Group #15777648 — subgroups

Subgroups — #15777648

Order 3328.

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

normal subgroups only

This group has 360 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	#13117085	C₂	1	2	yes	maximal, complemented
3	#13114451	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112753	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1709	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#494	C₂²	1	1	yes	characteristic
7	C₁₃×#493	C₂²	1	1	yes	characteristic
8	C₁₃×#486	—	2	2	no	—
9	C₂×#88068	—	2	4	no	—
10	C₂×#88066	—	2	4	no	—
11	#11718356	—	2	2	no	—
12	#11718356	C₂²	1	2	yes	—
13	C₈²×C₁₃	C₂²	1	1	yes	characteristic
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	C₂×D₁₀₄	—	2	2	no	—
20	C₂×D₁₀₄	—	2	2	no	—
21	C₂×D₁₀₄	—	4	4	no	—
22	#88128	—	2	2	no	—
23	C₁₃×#102	—	4	4	no	—
24	C₄×C₈×C₁₃	D₄	1	2	yes	—
25	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	#88068	—	4	8	no	—
27	#88066	—	4	8	no	—
28	C₂×#6090	D₄	1	4	yes	—
29	#12519	—	13	1	no	maximal, Sylow
30	C₂×Q₈×C₁₃	—	4	1	no	—
31	C₂×D₄×C₁₃	—	4	1	no	—
32	C₂×D₄×C₁₃	—	4	2	no	—
33	C₂×D₅₂	—	4	2	no	—
34	C₂×D₅₂	—	4	4	no	—
35	C₁₃×Q₁₆	—	4	4	no	—
36	C₁₃×D₈	—	4	4	no	—
37	C₂×C₈×C₁₃	D₈	1	4	yes	—
38	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
39	C₁₃×C₄⋊C₄	—	4	2	no	—
40	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
41	D₁₀₄	—	4	4	no	—
42	D₁₀₄	—	4	4	no	—
43	D₁₀₄	—	8	8	no	—
44	#6090	D₈	1	8	yes	complemented
45	#2248	—	13	2	no	—
46	#2246	—	13	2	no	—
47	#1712	—	13	1	no	—
48	#1709	D₁₃	1	1	yes	characteristic, complemented
49	#1311	—	13	1	no	—
50	C₂³×C₁₃	—	8	1	no	—
51	C₂²×D₁₃	—	8	2	no	—
52	Q₈×C₁₃	—	8	2	no	—
53	D₄×C₁₃	—	8	2	no	—
54	D₄×C₁₃	—	8	4	no	—
55	C₂×C₄×C₁₃	—	8	1	no	—
56	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
58	D₅₂	—	8	2	no	—
59	D₅₂	—	8	2	no	—
60	D₅₂	—	8	8	no	—
61	C₈×C₁₃	—	2	4	no	—
62	C₈×C₁₃	C₂×D₈	1	4	yes	—
63	C₂×SD₃₂	—	26	4	no	—
64	C₂×D₁₆	—	26	4	no	—
65	#494	C₂×D₁₃	1	1	yes	characteristic
66	#493	—	13	2	no	—
67	#493	—	26	2	no	—
68	#493	C₂×D₁₃	1	1	yes	characteristic
69	#486	—	2	2	no	—
70	C₄×C₁₆	—	13	2	no	—
71	C₈²	C₂×D₁₃	1	1	yes	characteristic
72	C₂²×C₁₃	—	16	1	no	—
73	C₂²×C₁₃	—	16	2	no	—
74	C₂²×C₁₃	#493	1	1	yes	characteristic
75	C₂×D₁₃	—	16	2	no	—
76	C₂×D₁₃	—	16	4	no	—
77	C₄×C₁₃	—	16	1	no	—
78	C₄×C₁₃	#493	1	2	yes	—
79	C₄×C₁₃	#509	1	4	yes	—
80	C₂×Q₁₆	—	2	2	no	—
81	C₂×D₈	—	2	2	no	—
82	C₂×D₈	—	26	2	no	—
83	C₂×D₈	—	26	2	no	—
84	C₂×D₈	—	52	4	no	—
85	C₄⋊Q₈	—	2	1	no	—
86	#127	—	2	1	no	—
87	#127	—	26	2	no	—
88	SD₃₂	—	52	8	no	—
89	D₁₆	—	52	8	no	—
90	C₂×C₁₆	—	13	4	no	—
91	#102	—	4	4	no	—
92	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
93	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
94	C₂×C₁₃	—	32	1	no	—
95	C₂×C₁₃	#1712	1	1	yes	characteristic
96	C₂×C₁₃	#2250	1	2	yes	—
97	D₁₃	—	32	2	no	—
98	C₂×Q₈	—	4	1	no	—
99	C₂×D₄	—	4	2	no	—
100	C₂×D₄	—	4	1	no	—
101	C₂×D₄	—	52	4	no	—
102	C₂×D₄	—	52	2	no	—
103	Q₁₆	—	4	4	no	—
104	D₈	—	4	4	no	—
105	D₈	—	52	4	no	—
106	D₈	—	52	4	no	—
107	D₈	—	104	8	no	—
108	C₂×C₈	#6104	1	4	yes	—
109	C₂×C₈	D₄×D₁₃	1	2	yes	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
112	C₁₆	—	13	8	no	—
113	C₁₃	#12519	1	1	yes	characteristic, complemented, Sylow
114	C₂³	—	8	1	no	—
115	C₂³	—	104	2	no	—
116	Q₈	—	8	2	no	—
117	D₄	—	8	2	no	—
118	D₄	—	8	4	no	—
119	D₄	—	104	2	no	—
120	D₄	—	104	2	no	—
121	D₄	—	104	8	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	C₂×#6104	1	2	yes	—
124	C₂×C₄	#88194	1	1	yes	characteristic
125	C₈	—	2	4	no	—
126	C₈	D₈×D₁₃	1	4	yes	—
127	C₂²	—	16	1	no	—
128	C₂²	—	16	2	no	—
129	C₂²	—	208	2	no	—
130	C₂²	—	208	4	no	—
131	C₂²	#11718734	1	1	yes	characteristic
132	C₄	—	16	1	no	—
133	C₄	#11718812	1	2	yes	—
134	C₄	#11718854	1	4	yes	—
135	C₂	—	32	1	no	—
136	C₂	—	416	2	no	—
137	C₂	#13114450	1	1	yes	characteristic
138	C₂	#13117103	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13117093

C₂

yes

maximal, complemented

#13117085

C₂

yes

maximal, complemented

#13114451

C₂

yes

maximal, characteristic, complemented

#13112753

C₂

yes

maximal, characteristic, complemented

C₁₃×#1709

C₂

yes

maximal, characteristic, complemented

C₁₃×#494

C₂²

yes

characteristic

C₁₃×#493

C₂²

yes

characteristic

C₁₃×#486

—

—

—

—

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

C₄×#6090

C₂²

yes

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×D₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×#127

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

#88128

—

C₁₃×#102

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

—

D₄

yes

—

#12519

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₁₃×Q₁₆

—

C₁₃×D₈

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

D₁₀₄

—

D₁₀₄

—

D₁₀₄

—

#6090

D₈

yes

—

—

—

D₁₃

yes

characteristic, complemented

#1311

—

C₂³×C₁₃

—

C₂²×D₁₃

—

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

D₅₂

—

D₅₂

—

D₅₂

—

C₈×C₁₃

—

C₈×C₁₃

C₂×D₈

yes

—

C₂×SD₃₂

—

C₂×D₁₆

—

#494

C₂×D₁₃

yes

—

—

C₂×D₁₃

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#493

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#493

yes

—

C₄×C₁₃

#509

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#127

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

#102

—

C₄×C₈

C₂²⋊D₁₃

yes

—

C₄×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

#1712

yes

characteristic

C₂×C₁₃

#2250

yes

—

D₁₃

—

C₂×Q₈

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

Q₁₆

—

104

D₈

—

105

D₈

—

106

D₈

—

107

D₈

—

104

—

108

C₂×C₈

#6104

yes

—

109

C₂×C₈

D₄×D₁₃

yes

—

110

C₄⋊C₄

—

111

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

112

C₁₆

—

113

C₁₃

#12519

yes

characteristic, complemented, Sylow

114

C₂³

—

115

C₂³

—

104

—

116

Q₈

—

117

D₄

—

118

D₄

—

119

D₄

—

104

—

120

D₄

—

104

—

121

D₄

—

104

—

122

C₂×C₄

—

123

C₂×C₄

C₂×#6104

yes

—

124

C₂×C₄

#88194

yes

characteristic

125

C₈

—

126

C₈

D₈×D₁₃

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

208

—

130

C₂²

—

208

—

131

C₂²

#11718734

yes

characteristic

132

C₄

—

133

C₄

#11718812

yes

—

134

C₄

#11718854

yes

—

135

C₂

—

136

C₂

—

416

—

137

C₂

#13114450

yes

characteristic

138

C₂

#13117103

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
hypercenter	#1709	D₁₃
hypocenter	C₁₃	#12519
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

C₂²×C₁₃

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #13117093	C₂	semidirect
2. #13117085	C₂	semidirect
3. #13114451	C₂	semidirect
4. #13112753	C₂	semidirect
5. C₁₃×#1709	C₂	semidirect
44. #6090	D₈	semidirect
48. #1709	D₁₃	semidirect
113. C₁₃	#12519	semidirect

H ≅ G/N

type

1. #13117093

C₂

semidirect

2. #13117085

C₂

semidirect

3. #13114451

C₂

semidirect

4. #13112753

C₂

semidirect

5. C₁₃×#1709

C₂

semidirect

44. #6090

D₈

semidirect

48. #1709

D₁₃

semidirect

113. C₁₃

#12519

semidirect

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

subgroup	index
#13117093	2
#13117085	2
#13114451	2
#13112753	2
C₁₃×#1709	2
#12519	13

subgroup

index

C₁₃×#1709

Subgroups — #15777648

Conjugacy classes of subgroups

Complements

Maximal subgroups