Group #15777591 — subgroups

Subgroups — #15777591

Order 3328.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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	#13117118	C₂	1	1	yes	maximal, characteristic
2	#13117063	C₂	1	2	yes	maximal
3	#13114430	C₂	1	2	yes	maximal
4	#13112752	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2247	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×Q₃₂	—	2	2	no	—
7	C₁₃×#494	C₂²	1	2	yes	—
8	#11718754	—	2	2	no	—
9	#11718740	—	2	2	no	—
10	#11718343	C₂²	1	2	yes	—
11	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	#11718156	—	2	4	no	—
13	#11718097	—	2	2	no	—
14	C₄×#6090	C₂²	1	1	yes	characteristic
15	C₈×#1055	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×Q₁₆	—	2	2	no	—
17	C₂×C₁₃×Q₁₆	—	2	2	no	—
18	C₁₃×C₄⋊Q₈	—	2	2	no	—
19	#88123	—	2	2	no	—
20	C₁₃×Q₃₂	—	4	4	no	—
21	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
22	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	#88058	—	2	2	no	—
24	#88058	—	2	2	no	—
25	C₂×#6090	D₄	1	2	yes	—
26	#88050	—	4	4	no	—
27	#88048	—	4	4	no	—
28	C₄×#1055	D₄	1	2	yes	—
29	#12507	—	13	1	no	maximal, Sylow
30	C₂×Q₈×C₁₃	—	4	2	no	—
31	C₂×Dic₂₆	—	4	2	no	—
32	C₁₃×Q₁₆	—	4	4	no	—
33	C₁₃×Q₁₆	—	4	4	no	—
34	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₁₃×C₄⋊C₄	—	4	2	no	—
37	C₁₃×C₄⋊C₄	—	4	2	no	—
38	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₁₃	—	4	2	no	—
40	C₄⋊Dic₁₃	—	4	2	no	—
41	C₂×#1055	D₈	1	4	yes	—
42	C₁₃×C₁₆	SD₁₆	1	4	yes	—
43	#6090	—	2	2	no	—
44	#2254	—	13	1	no	—
45	#2247	D₁₃	1	1	yes	characteristic
46	#2241	—	13	2	no	—
47	#1707	—	13	2	no	—
48	#1311	—	13	1	no	—
49	Q₈×C₁₃	—	8	2	no	—
50	Q₈×C₁₃	—	8	2	no	—
51	C₂×C₄×C₁₃	—	8	2	no	—
52	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
54	C₂×Dic₁₃	—	8	2	no	—
55	Dic₂₆	—	8	4	no	—
56	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
57	C₈×C₁₃	#135	1	2	yes	—
58	#1055	—	2	4	no	—
59	C₂×Q₃₂	—	2	2	no	—
60	#500	—	13	2	no	—
61	#498	—	26	2	no	—
62	#494	C₂×D₁₃	1	2	yes	—
63	#487	—	26	2	no	—
64	#367	—	26	2	no	—
65	#358	—	26	4	no	—
66	C₄×C₁₆	—	13	1	no	—
67	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
68	C₈²	—	13	1	no	—
69	C₂²×C₁₃	#493	1	1	yes	characteristic
70	C₄×C₁₃	—	16	2	no	—
71	C₄×C₁₃	#492	1	1	yes	characteristic
72	C₄×C₁₃	#495	1	1	yes	characteristic
73	C₄×C₁₃	#510	1	4	yes	—
74	Dic₁₃	—	16	2	no	—
75	C₂×Q₁₆	—	2	2	no	—
76	C₂×Q₁₆	—	2	2	no	—
77	C₄⋊Q₈	—	2	2	no	—
78	C₄⋊Q₈	—	26	2	no	—
79	Q₃₂	—	4	4	no	—
80	C₂×C₁₆	—	13	2	no	—
81	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
82	#107	—	26	2	no	—
83	#107	—	26	2	no	—
84	#106	—	52	4	no	—
85	#103	—	52	4	no	—
86	C₄×C₈	—	13	2	no	—
87	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
88	C₂×C₁₃	#1706	1	1	yes	characteristic
89	C₂×C₁₃	#2251	1	2	yes	—
90	C₂×Q₈	—	4	2	no	—
91	C₂×Q₈	—	52	2	no	—
92	Q₁₆	—	4	4	no	—
93	Q₁₆	—	4	4	no	—
94	C₂×C₈	—	13	4	no	—
95	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
96	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	52	2	no	—
100	C₄⋊C₄	—	52	2	no	—
101	C₄²	D₄×D₁₃	1	1	yes	characteristic
102	C₁₆	—	26	2	no	—
103	C₁₆	#6105	1	4	yes	—
104	C₁₃	#12507	1	1	yes	characteristic, complemented, Sylow
105	Q₈	—	8	2	no	—
106	Q₈	—	8	2	no	—
107	Q₈	—	104	4	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	—	104	2	no	—
110	C₂×C₄	D₈×D₁₃	1	2	yes	—
111	C₂×C₄	#88194	1	1	yes	characteristic
112	C₈	—	26	4	no	—
113	C₈	#88166	1	2	yes	—
114	C₈	C₂×#6105	1	2	yes	—
115	C₂²	#11718812	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	208	2	no	—
118	C₄	#11718579	1	4	yes	—
119	C₄	#11718737	1	1	yes	characteristic
120	C₄	#11718816	1	1	yes	characteristic
121	C₂	#13114425	1	1	yes	characteristic
122	C₂	#13117104	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13117118

C₂

yes

maximal, characteristic

#13117063

C₂

yes

maximal

#13114430

C₂

yes

maximal

#13112752

C₂

yes

maximal, characteristic

C₁₃×#2247

C₂

yes

maximal, characteristic

C₂×C₁₃×Q₃₂

—

C₁₃×#494

C₂²

yes

—

#11718754

—

#11718740

—

#11718343

C₂²

yes

—

C₄×C₁₃×C₁₆

C₂²

yes

—

—

C₂²

yes

characteristic

C₈×#1055

C₂²

yes

characteristic

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₁₃×C₄⋊Q₈

—

#88123

—

C₁₃×Q₃₂

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

—

D₄

yes

—

#88050

—

#88048

—

C₄×#1055

D₄

yes

—

#12507

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×Dic₂₆

—

C₁₃×Q₁₆

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

D₈

yes

—

C₁₃×C₁₆

SD₁₆

yes

—

#6090

—

#2254

—

#2247

D₁₃

yes

—

—

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

C₂×SD₁₆

yes

—

C₈×C₁₃

#135

yes

—

#1055

—

C₂×Q₃₂

—

#500

—

#498

—

#494

C₂×D₁₃

yes

—

#487

—

#367

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

C₈²

—

C₂²×C₁₃

#493

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₃

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

D₄×D₁₃

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

D₄×D₁₃

yes

characteristic

102

C₁₆

—

103

C₁₆

#6105

yes

—

104

C₁₃

#12507

yes

characteristic, complemented, Sylow

105

Q₈

—

106

Q₈

—

107

Q₈

—

104

—

108

C₂×C₄

—

109

C₂×C₄

—

104

—

110

C₂×C₄

D₈×D₁₃

yes

—

111

C₂×C₄

#88194

yes

characteristic

112

C₈

—

113

C₈

#88166

yes

—

114

C₈

C₂×#6105

yes

—

115

C₂²

#11718812

yes

characteristic

116

C₄

—

117

C₄

—

208

—

118

C₄

#11718579

yes

—

119

C₄

#11718737

yes

characteristic

120

C₄

#11718816

yes

characteristic

121

C₂

#13114425

yes

characteristic

122

C₂

#13117104

yes

—

	subgroup	G/H
center	C₂²	#11718812
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2247	C₂
socle, prime radical	C₂²×C₁₃	#493
hypercenter	#2247	D₁₃
hypocenter	C₁₃	#12507
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#11718812

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#2247

C₂

C₂²×C₁₃

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
104. C₁₃	#12507	semidirect

H ≅ G/N

type

104. C₁₃

#12507

semidirect

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

subgroup	index
#13117118	2
#13117063	2
#13114430	2
#13112752	2
C₁₃×#2247	2
#12507	13

subgroup

index

C₁₃×#2247

Subgroups — #15777591

Conjugacy classes of subgroups

Complements

Maximal subgroups