Group #13056708 — subgroups

Subgroups — #13056708

Order 1344.

220 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 67 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×C₇×#127	C₂	1	1	yes	maximal, characteristic
2	#10621353	C₂	1	1	yes	maximal, characteristic, complemented
3	#10620671	C₂	1	4	yes	maximal, complemented
4	#10620642	C₂	1	1	yes	maximal, characteristic, complemented
5	#89843	—	3	1	no	maximal, Hall
6	C₂×C₃×C₇×D₄	—	2	2	no	—
7	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
8	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₁	C₂²	1	2	yes	—
10	#66629	—	2	2	no	—
11	C₂×C₃⋊(C₇⋊C₈)	C₂²	1	2	yes	—
12	C₇×#127	D₃	1	1	yes	characteristic
13	#6433	—	3	1	no	—
14	#6406	—	3	4	no	—
15	#6377	—	3	1	no	—
16	#5071	—	7	1	no	maximal, Hall
17	C₂³×C₃×C₇	—	2	2	no	—
18	C₃×C₇×D₄	—	2	4	no	—
19	C₃×C₇×D₄	—	4	4	no	—
20	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
21	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂×Dic₂₁	—	2	2	no	—
23	C₃⋊(C₇⋊C₈)	—	2	2	no	—
24	C₂×C₇×D₄	—	2	2	no	—
25	C₂×C₇×D₄	C₂×D₃	1	2	yes	—
26	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
27	C₄⋊Dic₇	—	3	2	no	—
28	Dic₇⋊C₄	—	6	2	no	—
29	C₂×C₇⋊C₈	—	3	2	no	—
30	C₃×#127	D₇	1	1	yes	characteristic
31	#870	—	7	1	no	—
32	#832	—	7	4	no	—
33	#803	—	7	1	no	—
34	C₂²×C₃×C₇	—	4	2	no	—
35	C₂²×C₃×C₇	—	4	4	no	—
36	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
37	C₃×C₄×C₇	—	2	2	no	—
38	C₃×C₄×C₇	#41	1	2	yes	—
39	Dic₂₁	—	4	2	no	—
40	#490	—	21	1	no	Sylow
41	C₂³×C₇	—	2	2	no	—
42	C₇×D₄	—	2	4	no	—
43	C₇×D₄	—	4	4	no	—
44	C₂×C₄×C₇	#67	1	2	yes	—
45	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
46	C₂×Dic₇	—	6	2	no	—
47	C₇⋊C₈	—	6	2	no	—
48	C₂×C₃×D₄	—	2	2	no	—
49	C₂×C₃×D₄	C₂×D₇	1	2	yes	—
50	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
51	C₄⋊Dic₃	—	7	2	no	—
52	Dic₃⋊C₄	—	14	2	no	—
53	C₂×C₃⋊C₈	—	7	2	no	—
54	C₂×C₃×C₇	—	8	2	no	—
55	C₂×C₃×C₇	#124	1	1	yes	characteristic
56	C₂×C₃×C₇	AGL(1,ℤ/8)	1	2	yes	—
57	#127	D₂₁	1	1	yes	characteristic
58	#125	—	21	1	no	—
59	#102	—	21	4	no	—
60	#97	—	21	1	no	—
61	C₂²×C₇	—	4	4	no	—
62	C₂²×C₇	—	4	2	no	—
63	C₂²×C₇	C₂×#67	1	1	yes	characteristic
64	C₄×C₇	—	2	2	no	—
65	C₄×C₇	#237	1	2	yes	—
66	Dic₇	—	12	2	no	—
67	C₂³×C₃	—	2	2	no	—
68	C₃×D₄	—	2	4	no	—
69	C₃×D₄	—	4	4	no	—
70	C₂×C₃×C₄	C₂²⋊D₇	1	2	yes	—
71	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
72	C₂×Dic₃	—	14	2	no	—
73	C₃⋊C₈	—	14	2	no	—
74	C₃×C₇	#490	1	1	yes	characteristic, complemented, Hall
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	C₂×D₂₁	1	2	yes	—
77	C₂×C₈	—	21	2	no	—
78	C₄⋊C₄	—	21	2	no	—
79	C₄⋊C₄	—	42	2	no	—
80	C₄²	C₂×D₂₁	1	1	yes	characteristic
81	C₂×C₇	—	8	2	no	—
82	C₂×C₇	#932	1	2	yes	—
83	C₂×C₇	#936	1	1	yes	characteristic
84	C₂²×C₃	—	4	4	no	—
85	C₂²×C₃	—	4	2	no	—
86	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
87	C₃×C₄	—	2	2	no	—
88	C₃×C₄	#1159	1	2	yes	—
89	Dic₃	—	28	2	no	—
90	C₂³	—	2	2	no	—
91	D₄	—	2	4	no	—
92	D₄	—	4	4	no	—
93	C₂×C₄	—	42	2	no	—
94	C₂×C₄	#4268	1	2	yes	—
95	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
96	C₈	—	42	2	no	—
97	C₇	#5071	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₃	—	8	2	no	—
99	C₂×C₃	#6495	1	2	yes	—
100	C₂×C₃	#6499	1	1	yes	characteristic
101	C₂²	—	4	4	no	—
102	C₂²	—	4	2	no	—
103	C₂²	C₂×#4268	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	84	2	no	—
106	C₄	#66730	1	2	yes	—
107	C₃	#89843	1	1	yes	characteristic, complemented, Sylow
108	C₂	—	8	2	no	—
109	C₂	#10621415	1	2	yes	—
110	C₂	#10621419	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₇×#127

C₂

yes

maximal, characteristic

#10621353

C₂

yes

maximal, characteristic, complemented

#10620671

C₂

yes

maximal, complemented

#10620642

C₂

yes

maximal, characteristic, complemented

#89843

—

maximal, Hall

C₂×C₃×C₇×D₄

—

C₂×C₃×C₇×D₄

C₂²

yes

—

C₃×C₄²×C₇

C₂²

yes

characteristic

C₄⋊Dic₂₁

C₂²

yes

—

#66629

—

C₂×C₃⋊(C₇⋊C₈)

C₂²

yes

—

C₇×#127

D₃

yes

—

—

—

—

C₂³×C₃×C₇

—

C₃×C₇×D₄

—

C₃×C₇×D₄

—

C₂×C₃×C₄×C₇

D₄

yes

—

C₂×C₃×C₄×C₇

C₂³

yes

characteristic

C₂×Dic₂₁

—

C₃⋊(C₇⋊C₈)

—

C₂×C₇×D₄

—

C₂×C₇×D₄

C₂×D₃

yes

—

C₄²×C₇

C₂×D₃

yes

characteristic

C₄⋊Dic₇

—

Dic₇⋊C₄

—

C₂×C₇⋊C₈

—

C₃×#127

D₇

yes

—

—

—

C₂²×C₃×C₇

—

C₂²×C₃×C₇

—

C₂²×C₃×C₇

C₂×D₄

yes

characteristic

C₃×C₄×C₇

—

C₃×C₄×C₇

#41

yes

—

Dic₂₁

—

#490

—

Sylow

C₂³×C₇

—

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

#67

yes

—

C₂×C₄×C₇

C₂²×D₃

yes

characteristic

C₂×Dic₇

—

C₇⋊C₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

C₂×D₇

yes

—

C₃×C₄²

C₂×D₇

yes

characteristic

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

—

C₂×C₃×C₇

—

C₂×C₃×C₇

#124

yes

characteristic

C₂×C₃×C₇

AGL(1,ℤ/8)

yes

—

#127

D₂₁

yes

—

—

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂×#67

yes

characteristic

C₄×C₇

—

C₄×C₇

#237

yes

—

Dic₇

—

C₂³×C₃

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

C₂²⋊D₇

yes

—

C₂×C₃×C₄

C₂²×D₇

yes

characteristic

C₂×Dic₃

—

C₃⋊C₈

—

C₃×C₇

#490

yes

characteristic, complemented, Hall

C₂×D₄

—

C₂×D₄

C₂×D₂₁

yes

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₂₁

yes

characteristic

C₂×C₇

—

C₂×C₇

#932

yes

—

C₂×C₇

#936

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₂²⋊D₇

yes

characteristic

C₃×C₄

—

C₃×C₄

#1159

yes

—

Dic₃

—

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#4268

yes

—

C₂×C₄

C₂²×D₂₁

yes

characteristic

C₈

—

C₇

#5071

yes

characteristic, complemented, Sylow

C₂×C₃

—

C₂×C₃

#6495

yes

—

100

C₂×C₃

#6499

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

C₂×#4268

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

#66730

yes

—

107

C₃

#89843

yes

characteristic, complemented, Sylow

108

C₂

—

109

C₂

#10621415

yes

—

110

C₂

#10621419

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#4268
commutator, cosocle	C₂×C₃×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₁
Fitting, prime radical	C₃×C₇×#127	C₂
socle	C₂²×C₃×C₇	C₂×D₄
hypercenter	#127	D₂₁
hypocenter	C₃×C₇	#490
LCS	G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

subgroup

G/H

center

C₂²

C₂×#4268

commutator, cosocle

C₂×C₃×C₄×C₇

C₂³

Frattini

C₂×C₄

C₂²×D₂₁

Fitting, prime radical

C₃×C₇×#127

C₂

socle

C₂²×C₃×C₇

C₂×D₄

D₂₁

C₃×C₇

G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #127

N	H ≅ G/N	type
2. #10621353	C₂	semidirect
3. #10620671	C₂	semidirect
4. #10620642	C₂	semidirect
74. C₃×C₇	#490	semidirect
97. C₇	#5071	semidirect
107. C₃	#89843	semidirect

H ≅ G/N

type

2. #10621353

C₂

semidirect

3. #10620671

C₂

semidirect

4. #10620642

C₂

semidirect

74. C₃×C₇

#490

semidirect

97. C₇

#5071

semidirect

107. C₃

#89843

semidirect

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

subgroup	index
C₃×C₇×#127	2
#10621353	2
#10620671	2
#10620642	2
#89843	3
#5071	7

subgroup

index

C₃×C₇×#127

Subgroups — #13056708

Conjugacy classes of subgroups

Complements

Maximal subgroups