Group #15756590 — subgroups

Subgroups — #15756590

Order 3328.

362 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 179 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	#13114988	C₂	1	2	yes	maximal
2	#13112671	C₂	1	4	yes	maximal
3	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
5	C₄³×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊#1055	—	2	2	no	—
7	C₂×C₄⋊#1055	—	2	2	no	—
8	C₂×C₄⋊#1055	C₂²	1	4	yes	—
9	C₁₃×#336	C₄	1	4	yes	—
10	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
11	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
12	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
13	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
15	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
16	C₂²×#1055	—	2	4	no	—
17	C₄×C₈×C₁₃	—	2	4	no	—
18	C₄⋊#1055	—	2	8	no	—
19	C₄⋊#1055	—	4	4	no	—
20	C₄⋊#1055	—	4	4	no	—
21	#8787	—	13	1	no	maximal, Sylow
22	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
23	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
24	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₈×C₁₃	—	2	4	no	—
28	C₂×C₈×C₁₃	—	2	8	no	—
29	C₄²×C₁₃	—	2	4	no	—
30	C₄²×C₁₃	—	4	2	no	—
31	C₄²×C₁₃	D₈	1	2	yes	—
32	C₄²×C₁₃	SD₁₆	1	4	yes	—
33	C₄²×C₁₃	Q₁₆	1	2	yes	—
34	C₄²×C₁₃	#41	1	4	yes	—
35	C₂×#1055	—	4	4	no	—
36	C₂×#1055	—	4	4	no	—
37	C₂×#1055	—	4	4	no	—
38	#1842	—	13	2	no	—
39	#1744	D₁₃	1	1	yes	characteristic
40	#1277	—	13	4	no	—
41	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃	—	2	2	no	—
43	C₂×C₄×C₁₃	—	2	2	no	—
44	C₂×C₄×C₁₃	—	2	2	no	—
45	C₂×C₄×C₁₃	—	2	2	no	—
46	C₂×C₄×C₁₃	—	2	2	no	—
47	C₂×C₄×C₁₃	#106	1	4	yes	—
48	C₂×C₄×C₁₃	#107	1	4	yes	—
49	C₂×C₄×C₁₃	#117	1	2	yes	—
50	C₂×C₄×C₁₃	#124	1	2	yes	—
51	C₂×C₄×C₁₃	#125	1	2	yes	—
52	C₂×C₄×C₁₃	#131	1	4	yes	—
53	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
55	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
56	C₈×C₁₃	—	4	4	no	—
57	#1055	—	8	4	no	—
58	C₂×C₄⋊C₈	—	13	4	no	—
59	C₂×C₄⋊C₈	—	26	2	no	—
60	C₂×C₄⋊C₈	—	26	2	no	—
61	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
62	C₄³	C₂×D₁₃	1	1	yes	characteristic
63	#336	Dic₁₃	1	4	yes	—
64	C₂²×C₁₃	#382	1	1	yes	characteristic
65	C₂²×C₁₃	#424	1	2	yes	—
66	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
67	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
68	C₂²×C₁₃	#430	1	2	yes	—
69	C₄×C₁₃	—	2	4	no	—
70	C₄×C₁₃	—	2	4	no	—
71	C₄×C₁₃	—	2	2	no	—
72	C₄×C₁₃	#433	1	4	yes	—
73	C₄×C₁₃	#486	1	2	yes	—
74	C₄×C₁₃	#487	1	2	yes	—
75	C₂²×C₈	—	26	4	no	—
76	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
77	C₂×C₄²	Dic₂₆	1	2	yes	—
78	C₂×C₄²	D₅₂	1	1	yes	characteristic
79	C₂×C₄²	D₅₂	1	1	yes	characteristic
80	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
81	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
82	C₄⋊C₈	—	26	8	no	—
83	C₄⋊C₈	—	52	4	no	—
84	C₄⋊C₈	—	52	4	no	—
85	C₄×C₈	—	2	4	no	—
86	C₂×C₁₃	#1570	1	2	yes	—
87	C₂×C₁₃	#1571	1	2	yes	—
88	C₂×C₁₃	#1832	1	1	yes	characteristic
89	C₂×C₁₃	#1834	1	1	yes	characteristic
90	C₂×C₁₃	#1838	1	1	yes	characteristic
91	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
92	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
93	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
95	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	8	no	—
98	C₂×C₈	—	52	4	no	—
99	C₂×C₈	—	52	4	no	—
100	C₂×C₈	—	52	4	no	—
101	C₄²	—	2	4	no	—
102	C₄²	—	4	2	no	—
103	C₄²	#6095	1	4	yes	—
104	C₄²	D₁₀₄	1	2	yes	—
105	C₄²	Dic₅₂	1	2	yes	—
106	C₄²	#6127	1	4	yes	—
107	C₁₃	#8787	1	1	yes	characteristic, complemented, Sylow
108	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	#88057	1	4	yes	—
115	C₂×C₄	#88058	1	4	yes	—
116	C₂×C₄	#88124	1	2	yes	—
117	C₂×C₄	#88129	1	2	yes	—
118	C₂×C₄	C₂×#6095	1	2	yes	—
119	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
121	C₂×C₄	#88180	1	2	yes	—
122	C₂×C₄	#88202	1	4	yes	—
123	C₈	—	4	4	no	—
124	C₈	—	104	4	no	—
125	C₂²	#11718597	1	1	yes	characteristic
126	C₂²	#11718660	1	2	yes	—
127	C₂²	C₂×#88057	1	1	yes	characteristic
128	C₂²	C₂×#88058	1	1	yes	characteristic
129	C₂²	#11718786	1	2	yes	—
130	C₄	—	2	4	no	—
131	C₄	—	2	4	no	—
132	C₄	—	2	2	no	—
133	C₄	#11718342	1	2	yes	—
134	C₄	#11718355	1	2	yes	—
135	C₄	#11718661	1	4	yes	—
136	C₂	#13113607	1	2	yes	—
137	C₂	#13113611	1	2	yes	—
138	C₂	#13114956	1	1	yes	characteristic
139	C₂	#13114964	1	1	yes	characteristic
140	C₂	#13114977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114988

C₂

yes

maximal

#13112671

C₂

yes

maximal

C₁₃×#1744

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₄³×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊#1055

—

C₂×C₄⋊#1055

—

C₂×C₄⋊#1055

C₂²

yes

—

C₁₃×#336

C₄

yes

—

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

Q₈

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

—

C₄×C₈×C₁₃

—

—

—

—

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

D₈

yes

—

C₄²×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

Q₁₆

yes

—

C₄²×C₁₃

#41

yes

—

—

—

—

—

D₁₃

yes

characteristic

#1277

—

C₂³×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#106

yes

—

C₂×C₄×C₁₃

#107

yes

—

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#124

yes

—

C₂×C₄×C₁₃

#125

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×Q₁₆

yes

characteristic

C₈×C₁₃

—

#1055

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

C₄³

C₂×D₁₃

yes

characteristic

#336

Dic₁₃

yes

—

C₂²×C₁₃

#382

yes

characteristic

C₂²×C₁₃

#424

yes

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#433

yes

—

C₄×C₁₃

#486

yes

—

C₄×C₁₃

#487

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₁₃

yes

—

C₂×C₄²

Dic₂₆

yes

—

C₂×C₄²

D₅₂

yes

characteristic

C₂×C₄²

D₅₂

yes

characteristic

C₂×C₄²

C₂×Dic₁₃

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₂×C₁₃

#1570

yes

—

C₂×C₁₃

#1571

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂²×C₄

C₄⋊Dic₁₃

yes

—

C₂²×C₄

C₄⋊Dic₁₃

yes

—

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

C₂²×C₄

C₂×D₅₂

yes

characteristic

C₂²×C₄

C₂²×Dic₁₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄²

—

102

C₄²

—

103

C₄²

#6095

yes

—

104

C₄²

D₁₀₄

yes

—

105

C₄²

Dic₅₂

yes

—

106

C₄²

#6127

yes

—

107

C₁₃

#8787

yes

characteristic, complemented, Sylow

108

C₂³

C₂×C₄⋊Dic₁₃

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#88057

yes

—

115

C₂×C₄

#88058

yes

—

116

C₂×C₄

#88124

yes

—

117

C₂×C₄

#88129

yes

—

118

C₂×C₄

C₂×#6095

yes

—

119

C₂×C₄

C₂×D₁₀₄

yes

characteristic

120

C₂×C₄

C₂×Dic₅₂

yes

characteristic

121

C₂×C₄

#88180

yes

—

122

C₂×C₄

#88202

yes

—

123

C₈

—

124

C₈

—

104

—

125

C₂²

#11718597

yes

characteristic

126

C₂²

#11718660

yes

—

127

C₂²

C₂×#88057

yes

characteristic

128

C₂²

C₂×#88058

yes

characteristic

129

C₂²

#11718786

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#11718342

yes

—

134

C₄

#11718355

yes

—

135

C₄

#11718661

yes

—

136

C₂

#13113607

yes

—

137

C₂

#13113611

yes

—

138

C₂

#13114956

yes

characteristic

139

C₂

#13114964

yes

characteristic

140

C₂

#13114977

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1744	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1744	D₁₃
hypocenter	C₁₃	#8787
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₅₂

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1744

C₂

socle, prime radical

C₂³×C₁₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
107. C₁₃	#8787	semidirect

H ≅ G/N

type

107. C₁₃

#8787

semidirect

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

subgroup	index
#13114988	2
#13112671	2
C₁₃×#1744	2
#8787	13

subgroup

index

#13114988

#13112671

C₁₃×#1744

#8787

Subgroups — #15756590

Conjugacy classes of subgroups

Complements

Maximal subgroups