Group #15777632 — subgroups

Subgroups — #15777632

Order 3328.

310 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 95 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	#13117089	C₂	1	1	yes	maximal, characteristic
2	#13117088	C₂	1	2	yes	maximal
3	#13114452	C₂	1	2	yes	maximal
4	C₁₆×#1055	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	C₂×#88069	—	2	4	no	—
9	#11718739	—	2	4	no	—
10	C₂×Dic₁₀₄	—	2	2	no	—
11	#11718360	C₂²	1	2	yes	—
12	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₄×#6090	C₂²	1	1	yes	characteristic
14	C₈×#1055	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×Q₁₆	—	2	2	no	—
16	C₂×C₁₃×Q₁₆	—	2	2	no	—
17	C₁₃×C₄⋊Q₈	—	2	2	no	—
18	C₂×Dic₅₂	—	2	2	no	—
19	C₂×Dic₅₂	—	2	2	no	—
20	#88123	—	2	2	no	—
21	C₁₃×Q₃₂	—	4	4	no	—
22	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
23	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
24	#88069	—	4	8	no	—
25	C₂×#6090	D₄	1	2	yes	—
26	#88047	—	4	8	no	—
27	C₄×#1055	D₄	1	2	yes	—
28	Dic₁₀₄	—	4	4	no	—
29	#12516	—	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	4	no	—
37	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊Dic₁₃	—	4	4	no	—
39	C₂×#1055	D₈	1	4	yes	—
40	Dic₅₂	—	4	4	no	—
41	Dic₅₂	—	4	4	no	—
42	C₁₃×C₁₆	D₈	1	4	yes	—
43	#6090	D₈	1	4	yes	—
44	#2247	—	13	2	no	—
45	#2247	—	13	1	no	—
46	#2247	D₁₃	1	1	yes	characteristic
47	#1713	—	13	2	no	—
48	C₈×C₁₆	—	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	2	no	—
56	Dic₂₆	—	8	2	no	—
57	C₈×C₁₃	C₂×D₈	1	2	yes	—
58	C₈×C₁₃	C₂×D₈	1	2	yes	—
59	#1055	Q₃₂	1	8	yes	complemented
60	C₂×Q₃₂	—	2	2	no	—
61	C₂×Q₃₂	—	26	4	no	—
62	C₂×Q₃₂	—	26	2	no	—
63	#500	—	26	4	no	—
64	#494	—	13	2	no	—
65	#494	C₂×D₁₃	1	2	yes	—
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	#493	1	1	yes	characteristic
72	C₄×C₁₃	#493	1	1	yes	characteristic
73	C₄×C₁₃	C₂×Q₃₂	1	4	yes	—
74	Dic₁₃	—	16	2	no	—
75	C₂×Q₁₆	—	2	2	no	—
76	C₂×Q₁₆	—	2	2	no	—
77	C₂×Q₁₆	—	26	2	no	—
78	C₂×Q₁₆	—	26	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	—	26	2	no	—
81	Q₃₂	—	4	4	no	—
82	Q₃₂	—	52	8	no	—
83	Q₃₂	—	52	4	no	—
84	C₂×C₁₆	—	13	2	no	—
85	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
86	#107	—	52	8	no	—
87	C₄×C₈	—	13	2	no	—
88	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
89	C₂×C₁₃	#1712	1	1	yes	characteristic
90	C₂×C₁₃	#2247	1	2	yes	—
91	C₂×Q₈	—	4	2	no	—
92	C₂×Q₈	—	52	2	no	—
93	Q₁₆	—	4	4	no	—
94	Q₁₆	—	4	4	no	—
95	Q₁₆	—	52	4	no	—
96	Q₁₆	—	52	4	no	—
97	C₂×C₈	—	13	4	no	—
98	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
99	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
100	C₄⋊C₄	—	4	4	no	—
101	C₄⋊C₄	—	52	4	no	—
102	C₄²	D₄×D₁₃	1	1	yes	characteristic
103	C₁₆	—	13	4	no	—
104	C₁₆	#6104	1	4	yes	—
105	C₁₃	#12516	1	1	yes	characteristic, complemented, Sylow
106	Q₈	—	8	2	no	—
107	Q₈	—	8	2	no	—
108	Q₈	—	104	2	no	—
109	Q₈	—	104	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	104	2	no	—
112	C₂×C₄	D₈×D₁₃	1	2	yes	—
113	C₂×C₄	#88194	1	1	yes	characteristic
114	C₈	—	13	8	no	—
115	C₈	D₈×D₁₃	1	2	yes	—
116	C₈	C₂×#6104	1	2	yes	—
117	C₂²	#11718812	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	208	2	no	—
120	C₄	D₁₃×Q₃₂	1	4	yes	—
121	C₄	#11718734	1	1	yes	characteristic
122	C₄	#11718812	1	1	yes	characteristic
123	C₂	#13114450	1	1	yes	characteristic
124	C₂	#13117087	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13117089

C₂

yes

maximal, characteristic

#13117088

C₂

yes

maximal

#13114452

C₂

yes

maximal

C₁₆×#1055

C₂

yes

maximal, characteristic

C₁₃×#2247

C₂

yes

maximal, characteristic

C₂×C₁₃×Q₃₂

—

C₁₃×#494

C₂²

yes

—

C₂×#88069

—

#11718739

—

C₂×Dic₁₀₄

—

#11718360

C₂²

yes

—

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

C₄×#6090

C₂²

yes

characteristic

C₈×#1055

C₂²

yes

characteristic

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₁₃×C₄⋊Q₈

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

#88123

—

C₁₃×Q₃₂

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88069

—

C₂×#6090

D₄

yes

—

#88047

—

C₄×#1055

D₄

yes

—

Dic₁₀₄

—

#12516

—

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

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₂×#1055

D₈

yes

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

D₈

yes

—

#6090

D₈

yes

—

#2247

—

#2247

—

#2247

D₁₃

yes

characteristic

#1713

—

C₈×C₁₆

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

C₂×D₈

yes

—

C₈×C₁₃

C₂×D₈

yes

—

#1055

Q₃₂

yes

complemented

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

#500

—

#494

—

#494

C₂×D₁₃

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

C₈²

—

C₂²×C₁₃

#493

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂×Q₃₂

yes

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₃

yes

—

#107

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

D₄×D₁₃

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

D₄×D₁₃

yes

characteristic

103

C₁₆

—

104

C₁₆

#6104

yes

—

105

C₁₃

#12516

yes

characteristic, complemented, Sylow

106

Q₈

—

107

Q₈

—

108

Q₈

—

104

—

109

Q₈

—

104

—

110

C₂×C₄

—

111

C₂×C₄

—

104

—

112

C₂×C₄

D₈×D₁₃

yes

—

113

C₂×C₄

#88194

yes

characteristic

114

C₈

—

115

C₈

D₈×D₁₃

yes

—

116

C₈

C₂×#6104

yes

—

117

C₂²

#11718812

yes

characteristic

118

C₄

—

119

C₄

—

208

—

120

C₄

D₁₃×Q₃₂

yes

—

121

C₄

#11718734

yes

characteristic

122

C₄

#11718812

yes

characteristic

123

C₂

#13114450

yes

characteristic

124

C₂

#13117087

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	#12516
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
59. #1055	Q₃₂	semidirect
105. C₁₃	#12516	semidirect

H ≅ G/N

type

59. #1055

Q₃₂

semidirect

105. C₁₃

#12516

semidirect

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

subgroup	index
#13117089	2
#13117088	2
#13114452	2
C₁₆×#1055	2
C₁₃×#2247	2
#12516	13

subgroup

index

#13117089

#13117088

#13114452

C₁₆×#1055

C₁₃×#2247

#12516

Subgroups — #15777632

Conjugacy classes of subgroups

Complements

Maximal subgroups