Group #27196654 — subgroups

Subgroups — #27196654

Order 13184.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#20267834	C₂	1	2	yes	maximal
2	C₂×#15692953	C₂	1	1	yes	maximal, characteristic
3	#20267559	C₂	1	2	yes	maximal
4	#20267556	C₂	1	1	yes	maximal, characteristic
5	#424×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13110089	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
8	#15692954	—	2	2	no	—
9	#15692953	—	2	2	no	—
10	#15692952	—	2	2	no	—
11	C₂×M₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
13	#117×C₁₀₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₀₃	—	2	2	no	—
15	C₂²×Dic₁₀₃	—	2	2	no	—
16	#13110089	—	2	2	no	—
17	#13110089	Q₈	1	4	yes	—
18	C₂×C₁₀₃⋊C₈	—	2	2	no	—
19	C₄⋊Dic₁₀₃	—	2	2	no	—
20	C₄⋊Dic₁₀₃	—	2	2	no	—
21	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₀₃	—	2	2	no	—
23	C₂×C₈×C₁₀₃	—	2	2	no	—
24	C₂×C₈×C₁₀₃	—	2	1	no	—
25	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
26	C₄⋊C₄×C₁₀₃	D₄	1	2	yes	—
27	C₂²⋊C₄×C₁₀₃	—	2	1	no	—
28	C₄²×C₁₀₃	—	2	1	no	—
29	C₂³×C₁₀₃	#41	1	1	yes	characteristic
30	C₂×C₄×C₁₀₃	—	2	2	no	—
31	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₃	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₁₀₃	#41	1	2	yes	—
35	C₂×Dic₁₀₃	—	4	2	no	—
36	C₂×Dic₁₀₃	—	8	2	no	—
37	C₈×C₁₀₃	—	2	2	no	—
38	C₈×C₁₀₃	—	4	1	no	—
39	C₁₀₃⋊C₈	—	2	4	no	—
40	C₂²×C₁₀₃	—	2	2	no	—
41	C₂²×C₁₀₃	#121	1	1	yes	characteristic
42	C₂²×C₁₀₃	#125	1	2	yes	—
43	C₄×C₁₀₃	—	4	2	no	—
44	C₄×C₁₀₃	C₂²⋊Q₈	1	2	yes	—
45	C₄×C₁₀₃	#123	1	1	yes	characteristic
46	C₄×C₁₀₃	C₄⋊Q₈	1	1	yes	characteristic
47	Dic₁₀₃	—	8	2	no	—
48	C₂×C₁₀₃	—	2	2	no	—
49	C₂×C₁₀₃	#398	1	1	yes	characteristic
50	C₂×C₁₀₃	#471	1	2	yes	—
51	#2090	—	103	1	no	maximal, Sylow
52	C₁₀₃	#2090	1	1	yes	characteristic, complemented, Sylow
53	#428	—	103	2	no	—
54	C₂×#106	—	103	1	no	—
55	#424	D₁₀₃	1	1	yes	characteristic
56	#343	—	103	2	no	—
57	#341	—	103	1	no	—
58	C₂×M₁₆	—	103	2	no	—
59	C₂×M₁₆	C₂×D₁₀₃	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
61	#117	C₂×D₁₀₃	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	103	2	no	—
63	#107	—	206	2	no	—
64	#106	—	206	2	no	—
65	#106	—	206	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²×C₄	—	206	2	no	—
68	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	103	4	no	—
71	M₁₆	—	206	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	206	2	no	—
75	C₂×C₈	C₂²⋊D₁₀₃	1	2	yes	—
76	C₄⋊C₄	—	206	2	no	—
77	C₄⋊C₄	—	206	2	no	—
78	C₄⋊C₄	D₄₁₂	1	2	yes	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₂³	#13110105	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	824	2	no	—
84	C₂×C₄	—	412	2	no	—
85	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
86	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₁₀₃	1	2	yes	—
88	C₂×C₄	#13110104	1	2	yes	—
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	206	4	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#15692995	1	1	yes	characteristic
94	C₂²	#15693011	1	2	yes	—
95	C₄	—	4	2	no	—
96	C₄	—	824	2	no	—
97	C₄	#15693001	1	2	yes	—
98	C₄	#15693002	1	1	yes	characteristic
99	C₄	#15693019	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#20267721	1	1	yes	characteristic
102	C₂	#20268054	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20267834

C₂

yes

maximal

C₂×#15692953

C₂

yes

maximal, characteristic

#20267559

C₂

yes

maximal

#20267556

C₂

yes

maximal, characteristic

#424×C₁₀₃

C₂

yes

maximal, characteristic

C₂×#13110089

C₂²

yes

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

—

#15692954

—

#15692953

—

#15692952

—

C₂×M₁₆×C₁₀₃

C₂²

yes

characteristic

C₂²×C₈×C₁₀₃

C₂²

yes

characteristic

#117×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₃

—

C₂²×Dic₁₀₃

—

#13110089

—

#13110089

Q₈

yes

—

C₂×C₁₀₃⋊C₈

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

M₁₆×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₄⋊C₄×C₁₀₃

D₄

yes

—

C₂²⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

—

C₂³×C₁₀₃

#41

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×Q₈

yes

—

C₂×C₄×C₁₀₃

#41

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₈×C₁₀₃

—

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#121

yes

characteristic

C₂²×C₁₀₃

#125

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂²⋊Q₈

yes

—

C₄×C₁₀₃

#123

yes

characteristic

C₄×C₁₀₃

C₄⋊Q₈

yes

characteristic

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#398

yes

characteristic

C₂×C₁₀₃

#471

yes

—

#2090

—

103

maximal, Sylow

C₁₀₃

#2090

yes

characteristic, complemented, Sylow

#428

—

103

—

C₂×#106

—

103

—

#424

D₁₀₃

yes

characteristic

#343

—

103

—

#341

—

103

—

C₂×M₁₆

—

103

—

C₂×M₁₆

C₂×D₁₀₃

yes

characteristic

C₂²×C₈

C₂×D₁₀₃

yes

characteristic

#117

C₂×D₁₀₃

yes

characteristic

C₂×C₄⋊C₄

—

103

—

#107

—

206

—

#106

—

206

—

#106

—

206

—

C₄⋊C₈

—

C₂²×C₄

—

206

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

M₁₆

—

M₁₆

—

103

—

M₁₆

—

206

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

206

—

C₂×C₈

C₂²⋊D₁₀₃

yes

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄⋊C₄

D₄₁₂

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#13110105

yes

characteristic

C₂×C₄

—

C₂×C₄

—

824

—

C₂×C₄

—

412

—

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

C₂×C₄

C₂×D₄₁₂

yes

characteristic

C₂×C₄

Q₈×D₁₀₃

yes

—

C₂×C₄

#13110104

yes

—

C₈

—

C₈

—

C₈

—

206

—

C₂²

—

C₂²

#15692995

yes

characteristic

C₂²

#15693011

yes

—

C₄

—

C₄

—

824

—

C₄

#15693001

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#20267721

yes

characteristic

102

C₂

#20268054

yes

—

	subgroup	G/H
center	C₂²	#15692995
commutator, cosocle	C₂²×C₄×C₁₀₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#424×C₁₀₃	C₂
socle	C₂²×C₁₀₃	#121
prime radical	C₂³×C₁₀₃	#41
hypercenter	#424	D₁₀₃
hypocenter	C₁₀₃	#2090
LCS	G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#15692995

commutator, cosocle

C₂²×C₄×C₁₀₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#424×C₁₀₃

C₂

C₂²×C₁₀₃

C₂³×C₁₀₃

D₁₀₃

C₁₀₃

G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
52. C₁₀₃	#2090	semidirect

H ≅ G/N

type

52. C₁₀₃

#2090

semidirect

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

subgroup	index
#20267834	2
C₂×#15692953	2
#20267559	2
#20267556	2
#424×C₁₀₃	2
#2090	103

subgroup

index

#424×C₁₀₃

103

Subgroups — #27196654

Conjugacy classes of subgroups

Complements

Maximal subgroups