Group #10709890 — subgroups

Subgroups — #10709890

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×#1712	C₂	1	1	yes	maximal, characteristic
2	#71288	C₂	1	4	yes	maximal, complemented
3	#68064	C₂	1	1	yes	maximal, characteristic, complemented
4	#67437	C₂	1	1	yes	maximal, characteristic, complemented
5	#12513	—	3	1	no	maximal, Sylow
6	C₃×#493	—	2	2	no	—
7	C₃×#493	C₂²	1	2	yes	—
8	#4682	C₂²	1	2	yes	—
9	#4681	—	2	2	no	—
10	C₃×C₈²	C₂²	1	1	yes	characteristic
11	#4562	—	2	8	no	—
12	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
13	#2240	—	3	4	no	—
14	#1712	D₃	1	1	yes	characteristic
15	#1710	—	3	1	no	—
16	#1311	—	3	1	no	—
17	C₂×C₃×D₈	—	2	4	no	—
18	C₂×C₃×D₈	—	4	4	no	—
19	C₃×#127	—	2	2	no	—
20	#869	—	2	2	no	—
21	C₃×C₄×C₈	D₄	1	2	yes	—
22	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
23	#818	—	2	4	no	—
24	#816	—	4	4	no	—
25	C₂×C₃⋊C₁₆	D₄	1	4	yes	—
26	#500	—	3	2	no	—
27	#493	—	2	2	no	—
28	#493	C₂×D₃	1	2	yes	—
29	#487	—	6	2	no	—
30	#357	—	6	8	no	—
31	C₄×C₁₆	—	3	2	no	—
32	C₈²	C₂×D₃	1	1	yes	characteristic
33	C₂×C₃×D₄	—	4	2	no	—
34	C₂×C₃×D₄	—	4	4	no	—
35	C₂×Dic₆	—	4	2	no	—
36	C₃×D₈	—	4	8	no	—
37	C₃×D₈	—	8	8	no	—
38	C₂×C₃×C₈	D₈	1	4	yes	—
39	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
40	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
41	C₄⋊Dic₃	—	4	2	no	—
42	C₄⋊Dic₃	—	4	2	no	—
43	C₃⋊C₁₆	—	2	4	no	—
44	C₂×D₈	—	2	4	no	—
45	C₂×D₈	—	4	4	no	—
46	C₄⋊Q₈	—	6	2	no	—
47	#127	—	2	2	no	—
48	C₂×C₁₆	—	3	4	no	—
49	#107	—	6	4	no	—
50	#103	—	12	4	no	—
51	C₄×C₈	#67	1	2	yes	—
52	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
53	C₂³×C₃	—	8	2	no	—
54	C₃×D₄	—	8	4	no	—
55	C₃×D₄	—	8	8	no	—
56	C₂×C₃×C₄	#127	1	1	yes	characteristic
57	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
58	C₂×Dic₃	—	8	2	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	—	4	4	no	—
64	C₂×D₄	—	4	2	no	—
65	D₈	—	4	8	no	—
66	D₈	—	8	8	no	—
67	C₂×C₈	#213	1	4	yes	—
68	C₂×C₈	D₃×D₄	1	2	yes	—
69	C₄⋊C₄	—	12	2	no	—
70	C₄⋊C₄	—	12	2	no	—
71	C₄²	C₂×#67	1	1	yes	characteristic
72	C₁₆	—	6	4	no	—
73	C₂²×C₃	—	16	2	no	—
74	C₂²×C₃	—	16	4	no	—
75	C₂²×C₃	#493	1	1	yes	characteristic
76	C₃×C₄	#495	1	2	yes	—
77	C₃×C₄	#509	1	4	yes	—
78	Dic₃	—	16	2	no	—
79	C₂³	—	8	2	no	—
80	Q₈	—	24	4	no	—
81	D₄	—	8	8	no	—
82	D₄	—	8	4	no	—
83	C₂×C₄	—	24	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	#912	1	4	yes	—
88	C₂×C₃	—	32	2	no	—
89	C₂×C₃	#1709	1	1	yes	characteristic
90	C₂×C₃	#2250	1	2	yes	—
91	C₂²	—	16	2	no	—
92	C₂²	—	16	4	no	—
93	C₂²	#5073	1	1	yes	characteristic
94	C₄	—	48	2	no	—
95	C₄	#5147	1	4	yes	—
96	C₄	#5155	1	2	yes	—
97	C₃	#12513	1	1	yes	characteristic, complemented, Sylow
98	C₂	—	32	2	no	—
99	C₂	#71275	1	1	yes	characteristic
100	C₂	#71292	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#1712

C₂

yes

maximal, characteristic

#71288

C₂

yes

maximal, complemented

#68064

C₂

yes

maximal, characteristic, complemented

#67437

C₂

yes

maximal, characteristic, complemented

—

—

C₂²

yes

—

#4682

C₂²

yes

—

#4681

—

C₃×C₈²

C₂²

yes

characteristic

#4562

—

C₄×C₃⋊C₁₆

C₂²

yes

—

#2240

—

#1712

D₃

yes

characteristic

#1710

—

#1311

—

C₂×C₃×D₈

—

C₂×C₃×D₈

—

C₃×#127

—

#869

—

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

C₂³

yes

characteristic

#818

—

#816

—

C₂×C₃⋊C₁₆

D₄

yes

—

#500

—

#493

—

#493

C₂×D₃

yes

—

#487

—

#357

—

C₄×C₁₆

—

C₈²

C₂×D₃

yes

characteristic

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×Dic₆

—

C₃×D₈

—

C₃×D₈

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₃⋊C₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

#107

—

#103

—

C₄×C₈

#67

yes

—

C₄×C₈

C₂²×D₃

yes

characteristic

C₂³×C₃

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

—

C₂×Dic₃

—

Dic₆

—

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

—

C₂²×C₃

—

C₂²×C₃

#493

yes

characteristic

C₃×C₄

#495

yes

—

C₃×C₄

#509

yes

—

Dic₃

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂×#213

yes

—

C₂×C₄

#940

yes

characteristic

C₈

—

C₈

#912

yes

—

C₂×C₃

—

C₂×C₃

#1709

yes

characteristic

C₂×C₃

#2250

yes

—

C₂²

—

C₂²

—

C₂²

#5073

yes

characteristic

C₄

—

C₄

#5147

yes

—

C₄

#5155

yes

—

C₃

#12513

yes

characteristic, complemented, Sylow

C₂

—

C₂

#71275

yes

characteristic

100

C₂

#71292

yes

—

	subgroup	G/H
center	C₂²	#5073
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting, prime radical	C₃×#1712	C₂
socle	C₂²×C₃	#493
hypercenter	#1712	D₃
hypocenter	C₃	#12513
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#5073

commutator, cosocle

C₃×C₄×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₃

Fitting, prime radical

C₂

C₂²×C₃

D₃

C₃

G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
2. #71288	C₂	semidirect
3. #68064	C₂	semidirect
4. #67437	C₂	semidirect
97. C₃	#12513	semidirect

H ≅ G/N

type

2. #71288

C₂

semidirect

3. #68064

C₂

semidirect

4. #67437

C₂

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×#1712	2
#71288	2
#68064	2
#67437	2
#12513	3

subgroup

index

Subgroups — #10709890

Conjugacy classes of subgroups

Complements

Maximal subgroups