Group #16807817 — subgroups

Subgroups — #261×#564 #16807817

Order 3328.

1800 conjugacy classes of subgroups; only the 776 normal subgroups are shown.

normal subgroups only

This group has 1800 conjugacy classes of subgroups — too many to draw. Only its 776 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	#13131382	C₂	1	15	yes	maximal
2	#125×#261	C₂	1	15	yes	maximal
3	D₁₃×#564	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#564	C₄	1	1	yes	characteristic, complemented
5	D₁₃×#125	C₂²	1	15	yes	—
6	C₁₃⋊#564	C₄	1	15	yes	complemented
7	#11719083	C₂²	1	30	yes	—
8	#11719082	C₂²	1	60	yes	—
9	C₄⋊C₄×#261	C₂²	1	30	yes	—
10	#11719060	C₂²	1	15	yes	—
11	C₄²×#261	C₂²	1	5	yes	—
12	C₂×C₄⋊#261	C₂³	1	30	yes	—
13	C₂×C₄×#261	C₂³	1	15	yes	—
14	C₂×C₄×#261	C₂³	1	15	yes	—
15	C₁₃×#125	C₂×C₄	1	15	yes	—
16	C₄⋊C₄×D₁₃	C₂³	1	30	yes	—
17	#88144	C₂×C₄	1	30	yes	—
18	#88143	C₂×C₄	1	60	yes	—
19	C₄²×D₁₃	C₂³	1	5	yes	—
20	#88124	C₂×C₄	1	15	yes	—
21	(C₂×#261)⋊C₄	C₂³	1	60	yes	—
22	C₄×#564	—	13	1	no	maximal, Sylow
23	C₂²×#261	C₂⁴	1	15	yes	—
24	C₂²×#261	C₂⁴	1	1	yes	characteristic
25	C₂×C₄×D₁₃	C₂⁴	1	15	yes	—
26	C₄⋊#261	—	4	30	no	—
27	C₄×#261	—	2	30	no	—
28	C₄×#261	—	4	15	no	—
29	C₁₃×C₄⋊C₄	C₂²×C₄	1	30	yes	—
30	C₄²×C₁₃	C₂²×C₄	1	5	yes	—
31	C₄⋊Dic₁₃	C₂²×C₄	1	30	yes	—
32	#6101	C₂²×C₄	1	60	yes	—
33	C₄×Dic₁₃	C₂²×C₄	1	15	yes	—
34	C₂×#564	—	13	1	no	—
35	#2382	—	13	15	no	—
36	C₄×#125	—	13	15	no	—
37	C₂²×D₁₃	C₂⁵	1	1	yes	characteristic
38	C₂×#261	—	2	15	no	—
39	C₂×#261	—	2	30	no	—
40	C₂×#261	2₋¹⁺⁴	1	6	yes	—
41	C₂×C₄×C₁₃	C₂³×C₄	1	15	yes	—
42	C₂×Dic₁₃	C₂³×C₄	1	15	yes	—
43	C₄×D₁₃	—	2	15	no	—
44	C₄×D₁₃	—	2	15	no	—
45	#564	—	13	15	no	—
46	#564	#261	1	1	yes	characteristic, direct product factor
47	C₂×#125	—	13	15	no	—
48	#389	—	13	30	no	—
49	#387	—	13	60	no	—
50	#382	—	13	15	no	—
51	C₄×C₄⋊C₄	—	13	30	no	—
52	C₄³	—	13	5	no	—
53	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
54	C₂×D₁₃	C₂×2₋¹⁺⁴	1	3	yes	—
55	C₂×D₁₃	#585	1	3	yes	—
56	#261	—	4	15	no	—
57	#261	#564	1	4	yes	direct product factor
58	C₄×C₁₃	—	2	15	no	—
59	Dic₁₃	—	2	15	no	—
60	#125	—	13	60	no	—
61	#125	—	13	30	no	—
62	#125	—	13	15	no	—
63	#125	C₂×#261	1	15	yes	—
64	C₂×C₄⋊C₄	—	13	30	no	—
65	C₂×C₄⋊C₄	—	13	30	no	—
66	C₂×C₄²	—	13	5	no	—
67	C₂×C₄²	—	13	15	no	—
68	C₂×C₄²	—	13	15	no	—
69	#95	—	13	60	no	—
70	C₂×C₁₃	C₄×2₋¹⁺⁴	1	3	yes	—
71	D₁₃	C₂×#564	1	1	yes	characteristic
72	D₁₃	#3531	1	3	yes	—
73	C₂²×C₄	—	13	1	no	—
74	C₂²×C₄	—	13	15	no	—
75	C₂²×C₄	—	13	15	no	—
76	C₄⋊C₄	—	13	60	no	—
77	C₄⋊C₄	—	13	30	no	—
78	C₄⋊C₄	—	52	30	no	—
79	C₄⋊C₄	C₂²×#261	1	30	yes	—
80	C₄²	—	13	15	no	—
81	C₄²	—	26	30	no	—
82	C₄²	—	52	15	no	—
83	C₄²	C₂²×#261	1	5	yes	—
84	C₁₃	C₄×#564	1	1	yes	characteristic, complemented, Sylow
85	C₂³	—	13	1	no	—
86	C₂×C₄	—	13	6	no	—
87	C₂×C₄	—	13	15	no	—
88	C₂×C₄	—	26	15	no	—
89	C₂×C₄	—	26	15	no	—
90	C₂×C₄	—	26	15	no	—
91	C₂×C₄	—	26	30	no	—
92	C₂×C₄	C₂³×#261	1	15	yes	—
93	C₂²	—	13	3	no	—
94	C₂²	—	13	3	no	—
95	C₂²	C₂⁴×#261	1	1	yes	characteristic
96	C₄	—	2	15	no	—
97	C₄	—	13	4	no	—
98	C₄	—	26	15	no	—
99	C₄	—	52	15	no	—
100	C₂	—	13	1	no	—
101	C₂	—	13	3	no	—
102	C₂	2₋¹⁺⁴×#261	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13131382

C₂

yes

maximal

#125×#261

C₂

yes

maximal

D₁₃×#564

C₂

yes

maximal, characteristic

C₁₃×#564

C₄

yes

characteristic, complemented

D₁₃×#125

C₂²

yes

—

C₁₃⋊#564

C₄

yes

complemented

#11719083

C₂²

yes

—

#11719082

C₂²

yes

—

C₄⋊C₄×#261

C₂²

yes

—

#11719060

C₂²

yes

—

C₄²×#261

C₂²

yes

—

C₂×C₄⋊#261

C₂³

yes

—

C₂×C₄×#261

C₂³

yes

—

C₂×C₄×#261

C₂³

yes

—

C₁₃×#125

C₂×C₄

yes

—

C₄⋊C₄×D₁₃

C₂³

yes

—

#88144

C₂×C₄

yes

—

#88143

C₂×C₄

yes

—

C₄²×D₁₃

C₂³

yes

—

#88124

C₂×C₄

yes

—

(C₂×#261)⋊C₄

C₂³

yes

—

C₄×#564

—

maximal, Sylow

C₂²×#261

C₂⁴

yes

—

C₂²×#261

C₂⁴

yes

characteristic

C₂×C₄×D₁₃

C₂⁴

yes

—

C₄⋊#261

—

C₄×#261

—

C₄×#261

—

C₁₃×C₄⋊C₄

C₂²×C₄

yes

—

C₄²×C₁₃

C₂²×C₄

yes

—

C₄⋊Dic₁₃

C₂²×C₄

yes

—

#6101

C₂²×C₄

yes

—

C₄×Dic₁₃

C₂²×C₄

yes

—

C₂×#564

—

#2382

—

C₄×#125

—

C₂²×D₁₃

C₂⁵

yes

—

—

2₋¹⁺⁴

yes

—

C₂×C₄×C₁₃

C₂³×C₄

yes

—

C₂×Dic₁₃

C₂³×C₄

yes

—

C₄×D₁₃

—

C₄×D₁₃

—

#564

—

#564

#261

yes

characteristic, direct product factor

—

—

—

—

C₄×C₄⋊C₄

—

C₄³

—

C₂²×C₁₃

C₂⁴×C₄

yes

characteristic

C₂×D₁₃

C₂×2₋¹⁺⁴

yes

—

C₂×D₁₃

#585

yes

—

#261

—

#261

#564

yes

direct product factor

C₄×C₁₃

—

Dic₁₃

—

—

—

—

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

#95

—

C₂×C₁₃

C₄×2₋¹⁺⁴

yes

—

D₁₃

C₂×#564

yes

characteristic

D₁₃

#3531

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×#261

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×#261

yes

—

C₁₃

C₄×#564

yes

characteristic, complemented, Sylow

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂³×#261

yes

—

C₂²

—

C₂²

—

C₂²

C₂⁴×#261

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

100

C₂

—

101

C₂

—

102

C₂

2₋¹⁺⁴×#261

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×#261
commutator, socle	C₂²×C₁₃	C₂⁴×C₄
Fitting	C₁₃×#564	C₄
prime radical, cosocle	C₂²×D₁₃	C₂⁵
hypercenter	#564	#261
hypocenter	C₁₃	C₄×#564
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×#261

commutator, socle

C₂²×C₁₃

C₂⁴×C₄

Fitting

C₁₃×#564

C₄

prime radical, cosocle

C₂²×D₁₃

C₂⁵

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
4. C₁₃×#564	C₄	semidirect
6. C₁₃⋊#564	C₄	semidirect
46. #564	#261	direct (N × H)
57. #261	#564	direct (N × H)
84. C₁₃	C₄×#564	semidirect

H ≅ G/N

type

4. C₁₃×#564

C₄

semidirect

6. C₁₃⋊#564

C₄

semidirect

46. #564

#261

direct (N × H)

57. #261

#564

direct (N × H)

84. C₁₃

C₄×#564

semidirect

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

subgroup	index
#13131382	2
#125×#261	2
D₁₃×#564	2
C₄×#564	13

subgroup

index

#13131382

#125×#261

D₁₃×#564

C₄×#564

Subgroups — #261×#564 #16807817

Conjugacy classes of subgroups

Complements

Maximal subgroups