Group #10709892 — subgroups

Subgroups — #10709892

Order 768.

290 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
2	#71381	C₂	1	4	yes	maximal, complemented
3	#68062	C₂	1	1	yes	maximal, characteristic
4	#67437	C₂	1	1	yes	maximal, characteristic, complemented
5	#12513	—	3	1	no	maximal, Sylow
6	C₃×#500	C₂²	1	2	yes	—
7	C₃×#487	—	2	2	no	—
8	#4695	—	2	2	no	—
9	#4695	C₂²	1	2	yes	—
10	C₃×C₈²	C₂²	1	1	yes	characteristic
11	#4491	—	2	8	no	—
12	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
13	#2240	—	3	4	no	—
14	#1712	—	3	1	no	—
15	#1710	D₃	1	1	yes	characteristic, complemented
16	#1311	—	3	1	no	—
17	C₃×C₄⋊Q₈	—	2	2	no	—
18	C₂×D₂₄	—	2	4	no	—
19	C₂×D₂₄	—	4	4	no	—
20	#874	—	2	2	no	—
21	C₃×#107	—	2	4	no	—
22	C₃×#103	—	4	4	no	—
23	C₃×C₄×C₈	D₄	1	2	yes	—
24	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₃⋊C₁₆	D₄	1	4	yes	—
26	#500	C₂×D₃	1	2	yes	—
27	#493	—	3	2	no	—
28	#493	—	6	2	no	—
29	#487	—	2	2	no	—
30	#357	—	6	8	no	—
31	C₄×C₁₆	—	3	2	no	—
32	C₈²	C₂×D₃	1	1	yes	characteristic
33	C₂×C₃×Q₈	—	4	2	no	—
34	C₂×D₁₂	—	4	2	no	—
35	C₂×D₁₂	—	4	4	no	—
36	C₂×C₃×C₈	D₈	1	4	yes	—
37	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
38	C₃×C₄⋊C₄	—	4	2	no	—
39	C₃×C₄⋊C₄	—	4	2	no	—
40	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
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	—	2	2	no	—
47	#127	—	6	2	no	—
48	C₂×C₁₆	—	3	4	no	—
49	#107	—	2	4	no	—
50	#103	—	4	4	no	—
51	C₄×C₈	#67	1	2	yes	—
52	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
53	C₂²×D₃	—	8	2	no	—
54	C₃×Q₈	—	8	4	no	—
55	C₂×C₃×C₄	—	8	2	no	—
56	C₂×C₃×C₄	#127	1	1	yes	characteristic
57	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
58	D₁₂	—	8	4	no	—
59	D₁₂	—	8	8	no	—
60	C₃×C₈	—	2	4	no	—
61	C₃×C₈	#135	1	4	yes	—
62	C₂×Q₈	—	4	2	no	—
63	C₂×D₄	—	12	4	no	—
64	C₂×D₄	—	12	2	no	—
65	D₈	—	12	8	no	—
66	D₈	—	24	8	no	—
67	C₂×C₈	#213	1	4	yes	—
68	C₂×C₈	D₃×D₄	1	2	yes	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	C₂×#67	1	1	yes	characteristic
72	C₁₆	—	6	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	—	16	2	no	—
77	C₃×C₄	#495	1	2	yes	—
78	C₃×C₄	#509	1	4	yes	—
79	C₂³	—	24	2	no	—
80	Q₈	—	8	4	no	—
81	D₄	—	24	4	no	—
82	D₄	—	24	8	no	—
83	C₂×C₄	—	8	2	no	—
84	C₂×C₄	C₂×#213	1	2	yes	—
85	C₂×C₄	#940	1	1	yes	characteristic
86	C₈	—	2	4	no	—
87	C₈	#919	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	2	no	—
92	C₂²	—	48	4	no	—
93	C₂²	#5073	1	1	yes	characteristic
94	C₄	—	16	2	no	—
95	C₄	#5191	1	2	yes	—
96	C₄	#5193	1	4	yes	—
97	C₃	#12513	1	1	yes	characteristic, complemented, Sylow
98	C₂	—	96	2	no	—
99	C₂	#71297	1	1	yes	characteristic
100	C₂	#71321	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#1710

C₂

yes

maximal, characteristic, complemented

#71381

C₂

yes

maximal, complemented

#68062

C₂

yes

maximal, characteristic

#67437

C₂

yes

maximal, characteristic, complemented

#12513

—

maximal, Sylow

C₃×#500

C₂²

yes

—

C₃×#487

—

#4695

—

#4695

C₂²

yes

—

C₃×C₈²

C₂²

yes

characteristic

#4491

—

C₄×C₃⋊C₁₆

C₂²

yes

—

#2240

—

#1712

—

#1710

D₃

yes

characteristic, complemented

#1311

—

C₃×C₄⋊Q₈

—

C₂×D₂₄

—

C₂×D₂₄

—

#874

—

C₃×#107

—

C₃×#103

—

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

C₂³

yes

characteristic

C₂×C₃⋊C₁₆

D₄

yes

—

#500

C₂×D₃

yes

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×D₄

yes

characteristic

D₂₄

—

D₂₄

—

C₃⋊C₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

#107

—

#103

—

C₄×C₈

#67

yes

—

C₄×C₈

C₂²×D₃

yes

characteristic

C₂²×D₃

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

—

D₁₂

—

D₁₂

—

C₃×C₈

—

C₃×C₈

#135

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

C₂×C₈

#213

yes

—

C₂×C₈

D₃×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×#67

yes

characteristic

C₁₆

—

C₂²×C₃

#493

yes

characteristic

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

#495

yes

—

C₃×C₄

#509

yes

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂×#213

yes

—

C₂×C₄

#940

yes

characteristic

C₈

—

C₈

#919

yes

—

C₂×C₃

#1709

yes

characteristic

C₂×C₃

#2250

yes

—

D₃

—

C₂²

—

C₂²

—

C₂²

#5073

yes

characteristic

C₄

—

C₄

#5191

yes

—

C₄

#5193

yes

—

C₃

#12513

yes

characteristic, complemented, Sylow

C₂

—

C₂

#71297

yes

characteristic

100

C₂

#71321

yes

—

	subgroup	G/H
center	C₂²	#5073
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#1710	C₂
socle	C₂²×C₃	#493
prime radical	#68062	C₂
hypercenter	#1710	D₃
hypocenter	C₃	#12513
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#5073

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 ◁ #1710

N	H ≅ G/N	type
1. C₃×#1710	C₂	semidirect
2. #71381	C₂	semidirect
4. #67437	C₂	semidirect
15. #1710	D₃	semidirect
97. C₃	#12513	semidirect

H ≅ G/N

type

1. C₃×#1710

C₂

semidirect

2. #71381

C₂

semidirect

4. #67437

C₂

semidirect

15. #1710

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
C₃×#1710	2
#71381	2
#68062	2
#67437	2
#12513	3

subgroup

index

Subgroups — #10709892

Conjugacy classes of subgroups

Complements

Maximal subgroups