Group #10716161 — subgroups

Subgroups — #10716161

Order 768.

182 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 75 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×#2252	C₂	1	1	yes	maximal, characteristic
2	#69066	C₂	1	1	yes	maximal, characteristic
3	#67578	C₂	1	4	yes	maximal
4	C₄×C₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
5	#13720	—	3	1	no	maximal, Sylow
6	C₃×#500	—	2	1	no	—
7	C₂×Dic₂₄	—	2	1	no	—
8	C₂×Dic₂₄	C₂²	1	2	yes	—
9	#4699	—	2	1	no	—
10	C₃×#366	—	2	1	no	—
11	C₃×#366	C₂²	1	2	yes	—
12	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊C₃₂	C₂²	1	2	yes	—
14	#2252	D₃	1	1	yes	characteristic
15	#2247	—	3	1	no	—
16	#1416	—	3	4	no	—
17	C₄×C₃₂	—	3	1	no	—
18	C₃×C₄⋊Q₈	—	4	1	no	—
19	C₂×Dic₁₂	—	2	2	no	—
20	C₂×Dic₁₂	—	4	1	no	—
21	#869	—	4	1	no	—
22	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
23	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
24	C₃×#107	—	2	2	no	—
25	C₃×#107	—	4	1	no	—
26	C₃×C₄×C₈	D₄	1	1	yes	characteristic
27	Dic₂₄	—	2	4	no	—
28	Dic₂₄	—	4	2	no	—
29	C₃⋊C₃₂	—	2	2	no	—
30	C₂×Q₃₂	—	3	2	no	—
31	C₂×Q₃₂	—	6	1	no	—
32	#500	—	2	1	no	—
33	#494	—	6	1	no	—
34	C₂×C₃₂	—	3	2	no	—
35	#366	—	2	1	no	—
36	#366	C₂×D₃	1	2	yes	—
37	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
38	C₂×C₃×Q₈	—	8	1	no	—
39	C₂×Dic₆	—	4	2	no	—
40	C₂×C₃×C₈	D₈	1	1	yes	characteristic
41	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
42	C₃×C₄⋊C₄	—	4	2	no	—
43	C₃×C₄⋊C₄	—	8	1	no	—
44	C₃×C₄²	D₈	1	1	yes	characteristic
45	C₄⋊Dic₃	—	8	1	no	—
46	C₄⋊Dic₃	—	8	1	no	—
47	Dic₁₂	—	4	2	no	—
48	Dic₁₂	—	4	2	no	—
49	Dic₁₂	—	8	2	no	—
50	C₃×C₁₆	SD₁₆	1	2	yes	—
51	C₃×C₁₆	#41	1	2	yes	—
52	C₂×Q₁₆	—	6	2	no	—
53	C₂×Q₁₆	—	12	1	no	—
54	C₄⋊Q₈	—	4	1	no	—
55	C₄⋊Q₈	—	12	1	no	—
56	Q₃₂	—	6	4	no	—
57	Q₃₂	—	12	2	no	—
58	C₂×C₁₆	#67	1	1	yes	characteristic
59	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
60	#107	—	2	2	no	—
61	#107	—	4	1	no	—
62	C₄×C₈	#67	1	1	yes	characteristic
63	C₃₂	—	6	2	no	—
64	C₃×Q₈	—	16	2	no	—
65	C₂×C₃×C₄	—	8	2	no	—
66	C₂×C₃×C₄	D₁₆	1	1	yes	characteristic
67	C₂×C₃×C₄	D₁₆	1	1	yes	characteristic
68	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
69	C₂×Dic₃	—	8	2	no	—
70	Dic₆	—	8	2	no	—
71	Dic₆	—	8	2	no	—
72	C₃×C₈	SD₃₂	1	2	yes	—
73	C₃×C₈	#124	1	1	yes	characteristic
74	C₃×C₈	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×Q₈	—	8	1	no	—
76	C₂×Q₈	—	12	2	no	—
77	Q₁₆	—	12	2	no	—
78	Q₁₆	—	12	2	no	—
79	Q₁₆	—	24	2	no	—
80	C₂×C₈	#213	1	1	yes	characteristic
81	C₂×C₈	C₂×#67	1	1	yes	characteristic
82	C₄⋊C₄	—	4	2	no	—
83	C₄⋊C₄	—	8	1	no	—
84	C₄⋊C₄	—	24	1	no	—
85	C₄⋊C₄	—	24	1	no	—
86	C₄²	#213	1	1	yes	characteristic
87	C₁₆	#215	1	2	yes	—
88	C₁₆	#239	1	2	yes	—
89	C₂²×C₃	C₂×D₁₆	1	1	yes	characteristic
90	C₃×C₄	—	16	2	no	—
91	C₃×C₄	SD₆₄	1	2	yes	—
92	C₃×C₄	Q₆₄	1	2	yes	—
93	C₃×C₄	#486	1	1	yes	characteristic
94	C₃×C₄	C₂×SD₃₂	1	1	yes	characteristic
95	Dic₃	—	16	2	no	—
96	Q₈	—	16	2	no	—
97	Q₈	—	24	2	no	—
98	Q₈	—	24	2	no	—
99	C₂×C₄	—	8	2	no	—
100	C₂×C₄	—	24	2	no	—
101	C₂×C₄	#826	1	1	yes	characteristic
102	C₂×C₄	#826	1	1	yes	characteristic
103	C₂×C₄	C₂×#213	1	1	yes	characteristic
104	C₈	#828	1	2	yes	—
105	C₈	C₂×#215	1	1	yes	characteristic
106	C₈	#947	1	1	yes	characteristic
107	C₂×C₃	#2240	1	1	yes	characteristic
108	C₂×C₃	C₂×SD₆₄	1	1	yes	characteristic
109	C₂×C₃	C₂×Q₆₄	1	1	yes	characteristic
110	C₂²	C₂×#826	1	1	yes	characteristic
111	C₄	—	16	2	no	—
112	C₄	—	48	2	no	—
113	C₄	#4520	1	2	yes	—
114	C₄	#4522	1	2	yes	—
115	C₄	#5088	1	1	yes	characteristic
116	C₄	C₂×#828	1	1	yes	characteristic
117	C₃	#13720	1	1	yes	characteristic, complemented, Sylow
118	C₂	#71381	1	1	yes	characteristic
119	C₂	C₂×#4520	1	1	yes	characteristic
120	C₂	C₂×#4522	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#2252

C₂

yes

maximal, characteristic

#69066

C₂

yes

maximal, characteristic

#67578

C₂

yes

maximal

C₄×C₃⋊C₃₂

C₂

yes

maximal, characteristic

#13720

—

maximal, Sylow

C₃×#500

—

C₂×Dic₂₄

—

C₂×Dic₂₄

C₂²

yes

—

#4699

—

C₃×#366

—

C₃×#366

C₂²

yes

—

C₃×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₃⋊C₃₂

C₂²

yes

—

#2252

D₃

yes

characteristic

#2247

—

#1416

—

C₄×C₃₂

—

C₃×C₄⋊Q₈

—

C₂×Dic₁₂

—

C₂×Dic₁₂

—

#869

—

C₂×C₃×C₁₆

D₄

yes

characteristic

C₂×C₃×C₁₆

C₂³

yes

characteristic

C₃×#107

—

C₃×#107

—

C₃×C₄×C₈

D₄

yes

characteristic

Dic₂₄

—

Dic₂₄

—

C₃⋊C₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

#500

—

#494

—

C₂×C₃₂

—

#366

—

#366

C₂×D₃

yes

—

C₄×C₁₆

C₂×D₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×C₃×C₈

D₈

yes

characteristic

C₂×C₃×C₈

C₂×D₄

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

D₈

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₁₂

—

Dic₁₂

—

Dic₁₂

—

C₃×C₁₆

SD₁₆

yes

—

C₃×C₁₆

#41

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

#67

yes

characteristic

C₂×C₁₆

C₂²×D₃

yes

characteristic

#107

—

#107

—

C₄×C₈

#67

yes

characteristic

C₃₂

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₁₆

yes

characteristic

C₂×C₃×C₄

D₁₆

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

characteristic

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

SD₃₂

yes

—

C₃×C₈

#124

yes

characteristic

C₃×C₈

C₂×SD₁₆

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

#213

yes

characteristic

C₂×C₈

C₂×#67

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

#213

yes

characteristic

C₁₆

#215

yes

—

C₁₆

#239

yes

—

C₂²×C₃

C₂×D₁₆

yes

characteristic

C₃×C₄

—

C₃×C₄

SD₆₄

yes

—

C₃×C₄

Q₆₄

yes

—

C₃×C₄

#486

yes

characteristic

C₃×C₄

C₂×SD₃₂

yes

characteristic

Dic₃

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#826

yes

characteristic

102

C₂×C₄

#826

yes

characteristic

103

C₂×C₄

C₂×#213

yes

characteristic

104

C₈

#828

yes

—

105

C₈

C₂×#215

yes

characteristic

106

C₈

#947

yes

characteristic

107

C₂×C₃

#2240

yes

characteristic

108

C₂×C₃

C₂×SD₆₄

yes

characteristic

109

C₂×C₃

C₂×Q₆₄

yes

characteristic

110

C₂²

C₂×#826

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

#4520

yes

—

114

C₄

#4522

yes

—

115

C₄

#5088

yes

characteristic

116

C₄

C₂×#828

yes

characteristic

117

C₃

#13720

yes

characteristic, complemented, Sylow

118

C₂

#71381

yes

characteristic

119

C₂

C₂×#4520

yes

characteristic

120

C₂

C₂×#4522

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#826
commutator, cosocle	C₂×C₃×C₁₆	C₂³
Frattini	C₂×C₁₆	C₂²×D₃
Fitting	C₃×#2252	C₂
socle, prime radical	C₂²×C₃	C₂×D₁₆
hypercenter	#2252	D₃
hypocenter	C₃	#13720
LCS	G ▷ C₂×C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

subgroup

G/H

center

C₂²

C₂×#826

commutator, cosocle

C₂×C₃×C₁₆

C₂³

Frattini

C₂×C₁₆

C₂²×D₃

Fitting

C₃×#2252

C₂

socle, prime radical

C₂²×C₃

C₂×D₁₆

D₃

C₃

G ▷ C₂×C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

N	H ≅ G/N	type
117. C₃	#13720	semidirect

H ≅ G/N

type

117. C₃

#13720

semidirect

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

subgroup	index
C₃×#2252	2
#69066	2
#67578	2
C₄×C₃⋊C₃₂	2
#13720	3

subgroup

index

C₄×C₃⋊C₃₂

Subgroups — #10716161

Conjugacy classes of subgroups

Complements

Maximal subgroups