Group #13115677 — subgroups

Subgroups — #13115677

Order 1664.

396 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×C₁₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×#88073	C₂	1	4	yes	maximal, complemented
3	#11718585	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#6099	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄×C₁₃	C₄	1	4	yes	—
6	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
7	C₁₃×#127	—	2	1	no	—
8	C₁₃×#127	—	2	1	no	—
9	C₁₃×#127	—	2	2	no	—
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#6099	C₂²	1	2	yes	—
12	C₂×#6099	C₂²	1	2	yes	—
13	#88073	—	2	8	no	—
14	C₄⋊#1055	—	2	2	no	—
15	#88043	—	2	1	no	—
16	C₄×#1055	—	2	1	no	—
17	C₂⁴×C₁₃	C₂×C₄	1	4	yes	—
18	C₂×D₄×C₁₃	—	2	4	no	—
19	C₂×D₄×C₁₃	—	2	4	no	—
20	C₂×D₄×C₁₃	—	2	4	no	—
21	C₂×D₄×C₁₃	—	2	2	no	—
22	C₂×D₄×C₁₃	—	2	2	no	—
23	C₂×D₄×C₁₃	—	2	4	no	—
24	C₂×D₄×C₁₃	—	4	8	no	—
25	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₄²×C₁₃	D₄	1	1	yes	characteristic
28	C₄²×C₁₃	D₄	1	1	yes	characteristic
29	C₄²×C₁₃	D₄	1	2	yes	—
30	#6099	—	2	2	no	—
31	#6099	—	2	4	no	—
32	#6099	D₄	1	4	yes	complemented
33	C₂×#1055	—	2	2	no	—
34	C₂×#1055	—	2	2	no	—
35	#1965	—	13	1	no	maximal, Sylow
36	C₂³×C₁₃	—	2	4	no	—
37	C₂³×C₁₃	—	2	8	no	—
38	C₂³×C₁₃	—	4	8	no	—
39	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	D₄×C₁₃	—	4	8	no	—
41	D₄×C₁₃	—	4	8	no	—
42	D₄×C₁₃	—	4	4	no	—
43	D₄×C₁₃	—	4	2	no	—
44	D₄×C₁₃	—	4	2	no	—
45	C₂×C₄×C₁₃	—	2	4	no	—
46	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃	#41	1	2	yes	—
52	#1055	—	2	4	no	—
53	#1055	—	4	2	no	—
54	C₂×#127	D₁₃	1	1	yes	characteristic
55	#423	—	13	1	no	—
56	C₂×#100	—	13	4	no	—
57	C₄×M₁₆	—	13	1	no	—
58	C₂²×C₁₃	—	2	2	no	—
59	C₂²×C₁₃	—	4	4	no	—
60	C₂²×C₁₃	—	4	8	no	—
61	C₂²×C₁₃	—	8	8	no	—
62	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₁₃	C₄×D₄	1	2	yes	—
64	C₄×C₁₃	—	2	2	no	—
65	C₄×C₁₃	#100	1	4	yes	complemented
66	C₄×C₁₃	#121	1	2	yes	—
67	C₄×C₁₃	#124	1	1	yes	characteristic
68	C₄×C₁₃	#127	1	1	yes	characteristic
69	C₂²×D₄	Dic₁₃	1	4	yes	—
70	C₂²×D₄	C₂×D₁₃	1	2	yes	—
71	C₂×M₁₆	—	13	2	no	—
72	C₂×M₁₆	—	13	2	no	—
73	#127	—	2	1	no	—
74	#127	—	2	1	no	—
75	#127	—	2	2	no	—
76	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
77	C₄⋊C₈	—	26	2	no	—
78	#100	—	26	8	no	—
79	#97	—	26	1	no	—
80	C₄×C₈	—	26	1	no	—
81	C₂×C₁₃	—	2	2	no	—
82	C₂×C₁₃	—	8	4	no	—
83	C₂×C₁₃	#390	1	1	yes	characteristic
84	C₂×C₁₃	C₂×#100	1	2	yes	—
85	C₂⁴	C₂×Dic₁₃	1	4	yes	—
86	C₂×D₄	—	2	4	no	—
87	C₂×D₄	—	2	4	no	—
88	C₂×D₄	—	2	4	no	—
89	C₂×D₄	—	2	4	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	4	8	no	—
93	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
94	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
95	M₁₆	—	13	4	no	—
96	M₁₆	—	26	2	no	—
97	M₁₆	—	26	4	no	—
98	C₂×C₈	—	26	2	no	—
99	C₂×C₈	—	26	2	no	—
100	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
101	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
102	C₄²	C₂²⋊D₁₃	1	2	yes	—
103	C₁₃	#1965	1	1	yes	characteristic, complemented, Sylow
104	C₂³	—	2	4	no	—
105	C₂³	—	2	8	no	—
106	C₂³	—	4	8	no	—
107	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
108	D₄	—	4	8	no	—
109	D₄	—	4	8	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	4	no	—
112	D₄	—	4	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	C₂²⋊Dic₁₃	1	2	yes	—
115	C₂×C₄	C₂²⋊Dic₁₃	1	2	yes	—
116	C₂×C₄	D₄×D₁₃	1	2	yes	—
117	C₂×C₄	#6129	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
120	C₈	—	26	4	no	—
121	C₈	—	52	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	4	8	no	—
124	C₂²	—	4	4	no	—
125	C₂²	—	8	8	no	—
126	C₂²	D₄×Dic₁₃	1	2	yes	—
127	C₂²	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	#88073	1	4	yes	complemented
130	C₄	#88190	1	1	yes	characteristic
131	C₄	#88192	1	2	yes	—
132	C₄	#88194	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	—	8	4	no	—
135	C₂	C₂×#88073	1	2	yes	—
136	C₂	#11718882	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#127

C₂

yes

maximal, characteristic

C₂×#88073

C₂

yes

maximal, complemented

#11718585

C₂

yes

maximal, characteristic, complemented

C₄×#6099

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₃

C₄

yes

—

C₂²×D₄×C₁₃

C₂²

yes

—

C₁₃×#127

—

C₁₃×#127

—

C₁₃×#127

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6099

C₂²

yes

—

C₂×#6099

C₂²

yes

—

—

—

—

—

C₂⁴×C₁₃

C₂×C₄

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₄²×C₁₃

D₄

yes

characteristic

C₄²×C₁₃

D₄

yes

characteristic

C₄²×C₁₃

D₄

yes

—

#6099

—

#6099

—

#6099

D₄

yes

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂²×C₄

yes

characteristic

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

#1055

—

#1055

—

C₂×#127

D₁₃

yes

characteristic

#423

—

C₂×#100

—

C₄×M₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

—

C₄×C₁₃

#100

yes

complemented

C₄×C₁₃

#121

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂²×D₄

Dic₁₃

yes

—

C₂²×D₄

C₂×D₁₃

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

#127

—

#127

—

#127

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

#100

—

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

#390

yes

characteristic

C₂×C₁₃

C₂×#100

yes

—

C₂⁴

C₂×Dic₁₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂×Dic₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₄²

C₂²⋊D₁₃

yes

characteristic

101

C₄²

C₂²⋊D₁₃

yes

characteristic

102

C₄²

C₂²⋊D₁₃

yes

—

103

C₁₃

#1965

yes

characteristic, complemented, Sylow

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

C₂²×Dic₁₃

yes

characteristic

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²⋊Dic₁₃

yes

—

115

C₂×C₄

C₂²⋊Dic₁₃

yes

—

116

C₂×C₄

D₄×D₁₃

yes

—

117

C₂×C₄

#6129

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

120

C₈

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

D₄×Dic₁₃

yes

—

127

C₂²

C₂×C₂²⋊Dic₁₃

yes

characteristic

128

C₄

—

129

C₄

#88073

yes

complemented

130

C₄

#88190

yes

characteristic

131

C₄

#88192

yes

—

132

C₄

#88194

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

C₂×#88073

yes

—

136

C₂

#11718882

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₃
commutator	C₂³×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting, prime radical	C₂×C₁₃×#127	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₂×#127	D₁₃
hypocenter	C₁₃	#1965
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₃

commutator

C₂³×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting, prime radical

C₂×C₁₃×#127

C₂

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
2. C₂×#88073	C₂	semidirect
3. #11718585	C₂	semidirect
4. C₄×#6099	C₂	semidirect
32. #6099	D₄	semidirect
65. C₄×C₁₃	#100	semidirect
103. C₁₃	#1965	semidirect
129. C₄	#88073	semidirect

H ≅ G/N

type

2. C₂×#88073

C₂

semidirect

3. #11718585

C₂

semidirect

4. C₄×#6099

C₂

semidirect

32. #6099

D₄

semidirect

65. C₄×C₁₃

#100

semidirect

103. C₁₃

#1965

semidirect

129. C₄

#88073

semidirect

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

subgroup	index
C₂×C₁₃×#127	2
C₂×#88073	2
#11718585	2
C₄×#6099	2
#1965	13

subgroup

index

C₂×C₁₃×#127

Subgroups — #13115677

Conjugacy classes of subgroups

Complements

Maximal subgroups