Group #10709925 — subgroups

Subgroups — #10709925

Order 768.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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
2	#71385	C₂	1	2	yes	maximal
3	#71384	C₂	1	2	yes	maximal
4	#68064	C₂	1	1	yes	maximal, characteristic
5	C₈×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#12522	—	3	1	no	maximal, Sylow
7	C₃×#500	C₂²	1	2	yes	—
8	C₃×#487	—	2	2	no	—
9	#4682	C₂²	1	2	yes	—
10	#4681	—	2	2	no	—
11	C₃×C₈²	C₂²	1	1	yes	characteristic
12	#4490	—	2	4	no	—
13	#4489	—	2	4	no	—
14	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	#2254	—	3	2	no	—
17	#2252	—	3	2	no	—
18	#1710	—	3	1	no	—
19	#1710	D₃	1	1	yes	characteristic
20	C₈×C₁₆	—	3	1	no	—
21	C₃×C₄⋊Q₈	—	2	2	no	—
22	#869	—	2	2	no	—
23	C₃×#107	—	2	2	no	—
24	C₃×#107	—	2	2	no	—
25	C₃×#103	—	4	4	no	—
26	C₃×C₄×C₈	D₄	1	2	yes	—
27	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
28	#818	—	2	2	no	—
29	#818	—	2	2	no	—
30	#816	—	4	4	no	—
31	C₂×C₃⋊C₁₆	D₄	1	2	yes	—
32	C₂×C₃⋊C₁₆	D₄	1	2	yes	—
33	#500	—	3	2	no	—
34	#500	C₂×D₃	1	2	yes	—
35	#487	—	2	2	no	—
36	#487	—	6	2	no	—
37	#367	—	6	4	no	—
38	#366	—	6	4	no	—
39	C₄×C₁₆	—	3	1	no	—
40	C₄×C₁₆	—	3	1	no	—
41	C₈²	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃×Q₈	—	4	2	no	—
43	C₂×Dic₆	—	4	2	no	—
44	C₂×C₃×C₈	D₈	1	4	yes	—
45	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
47	C₃×C₄⋊C₄	—	4	2	no	—
48	C₃×C₄⋊C₄	—	4	2	no	—
49	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
50	C₄⋊Dic₃	—	4	2	no	—
51	C₄⋊Dic₃	—	4	2	no	—
52	C₃⋊C₁₆	Q₁₆	1	4	yes	—
53	C₃⋊C₁₆	Q₁₆	1	4	yes	—
54	C₄⋊Q₈	—	2	2	no	—
55	C₄⋊Q₈	—	6	2	no	—
56	C₂×C₁₆	—	3	2	no	—
57	C₂×C₁₆	—	3	2	no	—
58	#107	—	2	2	no	—
59	#107	—	2	2	no	—
60	#107	—	6	2	no	—
61	#107	—	6	2	no	—
62	#103	—	4	4	no	—
63	#103	—	12	4	no	—
64	C₄×C₈	#67	1	2	yes	—
65	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
66	C₃×Q₈	—	8	4	no	—
67	C₂×C₃×C₄	—	8	2	no	—
68	C₂×C₃×C₄	#127	1	1	yes	characteristic
69	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
70	C₂×Dic₃	—	8	2	no	—
71	Dic₆	—	8	4	no	—
72	C₃×C₈	—	2	4	no	—
73	C₃×C₈	C₂×Q₁₆	1	2	yes	—
74	C₃×C₈	C₂×Q₁₆	1	2	yes	—
75	C₂×Q₈	—	4	2	no	—
76	C₂×Q₈	—	12	2	no	—
77	C₂×C₈	#213	1	4	yes	—
78	C₂×C₈	D₃×D₄	1	1	yes	characteristic
79	C₂×C₈	D₃×D₄	1	1	yes	characteristic
80	C₄⋊C₄	—	4	2	no	—
81	C₄⋊C₄	—	4	2	no	—
82	C₄⋊C₄	—	12	2	no	—
83	C₄⋊C₄	—	12	2	no	—
84	C₄²	C₂×#67	1	1	yes	characteristic
85	C₁₆	—	3	4	no	—
86	C₁₆	—	3	4	no	—
87	C₂²×C₃	#493	1	1	yes	characteristic
88	C₃×C₄	—	16	2	no	—
89	C₃×C₄	#494	1	1	yes	characteristic
90	C₃×C₄	#494	1	1	yes	characteristic
91	C₃×C₄	#508	1	4	yes	—
92	Dic₃	—	16	2	no	—
93	Q₈	—	8	4	no	—
94	Q₈	—	24	4	no	—
95	C₂×C₄	—	8	2	no	—
96	C₂×C₄	—	24	2	no	—
97	C₂×C₄	C₂×#213	1	2	yes	—
98	C₂×C₄	#940	1	1	yes	characteristic
99	C₈	—	2	4	no	—
100	C₈	D₃×Q₁₆	1	2	yes	—
101	C₈	D₃×Q₁₆	1	2	yes	—
102	C₂×C₃	#1713	1	1	yes	characteristic
103	C₂×C₃	#2249	1	2	yes	—
104	C₂²	#5073	1	1	yes	characteristic
105	C₄	—	16	2	no	—
106	C₄	—	48	2	no	—
107	C₄	#5183	1	1	yes	characteristic
108	C₄	#5183	1	1	yes	characteristic
109	C₄	#5194	1	4	yes	—
110	C₃	#12522	1	1	yes	characteristic, complemented, Sylow
111	C₂	#71298	1	1	yes	characteristic
112	C₂	#71320	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#1710

C₂

yes

maximal, characteristic

#71385

C₂

yes

maximal

#71384

C₂

yes

maximal

#68064

C₂

yes

maximal, characteristic

C₈×C₃⋊C₁₆

C₂

yes

maximal, characteristic

#12522

—

maximal, Sylow

C₃×#500

C₂²

yes

—

C₃×#487

—

#4682

C₂²

yes

—

#4681

—

C₃×C₈²

C₂²

yes

characteristic

#4490

—

#4489

—

C₄×C₃⋊C₁₆

C₂²

yes

characteristic

C₄×C₃⋊C₁₆

C₂²

yes

—

—

—

D₃

yes

characteristic

C₈×C₁₆

—

C₃×C₄⋊Q₈

—

—

—

—

—

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

C₂³

yes

—

—

—

C₂×C₃⋊C₁₆

D₄

yes

—

C₂×C₃⋊C₁₆

D₄

yes

—

#500

—

#500

C₂×D₃

yes

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

C₂×D₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

characteristic

C₂×C₃×C₈

C₂×D₄

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₃⋊C₁₆

Q₁₆

yes

—

C₃⋊C₁₆

Q₁₆

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

C₄×C₈

#67

yes

—

C₄×C₈

C₂²×D₃

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

—

C₂×Dic₃

—

Dic₆

—

C₃×C₈

—

C₃×C₈

C₂×Q₁₆

yes

—

C₃×C₈

C₂×Q₁₆

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

#213

yes

—

C₂×C₈

D₃×D₄

yes

characteristic

C₂×C₈

D₃×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×#67

yes

characteristic

C₁₆

—

C₁₆

—

C₂²×C₃

#493

yes

characteristic

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

C₃×C₄

yes

—

Dic₃

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#213

yes

—

C₂×C₄

#940

yes

characteristic

C₈

—

100

C₈

D₃×Q₁₆

yes

—

101

C₈

D₃×Q₁₆

yes

—

102

C₂×C₃

#1713

yes

characteristic

103

C₂×C₃

#2249

yes

—

104

C₂²

#5073

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

#5183

yes

characteristic

108

C₄

#5183

yes

characteristic

109

C₄

#5194

yes

—

110

C₃

#12522

yes

characteristic, complemented, Sylow

111

C₂

#71298

yes

characteristic

112

C₂

#71320

yes

—

	subgroup	G/H
center	C₂²	#5073
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#1710	C₂
socle, prime radical	C₂²×C₃	#493
hypercenter	#1710	D₃
hypocenter	C₃	#12522
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

D₃

C₃

G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
110. C₃	#12522	semidirect

H ≅ G/N

type

110. C₃

#12522

semidirect

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

subgroup	index
C₃×#1710	2
#71385	2
#71384	2
#68064	2
C₈×C₃⋊C₁₆	2
#12522	3

subgroup

index

C₈×C₃⋊C₁₆

Subgroups — #10709925

Conjugacy classes of subgroups

Complements

Maximal subgroups