Group #13114464 — subgroups

Subgroups — #13114464

Order 1664.

220 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 69 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	#11718356	C₂	1	1	yes	maximal, characteristic
2	#11718355	C₂	1	2	yes	maximal, complemented
3	#11718354	C₂	1	2	yes	maximal, complemented
4	#11718341	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₀₄	—	2	2	no	—
7	C₂×#6095	—	2	4	no	—
8	#88128	C₂²	1	1	yes	characteristic
9	#88128	C₂²	1	1	yes	characteristic
10	#88123	C₂²	1	2	yes	—
11	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₁₃	C₂²	1	2	yes	—
13	#88061	—	2	4	no	—
14	#88057	—	2	2	no	—
15	C₂×D₅₂	—	2	2	no	—
16	C₂×D₅₂	—	2	1	no	—
17	C₂×D₅₂	—	2	1	no	—
18	C₂×D₅₂	—	2	2	no	—
19	C₂×Dic₂₆	—	2	2	no	—
20	C₂×C₈×C₁₃	D₄	1	4	yes	—
21	C₂×C₈×C₁₃	D₄	1	2	yes	—
22	C₄²×C₁₃	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₁₃	—	2	2	no	—
24	C₄⋊Dic₁₃	—	2	2	no	—
25	D₁₀₄	—	4	4	no	—
26	#6095	—	4	8	no	—
27	#1715	—	13	1	no	maximal, Sylow
28	C₂²×D₁₃	—	4	1	no	—
29	C₂²×D₁₃	—	4	1	no	—
30	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
32	C₂×Dic₁₃	—	4	2	no	—
33	D₅₂	—	4	2	no	—
34	D₅₂	—	4	2	no	—
35	D₅₂	—	4	4	no	—
36	D₅₂	—	4	4	no	—
37	Dic₂₆	—	4	4	no	—
38	C₈×C₁₃	—	2	4	no	—
39	C₈×C₁₃	SD₁₆	1	4	yes	complemented
40	#498	—	13	1	no	—
41	#493	—	13	1	no	—
42	#492	—	13	2	no	—
43	#486	—	13	2	no	—
44	#322	D₁₃	1	1	yes	characteristic, complemented
45	C₂²×C₁₃	#127	1	1	yes	characteristic
46	C₂×D₁₃	—	8	1	no	—
47	C₂×D₁₃	—	8	1	no	—
48	C₂×D₁₃	—	8	2	no	—
49	C₂×D₁₃	—	8	2	no	—
50	C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
51	C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
52	Dic₁₃	—	8	2	no	—
53	C₂×SD₁₆	—	26	4	no	—
54	C₂×D₈	—	26	2	no	—
55	C₄⋊Q₈	—	13	2	no	—
56	#127	—	13	1	no	—
57	#127	—	13	1	no	—
58	#106	—	26	2	no	—
59	#102	—	26	4	no	—
60	C₄×C₈	C₂×D₁₃	1	2	yes	—
61	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
62	C₂×C₁₃	#492	1	1	yes	characteristic
63	C₂×C₁₃	#496	1	2	yes	—
64	D₁₃	—	16	1	no	—
65	D₁₃	—	16	1	no	—
66	C₂×Q₈	—	26	2	no	—
67	C₂×D₄	—	26	2	no	—
68	C₂×D₄	—	26	1	no	—
69	C₂×D₄	—	26	1	no	—
70	C₂×D₄	—	26	2	no	—
71	SD₁₆	—	52	8	no	—
72	D₈	—	52	4	no	—
73	C₂×C₈	D₅₂	1	4	yes	—
74	C₂×C₈	D₅₂	1	2	yes	—
75	C₄⋊C₄	—	26	2	no	—
76	C₄⋊C₄	—	26	2	no	—
77	C₄²	C₂²×D₁₃	1	1	yes	characteristic
78	C₁₃	#1715	1	1	yes	characteristic, complemented, Sylow
79	C₂³	—	52	1	no	—
80	C₂³	—	52	1	no	—
81	Q₈	—	52	4	no	—
82	D₄	—	52	2	no	—
83	D₄	—	52	2	no	—
84	D₄	—	52	4	no	—
85	D₄	—	52	4	no	—
86	C₂×C₄	—	52	2	no	—
87	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
88	C₂×C₄	C₂×D₅₂	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	#6095	1	4	yes	complemented
91	C₂²	—	104	1	no	—
92	C₂²	—	104	1	no	—
93	C₂²	—	104	2	no	—
94	C₂²	—	104	2	no	—
95	C₂²	#88128	1	1	yes	characteristic
96	C₄	—	104	2	no	—
97	C₄	C₂×#6095	1	2	yes	—
98	C₄	#88162	1	4	yes	—
99	C₂	—	208	1	no	—
100	C₂	—	208	1	no	—
101	C₂	#11718354	1	1	yes	characteristic
102	C₂	#11718373	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11718356

C₂

yes

maximal, characteristic

#11718355

C₂

yes

maximal, complemented

#11718354

C₂

yes

maximal, complemented

#11718341

C₂

yes

maximal, characteristic, complemented

C₁₃×#322

C₂

yes

maximal, characteristic, complemented

C₂×D₁₀₄

—

C₂×#6095

—

#88128

C₂²

yes

characteristic

#88128

C₂²

yes

characteristic

#88123

C₂²

yes

—

C₄×C₈×C₁₃

C₂²

yes

characteristic

C₄×C₈×C₁₃

C₂²

yes

—

#88061

—

#88057

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

D₄

yes

—

C₂×C₈×C₁₃

D₄

yes

—

C₄²×C₁₃

C₂³

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

D₁₀₄

—

#6095

—

#1715

—

maximal, Sylow

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

SD₁₆

yes

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×C₁₃

#127

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

C₂×SD₁₆

yes

—

C₄×C₁₃

AGL(1,ℤ/8)

yes

—

Dic₁₃

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

—

—

—

—

C₄×C₈

C₂×D₁₃

yes

—

C₄×C₈

C₂×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

D₁₃

—

D₁₃

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

C₂×C₈

D₅₂

yes

—

C₂×C₈

D₅₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₁₃

yes

characteristic

C₁₃

#1715

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂×D₅₂

yes

characteristic

C₂×C₄

C₂×D₅₂

yes

—

C₈

—

C₈

#6095

yes

complemented

C₂²

—

104

—

C₂²

—

104

—

C₂²

—

104

—

C₂²

—

104

—

C₂²

#88128

yes

characteristic

C₄

—

104

—

C₄

C₂×#6095

yes

—

C₄

#88162

yes

—

C₂

—

208

—

100

C₂

—

208

—

101

C₂

#11718354

yes

characteristic

102

C₂

#11718373

yes

—

	subgroup	G/H
center	C₂²	#88128
commutator, cosocle	C₄²×C₁₃	C₂³
Frattini	C₄²	C₂²×D₁₃
Fitting	C₁₃×#322	C₂
socle	C₂²×C₁₃	#127
prime radical	#11718356	C₂
hypercenter	#322	D₁₃
hypocenter	C₁₃	#1715
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#88128

commutator, cosocle

C₄²×C₁₃

C₂³

Frattini

C₄²

C₂²×D₁₃

Fitting

C₁₃×#322

C₂

C₂²×C₁₃

C₂

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
2. #11718355	C₂	semidirect
3. #11718354	C₂	semidirect
4. #11718341	C₂	semidirect
5. C₁₃×#322	C₂	semidirect
39. C₈×C₁₃	SD₁₆	semidirect
44. #322	D₁₃	semidirect
78. C₁₃	#1715	semidirect
90. C₈	#6095	semidirect

H ≅ G/N

type

2. #11718355

C₂

semidirect

3. #11718354

C₂

semidirect

4. #11718341

C₂

semidirect

5. C₁₃×#322

C₂

semidirect

39. C₈×C₁₃

SD₁₆

semidirect

44. #322

D₁₃

semidirect

78. C₁₃

#1715

semidirect

90. C₈

#6095

semidirect

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

subgroup	index
#11718356	2
#11718355	2
#11718354	2
#11718341	2
C₁₃×#322	2
#1715	13

subgroup

index

C₁₃×#322

Subgroups — #13114464

Conjugacy classes of subgroups

Complements

Maximal subgroups