Group #10681908 — subgroups

Subgroups — #10681908

Order 768.

152 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 51 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×#1709	C₂	1	1	yes	maximal, characteristic
2	#67439	C₂	1	2	yes	maximal, complemented
3	#7092	—	3	1	no	maximal, Sylow
4	C₃×#494	C₄	1	1	yes	characteristic
5	C₃×#493	C₄	1	1	yes	characteristic
6	C₃×#486	—	2	2	no	—
7	C₃×C₈²	C₂²	1	1	yes	characteristic
8	C₄⋊(C₃⋊C₁₆)	—	2	2	no	—
9	#1709	D₃	1	1	yes	characteristic
10	#1367	—	3	2	no	—
11	C₂×C₃×Q₁₆	—	4	1	no	—
12	C₂×C₃×D₈	—	4	1	no	—
13	C₃×C₄⋊Q₈	—	2	1	no	—
14	C₃×#127	—	2	1	no	—
15	C₃×#102	—	4	2	no	—
16	C₃×#102	—	4	2	no	—
17	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	C₃×C₄×C₈	D₄	1	2	yes	—
19	C₂×C₃⋊C₁₆	—	4	2	no	—
20	#494	Dic₃	1	1	yes	characteristic
21	#493	Dic₃	1	1	yes	characteristic
22	#486	—	2	2	no	—
23	C₄⋊C₁₆	—	6	2	no	—
24	C₈²	C₂×D₃	1	1	yes	characteristic
25	C₂×C₃×Q₈	—	4	1	no	—
26	C₂×C₃×D₄	—	4	1	no	—
27	C₂×C₃×D₄	—	4	2	no	—
28	C₃×Q₁₆	—	8	2	no	—
29	C₃×D₈	—	8	2	no	—
30	C₂×C₃×C₈	—	2	1	no	—
31	C₂×C₃×C₈	D₈	1	2	yes	—
32	C₂×C₃×C₈	SD₁₆	1	2	yes	—
33	C₃×C₄⋊C₄	—	4	2	no	—
34	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
35	C₃⋊C₁₆	—	8	2	no	—
36	C₂×Q₁₆	—	4	1	no	—
37	C₂×D₈	—	4	1	no	—
38	C₄⋊Q₈	—	2	1	no	—
39	#127	—	2	1	no	—
40	C₂×C₁₆	—	12	2	no	—
41	#102	—	4	2	no	—
42	#102	—	4	2	no	—
43	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
44	C₄×C₈	#67	1	2	yes	—
45	C₂³×C₃	—	8	1	no	—
46	C₃×Q₈	—	8	2	no	—
47	C₃×D₄	—	8	2	no	—
48	C₃×D₄	—	8	2	no	—
49	C₃×D₄	—	8	2	no	—
50	C₂×C₃×C₄	—	8	1	no	—
51	C₂×C₃×C₄	#100	1	1	yes	characteristic
52	C₂×C₃×C₄	#102	1	2	yes	—
53	C₃×C₈	—	2	2	no	—
54	C₃×C₈	—	2	2	no	—
55	C₃×C₈	—	2	2	no	—
56	C₂×Q₈	—	4	1	no	—
57	C₂×D₄	—	4	2	no	—
58	C₂×D₄	—	4	1	no	—
59	Q₁₆	—	8	2	no	—
60	D₈	—	8	2	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	#213	1	2	yes	—
63	C₂×C₈	#214	1	2	yes	—
64	C₄⋊C₄	—	4	2	no	—
65	C₄²	#217	1	1	yes	characteristic
66	C₁₆	—	24	2	no	—
67	C₂²×C₃	—	16	1	no	—
68	C₂²×C₃	—	16	2	no	—
69	C₂²×C₃	#331	1	1	yes	characteristic
70	C₃×C₄	—	2	1	no	—
71	C₃×C₄	—	16	1	no	—
72	C₃×C₄	#359	1	2	yes	—
73	C₃×C₄	#360	1	2	yes	—
74	C₂³	—	8	1	no	—
75	Q₈	—	8	2	no	—
76	D₄	—	8	2	no	—
77	D₄	—	8	2	no	—
78	D₄	—	8	2	no	—
79	C₂×C₄	—	8	1	no	—
80	C₂×C₄	#832	1	2	yes	—
81	C₂×C₄	#833	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	—	2	2	no	—
84	C₈	—	2	2	no	—
85	C₂×C₃	—	32	1	no	—
86	C₂×C₃	#1357	1	1	yes	characteristic
87	C₂×C₃	#1360	1	2	yes	—
88	C₂²	—	16	1	no	—
89	C₂²	—	16	2	no	—
90	C₂²	#4544	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	—	16	1	no	—
93	C₄	#4566	1	2	yes	—
94	C₄	#4567	1	2	yes	—
95	C₃	#7092	1	1	yes	characteristic, complemented, Sylow
96	C₂	—	32	1	no	—
97	C₂	#67791	1	1	yes	characteristic
98	C₂	#67796	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#1709

C₂

yes

maximal, characteristic

#67439

C₂

yes

maximal, complemented

#7092

—

maximal, Sylow

C₃×#494

C₄

yes

characteristic

C₃×#493

C₄

yes

characteristic

C₃×#486

—

C₃×C₈²

C₂²

yes

characteristic

C₄⋊(C₃⋊C₁₆)

—

#1709

D₃

yes

characteristic

#1367

—

C₂×C₃×Q₁₆

—

C₂×C₃×D₈

—

C₃×C₄⋊Q₈

—

C₃×#127

—

C₃×#102

—

C₃×#102

—

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

—

C₂×C₃⋊C₁₆

—

#494

Dic₃

yes

characteristic

#493

Dic₃

yes

characteristic

#486

—

C₄⋊C₁₆

—

C₈²

C₂×D₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₃×Q₁₆

—

C₃×D₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

SD₁₆

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²⋊C₄

yes

characteristic

C₃⋊C₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

#102

—

#102

—

C₄×C₈

C₂×Dic₃

yes

characteristic

C₄×C₈

#67

yes

—

C₂³×C₃

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#100

yes

characteristic

C₂×C₃×C₄

#102

yes

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

#213

yes

—

C₂×C₈

#214

yes

—

C₄⋊C₄

—

C₄²

#217

yes

characteristic

C₁₆

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#331

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#359

yes

—

C₃×C₄

#360

yes

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#832

yes

—

C₂×C₄

#833

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

#1357

yes

characteristic

C₂×C₃

#1360

yes

—

C₂²

—

C₂²

—

C₂²

#4544

yes

characteristic

C₄

—

C₄

—

C₄

#4566

yes

—

C₄

#4567

yes

—

C₃

#7092

yes

characteristic, complemented, Sylow

C₂

—

C₂

#67791

yes

characteristic

C₂

#67796

yes

—

	subgroup	G/H
center	C₂²	#4544
commutator	C₃×C₄×C₈	C₂×C₄
Frattini	C₈²	C₂×D₃
Fitting	C₃×#1709	C₂
socle	C₂²×C₃	#331
prime radical	C₃×#493	C₄
cosocle	C₃×C₈²	C₂²
hypercenter	#1709	D₃
hypocenter	C₃	#7092
LCS	G ▷ C₃×C₄×C₈ ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#4544

commutator

C₃×C₄×C₈

C₂×C₄

Frattini

C₈²

C₂×D₃

C₂

C₂²×C₃

C₄

C₃×C₈²

C₂²

D₃

C₃

G ▷ C₃×C₄×C₈ ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
2. #67439	C₂	semidirect
95. C₃	#7092	semidirect

H ≅ G/N

type

2. #67439

C₂

semidirect

95. C₃

#7092

semidirect

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

subgroup	index
C₃×#1709	2
#67439	2
#7092	3

subgroup

index

C₃×#1709

#67439

#7092

Subgroups — #10681908

Conjugacy classes of subgroups

Complements

Maximal subgroups