Group #15756737 — subgroups

Subgroups — #15756737

Order 3328.

306 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 155 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	#13114974	C₂	1	2	yes	maximal
2	#13113595	C₂	1	2	yes	maximal
3	#13113592	C₂	1	2	yes	maximal
4	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	#11718585	C₂²	1	4	yes	—
9	C₁₃×#322	C₄	1	4	yes	—
10	#11718144	—	2	4	no	—
11	#11718050	—	2	4	no	—
12	#11718049	—	2	4	no	—
13	C₂²×C₈×C₁₃	D₄	1	2	yes	—
14	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
15	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×#6099	—	2	4	no	—
18	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₁₃	D₄	1	2	yes	—
22	C₄×C₈×C₁₃	D₄	1	2	yes	—
23	C₄×C₈×C₁₃	Q₈	1	2	yes	—
24	C₄⋊#1055	—	2	4	no	—
25	C₄⋊#1055	—	2	4	no	—
26	#8827	—	13	1	no	maximal, Sylow
27	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₁₃	—	2	4	no	—
31	C₂×C₈×C₁₃	—	4	2	no	—
32	C₂×C₈×C₁₃	—	4	2	no	—
33	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
34	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
36	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
40	#6099	—	4	8	no	—
41	C₂×#1055	—	4	4	no	—
42	#1837	—	13	2	no	—
43	#1567	—	13	2	no	—
44	#1566	—	13	2	no	—
45	C₂×#322	D₁₃	1	1	yes	characteristic
46	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₃	—	2	2	no	—
48	C₂×C₄×C₁₃	—	2	2	no	—
49	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	#135	1	4	yes	—
56	C₈×C₁₃	—	2	4	no	—
57	C₈×C₁₃	—	2	4	no	—
58	C₈×C₁₃	—	2	4	no	—
59	#1055	—	8	4	no	—
60	#423	—	13	4	no	—
61	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
64	#341	—	26	4	no	—
65	#335	—	26	4	no	—
66	#334	—	26	4	no	—
67	#322	Dic₁₃	1	4	yes	—
68	C₂²×C₁₃	—	2	1	no	—
69	C₂²×C₁₃	—	2	1	no	—
70	C₂²×C₁₃	#384	1	1	yes	characteristic
71	C₂²×C₁₃	#499	1	2	yes	—
72	C₄×C₁₃	—	2	1	no	—
73	C₄×C₁₃	—	2	1	no	—
74	C₄×C₁₃	#427	1	2	yes	—
75	C₄×C₁₃	#430	1	2	yes	—
76	C₄×C₁₃	#430	1	2	yes	—
77	C₄×C₁₃	#495	1	2	yes	—
78	C₂×M₁₆	—	26	4	no	—
79	C₂²×C₈	Dic₂₆	1	2	yes	—
80	C₂²×C₈	Dic₂₆	1	2	yes	—
81	C₂²×C₈	D₅₂	1	2	yes	—
82	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
83	C₄⋊C₈	—	26	4	no	—
84	C₄⋊C₈	—	26	4	no	—
85	C₄×C₈	Dic₂₆	1	2	yes	—
86	C₄×C₈	D₅₂	1	2	yes	—
87	C₄×C₈	D₅₂	1	2	yes	—
88	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
89	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
90	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
91	C₂×C₁₃	—	2	2	no	—
92	C₂×C₁₃	#1846	1	1	yes	characteristic
93	C₂×C₁₃	#1849	1	1	yes	characteristic
94	C₂×C₁₃	#1849	1	1	yes	characteristic
95	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
96	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
97	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
98	M₁₆	—	52	8	no	—
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	4	2	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	52	4	no	—
103	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
104	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
105	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
106	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
107	C₄²	C₂×D₅₂	1	1	yes	characteristic
108	C₄²	C₂×D₅₂	1	1	yes	characteristic
109	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
110	C₁₃	#8827	1	1	yes	characteristic, complemented, Sylow
111	C₂³	#88123	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	#88123	1	1	yes	characteristic
115	C₂×C₄	#88123	1	1	yes	characteristic
116	C₂×C₄	#88128	1	1	yes	characteristic
117	C₂×C₄	#88158	1	4	yes	—
118	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
119	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
120	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₈	—	2	4	no	—
123	C₈	—	2	4	no	—
124	C₈	—	104	4	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#11718344	1	2	yes	—
128	C₂²	#11718596	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	#11718357	1	2	yes	—
132	C₄	#11718767	1	2	yes	—
133	C₄	#11718786	1	2	yes	—
134	C₄	#11718786	1	2	yes	—
135	C₂	—	2	2	no	—
136	C₂	#13115000	1	1	yes	characteristic
137	C₂	#13115011	1	1	yes	characteristic
138	C₂	#13115011	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114974

C₂

yes

maximal

#13113595

C₂

yes

maximal

#13113592

C₂

yes

maximal

C₂×C₁₃×#322

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

#11718585

C₂²

yes

—

C₁₃×#322

C₄

yes

—

#11718144

—

#11718050

—

#11718049

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

Q₈

yes

—

C₂²×C₈×C₁₃

Q₈

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

—

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

—

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

#135

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

#423

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

—

Dic₁₃

yes

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#384

yes

characteristic

C₂²×C₁₃

#499

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#427

yes

—

C₄×C₁₃

#430

yes

—

C₄×C₁₃

#430

yes

—

C₄×C₁₃

#495

yes

—

C₂×M₁₆

—

C₂²×C₈

Dic₂₆

yes

—

C₂²×C₈

Dic₂₆

yes

—

C₂²×C₈

D₅₂

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

Dic₂₆

yes

—

C₄×C₈

D₅₂

yes

—

C₄×C₈

D₅₂

yes

—

C₄×C₈

C₂×Dic₁₃

yes

—

C₄×C₈

C₂×Dic₁₃

yes

—

C₄×C₈

C₂×Dic₁₃

yes

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

C₂²×C₄

C₂×D₅₂

yes

characteristic

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₄⋊Dic₁₃

yes

—

104

C₂×C₈

C₄⋊Dic₁₃

yes

—

105

C₂×C₈

C₄⋊Dic₁₃

yes

—

106

C₄²

C₂×Dic₂₆

yes

characteristic

107

C₄²

C₂×D₅₂

yes

characteristic

108

C₄²

C₂×D₅₂

yes

characteristic

109

C₄²

C₂²×Dic₁₃

yes

characteristic

110

C₁₃

#8827

yes

characteristic, complemented, Sylow

111

C₂³

#88123

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#88123

yes

characteristic

115

C₂×C₄

#88123

yes

characteristic

116

C₂×C₄

#88128

yes

characteristic

117

C₂×C₄

#88158

yes

—

118

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

119

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

120

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

104

—

125

C₂²

—

126

C₂²

—

127

C₂²

#11718344

yes

—

128

C₂²

#11718596

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

#11718357

yes

—

132

C₄

#11718767

yes

—

133

C₄

#11718786

yes

—

134

C₄

#11718786

yes

—

135

C₂

—

136

C₂

#13115000

yes

characteristic

137

C₂

#13115011

yes

characteristic

138

C₂

#13115011

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88128
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle	C₂²×C₁₃	#384
prime radical	C₂³×C₁₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#8827
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#88128

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₂×C₁₃×#322

C₂

socle

C₂²×C₁₃

#384

prime radical

C₂³×C₁₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
110. C₁₃	#8827	semidirect

H ≅ G/N

type

110. C₁₃

#8827

semidirect

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

subgroup	index
#13114974	2
#13113595	2
#13113592	2
C₂×C₁₃×#322	2
#8827	13

subgroup

index

C₂×C₁₃×#322

Subgroups — #15756737

Conjugacy classes of subgroups

Complements

Maximal subgroups