Group #13115681 — subgroups

Subgroups — #13115681

Order 1664.

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

normal subgroups only

This group has 292 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#88076	C₂	1	4	yes	maximal
3	#11718585	C₂	1	1	yes	maximal, characteristic
4	C₄×#6099	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
6	C₁₃×C₄⋊Q₈	—	2	1	no	—
7	C₁₃×C₄⋊Q₈	—	2	1	no	—
8	C₁₃×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₁₃×C₄⋊C₄	C₄	1	4	yes	—
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#6099	C₂²	1	2	yes	—
12	C₂×#6099	C₂²	1	2	yes	—
13	#88076	—	2	8	no	—
14	C₄⋊#1055	—	2	2	no	—
15	#88043	—	2	1	no	—
16	C₄×#1055	—	2	1	no	—
17	C₂×Q₈×C₁₃	—	2	4	no	—
18	C₂×Q₈×C₁₃	—	2	2	no	—
19	C₂×Q₈×C₁₃	—	2	2	no	—
20	C₂×Q₈×C₁₃	—	2	4	no	—
21	C₂²×C₄×C₁₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₁₃×C₄⋊C₄	—	2	4	no	—
25	C₁₃×C₄⋊C₄	—	2	4	no	—
26	C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₄²×C₁₃	D₄	1	1	yes	characteristic
28	C₄²×C₁₃	D₄	1	2	yes	—
29	#6099	—	2	2	no	—
30	#6099	—	2	4	no	—
31	#6099	D₄	1	4	yes	—
32	C₂×#1055	—	2	2	no	—
33	C₂×#1055	—	2	2	no	—
34	#1966	—	13	1	no	maximal, Sylow
35	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	Q₈×C₁₃	—	4	4	no	—
37	Q₈×C₁₃	—	4	2	no	—
38	Q₈×C₁₃	—	4	2	no	—
39	C₂×C₄×C₁₃	—	2	4	no	—
40	C₂×C₄×C₁₃	—	2	4	no	—
41	C₂×C₄×C₁₃	—	2	8	no	—
42	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃	#41	1	2	yes	—
48	#1055	—	2	4	no	—
49	#1055	—	4	2	no	—
50	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
51	#423	—	13	1	no	—
52	C₂×#101	—	13	4	no	—
53	C₄×M₁₆	—	13	1	no	—
54	C₂²×C₁₃	—	2	2	no	—
55	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₃	C₄×D₄	1	2	yes	—
57	C₄×C₁₃	—	2	2	no	—
58	C₄×C₁₃	—	4	4	no	—
59	C₄×C₁₃	#101	1	4	yes	complemented
60	C₄×C₁₃	#121	1	2	yes	—
61	C₄×C₁₃	#124	1	1	yes	characteristic
62	C₄×C₁₃	#127	1	1	yes	characteristic
63	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
64	C₂×M₁₆	—	13	2	no	—
65	C₂×M₁₆	—	13	2	no	—
66	C₄⋊Q₈	—	2	1	no	—
67	C₄⋊Q₈	—	2	1	no	—
68	C₄⋊Q₈	—	2	2	no	—
69	C₂×C₄⋊C₄	Dic₁₃	1	4	yes	—
70	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
71	C₄⋊C₈	—	26	2	no	—
72	#101	—	26	8	no	—
73	#97	—	26	1	no	—
74	C₄×C₈	—	26	1	no	—
75	C₂×C₁₃	—	2	2	no	—
76	C₂×C₁₃	#390	1	1	yes	characteristic
77	C₂×C₁₃	C₂×#101	1	2	yes	—
78	C₂×Q₈	—	2	4	no	—
79	C₂×Q₈	—	2	4	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂²×C₄	C₂×Dic₁₃	1	4	yes	—
83	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
84	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
85	M₁₆	—	13	4	no	—
86	M₁₆	—	26	2	no	—
87	M₁₆	—	26	4	no	—
88	C₂×C₈	—	26	2	no	—
89	C₂×C₈	—	26	2	no	—
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
93	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
94	C₄²	C₂²⋊D₁₃	1	2	yes	—
95	C₁₃	#1966	1	1	yes	characteristic, complemented, Sylow
96	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
97	Q₈	—	4	2	no	—
98	Q₈	—	4	4	no	—
99	Q₈	—	4	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	8	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	C₂²⋊Dic₁₃	1	2	yes	—
104	C₂×C₄	C₂²⋊Dic₁₃	1	2	yes	—
105	C₂×C₄	D₄×D₁₃	1	2	yes	—
106	C₂×C₄	#6129	1	2	yes	—
107	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
109	C₈	—	26	4	no	—
110	C₈	—	52	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	D₄×Dic₁₃	1	2	yes	—
113	C₂²	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	#88076	1	4	yes	complemented
117	C₄	#88190	1	1	yes	characteristic
118	C₄	#88192	1	2	yes	—
119	C₄	#88194	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#11718882	1	1	yes	characteristic
122	C₂	C₂×#88076	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#88076

C₂

yes

maximal

#11718585

C₂

yes

maximal, characteristic

C₄×#6099

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃

C₂²

yes

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊C₄

C₄

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×#6099

C₂²

yes

—

C₂×#6099

C₂²

yes

—

—

—

—

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

characteristic

C₄²×C₁₃

D₄

yes

characteristic

C₄²×C₁₃

D₄

yes

—

#6099

—

#6099

—

#6099

D₄

yes

—

—

—

—

C₂³×C₁₃

C₂²×C₄

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×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×C₄⋊Q₈

D₁₃

yes

characteristic

#423

—

C₂×#101

—

C₄×M₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#101

yes

complemented

C₄×C₁₃

#121

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂²×Q₈

C₂×D₁₃

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

Dic₁₃

yes

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₄⋊C₈

—

#101

—

#97

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

#390

yes

characteristic

C₂×C₁₃

C₂×#101

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₁₃

yes

—

C₂²×C₄

C₂×Dic₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₁₃

yes

characteristic

C₄²

C₂²⋊D₁₃

yes

characteristic

C₄²

C₂²⋊D₁₃

yes

—

C₁₃

#1966

yes

characteristic, complemented, Sylow

C₂³

C₂²×Dic₁₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²⋊Dic₁₃

yes

—

104

C₂×C₄

C₂²⋊Dic₁₃

yes

—

105

C₂×C₄

D₄×D₁₃

yes

—

106

C₂×C₄

#6129

yes

—

107

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

109

C₈

—

110

C₈

—

111

C₂²

—

112

C₂²

D₄×Dic₁₃

yes

—

113

C₂²

C₂×C₂²⋊Dic₁₃

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

#88076

yes

complemented

117

C₄

#88190

yes

characteristic

118

C₄

#88192

yes

—

119

C₄

#88194

yes

characteristic

120

C₂

—

121

C₂

#11718882

yes

characteristic

122

C₂

C₂×#88076

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₃
commutator, prime radical	C₂³×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#1966
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₃

commutator, prime radical

C₂³×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₂×C₁₃×C₄⋊Q₈

C₂

socle

C₂²×C₁₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₃

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₃

hypocenter

C₁₃

#1966

LCS

G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
59. C₄×C₁₃	#101	semidirect
95. C₁₃	#1966	semidirect
116. C₄	#88076	semidirect

H ≅ G/N

type

59. C₄×C₁₃

#101

semidirect

95. C₁₃

#1966

semidirect

116. C₄

#88076

semidirect

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

subgroup	index
C₂×C₁₃×C₄⋊Q₈	2
C₂×#88076	2
#11718585	2
C₄×#6099	2
#1966	13

subgroup

index

C₂×C₁₃×C₄⋊Q₈

Subgroups — #13115681

Conjugacy classes of subgroups

Complements

Maximal subgroups