Group #10706826 — subgroups

Subgroups — #10706826

Order 768.

216 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 115 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×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#69732	C₂	1	2	yes	maximal
3	#67513	C₂	1	4	yes	maximal, complemented
4	#12021	—	3	1	no	maximal, Sylow
5	C₂×C₃×M₃₂	C₄	1	2	yes	—
6	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
7	#4926	—	2	2	no	—
8	C₃×#346	C₂²	1	2	yes	—
9	C₃×C₄×C₁₆	C₄	1	1	yes	characteristic
10	C₃×C₄×C₁₆	C₄	1	1	yes	characteristic
11	#4458	—	2	2	no	—
12	#4457	C₂²	1	4	yes	—
13	C₄×M₃₂	D₃	1	1	yes	characteristic
14	#1568	—	3	2	no	—
15	#1368	—	3	4	no	—
16	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₃×C₈	Q₈	1	1	yes	characteristic
18	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
19	C₃×M₃₂	—	2	4	no	—
20	C₂×C₃×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₃×C₁₆	C₂×C₄	1	2	yes	—
22	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₃×C₄×C₈	D₄	1	1	yes	characteristic
24	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
25	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
26	#830	—	4	2	no	—
27	C₄⋊(C₃⋊C₈)	—	2	4	no	—
28	#803	—	4	2	no	—
29	C₂×M₃₂	Dic₃	1	2	yes	—
30	#432	—	6	2	no	—
31	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
32	#346	C₂×D₃	1	2	yes	—
33	C₄×C₁₆	Dic₃	1	1	yes	characteristic
34	C₄×C₁₆	Dic₃	1	1	yes	characteristic
35	#335	—	6	2	no	—
36	#334	—	3	4	no	—
37	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₃×C₄	SD₁₆	1	2	yes	—
39	C₂×C₃×C₈	—	2	2	no	—
40	C₂×C₃×C₈	—	2	2	no	—
41	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
45	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
46	C₃×C₄²	SD₁₆	1	2	yes	—
47	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
48	C₂×C₃⋊C₈	—	4	4	no	—
49	C₃×C₁₆	—	2	2	no	—
50	C₃×C₁₆	M₁₆	1	4	yes	—
51	C₂²×C₈	Dic₆	1	1	yes	characteristic
52	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
53	C₂×C₄²	D₁₂	1	1	yes	characteristic
54	M₃₂	—	2	4	no	—
55	C₂×C₁₆	C₂×Dic₃	1	2	yes	—
56	C₂×C₁₆	C₂×Dic₃	1	2	yes	—
57	C₄⋊C₈	—	6	4	no	—
58	C₂²⋊C₈	—	12	2	no	—
59	#97	—	12	2	no	—
60	C₄×C₈	Dic₆	1	1	yes	characteristic
61	C₄×C₈	D₁₂	1	1	yes	characteristic
62	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
63	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
64	C₂³×C₃	#106	1	1	yes	characteristic
65	C₂×C₃×C₄	—	2	1	no	—
66	C₂×C₃×C₄	—	2	1	no	—
67	C₂×C₃×C₄	—	4	2	no	—
68	C₂×C₃×C₄	#106	1	2	yes	—
69	C₂×C₃×C₄	#106	1	1	yes	characteristic
70	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
72	C₃×C₈	—	2	1	no	—
73	C₃×C₈	—	2	1	no	—
74	C₃×C₈	#108	1	2	yes	—
75	C₃×C₈	C₂×M₁₆	1	2	yes	—
76	C₃⋊C₈	—	8	4	no	—
77	C₂²×C₄	#204	1	2	yes	—
78	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	12	4	no	—
82	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
83	C₂×C₈	C₄⋊Dic₃	1	1	yes	characteristic
84	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
85	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
86	C₄²	#204	1	2	yes	—
87	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
88	C₄²	C₂×D₁₂	1	1	yes	characteristic
89	C₁₆	—	2	2	no	—
90	C₁₆	#208	1	4	yes	—
91	C₂²×C₃	—	2	1	no	—
92	C₂²×C₃	—	2	2	no	—
93	C₂²×C₃	C₂×#106	1	1	yes	characteristic
94	C₃×C₄	—	2	1	no	—
95	C₃×C₄	—	2	2	no	—
96	C₃×C₄	—	2	2	no	—
97	C₃×C₄	#423	1	1	yes	characteristic
98	C₃×C₄	C₂×#108	1	1	yes	characteristic
99	C₂³	#817	1	1	yes	characteristic
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	#817	1	1	yes	characteristic
104	C₂×C₄	#817	1	2	yes	—
105	C₂×C₄	C₂×#204	1	2	yes	—
106	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	2	1	no	—
109	C₈	—	24	4	no	—
110	C₈	#819	1	2	yes	—
111	C₈	C₂×#208	1	2	yes	—
112	C₂×C₃	—	4	1	no	—
113	C₂×C₃	#1566	1	1	yes	characteristic
114	C₂×C₃	#2155	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	—	2	1	no	—
117	C₂²	C₂×#817	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	1	no	—
121	C₄	#4924	1	1	yes	characteristic
122	C₄	C₂×#819	1	1	yes	characteristic
123	C₃	#12021	1	1	yes	characteristic, complemented, Sylow
124	C₂	—	4	1	no	—
125	C₂	#69730	1	1	yes	characteristic
126	C₂	#71427	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₄×M₃₂

C₂

yes

maximal, characteristic

#69732

C₂

yes

maximal

#67513

C₂

yes

maximal, complemented

#12021

—

maximal, Sylow

C₂×C₃×M₃₂

C₄

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

#4926

—

C₃×#346

C₂²

yes

—

C₃×C₄×C₁₆

C₄

yes

characteristic

C₃×C₄×C₁₆

C₄

yes

characteristic

#4458

—

#4457

C₂²

yes

—

C₄×M₃₂

D₃

yes

characteristic

#1568

—

#1368

—

C₂²×C₃×C₈

C₂×C₄

yes

characteristic

C₂²×C₃×C₈

Q₈

yes

characteristic

C₂×C₃×C₄²

D₄

yes

characteristic

C₃×M₃₂

—

C₂×C₃×C₁₆

C₂×C₄

yes

—

C₂×C₃×C₁₆

C₂×C₄

yes

—

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

Q₈

yes

characteristic

C₃×C₄×C₈

C₂³

yes

characteristic

#830

—

C₄⋊(C₃⋊C₈)

—

#803

—

C₂×M₃₂

Dic₃

yes

—

#432

—

C₂×C₄×C₈

C₂×D₃

yes

characteristic

#346

C₂×D₃

yes

—

C₄×C₁₆

Dic₃

yes

characteristic

C₄×C₁₆

Dic₃

yes

characteristic

#335

—

#334

—

C₂²×C₃×C₄

C₄⋊C₄

yes

characteristic

C₂²×C₃×C₄

SD₁₆

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄⋊C₄

yes

characteristic

C₂×C₃×C₈

C₄⋊C₄

yes

characteristic

C₂×C₃×C₈

C₂²×C₄

yes

characteristic

C₂×C₃×C₈

C₂×Q₈

yes

characteristic

C₃×C₄²

C₄⋊C₄

yes

characteristic

C₃×C₄²

SD₁₆

yes

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₃×C₁₆

M₁₆

yes

—

C₂²×C₈

Dic₆

yes

characteristic

C₂²×C₈

C₂×Dic₃

yes

characteristic

C₂×C₄²

D₁₂

yes

characteristic

M₃₂

—

C₂×C₁₆

C₂×Dic₃

yes

—

C₂×C₁₆

C₂×Dic₃

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

Dic₆

yes

characteristic

C₄×C₈

D₁₂

yes

characteristic

C₄×C₈

C₂×Dic₃

yes

characteristic

C₄×C₈

C₂²×D₃

yes

characteristic

C₂³×C₃

#106

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#106

yes

—

C₂×C₃×C₄

#106

yes

characteristic

C₂×C₃×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

C₂×SD₁₆

yes

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

#108

yes

—

C₃×C₈

C₂×M₁₆

yes

—

C₃⋊C₈

—

C₂²×C₄

#204

yes

—

C₂²×C₄

C₄⋊Dic₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₃

yes

characteristic

C₂×C₈

C₄⋊Dic₃

yes

characteristic

C₂×C₈

C₂×Dic₆

yes

characteristic

C₂×C₈

C₂²×Dic₃

yes

characteristic

C₄²

#204

yes

—

C₄²

C₄⋊Dic₃

yes

characteristic

C₄²

C₂×D₁₂

yes

characteristic

C₁₆

—

C₁₆

#208

yes

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×#106

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

C₂³

yes

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#817

yes

characteristic

104

C₂×C₄

#817

yes

—

105

C₂×C₄

C₂×#204

yes

—

106

C₂×C₄

C₂×C₄⋊Dic₃

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

#819

yes

—

111

C₈

C₂×#208

yes

—

112

C₂×C₃

—

113

C₂×C₃

#1566

yes

characteristic

114

C₂×C₃

#2155

yes

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂×#817

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#4924

yes

characteristic

122

C₄

C₂×#819

yes

characteristic

123

C₃

#12021

yes

characteristic, complemented, Sylow

124

C₂

—

125

C₂

#69730

yes

characteristic

126

C₂

#71427

yes

—

	subgroup	G/H
center	C₂²	C₂×#817
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×C₄×M₃₂	C₂
socle	C₂²×C₃	C₂×#106
prime radical	C₂³×C₃	#106
cosocle	C₃×C₄×C₈	C₂³
hypercenter	C₄×M₃₂	D₃
hypocenter	C₃	#12021
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

subgroup

G/H

center

C₂²

C₂×#817

commutator

C₂×C₃×C₈

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₃

Fitting

C₃×C₄×M₃₂

C₂

C₂²×C₃

C₂³×C₃

C₃×C₄×C₈

C₂³

hypercenter

C₄×M₃₂

D₃

hypocenter

C₃

#12021

LCS

G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

N	H ≅ G/N	type
3. #67513	C₂	semidirect
123. C₃	#12021	semidirect

H ≅ G/N

type

3. #67513

C₂

semidirect

123. C₃

#12021

semidirect

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

subgroup	index
C₃×C₄×M₃₂	2
#69732	2
#67513	2
#12021	3

subgroup

index

C₃×C₄×M₃₂

#69732

#67513

#12021

Subgroups — #10706826

Conjugacy classes of subgroups

Complements

Maximal subgroups