Group #16778187 — subgroups

Subgroups — #16778187

Order 3328.

186 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 22 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	#13129968	C₂	1	2	yes	maximal, complemented
2	#13114450	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#493	C₄	1	1	yes	characteristic, complemented
4	#11718734	—	2	2	no	—
5	#11718356	C₂²	1	1	yes	characteristic
6	#11718225	—	2	2	no	—
7	C₈×#1055	C₄	1	1	yes	characteristic, complemented
8	C₂×C₁₃×D₈	—	4	1	no	—
9	C₁₃×#127	—	2	1	no	—
10	C₂×#6104	—	4	4	no	—
11	C₂×D₁₀₄	—	4	1	no	—
12	#88128	D₄	1	2	yes	—
13	#88107	—	4	2	no	—
14	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₄×#1055	—	2	1	no	—
16	#7529	—	13	1	no	maximal, Sylow
17	C₂×D₄×C₁₃	—	4	2	no	—
18	C₂×D₄×C₁₃	—	4	1	no	—
19	C₂×D₅₂	—	2	2	no	—
20	C₂×D₅₂	—	4	2	no	—
21	C₂²⋊#261	—	8	2	no	—
22	C₁₃×D₈	—	8	2	no	—
23	C₂×C₈×C₁₃	—	2	1	no	—
24	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
25	#6104	—	8	8	no	—
26	C₂×#1055	—	2	2	no	—
27	D₁₀₄	—	8	2	no	—
28	#1712	—	13	1	no	—
29	#1408	—	13	2	no	—
30	C₂³×C₁₃	—	8	1	no	—
31	C₂²×D₁₃	—	4	2	no	—
32	C₂×#261	—	16	2	no	—
33	D₄×C₁₃	—	8	4	no	—
34	D₄×C₁₃	—	8	2	no	—
35	C₂×C₄×C₁₃	—	2	1	no	—
36	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
37	D₅₂	—	8	2	no	—
38	D₅₂	—	8	4	no	—
39	C₈×C₁₃	—	2	2	no	—
40	#1055	—	2	4	no	—
41	#493	—	13	1	no	—
42	#493	—	26	2	no	—
43	#493	#261	1	1	yes	characteristic, complemented
44	#353	—	26	2	no	—
45	C₈²	—	13	1	no	—
46	C₂²×C₁₃	—	16	2	no	—
47	C₂²×C₁₃	—	16	1	no	—
48	C₂²×C₁₃	#353	1	1	yes	characteristic
49	C₂×D₁₃	—	8	2	no	—
50	C₂×D₁₃	—	16	2	no	—
51	#261	—	32	2	no	—
52	C₄×C₁₃	—	2	2	no	—
53	C₄×C₁₃	—	2	1	no	—
54	C₂×D₈	—	4	1	no	—
55	C₂×D₈	—	52	1	no	—
56	C₂×D₈	—	52	4	no	—
57	#127	—	2	1	no	—
58	#127	—	13	2	no	—
59	#99	—	52	2	no	—
60	C₄×C₈	—	26	1	no	—
61	C₄×C₈	C₂×#261	1	1	yes	characteristic
62	C₂×C₁₃	—	2	1	no	—
63	C₂×C₁₃	—	32	1	no	—
64	C₂×C₁₃	#1408	1	1	yes	characteristic
65	D₁₃	—	16	2	no	—
66	C₂×D₄	—	4	2	no	—
67	C₂×D₄	—	4	1	no	—
68	C₂×D₄	—	26	2	no	—
69	C₂×D₄	—	52	2	no	—
70	D₈	—	8	2	no	—
71	D₈	—	104	2	no	—
72	D₈	—	104	8	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	26	2	no	—
75	C₂²⋊C₄	—	104	2	no	—
76	C₄²	C₂²⋊#261	1	1	yes	characteristic
77	C₁₃	#7529	1	1	yes	characteristic, complemented, Sylow
78	C₂³	—	8	1	no	—
79	C₂³	—	52	2	no	—
80	D₄	—	8	2	no	—
81	D₄	—	8	4	no	—
82	D₄	—	104	2	no	—
83	D₄	—	104	4	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	208	2	no	—
86	C₂×C₄	#88119	1	1	yes	characteristic
87	C₈	—	2	2	no	—
88	C₈	—	26	4	no	—
89	C₂²	—	16	2	no	—
90	C₂²	—	16	1	no	—
91	C₂²	—	104	2	no	—
92	C₂²	—	208	2	no	—
93	C₂²	#11718294	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	416	2	no	—
97	C₂	—	2	1	no	—
98	C₂	—	32	1	no	—
99	C₂	—	208	2	no	—
100	C₂	#13129967	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13129968

C₂

yes

maximal, complemented

#13114450

C₂

yes

maximal, characteristic

C₁₃×#493

C₄

yes

characteristic, complemented

#11718734

—

#11718356

C₂²

yes

characteristic

#11718225

—

C₈×#1055

C₄

yes

characteristic, complemented

C₂×C₁₃×D₈

—

C₁₃×#127

—

C₂×#6104

—

C₂×D₁₀₄

—

#88128

D₄

yes

—

#88107

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂²⋊#261

—

C₁₃×D₈

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

C₂²⋊C₄

yes

characteristic

#6104

—

C₂×#1055

—

D₁₀₄

—

#1712

—

#1408

—

C₂³×C₁₃

—

C₂²×D₁₃

—

C₂×#261

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#99

yes

characteristic

D₅₂

—

D₅₂

—

C₈×C₁₃

—

—

—

—

yes

characteristic, complemented

#353

—

C₈²

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#353

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

#261

—

C₄×C₁₃

—

C₄×C₁₃

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

#127

—

#127

—

#99

—

C₄×C₈

—

C₄×C₈

C₂×#261

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

#1408

yes

characteristic

D₁₃

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

104

—

D₈

—

104

—

C₂×C₈

—

C₂×C₈

—

C₂²⋊C₄

—

104

—

C₄²

C₂²⋊#261

yes

characteristic

C₁₃

#7529

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

D₄

—

D₄

—

D₄

—

104

—

D₄

—

104

—

C₂×C₄

—

C₂×C₄

—

208

—

C₂×C₄

#88119

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

104

—

C₂²

—

208

—

C₂²

#11718294

yes

characteristic

C₄

—

C₄

—

C₄

—

416

—

C₂

—

C₂

—

C₂

—

208

—

100

C₂

#13129967

yes

characteristic

	subgroup	G/H
center	C₂	#13129967
commutator	C₄×C₈×C₁₃	C₂×C₄
Frattini	C₄×C₈	C₂×#261
Fitting	C₁₃×#493	C₄
socle	C₂×C₁₃	#1408
prime radical	#13114450	C₂
cosocle	#11718356	C₂²
hypercenter	#493	#261
hypocenter	C₁₃	#7529
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂

#13129967

commutator

C₄×C₈×C₁₃

C₂×C₄

Frattini

C₄×C₈

C₂×#261

Fitting

C₁₃×#493

C₄

C₂×C₁₃

C₂

C₂²

C₁₃

G ▷ C₄×C₈×C₁₃ ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. #13129968	C₂	semidirect
3. C₁₃×#493	C₄	semidirect
7. C₈×#1055	C₄	semidirect
43. #493	#261	semidirect
77. C₁₃	#7529	semidirect

H ≅ G/N

type

1. #13129968

C₂

semidirect

3. C₁₃×#493

C₄

semidirect

7. C₈×#1055

C₄

semidirect

43. #493

#261

semidirect

77. C₁₃

#7529

semidirect

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

subgroup	index
#13129968	2
#13114450	2
#7529	13

subgroup

index

#13129968

#13114450

#7529

Subgroups — #16778187

Conjugacy classes of subgroups

Complements

Maximal subgroups