Group #15749641 — subgroups

Subgroups — #15749641

Order 3328.

130 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 49 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	#13114446	C₂	1	1	yes	maximal, characteristic
2	#13112900	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1367	C₂	1	1	yes	maximal, characteristic
4	#11718343	C₄	1	1	yes	characteristic
5	#11718343	C₄	1	1	yes	characteristic
6	#11718342	—	2	1	no	—
7	#11718342	—	2	1	no	—
8	C₁₃×C₄⋊C₁₆	—	2	1	no	—
9	C₈²×C₁₃	C₂²	1	1	yes	characteristic
10	C₄⋊#6090	—	2	1	no	—
11	#88123	—	2	1	no	—
12	#88123	—	2	1	no	—
13	C₂×C₁₃×C₁₆	—	4	1	no	—
14	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
16	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
17	#88058	—	4	1	no	—
18	#88058	—	4	1	no	—
19	#88056	—	4	1	no	—
20	#88056	—	4	1	no	—
21	#88056	—	4	1	no	—
22	#88056	—	4	1	no	—
23	C₂×#6090	—	4	1	no	—
24	#7093	—	13	1	no	maximal, Sylow
25	C₂×Dic₂₆	—	4	1	no	—
26	C₂×Dic₂₆	—	4	1	no	—
27	C₂×C₈×C₁₃	—	2	1	no	—
28	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
29	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
30	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
31	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
32	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₄⋊Dic₁₃	—	4	1	no	—
34	C₄⋊Dic₁₃	—	4	1	no	—
35	C₄⋊Dic₁₃	—	4	1	no	—
36	C₄⋊Dic₁₃	—	4	1	no	—
37	C₁₃×C₁₆	—	8	1	no	—
38	#6090	—	8	1	no	—
39	#1710	—	13	1	no	—
40	#1367	—	13	1	no	—
41	#1367	D₁₃	1	1	yes	characteristic
42	C₂×C₄×C₁₃	#100	1	1	yes	characteristic
43	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
44	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
45	C₂×Dic₁₃	—	8	1	no	—
46	C₂×Dic₁₃	—	8	1	no	—
47	Dic₂₆	—	8	1	no	—
48	Dic₂₆	—	8	1	no	—
49	Dic₂₆	—	8	1	no	—
50	Dic₂₆	—	8	1	no	—
51	C₈×C₁₃	—	2	2	no	—
52	C₈×C₁₃	—	2	1	no	—
53	C₈×C₁₃	—	2	1	no	—
54	C₈×C₁₃	—	2	1	no	—
55	C₈×C₁₃	—	2	1	no	—
56	#500	—	13	1	no	—
57	#500	—	13	1	no	—
58	#487	—	26	1	no	—
59	#487	—	26	1	no	—
60	C₄⋊C₁₆	—	2	1	no	—
61	C₄⋊C₁₆	—	26	1	no	—
62	C₈²	C₂×D₁₃	1	1	yes	characteristic
63	C₂²×C₁₃	#331	1	1	yes	characteristic
64	C₄×C₁₃	—	2	1	no	—
65	C₄×C₁₃	#359	1	1	yes	characteristic
66	C₄×C₁₃	#359	1	1	yes	characteristic
67	C₄×C₁₃	#360	1	1	yes	characteristic
68	C₄×C₁₃	#360	1	1	yes	characteristic
69	Dic₁₃	—	16	1	no	—
70	Dic₁₃	—	16	1	no	—
71	C₄⋊Q₈	—	26	1	no	—
72	C₄⋊Q₈	—	26	1	no	—
73	C₂×C₁₆	—	4	1	no	—
74	C₂×C₁₆	—	52	1	no	—
75	#107	—	52	1	no	—
76	#107	—	52	1	no	—
77	#103	—	52	1	no	—
78	#103	—	52	1	no	—
79	#103	—	52	1	no	—
80	#103	—	52	1	no	—
81	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
82	C₄×C₈	D₅₂	1	1	yes	characteristic
83	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
84	C₂×C₁₃	#1358	1	1	yes	characteristic
85	C₂×C₁₃	#1360	1	1	yes	characteristic
86	C₂×C₁₃	#1360	1	1	yes	characteristic
87	C₂×Q₈	—	52	1	no	—
88	C₂×Q₈	—	52	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	#6095	1	1	yes	characteristic
91	C₂×C₈	D₁₀₄	1	1	yes	characteristic
92	C₂×C₈	#6104	1	1	yes	characteristic
93	C₂×C₈	#6106	1	1	yes	characteristic
94	C₄⋊C₄	—	52	1	no	—
95	C₄⋊C₄	—	52	1	no	—
96	C₄⋊C₄	—	52	1	no	—
97	C₄⋊C₄	—	52	1	no	—
98	C₄²	#6103	1	1	yes	characteristic
99	C₁₆	—	8	1	no	—
100	C₁₆	—	104	1	no	—
101	C₁₃	#7093	1	1	yes	characteristic, complemented, Sylow
102	Q₈	—	104	1	no	—
103	Q₈	—	104	1	no	—
104	Q₈	—	104	1	no	—
105	Q₈	—	104	1	no	—
106	C₂×C₄	—	104	1	no	—
107	C₂×C₄	—	104	1	no	—
108	C₂×C₄	#88049	1	1	yes	characteristic
109	C₂×C₄	#88061	1	1	yes	characteristic
110	C₂×C₄	#88063	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	1	no	—
113	C₈	—	2	1	no	—
114	C₈	—	2	1	no	—
115	C₈	—	2	1	no	—
116	C₂²	#11718077	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	208	1	no	—
119	C₄	—	208	1	no	—
120	C₄	#11718080	1	1	yes	characteristic
121	C₄	#11718089	1	1	yes	characteristic
122	C₄	#11718102	1	1	yes	characteristic
123	C₄	#11718110	1	1	yes	characteristic
124	C₂	#13112874	1	1	yes	characteristic
125	C₂	#13112878	1	1	yes	characteristic
126	C₂	#13112880	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114446

C₂

yes

maximal, characteristic

#13112900

C₂

yes

maximal, characteristic

C₁₃×#1367

C₂

yes

maximal, characteristic

#11718343

C₄

yes

characteristic

#11718343

C₄

yes

characteristic

#11718342

—

#11718342

—

C₁₃×C₄⋊C₁₆

—

C₈²×C₁₃

C₂²

yes

—

—

—

C₂×C₁₃×C₁₆

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

—

—

—

—

—

—

—

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

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

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₁₃×C₁₆

—

—

—

—

D₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#331

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

C₄⋊Q₈

—

C₄⋊Q₈

—

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

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

#6095

yes

characteristic

C₂×C₈

D₁₀₄

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

#6103

yes

characteristic

C₁₆

—

100

C₁₆

—

104

—

101

C₁₃

#7093

yes

characteristic, complemented, Sylow

102

Q₈

—

104

—

103

Q₈

—

104

—

104

Q₈

—

104

—

105

Q₈

—

104

—

106

C₂×C₄

—

104

—

107

C₂×C₄

—

104

—

108

C₂×C₄

#88049

yes

characteristic

109

C₂×C₄

#88061

yes

characteristic

110

C₂×C₄

#88063

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₂²

#11718077

yes

characteristic

117

C₄

—

118

C₄

—

208

—

119

C₄

—

208

—

120

C₄

#11718080

yes

characteristic

121

C₄

#11718089

yes

characteristic

122

C₄

#11718102

yes

characteristic

123

C₄

#11718110

yes

characteristic

124

C₂

#13112874

yes

characteristic

125

C₂

#13112878

yes

characteristic

126

C₂

#13112880

yes

characteristic

	subgroup	G/H
center	C₂²	#11718077
commutator	C₄×C₈×C₁₃	C₂×C₄
Frattini	C₈²	C₂×D₁₃
Fitting	C₁₃×#1367	C₂
socle, prime radical	C₂²×C₁₃	#331
cosocle	C₈²×C₁₃	C₂²
hypercenter	#1367	D₁₃
hypocenter	C₁₃	#7093
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂²

#11718077

commutator

C₄×C₈×C₁₃

C₂×C₄

Frattini

C₈²

C₂×D₁₃

Fitting

C₁₃×#1367

C₂

socle, prime radical

C₂²×C₁₃

#331

cosocle

C₈²×C₁₃

C₂²

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

N	H ≅ G/N	type
101. C₁₃	#7093	semidirect

H ≅ G/N

type

101. C₁₃

#7093

semidirect

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

subgroup	index
#13114446	2
#13112900	2
C₁₃×#1367	2
#7093	13

subgroup

index

#13114446

#13112900

C₁₃×#1367

#7093

Subgroups — #15749641

Conjugacy classes of subgroups

Complements

Maximal subgroups