
139	C₂	—	8	4	no	—
140	C₂	#13118013	1	1	yes	characteristic
141	C₂	#13123877	1	3	yes	—
142	C₂	#13123907	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115134

C₂

yes

maximal, complemented

#13114979

C₂

yes

maximal, complemented

C₄×C₁₃×#127

C₂

yes

maximal, characteristic

C₂×C₁₃×#127

C₄

yes

characteristic

C₂×C₄×D₄×C₁₃

C₂²

yes

—

C₂²×#6099

C₂²

yes

—

C₁₃×#390

C₄

yes

—

C₁₃×#384

C₄

yes

characteristic

C₄³×C₁₃

C₂²

yes

characteristic

C₂×C₂²⋊#1055

C₂²

yes

—

#11718116

C₂²

yes

—

C₂²×D₄×C₁₃

C₂×C₄

yes

—

C₂³×C₄×C₁₃

C₂³

yes

—

C₁₃×#127

—

C₄×D₄×C₁₃

—

C₄×D₄×C₁₃

—

C₂×C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₁₃×C₂²⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

—

C₂×#6099

—

C₂²×#1055

C₂³

yes

—

C₂²⋊#1055

—

#23357

—

maximal, Sylow

C₂⁴×C₁₃

C₂²×C₄

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

—

C₂²×C₄×C₁₃

C₂⁴

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

M₁₆

yes

—

—

D₁₃

yes

characteristic

#1870

—

#1839

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂³×C₄

yes

characteristic

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×C₄×C₁₃

H(5,𝔽₂)

yes

—

#1055

—

C₂²×M₁₆

—

C₂×#127

Dic₁₃

yes

characteristic

C₂×C₄×D₄

C₂×D₁₃

yes

—

C₂×C₂²⋊C₈

—

#390

Dic₁₃

yes

—

#384

Dic₁₃

yes

characteristic

C₄³

C₂×D₁₃

yes

characteristic

#336

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#518

yes

—

C₂²×C₁₃

C₂²×M₁₆

yes

characteristic

C₂²×C₁₃

#568

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#560

yes

—

C₂²×D₄

C₂×Dic₁₃

yes

—

C₂³×C₄

C₂²×D₁₃

yes

—

C₂×M₁₆

—

C₂²×C₈

—

#127

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

C₂×Dic₁₃

yes

—

C₂×C₂²⋊C₄

C₂×Dic₁₃

yes

—

C₂×C₄²

C₂×Dic₁₃

yes

—

C₂×C₄²

C₂×Dic₁₃

yes

characteristic

C₂×C₄²

C₂²×D₁₃

yes

—

C₂²⋊C₈

—

C₂×C₁₃

—

C₂×C₁₃

#2380

yes

characteristic

C₂×C₁₃

#2975

yes

—

C₂×C₁₃

#2980

yes

—

C₂⁴

C₂²×Dic₁₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×Dic₁₃

yes

—

103

C₂²×C₄

C₂²×Dic₁₃

yes

—

104

C₂²×C₄

C₂²×Dic₁₃

yes

—

105

C₂²×C₄

C₂³×D₁₃

yes

characteristic

106

M₁₆

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₄²

#6099

yes

complemented

113

C₁₃

#23357

yes

characteristic, complemented, Sylow

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂³×Dic₁₃

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×#6099

yes

—

127

C₂×C₄

#88251

yes

—

128

C₈

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂²×#6099

yes

characteristic

133

C₂²

#11719503

yes

—

134

C₂²

#11719524

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#11719425

yes

—

139

C₂

—

140

C₂

#13118013

yes

characteristic

141

C₂

#13123877

yes

—

142

C₂

#13123907

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle	C₂³×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×#127	C₂
prime radical	C₂×C₁₃×#127	C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#127	D₁₃
hypocenter	C₁₃	#23357
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#127

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₁₃

commutator, socle

C₂³×C₁₃

C₂³×C₄

Fitting

C₄×C₁₃×#127

C₂

prime radical

C₂×C₁₃×#127

C₄

cosocle

C₂²×C₄×C₁₃

C₂⁴

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂²×C₄ ◁ C₄×#127

N	H ≅ G/N	type
1. #13115134	C₂	semidirect
2. #13114979	C₂	semidirect
40. C₄²×C₁₃	M₁₆	semidirect
112. C₄²	#6099	semidirect
113. C₁₃	#23357	semidirect

H ≅ G/N

type

1. #13115134

C₂

semidirect

2. #13114979

C₂

semidirect

40. C₄²×C₁₃

M₁₆

semidirect

112. C₄²

#6099

semidirect

113. C₁₃

#23357

semidirect

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

subgroup	index
#13115134	2
#13114979	2
C₄×C₁₃×#127	2
#23357	13

subgroup

index

#13115134

#13114979

C₄×C₁₃×#127

#23357

Subgroups — #15889474

Conjugacy classes of subgroups

Complements

Maximal subgroups