Group #15750377 — subgroups

Subgroups — #15750377

Order 3328.

124 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 51 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	C₄×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic
2	#13112970	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1372	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₀₄	C₄	1	1	yes	characteristic
5	C₄×Dic₅₂	—	2	1	no	—
6	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₁₃×#335	—	2	1	no	—
8	#11718096	C₄	1	1	yes	characteristic
9	#11718095	—	2	1	no	—
10	C₂×#88034	—	2	1	no	—
11	C₂×Dic₅₂	—	2	1	no	—
12	C₄×Dic₂₆	—	4	1	no	—
13	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
15	C₁₃×C₄⋊C₈	—	4	1	no	—
16	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
17	#88058	—	2	1	no	—
18	#88056	—	4	1	no	—
19	Dic₁₀₄	—	2	1	no	—
20	Dic₁₀₄	C₈	1	2	yes	complemented
21	#88034	—	2	2	no	—
22	#7386	—	13	1	no	maximal, Sylow
23	C₂×Dic₂₆	—	4	1	no	—
24	C₂×C₈×C₁₃	—	8	1	no	—
25	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
27	C₄²×C₁₃	D₈	1	1	yes	characteristic
28	C₄⋊Dic₁₃	—	4	1	no	—
29	#6101	—	8	1	no	—
30	C₄×Dic₁₃	—	8	1	no	—
31	Dic₅₂	—	2	2	no	—
32	Dic₅₂	—	4	1	no	—
33	C₁₃×C₁₆	—	2	1	no	—
34	C₁₃×C₁₆	C₂×C₈	1	1	yes	characteristic
35	C₁₃×C₁₆	M₁₆	1	1	yes	characteristic
36	C₄×Q₃₂	—	13	1	no	—
37	#1397	—	13	1	no	—
38	#1372	D₁₃	1	1	yes	characteristic
39	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
40	C₂×C₄×C₁₃	D₁₆	1	1	yes	characteristic
41	C₂×C₄×C₁₃	SD₃₂	1	1	yes	characteristic
42	C₂×Dic₁₃	—	8	1	no	—
43	C₂×Dic₁₃	—	8	1	no	—
44	Dic₂₆	—	4	2	no	—
45	Dic₂₆	—	8	1	no	—
46	C₈×C₁₃	—	2	1	no	—
47	C₈×C₁₃	—	16	1	no	—
48	C₈×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
49	C₈×C₁₃	#104	1	1	yes	characteristic
50	C₂×Q₃₂	—	13	1	no	—
51	C₄×Q₁₆	—	26	1	no	—
52	C₂×C₃₂	—	26	1	no	—
53	#366	—	13	1	no	—
54	#358	—	26	1	no	—
55	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
56	#335	—	2	1	no	—
57	C₂²×C₁₃	#357	1	1	yes	characteristic
58	C₄×C₁₃	—	2	1	no	—
59	C₄×C₁₃	—	2	1	no	—
60	C₄×C₁₃	D₄⋊C₈	1	1	yes	characteristic
61	C₄×C₁₃	#359	1	1	yes	characteristic
62	Dic₁₃	—	8	2	no	—
63	Dic₁₃	—	16	1	no	—
64	C₂×Q₁₆	—	26	1	no	—
65	C₄×Q₈	—	52	1	no	—
66	Q₃₂	—	13	2	no	—
67	Q₃₂	—	26	1	no	—
68	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
69	C₂×C₁₆	D₅₂	1	1	yes	characteristic
70	#107	—	26	1	no	—
71	C₄⋊C₈	—	4	1	no	—
72	#103	—	52	1	no	—
73	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
74	C₃₂	—	26	2	no	—
75	C₂×C₁₃	#1331	1	1	yes	characteristic
76	C₂×C₁₃	#1418	1	1	yes	characteristic
77	C₂×C₁₃	#1420	1	1	yes	characteristic
78	C₂×Q₈	—	52	1	no	—
79	Q₁₆	—	26	2	no	—
80	Q₁₆	—	52	1	no	—
81	C₂×C₈	—	8	1	no	—
82	C₂×C₈	#6103	1	1	yes	characteristic
83	C₂×C₈	#6106	1	1	yes	characteristic
84	C₄⋊C₄	—	52	1	no	—
85	C₄⋊C₄	—	104	1	no	—
86	C₄²	—	104	1	no	—
87	C₄²	#6104	1	1	yes	characteristic
88	C₁₆	—	2	1	no	—
89	C₁₆	C₈×D₁₃	1	1	yes	characteristic
90	C₁₆	#6094	1	1	yes	characteristic
91	C₁₃	#7386	1	1	yes	characteristic, complemented, Sylow
92	Q₈	—	52	2	no	—
93	Q₈	—	104	1	no	—
94	C₂×C₄	—	104	1	no	—
95	C₂×C₄	—	104	1	no	—
96	C₂×C₄	#88049	1	1	yes	characteristic
97	C₂×C₄	#88066	1	1	yes	characteristic
98	C₂×C₄	#88068	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	16	1	no	—
101	C₈	#88060	1	1	yes	characteristic
102	C₈	#88065	1	1	yes	characteristic
103	C₂²	#11718083	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	104	2	no	—
107	C₄	—	208	1	no	—
108	C₄	#11718075	1	1	yes	characteristic
109	C₄	#11718089	1	1	yes	characteristic
110	C₂	#13112796	1	1	yes	characteristic
111	C₂	#13113022	1	1	yes	characteristic
112	C₂	#13113028	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₁₀₄

C₂

yes

maximal, characteristic

#13112970

C₂

yes

maximal, characteristic

C₁₃×#1372

C₂

yes

maximal, characteristic

C₂×Dic₁₀₄

C₄

yes

characteristic

C₄×Dic₅₂

—

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

C₁₃×#335

—

#11718096

C₄

yes

characteristic

#11718095

—

C₂×#88034

—

C₂×Dic₅₂

—

C₄×Dic₂₆

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

D₄

yes

characteristic

#88058

—

#88056

—

Dic₁₀₄

—

Dic₁₀₄

C₈

yes

—

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₃

SD₁₆

yes

characteristic

C₄²×C₁₃

D₈

yes

characteristic

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

—

C₁₃×C₁₆

C₂×C₈

yes

characteristic

C₁₃×C₁₆

M₁₆

yes

characteristic

C₄×Q₃₂

—

#1397

—

#1372

D₁₃

yes

characteristic

C₂×C₄×C₁₃

#102

yes

characteristic

C₂×C₄×C₁₃

D₁₆

yes

characteristic

C₂×C₄×C₁₃

SD₃₂

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂²⋊C₈

yes

characteristic

C₈×C₁₃

#104

yes

characteristic

C₂×Q₃₂

—

C₄×Q₁₆

—

C₂×C₃₂

—

#366

—

#358

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

#335

—

C₂²×C₁₃

#357

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

D₄⋊C₈

yes

characteristic

C₄×C₁₃

#359

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

C₄×D₁₃

yes

characteristic

C₂×C₁₆

D₅₂

yes

characteristic

#107

—

C₄⋊C₈

—

#103

—

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

C₃₂

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

104

—

C₄²

—

104

—

C₄²

#6104

yes

characteristic

C₁₆

—

C₁₆

C₈×D₁₃

yes

C₁₆

yes

C₁₃

yes

characteristic, complemented, Sylow

Q₈

—

Q₈

—

104

—

C₂×C₄

—

104

—

C₂×C₄

—

104

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

100

C₈

—

101

C₈

#88060

yes

characteristic

102

C₈

#88065

yes

characteristic

103

C₂²

#11718083

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

104

—

107

C₄

—

208

—

108

C₄

#11718075

yes

characteristic

109

C₄

#11718089

yes

characteristic

110

C₂

#13112796

yes

characteristic

111

C₂

#13113022

yes

characteristic

112

C₂

#13113028

yes

characteristic

	subgroup	G/H
center	C₂²	#11718083
commutator	C₁₃×C₁₆	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₁₃
Fitting	C₁₃×#1372	C₂
socle, prime radical	C₂²×C₁₃	#357
cosocle	C₄×C₁₃×C₁₆	C₂²
hypercenter	#1372	D₁₃
hypocenter	C₁₃	#7386
LCS	G ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1372

subgroup

G/H

center

C₂²

#11718083

commutator

C₁₃×C₁₆

C₂×C₈

Frattini

C₄×C₁₆

C₂×D₁₃

Fitting

C₁₃×#1372

C₂

socle, prime radical

C₂²×C₁₃

#357

cosocle

C₄×C₁₃×C₁₆

C₂²

D₁₃

C₁₃

G ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1372

N	H ≅ G/N	type
20. Dic₁₀₄	C₈	semidirect
91. C₁₃	#7386	semidirect

H ≅ G/N

type

20. Dic₁₀₄

C₈

semidirect

91. C₁₃

#7386

semidirect

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

subgroup	index
C₄×Dic₁₀₄	2
#13112970	2
C₁₃×#1372	2
#7386	13

subgroup

index

C₄×Dic₁₀₄

#13112970

C₁₃×#1372

#7386

Subgroups — #15750377

Conjugacy classes of subgroups

Complements

Maximal subgroups