Group #15749751 — subgroups

Subgroups — #15749751

Order 3328.

278 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 151 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×#88052	C₂	1	2	yes	maximal
2	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
4	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
5	C₂×#88052	C₄	1	4	yes	—
6	C₈²×C₁₃	—	2	2	no	—
7	#11718053	—	2	2	no	—
8	C₄×#6090	—	2	2	no	—
9	C₂²×C₈×C₁₃	—	2	2	no	—
10	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
11	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₁₃	—	2	2	no	—
13	C₄×C₈×C₁₃	C₈	1	4	yes	—
14	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
15	C₄×C₈×C₁₃	D₄	1	2	yes	—
16	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
17	#88052	—	2	4	no	—
18	#88052	C₈	1	8	yes	complemented
19	C₂×#6090	—	2	4	no	—
20	#7138	—	13	1	no	maximal, Sylow
21	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
22	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
23	C₂×C₈×C₁₃	—	2	8	no	—
24	C₂×C₈×C₁₃	—	2	4	no	—
25	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
26	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
28	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
29	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
31	C₄²×C₁₃	C₂×C₈	1	2	yes	—
32	#6090	—	2	8	no	—
33	C₄×M₃₂	—	13	2	no	—
34	C₂×C₈²	D₁₃	1	1	yes	characteristic
35	C₂³×C₁₃	#97	1	1	yes	characteristic
36	C₂×C₄×C₁₃	—	2	4	no	—
37	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
38	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
39	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
40	C₂×C₄×C₁₃	#97	1	2	yes	—
41	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
42	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
43	C₈×C₁₃	—	2	8	no	—
44	C₈×C₁₃	C₂²⋊C₈	1	4	yes	—
45	C₈×C₁₃	C₄⋊C₈	1	4	yes	—
46	C₂×M₃₂	—	13	4	no	—
47	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
48	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
49	#346	—	26	2	no	—
50	C₄×C₁₆	—	26	2	no	—
51	C₈²	—	2	2	no	—
52	C₂²×C₁₃	—	2	2	no	—
53	C₂²×C₁₃	#322	1	2	yes	—
54	C₂²×C₁₃	#336	1	1	yes	characteristic
55	C₄×C₁₃	—	2	2	no	—
56	C₄×C₁₃	C₈²	1	1	yes	characteristic
57	C₄×C₁₃	#322	1	1	yes	characteristic
58	C₄×C₁₃	#336	1	2	yes	—
59	C₄×C₁₃	#364	1	4	yes	—
60	C₂²×C₈	—	2	2	no	—
61	C₂²×C₈	C₄×D₁₃	1	2	yes	—
62	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
63	M₃₂	—	13	8	no	—
64	M₃₂	—	26	4	no	—
65	C₂×C₁₆	—	26	4	no	—
66	C₄×C₈	—	2	2	no	—
67	C₄×C₈	#1055	1	4	yes	—
68	C₄×C₈	Dic₂₆	1	1	yes	characteristic
69	C₄×C₈	D₅₂	1	1	yes	characteristic
70	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
71	C₂×C₁₃	—	2	2	no	—
72	C₂×C₁₃	#1273	1	1	yes	characteristic
73	C₂×C₁₃	#1376	1	2	yes	—
74	C₂²×C₄	#6099	1	2	yes	—
75	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
76	C₂×C₈	—	2	8	no	—
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	C₈×D₁₃	1	4	yes	—
79	C₂×C₈	#6094	1	4	yes	—
80	C₂×C₈	#6101	1	2	yes	—
81	C₂×C₈	#6103	1	2	yes	—
82	C₄²	C₂×#1055	1	2	yes	—
83	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
84	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
85	C₁₆	—	26	8	no	—
86	C₁₃	#7138	1	1	yes	characteristic, complemented, Sylow
87	C₂³	#88043	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	C₄×#1055	1	1	yes	characteristic
90	C₂×C₄	C₄⋊#1055	1	2	yes	—
91	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
92	C₂×C₄	#88055	1	2	yes	—
93	C₂×C₄	C₂²⋊#1055	1	2	yes	—
94	C₂×C₄	#88071	1	1	yes	characteristic
95	C₈	—	2	8	no	—
96	C₈	#88054	1	4	yes	—
97	C₈	#88060	1	4	yes	—
98	C₂²	—	2	2	no	—
99	C₂²	#11718046	1	2	yes	—
100	C₂²	#11718116	1	1	yes	characteristic
101	C₄	—	2	2	no	—
102	C₄	C₈×#1055	1	1	yes	characteristic
103	C₄	#11718047	1	1	yes	characteristic
104	C₄	#11718055	1	4	yes	—
105	C₄	#11718142	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	#13112663	1	1	yes	characteristic
108	C₂	#13112918	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#88052

C₂

yes

maximal

C₂×C₈²×C₁₃

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₄

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88052

C₄

yes

—

C₈²×C₁₃

—

#11718053

—

C₄×#6090

—

C₂²×C₈×C₁₃

—

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

C₈

yes

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

characteristic

#88052

—

#88052

C₈

yes

—

—

C₂²×C₄×C₁₃

C₄²

yes

characteristic

C₂²×C₄×C₁₃

M₁₆

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₂×C₈

yes

—

C₂×C₈×C₁₃

M₁₆

yes

—

C₄²×C₁₃

C₂²⋊C₄

yes

characteristic

C₄²×C₁₃

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

C₂×C₈

yes

—

#6090

—

C₄×M₃₂

—

C₂×C₈²

D₁₃

yes

characteristic

C₂³×C₁₃

#97

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#95

yes

characteristic

C₂×C₄×C₁₃

C₄×C₈

yes

characteristic

C₂×C₄×C₁₃

C₄×C₈

yes

—

C₂×C₄×C₁₃

#97

yes

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃

C₄⋊C₈

yes

—

C₈×C₁₃

—

C₈×C₁₃

C₂²⋊C₈

yes

—

C₈×C₁₃

C₄⋊C₈

yes

—

C₂×M₃₂

—

C₂×C₄×C₈

Dic₁₃

yes

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#346

—

C₄×C₁₆

—

C₈²

—

C₂²×C₁₃

—

C₂²×C₁₃

#322

yes

—

C₂²×C₁₃

#336

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

C₈²

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

C₄×C₁₃

#364

yes

—

C₂²×C₈

—

C₂²×C₈

C₄×D₁₃

yes

—

C₂×C₄²

C₂×Dic₁₃

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

#1055

yes

—

C₄×C₈

Dic₂₆

yes

characteristic

C₄×C₈

D₅₂

yes

characteristic

C₄×C₈

C₂²⋊D₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

#1273

yes

characteristic

C₂×C₁₃

#1376

yes

—

C₂²×C₄

#6099

yes

—

C₂²×C₄

C₄×Dic₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₁₃

yes

—

C₂×C₈

#6094

yes

—

C₂×C₈

#6101

yes

—

C₂×C₈

#6103

yes

—

C₄²

C₂×#1055

yes

—

C₄²

C₄⋊Dic₁₃

yes

characteristic

C₄²

C₂²⋊Dic₁₃

yes

characteristic

C₁₆

—

C₁₃

#7138

yes

characteristic, complemented, Sylow

C₂³

#88043

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×#1055

yes

characteristic

C₂×C₄

C₄⋊#1055

yes

—

C₂×C₄

C₈×Dic₁₃

yes

—

C₂×C₄

#88055

yes

—

C₂×C₄

C₂²⋊#1055

yes

—

C₂×C₄

#88071

yes

characteristic

C₈

—

C₈

#88054

yes

—

C₈

#88060

yes

—

C₂²

—

C₂²

#11718046

yes

—

100

C₂²

#11718116

yes

characteristic

101

C₄

—

102

C₄

C₈×#1055

yes

characteristic

103

C₄

#11718047

yes

characteristic

104

C₄

#11718055

yes

—

105

C₄

#11718142

yes

—

106

C₂

—

107

C₂

#13112663

yes

characteristic

108

C₂

#13112918

yes

—

	subgroup	G/H
center	C₄×C₈	D₅₂
commutator	C₄×C₁₃	C₈²
Frattini	C₂×C₄×C₈	C₂×D₁₃
Fitting	C₂×C₈²×C₁₃	C₂
socle	C₂²×C₁₃	#336
prime radical	C₂³×C₁₃	#97
cosocle	C₂×C₄×C₈×C₁₃	C₂²
hypercenter	C₂×C₈²	D₁₃
hypocenter	C₁₃	#7138
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

subgroup

G/H

center

C₄×C₈

D₅₂

commutator

C₄×C₁₃

C₈²

Frattini

C₂×C₄×C₈

C₂×D₁₃

Fitting

C₂×C₈²×C₁₃

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄×C₈×C₁₃

C₂²

hypercenter

C₂×C₈²

D₁₃

hypocenter

C₁₃

#7138

LCS

G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

N	H ≅ G/N	type
18. #88052	C₈	semidirect
86. C₁₃	#7138	semidirect

H ≅ G/N

type

18. #88052

C₈

semidirect

86. C₁₃

#7138

semidirect

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

subgroup	index
C₄×#88052	2
C₂×C₈²×C₁₃	2
#7138	13

subgroup

index

C₄×#88052

C₂×C₈²×C₁₃

#7138

Subgroups — #15749751

Conjugacy classes of subgroups

Complements

Maximal subgroups