Group #27195002 — subgroups

Subgroups — #27195002

Order 13184.

260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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×Dic₁₀₃⋊C₈	C₂	1	2	yes	maximal
2	C₂×#15692932	C₂	1	1	yes	maximal, characteristic
3	#20267637	C₂	1	1	yes	maximal, characteristic
4	#20267545	C₂	1	2	yes	maximal
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	1	yes	characteristic
8	C₂×C₄×Dic₁₀₃	C₂²	1	1	yes	characteristic
9	Dic₁₀₃⋊C₈	C₄	1	8	yes	complemented
10	#15692932	—	2	2	no	—
11	C₁₀₃⋊#95	C₄	1	4	yes	—
12	C₂²×C₈×C₁₀₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₀₃	—	2	2	no	—
15	C₂²×Dic₁₀₃	C₂×C₄	1	2	yes	—
16	C₂²×Dic₁₀₃	C₂×C₄	1	2	yes	—
17	C₂×C₁₀₃⋊C₈	—	2	2	no	—
18	C₂×C₁₀₃⋊C₈	—	2	2	no	—
19	C₂×C₁₀₃⋊C₈	C₂×C₄	1	4	yes	—
20	C₄×Dic₁₀₃	—	2	2	no	—
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	—	2	2	no	—
25	C₂×C₈×C₁₀₃	—	2	2	no	—
26	C₂×C₈×C₁₀₃	C₂×C₄	1	4	yes	—
27	C₄²×C₁₀₃	D₄	1	2	yes	—
28	C₄²×C₁₀₃	Q₈	1	2	yes	—
29	C₂³×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₃	—	2	4	no	—
31	C₂×C₄×C₁₀₃	C₄⋊C₄	1	4	yes	—
32	C₂×C₄×C₁₀₃	C₂²×C₄	1	2	yes	—
33	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₁₀₃	#41	1	2	yes	—
36	C₂×Dic₁₀₃	—	2	4	no	—
37	C₂×Dic₁₀₃	—	2	2	no	—
38	C₂×Dic₁₀₃	—	2	4	no	—
39	C₂×Dic₁₀₃	C₄²	1	4	yes	—
40	C₈×C₁₀₃	—	2	4	no	—
41	C₁₀₃⋊C₈	—	2	4	no	—
42	C₂²×C₁₀₃	C₂×C₄²	1	1	yes	characteristic
43	C₂²×C₁₀₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₁₀₃	#117	1	1	yes	characteristic
45	C₂²×C₁₀₃	#131	1	4	yes	—
46	C₄×C₁₀₃	—	2	4	no	—
47	C₄×C₁₀₃	C₄×D₄	1	2	yes	—
48	C₄×C₁₀₃	C₄×Q₈	1	2	yes	—
49	Dic₁₀₃	—	2	4	no	—
50	Dic₁₀₃	—	4	2	no	—
51	C₂×C₁₀₃	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₁₀₃	#405	1	2	yes	—
53	C₂×C₁₀₃	#424	1	2	yes	—
54	C₂×C₁₀₃	#433	1	2	yes	—
55	#1768	—	103	1	no	maximal, Sylow
56	C₁₀₃	#1768	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₄⋊C₈	—	103	2	no	—
58	C₂×#97	—	103	1	no	—
59	C₂×C₄×C₈	D₁₀₃	1	1	yes	characteristic
60	#376	—	103	1	no	—
61	#336	—	103	2	no	—
62	C₂²×C₈	—	103	2	no	—
63	C₂²×C₈	C₂×D₁₀₃	1	2	yes	—
64	C₂×C₄²	—	103	1	no	—
65	C₂×C₄²	—	103	1	no	—
66	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
67	C₄⋊C₈	—	103	8	no	—
68	#97	—	206	2	no	—
69	C₄×C₈	—	2	2	no	—
70	#95	—	103	4	no	—
71	C₂²×C₄	—	103	2	no	—
72	C₂²×C₄	—	103	2	no	—
73	C₂²×C₄	C₄×D₁₀₃	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	103	4	no	—
78	C₂×C₈	—	206	2	no	—
79	C₂×C₈	—	206	2	no	—
80	C₂×C₈	C₄×D₁₀₃	1	4	yes	—
81	C₄²	—	103	4	no	—
82	C₄²	—	206	2	no	—
83	C₄²	Dic₂₀₆	1	2	yes	—
84	C₄²	C₂²⋊D₁₀₃	1	2	yes	—
85	C₂³	C₂×C₄×D₁₀₃	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	103	4	no	—
88	C₂×C₄	—	206	2	no	—
89	C₂×C₄	—	206	4	no	—
90	C₂×C₄	—	206	4	no	—
91	C₂×C₄	Dic₁₀₃⋊C₄	1	4	yes	—
92	C₂×C₄	C₂×C₄×D₁₀₃	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₂₀₆	1	1	yes	characteristic
95	C₂×C₄	#13110105	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	206	4	no	—
98	C₂²	C₄²×D₁₀₃	1	1	yes	characteristic
99	C₂²	C₂×Dic₁₀₃⋊C₄	1	1	yes	characteristic
100	C₂²	#15692978	1	1	yes	characteristic
101	C₂²	#15693027	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	412	2	no	—
104	C₄	—	206	4	no	—
105	C₄	C₄×C₂²⋊D₁₀₃	1	2	yes	—
106	C₄	C₄×Dic₂₀₆	1	2	yes	—
107	C₂	C₄×Dic₁₀₃⋊C₄	1	1	yes	characteristic
108	C₂	#20267745	1	2	yes	—
109	C₂	#20267817	1	2	yes	—
110	C₂	#20267850	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₀₃⋊C₈

C₂

yes

maximal

C₂×#15692932

C₂

yes

maximal, characteristic

#20267637

C₂

yes

maximal, characteristic

#20267545

C₂

yes

maximal

C₂×C₄×C₈×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×C₁₀₃⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₀₃

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₃

C₂²

yes

characteristic

Dic₁₀₃⋊C₈

C₄

yes

complemented

#15692932

—

C₁₀₃⋊#95

C₄

yes

—

C₂²×C₈×C₁₀₃

C₂²

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₄×C₈×C₁₀₃

—

C₂²×Dic₁₀₃

C₂×C₄

yes

—

C₂²×Dic₁₀₃

C₂×C₄

yes

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₀₃

—

C₄×Dic₁₀₃

C₂×C₄

yes

—

C₂²×C₄×C₁₀₃

C₂×C₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

C₂×C₄

yes

—

C₄²×C₁₀₃

D₄

yes

—

C₄²×C₁₀₃

Q₈

yes

—

C₂³×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

C₄²

yes

—

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

C₂×C₄²

yes

characteristic

C₂²×C₁₀₃

C₂×C₄⋊C₄

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₄×D₄

yes

—

C₄×C₁₀₃

C₄×Q₈

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₃

#405

yes

—

C₂×C₁₀₃

#424

yes

—

C₂×C₁₀₃

#433

yes

—

#1768

—

103

maximal, Sylow

C₁₀₃

#1768

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

103

—

C₂×#97

—

103

—

C₂×C₄×C₈

D₁₀₃

yes

characteristic

#376

—

103

—

#336

—

103

—

C₂²×C₈

—

103

—

C₂²×C₈

C₂×D₁₀₃

yes

—

C₂×C₄²

—

103

—

C₂×C₄²

—

103

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

C₄⋊C₈

—

103

—

#97

—

206

—

C₄×C₈

—

#95

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

C₄×D₁₀₃

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

103

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₂×C₈

C₄×D₁₀₃

yes

—

C₄²

—

103

—

C₄²

—

206

—

C₄²

Dic₂₀₆

yes

—

C₄²

C₂²⋊D₁₀₃

yes

—

C₂³

C₂×C₄×D₁₀₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

103

—

C₂×C₄

—

206

—

C₂×C₄

—

206

—

C₂×C₄

—

206

—

C₂×C₄

Dic₁₀₃⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₀₃

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

C₂×C₄

C₂×Dic₂₀₆

yes

characteristic

C₂×C₄

#13110105

yes

—

C₈

—

C₈

—

206

—

C₂²

C₄²×D₁₀₃

yes

characteristic

C₂²

C₂×Dic₁₀₃⋊C₄

yes

characteristic

100

C₂²

#15692978

yes

characteristic

101

C₂²

#15693027

yes

—

102

C₄

—

103

C₄

—

412

—

104

C₄

—

206

—

105

C₄

C₄×C₂²⋊D₁₀₃

yes

—

106

C₄

C₄×Dic₂₀₆

yes

—

107

C₂

C₄×Dic₁₀₃⋊C₄

yes

characteristic

108

C₂

#20267745

yes

—

109

C₂

#20267817

yes

—

110

C₂

#20267850

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₃
commutator	C₂²×C₁₀₃	C₂×C₄²
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	#1768
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₃

commutator

C₂²×C₁₀₃

C₂×C₄²

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 ▷ C₂²×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. Dic₁₀₃⋊C₈	C₄	semidirect
56. C₁₀₃	#1768	semidirect

H ≅ G/N

type

9. Dic₁₀₃⋊C₈

C₄

semidirect

56. C₁₀₃

#1768

semidirect

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

subgroup	index
C₂×Dic₁₀₃⋊C₈	2
C₂×#15692932	2
#20267637	2
#20267545	2
C₂×C₄×C₈×C₁₀₃	2
#1768	103

subgroup

index

C₂×Dic₁₀₃⋊C₈

C₂×C₄×C₈×C₁₀₃

103

Subgroups — #27195002

Conjugacy classes of subgroups

Complements

Maximal subgroups