Group #10709891 — subgroups

Subgroups — #10709891

Order 768.

290 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 83 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	#69058	C₂	1	4	yes	maximal, complemented
2	#68064	C₂	1	1	yes	maximal, characteristic, complemented
3	#68062	C₂	1	1	yes	maximal, characteristic
4	C₃×#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	#12513	—	3	1	no	maximal, Sylow
6	#4695	—	2	2	no	—
7	#4695	C₂²	1	2	yes	—
8	#4682	C₂²	1	2	yes	—
9	#4681	—	2	2	no	—
10	C₃×C₄×C₁₆	C₂²	1	2	yes	—
11	C₃×C₈²	C₂²	1	1	yes	characteristic
12	#4509	—	2	8	no	—
13	#2240	—	3	4	no	—
14	#1712	—	3	1	no	—
15	#1710	—	3	1	no	—
16	#1311	D₃	1	1	yes	characteristic, complemented
17	C₂×D₂₄	—	2	4	no	—
18	C₂×D₂₄	—	4	4	no	—
19	#874	—	2	2	no	—
20	#869	—	2	2	no	—
21	C₂×C₃×C₁₆	D₄	1	4	yes	—
22	C₃×C₄×C₈	D₄	1	2	yes	—
23	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
24	#818	—	2	4	no	—
25	#816	—	4	4	no	—
26	#500	—	3	2	no	—
27	#493	—	3	2	no	—
28	#493	—	6	2	no	—
29	#487	—	6	2	no	—
30	#357	—	6	8	no	—
31	C₄×C₁₆	C₂×D₃	1	2	yes	—
32	C₈²	C₂×D₃	1	1	yes	characteristic
33	C₂×D₁₂	—	4	2	no	—
34	C₂×D₁₂	—	4	4	no	—
35	C₂×Dic₆	—	4	2	no	—
36	C₂×C₃×C₈	D₈	1	4	yes	—
37	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
38	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₃	—	4	2	no	—
40	C₄⋊Dic₃	—	4	2	no	—
41	D₂₄	—	4	8	no	—
42	D₂₄	—	8	8	no	—
43	C₃×C₁₆	—	2	4	no	—
44	C₂×D₈	—	6	4	no	—
45	C₂×D₈	—	12	4	no	—
46	C₄⋊Q₈	—	6	2	no	—
47	#127	—	6	2	no	—
48	C₂×C₁₆	D₁₂	1	4	yes	—
49	#107	—	6	4	no	—
50	#103	—	12	4	no	—
51	C₄×C₈	D₁₂	1	2	yes	—
52	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
53	C₂²×D₃	—	8	2	no	—
54	C₂×C₃×C₄	#127	1	1	yes	characteristic
55	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
56	C₂×Dic₃	—	8	2	no	—
57	D₁₂	—	8	4	no	—
58	D₁₂	—	8	8	no	—
59	Dic₆	—	8	4	no	—
60	C₃×C₈	—	2	4	no	—
61	C₃×C₈	#135	1	4	yes	—
62	C₂×Q₈	—	12	2	no	—
63	C₂×D₄	—	12	2	no	—
64	C₂×D₄	—	12	4	no	—
65	D₈	—	12	8	no	—
66	D₈	—	24	8	no	—
67	C₂×C₈	D₂₄	1	4	yes	—
68	C₂×C₈	C₂×D₁₂	1	2	yes	—
69	C₄⋊C₄	—	12	2	no	—
70	C₄⋊C₄	—	12	2	no	—
71	C₄²	C₂×D₁₂	1	1	yes	characteristic
72	C₁₆	—	2	4	no	—
73	C₂²×C₃	#493	1	1	yes	characteristic
74	C₂×D₃	—	16	2	no	—
75	C₂×D₃	—	16	4	no	—
76	C₃×C₄	#495	1	2	yes	—
77	C₃×C₄	#509	1	4	yes	—
78	Dic₃	—	16	2	no	—
79	C₂³	—	24	2	no	—
80	Q₈	—	24	4	no	—
81	D₄	—	24	8	no	—
82	D₄	—	24	4	no	—
83	C₂×C₄	—	24	2	no	—
84	C₂×C₄	#874	1	1	yes	characteristic
85	C₂×C₄	C₂×D₂₄	1	2	yes	—
86	C₈	—	2	4	no	—
87	C₈	#904	1	4	yes	—
88	C₂×C₃	#1709	1	1	yes	characteristic
89	C₂×C₃	#2250	1	2	yes	—
90	D₃	—	32	2	no	—
91	C₂²	—	48	4	no	—
92	C₂²	—	48	2	no	—
93	C₂²	#4695	1	1	yes	characteristic
94	C₄	—	48	2	no	—
95	C₄	#4696	1	2	yes	—
96	C₄	#4908	1	4	yes	—
97	C₃	#12513	1	1	yes	characteristic, complemented, Sylow
98	C₂	—	96	2	no	—
99	C₂	#68060	1	1	yes	characteristic
100	C₂	#69081	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#69058

C₂

yes

maximal, complemented

#68064

C₂

yes

maximal, characteristic, complemented

#68062

C₂

yes

maximal, characteristic

C₃×#1311

C₂

yes

maximal, characteristic, complemented

—

—

C₂²

yes

—

#4682

C₂²

yes

—

#4681

—

C₃×C₄×C₁₆

C₂²

yes

—

C₃×C₈²

C₂²

yes

—

—

—

—

D₃

yes

characteristic, complemented

C₂×D₂₄

—

C₂×D₂₄

—

#874

—

#869

—

C₂×C₃×C₁₆

D₄

yes

—

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

C₂³

yes

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₃

yes

—

C₈²

C₂×D₃

yes

characteristic

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

D₂₄

—

D₂₄

—

C₃×C₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₁₂

yes

—

#107

—

#103

—

C₄×C₈

D₁₂

yes

—

C₄×C₈

C₂²×D₃

yes

characteristic

C₂²×D₃

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

Dic₆

—

C₃×C₈

—

C₃×C₈

#135

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

C₂×C₈

D₂₄

yes

—

C₂×C₈

C₂×D₁₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₁₂

yes

characteristic

C₁₆

—

C₂²×C₃

#493

yes

characteristic

C₂×D₃

—

C₂×D₃

—

C₃×C₄

#495

yes

—

C₃×C₄

#509

yes

—

Dic₃

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#874

yes

characteristic

C₂×C₄

C₂×D₂₄

yes

—

C₈

—

C₈

#904

yes

—

C₂×C₃

#1709

yes

characteristic

C₂×C₃

#2250

yes

—

D₃

—

C₂²

—

C₂²

—

C₂²

#4695

yes

characteristic

C₄

—

C₄

#4696

yes

—

C₄

#4908

yes

—

C₃

#12513

yes

characteristic, complemented, Sylow

C₂

—

C₂

#68060

yes

characteristic

100

C₂

#69081

yes

—

	subgroup	G/H
center	C₂²	#4695
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#1311	C₂
socle	C₂²×C₃	#493
prime radical	#68062	C₂
hypercenter	#1311	D₃
hypocenter	C₃	#12513
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#4695

commutator, cosocle

C₃×C₄×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₃

C₂

C₂²×C₃

C₂

D₃

C₃

G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #69058	C₂	semidirect
2. #68064	C₂	semidirect
4. C₃×#1311	C₂	semidirect
16. #1311	D₃	semidirect
97. C₃	#12513	semidirect

H ≅ G/N

type

1. #69058

C₂

semidirect

2. #68064

C₂

semidirect

4. C₃×#1311

C₂

semidirect

16. #1311

D₃

semidirect

97. C₃

#12513

semidirect

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

subgroup	index
#69058	2
#68064	2
#68062	2
C₃×#1311	2
#12513	3

subgroup

index

Subgroups — #10709891

Conjugacy classes of subgroups

Complements

Maximal subgroups