Group #15756626 — subgroups

Subgroups — #15756626

Order 3328.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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	#13113599	C₂	1	2	yes	maximal
2	#13112698	C₂	1	4	yes	maximal
3	C₁₃×#1450	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#431	C₂²	1	2	yes	—
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	#11718587	C₂²	1	4	yes	—
7	C₁₃×#322	C₄	1	2	yes	—
8	C₈²×C₁₃	C₄	1	2	yes	—
9	#11718050	—	2	2	no	—
10	#11718049	—	2	2	no	—
11	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
12	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
13	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
14	C₁₃×C₄⋊C₈	—	2	2	no	—
15	C₁₃×C₂²⋊C₈	—	2	2	no	—
16	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂²⋊#1055	—	2	4	no	—
22	C₄⋊#1055	—	2	4	no	—
23	#88043	—	2	4	no	—
24	#8796	—	13	1	no	maximal, Sylow
25	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	#41	1	2	yes	—
27	C₂×C₈×C₁₃	—	2	2	no	—
28	C₂×C₈×C₁₃	—	2	2	no	—
29	C₂×C₈×C₁₃	—	2	4	no	—
30	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
34	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₁₃	#41	1	2	yes	—
37	C₂×#1055	—	4	4	no	—
38	#1568	—	13	2	no	—
39	#1450	D₁₃	1	1	yes	characteristic
40	#1287	—	13	4	no	—
41	C₂³×C₁₃	#125	1	1	yes	characteristic
42	C₂×C₄×C₁₃	—	2	1	no	—
43	C₂×C₄×C₁₃	—	2	1	no	—
44	C₂×C₄×C₁₃	—	4	2	no	—
45	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃	#117	1	2	yes	—
47	C₂×C₄×C₁₃	#124	1	2	yes	—
48	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
49	C₈×C₁₃	—	2	1	no	—
50	C₈×C₁₃	—	2	1	no	—
51	C₈×C₁₃	—	4	4	no	—
52	C₈×C₁₃	#108	1	2	yes	—
53	C₈×C₁₃	#108	1	2	yes	—
54	#1055	—	8	4	no	—
55	#432	—	13	4	no	—
56	#431	C₂×D₁₃	1	2	yes	—
57	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
58	#335	—	26	2	no	—
59	#334	—	26	2	no	—
60	#322	Dic₁₃	1	2	yes	—
61	C₈²	Dic₁₃	1	2	yes	—
62	C₂²×C₁₃	—	2	2	no	—
63	C₂²×C₁₃	—	2	1	no	—
64	C₂²×C₁₃	#382	1	1	yes	characteristic
65	C₄×C₁₃	—	2	2	no	—
66	C₄×C₁₃	—	2	2	no	—
67	C₄×C₁₃	—	2	1	no	—
68	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
69	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
70	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
71	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
72	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
73	C₄⋊C₈	—	2	2	no	—
74	C₄⋊C₈	—	26	4	no	—
75	C₂²⋊C₈	—	2	2	no	—
76	C₂²⋊C₈	—	26	4	no	—
77	#97	—	26	4	no	—
78	C₄×C₈	D₅₂	1	1	yes	characteristic
79	C₄×C₈	D₅₂	1	1	yes	characteristic
80	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
81	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
82	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
83	C₂×C₁₃	—	4	1	no	—
84	C₂×C₁₃	#1833	1	2	yes	—
85	C₂×C₁₃	#1837	1	1	yes	characteristic
86	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
87	C₂²×C₄	#6127	1	2	yes	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	52	4	no	—
92	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
93	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
94	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
95	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
96	C₄²	C₂×D₅₂	1	1	yes	characteristic
97	C₄²	#6127	1	2	yes	—
98	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
99	C₁₃	#8796	1	1	yes	characteristic, complemented, Sylow
100	C₂³	#88124	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	#88124	1	1	yes	characteristic
105	C₂×C₄	#88129	1	2	yes	—
106	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
107	C₂×C₄	#88180	1	2	yes	—
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	4	4	no	—
111	C₈	—	104	4	no	—
112	C₈	#88059	1	2	yes	—
113	C₈	#88059	1	2	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	1	no	—
116	C₂²	#11718597	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	C₂×#88059	1	1	yes	characteristic
121	C₄	C₂×#88059	1	1	yes	characteristic
122	C₂	—	4	1	no	—
123	C₂	#13114960	1	2	yes	—
124	C₂	#13114974	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13113599

C₂

yes

maximal

#13112698

C₂

yes

maximal

C₁₃×#1450

C₂

yes

maximal, characteristic

C₁₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

#11718587

C₂²

yes

—

C₁₃×#322

C₄

yes

—

C₈²×C₁₃

C₄

yes

—

#11718050

—

#11718049

—

C₂²×C₈×C₁₃

Q₈

yes

characteristic

C₂²×C₈×C₁₃

Q₈

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₂²⋊#1055

—

—

—

—

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₂×#1055

—

#1568

—

#1450

D₁₃

yes

characteristic

#1287

—

C₂³×C₁₃

#125

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#124

yes

—

C₂×C₄×C₁₃

#125

yes

characteristic

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

#108

yes

—

C₈×C₁₃

#108

yes

—

#1055

—

#432

—

#431

C₂×D₁₃

yes

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

Dic₁₃

yes

—

C₈²

Dic₁₃

yes

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#382

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂²×C₈

Dic₂₆

yes

characteristic

C₂²×C₈

Dic₂₆

yes

characteristic

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

D₅₂

yes

characteristic

C₄×C₈

D₅₂

yes

characteristic

C₄×C₈

C₂×Dic₁₃

yes

—

C₄×C₈

C₂×Dic₁₃

yes

characteristic

C₄×C₈

C₂×Dic₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

#1833

yes

—

C₂×C₁₃

#1837

yes

characteristic

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

C₂²×C₄

#6127

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₁₃

yes

characteristic

C₂×C₈

C₄⋊Dic₁₃

yes

characteristic

C₂×C₈

C₄⋊Dic₁₃

yes

characteristic

C₂×C₈

C₄⋊Dic₁₃

yes

characteristic

C₄²

C₂×D₅₂

yes

characteristic

C₄²

#6127

yes

—

C₄²

C₂²×Dic₁₃

yes

characteristic

C₁₃

#8796

yes

characteristic, complemented, Sylow

100

C₂³

#88124

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

#88124

yes

characteristic

105

C₂×C₄

#88129

yes

—

106

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

107

C₂×C₄

#88180

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

104

—

112

C₈

#88059

yes

—

113

C₈

#88059

yes

—

114

C₂²

—

115

C₂²

—

116

C₂²

#11718597

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂×#88059

yes

characteristic

121

C₄

C₂×#88059

yes

characteristic

122

C₂

—

123

C₂

#13114960

yes

—

124

C₂

#13114974

yes

characteristic

	subgroup	G/H
center	C₂²	#11718597
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1450	C₂
socle	C₂²×C₁₃	#382
prime radical	C₂³×C₁₃	#125
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1450	D₁₃
hypocenter	C₁₃	#8796
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#11718597

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1450

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
99. C₁₃	#8796	semidirect

H ≅ G/N

type

99. C₁₃

#8796

semidirect

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

subgroup	index
#13113599	2
#13112698	2
C₁₃×#1450	2
#8796	13

subgroup

index

#13113599

#13112698

C₁₃×#1450

#8796

Subgroups — #15756626

Conjugacy classes of subgroups

Complements

Maximal subgroups