Group #15777612 — subgroups

Subgroups — #15777612

Order 3328.

260 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 79 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	#13117063	C₂	1	2	yes	maximal, complemented
2	#13117059	C₂	1	2	yes	maximal
3	#13114446	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#6090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1709	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#494	C₂²	1	1	yes	characteristic
7	C₁₃×#493	C₂²	1	1	yes	characteristic
8	C₁₃×#486	—	2	2	no	—
9	#11718343	C₂²	1	2	yes	—
10	#11718342	—	2	2	no	—
11	C₈²×C₁₃	C₂²	1	1	yes	characteristic
12	#11718156	—	2	4	no	—
13	#11718154	—	2	4	no	—
14	C₄×#6090	C₂²	1	2	yes	—
15	C₂×C₁₃×Q₁₆	—	2	2	no	—
16	C₂×C₁₃×D₈	—	2	2	no	—
17	C₁₃×C₄⋊Q₈	—	2	1	no	—
18	C₁₃×#127	—	2	1	no	—
19	#88123	—	2	2	no	—
20	C₁₃×#102	—	4	4	no	—
21	C₄×C₈×C₁₃	D₄	1	2	yes	—
22	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	#88058	—	2	2	no	—
24	#88058	—	2	2	no	—
25	#88056	—	4	4	no	—
26	C₂×#6090	D₄	1	4	yes	—
27	#12511	—	13	1	no	maximal, Sylow
28	C₂×Q₈×C₁₃	—	4	1	no	—
29	C₂×D₄×C₁₃	—	4	1	no	—
30	C₂×D₄×C₁₃	—	4	2	no	—
31	C₂×Dic₂₆	—	4	2	no	—
32	C₁₃×Q₁₆	—	4	4	no	—
33	C₁₃×D₈	—	4	4	no	—
34	C₂×C₈×C₁₃	D₈	1	4	yes	—
35	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
36	C₁₃×C₄⋊C₄	—	4	2	no	—
37	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊Dic₁₃	—	4	2	no	—
39	C₄⋊Dic₁₃	—	4	2	no	—
40	#6090	—	2	4	no	—
41	#2241	—	13	2	no	—
42	#2240	—	13	2	no	—
43	#1710	—	13	1	no	—
44	#1709	D₁₃	1	1	yes	characteristic
45	C₈×C₁₆	—	13	1	no	—
46	C₂³×C₁₃	—	8	1	no	—
47	Q₈×C₁₃	—	8	2	no	—
48	D₄×C₁₃	—	8	2	no	—
49	D₄×C₁₃	—	8	4	no	—
50	C₂×C₄×C₁₃	—	8	1	no	—
51	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
52	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
53	C₂×Dic₁₃	—	8	2	no	—
54	Dic₂₆	—	8	4	no	—
55	C₈×C₁₃	—	2	4	no	—
56	C₈×C₁₃	#135	1	4	yes	—
57	#500	—	13	2	no	—
58	#494	C₂×D₁₃	1	1	yes	characteristic
59	#493	C₂×D₁₃	1	1	yes	characteristic
60	#487	—	26	2	no	—
61	#486	—	2	2	no	—
62	#358	—	26	4	no	—
63	#357	—	26	4	no	—
64	C₄×C₁₆	—	13	2	no	—
65	C₈²	C₂×D₁₃	1	1	yes	characteristic
66	C₂²×C₁₃	—	16	1	no	—
67	C₂²×C₁₃	—	16	2	no	—
68	C₂²×C₁₃	#493	1	1	yes	characteristic
69	C₄×C₁₃	—	16	1	no	—
70	C₄×C₁₃	#495	1	2	yes	—
71	C₄×C₁₃	#508	1	4	yes	—
72	Dic₁₃	—	16	2	no	—
73	C₂×Q₁₆	—	2	2	no	—
74	C₂×D₈	—	2	2	no	—
75	C₄⋊Q₈	—	2	1	no	—
76	C₄⋊Q₈	—	26	2	no	—
77	#127	—	2	1	no	—
78	C₂×C₁₆	—	13	4	no	—
79	#107	—	26	2	no	—
80	#107	—	26	2	no	—
81	#103	—	52	4	no	—
82	#102	—	4	4	no	—
83	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
84	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
85	C₂×C₁₃	—	32	1	no	—
86	C₂×C₁₃	#1709	1	1	yes	characteristic
87	C₂×C₁₃	#2249	1	2	yes	—
88	C₂×Q₈	—	4	1	no	—
89	C₂×Q₈	—	52	2	no	—
90	C₂×D₄	—	4	2	no	—
91	C₂×D₄	—	4	1	no	—
92	Q₁₆	—	4	4	no	—
93	D₈	—	4	4	no	—
94	C₂×C₈	#6104	1	4	yes	—
95	C₂×C₈	D₄×D₁₃	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄⋊C₄	—	52	2	no	—
98	C₄⋊C₄	—	52	2	no	—
99	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
100	C₁₆	—	26	4	no	—
101	C₁₃	#12511	1	1	yes	characteristic, complemented, Sylow
102	C₂³	—	8	1	no	—
103	Q₈	—	8	2	no	—
104	Q₈	—	104	4	no	—
105	D₄	—	8	2	no	—
106	D₄	—	8	4	no	—
107	C₂×C₄	—	8	1	no	—
108	C₂×C₄	—	104	2	no	—
109	C₂×C₄	C₂×#6104	1	2	yes	—
110	C₂×C₄	#88194	1	1	yes	characteristic
111	C₈	—	2	4	no	—
112	C₈	#88166	1	4	yes	—
113	C₂²	—	16	1	no	—
114	C₂²	—	16	2	no	—
115	C₂²	#11718734	1	1	yes	characteristic
116	C₄	—	16	1	no	—
117	C₄	—	208	2	no	—
118	C₄	#11718816	1	2	yes	—
119	C₄	#11718855	1	4	yes	—
120	C₂	—	32	1	no	—
121	C₂	#13114441	1	1	yes	characteristic
122	C₂	#13117095	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13117063

C₂

yes

maximal, complemented

#13117059

C₂

yes

maximal

#13114446

C₂

yes

maximal, characteristic, complemented

C₈×#6090

C₂

yes

maximal, characteristic, complemented

C₁₃×#1709

C₂

yes

maximal, characteristic

C₁₃×#494

C₂²

yes

characteristic

C₁₃×#493

C₂²

yes

characteristic

C₁₃×#486

—

#11718343

C₂²

yes

—

#11718342

—

C₈²×C₁₃

C₂²

yes

—

—

C₂²

yes

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×D₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×#127

—

#88123

—

C₁₃×#102

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

D₄

yes

—

#12511

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×Dic₂₆

—

C₁₃×Q₁₆

—

C₁₃×D₈

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

—

D₁₃

yes

characteristic

C₈×C₁₆

—

C₂³×C₁₃

—

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

#135

yes

—

#500

—

#494

C₂×D₁₃

yes

characteristic

#493

C₂×D₁₃

yes

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#493

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

#495

yes

—

C₄×C₁₃

#508

yes

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

—

—

—

—

C₄×C₈

C₂²⋊D₁₃

yes

—

C₄×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

#1709

yes

characteristic

C₂×C₁₃

#2249

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

#6104

yes

—

C₂×C₈

D₄×D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

100

C₁₆

—

101

C₁₃

#12511

yes

characteristic, complemented, Sylow

102

C₂³

—

103

Q₈

—

104

Q₈

—

104

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

104

—

109

C₂×C₄

C₂×#6104

yes

—

110

C₂×C₄

#88194

yes

characteristic

111

C₈

—

112

C₈

#88166

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

#11718734

yes

characteristic

116

C₄

—

117

C₄

—

208

—

118

C₄

#11718816

yes

—

119

C₄

#11718855

yes

—

120

C₂

—

121

C₂

#13114441

yes

characteristic

122

C₂

#13117095

yes

—

	subgroup	G/H
center	C₂²	#11718734
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1709	C₂
socle	C₂²×C₁₃	#493
prime radical	C₁₃×#493	C₂²
hypercenter	#1709	D₁₃
hypocenter	C₁₃	#12511
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#11718734

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1709

C₂

socle

C₂²×C₁₃

#493

prime radical

C₁₃×#493

C₂²

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #13117063	C₂	semidirect
3. #13114446	C₂	semidirect
4. C₈×#6090	C₂	semidirect
101. C₁₃	#12511	semidirect

H ≅ G/N

type

1. #13117063

C₂

semidirect

3. #13114446

C₂

semidirect

4. C₈×#6090

C₂

semidirect

101. C₁₃

#12511

semidirect

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

subgroup	index
#13117063	2
#13117059	2
#13114446	2
C₈×#6090	2
C₁₃×#1709	2
#12511	13

subgroup

index

C₁₃×#1709

Subgroups — #15777612

Conjugacy classes of subgroups

Complements

Maximal subgroups