Group #15777643 — subgroups

Subgroups — #15777643

Order 3328.

330 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 87 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	#13117093	C₂	1	2	yes	maximal, complemented
2	#13117086	C₂	1	1	yes	maximal, characteristic
3	#13114443	C₂	1	2	yes	maximal, complemented
4	C₁₆×#1055	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×Q₃₂	—	2	2	no	—
7	C₁₃×#494	C₂²	1	2	yes	—
8	C₂×#88068	—	2	4	no	—
9	#11718749	—	2	4	no	—
10	C₂×D₂₀₈	—	2	2	no	—
11	#11718356	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×D₁₀₄	—	2	2	no	—
19	C₂×D₁₀₄	—	2	2	no	—
20	#88128	—	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	#88068	—	4	8	no	—
25	C₂×#6090	D₄	1	2	yes	—
26	#88049	—	4	8	no	—
27	C₄×#1055	D₄	1	2	yes	—
28	D₂₀₈	—	4	4	no	—
29	#12518	—	13	1	no	maximal, Sylow
30	C₂×Q₈×C₁₃	—	4	2	no	—
31	C₂×D₅₂	—	4	2	no	—
32	C₂×D₅₂	—	4	4	no	—
33	C₁₃×Q₁₆	—	4	4	no	—
34	C₁₃×Q₁₆	—	4	4	no	—
35	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₁₃×C₄⋊C₄	—	4	4	no	—
38	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂×#1055	D₈	1	4	yes	—
40	D₁₀₄	—	4	4	no	—
41	D₁₀₄	—	4	4	no	—
42	C₁₃×C₁₆	D₈	1	4	yes	—
43	#6090	D₈	1	4	yes	—
44	#2248	—	13	2	no	—
45	#2247	D₁₃	1	1	yes	characteristic, complemented
46	#2246	—	13	1	no	—
47	#1709	—	13	2	no	—
48	C₈×C₁₆	—	13	1	no	—
49	C₂²×D₁₃	—	8	2	no	—
50	Q₈×C₁₃	—	8	2	no	—
51	Q₈×C₁₃	—	8	2	no	—
52	C₂×C₄×C₁₃	—	8	2	no	—
53	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
55	D₅₂	—	8	2	no	—
56	D₅₂	—	8	2	no	—
57	D₅₂	—	8	8	no	—
58	C₈×C₁₃	C₂×D₈	1	2	yes	—
59	C₈×C₁₃	C₂×D₈	1	2	yes	—
60	#1055	—	2	4	no	—
61	C₂×Q₃₂	—	2	2	no	—
62	C₂×SD₃₂	—	26	4	no	—
63	C₂×D₁₆	—	26	2	no	—
64	#494	C₂×D₁₃	1	2	yes	—
65	#493	—	13	2	no	—
66	#486	—	26	4	no	—
67	C₄×C₁₆	—	13	1	no	—
68	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
69	C₈²	—	13	1	no	—
70	C₂²×C₁₃	#493	1	1	yes	characteristic
71	C₂×D₁₃	—	16	2	no	—
72	C₂×D₁₃	—	16	4	no	—
73	C₄×C₁₃	—	16	2	no	—
74	C₄×C₁₃	#493	1	1	yes	characteristic
75	C₄×C₁₃	#493	1	1	yes	characteristic
76	C₄×C₁₃	#508	1	4	yes	—
77	C₂×Q₁₆	—	2	2	no	—
78	C₂×Q₁₆	—	2	2	no	—
79	C₂×D₈	—	26	2	no	—
80	C₂×D₈	—	26	2	no	—
81	C₄⋊Q₈	—	2	2	no	—
82	#127	—	26	2	no	—
83	Q₃₂	—	4	4	no	—
84	SD₃₂	—	52	8	no	—
85	D₁₆	—	52	4	no	—
86	C₂×C₁₆	—	13	2	no	—
87	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
88	#102	—	52	8	no	—
89	C₄×C₈	—	13	2	no	—
90	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
91	C₂×C₁₃	#1712	1	1	yes	characteristic
92	C₂×C₁₃	#2249	1	2	yes	—
93	D₁₃	—	32	2	no	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×D₄	—	52	4	no	—
96	C₂×D₄	—	52	2	no	—
97	Q₁₆	—	4	4	no	—
98	Q₁₆	—	4	4	no	—
99	D₈	—	52	4	no	—
100	D₈	—	52	4	no	—
101	C₂×C₈	—	13	4	no	—
102	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
103	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
104	C₄⋊C₄	—	4	4	no	—
105	C₄²	D₄×D₁₃	1	1	yes	characteristic
106	C₁₆	—	13	4	no	—
107	C₁₆	#6104	1	4	yes	—
108	C₁₃	#12518	1	1	yes	characteristic, complemented, Sylow
109	C₂³	—	104	2	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	8	2	no	—
112	D₄	—	104	2	no	—
113	D₄	—	104	2	no	—
114	D₄	—	104	8	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	D₈×D₁₃	1	2	yes	—
117	C₂×C₄	#88194	1	1	yes	characteristic
118	C₈	—	26	4	no	—
119	C₈	D₈×D₁₃	1	2	yes	—
120	C₈	C₂×#6104	1	2	yes	—
121	C₂²	—	208	2	no	—
122	C₂²	—	208	4	no	—
123	C₂²	#11718812	1	1	yes	characteristic
124	C₄	—	16	2	no	—
125	C₄	#11718580	1	4	yes	—
126	C₄	#11718734	1	1	yes	characteristic
127	C₄	#11718812	1	1	yes	characteristic
128	C₂	—	416	2	no	—
129	C₂	#13114450	1	1	yes	characteristic
130	C₂	#13117098	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13117093

C₂

yes

maximal, complemented

#13117086

C₂

yes

maximal, characteristic

#13114443

C₂

yes

maximal, complemented

C₁₆×#1055

C₂

yes

maximal, characteristic, complemented

C₁₃×#2247

C₂

yes

maximal, characteristic, complemented

C₂×C₁₃×Q₃₂

—

C₁₃×#494

C₂²

yes

—

C₂×#88068

—

#11718749

—

C₂×D₂₀₈

—

#11718356

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×D₁₀₄

—

C₂×D₁₀₄

—

#88128

—

C₁₃×Q₃₂

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88068

—

C₂×#6090

D₄

yes

—

#88049

—

C₄×#1055

D₄

yes

—

D₂₀₈

—

#12518

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

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

C₂×#1055

D₈

yes

—

D₁₀₄

—

D₁₀₄

—

C₁₃×C₁₆

D₈

yes

—

#6090

D₈

yes

—

#2248

—

#2247

D₁₃

yes

characteristic, complemented

#2246

—

#1709

—

C₈×C₁₆

—

C₂²×D₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

D₅₂

—

D₅₂

—

D₅₂

—

C₈×C₁₃

C₂×D₈

yes

—

C₈×C₁₃

C₂×D₈

yes

—

#1055

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

#494

C₂×D₁₃

yes

—

#493

—

#486

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

C₈²

—

C₂²×C₁₃

#493

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₃

yes

—

#102

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

D₁₃

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

D₈

—

100

D₈

—

101

C₂×C₈

—

102

C₂×C₈

D₄×D₁₃

yes

characteristic

103

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

104

C₄⋊C₄

—

105

C₄²

D₄×D₁₃

yes

characteristic

106

C₁₆

—

107

C₁₆

#6104

yes

—

108

C₁₃

#12518

yes

characteristic, complemented, Sylow

109

C₂³

—

104

—

110

Q₈

—

111

Q₈

—

112

D₄

—

104

—

113

D₄

—

104

—

114

D₄

—

104

—

115

C₂×C₄

—

116

C₂×C₄

D₈×D₁₃

yes

—

117

C₂×C₄

#88194

yes

characteristic

118

C₈

—

119

C₈

D₈×D₁₃

yes

—

120

C₈

C₂×#6104

yes

—

121

C₂²

—

208

—

122

C₂²

—

208

—

123

C₂²

#11718812

yes

characteristic

124

C₄

—

125

C₄

#11718580

yes

—

126

C₄

#11718734

yes

characteristic

127

C₄

#11718812

yes

characteristic

128

C₂

—

416

—

129

C₂

#13114450

yes

characteristic

130

C₂

#13117098

yes

—

	subgroup	G/H
center	C₂²	#11718812
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2247	C₂
socle	C₂²×C₁₃	#493
prime radical	#13117086	C₂
hypercenter	#2247	D₁₃
hypocenter	C₁₃	#12518
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

C₂

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
1. #13117093	C₂	semidirect
3. #13114443	C₂	semidirect
4. C₁₆×#1055	C₂	semidirect
5. C₁₃×#2247	C₂	semidirect
45. #2247	D₁₃	semidirect
108. C₁₃	#12518	semidirect

H ≅ G/N

type

1. #13117093

C₂

semidirect

3. #13114443

C₂

semidirect

4. C₁₆×#1055

C₂

semidirect

5. C₁₃×#2247

C₂

semidirect

45. #2247

D₁₃

semidirect

108. C₁₃

#12518

semidirect

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

subgroup	index
#13117093	2
#13117086	2
#13114443	2
C₁₆×#1055	2
C₁₃×#2247	2
#12518	13

subgroup

index

#13117093

#13117086

#13114443

C₁₆×#1055

C₁₃×#2247

#12518

Subgroups — #15777643

Conjugacy classes of subgroups

Complements

Maximal subgroups