Group #15756720 — subgroups

Subgroups — #15756720

Order 3328.

354 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 203 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	#13115008	C₂	1	2	yes	maximal
2	#13113592	C₂	1	4	yes	maximal
3	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×#88059	—	2	4	no	—
7	#11718585	C₂²	1	4	yes	—
8	C₈²×C₁₃	C₄	1	1	yes	characteristic
9	C₈²×C₁₃	C₄	1	2	yes	—
10	C₈²×C₁₃	C₄	1	1	yes	characteristic
11	#11718049	—	2	8	no	—
12	C₂²×C₈×C₁₃	D₄	1	2	yes	—
13	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
14	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
15	C₂×#6099	—	2	4	no	—
16	C₄×C₈×C₁₃	C₂×C₄	1	2	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	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₁₃	D₄	1	4	yes	—
21	C₄×C₈×C₁₃	Q₈	1	2	yes	—
22	#88059	—	4	8	no	—
23	C₄⋊#1055	—	2	8	no	—
24	#8822	—	13	1	no	maximal, Sylow
25	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
27	C₂×C₈×C₁₃	—	2	8	no	—
28	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
29	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
32	C₂×C₈×C₁₃	SD₁₆	1	8	yes	—
33	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₄²×C₁₃	C₂×D₄	1	2	yes	—
35	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
36	#6099	—	4	8	no	—
37	C₂×#1055	—	4	4	no	—
38	#1848	—	13	2	no	—
39	#1566	—	13	4	no	—
40	C₂×C₈²	D₁₃	1	1	yes	characteristic
41	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₃	—	2	4	no	—
43	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
46	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
47	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
48	C₈×C₁₃	—	2	4	no	—
49	C₈×C₁₃	#106	1	4	yes	—
50	C₈×C₁₃	#106	1	4	yes	—
51	C₈×C₁₃	#108	1	8	yes	complemented
52	#1055	—	8	4	no	—
53	C₂×#108	—	26	4	no	—
54	#423	—	13	4	no	—
55	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
57	#334	—	26	8	no	—
58	C₈²	Dic₁₃	1	1	yes	characteristic
59	C₈²	Dic₁₃	1	2	yes	—
60	C₈²	Dic₁₃	1	1	yes	characteristic
61	C₂²×C₁₃	—	2	2	no	—
62	C₂²×C₁₃	#384	1	1	yes	characteristic
63	C₂²×C₁₃	#498	1	2	yes	—
64	C₄×C₁₃	—	2	2	no	—
65	C₄×C₁₃	C₂×#106	1	2	yes	—
66	C₄×C₁₃	C₂×#108	1	4	yes	—
67	C₄×C₁₃	#492	1	2	yes	—
68	C₂×M₁₆	—	26	4	no	—
69	C₂²×C₈	Dic₂₆	1	4	yes	—
70	C₂²×C₈	D₅₂	1	2	yes	—
71	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
72	#108	—	52	8	no	—
73	C₄⋊C₈	—	26	8	no	—
74	C₄×C₈	Dic₂₆	1	2	yes	—
75	C₄×C₈	D₅₂	1	4	yes	—
76	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
77	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
78	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
79	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
80	C₂×C₁₃	—	2	2	no	—
81	C₂×C₁₃	#1844	1	1	yes	characteristic
82	C₂×C₁₃	#1848	1	2	yes	—
83	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
84	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
85	M₁₆	—	52	8	no	—
86	C₂×C₈	—	2	8	no	—
87	C₂×C₈	—	52	4	no	—
88	C₂×C₈	#6095	1	8	yes	—
89	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
90	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
91	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
92	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
93	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
94	C₄²	C₂×D₅₂	1	2	yes	—
95	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
96	C₁₃	#8822	1	1	yes	characteristic, complemented, Sylow
97	C₂³	#88123	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	#88123	1	2	yes	—
100	C₂×C₄	#88128	1	1	yes	characteristic
101	C₂×C₄	C₂×#6095	1	4	yes	—
102	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
103	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
104	C₈	—	2	4	no	—
105	C₈	—	104	4	no	—
106	C₈	#88057	1	4	yes	—
107	C₈	#88057	1	4	yes	—
108	C₈	#88059	1	8	yes	complemented
109	C₂²	—	2	2	no	—
110	C₂²	#11718341	1	2	yes	—
111	C₂²	#11718596	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	#11718354	1	2	yes	—
114	C₄	C₂×#88057	1	2	yes	—
115	C₄	C₂×#88059	1	4	yes	—
116	C₂	—	2	2	no	—
117	C₂	#13114993	1	1	yes	characteristic
118	C₂	#13115008	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115008

C₂

yes

maximal

#13113592

C₂

yes

maximal

C₂×C₈²×C₁₃

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88059

—

#11718585

C₂²

yes

—

C₈²×C₁₃

C₄

yes

characteristic

C₈²×C₁₃

C₄

yes

—

C₈²×C₁₃

C₄

yes

characteristic

#11718049

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

Q₈

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

—

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×Q₈

yes

—

—

—

—

C₂×C₈²

D₁₃

yes

characteristic

C₂³×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

#106

yes

—

C₈×C₁₃

#106

yes

—

C₈×C₁₃

yes

—

—

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄×C₈

C₂×D₁₃

yes

—

#334

—

C₈²

Dic₁₃

yes

characteristic

C₈²

Dic₁₃

yes

—

C₈²

Dic₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

#384

yes

characteristic

C₂²×C₁₃

#498

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×#106

yes

—

C₄×C₁₃

C₂×#108

yes

—

C₄×C₁₃

#492

yes

—

C₂×M₁₆

—

C₂²×C₈

Dic₂₆

yes

—

C₂²×C₈

D₅₂

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₄×C₈

Dic₂₆

yes

—

C₄×C₈

D₅₂

yes

—

C₄×C₈

C₂×Dic₁₃

yes

—

C₄×C₈

C₂×Dic₁₃

yes

characteristic

C₄×C₈

C₂×Dic₁₃

yes

characteristic

C₄×C₈

C₂×Dic₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

#1844

yes

characteristic

C₂×C₁₃

#1848

yes

—

C₂²×C₄

C₂×Dic₂₆

yes

—

C₂²×C₄

C₂×D₅₂

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#6095

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₄²

C₂×Dic₂₆

yes

characteristic

C₄²

C₂×D₅₂

yes

—

C₄²

C₂²×Dic₁₃

yes

characteristic

C₁₃

#8822

yes

characteristic, complemented, Sylow

C₂³

#88123

yes

characteristic

C₂×C₄

—

C₂×C₄

#88123

yes

—

100

C₂×C₄

#88128

yes

characteristic

101

C₂×C₄

C₂×#6095

yes

—

102

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

103

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

—

104

C₈

—

105

C₈

—

104

—

106

C₈

#88057

yes

—

107

C₈

#88057

yes

—

108

C₈

#88059

yes

complemented

109

C₂²

—

110

C₂²

#11718341

yes

—

111

C₂²

#11718596

yes

characteristic

112

C₄

—

113

C₄

#11718354

yes

—

114

C₄

C₂×#88057

yes

—

115

C₄

C₂×#88059

yes

—

116

C₂

—

117

C₂

#13114993

yes

characteristic

118

C₂

#13115008

yes

—

	subgroup	G/H
center	C₂×C₄	#88128
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₈²×C₁₃	C₂
socle	C₂²×C₁₃	#384
prime radical	C₂³×C₁₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×C₈²	D₁₃
hypocenter	C₁₃	#8822
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂×C₄

#88128

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₂×C₈²×C₁₃

C₂

socle

C₂²×C₁₃

#384

prime radical

C₂³×C₁₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₃

C₂³

hypercenter

C₂×C₈²

D₁₃

hypocenter

C₁₃

#8822

LCS

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
51. C₈×C₁₃	#108	semidirect
96. C₁₃	#8822	semidirect
108. C₈	#88059	semidirect

H ≅ G/N

type

51. C₈×C₁₃

#108

semidirect

96. C₁₃

#8822

semidirect

108. C₈

#88059

semidirect

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

subgroup	index
#13115008	2
#13113592	2
C₂×C₈²×C₁₃	2
#8822	13

subgroup

index

#13115008

#13113592

C₂×C₈²×C₁₃

#8822

Subgroups — #15756720

Conjugacy classes of subgroups

Complements

Maximal subgroups