Group #15778355 — subgroups

Subgroups — #15778355

Order 3328.

958 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 219 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×C₂²⋊#6090	C₂	1	6	yes	maximal, complemented
2	C₂²×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₃×#41	C₄	1	1	yes	characteristic
4	C₂×C₁₃×#131	—	2	12	no	—
5	C₂²×C₁₃×M₁₆	C₄	1	3	yes	—
6	C₂³×C₈×C₁₃	C₂²	1	3	yes	—
7	C₂²×#6090	C₂²	1	4	yes	—
8	C₂²⋊#6090	—	2	12	no	—
9	C₂×C₁₃×#41	—	2	6	no	—
10	C₂×C₁₃×#41	—	2	6	no	—
11	C₂²×Q₈×C₁₃	C₈	1	1	yes	characteristic
12	C₂²×D₄×C₁₃	C₈	1	3	yes	—
13	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
14	C₁₃×#131	—	2	32	no	—
15	C₂×C₁₃×M₁₆	—	2	6	no	—
16	C₂×C₁₃×M₁₆	—	2	12	no	—
17	C₂²×C₈×C₁₃	—	2	12	no	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
19	C₂²×C₈×C₁₃	D₄	1	12	yes	—
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×#6090	—	2	12	no	—
22	#12680	—	13	1	no	maximal, Sylow
23	C₂⁴×C₁₃	C₂×C₈	1	3	yes	—
24	C₁₃×#41	—	2	8	no	—
25	C₁₃×#41	—	2	24	no	—
26	C₂×Q₈×C₁₃	—	2	6	no	—
27	C₂×D₄×C₁₃	—	2	12	no	—
28	C₂×D₄×C₁₃	—	2	6	no	—
29	C₂²×C₄×C₁₃	—	2	6	no	—
30	C₂²×C₄×C₁₃	—	2	6	no	—
31	C₂²×C₄×C₁₃	C₂²⋊C₄	1	6	yes	—
32	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
33	C₂²×C₄×C₁₃	M₁₆	1	6	yes	—
34	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₁₃×M₁₆	—	2	24	no	—
36	C₂×C₈×C₁₃	—	2	6	no	—
37	C₂×C₈×C₁₃	—	2	24	no	—
38	C₂×C₈×C₁₃	C₂²⋊C₄	1	6	yes	—
39	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
40	#6090	—	4	4	no	—
41	C₂²×#131	D₁₃	1	1	yes	characteristic
42	C₂×C₂²⋊C₁₆	—	13	6	no	—
43	C₂³×C₁₃	—	2	6	no	—
44	C₂³×C₁₃	—	2	12	no	—
45	C₂³×C₁₃	C₂²⋊C₈	1	6	yes	—
46	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
47	Q₈×C₁₃	—	2	8	no	—
48	D₄×C₁₃	—	2	24	no	—
49	C₂×C₄×C₁₃	—	2	6	no	—
50	C₂×C₄×C₁₃	—	2	6	no	—
51	C₂×C₄×C₁₃	—	2	6	no	—
52	C₂×C₄×C₁₃	—	2	12	no	—
53	C₂×C₄×C₁₃	C₂²⋊C₈	1	6	yes	—
54	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	3	yes	—
55	C₂×C₄×C₁₃	C₂×M₁₆	1	3	yes	—
56	C₈×C₁₃	—	2	6	no	—
57	C₈×C₁₃	#120	1	4	yes	—
58	C₂²×#41	Dic₁₃	1	1	yes	characteristic
59	C₂×#131	—	2	12	no	—
60	C₂²×M₁₆	Dic₁₃	1	3	yes	—
61	C₂³×C₈	C₂×D₁₃	1	3	yes	—
62	C₂²×C₁₆	—	13	4	no	—
63	C₂²⋊C₁₆	—	26	12	no	—
64	C₂²×C₁₃	—	2	12	no	—
65	C₂²×C₁₃	—	2	6	no	—
66	C₂²×C₁₃	—	4	12	no	—
67	C₂²×C₁₃	C₂×C₂²⋊C₈	1	3	yes	—
68	C₂²×C₁₃	#504	1	4	yes	—
69	C₄×C₁₃	—	2	6	no	—
70	C₄×C₁₃	#379	1	1	yes	characteristic
71	C₄×C₁₃	#407	1	3	yes	—
72	C₂×#41	—	2	6	no	—
73	C₂×#41	—	2	6	no	—
74	C₂²×Q₈	#1055	1	1	yes	characteristic
75	C₂²×D₄	#1055	1	3	yes	—
76	C₂³×C₄	C₂×Dic₁₃	1	3	yes	—
77	#131	—	2	32	no	—
78	C₂×M₁₆	—	2	6	no	—
79	C₂×M₁₆	—	2	12	no	—
80	C₂²×C₈	—	2	12	no	—
81	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
82	C₂²×C₈	C₂²⋊D₁₃	1	12	yes	—
83	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
84	C₂×C₁₆	—	26	12	no	—
85	C₂×C₁₃	—	4	6	no	—
86	C₂×C₁₃	#1779	1	1	yes	characteristic
87	C₂×C₁₃	#2113	1	6	yes	—
88	C₂⁴	C₂×#1055	1	3	yes	—
89	#41	—	2	8	no	—
90	#41	—	2	24	no	—
91	C₂×Q₈	—	2	6	no	—
92	C₂×D₄	—	2	12	no	—
93	C₂×D₄	—	2	6	no	—
94	C₂²×C₄	—	2	6	no	—
95	C₂²×C₄	—	2	6	no	—
96	C₂²×C₄	C₂×#1055	1	3	yes	—
97	C₂²×C₄	#6099	1	6	yes	—
98	C₂²×C₄	C₂²⋊Dic₁₃	1	6	yes	—
99	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
100	M₁₆	—	2	24	no	—
101	C₂×C₈	—	2	6	no	—
102	C₂×C₈	—	2	24	no	—
103	C₂×C₈	C₂²⋊Dic₁₃	1	6	yes	—
104	C₂×C₈	C₂×C₂²⋊D₁₃	1	6	yes	—
105	C₁₆	—	52	4	no	—
106	C₁₃	#12680	1	1	yes	characteristic, complemented, Sylow
107	C₂³	—	2	12	no	—
108	C₂³	—	2	6	no	—
109	C₂³	C₂²⋊#1055	1	6	yes	—
110	C₂³	C₂²×#1055	1	1	yes	characteristic
111	Q₈	—	2	8	no	—
112	D₄	—	2	24	no	—
113	C₂×C₄	—	2	6	no	—
114	C₂×C₄	—	2	12	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	2	6	no	—
117	C₂×C₄	C₂²⋊#1055	1	6	yes	—
118	C₂×C₄	C₂×#6099	1	3	yes	—
119	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	3	yes	—
120	C₈	—	2	6	no	—
121	C₈	#88207	1	4	yes	—
122	C₂²	—	2	12	no	—
123	C₂²	—	2	6	no	—
124	C₂²	—	4	12	no	—
125	C₂²	#11718801	1	4	yes	—
126	C₂²	C₂×C₂²⋊#1055	1	3	yes	—
127	C₄	—	2	6	no	—
128	C₄	#11718868	1	3	yes	—
129	C₄	#11718908	1	1	yes	characteristic
130	C₂	—	4	6	no	—
131	C₂	#13114746	1	1	yes	characteristic
132	C₂	#13116450	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊#6090

C₂

yes

maximal, complemented

C₂²×C₁₃×#131

C₂

yes

maximal, characteristic

C₂²×C₁₃×#41

C₄

yes

characteristic

C₂×C₁₃×#131

—

C₂²×C₁₃×M₁₆

C₄

yes

—

C₂³×C₈×C₁₃

C₂²

yes

—

C₂²×#6090

C₂²

yes

—

C₂²⋊#6090

—

C₂×C₁₃×#41

—

C₂×C₁₃×#41

—

C₂²×Q₈×C₁₃

C₈

yes

characteristic

C₂²×D₄×C₁₃

C₈

yes

—

C₂³×C₄×C₁₃

C₂×C₄

yes

—

C₁₃×#131

—

C₂×C₁₃×M₁₆

—

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

—

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

C₂³

yes

—

—

C₂⁴×C₁₃

C₂×C₈

yes

—

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₈

yes

—

C₂²×C₄×C₁₃

M₁₆

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

#6090

—

C₂²×#131

D₁₃

yes

characteristic

C₂×C₂²⋊C₁₆

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂²⋊C₈

yes

—

C₂³×C₁₃

C₂²×C₈

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

#120

yes

—

C₂²×#41

Dic₁₃

yes

characteristic

C₂×#131

—

C₂²×M₁₆

Dic₁₃

yes

—

C₂³×C₈

C₂×D₁₃

yes

—

C₂²×C₁₆

—

C₂²⋊C₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₈

yes

—

C₂²×C₁₃

#504

yes

—

C₄×C₁₃

—

C₄×C₁₃

#379

yes

characteristic

C₄×C₁₃

#407

yes

—

C₂×#41

—

C₂×#41

—

C₂²×Q₈

#1055

yes

characteristic

C₂²×D₄

#1055

yes

—

C₂³×C₄

C₂×Dic₁₃

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₁₃

yes

—

C₂²×C₈

C₂²⋊D₁₃

yes

—

C₂²×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₃

—

C₂×C₁₃

#1779

yes

characteristic

C₂×C₁₃

#2113

yes

—

C₂⁴

C₂×#1055

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#1055

yes

—

C₂²×C₄

#6099

yes

—

C₂²×C₄

C₂²⋊Dic₁₃

yes

—

C₂²×C₄

C₂²×Dic₁₃

yes

characteristic

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₂²⋊Dic₁₃

yes

—

104

C₂×C₈

C₂×C₂²⋊D₁₃

yes

—

105

C₁₆

—

106

C₁₃

#12680

yes

characteristic, complemented, Sylow

107

C₂³

—

108

C₂³

—

109

C₂³

C₂²⋊#1055

yes

—

110

C₂³

C₂²×#1055

yes

characteristic

111

Q₈

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²⋊#1055

yes

—

118

C₂×C₄

C₂×#6099

yes

—

119

C₂×C₄

C₂×C₂²⋊Dic₁₃

yes

—

120

C₈

—

121

C₈

#88207

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

#11718801

yes

—

126

C₂²

C₂×C₂²⋊#1055

yes

—

127

C₄

—

128

C₄

#11718868

yes

—

129

C₄

#11718908

yes

characteristic

130

C₂

—

131

C₂

#13114746

yes

characteristic

132

C₂

#13116450

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator, socle	C₂³×C₁₃	C₂²×C₈
Fitting	C₂²×C₁₃×#131	C₂
prime radical	C₂²×C₁₃×#41	C₄
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂²×#131	D₁₃
hypocenter	C₁₃	#12680
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×#131

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₁₃

commutator, socle

C₂³×C₁₃

C₂²×C₈

Fitting

C₂²×C₁₃×#131

C₂

prime radical

C₂²×C₁₃×#41

C₄

cosocle

C₂²×C₈×C₁₃

C₂³

hypercenter

C₂²×#131

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂²×C₈ ◁ C₂²×#131

N	H ≅ G/N	type
1. C₂×C₂²⋊#6090	C₂	semidirect
106. C₁₃	#12680	semidirect

H ≅ G/N

type

1. C₂×C₂²⋊#6090

C₂

semidirect

106. C₁₃

#12680

semidirect

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

subgroup	index
C₂×C₂²⋊#6090	2
C₂²×C₁₃×#131	2
#12680	13

subgroup

index

C₂×C₂²⋊#6090

C₂²×C₁₃×#131

#12680

Subgroups — #15778355

Conjugacy classes of subgroups

Complements

Maximal subgroups