Group #15749649 — subgroups

Subgroups — #15749649

Order 3328.

186 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 57 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	#13114451	C₂	1	1	yes	maximal, characteristic
2	#13112906	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#1370	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718356	—	2	1	no	—
5	#11718356	—	2	1	no	—
6	#11718356	C₄	1	2	yes	—
7	C₁₃×C₄⋊C₁₆	—	2	1	no	—
8	C₈²×C₁₃	C₂²	1	1	yes	characteristic
9	C₄⋊#6090	—	2	1	no	—
10	C₂×D₁₀₄	—	4	2	no	—
11	C₂×D₁₀₄	—	4	1	no	—
12	C₂×D₁₀₄	—	4	1	no	—
13	C₂×D₁₀₄	—	4	1	no	—
14	C₂×D₁₀₄	—	4	1	no	—
15	#88128	—	2	2	no	—
16	C₂×C₁₃×C₁₆	—	4	1	no	—
17	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
19	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂×#6090	—	4	1	no	—
21	#7096	—	13	1	no	maximal, Sylow
22	C₂×D₅₂	—	4	2	no	—
23	C₂×D₅₂	—	4	2	no	—
24	C₂×D₅₂	—	4	2	no	—
25	C₂×C₈×C₁₃	—	2	1	no	—
26	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
27	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
28	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
29	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
30	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
31	D₁₀₄	—	8	4	no	—
32	D₁₀₄	—	8	2	no	—
33	D₁₀₄	—	8	2	no	—
34	D₁₀₄	—	8	1	no	—
35	D₁₀₄	—	8	1	no	—
36	D₁₀₄	—	8	1	no	—
37	D₁₀₄	—	8	1	no	—
38	C₁₃×C₁₆	—	8	1	no	—
39	#6090	—	8	1	no	—
40	#1712	—	13	1	no	—
41	#1370	—	13	1	no	—
42	#1370	D₁₃	1	1	yes	characteristic, complemented
43	C₂²×D₁₃	—	8	2	no	—
44	C₂×C₄×C₁₃	#100	1	1	yes	characteristic
45	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
46	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
47	D₅₂	—	8	4	no	—
48	D₅₂	—	8	2	no	—
49	D₅₂	—	8	2	no	—
50	D₅₂	—	8	2	no	—
51	D₅₂	—	8	2	no	—
52	C₈×C₁₃	—	2	2	no	—
53	C₈×C₁₃	—	2	1	no	—
54	C₈×C₁₃	—	2	1	no	—
55	C₈×C₁₃	D₁₆	1	1	yes	characteristic
56	C₈×C₁₃	D₁₆	1	1	yes	characteristic
57	C₈×C₁₃	SD₃₂	1	1	yes	characteristic
58	C₈×C₁₃	SD₃₂	1	1	yes	characteristic
59	#493	—	13	2	no	—
60	#493	—	26	1	no	—
61	#493	—	26	1	no	—
62	C₄⋊C₁₆	—	2	1	no	—
63	C₄⋊C₁₆	—	26	1	no	—
64	C₈²	C₂×D₁₃	1	1	yes	characteristic
65	C₂²×C₁₃	#331	1	1	yes	characteristic
66	C₂×D₁₃	—	16	2	no	—
67	C₂×D₁₃	—	16	2	no	—
68	C₂×D₁₃	—	16	2	no	—
69	C₄×C₁₃	—	2	1	no	—
70	C₄×C₁₃	#357	1	1	yes	characteristic
71	C₄×C₁₃	#357	1	1	yes	characteristic
72	C₄×C₁₃	#361	1	1	yes	characteristic
73	C₄×C₁₃	#361	1	1	yes	characteristic
74	C₂×D₈	—	52	2	no	—
75	C₂×D₈	—	52	1	no	—
76	C₂×D₈	—	52	1	no	—
77	C₂×D₈	—	52	1	no	—
78	C₂×D₈	—	52	1	no	—
79	#127	—	26	2	no	—
80	C₂×C₁₆	—	4	1	no	—
81	C₂×C₁₆	—	52	1	no	—
82	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
83	C₄×C₈	D₅₂	1	1	yes	characteristic
84	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
85	C₂×C₁₃	#1357	1	1	yes	characteristic
86	C₂×C₁₃	#1361	1	1	yes	characteristic
87	C₂×C₁₃	#1361	1	1	yes	characteristic
88	D₁₃	—	32	2	no	—
89	C₂×D₄	—	52	2	no	—
90	C₂×D₄	—	52	2	no	—
91	C₂×D₄	—	52	2	no	—
92	D₈	—	104	4	no	—
93	D₈	—	104	2	no	—
94	D₈	—	104	2	no	—
95	D₈	—	104	1	no	—
96	D₈	—	104	1	no	—
97	D₈	—	104	1	no	—
98	D₈	—	104	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	#6095	1	1	yes	characteristic
101	C₂×C₈	D₁₀₄	1	1	yes	characteristic
102	C₂×C₈	#6104	1	1	yes	characteristic
103	C₂×C₈	#6106	1	1	yes	characteristic
104	C₄²	#6103	1	1	yes	characteristic
105	C₁₆	—	8	1	no	—
106	C₁₆	—	104	1	no	—
107	C₁₃	#7096	1	1	yes	characteristic, complemented, Sylow
108	C₂³	—	104	2	no	—
109	D₄	—	104	4	no	—
110	D₄	—	104	2	no	—
111	D₄	—	104	2	no	—
112	D₄	—	104	2	no	—
113	D₄	—	104	2	no	—
114	C₂×C₄	#88049	1	1	yes	characteristic
115	C₂×C₄	#88061	1	1	yes	characteristic
116	C₂×C₄	#88063	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	1	no	—
119	C₈	—	2	1	no	—
120	C₈	D₂₀₈	1	1	yes	characteristic
121	C₈	#88040	1	1	yes	characteristic
122	C₈	#88066	1	1	yes	characteristic
123	C₈	#88068	1	1	yes	characteristic
124	C₂²	—	208	2	no	—
125	C₂²	—	208	2	no	—
126	C₂²	—	208	2	no	—
127	C₂²	#11718077	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	#11718083	1	1	yes	characteristic
130	C₄	#11718087	1	1	yes	characteristic
131	C₄	#11718101	1	1	yes	characteristic
132	C₄	#11718108	1	1	yes	characteristic
133	C₂	—	416	2	no	—
134	C₂	#13112872	1	1	yes	characteristic
135	C₂	#13112881	1	1	yes	characteristic
136	C₂	#13112883	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114451

C₂

yes

maximal, characteristic

#13112906

C₂

yes

maximal, characteristic, complemented

C₁₃×#1370

C₂

yes

maximal, characteristic, complemented

#11718356

—

#11718356

—

#11718356

C₄

yes

—

C₁₃×C₄⋊C₁₆

—

C₈²×C₁₃

C₂²

yes

characteristic

C₄⋊#6090

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

#88128

—

C₂×C₁₃×C₁₆

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

—

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₈

yes

characteristic

C₂×C₈×C₁₃

D₈

yes

characteristic

C₂×C₈×C₁₃

SD₁₆

yes

characteristic

C₂×C₈×C₁₃

SD₁₆

yes

characteristic

C₄²×C₁₃

C₂²⋊C₄

yes

characteristic

D₁₀₄

—

D₁₀₄

—

D₁₀₄

—

D₁₀₄

—

D₁₀₄

—

D₁₀₄

—

D₁₀₄

—

C₁₃×C₁₆

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×D₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

D₁₆

yes

characteristic

C₈×C₁₃

D₁₆

yes

characteristic

C₈×C₁₃

SD₃₂

yes

characteristic

C₈×C₁₃

SD₃₂

yes

—

—

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#331

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

#127

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

C₄×D₁₃

yes

characteristic

C₄×C₈

D₅₂

yes

characteristic

C₄×C₈

C₂²⋊D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

D₁₃

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

104

—

D₈

—

104

—

D₈

—

104

—

D₈

—

104

—

D₈

—

104

—

D₈

—

104

—

D₈

—

104

—

C₂×C₈

—

100

C₂×C₈

#6095

yes

characteristic

101

C₂×C₈

D₁₀₄

yes

characteristic

102

C₂×C₈

#6104

yes

characteristic

103

C₂×C₈

#6106

yes

characteristic

104

C₄²

#6103

yes

characteristic

105

C₁₆

—

106

C₁₆

—

104

—

107

C₁₃

#7096

yes

characteristic, complemented, Sylow

108

C₂³

—

104

—

109

D₄

—

104

—

110

D₄

—

104

—

111

D₄

—

104

—

112

D₄

—

104

—

113

D₄

—

104

—

114

C₂×C₄

#88049

yes

characteristic

115

C₂×C₄

#88061

yes

characteristic

116

C₂×C₄

#88063

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

D₂₀₈

yes

characteristic

121

C₈

#88040

yes

characteristic

122

C₈

#88066

yes

characteristic

123

C₈

#88068

yes

characteristic

124

C₂²

—

208

—

125

C₂²

—

208

—

126

C₂²

—

208

—

127

C₂²

#11718077

yes

characteristic

128

C₄

—

129

C₄

#11718083

yes

characteristic

130

C₄

#11718087

yes

characteristic

131

C₄

#11718101

yes

characteristic

132

C₄

#11718108

yes

characteristic

133

C₂

—

416

—

134

C₂

#13112872

yes

characteristic

135

C₂

#13112881

yes

characteristic

136

C₂

#13112883

yes

characteristic

	subgroup	G/H
center	C₂²	#11718077
commutator	C₄×C₈×C₁₃	C₂×C₄
Frattini	C₈²	C₂×D₁₃
Fitting	C₁₃×#1370	C₂
socle	C₂²×C₁₃	#331
prime radical	#13114451	C₂
cosocle	C₈²×C₁₃	C₂²
hypercenter	#1370	D₁₃
hypocenter	C₁₃	#7096
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

subgroup

G/H

center

C₂²

#11718077

commutator

C₄×C₈×C₁₃

C₂×C₄

Frattini

C₈²

C₂×D₁₃

Fitting

C₁₃×#1370

C₂

C₂²×C₁₃

C₂

C₈²×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₄×C₈×C₁₃ ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

N	H ≅ G/N	type
2. #13112906	C₂	semidirect
3. C₁₃×#1370	C₂	semidirect
42. #1370	D₁₃	semidirect
107. C₁₃	#7096	semidirect

H ≅ G/N

type

2. #13112906

C₂

semidirect

3. C₁₃×#1370

C₂

semidirect

42. #1370

D₁₃

semidirect

107. C₁₃

#7096

semidirect

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

subgroup	index
#13114451	2
#13112906	2
C₁₃×#1370	2
#7096	13

subgroup

index

#13114451

#13112906

C₁₃×#1370

#7096

Subgroups — #15749649

Conjugacy classes of subgroups

Complements

Maximal subgroups