Group #15777609 — subgroups

Subgroups — #15777609

Order 3328.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#13117064	C₂	1	4	yes	maximal
2	#13114454	C₂	1	1	yes	maximal, characteristic
3	#13114454	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
5	#11718360	—	2	1	no	—
6	#11718360	C₂²	1	2	yes	—
7	#11718343	—	2	1	no	—
8	#11718343	—	2	1	no	—
9	#11718343	—	2	1	no	—
10	#11718343	C₂²	1	2	yes	—
11	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
12	C₈²×C₁₃	C₂²	1	1	yes	characteristic
13	#11718095	—	2	8	no	—
14	C₂×Dic₅₂	—	2	4	no	—
15	C₂×Dic₅₂	—	4	2	no	—
16	#88123	—	2	2	no	—
17	#88123	—	2	2	no	—
18	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
19	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	#88058	—	2	4	no	—
23	#88058	—	4	2	no	—
24	#88058	—	4	2	no	—
25	#88058	—	4	2	no	—
26	#12510	—	13	1	no	maximal, Sylow
27	C₂×Dic₂₆	—	4	2	no	—
28	C₂×Dic₂₆	—	4	2	no	—
29	C₂×C₈×C₁₃	D₈	1	2	yes	—
30	C₂×C₈×C₁₃	D₈	1	2	yes	—
31	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
32	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
33	C₄⋊Dic₁₃	—	4	2	no	—
34	C₄⋊Dic₁₃	—	4	2	no	—
35	C₄⋊Dic₁₃	—	4	2	no	—
36	C₄⋊Dic₁₃	—	4	2	no	—
37	Dic₅₂	—	4	8	no	—
38	Dic₅₂	—	8	4	no	—
39	C₁₃×C₁₆	—	2	4	no	—
40	#2241	—	13	4	no	—
41	#1713	—	13	1	no	—
42	#1713	—	13	1	no	—
43	C₈×C₁₆	D₁₃	1	1	yes	characteristic
44	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
45	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
47	C₂×Dic₁₃	—	8	2	no	—
48	C₂×Dic₁₃	—	8	2	no	—
49	Dic₂₆	—	8	4	no	—
50	Dic₂₆	—	8	4	no	—
51	C₈×C₁₃	SD₃₂	1	4	yes	—
52	C₈×C₁₃	Q₃₂	1	4	yes	—
53	C₈×C₁₃	#135	1	4	yes	—
54	#500	—	13	2	no	—
55	#500	—	26	1	no	—
56	#500	—	26	1	no	—
57	#500	—	26	1	no	—
58	#494	—	13	2	no	—
59	#494	—	26	1	no	—
60	#358	—	26	8	no	—
61	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
62	C₈²	C₂×D₁₃	1	1	yes	characteristic
63	C₂²×C₁₃	#493	1	1	yes	characteristic
64	C₄×C₁₃	#495	1	2	yes	—
65	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
66	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
67	Dic₁₃	—	16	2	no	—
68	Dic₁₃	—	16	2	no	—
69	C₂×Q₁₆	—	26	4	no	—
70	C₂×Q₁₆	—	52	2	no	—
71	C₄⋊Q₈	—	26	2	no	—
72	C₄⋊Q₈	—	26	2	no	—
73	C₂×C₁₆	D₅₂	1	4	yes	—
74	#107	—	26	4	no	—
75	#107	—	52	2	no	—
76	#107	—	52	2	no	—
77	#107	—	52	2	no	—
78	C₄×C₈	D₅₂	1	1	yes	characteristic
79	C₄×C₈	D₅₂	1	1	yes	characteristic
80	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
81	C₂×C₁₃	#1709	1	1	yes	characteristic
82	C₂×C₁₃	#2247	1	1	yes	characteristic
83	C₂×C₁₃	#2248	1	1	yes	characteristic
84	C₂×Q₈	—	52	2	no	—
85	C₂×Q₈	—	52	2	no	—
86	Q₁₆	—	52	8	no	—
87	Q₁₆	—	104	4	no	—
88	C₂×C₈	D₁₀₄	1	2	yes	—
89	C₂×C₈	D₁₀₄	1	2	yes	—
90	C₂×C₈	C₂×D₅₂	1	2	yes	—
91	C₄⋊C₄	—	52	2	no	—
92	C₄⋊C₄	—	52	2	no	—
93	C₄⋊C₄	—	52	2	no	—
94	C₄⋊C₄	—	52	2	no	—
95	C₄²	C₂×D₅₂	1	1	yes	characteristic
96	C₁₆	—	2	4	no	—
97	C₁₃	#12510	1	1	yes	characteristic, complemented, Sylow
98	Q₈	—	104	4	no	—
99	Q₈	—	104	4	no	—
100	C₂×C₄	—	104	2	no	—
101	C₂×C₄	—	104	2	no	—
102	C₂×C₄	#88128	1	1	yes	characteristic
103	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
104	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
105	C₈	#88040	1	4	yes	—
106	C₈	Dic₁₀₄	1	4	yes	—
107	C₈	#88158	1	4	yes	—
108	C₂²	#11718356	1	1	yes	characteristic
109	C₄	—	208	2	no	—
110	C₄	—	208	2	no	—
111	C₄	#11718357	1	2	yes	—
112	C₄	C₂×#88040	1	2	yes	—
113	C₄	C₂×Dic₁₀₄	1	2	yes	—
114	C₂	#13114442	1	1	yes	characteristic
115	C₂	#13117089	1	1	yes	characteristic
116	C₂	#13117092	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117064

C₂

yes

maximal

#13114454

C₂

yes

maximal, characteristic

#13114454

C₂

yes

maximal, characteristic

C₈×C₁₃×C₁₆

C₂

yes

maximal, characteristic

#11718360

—

#11718360

C₂²

yes

—

—

—

—

C₂²

yes

—

C₄×C₁₃×C₁₆

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

#11718095

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

#88123

—

#88123

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

—

#2241

—

#1713

—

#1713

—

C₈×C₁₆

D₁₃

yes

characteristic

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

SD₃₂

yes

—

C₈×C₁₃

Q₃₂

yes

—

C₈×C₁₃

#135

yes

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₁₃

yes

—

C₈²

C₂×D₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

yes

—

C₄×C₁₃

C₂×SD₃₂

yes

—

C₄×C₁₃

C₂×Q₃₂

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

D₅₂

yes

—

—

—

—

—

C₄×C₈

D₅₂

yes

characteristic

C₄×C₈

D₅₂

yes

characteristic

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

104

—

C₂×C₈

D₁₀₄

yes

—

C₂×C₈

D₁₀₄

yes

—

C₂×C₈

C₂×D₅₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₅₂

yes

characteristic

C₁₆

—

C₁₃

#12510

yes

characteristic, complemented, Sylow

Q₈

—

104

—

Q₈

—

104

—

100

C₂×C₄

—

104

—

101

C₂×C₄

—

104

—

102

C₂×C₄

#88128

yes

characteristic

103

C₂×C₄

C₂×D₁₀₄

yes

characteristic

104

C₂×C₄

C₂×D₁₀₄

yes

characteristic

105

C₈

#88040

yes

—

106

C₈

Dic₁₀₄

yes

—

107

C₈

#88158

yes

—

108

C₂²

#11718356

yes

characteristic

109

C₄

—

208

—

110

C₄

—

208

—

111

C₄

#11718357

yes

—

112

C₄

C₂×#88040

yes

—

113

C₄

C₂×Dic₁₀₄

yes

—

114

C₂

#13114442

yes

characteristic

115

C₂

#13117089

yes

characteristic

116

C₂

#13117092

yes

characteristic

	subgroup	G/H
center	C₂²	#11718356
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₈×C₁₃×C₁₆	C₂
socle, prime radical	C₂²×C₁₃	#493
hypercenter	C₈×C₁₆	D₁₃
hypocenter	C₁₃	#12510
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#11718356

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₈×C₁₃×C₁₆

C₂

socle, prime radical

C₂²×C₁₃

#493

hypercenter

C₈×C₁₆

D₁₃

hypocenter

C₁₃

#12510

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
97. C₁₃	#12510	semidirect

H ≅ G/N

type

97. C₁₃

#12510

semidirect

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

subgroup	index
#13117064	2
#13114454	2
C₈×C₁₃×C₁₆	2
#12510	13

subgroup

index

#13117064

#13114454

C₈×C₁₃×C₁₆

#12510

Subgroups — #15777609

Conjugacy classes of subgroups

Complements

Maximal subgroups