Group #27193919 — subgroups

Subgroups — #27193919

Order 13184.

168 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20267815	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₁₀₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20267544	C₂	1	4	yes	maximal, complemented
4	#431×C₁₀₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13110089	—	2	1	no	—
6	C₂²×C₁₀₃⋊C₈	—	2	1	no	—
7	C₄⋊(C₁₀₃⋊C₈)	—	2	1	no	—
8	C₄⋊(C₁₀₃⋊C₈)	—	2	1	no	—
9	C₄⋊(C₁₀₃⋊C₈)	C₂²	1	2	yes	—
10	C₄⋊(C₁₀₃⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₀₃	C₄	1	2	yes	—
13	C₂²⋊C₈×C₁₀₃	C₄	1	2	yes	—
14	C₄×C₈×C₁₀₃	C₂²	1	2	yes	—
15	#13110089	—	4	2	no	—
16	C₂×C₁₀₃⋊C₈	—	2	2	no	—
17	C₂×C₁₀₃⋊C₈	—	2	2	no	—
18	C₂×C₁₀₃⋊C₈	—	4	2	no	—
19	C₂²×C₄×C₁₀₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₀₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₃	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₁₀₃	C₂×C₄	1	4	yes	—
23	C₄²×C₁₀₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₀₃	D₄	1	1	yes	characteristic
25	C₄²×C₁₀₃	Q₈	1	1	yes	characteristic
26	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
27	C₂³×C₁₀₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₃	—	2	2	no	—
29	C₂×C₄×C₁₀₃	—	4	1	no	—
30	C₂×C₄×C₁₀₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₃	D₈	1	2	yes	—
34	C₂×C₄×C₁₀₃	Q₁₆	1	2	yes	—
35	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
38	C₈×C₁₀₃	—	2	4	no	—
39	C₁₀₃⋊C₈	—	4	2	no	—
40	C₁₀₃⋊C₈	—	4	2	no	—
41	C₂²×C₁₀₃	—	2	1	no	—
42	C₂²×C₁₀₃	—	2	1	no	—
43	C₂²×C₁₀₃	#107	1	2	yes	—
44	C₂²×C₁₀₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₁₀₃	—	2	1	no	—
46	C₄×C₁₀₃	—	2	1	no	—
47	C₄×C₁₀₃	—	2	1	no	—
48	C₄×C₁₀₃	#107	1	2	yes	—
49	C₄×C₁₀₃	#131	1	2	yes	—
50	C₄×C₁₀₃	C₂×D₈	1	1	yes	characteristic
51	C₄×C₁₀₃	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×C₁₀₃	—	2	2	no	—
53	C₂×C₁₀₃	#424	1	1	yes	characteristic
54	C₂×C₁₀₃	C₂×#107	1	1	yes	characteristic
55	C₂×C₁₀₃	#430	1	1	yes	characteristic
56	#1571	—	103	1	no	maximal, Sylow
57	C₁₀₃	#1571	1	1	yes	characteristic, complemented, Sylow
58	#431	D₁₀₃	1	1	yes	characteristic
59	#423	—	103	1	no	—
60	C₂×C₄⋊C₈	—	103	1	no	—
61	#335	—	103	4	no	—
62	C₂×M₁₆	—	206	1	no	—
63	C₂²×C₈	—	206	1	no	—
64	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
65	C₄⋊C₈	—	103	2	no	—
66	C₄⋊C₈	—	103	2	no	—
67	C₄⋊C₈	—	206	1	no	—
68	C₄⋊C₈	—	206	1	no	—
69	C₄⋊C₈	Dic₁₀₃	1	2	yes	—
70	C₂²⋊C₈	Dic₁₀₃	1	2	yes	—
71	C₄×C₈	C₂×D₁₀₃	1	2	yes	—
72	C₂²×C₄	Dic₂₀₆	1	1	yes	characteristic
73	C₂²×C₄	D₄₁₂	1	1	yes	characteristic
74	C₂²×C₄	C₂×Dic₁₀₃	1	1	yes	characteristic
75	M₁₆	—	412	2	no	—
76	C₂×C₈	—	412	2	no	—
77	C₂×C₈	—	206	2	no	—
78	C₂×C₈	—	206	2	no	—
79	C₂×C₈	C₂×Dic₁₀₃	1	4	yes	—
80	C₄²	Dic₂₀₆	1	1	yes	characteristic
81	C₄²	D₄₁₂	1	1	yes	characteristic
82	C₄²	C₂×Dic₁₀₃	1	1	yes	characteristic
83	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
84	C₂³	C₄⋊Dic₁₀₃	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	C₄⋊Dic₁₀₃	1	1	yes	characteristic
88	C₂×C₄	C₄⋊Dic₁₀₃	1	1	yes	characteristic
89	C₂×C₄	C₄⋊Dic₁₀₃	1	1	yes	characteristic
90	C₂×C₄	D₈₂₄	1	2	yes	—
91	C₂×C₄	Dic₄₁₂	1	2	yes	—
92	C₂×C₄	C₂²×Dic₁₀₃	1	1	yes	characteristic
93	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₂₀₆	1	1	yes	characteristic
95	C₈	—	2	4	no	—
96	C₈	—	412	2	no	—
97	C₈	—	412	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#15692955	1	2	yes	—
101	C₂²	C₂×C₄⋊Dic₁₀₃	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#15692955	1	2	yes	—
106	C₄	#15693028	1	2	yes	—
107	C₄	C₂×D₈₂₄	1	1	yes	characteristic
108	C₄	C₂×Dic₄₁₂	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#20267818	1	1	yes	characteristic
111	C₂	C₂×#15692955	1	1	yes	characteristic
112	C₂	#20267840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20267815

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₁₀₃⋊C₈)

C₂

yes

maximal, characteristic

#20267544

C₂

yes

maximal, complemented

#431×C₁₀₃

C₂

yes

maximal, characteristic

C₂×#13110089

—

C₂²×C₁₀₃⋊C₈

—

C₄⋊(C₁₀₃⋊C₈)

—

C₄⋊(C₁₀₃⋊C₈)

—

C₄⋊(C₁₀₃⋊C₈)

C₂²

yes

—

C₄⋊(C₁₀₃⋊C₈)

C₂²

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₃

C₄

yes

—

C₂²⋊C₈×C₁₀₃

C₄

yes

—

C₄×C₈×C₁₀₃

C₂²

yes

—

#13110089

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂²×C₄×C₁₀₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₃

D₄

yes

characteristic

C₂²×C₄×C₁₀₃

Q₈

yes

characteristic

C₂×C₈×C₁₀₃

C₂×C₄

yes

—

C₄²×C₁₀₃

C₂×C₄

yes

characteristic

C₄²×C₁₀₃

D₄

yes

characteristic

C₄²×C₁₀₃

Q₈

yes

characteristic

C₄²×C₁₀₃

C₂³

yes

characteristic

C₂³×C₁₀₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₃

D₈

yes

—

C₂×C₄×C₁₀₃

Q₁₆

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×Q₈

yes

characteristic

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#107

yes

—

C₂²×C₁₀₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#107

yes

—

C₄×C₁₀₃

#131

yes

—

C₄×C₁₀₃

C₂×D₈

yes

characteristic

C₄×C₁₀₃

C₂×Q₁₆

yes

characteristic

C₂×C₁₀₃

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

103

maximal, Sylow

C₁₀₃

#1571

yes

characteristic, complemented, Sylow

#431

D₁₀₃

yes

characteristic

#423

—

103

—

C₂×C₄⋊C₈

—

103

—

#335

—

103

—

C₂×M₁₆

—

206

—

C₂²×C₈

—

206

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

C₄⋊C₈

—

103

—

C₄⋊C₈

—

103

—

C₄⋊C₈

—

206

—

C₄⋊C₈

—

206

—

C₄⋊C₈

Dic₁₀₃

yes

—

C₂²⋊C₈

Dic₁₀₃

yes

—

C₄×C₈

C₂×D₁₀₃

yes

—

C₂²×C₄

Dic₂₀₆

yes

characteristic

C₂²×C₄

D₄₁₂

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₃

yes

characteristic

M₁₆

—

412

—

C₂×C₈

—

412

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₂×C₈

C₂×Dic₁₀₃

yes

—

C₄²

Dic₂₀₆

yes

characteristic

C₄²

D₄₁₂

yes

characteristic

C₄²

C₂×Dic₁₀₃

yes

characteristic

C₄²

C₂²×D₁₀₃

yes

characteristic

C₂³

C₄⋊Dic₁₀₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₁₀₃

yes

characteristic

C₂×C₄

C₄⋊Dic₁₀₃

yes

characteristic

C₂×C₄

C₄⋊Dic₁₀₃

yes

characteristic

C₂×C₄

D₈₂₄

yes

—

C₂×C₄

Dic₄₁₂

yes

—

C₂×C₄

C₂²×Dic₁₀₃

yes

characteristic

C₂×C₄

C₂×D₄₁₂

yes

characteristic

C₂×C₄

C₂×Dic₂₀₆

yes

characteristic

C₈

—

C₈

—

412

—

C₈

—

412

—

C₂²

—

C₂²

—

100

C₂²

#15692955

yes

—

101

C₂²

C₂×C₄⋊Dic₁₀₃

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#15692955

yes

—

106

C₄

#15693028

yes

—

107

C₄

C₂×D₈₂₄

yes

characteristic

108

C₄

C₂×Dic₄₁₂

yes

characteristic

109

C₂

—

110

C₂

#20267818

yes

characteristic

111

C₂

C₂×#15692955

yes

characteristic

112

C₂

#20267840

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₁₂
commutator	C₂×C₄×C₁₀₃	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₃
Fitting	#431×C₁₀₃	C₂
socle	C₂²×C₁₀₃	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₃	C₄⋊C₄
cosocle	C₄²×C₁₀₃	C₂³
hypercenter	#431	D₁₀₃
hypocenter	C₁₀₃	#1571
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₄₁₂

commutator

C₂×C₄×C₁₀₃

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₃

Fitting

#431×C₁₀₃

C₂

socle

C₂²×C₁₀₃

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₃

C₄⋊C₄

cosocle

C₄²×C₁₀₃

C₂³

D₁₀₃

C₁₀₃

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #20267544	C₂	semidirect
57. C₁₀₃	#1571	semidirect

H ≅ G/N

type

3. #20267544

C₂

semidirect

57. C₁₀₃

#1571

semidirect

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

subgroup	index
#20267815	2
C₂×C₄⋊(C₁₀₃⋊C₈)	2
#20267544	2
#431×C₁₀₃	2
#1571	103

subgroup

index

#20267815

C₂×C₄⋊(C₁₀₃⋊C₈)

#20267544

#431×C₁₀₃

#1571

103

Subgroups — #27193919

Conjugacy classes of subgroups

Complements

Maximal subgroups