Group #13122215 — subgroups

Subgroups — #13122215

Order 1664.

520 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₁₃×#392	C₂	1	1	yes	maximal, characteristic
2	#11718881	C₂	1	6	yes	maximal, complemented
3	#11718608	C₂	1	3	yes	maximal, complemented
4	#11718604	C₂	1	3	yes	maximal, complemented
5	C₁₃⋊#401	C₂	1	2	yes	maximal, complemented
6	C₂²×D₄×C₁₃	C₂²	1	3	yes	—
7	C₂³×Dic₁₃	C₂²	1	3	yes	—
8	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
10	C₂×C₂²⋊Dic₁₃	C₂²	1	3	yes	—
11	C₂×C₂²⋊Dic₁₃	C₂²	1	6	yes	—
12	C₁₃×#95	C₂²	1	1	yes	characteristic
13	#88071	C₂²	1	6	yes	—
14	#88071	C₂²	1	6	yes	—
15	#88071	C₂²	1	3	yes	—
16	C₂⁴×C₁₃	C₂³	1	3	yes	—
17	C₂×D₄×C₁₃	—	2	12	no	—
18	C₂×D₄×C₁₃	—	4	3	no	—
19	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
20	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₁₃	—	2	6	no	—
22	C₂²×Dic₁₃	—	4	3	no	—
23	C₂²×Dic₁₃	—	4	3	no	—
24	C₂²×Dic₁₃	C₂³	1	6	yes	—
25	C₂²×Dic₁₃	C₂³	1	2	yes	—
26	C₂×Dic₂₆	—	4	3	no	—
27	C₁₃×C₂²⋊C₄	—	4	3	no	—
28	C₂²⋊Dic₁₃	—	4	6	no	—
29	C₂²⋊Dic₁₃	—	4	3	no	—
30	#2760	—	13	1	no	maximal, Sylow
31	C₂³×C₁₃	—	2	6	no	—
32	C₂³×C₁₃	—	2	3	no	—
33	C₂³×C₁₃	—	2	6	no	—
34	C₂³×C₁₃	—	2	3	no	—
35	C₂³×C₁₃	#41	1	6	yes	—
36	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
37	D₄×C₁₃	—	4	12	no	—
38	C₂×C₄×C₁₃	—	2	6	no	—
39	C₂×C₄×C₁₃	—	2	3	no	—
40	C₂×C₄×C₁₃	—	2	3	no	—
41	C₂×Dic₁₃	—	2	6	no	—
42	C₂×Dic₁₃	—	2	6	no	—
43	C₂×Dic₁₃	—	2	6	no	—
44	C₂×Dic₁₃	—	2	6	no	—
45	C₂×Dic₁₃	—	4	12	no	—
46	Dic₂₆	—	4	4	no	—
47	#401	—	13	2	no	—
48	#393	—	13	3	no	—
49	#392	D₁₃	1	1	yes	characteristic
50	#386	—	13	6	no	—
51	#381	—	13	3	no	—
52	C₂²×C₁₃	—	2	6	no	—
53	C₂²×C₁₃	—	2	12	no	—
54	C₂²×C₁₃	—	4	6	no	—
55	C₂²×C₁₃	—	4	3	no	—
56	C₂²×C₁₃	—	4	3	no	—
57	C₂²×C₁₃	C₂×#41	1	3	yes	—
58	C₂²×C₁₃	H(5,𝔽₂)	1	3	yes	—
59	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
60	C₄×C₁₃	—	4	3	no	—
61	C₄×C₁₃	—	4	1	no	—
62	Dic₁₃	—	4	6	no	—
63	Dic₁₃	—	4	2	no	—
64	C₂²×Q₈	—	13	1	no	—
65	C₂²×D₄	C₂×D₁₃	1	3	yes	—
66	C₂³×C₄	—	13	3	no	—
67	C₂×C₂²⋊C₄	—	13	3	no	—
68	C₂×C₂²⋊C₄	—	13	6	no	—
69	C₂×C₂²⋊C₄	C₂×D₁₃	1	3	yes	—
70	#95	—	13	3	no	—
71	#95	—	13	6	no	—
72	#95	—	13	6	no	—
73	#95	C₂×D₁₃	1	1	yes	characteristic
74	C₂×C₁₃	—	4	6	no	—
75	C₂×C₁₃	#538	1	3	yes	—
76	C₂×C₁₃	#551	1	3	yes	—
77	C₂×C₁₃	#560	1	1	yes	characteristic
78	C₂⁴	C₂²×D₁₃	1	3	yes	—
79	C₂×Q₈	—	52	3	no	—
80	C₂×D₄	—	2	12	no	—
81	C₂×D₄	—	4	3	no	—
82	C₂²×C₄	—	13	6	no	—
83	C₂²×C₄	—	13	2	no	—
84	C₂²×C₄	—	26	6	no	—
85	C₂²×C₄	—	52	3	no	—
86	C₂²×C₄	—	52	3	no	—
87	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
88	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
89	C₂²⋊C₄	—	4	3	no	—
90	C₂²⋊C₄	—	52	6	no	—
91	C₂²⋊C₄	—	52	3	no	—
92	C₁₃	#2760	1	1	yes	characteristic, complemented, Sylow
93	C₂³	—	2	6	no	—
94	C₂³	—	2	3	no	—
95	C₂³	—	2	6	no	—
96	C₂³	—	2	3	no	—
97	C₂³	#6129	1	6	yes	—
98	C₂³	C₂³×D₁₃	1	1	yes	characteristic
99	Q₈	—	52	4	no	—
100	D₄	—	4	12	no	—
101	C₂×C₄	—	2	6	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	26	6	no	—
105	C₂×C₄	—	26	6	no	—
106	C₂×C₄	—	26	6	no	—
107	C₂×C₄	—	26	6	no	—
108	C₂×C₄	—	52	12	no	—
109	C₂²	—	2	12	no	—
110	C₂²	—	2	6	no	—
111	C₂²	—	4	3	no	—
112	C₂²	—	4	3	no	—
113	C₂²	—	4	6	no	—
114	C₂²	C₂×#6129	1	3	yes	—
115	C₂²	#88251	1	3	yes	—
116	C₂²	#88256	1	1	yes	characteristic
117	C₄	—	4	1	no	—
118	C₄	—	4	3	no	—
119	C₄	—	52	2	no	—
120	C₄	—	52	6	no	—
121	C₂	—	4	6	no	—
122	C₂	#11719292	1	3	yes	—
123	C₂	#11719322	1	3	yes	—
124	C₂	#11719425	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#392

C₂

yes

maximal, characteristic

#11718881

C₂

yes

maximal, complemented

#11718608

C₂

yes

maximal, complemented

#11718604

C₂

yes

maximal, complemented

C₁₃⋊#401

C₂

yes

maximal, complemented

C₂²×D₄×C₁₃

C₂²

yes

—

C₂³×Dic₁₃

C₂²

yes

—

C₂²×Dic₂₆

C₂²

yes

characteristic

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

C₁₃×#95

C₂²

yes

characteristic

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

—

C₁₃×C₂²⋊C₄

—

C₂²⋊Dic₁₃

—

C₂²⋊Dic₁₃

—

#2760

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

#41

yes

—

C₂³×C₁₃

C₂⁴

yes

characteristic

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

#401

—

#393

—

#392

D₁₃

yes

characteristic

#386

—

#381

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×#41

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×D₄

C₂×D₁₃

yes

—

C₂³×C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

—

—

—

C₂×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

#538

yes

—

C₂×C₁₃

#551

yes

—

C₂×C₁₃

#560

yes

characteristic

C₂⁴

C₂²×D₁₃

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₁₃

#2760

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#6129

yes

—

C₂³

C₂³×D₁₃

yes

characteristic

Q₈

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×#6129

yes

—

115

C₂²

#88251

yes

—

116

C₂²

#88256

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₂

—

122

C₂

#11719292

yes

—

123

C₂

#11719322

yes

—

124

C₂

#11719425

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting, prime radical	C₁₃×#392	C₂
hypercenter	#392	D₁₃
hypocenter	C₁₃	#2760
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #392

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting, prime radical

C₁₃×#392

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #392

N	H ≅ G/N	type
2. #11718881	C₂	semidirect
3. #11718608	C₂	semidirect
4. #11718604	C₂	semidirect
5. C₁₃⋊#401	C₂	semidirect
92. C₁₃	#2760	semidirect

H ≅ G/N

type

2. #11718881

C₂

semidirect

3. #11718608

C₂

semidirect

4. #11718604

C₂

semidirect

5. C₁₃⋊#401

C₂

semidirect

92. C₁₃

#2760

semidirect

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

subgroup	index
C₁₃×#392	2
#11718881	2
#11718608	2
#11718604	2
C₁₃⋊#401	2
#2760	13

subgroup

index

C₁₃×#392

C₁₃⋊#401

Subgroups — #13122215

Conjugacy classes of subgroups

Complements

Maximal subgroups