Group #27194899 — subgroups

Subgroups — #27194899

Order 13184.

200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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	#20267746	C₂	1	2	yes	maximal
2	C₁₀₃⋊#338	C₂	1	1	yes	maximal, characteristic
3	#20267548	C₂	1	1	yes	maximal, characteristic
4	#20267547	C₂	1	2	yes	maximal
5	#405×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13110089	C₂²	1	2	yes	—
7	#15692980	C₂²	1	2	yes	—
8	#15692931	C₄	1	4	yes	complemented
9	C₈×Dic₁₀₃	C₄	1	4	yes	complemented
10	C₂×M₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
12	#117×C₁₀₃	C₂²	1	1	yes	characteristic
13	#97×C₁₀₃	C₄	1	2	yes	complemented
14	C₄×C₈×C₁₀₃	C₄	1	2	yes	complemented
15	#13110089	—	2	2	no	—
16	#13110089	—	2	2	no	—
17	C₂×C₁₀₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₁₀₃	—	2	2	no	—
19	C₂²⋊Dic₁₀₃	—	2	2	no	—
20	C₄×Dic₁₀₃	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₀₃	—	2	2	no	—
23	C₂×C₈×C₁₀₃	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₀₃	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₁₀₃	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₀₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₀₃	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₃	—	2	1	no	—
29	C₂²⋊C₄×C₁₀₃	—	2	1	no	—
30	C₄²×C₁₀₃	C₂×C₄	1	2	yes	—
31	C₂³×C₁₀₃	#41	1	1	yes	characteristic
32	C₂×C₄×C₁₀₃	—	2	2	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	1	yes	characteristic
36	C₂×C₄×C₁₀₃	#41	1	2	yes	—
37	C₂×Dic₁₀₃	—	2	4	no	—
38	C₈×C₁₀₃	—	2	2	no	—
39	C₈×C₁₀₃	C₄²	1	2	yes	—
40	C₈×C₁₀₃	C₄²	1	1	yes	characteristic
41	C₈×C₁₀₃	C₄²	1	1	yes	characteristic
42	C₁₀₃⋊C₈	—	2	4	no	—
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	2	no	—
47	C₄×C₁₀₃	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₀₃	#117	1	2	yes	—
49	C₄×C₁₀₃	#117	1	1	yes	characteristic
50	Dic₁₀₃	—	4	4	no	—
51	C₂×C₁₀₃	—	2	2	no	—
52	C₂×C₁₀₃	—	2	1	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	2	no	—
57	#405	D₁₀₃	1	1	yes	characteristic
58	#338	—	103	1	no	—
59	H(3,ℤ/4)	—	103	1	no	—
60	H(3,ℤ/4)	—	103	2	no	—
61	C₂×M₁₆	—	103	2	no	—
62	C₂×M₁₆	C₂×D₁₀₃	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
64	#117	—	103	2	no	—
65	#117	C₂×D₁₀₃	1	1	yes	characteristic
66	#97	—	103	4	no	—
67	#97	Dic₁₀₃	1	2	yes	complemented
68	C₄×C₈	—	103	4	no	—
69	C₄×C₈	Dic₁₀₃	1	2	yes	complemented
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	4	no	—
75	C₂×C₈	C₄×D₁₀₃	1	2	yes	—
76	C₂×C₈	C₄×D₁₀₃	1	2	yes	—
77	C₂×C₈	C₂×Dic₁₀₃	1	2	yes	—
78	C₂×C₈	C₂×Dic₁₀₃	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₁₀₃	1	1	yes	characteristic
80	C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₄	—	206	2	no	—
82	C₂²⋊C₄	—	2	1	no	—
83	C₂²⋊C₄	—	206	2	no	—
84	C₄²	—	103	4	no	—
85	C₄²	C₂×Dic₁₀₃	1	2	yes	—
86	C₂³	#13110105	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	206	4	no	—
89	C₂×C₄	C₂×C₄×D₁₀₃	1	2	yes	—
90	C₂×C₄	C₂²×Dic₁₀₃	1	1	yes	characteristic
91	C₂×C₄	#13110105	1	1	yes	characteristic
92	C₂×C₄	#13110105	1	2	yes	—
93	C₈	—	2	2	no	—
94	C₈	—	206	4	no	—
95	C₈	C₄×Dic₁₀₃	1	1	yes	characteristic
96	C₈	C₄×Dic₁₀₃	1	2	yes	—
97	C₈	C₄×Dic₁₀₃	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	#15692978	1	2	yes	—
100	C₂²	#15692980	1	1	yes	characteristic
101	C₄	—	4	2	no	—
102	C₄	—	412	4	no	—
103	C₄	C₂×C₄×Dic₁₀₃	1	1	yes	characteristic
104	C₄	#15692978	1	2	yes	—
105	C₄	#15692980	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	2	2	no	—
108	C₂	#20267637	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20267746

C₂

yes

maximal

C₁₀₃⋊#338

C₂

yes

maximal, characteristic

#20267548

C₂

yes

maximal, characteristic

#20267547

C₂

yes

maximal

#405×C₁₀₃

C₂

yes

maximal, characteristic

C₂×#13110089

C₂²

yes

—

#15692980

C₂²

yes

—

#15692931

C₄

yes

complemented

C₈×Dic₁₀₃

C₄

yes

complemented

C₂×M₁₆×C₁₀₃

C₂²

yes

characteristic

C₂²×C₈×C₁₀₃

C₂²

yes

characteristic

#117×C₁₀₃

C₂²

yes

characteristic

#97×C₁₀₃

C₄

yes

complemented

C₄×C₈×C₁₀₃

C₄

yes

complemented

#13110089

—

#13110089

—

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

—

C₂×C₈×C₁₀₃

C₂×C₄

yes

—

C₂×C₈×C₁₀₃

C₂×C₄

yes

—

C₂×C₈×C₁₀₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₃

C₂×C₄

yes

characteristic

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

C₂×C₄×C₁₀₃

yes

C₂×C₄×C₁₀₃

yes

—

C₂×Dic₁₀₃

—

C₈×C₁₀₃

—

C₈×C₁₀₃

C₄²

yes

—

C₈×C₁₀₃

C₄²

yes

characteristic

C₈×C₁₀₃

C₄²

yes

characteristic

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#117

yes

—

C₂²×C₁₀₃

#117

yes

characteristic

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂×C₄²

yes

characteristic

C₄×C₁₀₃

#117

yes

—

C₄×C₁₀₃

#117

yes

characteristic

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#376

yes

characteristic

#1747

—

103

maximal, Sylow

C₁₀₃

#1747

yes

characteristic, complemented, Sylow

#405

—

103

—

#405

D₁₀₃

yes

characteristic

#338

—

103

—

H(3,ℤ/4)

—

103

—

H(3,ℤ/4)

—

103

—

C₂×M₁₆

—

103

—

C₂×M₁₆

C₂×D₁₀₃

yes

characteristic

C₂²×C₈

C₂×D₁₀₃

yes

characteristic

#117

—

103

—

#117

C₂×D₁₀₃

yes

characteristic

#97

—

103

—

#97

Dic₁₀₃

yes

complemented

C₄×C₈

—

103

—

C₄×C₈

Dic₁₀₃

yes

complemented

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

M₁₆

—

M₁₆

—

206

—

M₁₆

—

206

—

C₂×C₈

—

103

—

C₂×C₈

C₄×D₁₀₃

yes

—

C₂×C₈

C₄×D₁₀₃

yes

—

C₂×C₈

C₂×Dic₁₀₃

yes

—

C₂×C₈

C₂×Dic₁₀₃

yes

characteristic

C₂×C₈

C₂×Dic₁₀₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

206

—

C₂²⋊C₄

—

C₂²⋊C₄

—

206

—

C₄²

—

103

—

C₄²

C₂×Dic₁₀₃

yes

—

C₂³

#13110105

yes

characteristic

C₂×C₄

—

C₂×C₄

—

206

—

C₂×C₄

C₂×C₄×D₁₀₃

yes

—

C₂×C₄

C₂²×Dic₁₀₃

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

206

—

C₈

C₄×Dic₁₀₃

yes

characteristic

C₈

C₄×Dic₁₀₃

yes

—

C₈

C₄×Dic₁₀₃

yes

characteristic

C₂²

—

C₂²

#15692978

yes

—

100

C₂²

#15692980

yes

characteristic

101

C₄

—

102

C₄

—

412

—

103

C₄

C₂×C₄×Dic₁₀₃

yes

characteristic

104

C₄

#15692978

yes

—

105

C₄

#15692980

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

#20267637

yes

characteristic

	subgroup	G/H
center	C₈	C₄×Dic₁₀₃
commutator	C₄×C₁₀₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#405×C₁₀₃	C₂
socle	C₂×C₁₀₃	#376
prime radical	C₂³×C₁₀₃	#41
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	#405	D₁₀₃
hypocenter	C₁₀₃	#1747
LCS	G ▷ C₄×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₈ ◁ #405

subgroup

G/H

center

C₈

C₄×Dic₁₀₃

commutator

C₄×C₁₀₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#405×C₁₀₃

C₂

C₂×C₁₀₃

C₂³×C₁₀₃

C₂²×C₄×C₁₀₃

C₂³

D₁₀₃

C₁₀₃

G ▷ C₄×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₈ ◁ #405

N	H ≅ G/N	type
8. #15692931	C₄	semidirect
9. C₈×Dic₁₀₃	C₄	semidirect
13. #97×C₁₀₃	C₄	semidirect
14. C₄×C₈×C₁₀₃	C₄	semidirect
55. C₁₀₃	#1747	semidirect
67. #97	Dic₁₀₃	semidirect
69. C₄×C₈	Dic₁₀₃	semidirect

H ≅ G/N

type

8. #15692931

C₄

semidirect

9. C₈×Dic₁₀₃

C₄

semidirect

13. #97×C₁₀₃

C₄

semidirect

14. C₄×C₈×C₁₀₃

C₄

semidirect

55. C₁₀₃

#1747

semidirect

67. #97

Dic₁₀₃

semidirect

69. C₄×C₈

Dic₁₀₃

semidirect

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

subgroup	index
#20267746	2
C₁₀₃⋊#338	2
#20267548	2
#20267547	2
#405×C₁₀₃	2
#1747	103

subgroup

index

C₁₀₃⋊#338

#405×C₁₀₃

103

Subgroups — #27194899

Conjugacy classes of subgroups

Complements

Maximal subgroups