Group #15756691 — subgroups

Subgroups — #15756691

Order 3328.

422 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 203 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	#13114993	C₂	1	1	yes	maximal, characteristic
2	#13114963	C₂	1	4	yes	maximal
3	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#106	—	2	2	no	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₁₃×#384	C₂²	1	2	yes	—
8	C₂×#88057	—	2	2	no	—
9	#11718596	C₂²	1	2	yes	—
10	C₂×C₄×#1055	C₂²	1	2	yes	—
11	#11718121	—	2	8	no	—
12	C₈×#1055	C₄	1	4	yes	complemented
13	C₂²×C₈×C₁₃	D₄	1	2	yes	—
14	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
15	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
16	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₃	—	2	2	no	—
18	C₂×C₄⋊Dic₁₃	—	2	4	no	—
19	C₂²×#1055	D₄	1	4	yes	—
20	C₁₃×#106	—	4	4	no	—
21	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₁₃	Q₈	1	2	yes	—
23	#88057	—	4	4	no	—
24	C₄×#1055	C₂×C₄	1	4	yes	—
25	C₄×#1055	Q₈	1	4	yes	—
26	#8815	—	13	1	no	maximal, Sylow
27	C₂²×C₄×C₁₃	—	4	2	no	—
28	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
29	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
30	C₂²×Dic₁₃	—	4	2	no	—
31	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
32	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
33	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
34	C₁₃×C₄⋊C₄	—	4	4	no	—
35	C₁₃×C₄⋊C₄	—	4	8	no	—
36	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
38	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
39	C₄⋊Dic₁₃	—	4	4	no	—
40	C₄⋊Dic₁₃	—	4	8	no	—
41	C₂×#1055	—	2	8	no	—
42	C₂×#1055	C₄⋊C₄	1	8	yes	—
43	#1844	—	13	1	no	—
44	#1844	D₁₃	1	1	yes	characteristic
45	#1834	—	13	4	no	—
46	C₂×C₈²	—	13	1	no	—
47	C₂³×C₁₃	#127	1	1	yes	characteristic
48	C₂×C₄×C₁₃	—	8	2	no	—
49	C₂×C₄×C₁₃	—	8	4	no	—
50	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
52	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
54	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
55	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
56	C₂×C₄×C₁₃	#135	1	8	yes	—
57	C₂×Dic₁₃	—	8	2	no	—
58	C₂×Dic₁₃	—	8	4	no	—
59	C₈×C₁₃	#106	1	8	yes	—
60	#1055	—	2	8	no	—
61	C₂×#106	—	2	2	no	—
62	C₂×#106	—	26	2	no	—
63	C₂×C₄×C₈	—	13	2	no	—
64	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
65	#384	—	13	2	no	—
66	#384	C₂×D₁₃	1	2	yes	—
67	#340	—	26	8	no	—
68	C₈²	—	13	4	no	—
69	C₂²×C₁₃	#384	1	1	yes	characteristic
70	C₂²×C₁₃	#492	1	1	yes	characteristic
71	C₂²×C₁₃	#492	1	1	yes	characteristic
72	C₂²×C₁₃	#495	1	4	yes	—
73	C₄×C₁₃	—	16	2	no	—
74	C₄×C₁₃	C₂×#106	1	2	yes	—
75	C₄×C₁₃	#427	1	4	yes	—
76	C₄×C₁₃	#498	1	2	yes	—
77	C₄×C₁₃	#499	1	4	yes	—
78	Dic₁₃	—	16	2	no	—
79	C₂²×C₈	—	13	4	no	—
80	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
81	C₂×C₄⋊C₄	—	2	4	no	—
82	C₂×C₄⋊C₄	—	2	2	no	—
83	C₂×C₄⋊C₄	—	26	4	no	—
84	C₂×C₄⋊C₄	—	26	2	no	—
85	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
86	#106	—	4	4	no	—
87	#106	—	52	4	no	—
88	C₄×C₈	—	13	4	no	—
89	C₄×C₈	—	13	4	no	—
90	C₄×C₈	Dic₂₆	1	2	yes	—
91	C₄×C₈	C₄×D₁₃	1	2	yes	—
92	C₂×C₁₃	#1708	1	2	yes	—
93	C₂×C₁₃	#1708	1	2	yes	—
94	C₂×C₁₃	#1844	1	1	yes	characteristic
95	C₂×C₁₃	#1846	1	2	yes	—
96	C₂²×C₄	—	4	2	no	—
97	C₂²×C₄	—	52	2	no	—
98	C₂²×C₄	D₄×D₁₃	1	2	yes	—
99	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
100	C₂×C₈	—	13	8	no	—
101	C₂×C₈	—	26	8	no	—
102	C₂×C₈	#6101	1	4	yes	—
103	C₂×C₈	#6105	1	4	yes	—
104	C₂×C₈	#6106	1	4	yes	—
105	C₄⋊C₄	—	4	8	no	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄⋊C₄	—	52	8	no	—
108	C₄⋊C₄	—	52	4	no	—
109	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
110	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
111	C₄²	Q₈×D₁₃	1	2	yes	—
112	C₁₃	#8815	1	1	yes	characteristic, complemented, Sylow
113	C₂³	#88194	1	1	yes	characteristic
114	C₂×C₄	—	8	2	no	—
115	C₂×C₄	—	8	4	no	—
116	C₂×C₄	—	104	2	no	—
117	C₂×C₄	—	104	4	no	—
118	C₂×C₄	#88142	1	2	yes	—
119	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
120	C₂×C₄	#88170	1	8	yes	—
121	C₂×C₄	C₂×#6101	1	1	yes	characteristic
122	C₂×C₄	C₂×#6105	1	2	yes	—
123	C₂×C₄	C₂×#6106	1	2	yes	—
124	C₂×C₄	#88198	1	1	yes	characteristic
125	C₈	—	26	8	no	—
126	C₈	#88048	1	8	yes	—
127	C₂²	#11718637	1	1	yes	characteristic
128	C₂²	#11718737	1	1	yes	characteristic
129	C₂²	#11718746	1	1	yes	characteristic
130	C₂²	#11718829	1	4	yes	—
131	C₄	—	16	2	no	—
132	C₄	—	208	2	no	—
133	C₄	#11718519	1	4	yes	—
134	C₄	#11718521	1	4	yes	—
135	C₄	C₂×#88048	1	2	yes	—
136	C₄	#11718740	1	2	yes	—
137	C₂	#13114435	1	2	yes	—
138	C₂	#13114438	1	2	yes	—
139	C₂	#13114992	1	1	yes	characteristic
140	C₂	#13114997	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13114993

C₂

yes

maximal, characteristic

#13114963

C₂

yes

maximal

C₂×C₈×#1055

C₂

yes

maximal, characteristic

C₁₃×#1844

C₂

yes

maximal, characteristic

C₂×C₁₃×#106

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₁₃×#384

C₂²

yes

—

C₂×#88057

—

#11718596

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

—

#11718121

—

C₈×#1055

C₄

yes

complemented

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂²×#1055

D₄

yes

—

C₁₃×#106

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

#88057

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

Q₈

yes

—

#8815

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

C₄⋊C₄

yes

—

#1844

—

#1844

D₁₃

yes

characteristic

#1834

—

C₂×C₈²

—

C₂³×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

#106

yes

—

#1055

—

C₂×#106

—

C₂×#106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#384

—

#384

C₂×D₁₃

yes

—

#340

—

C₈²

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×#106

yes

—

C₄×C₁₃

#427

yes

—

C₄×C₁₃

#498

yes

—

C₄×C₁₃

#499

yes

—

Dic₁₃

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₁₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

#106

—

#106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₂₆

yes

—

C₄×C₈

C₄×D₁₃

yes

—

C₂×C₁₃

#1708

yes

—

C₂×C₁₃

#1708

yes

—

C₂×C₁₃

#1844

yes

characteristic

C₂×C₁₃

#1846

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×D₁₃

yes

—

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

#6101

yes

—

103

C₂×C₈

#6105

yes

—

104

C₂×C₈

#6106

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

C₂×Dic₂₆

yes

characteristic

110

C₄²

C₂×C₄×D₁₃

yes

characteristic

111

C₄²

Q₈×D₁₃

yes

—

112

C₁₃

#8815

yes

characteristic, complemented, Sylow

113

C₂³

#88194

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

104

—

117

C₂×C₄

—

104

—

118

C₂×C₄

#88142

yes

—

119

C₂×C₄

C₄⋊C₄×D₁₃

yes

—

120

C₂×C₄

#88170

yes

—

121

C₂×C₄

C₂×#6101

yes

characteristic

122

C₂×C₄

C₂×#6105

yes

—

123

C₂×C₄

C₂×#6106

yes

—

124

C₂×C₄

#88198

yes

characteristic

125

C₈

—

126

C₈

#88048

yes

—

127

C₂²

#11718637

yes

characteristic

128

C₂²

#11718737

yes

characteristic

129

C₂²

#11718746

yes

characteristic

130

C₂²

#11718829

yes

—

131

C₄

—

132

C₄

—

208

—

133

C₄

#11718519

yes

—

134

C₄

#11718521

yes

—

135

C₄

C₂×#88048

yes

—

136

C₄

#11718740

yes

—

137

C₂

#13114435

yes

—

138

C₂

#13114438

yes

—

139

C₂

#13114992

yes

characteristic

140

C₂

#13114997

yes

—

	subgroup	G/H
center	C₂³	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1844	C₂
socle, prime radical	C₂³×C₁₃	#127
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1844	D₁₃
hypocenter	C₁₃	#8815
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#88194

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1844

C₂

socle, prime radical

C₂³×C₁₃

#127

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
12. C₈×#1055	C₄	semidirect
112. C₁₃	#8815	semidirect

H ≅ G/N

type

12. C₈×#1055

C₄

semidirect

112. C₁₃

#8815

semidirect

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

subgroup	index
#13114993	2
#13114963	2
C₂×C₈×#1055	2
C₁₃×#1844	2
#8815	13

subgroup

index

#13114993

#13114963

C₂×C₈×#1055

C₁₃×#1844

#8815

Subgroups — #15756691

Conjugacy classes of subgroups

Complements

Maximal subgroups