Group #27194830 — subgroups

Subgroups — #27194830

Order 13184.

328 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 205 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	C₂×#15692932	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692931	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₀₃	C₂	1	2	yes	maximal
4	#20267637	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₀₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₀₃	C₂²	1	2	yes	—
8	#15692932	C₄	1	4	yes	complemented
9	#15692931	C₄	1	8	yes	complemented
10	C₈×Dic₁₀₃	C₄	1	8	yes	complemented
11	C₁₀₃⋊#95	C₄	1	4	yes	—
12	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₀₃	C₄	1	4	yes	complemented
16	C₂²×Dic₁₀₃	C₂×C₄	1	4	yes	—
17	C₂×C₁₀₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₁₀₃⋊C₈	C₂×C₄	1	4	yes	—
19	C₂×C₁₀₃⋊C₈	C₂×C₄	1	4	yes	—
20	C₄×Dic₁₀₃	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₀₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₀₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₁₀₃	C₂×C₄	1	4	yes	—
25	C₂×C₈×C₁₀₃	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₁₀₃	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₁₀₃	C₂×C₄	1	2	yes	—
28	C₄²×C₁₀₃	C₂×C₄	1	2	yes	—
29	C₄²×C₁₀₃	C₂×C₄	1	2	yes	—
30	C₂³×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₃	—	2	4	no	—
32	C₂×C₄×C₁₀₃	C₄²	1	4	yes	—
33	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₃	C₂²×C₄	1	2	yes	—
35	C₂×C₄×C₁₀₃	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₁₀₃	—	2	8	no	—
38	C₂×Dic₁₀₃	C₄²	1	8	yes	—
39	C₈×C₁₀₃	C₄²	1	4	yes	—
40	C₈×C₁₀₃	C₄²	1	4	yes	—
41	C₁₀₃⋊C₈	C₄²	1	8	yes	—
42	C₂²×C₁₀₃	C₂×C₄²	1	2	yes	—
43	C₂²×C₁₀₃	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₁₀₃	#131	1	4	yes	—
45	C₄×C₁₀₃	—	2	4	no	—
46	C₄×C₁₀₃	C₂×C₄²	1	2	yes	—
47	C₄×C₁₀₃	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₀₃	C₂×C₄²	1	1	yes	characteristic
49	Dic₁₀₃	—	2	8	no	—
50	C₂×C₁₀₃	C₄³	1	1	yes	characteristic
51	C₂×C₁₀₃	#405	1	4	yes	—
52	C₂×C₁₀₃	#405	1	2	yes	—
53	#1726	—	103	1	no	maximal, Sylow
54	C₁₀₃	#1726	1	1	yes	characteristic, complemented, Sylow
55	C₂×#97	—	103	2	no	—
56	C₂×#97	—	103	1	no	—
57	C₂×C₄×C₈	—	103	2	no	—
58	C₂×C₄×C₈	D₁₀₃	1	1	yes	characteristic
59	#376	—	103	1	no	—
60	C₂²×C₈	—	103	2	no	—
61	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
62	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
63	C₂×C₄²	—	103	2	no	—
64	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
65	#97	—	103	8	no	—
66	#97	—	103	4	no	—
67	C₄×C₈	—	103	8	no	—
68	C₄×C₈	Dic₁₀₃	1	4	yes	complemented
69	#95	—	103	4	no	—
70	C₂²×C₄	—	103	4	no	—
71	C₂²×C₄	C₄×D₁₀₃	1	2	yes	—
72	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
73	C₂×C₈	—	103	4	no	—
74	C₂×C₈	—	103	4	no	—
75	C₂×C₈	—	103	4	no	—
76	C₂×C₈	C₄×D₁₀₃	1	4	yes	—
77	C₂×C₈	C₄×D₁₀₃	1	4	yes	—
78	C₂×C₈	C₂×Dic₁₀₃	1	2	yes	—
79	C₂×C₈	C₂×Dic₁₀₃	1	2	yes	—
80	C₄²	—	103	4	no	—
81	C₄²	—	103	4	no	—
82	C₄²	C₄×D₁₀₃	1	2	yes	—
83	C₄²	C₂×Dic₁₀₃	1	2	yes	—
84	C₂³	C₂×C₄×D₁₀₃	1	1	yes	characteristic
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	103	8	no	—
87	C₂×C₄	—	206	8	no	—
88	C₂×C₄	C₄×Dic₁₀₃	1	4	yes	—
89	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₁₀₃	1	2	yes	—
91	C₂×C₄	C₂×C₄×D₁₀₃	1	2	yes	—
92	C₂×C₄	C₂²×Dic₁₀₃	1	1	yes	characteristic
93	C₈	—	103	8	no	—
94	C₈	C₄×Dic₁₀₃	1	4	yes	—
95	C₈	C₄×Dic₁₀₃	1	4	yes	—
96	C₂²	C₄²×D₁₀₃	1	2	yes	—
97	C₂²	C₂×C₄×Dic₁₀₃	1	1	yes	characteristic
98	C₂²	#15693027	1	4	yes	—
99	C₄	—	2	4	no	—
100	C₄	—	206	8	no	—
101	C₄	C₄²×D₁₀₃	1	2	yes	—
102	C₄	C₂×C₄×Dic₁₀₃	1	1	yes	characteristic
103	C₄	C₂×C₄×Dic₁₀₃	1	1	yes	characteristic
104	C₂	C₄²×Dic₁₀₃	1	1	yes	characteristic
105	C₂	#20267745	1	4	yes	—
106	C₂	#20267746	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692932

C₂

yes

maximal, characteristic

C₂×#15692931

C₂

yes

maximal

C₂×C₈×Dic₁₀₃

C₂

yes

maximal

#20267637

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×C₁₀₃⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₀₃

C₂²

yes

—

#15692932

C₄

yes

complemented

#15692931

C₄

yes

complemented

C₈×Dic₁₀₃

C₄

yes

complemented

C₁₀₃⋊#95

C₄

yes

—

C₂²×C₈×C₁₀₃

C₂²

yes

characteristic

C₂²×C₈×C₁₀₃

C₂²

yes

characteristic

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₄×C₈×C₁₀₃

C₄

yes

complemented

C₂²×Dic₁₀₃

C₂×C₄

yes

—

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

—

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

—

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₀₃

C₂×C₄

yes

—

C₄×Dic₁₀₃

C₂×C₄

yes

—

C₂²×C₄×C₁₀₃

C₂×C₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₈×C₁₀₃

C₂×C₄

yes

—

C₂×C₈×C₁₀₃

C₂×C₄

yes

—

C₂×C₈×C₁₀₃

C₂×C₄

yes

—

C₂×C₈×C₁₀₃

C₂×C₄

yes

—

C₄²×C₁₀₃

C₂×C₄

yes

—

C₄²×C₁₀₃

C₂×C₄

yes

—

C₂³×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₄²

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂²×C₄

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

C₄²

yes

—

C₈×C₁₀₃

C₄²

yes

—

C₈×C₁₀₃

C₄²

yes

—

C₁₀₃⋊C₈

C₄²

yes

—

C₂²×C₁₀₃

C₂×C₄²

yes

—

C₂²×C₁₀₃

C₂×C₄²

yes

characteristic

C₂²×C₁₀₃

#131

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂×C₄²

yes

—

C₄×C₁₀₃

C₂×C₄²

yes

characteristic

C₄×C₁₀₃

C₂×C₄²

yes

characteristic

Dic₁₀₃

—

C₂×C₁₀₃

C₄³

yes

characteristic

C₂×C₁₀₃

#405

yes

—

C₂×C₁₀₃

#405

yes

—

#1726

—

103

maximal, Sylow

C₁₀₃

#1726

yes

characteristic, complemented, Sylow

C₂×#97

—

103

—

C₂×#97

—

103

—

C₂×C₄×C₈

—

103

—

C₂×C₄×C₈

D₁₀₃

yes

characteristic

#376

—

103

—

C₂²×C₈

—

103

—

C₂²×C₈

C₂×D₁₀₃

yes

characteristic

C₂²×C₈

C₂×D₁₀₃

yes

characteristic

C₂×C₄²

—

103

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

#97

—

103

—

#97

—

103

—

C₄×C₈

—

103

—

C₄×C₈

Dic₁₀₃

yes

complemented

#95

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

C₄×D₁₀₃

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

—

103

—

C₂×C₈

—

103

—

C₂×C₈

—

103

—

C₂×C₈

C₄×D₁₀₃

yes

—

C₂×C₈

C₄×D₁₀₃

yes

—

C₂×C₈

C₂×Dic₁₀₃

yes

—

C₂×C₈

C₂×Dic₁₀₃

yes

—

C₄²

—

103

—

C₄²

—

103

—

C₄²

C₄×D₁₀₃

yes

—

C₄²

C₂×Dic₁₀₃

yes

—

C₂³

C₂×C₄×D₁₀₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

103

—

C₂×C₄

—

206

—

C₂×C₄

C₄×Dic₁₀₃

yes

—

C₂×C₄

C₂×C₄×D₁₀₃

yes

characteristic

C₂×C₄

C₂×C₄×D₁₀₃

yes

—

C₂×C₄

C₂×C₄×D₁₀₃

yes

—

C₂×C₄

C₂²×Dic₁₀₃

yes

characteristic

C₈

—

103

—

C₈

C₄×Dic₁₀₃

yes

—

C₈

C₄×Dic₁₀₃

yes

—

C₂²

C₄²×D₁₀₃

yes

—

C₂²

C₂×C₄×Dic₁₀₃

yes

characteristic

C₂²

#15693027

yes

—

C₄

—

100

C₄

—

206

—

101

C₄

C₄²×D₁₀₃

yes

—

102

C₄

C₂×C₄×Dic₁₀₃

yes

characteristic

103

C₄

C₂×C₄×Dic₁₀₃

yes

characteristic

104

C₂

C₄²×Dic₁₀₃

yes

characteristic

105

C₂

#20267745

yes

—

106

C₂

#20267746

yes

—

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₀₃
commutator	C₂×C₁₀₃	C₄³
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	C₂×C₄×C₈×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	C₂×C₄×C₈	D₁₀₃
hypocenter	C₁₀₃	#1726
LCS	G ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₈

C₂×D₁₀₃

commutator

C₂×C₁₀₃

C₄³

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

C₂×C₄×C₈×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₂²×C₄

cosocle

C₂²×C₄×C₁₀₃

C₂³

hypercenter

C₂×C₄×C₈

D₁₀₃

C₁₀₃

G ▷ C₂×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
8. #15692932	C₄	semidirect
9. #15692931	C₄	semidirect
10. C₈×Dic₁₀₃	C₄	semidirect
15. C₄×C₈×C₁₀₃	C₄	semidirect
54. C₁₀₃	#1726	semidirect
68. C₄×C₈	Dic₁₀₃	semidirect

H ≅ G/N

type

8. #15692932

C₄

semidirect

9. #15692931

C₄

semidirect

10. C₈×Dic₁₀₃

C₄

semidirect

15. C₄×C₈×C₁₀₃

C₄

semidirect

54. C₁₀₃

#1726

semidirect

68. C₄×C₈

Dic₁₀₃

semidirect

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

subgroup	index
C₂×#15692932	2
C₂×#15692931	2
C₂×C₈×Dic₁₀₃	2
#20267637	2
C₂×C₄×C₈×C₁₀₃	2
#1726	103

subgroup

index

C₂×C₈×Dic₁₀₃

C₂×C₄×C₈×C₁₀₃

103

Subgroups — #27194830

Conjugacy classes of subgroups

Complements

Maximal subgroups