Group #10709927 — subgroups

Subgroups — #10709927

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×#2254	C₂	1	1	yes	maximal, characteristic
2	#71384	C₂	1	2	yes	maximal
3	#71361	C₂	1	2	yes	maximal
4	#69046	C₂	1	1	yes	maximal, characteristic
5	C₁₆×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#12522	—	3	1	no	maximal, Sylow
7	C₃×#500	C₂²	1	2	yes	—
8	#5093	—	2	2	no	—
9	#5079	—	2	2	no	—
10	#4682	C₂²	1	2	yes	—
11	C₃×#367	—	2	2	no	—
12	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#4505	—	2	2	no	—
14	#4489	—	2	4	no	—
15	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₈×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	#2254	—	3	1	no	—
18	#2254	D₃	1	1	yes	characteristic
19	#2252	—	3	2	no	—
20	#1710	—	3	2	no	—
21	C₈×C₁₆	—	3	1	no	—
22	C₃×C₄⋊Q₈	—	2	2	no	—
23	#869	—	2	2	no	—
24	C₂×C₃×C₁₆	D₄	1	2	yes	—
25	C₃×#107	—	2	2	no	—
26	C₃×#107	—	2	2	no	—
27	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
28	#835	—	4	4	no	—
29	#818	—	2	2	no	—
30	#818	—	2	2	no	—
31	C₂×C₃⋊C₁₆	D₄	1	2	yes	—
32	#810	—	4	4	no	—
33	C₄×C₃⋊C₈	D₄	1	2	yes	—
34	#500	—	3	2	no	—
35	#500	C₂×D₃	1	2	yes	—
36	#487	—	6	2	no	—
37	#487	—	6	2	no	—
38	#367	—	2	2	no	—
39	#367	—	6	2	no	—
40	#366	—	6	4	no	—
41	C₄×C₁₆	—	3	1	no	—
42	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
43	C₈²	—	3	1	no	—
44	C₂×C₃×Q₈	—	4	2	no	—
45	C₂×Dic₆	—	4	2	no	—
46	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
48	C₃×C₄⋊C₄	—	4	2	no	—
49	C₃×C₄⋊C₄	—	4	2	no	—
50	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
51	C₄⋊Dic₃	—	4	2	no	—
52	C₄⋊Dic₃	—	4	2	no	—
53	C₂×C₃⋊C₈	D₈	1	4	yes	—
54	C₃×C₁₆	Q₁₆	1	4	yes	—
55	C₃⋊C₁₆	Q₁₆	1	4	yes	—
56	C₄⋊Q₈	—	2	2	no	—
57	C₄⋊Q₈	—	6	2	no	—
58	C₂×C₁₆	—	3	2	no	—
59	C₂×C₁₆	#67	1	2	yes	—
60	#107	—	2	2	no	—
61	#107	—	2	2	no	—
62	#107	—	6	2	no	—
63	#107	—	6	2	no	—
64	#103	—	12	4	no	—
65	#103	—	12	4	no	—
66	C₄×C₈	—	3	2	no	—
67	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
68	C₃×Q₈	—	8	4	no	—
69	C₂×C₃×C₄	—	8	2	no	—
70	C₂×C₃×C₄	#127	1	1	yes	characteristic
71	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
72	C₂×Dic₃	—	8	2	no	—
73	Dic₆	—	8	4	no	—
74	C₃×C₈	C₂×Q₁₆	1	2	yes	—
75	C₃×C₈	C₂×Q₁₆	1	2	yes	—
76	C₃⋊C₈	—	2	4	no	—
77	C₂×Q₈	—	4	2	no	—
78	C₂×Q₈	—	12	2	no	—
79	C₂×C₈	—	3	4	no	—
80	C₂×C₈	D₃×D₄	1	1	yes	characteristic
81	C₂×C₈	C₂×#67	1	1	yes	characteristic
82	C₄⋊C₄	—	4	2	no	—
83	C₄⋊C₄	—	4	2	no	—
84	C₄⋊C₄	—	12	2	no	—
85	C₄⋊C₄	—	12	2	no	—
86	C₄²	D₃×D₄	1	1	yes	characteristic
87	C₁₆	—	3	4	no	—
88	C₁₆	#216	1	4	yes	—
89	C₂²×C₃	#493	1	1	yes	characteristic
90	C₃×C₄	—	16	2	no	—
91	C₃×C₄	#494	1	1	yes	characteristic
92	C₃×C₄	#494	1	1	yes	characteristic
93	C₃×C₄	#508	1	4	yes	—
94	Dic₃	—	16	2	no	—
95	Q₈	—	8	4	no	—
96	Q₈	—	24	4	no	—
97	C₂×C₄	—	8	2	no	—
98	C₂×C₄	—	24	2	no	—
99	C₂×C₄	D₃×D₈	1	2	yes	—
100	C₂×C₄	#940	1	1	yes	characteristic
101	C₈	—	6	4	no	—
102	C₈	D₃×Q₁₆	1	2	yes	—
103	C₈	C₂×#216	1	2	yes	—
104	C₂×C₃	#1713	1	1	yes	characteristic
105	C₂×C₃	#2249	1	2	yes	—
106	C₂²	#5151	1	1	yes	characteristic
107	C₄	—	16	2	no	—
108	C₄	—	48	2	no	—
109	C₄	#4916	1	4	yes	—
110	C₄	#5092	1	1	yes	characteristic
111	C₄	#5183	1	1	yes	characteristic
112	C₃	#12522	1	1	yes	characteristic, complemented, Sylow
113	C₂	#71362	1	1	yes	characteristic
114	C₂	#71480	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#2254

C₂

yes

maximal, characteristic

#71384

C₂

yes

maximal

#71361

C₂

yes

maximal

#69046

C₂

yes

maximal, characteristic

C₁₆×C₃⋊C₈

C₂

yes

maximal, characteristic

#12522

—

maximal, Sylow

C₃×#500

C₂²

yes

—

#5093

—

#5079

—

#4682

C₂²

yes

—

C₃×#367

—

C₃×C₄×C₁₆

C₂²

yes

characteristic

#4505

—

#4489

—

C₄×C₃⋊C₁₆

C₂²

yes

characteristic

C₈×C₃⋊C₈

C₂²

yes

characteristic

#2254

—

#2254

D₃

yes

characteristic

#2252

—

#1710

—

C₈×C₁₆

—

C₃×C₄⋊Q₈

—

#869

—

C₂×C₃×C₁₆

D₄

yes

—

C₃×#107

—

C₃×#107

—

C₃×C₄×C₈

C₂³

yes

—

—

—

C₂×C₃⋊C₁₆

D₄

yes

—

#810

—

C₄×C₃⋊C₈

D₄

yes

—

#500

—

#500

C₂×D₃

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₃

yes

characteristic

C₈²

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

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

—

C₂×C₃⋊C₈

D₈

yes

—

C₃×C₁₆

Q₁₆

yes

—

C₃⋊C₁₆

Q₁₆

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₂×C₁₆

#67

yes

—

—

—

—

—

—

—

C₄×C₈

—

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

C₂×Q₁₆

yes

—

C₃×C₈

C₂×Q₁₆

yes

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

D₃×D₄

yes

characteristic

C₂×C₈

C₂×#67

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₃×D₄

yes

characteristic

C₁₆

—

C₁₆

#216

yes

—

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

D₃×D₈

yes

—

100

C₂×C₄

#940

yes

characteristic

101

C₈

—

102

C₈

D₃×Q₁₆

yes

—

103

C₈

C₂×#216

yes

—

104

C₂×C₃

#1713

yes

characteristic

105

C₂×C₃

#2249

yes

—

106

C₂²

#5151

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

#4916

yes

—

110

C₄

#5092

yes

characteristic

111

C₄

#5183

yes

characteristic

112

C₃

#12522

yes

characteristic, complemented, Sylow

113

C₂

#71362

yes

characteristic

114

C₂

#71480

yes

—

	subgroup	G/H
center	C₂²	#5151
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#2254	C₂
socle, prime radical	C₂²×C₃	#493
hypercenter	#2254	D₃
hypocenter	C₃	#12522
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#5151

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 ◁ C₄×C₈ ◁ #2254

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

H ≅ G/N

type

112. C₃

#12522

semidirect

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

subgroup	index
C₃×#2254	2
#71384	2
#71361	2
#69046	2
C₁₆×C₃⋊C₈	2
#12522	3

subgroup

index

C₁₆×C₃⋊C₈

Subgroups — #10709927

Conjugacy classes of subgroups

Complements

Maximal subgroups