Group #10709929 — subgroups

Subgroups — #10709929

Order 768.

262 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 103 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×#1713	C₂	1	1	yes	maximal, characteristic
2	#71385	C₂	1	4	yes	maximal
3	#68065	C₂	1	1	yes	maximal, characteristic
4	C₈×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#12523	—	3	1	no	maximal, Sylow
6	C₃×#500	—	2	1	no	—
7	C₃×#500	C₂²	1	2	yes	—
8	C₃×#494	—	2	1	no	—
9	#4699	—	2	1	no	—
10	#4682	—	2	1	no	—
11	#4682	C₂²	1	2	yes	—
12	C₃×C₈²	C₂²	1	1	yes	characteristic
13	#4490	—	2	8	no	—
14	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
15	#2254	—	3	4	no	—
16	#1713	—	3	1	no	—
17	#1713	D₃	1	1	yes	characteristic
18	C₈×C₁₆	—	3	1	no	—
19	C₂×C₃×Q₁₆	—	4	2	no	—
20	C₃×C₄⋊Q₈	—	2	2	no	—
21	C₂×Dic₁₂	—	4	2	no	—
22	#869	—	2	2	no	—
23	C₃×#107	—	2	4	no	—
24	C₃×#107	—	4	2	no	—
25	C₃×C₄×C₈	D₄	1	1	yes	characteristic
26	C₃×C₄×C₈	D₄	1	1	yes	characteristic
27	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
28	#818	—	2	4	no	—
29	#818	—	4	2	no	—
30	C₂×C₃⋊C₁₆	D₄	1	4	yes	—
31	#500	—	2	1	no	—
32	#500	—	3	2	no	—
33	#500	—	6	1	no	—
34	#500	C₂×D₃	1	2	yes	—
35	#494	—	2	1	no	—
36	#494	—	6	1	no	—
37	#367	—	6	8	no	—
38	C₄×C₁₆	—	3	2	no	—
39	C₈²	C₂×D₃	1	1	yes	characteristic
40	C₂×C₃×Q₈	—	4	2	no	—
41	C₂×Dic₆	—	4	2	no	—
42	C₃×Q₁₆	—	8	4	no	—
43	C₂×C₃×C₈	D₈	1	2	yes	—
44	C₂×C₃×C₈	D₈	1	2	yes	—
45	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
46	C₃×C₄⋊C₄	—	4	2	no	—
47	C₃×C₄⋊C₄	—	4	2	no	—
48	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
49	C₄⋊Dic₃	—	4	2	no	—
50	C₄⋊Dic₃	—	4	2	no	—
51	Dic₁₂	—	8	4	no	—
52	C₃⋊C₁₆	Q₁₆	1	8	yes	complemented
53	C₂×Q₁₆	—	4	2	no	—
54	C₂×Q₁₆	—	12	2	no	—
55	C₄⋊Q₈	—	2	2	no	—
56	C₄⋊Q₈	—	6	2	no	—
57	C₂×C₁₆	—	3	4	no	—
58	#107	—	2	4	no	—
59	#107	—	4	2	no	—
60	#107	—	6	4	no	—
61	#107	—	12	2	no	—
62	C₄×C₈	#67	1	1	yes	characteristic
63	C₄×C₈	#67	1	1	yes	characteristic
64	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
65	C₃×Q₈	—	8	4	no	—
66	C₂×C₃×C₄	—	8	2	no	—
67	C₂×C₃×C₄	#127	1	1	yes	characteristic
68	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
69	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
70	C₂×Dic₃	—	8	2	no	—
71	Dic₆	—	8	4	no	—
72	C₃×C₈	SD₃₂	1	4	yes	—
73	C₃×C₈	SD₃₂	1	4	yes	—
74	C₃×C₈	C₂×Q₁₆	1	4	yes	—
75	C₂×Q₈	—	4	2	no	—
76	C₂×Q₈	—	12	2	no	—
77	Q₁₆	—	8	4	no	—
78	Q₁₆	—	24	4	no	—
79	C₂×C₈	#213	1	2	yes	—
80	C₂×C₈	#213	1	2	yes	—
81	C₂×C₈	D₃×D₄	1	2	yes	—
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	C₂×#67	1	1	yes	characteristic
87	C₁₆	—	3	8	no	—
88	C₂²×C₃	#493	1	1	yes	characteristic
89	C₃×C₄	—	16	2	no	—
90	C₃×C₄	#494	1	2	yes	—
91	C₃×C₄	C₂×SD₃₂	1	2	yes	—
92	C₃×C₄	C₂×SD₃₂	1	2	yes	—
93	Dic₃	—	16	2	no	—
94	Q₈	—	8	4	no	—
95	Q₈	—	24	4	no	—
96	C₂×C₄	—	8	2	no	—
97	C₂×C₄	—	24	2	no	—
98	C₂×C₄	C₂×#213	1	1	yes	characteristic
99	C₂×C₄	C₂×#213	1	1	yes	characteristic
100	C₂×C₄	#940	1	1	yes	characteristic
101	C₈	#827	1	4	yes	—
102	C₈	#828	1	4	yes	—
103	C₈	D₃×Q₁₆	1	4	yes	—
104	C₂×C₃	#1713	1	1	yes	characteristic
105	C₂×C₃	#2248	1	1	yes	characteristic
106	C₂×C₃	#2248	1	1	yes	characteristic
107	C₂²	#5073	1	1	yes	characteristic
108	C₄	—	16	2	no	—
109	C₄	—	48	2	no	—
110	C₄	C₂×#827	1	2	yes	—
111	C₄	C₂×#828	1	2	yes	—
112	C₄	#5183	1	2	yes	—
113	C₃	#12523	1	1	yes	characteristic, complemented, Sylow
114	C₂	#71291	1	1	yes	characteristic
115	C₂	#71298	1	1	yes	characteristic
116	C₂	#71313	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#1713

C₂

yes

maximal, characteristic

#71385

C₂

yes

maximal

#68065

C₂

yes

maximal, characteristic

C₈×C₃⋊C₁₆

C₂

yes

maximal, characteristic

—

—

C₂²

yes

—

—

—

—

C₂²

yes

—

C₃×C₈²

C₂²

yes

characteristic

#4490

—

C₄×C₃⋊C₁₆

C₂²

yes

—

#2254

—

#1713

—

#1713

D₃

yes

characteristic

C₈×C₁₆

—

C₂×C₃×Q₁₆

—

C₃×C₄⋊Q₈

—

C₂×Dic₁₂

—

#869

—

C₃×#107

—

C₃×#107

—

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

C₂³

yes

characteristic

#818

—

#818

—

C₂×C₃⋊C₁₆

D₄

yes

—

—

—

—

C₂×D₃

yes

—

#494

—

#494

—

#367

—

C₄×C₁₆

—

C₈²

C₂×D₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₁₂

—

C₃⋊C₁₆

Q₁₆

yes

complemented

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

—

—

—

—

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

C₂²×D₃

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

characteristic

C₂×Dic₃

—

Dic₆

—

C₃×C₈

SD₃₂

yes

—

C₃×C₈

SD₃₂

yes

—

C₃×C₈

C₂×Q₁₆

yes

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

#213

yes

—

C₂×C₈

#213

yes

—

C₂×C₈

D₃×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×#67

yes

characteristic

C₁₆

—

C₂²×C₃

#493

yes

characteristic

C₃×C₄

—

C₃×C₄

#494

yes

—

C₃×C₄

C₂×SD₃₂

yes

—

C₃×C₄

C₂×SD₃₂

yes

—

Dic₃

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#940

yes

characteristic

101

C₈

#827

yes

—

102

C₈

#828

yes

—

103

C₈

D₃×Q₁₆

yes

—

104

C₂×C₃

#1713

yes

characteristic

105

C₂×C₃

#2248

yes

characteristic

106

C₂×C₃

#2248

yes

characteristic

107

C₂²

#5073

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

C₂×#827

yes

—

111

C₄

C₂×#828

yes

—

112

C₄

#5183

yes

—

113

C₃

#12523

yes

characteristic, complemented, Sylow

114

C₂

#71291

yes

characteristic

115

C₂

#71298

yes

characteristic

116

C₂

#71313

yes

characteristic

	subgroup	G/H
center	C₂²	#5073
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#1713	C₂
socle, prime radical	C₂²×C₃	#493
hypercenter	#1713	D₃
hypocenter	C₃	#12523
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

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 ◁ #1713

N	H ≅ G/N	type
52. C₃⋊C₁₆	Q₁₆	semidirect
113. C₃	#12523	semidirect

H ≅ G/N

type

52. C₃⋊C₁₆

Q₁₆

semidirect

113. C₃

#12523

semidirect

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

subgroup	index
C₃×#1713	2
#71385	2
#68065	2
C₈×C₃⋊C₁₆	2
#12523	3

subgroup

index

C₈×C₃⋊C₁₆

Subgroups — #10709929

Conjugacy classes of subgroups

Complements

Maximal subgroups