Group #15777630 — subgroups

Subgroups — #15777630

Order 3328.

398 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 103 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	#13117085	C₂	1	4	yes	maximal, complemented
2	#13114451	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×#6090	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#493	—	2	2	no	—
6	C₁₃×#493	C₂²	1	2	yes	—
7	C₂×#88066	—	2	8	no	—
8	#11718356	—	2	2	no	—
9	#11718356	C₂²	1	2	yes	—
10	C₈²×C₁₃	C₂²	1	1	yes	characteristic
11	C₄×#6090	C₂²	1	2	yes	—
12	C₂×C₁₃×D₈	—	2	4	no	—
13	C₂×C₁₃×D₈	—	4	4	no	—
14	C₁₃×#127	—	2	2	no	—
15	C₂×D₁₀₄	—	2	4	no	—
16	C₂×D₁₀₄	—	4	4	no	—
17	#88128	—	2	2	no	—
18	C₄×C₈×C₁₃	D₄	1	2	yes	—
19	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	#88066	—	4	16	no	—
21	C₂×#6090	D₄	1	4	yes	—
22	#12515	—	13	1	no	maximal, Sylow
23	C₂×D₄×C₁₃	—	4	2	no	—
24	C₂×D₄×C₁₃	—	4	4	no	—
25	C₂×D₅₂	—	4	2	no	—
26	C₂×D₅₂	—	4	4	no	—
27	C₁₃×D₈	—	4	8	no	—
28	C₁₃×D₈	—	8	8	no	—
29	C₂×C₈×C₁₃	D₈	1	4	yes	—
30	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
31	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
32	D₁₀₄	—	4	8	no	—
33	D₁₀₄	—	8	8	no	—
34	#6090	D₈	1	8	yes	complemented
35	#2246	—	13	4	no	—
36	#1712	—	13	1	no	—
37	#1712	D₁₃	1	1	yes	characteristic, complemented
38	C₈×C₁₆	—	13	1	no	—
39	C₂³×C₁₃	—	8	2	no	—
40	C₂²×D₁₃	—	8	2	no	—
41	D₄×C₁₃	—	8	4	no	—
42	D₄×C₁₃	—	8	8	no	—
43	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
44	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
45	D₅₂	—	8	4	no	—
46	D₅₂	—	8	8	no	—
47	C₈×C₁₃	D₁₆	1	8	yes	complemented
48	C₈×C₁₃	C₂×D₈	1	4	yes	—
49	C₂×D₁₆	—	26	8	no	—
50	#493	—	2	2	no	—
51	#493	—	13	2	no	—
52	#493	—	26	2	no	—
53	#493	C₂×D₁₃	1	2	yes	—
54	C₄×C₁₆	—	13	2	no	—
55	C₈²	C₂×D₁₃	1	1	yes	characteristic
56	C₂²×C₁₃	—	16	2	no	—
57	C₂²×C₁₃	—	16	4	no	—
58	C₂²×C₁₃	#493	1	1	yes	characteristic
59	C₂×D₁₃	—	16	2	no	—
60	C₂×D₁₃	—	16	4	no	—
61	C₄×C₁₃	#493	1	2	yes	—
62	C₄×C₁₃	C₂×D₁₆	1	4	yes	—
63	C₂×D₈	—	2	4	no	—
64	C₂×D₈	—	4	4	no	—
65	C₂×D₈	—	26	4	no	—
66	C₂×D₈	—	52	4	no	—
67	#127	—	2	2	no	—
68	#127	—	26	2	no	—
69	D₁₆	—	52	16	no	—
70	C₂×C₁₆	—	13	4	no	—
71	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
72	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
73	C₂×C₁₃	—	32	2	no	—
74	C₂×C₁₃	#1712	1	1	yes	characteristic
75	C₂×C₁₃	#2246	1	2	yes	—
76	D₁₃	—	32	2	no	—
77	C₂×D₄	—	4	4	no	—
78	C₂×D₄	—	4	2	no	—
79	C₂×D₄	—	52	4	no	—
80	C₂×D₄	—	52	2	no	—
81	D₈	—	4	8	no	—
82	D₈	—	8	8	no	—
83	D₈	—	52	8	no	—
84	D₈	—	104	8	no	—
85	C₂×C₈	#6104	1	4	yes	—
86	C₂×C₈	D₄×D₁₃	1	2	yes	—
87	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
88	C₁₆	—	13	8	no	—
89	C₁₃	#12515	1	1	yes	characteristic, complemented, Sylow
90	C₂³	—	8	2	no	—
91	C₂³	—	104	2	no	—
92	D₄	—	8	8	no	—
93	D₄	—	8	4	no	—
94	D₄	—	104	8	no	—
95	D₄	—	104	4	no	—
96	C₂×C₄	C₂×#6104	1	2	yes	—
97	C₂×C₄	#88194	1	1	yes	characteristic
98	C₈	#88066	1	8	yes	complemented
99	C₈	D₈×D₁₃	1	4	yes	—
100	C₂²	—	16	2	no	—
101	C₂²	—	16	4	no	—
102	C₂²	—	208	2	no	—
103	C₂²	—	208	4	no	—
104	C₂²	#11718734	1	1	yes	characteristic
105	C₄	C₂×#88066	1	4	yes	—
106	C₄	#11718812	1	2	yes	—
107	C₂	—	32	2	no	—
108	C₂	—	416	2	no	—
109	C₂	#13114450	1	1	yes	characteristic
110	C₂	#13117085	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13117085

C₂

yes

maximal, complemented

#13114451

C₂

yes

maximal, characteristic, complemented

C₈×#6090

C₂

yes

maximal, characteristic, complemented

C₁₃×#1712

C₂

yes

maximal, characteristic, complemented

C₁₃×#493

—

C₁₃×#493

C₂²

yes

—

C₂×#88066

—

#11718356

—

#11718356

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

C₄×#6090

C₂²

yes

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₁₃×#127

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

#88128

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88066

—

C₂×#6090

D₄

yes

—

#12515

—

maximal, Sylow

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₁₃×D₈

—

C₁₃×D₈

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

D₁₀₄

—

D₁₀₄

—

#6090

D₈

yes

—

—

D₁₃

yes

characteristic, complemented

C₈×C₁₆

—

C₂³×C₁₃

—

C₂²×D₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

D₅₂

—

D₅₂

—

C₈×C₁₃

D₁₆

yes

complemented

C₈×C₁₃

C₂×D₈

yes

—

C₂×D₁₆

—

—

—

—

C₂×D₁₃

yes

—

C₄×C₁₆

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#493

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

#493

yes

—

C₄×C₁₃

C₂×D₁₆

yes

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

#127

—

#127

—

D₁₆

—

C₂×C₁₆

—

C₄×C₈

C₂²⋊D₁₃

yes

—

C₄×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

#1712

yes

characteristic

C₂×C₁₃

#2246

yes

—

D₁₃

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

D₈

—

104

—

C₂×C₈

#6104

yes

—

C₂×C₈

D₄×D₁₃

yes

—

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

C₁₆

—

C₁₃

#12515

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

104

—

D₄

—

D₄

—

D₄

—

104

—

D₄

—

104

—

C₂×C₄

C₂×#6104

yes

—

C₂×C₄

yes

C₈

yes

C₈

D₈×D₁₃

yes

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

208

—

103

C₂²

—

208

—

104

C₂²

#11718734

yes

characteristic

105

C₄

C₂×#88066

yes

—

106

C₄

#11718812

yes

—

107

C₂

—

108

C₂

—

416

—

109

C₂

#13114450

yes

characteristic

110

C₂

#13117085

yes

—

	subgroup	G/H
center	C₂²	#11718734
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1712	C₂
socle	C₂²×C₁₃	#493
hypercenter	#1712	D₁₃
hypocenter	C₁₃	#12515
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#11718734

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1712

C₂

C₂²×C₁₃

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #13117085	C₂	semidirect
2. #13114451	C₂	semidirect
3. C₈×#6090	C₂	semidirect
4. C₁₃×#1712	C₂	semidirect
34. #6090	D₈	semidirect
37. #1712	D₁₃	semidirect
47. C₈×C₁₃	D₁₆	semidirect
89. C₁₃	#12515	semidirect
98. C₈	#88066	semidirect

H ≅ G/N

type

1. #13117085

C₂

semidirect

2. #13114451

C₂

semidirect

3. C₈×#6090

C₂

semidirect

4. C₁₃×#1712

C₂

semidirect

34. #6090

D₈

semidirect

37. #1712

D₁₃

semidirect

47. C₈×C₁₃

D₁₆

semidirect

89. C₁₃

#12515

semidirect

98. C₈

#88066

semidirect

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

subgroup	index
#13117085	2
#13114451	2
C₈×#6090	2
C₁₃×#1712	2
#12515	13

subgroup

index

C₁₃×#1712

Subgroups — #15777630

Conjugacy classes of subgroups

Complements

Maximal subgroups