Group #27195049 — subgroups

Subgroups — #27195049

Order 13184.

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

normal subgroups only

This group has 220 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	C₂×#15692956	C₂	1	2	yes	maximal
2	M₁₆×Dic₁₀₃	C₂	1	2	yes	maximal
3	C₂×#15692931	C₂	1	1	yes	maximal, characteristic
4	#20267556	C₂	1	1	yes	maximal, characteristic
5	#341×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13110089	C₂²	1	2	yes	—
7	C₂²×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₃	C₂²	1	1	yes	characteristic
9	#15692956	C₄	1	8	yes	complemented
10	#15692931	—	2	2	no	—
11	#15692931	C₄	1	4	yes	—
12	C₈×Dic₁₀₃	—	2	2	no	—
13	C₂×M₁₆×C₁₀₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₃	Q₈	1	2	yes	—
16	#13110089	—	2	2	no	—
17	#13110089	C₂×C₄	1	4	yes	—
18	C₂×C₁₀₃⋊C₈	—	2	2	no	—
19	C₂×C₁₀₃⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₀₃⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₀₃	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₀₃	D₄	1	2	yes	—
23	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
24	M₁₆×C₁₀₃	—	2	2	no	—
25	M₁₆×C₁₀₃	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₁₀₃	—	2	2	no	—
27	C₂×C₈×C₁₀₃	—	2	2	no	—
28	C₂×C₈×C₁₀₃	C₂×C₄	1	2	yes	—
29	C₂³×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₃	C₂²×C₄	1	2	yes	—
32	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₃	#41	1	2	yes	—
34	C₂×Dic₁₀₃	—	4	2	no	—
35	C₂×Dic₁₀₃	C₄⋊C₄	1	4	yes	—
36	C₈×C₁₀₃	—	2	2	no	—
37	C₈×C₁₀₃	—	2	4	no	—
38	C₁₀₃⋊C₈	—	2	4	no	—
39	C₁₀₃⋊C₈	C₄²	1	4	yes	—
40	C₂²×C₁₀₃	—	2	2	no	—
41	C₂²×C₁₀₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₁₀₃	C₄×Q₈	1	2	yes	—
43	C₄×C₁₀₃	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₁₀₃	#117	1	1	yes	characteristic
45	C₄×C₁₀₃	C₄×D₄	1	2	yes	—
46	Dic₁₀₃	—	2	4	no	—
47	C₂×C₁₀₃	—	2	2	no	—
48	C₂×C₁₀₃	C₄×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₁₀₃	#430	1	2	yes	—
50	#1778	—	103	1	no	maximal, Sylow
51	C₁₀₃	#1778	1	1	yes	characteristic, complemented, Sylow
52	C₂×#108	—	103	2	no	—
53	C₄×M₁₆	—	103	2	no	—
54	C₂×#97	—	103	1	no	—
55	#341	—	103	1	no	—
56	#341	D₁₀₃	1	1	yes	characteristic
57	C₂×M₁₆	—	103	2	no	—
58	C₂×M₁₆	C₂×D₁₀₃	1	2	yes	—
59	C₂²×C₈	—	103	1	no	—
60	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
61	C₂×C₄²	—	103	1	no	—
62	#108	—	103	8	no	—
63	#97	—	103	4	no	—
64	#97	—	206	2	no	—
65	C₄×C₈	—	206	2	no	—
66	C₂²×C₄	—	103	2	no	—
67	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	103	4	no	—
70	M₁₆	—	206	2	no	—
71	M₁₆	C₄×D₁₀₃	1	4	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	103	2	no	—
75	C₂×C₈	—	103	4	no	—
76	C₂×C₈	—	206	2	no	—
77	C₂×C₈	C₄×D₁₀₃	1	2	yes	—
78	C₄²	—	103	2	no	—
79	C₄²	—	103	2	no	—
80	C₂³	Q₈×D₁₀₃	1	1	yes	characteristic
81	C₂×C₄	—	103	4	no	—
82	C₂×C₄	—	412	2	no	—
83	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄×D₁₀₃	1	2	yes	—
85	C₂×C₄	D₄×D₁₀₃	1	1	yes	characteristic
86	C₂×C₄	#13110104	1	2	yes	—
87	C₈	—	2	2	no	—
88	C₈	—	2	4	no	—
89	C₈	—	103	4	no	—
90	C₈	—	206	4	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₄⋊C₄×D₁₀₃	1	1	yes	characteristic
93	C₂²	#15692986	1	2	yes	—
94	C₄	—	206	4	no	—
95	C₄	C₄²×D₁₀₃	1	1	yes	characteristic
96	C₄	#15692977	1	1	yes	characteristic
97	C₄	(C₂²⋊D₁₀₃)⋊C₄	1	2	yes	—
98	C₂	—	2	2	no	—
99	C₂	#20267642	1	1	yes	characteristic
100	C₂	#20267838	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692956

C₂

yes

maximal

M₁₆×Dic₁₀₃

C₂

yes

maximal

C₂×#15692931

C₂

yes

maximal, characteristic

#20267556

C₂

yes

maximal, characteristic

#341×C₁₀₃

C₂

yes

maximal, characteristic

C₂×#13110089

C₂²

yes

—

C₂²×C₁₀₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₃

C₂²

yes

characteristic

#15692956

C₄

yes

complemented

#15692931

—

#15692931

C₄

yes

—

C₈×Dic₁₀₃

—

C₂×M₁₆×C₁₀₃

C₂²

yes

—

C₂²×C₈×C₁₀₃

C₂²

yes

characteristic

C₂²×Dic₁₀₃

Q₈

yes

—

#13110089

—

#13110089

C₂×C₄

yes

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

—

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₀₃

C₂×C₄

yes

—

C₄×Dic₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

M₁₆×C₁₀₃

—

M₁₆×C₁₀₃

C₂×C₄

yes

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

C₂×C₄

yes

—

C₂³×C₁₀₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂²×C₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

C₄⋊C₄

yes

—

C₈×C₁₀₃

—

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₁₀₃⋊C₈

C₄²

yes

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₃

C₄×Q₈

yes

—

C₄×C₁₀₃

C₂×C₄²

yes

characteristic

C₄×C₁₀₃

#117

yes

characteristic

C₄×C₁₀₃

C₄×D₄

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₃

#430

yes

—

#1778

—

103

maximal, Sylow

C₁₀₃

#1778

yes

characteristic, complemented, Sylow

C₂×#108

—

103

—

C₄×M₁₆

—

103

—

C₂×#97

—

103

—

#341

—

103

—

#341

D₁₀₃

yes

characteristic

C₂×M₁₆

—

103

—

C₂×M₁₆

C₂×D₁₀₃

yes

—

C₂²×C₈

—

103

—

C₂²×C₈

C₂×D₁₀₃

yes

characteristic

C₂×C₄²

—

103

—

#108

—

103

—

#97

—

103

—

#97

—

206

—

C₄×C₈

—

206

—

C₂²×C₄

—

103

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

M₁₆

—

M₁₆

—

103

—

M₁₆

—

206

—

M₁₆

C₄×D₁₀₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

103

—

C₂×C₈

—

103

—

C₂×C₈

—

206

—

C₂×C₈

C₄×D₁₀₃

yes

—

C₄²

—

103

—

C₄²

—

103

—

C₂³

Q₈×D₁₀₃

yes

characteristic

C₂×C₄

—

103

—

C₂×C₄

—

412

—

C₂×C₄

C₂×C₄×D₁₀₃

yes

characteristic

C₂×C₄

C₂×C₄×D₁₀₃

yes

—

C₂×C₄

D₄×D₁₀₃

yes

characteristic

C₂×C₄

#13110104

yes

—

C₈

—

C₈

—

C₈

—

103

—

C₈

—

206

—

C₂²

—

C₂²

C₄⋊C₄×D₁₀₃

yes

characteristic

C₂²

#15692986

yes

—

C₄

—

206

—

C₄

C₄²×D₁₀₃

yes

characteristic

C₄

#15692977

yes

characteristic

C₄

(C₂²⋊D₁₀₃)⋊C₄

yes

—

C₂

—

C₂

#20267642

yes

characteristic

100

C₂

#20267838

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₃
commutator	C₄×C₁₀₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#341×C₁₀₃	C₂
socle	C₂²×C₁₀₃	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	#341	D₁₀₃
hypocenter	C₁₀₃	#1778
LCS	G ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₃

commutator

C₄×C₁₀₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#341×C₁₀₃

C₂

socle

C₂²×C₁₀₃

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₃

C₂×Q₈

cosocle

C₂²×C₄×C₁₀₃

C₂³

D₁₀₃

C₁₀₃

G ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #15692956	C₄	semidirect
51. C₁₀₃	#1778	semidirect

H ≅ G/N

type

9. #15692956

C₄

semidirect

51. C₁₀₃

#1778

semidirect

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

subgroup	index
C₂×#15692956	2
M₁₆×Dic₁₀₃	2
C₂×#15692931	2
#20267556	2
#341×C₁₀₃	2
#1778	103

subgroup

index

M₁₆×Dic₁₀₃

#341×C₁₀₃

103

Subgroups — #27195049

Conjugacy classes of subgroups

Complements

Maximal subgroups