Group #27194898 — subgroups

Subgroups — #27194898

Order 13184.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20267749	C₂	1	1	yes	maximal, characteristic
2	#20267747	C₂	1	2	yes	maximal
3	C₁₀₃⋊#338	C₂	1	1	yes	maximal, characteristic
4	#20267547	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13110089	C₂²	1	2	yes	—
7	C₂²×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
8	#15692980	C₂²	1	1	yes	characteristic
9	#15692932	C₄	1	4	yes	complemented
10	#15692931	C₄	1	2	yes	complemented
11	C₄×C₁₀₃⋊C₈	C₄	1	4	yes	complemented
12	C₈×Dic₁₀₃	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
14	#117×C₁₀₃	C₂²	1	2	yes	—
15	#13110089	—	2	2	no	—
16	#13110089	—	2	2	no	—
17	C₂×C₁₀₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₁₀₃⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₀₃⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₀₃⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₀₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₁₀₃	—	2	1	no	—
23	C₂²⋊Dic₁₀₃	—	2	1	no	—
24	C₄×Dic₁₀₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₀₃	—	2	2	no	—
27	C₂×C₈×C₁₀₃	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₁₀₃	—	2	2	no	—
29	C₂²⋊C₄×C₁₀₃	—	2	2	no	—
30	C₄²×C₁₀₃	C₂×C₄	1	4	yes	—
31	C₂³×C₁₀₃	#41	1	1	yes	characteristic
32	C₂×C₄×C₁₀₃	—	2	4	no	—
33	C₂×C₄×C₁₀₃	C₂²×C₄	1	2	yes	—
34	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₃	#41	1	2	yes	—
36	C₂×C₄×C₁₀₃	#41	1	1	yes	characteristic
37	C₂×Dic₁₀₃	—	2	2	no	—
38	C₈×C₁₀₃	—	2	2	no	—
39	C₁₀₃⋊C₈	—	2	4	no	—
40	C₁₀₃⋊C₈	C₄²	1	1	yes	characteristic
41	C₁₀₃⋊C₈	C₄²	1	1	yes	characteristic
42	C₁₀₃⋊C₈	C₄²	1	2	yes	—
43	C₂²×C₁₀₃	—	4	1	no	—
44	C₂²×C₁₀₃	#117	1	2	yes	—
45	C₂²×C₁₀₃	#117	1	1	yes	characteristic
46	C₄×C₁₀₃	—	4	4	no	—
47	C₄×C₁₀₃	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₀₃	#117	1	1	yes	characteristic
49	C₄×C₁₀₃	#117	1	2	yes	—
50	Dic₁₀₃	—	4	2	no	—
51	C₂×C₁₀₃	—	2	1	no	—
52	C₂×C₁₀₃	—	2	2	no	—
53	C₂×C₁₀₃	#376	1	1	yes	characteristic
54	#1747	—	103	1	no	maximal, Sylow
55	C₁₀₃	#1747	1	1	yes	characteristic, complemented, Sylow
56	#405	—	103	1	no	—
57	#405	—	103	2	no	—
58	#338	—	103	1	no	—
59	H(3,ℤ/4)	—	103	2	no	—
60	H(3,ℤ/4)	D₁₀₃	1	1	yes	characteristic
61	C₂×M₁₆	—	103	2	no	—
62	C₂×M₁₆	C₂×D₁₀₃	1	1	yes	characteristic
63	C₂²×C₈	—	103	1	no	—
64	#117	—	103	1	no	—
65	#117	C₂×D₁₀₃	1	2	yes	—
66	#97	—	103	2	no	—
67	#97	—	103	4	no	—
68	C₄×C₈	—	103	2	no	—
69	C₄×C₈	—	103	4	no	—
70	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	206	2	no	—
73	M₁₆	—	206	2	no	—
74	C₂×C₈	—	103	1	no	—
75	C₂×C₈	—	103	2	no	—
76	C₂×C₈	—	103	1	no	—
77	C₂×C₈	—	103	2	no	—
78	C₂×C₈	—	103	4	no	—
79	C₂×C₈	C₄×D₁₀₃	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	206	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	206	1	no	—
84	C₄²	—	103	2	no	—
85	C₄²	C₄×D₁₀₃	1	4	yes	—
86	C₂³	#13110104	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	206	2	no	—
89	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₁₀₃	1	2	yes	—
91	C₂×C₄	#13110104	1	2	yes	—
92	C₂×C₄	#13110106	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	103	1	no	—
95	C₈	—	103	1	no	—
96	C₈	—	103	2	no	—
97	C₈	—	206	4	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#15692977	1	2	yes	—
100	C₂²	#15692979	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	412	2	no	—
103	C₄	C₄²×D₁₀₃	1	1	yes	characteristic
104	C₄	#15692977	1	1	yes	characteristic
105	C₄	#15692979	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#20267636	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20267749

C₂

yes

maximal, characteristic

#20267747

C₂

yes

maximal

C₁₀₃⋊#338

C₂

yes

maximal, characteristic

#20267547

C₂

yes

maximal

H(3,ℤ/4)×C₁₀₃

C₂

yes

maximal, characteristic

C₂×#13110089

C₂²

yes

—

C₂²×C₁₀₃⋊C₈

C₂²

yes

characteristic

#15692980

C₂²

yes

characteristic

#15692932

C₄

yes

complemented

#15692931

C₄

yes

complemented

C₄×C₁₀₃⋊C₈

C₄

yes

complemented

C₈×Dic₁₀₃

C₄

yes

complemented

C₂×M₁₆×C₁₀₃

C₂²

yes

characteristic

#117×C₁₀₃

C₂²

yes

—

#13110089

—

#13110089

—

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

—

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

—

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₀₃

—

C₂²⋊Dic₁₀₃

—

C₄×Dic₁₀₃

C₂×C₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

M₁₆×C₁₀₃

—

C₂×C₈×C₁₀₃

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₃

—

C₂²⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

C₂×C₄

yes

—

C₂³×C₁₀₃

#41

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×C₄×C₁₀₃

#41

yes

characteristic

C₂×Dic₁₀₃

—

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₁₀₃⋊C₈

C₄²

yes

characteristic

C₁₀₃⋊C₈

C₄²

yes

characteristic

C₁₀₃⋊C₈

C₄²

yes

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#117

yes

—

C₂²×C₁₀₃

#117

yes

characteristic

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂×C₄²

yes

C₄×C₁₀₃

yes

C₄×C₁₀₃

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#376

yes

characteristic

#1747

—

103

maximal, Sylow

C₁₀₃

#1747

yes

characteristic, complemented, Sylow

#405

—

103

—

#405

—

103

—

#338

—

103

—

H(3,ℤ/4)

—

103

—

H(3,ℤ/4)

D₁₀₃

yes

characteristic

C₂×M₁₆

—

103

—

C₂×M₁₆

C₂×D₁₀₃

yes

characteristic

C₂²×C₈

—

103

—

#117

—

103

—

#117

C₂×D₁₀₃

yes

—

#97

—

103

—

#97

—

103

—

C₄×C₈

—

103

—

C₄×C₈

—

103

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

M₁₆

—

M₁₆

—

206

—

M₁₆

—

206

—

C₂×C₈

—

103

—

C₂×C₈

—

103

—

C₂×C₈

—

103

—

C₂×C₈

—

103

—

C₂×C₈

—

103

—

C₂×C₈

C₄×D₁₀₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

206

—

C₂²⋊C₄

—

C₂²⋊C₄

—

206

—

C₄²

—

103

—

C₄²

C₄×D₁₀₃

yes

—

C₂³

#13110104

yes

characteristic

C₂×C₄

—

C₂×C₄

—

206

—

C₂×C₄

C₂×C₄×D₁₀₃

yes

characteristic

C₂×C₄

C₂×C₄×D₁₀₃

yes

—

C₂×C₄

#13110104

yes

—

C₂×C₄

#13110106

yes

characteristic

C₈

—

C₈

—

103

—

C₈

—

103

—

C₈

—

103

—

C₈

—

206

—

C₂²

—

C₂²

#15692977

yes

—

100

C₂²

#15692979

yes

characteristic

101

C₄

—

102

C₄

—

412

—

103

C₄

C₄²×D₁₀₃

yes

characteristic

104

C₄

#15692977

yes

characteristic

105

C₄

#15692979

yes

—

106

C₂

—

107

C₂

—

108

C₂

#20267636

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₁₀₃
commutator	C₄×C₁₀₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	H(3,ℤ/4)×C₁₀₃	C₂
socle	C₂×C₁₀₃	#376
prime radical	C₂³×C₁₀₃	#41
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	H(3,ℤ/4)	D₁₀₃
hypocenter	C₁₀₃	#1747
LCS	G ▷ C₄×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₁₀₃

commutator

C₄×C₁₀₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

H(3,ℤ/4)×C₁₀₃

C₂

C₂×C₁₀₃

C₂³×C₁₀₃

C₂²×C₄×C₁₀₃

C₂³

D₁₀₃

C₁₀₃

G ▷ C₄×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #15692932	C₄	semidirect
10. #15692931	C₄	semidirect
11. C₄×C₁₀₃⋊C₈	C₄	semidirect
12. C₈×Dic₁₀₃	C₄	semidirect
55. C₁₀₃	#1747	semidirect

H ≅ G/N

type

9. #15692932

C₄

semidirect

10. #15692931

C₄

semidirect

11. C₄×C₁₀₃⋊C₈

C₄

semidirect

12. C₈×Dic₁₀₃

C₄

semidirect

55. C₁₀₃

#1747

semidirect

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

subgroup	index
#20267749	2
#20267747	2
C₁₀₃⋊#338	2
#20267547	2
H(3,ℤ/4)×C₁₀₃	2
#1747	103

subgroup

index

C₁₀₃⋊#338

H(3,ℤ/4)×C₁₀₃

103

Subgroups — #27194898

Conjugacy classes of subgroups

Complements

Maximal subgroups