Group #13128238 — subgroups

Subgroups — #13128238

Order 1664.

996 conjugacy classes of subgroups; only the 509 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 509 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×#543	C₂	1	1	yes	maximal, characteristic
2	#11719328	C₂	1	12	yes	maximal, complemented
3	#11719327	C₂	1	12	yes	maximal, complemented
4	#11719188	C₂	1	3	yes	maximal
5	C₂×#88131	C₂	1	3	yes	maximal
6	C₂³×Dic₁₃	C₂²	1	1	yes	characteristic
7	C₁₃×C₂²⋊Q₈	C₂²	1	12	yes	—
8	C₂×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
9	C₂×C₂²⋊Dic₁₃	C₂²	1	3	yes	—
10	Q₈×Dic₁₃	C₂²	1	4	yes	complemented
11	#88198	C₂²	1	12	yes	complemented
12	#88178	C₂²	1	12	yes	—
13	C₂×#6101	C₂²	1	6	yes	—
14	C₂×C₄×Dic₁₃	C₂²	1	6	yes	—
15	#88143	C₂²	1	24	yes	complemented
16	#88142	C₂²	1	12	yes	complemented
17	#88141	C₂²	1	12	yes	complemented
18	#88132	C₂²	1	24	yes	—
19	#88131	C₂²	1	12	yes	—
20	#88131	C₂²	1	12	yes	—
21	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
23	C₂²×C₄×C₁₃	C₂³	1	6	yes	—
24	C₂²×Dic₁₃	Q₈	1	8	yes	complemented
25	C₂²×Dic₁₃	C₂³	1	6	yes	—
26	C₂²×Dic₁₃	C₂³	1	3	yes	—
27	C₂²×Dic₁₃	C₂³	1	3	yes	—
28	C₂×Dic₂₆	C₂³	1	12	yes	—
29	C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
30	C₁₃×C₂²⋊C₄	C₂³	1	12	yes	—
31	C₂²⋊Dic₁₃	C₂³	1	12	yes	—
32	C₄⋊Dic₁₃	C₂³	1	12	yes	—
33	#6101	C₂³	1	24	yes	—
34	#6101	C₂³	1	24	yes	—
35	C₄×Dic₁₃	C₂³	1	12	yes	—
36	C₄×Dic₁₃	C₂³	1	12	yes	—
37	#3458	—	13	1	no	maximal, Sylow
38	C₂³×C₁₃	—	4	2	no	—
39	C₂³×C₁₃	C₂×Q₈	1	4	yes	—
40	C₂³×C₁₃	C₂⁴	1	3	yes	—
41	Q₈×C₁₃	—	4	4	no	—
42	C₂×C₄×C₁₃	—	4	6	no	—
43	C₂×C₄×C₁₃	C₂⁴	1	12	yes	—
44	C₂×Dic₁₃	—	4	6	no	—
45	C₂×Dic₁₃	C₂×Q₈	1	12	yes	—
46	C₂×Dic₁₃	C₂×Q₈	1	12	yes	—
47	C₂×Dic₁₃	C₂⁴	1	12	yes	—
48	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₁₃	C₂⁴	1	3	yes	—
50	Dic₂₆	—	4	12	no	—
51	#544	—	13	12	no	—
52	#543	—	13	3	no	—
53	#543	D₁₃	1	1	yes	characteristic
54	#533	—	13	12	no	—
55	C₂×#117	—	13	3	no	—
56	C₂²×C₁₃	—	2	6	no	—
57	C₂²×C₁₃	—	4	4	no	—
58	C₂²×C₁₃	C₂²×Q₈	1	6	yes	—
59	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
60	C₄×C₁₃	—	2	12	no	—
61	Dic₁₃	—	2	12	no	—
62	Dic₁₃	C₂²×Q₈	1	2	yes	—
63	Dic₁₃	C₂²×Q₈	1	6	yes	—
64	C₂³×C₄	—	13	1	no	—
65	C₄⋊Q₈	—	13	12	no	—
66	C₄⋊Q₈	—	13	12	no	—
67	#125	—	13	24	no	—
68	C₂²⋊Q₈	—	13	12	no	—
69	C₂²⋊Q₈	—	13	24	no	—
70	C₂²⋊Q₈	C₂×D₁₃	1	12	yes	—
71	C₄×Q₈	—	13	4	no	—
72	C₄×Q₈	—	13	12	no	—
73	#117	—	13	12	no	—
74	#117	—	13	12	no	—
75	C₂×C₄⋊C₄	—	13	6	no	—
76	C₂×C₂²⋊C₄	—	13	3	no	—
77	C₂×C₂²⋊C₄	C₂×D₁₃	1	3	yes	—
78	C₂×C₄²	—	13	6	no	—
79	C₂×C₁₃	—	2	6	no	—
80	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
81	C₂×C₁₃	#585	1	2	yes	—
82	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
83	C₂×Q₈	—	13	12	no	—
84	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
85	C₂²×C₄	—	13	8	no	—
86	C₂²×C₄	—	13	3	no	—
87	C₂²×C₄	—	13	3	no	—
88	C₂²×C₄	—	13	6	no	—
89	C₂²×C₄	C₂²×D₁₃	1	6	yes	—
90	C₄⋊C₄	—	13	24	no	—
91	C₄⋊C₄	—	13	24	no	—
92	C₄⋊C₄	—	13	12	no	—
93	C₄⋊C₄	C₂²×D₁₃	1	12	yes	—
94	C₂²⋊C₄	—	13	12	no	—
95	C₂²⋊C₄	C₂²×D₁₃	1	12	yes	—
96	C₄²	—	13	12	no	—
97	C₄²	—	13	12	no	—
98	C₁₃	#3458	1	1	yes	characteristic, complemented, Sylow
99	C₂³	—	4	2	no	—
100	C₂³	Q₈×D₁₃	1	4	yes	—
101	C₂³	C₂³×D₁₃	1	3	yes	—
102	Q₈	—	4	4	no	—
103	Q₈	—	52	12	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	13	12	no	—
106	C₂×C₄	—	13	12	no	—
107	C₂×C₄	—	13	1	no	—
108	C₂×C₄	—	13	3	no	—
109	C₂×C₄	—	13	12	no	—
110	C₂×C₄	—	52	6	no	—
111	C₂×C₄	C₂³×D₁₃	1	12	yes	—
112	C₂²	—	2	6	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×Q₈×D₁₃	1	6	yes	—
115	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
116	C₄	—	2	12	no	—
117	C₄	—	13	2	no	—
118	C₄	—	13	6	no	—
119	C₄	—	26	12	no	—
120	C₂	—	2	6	no	—
121	C₂	C₂²×Q₈×D₁₃	1	1	yes	characteristic
122	C₂	#11719650	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×#543

C₂

yes

maximal, characteristic

#11719328

C₂

yes

maximal, complemented

#11719327

C₂

yes

maximal, complemented

#11719188

C₂

yes

maximal

C₂×#88131

C₂

yes

maximal

C₂³×Dic₁₃

C₂²

yes

characteristic

C₁₃×C₂²⋊Q₈

C₂²

yes

—

C₂×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

Q₈×Dic₁₃

C₂²

yes

complemented

#88198

C₂²

yes

complemented

#88178

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88143

C₂²

yes

complemented

#88142

C₂²

yes

complemented

#88141

C₂²

yes

complemented

#88132

C₂²

yes

—

#88131

C₂²

yes

—

#88131

C₂²

yes

—

C₂⁴×C₁₃

C₂³

yes

characteristic

C₂×Q₈×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

Q₈

yes

complemented

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×Dic₂₆

C₂³

yes

—

C₁₃×C₄⋊C₄

C₂³

yes

—

C₁₃×C₂²⋊C₄

C₂³

yes

—

C₂²⋊Dic₁₃

C₂³

yes

—

C₄⋊Dic₁₃

C₂³

yes

—

#6101

C₂³

yes

—

#6101

C₂³

yes

—

C₄×Dic₁₃

C₂³

yes

—

C₄×Dic₁₃

C₂³

yes

—

#3458

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

C₂×Q₈

yes

—

C₂³×C₁₃

C₂⁴

yes

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂⁴

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

C₂⁴

yes

—

C₂×Dic₁₃

C₂⁴

yes

characteristic

C₂×Dic₁₃

C₂⁴

yes

—

Dic₂₆

—

#544

—

#543

—

#543

D₁₃

yes

characteristic

#533

—

C₂×#117

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×Q₈

yes

—

C₂²×C₁₃

C₂⁵

yes

characteristic

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

C₂²×Q₈

yes

—

Dic₁₃

C₂²×Q₈

yes

—

C₂³×C₄

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂×D₁₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₁₃

—

C₂×C₁₃

C₂³×Q₈

yes

characteristic

C₂×C₁₃

#585

yes

—

C₂⁴

C₂²×D₁₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²×D₁₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₁₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂²×D₁₃

yes

—

C₄²

—

C₄²

—

C₁₃

#3458

yes

characteristic, complemented, Sylow

C₂³

—

100

C₂³

Q₈×D₁₃

yes

—

101

C₂³

C₂³×D₁₃

yes

—

102

Q₈

—

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂³×D₁₃

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×Q₈×D₁₃

yes

—

115

C₂²

C₂⁴×D₁₃

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₂

—

121

C₂

C₂²×Q₈×D₁₃

yes

characteristic

122

C₂

#11719650

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₁₃×#543	C₂
prime radical	C₂⁴×C₁₃	C₂³
hypercenter	#543	D₁₃
hypocenter	C₁₃	#3458
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #543

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₃

commutator, socle, cosocle

C₂²×C₁₃

C₂⁵

Fitting

C₁₃×#543

C₂

prime radical

C₂⁴×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂² ◁ #543

N	H ≅ G/N	type
2. #11719328	C₂	semidirect
3. #11719327	C₂	semidirect
10. Q₈×Dic₁₃	C₂²	semidirect
11. #88198	C₂²	semidirect
15. #88143	C₂²	semidirect
16. #88142	C₂²	semidirect
17. #88141	C₂²	semidirect
24. C₂²×Dic₁₃	Q₈	semidirect
98. C₁₃	#3458	semidirect

H ≅ G/N

type

2. #11719328

C₂

semidirect

3. #11719327

C₂

semidirect

10. Q₈×Dic₁₃

C₂²

semidirect

11. #88198

C₂²

semidirect

15. #88143

C₂²

semidirect

16. #88142

C₂²

semidirect

17. #88141

C₂²

semidirect

24. C₂²×Dic₁₃

Q₈

semidirect

98. C₁₃

#3458

semidirect

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

subgroup	index
C₁₃×#543	2
#11719328	2
#11719327	2
#11719188	2
C₂×#88131	2
#3458	13

subgroup

index

C₁₃×#543

Subgroups — #13128238

Conjugacy classes of subgroups

Complements

Maximal subgroups