Group #10709905 — subgroups

Subgroups — #10709905

Order 768.

354 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 115 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	#69061	C₂	1	4	yes	maximal, complemented
2	#68065	C₂	1	1	yes	maximal, characteristic, complemented
3	#68062	C₂	1	1	yes	maximal, characteristic
4	C₃×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#12517	—	3	1	no	maximal, Sylow
6	C₂×#800	—	2	8	no	—
7	#4699	C₂²	1	2	yes	—
8	#4695	—	2	2	no	—
9	#4695	C₂²	1	2	yes	—
10	#4682	—	2	2	no	—
11	C₃×C₄×C₁₆	C₂²	1	2	yes	—
12	C₃×C₈²	C₂²	1	1	yes	characteristic
13	#2248	—	3	4	no	—
14	#1713	—	3	1	no	—
15	#1712	—	3	1	no	—
16	C₈×C₁₆	D₃	1	1	yes	characteristic, complemented
17	C₂×Dic₁₂	—	2	4	no	—
18	C₂×D₂₄	—	2	4	no	—
19	C₂×D₂₄	—	4	4	no	—
20	#874	—	2	2	no	—
21	#869	—	2	2	no	—
22	C₂×C₃×C₁₆	D₄	1	4	yes	—
23	C₃×C₄×C₈	D₄	1	2	yes	—
24	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
25	#818	—	4	4	no	—
26	#800	—	4	16	no	—
27	C₂×SD₃₂	—	6	8	no	—
28	#500	—	6	2	no	—
29	#494	—	3	2	no	—
30	#493	—	3	2	no	—
31	#493	—	6	2	no	—
32	C₄×C₁₆	C₂×D₃	1	2	yes	—
33	C₈²	C₂×D₃	1	1	yes	characteristic
34	C₂×D₁₂	—	4	2	no	—
35	C₂×D₁₂	—	4	4	no	—
36	C₂×Dic₆	—	4	2	no	—
37	C₂×C₃×C₈	D₈	1	4	yes	—
38	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
39	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
40	C₄⋊Dic₃	—	4	4	no	—
41	Dic₁₂	—	4	8	no	—
42	D₂₄	—	4	8	no	—
43	D₂₄	—	8	8	no	—
44	C₃×C₁₆	D₈	1	8	yes	complemented
45	C₂×Q₁₆	—	6	4	no	—
46	C₂×D₈	—	6	4	no	—
47	C₂×D₈	—	12	4	no	—
48	C₄⋊Q₈	—	6	2	no	—
49	#127	—	6	2	no	—
50	SD₃₂	—	12	16	no	—
51	C₂×C₁₆	D₁₂	1	4	yes	—
52	#107	—	12	4	no	—
53	C₄×C₈	D₁₂	1	2	yes	—
54	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
55	C₂²×D₃	—	8	2	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	D₁₂	—	8	4	no	—
60	D₁₂	—	8	8	no	—
61	Dic₆	—	8	4	no	—
62	C₃×C₈	SD₃₂	1	8	yes	complemented
63	C₃×C₈	C₂×D₈	1	4	yes	—
64	C₂×Q₈	—	12	2	no	—
65	C₂×D₄	—	12	2	no	—
66	C₂×D₄	—	12	4	no	—
67	Q₁₆	—	12	8	no	—
68	D₈	—	12	8	no	—
69	D₈	—	24	8	no	—
70	C₂×C₈	D₂₄	1	4	yes	—
71	C₂×C₈	C₂×D₁₂	1	2	yes	—
72	C₄⋊C₄	—	12	4	no	—
73	C₄²	C₂×D₁₂	1	1	yes	characteristic
74	C₁₆	D₂₄	1	8	yes	complemented
75	C₂²×C₃	#493	1	1	yes	characteristic
76	C₂×D₃	—	16	2	no	—
77	C₂×D₃	—	16	4	no	—
78	C₃×C₄	#493	1	2	yes	—
79	C₃×C₄	C₂×SD₃₂	1	4	yes	—
80	Dic₃	—	16	2	no	—
81	C₂³	—	24	2	no	—
82	Q₈	—	24	4	no	—
83	D₄	—	24	8	no	—
84	D₄	—	24	4	no	—
85	C₂×C₄	—	24	2	no	—
86	C₂×C₄	#874	1	1	yes	characteristic
87	C₂×C₄	C₂×D₂₄	1	2	yes	—
88	C₈	#800	1	8	yes	complemented
89	C₈	C₂×D₂₄	1	4	yes	—
90	C₂×C₃	#1712	1	1	yes	characteristic
91	C₂×C₃	#2248	1	2	yes	—
92	D₃	—	32	2	no	—
93	C₂²	—	48	4	no	—
94	C₂²	—	48	2	no	—
95	C₂²	#4695	1	1	yes	characteristic
96	C₄	—	48	2	no	—
97	C₄	#4695	1	2	yes	—
98	C₄	C₂×#800	1	4	yes	—
99	C₃	#12517	1	1	yes	characteristic, complemented, Sylow
100	C₂	—	96	2	no	—
101	C₂	#68062	1	1	yes	characteristic
102	C₂	#69061	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#69061

C₂

yes

maximal, complemented

#68065

C₂

yes

maximal, characteristic, complemented

#68062

C₂

yes

maximal, characteristic

C₃×C₈×C₁₆

C₂

yes

maximal, characteristic, complemented

—

—

C₂²

yes

—

#4695

—

#4695

C₂²

yes

—

#4682

—

C₃×C₄×C₁₆

C₂²

yes

—

C₃×C₈²

C₂²

yes

—

—

—

C₈×C₁₆

D₃

yes

characteristic, complemented

C₂×Dic₁₂

—

C₂×D₂₄

—

C₂×D₂₄

—

#874

—

#869

—

C₂×C₃×C₁₆

D₄

yes

—

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

C₂³

yes

characteristic

#818

—

#800

—

C₂×SD₃₂

—

—

—

—

—

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

—

Dic₁₂

—

D₂₄

—

D₂₄

—

C₃×C₁₆

D₈

yes

complemented

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

SD₃₂

—

C₂×C₁₆

D₁₂

yes

—

#107

—

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

SD₃₂

yes

complemented

C₃×C₈

C₂×D₈

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

D₈

—

C₂×C₈

D₂₄

yes

—

C₂×C₈

C₂×D₁₂

yes

—

C₄⋊C₄

—

C₄²

C₂×D₁₂

yes

characteristic

C₁₆

D₂₄

yes

complemented

C₂²×C₃

#493

yes

characteristic

C₂×D₃

—

C₂×D₃

—

C₃×C₄

#493

yes

—

C₃×C₄

C₂×SD₃₂

yes

—

Dic₃

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#874

yes

characteristic

C₂×C₄

C₂×D₂₄

yes

—

C₈

#800

yes

complemented

C₈

C₂×D₂₄

yes

—

C₂×C₃

#1712

yes

characteristic

C₂×C₃

#2248

yes

—

D₃

—

C₂²

—

C₂²

—

C₂²

#4695

yes

characteristic

C₄

—

C₄

#4695

yes

—

C₄

C₂×#800

yes

—

C₃

#12517

yes

characteristic, complemented, Sylow

100

C₂

—

101

C₂

#68062

yes

characteristic

102

C₂

#69061

yes

—

	subgroup	G/H
center	C₂²	#4695
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×C₈×C₁₆	C₂
socle	C₂²×C₃	#493
prime radical	#68062	C₂
hypercenter	C₈×C₁₆	D₃
hypocenter	C₃	#12517
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#4695

commutator, cosocle

C₃×C₄×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₃

Fitting

C₃×C₈×C₁₆

C₂

C₂²×C₃

C₂

C₈×C₁₆

D₃

hypocenter

C₃

#12517

LCS

G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #69061	C₂	semidirect
2. #68065	C₂	semidirect
4. C₃×C₈×C₁₆	C₂	semidirect
16. C₈×C₁₆	D₃	semidirect
44. C₃×C₁₆	D₈	semidirect
62. C₃×C₈	SD₃₂	semidirect
74. C₁₆	D₂₄	semidirect
88. C₈	#800	semidirect
99. C₃	#12517	semidirect

H ≅ G/N

type

1. #69061

C₂

semidirect

2. #68065

C₂

semidirect

4. C₃×C₈×C₁₆

C₂

semidirect

16. C₈×C₁₆

D₃

semidirect

44. C₃×C₁₆

D₈

semidirect

62. C₃×C₈

SD₃₂

semidirect

74. C₁₆

D₂₄

semidirect

88. C₈

#800

semidirect

99. C₃

#12517

semidirect

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

subgroup	index
#69061	2
#68065	2
#68062	2
C₃×C₈×C₁₆	2
#12517	3

subgroup

index

C₃×C₈×C₁₆

Subgroups — #10709905

Conjugacy classes of subgroups

Complements

Maximal subgroups