Group #15756679 — subgroups

Subgroups — #15756679

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	#13114953	C₂	1	2	yes	maximal
2	C₂×#11718050	C₂	1	2	yes	maximal
3	C₂×#11718049	C₂	1	2	yes	maximal
4	C₁₃×#1744	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	C₄³×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#1055	C₂²	1	4	yes	—
9	C₁₃×#336	C₄	1	4	yes	—
10	#11718116	—	2	4	no	—
11	#11718050	—	2	4	no	—
12	#11718049	—	2	4	no	—
13	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₃	D₄	1	2	yes	—
17	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×#1055	—	2	4	no	—
20	C₄×C₈×C₁₃	D₄	1	2	yes	—
21	C₄×C₈×C₁₃	D₄	1	2	yes	—
22	C₄×C₈×C₁₃	Q₈	1	2	yes	—
23	C₄×C₈×C₁₃	Q₈	1	2	yes	—
24	C₄⋊#1055	—	2	4	no	—
25	C₄⋊#1055	—	2	4	no	—
26	#8811	—	13	1	no	maximal, Sylow
27	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
28	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₈×C₁₃	—	2	4	no	—
32	C₂×C₈×C₁₃	—	2	4	no	—
33	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
34	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
35	C₄²×C₁₃	—	2	4	no	—
36	C₄²×C₁₃	—	2	4	no	—
37	C₄²×C₁₃	—	4	2	no	—
38	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
41	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×#1055	—	4	4	no	—
43	C₂×#1055	—	4	8	no	—
44	#1831	—	13	2	no	—
45	#1744	D₁₃	1	1	yes	characteristic
46	C₂×#335	—	13	2	no	—
47	C₂×#334	—	13	2	no	—
48	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃	—	2	2	no	—
50	C₂×C₄×C₁₃	—	2	4	no	—
51	C₂×C₄×C₁₃	—	2	4	no	—
52	C₂×C₄×C₁₃	—	2	4	no	—
53	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#131	1	4	yes	—
60	C₂×C₄×C₁₃	#135	1	4	yes	—
61	C₈×C₁₃	—	2	4	no	—
62	C₈×C₁₃	—	2	4	no	—
63	#1055	—	8	4	no	—
64	C₂×C₄⋊C₈	—	13	4	no	—
65	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
67	C₄³	C₂×D₁₃	1	1	yes	characteristic
68	#336	—	26	4	no	—
69	#336	Dic₁₃	1	4	yes	—
70	#335	—	26	4	no	—
71	#334	—	26	4	no	—
72	C₂²×C₁₃	#384	1	1	yes	characteristic
73	C₂²×C₁₃	#424	1	2	yes	—
74	C₂²×C₁₃	#427	1	2	yes	—
75	C₂²×C₁₃	#430	1	2	yes	—
76	C₄×C₁₃	—	2	1	no	—
77	C₄×C₁₃	—	2	1	no	—
78	C₄×C₁₃	—	4	4	no	—
79	C₄×C₁₃	#424	1	2	yes	—
80	C₄×C₁₃	#424	1	2	yes	—
81	C₄×C₁₃	#495	1	2	yes	—
82	C₄×C₁₃	#499	1	2	yes	—
83	C₂²×C₈	—	26	4	no	—
84	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
85	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
86	C₂×C₄²	Dic₂₆	1	2	yes	—
87	C₂×C₄²	D₅₂	1	2	yes	—
88	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
89	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
90	C₄⋊C₈	—	26	4	no	—
91	C₄⋊C₈	—	26	4	no	—
92	C₄×C₈	Dic₂₆	1	2	yes	—
93	C₄×C₈	Dic₂₆	1	2	yes	—
94	C₄×C₈	D₅₂	1	2	yes	—
95	C₄×C₈	D₅₂	1	2	yes	—
96	C₂×C₁₃	#1572	1	4	yes	—
97	C₂×C₁₃	#1843	1	1	yes	characteristic
98	C₂×C₁₃	#1846	1	1	yes	characteristic
99	C₂×C₁₃	#1849	1	1	yes	characteristic
100	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
101	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
102	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
103	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	52	4	no	—
107	C₂×C₈	—	52	8	no	—
108	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
109	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
110	C₄²	—	2	4	no	—
111	C₄²	—	2	4	no	—
112	C₄²	—	4	2	no	—
113	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
114	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
115	C₄²	C₂×D₅₂	1	1	yes	characteristic
116	C₄²	C₂×D₅₂	1	1	yes	characteristic
117	C₁₃	#8811	1	1	yes	characteristic, complemented, Sylow
118	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	#88123	1	1	yes	characteristic
124	C₂×C₄	#88123	1	1	yes	characteristic
125	C₂×C₄	#88123	1	1	yes	characteristic
126	C₂×C₄	#88128	1	1	yes	characteristic
127	C₂×C₄	#88158	1	4	yes	—
128	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
129	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
130	C₂×C₄	#88202	1	4	yes	—
131	C₈	—	2	4	no	—
132	C₈	—	2	4	no	—
133	C₈	—	104	4	no	—
134	C₂²	#11718596	1	1	yes	characteristic
135	C₂²	#11718660	1	2	yes	—
136	C₂²	#11718767	1	2	yes	—
137	C₂²	#11718786	1	2	yes	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	4	no	—
141	C₄	#11718344	1	2	yes	—
142	C₄	#11718357	1	2	yes	—
143	C₄	#11718660	1	2	yes	—
144	C₄	#11718660	1	2	yes	—
145	C₂	#13113616	1	4	yes	—
146	C₂	#13114990	1	1	yes	characteristic
147	C₂	#13115000	1	1	yes	characteristic
148	C₂	#13115011	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114953

C₂

yes

maximal

C₂×#11718050

C₂

yes

maximal

C₂×#11718049

C₂

yes

maximal

C₁₃×#1744

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₄³×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊#1055

C₂²

yes

—

C₁₃×#336

C₄

yes

—

#11718116

—

#11718050

—

#11718049

—

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

Q₈

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

C₄×C₈×C₁₃

Q₈

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

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

—

—

—

D₁₃

yes

characteristic

C₂×#335

—

C₂×#334

—

C₂³×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

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

#127

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₄³

C₂×D₁₃

yes

characteristic

#336

—

#336

Dic₁₃

yes

—

#335

—

#334

—

C₂²×C₁₃

#384

yes

characteristic

C₂²×C₁₃

#424

yes

—

C₂²×C₁₃

#427

yes

—

C₂²×C₁₃

#430

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#424

yes

—

C₄×C₁₃

#424

yes

—

C₄×C₁₃

#495

yes

—

C₄×C₁₃

#499

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₁₃

yes

—

C₂²×C₈

C₂×Dic₁₃

yes

—

C₂×C₄²

Dic₂₆

yes

—

C₂×C₄²

D₅₂

yes

—

C₂×C₄²

C₂×Dic₁₃

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

Dic₂₆

yes

—

C₄×C₈

Dic₂₆

yes

—

C₄×C₈

D₅₂

yes

—

C₄×C₈

D₅₂

yes

—

C₂×C₁₃

#1572

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

100

C₂²×C₄

C₄⋊Dic₁₃

yes

—

101

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

102

C₂²×C₄

C₂×D₅₂

yes

characteristic

103

C₂²×C₄

C₂²×Dic₁₃

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₄⋊Dic₁₃

yes

—

109

C₂×C₈

C₄⋊Dic₁₃

yes

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

C₂×Dic₂₆

yes

characteristic

114

C₄²

C₂×Dic₂₆

yes

characteristic

115

C₄²

C₂×D₅₂

yes

characteristic

116

C₄²

C₂×D₅₂

yes

characteristic

117

C₁₃

#8811

yes

characteristic, complemented, Sylow

118

C₂³

C₂×C₄⋊Dic₁₃

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

#88123

yes

characteristic

124

C₂×C₄

#88123

yes

characteristic

125

C₂×C₄

#88123

yes

characteristic

126

C₂×C₄

#88128

yes

characteristic

127

C₂×C₄

#88158

yes

—

128

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

129

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

130

C₂×C₄

#88202

yes

—

131

C₈

—

132

C₈

—

133

C₈

—

104

—

134

C₂²

#11718596

yes

characteristic

135

C₂²

#11718660

yes

—

136

C₂²

#11718767

yes

—

137

C₂²

#11718786

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#11718344

yes

—

142

C₄

#11718357

yes

—

143

C₄

#11718660

yes

—

144

C₄

#11718660

yes

—

145

C₂

#13113616

yes

—

146

C₂

#13114990

yes

characteristic

147

C₂

#13115000

yes

characteristic

148

C₂

#13115011

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	#8811
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
117. C₁₃	#8811	semidirect

H ≅ G/N

type

117. C₁₃

#8811

semidirect

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

subgroup	index
#13114953	2
C₂×#11718050	2
C₂×#11718049	2
C₁₃×#1744	2
#8811	13

subgroup

index

C₁₃×#1744

Subgroups — #15756679

Conjugacy classes of subgroups

Complements

Maximal subgroups