Group #27194997 — subgroups

Subgroups — #27194997

Order 13184.

292 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₄⋊(C₁₀₃⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#15692932	C₂	1	2	yes	maximal
3	#20267546	C₂	1	2	yes	maximal
4	C₄³×C₁₀₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₀₃⋊C₈	C₂²	1	4	yes	—
6	C₄⋊(C₁₀₃⋊C₈)	C₄	1	8	yes	complemented
7	#15692932	—	2	4	no	—
8	C₂×C₄²×C₁₀₃	C₄	1	2	yes	—
9	C₂×C₄²×C₁₀₃	C₄	1	2	yes	—
10	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₀₃⋊C₈	—	2	4	no	—
14	C₂×C₁₀₃⋊C₈	—	2	4	no	—
15	C₂×C₁₀₃⋊C₈	C₂×C₄	1	8	yes	—
16	C₂²×C₄×C₁₀₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₀₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₁₀₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₁₀₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
21	C₄²×C₁₀₃	—	2	4	no	—
22	C₄²×C₁₀₃	—	2	4	no	—
23	C₄²×C₁₀₃	—	2	2	no	—
24	C₄²×C₁₀₃	C₂×C₄	1	4	yes	—
25	C₄²×C₁₀₃	D₄	1	2	yes	—
26	C₄²×C₁₀₃	Q₈	1	2	yes	—
27	C₂³×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₃	—	2	2	no	—
29	C₂×C₄×C₁₀₃	—	2	2	no	—
30	C₂×C₄×C₁₀₃	—	2	2	no	—
31	C₂×C₄×C₁₀₃	—	2	2	no	—
32	C₂×C₄×C₁₀₃	—	2	2	no	—
33	C₂×C₄×C₁₀₃	C₄²	1	2	yes	—
34	C₂×C₄×C₁₀₃	C₄²	1	2	yes	—
35	C₂×C₄×C₁₀₃	C₄⋊C₄	1	4	yes	—
36	C₂×C₄×C₁₀₃	M₁₆	1	4	yes	—
37	C₂×C₄×C₁₀₃	M₁₆	1	4	yes	—
38	C₂×C₄×C₁₀₃	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₀₃	#41	1	2	yes	—
42	C₁₀₃⋊C₈	—	2	8	no	—
43	C₂²×C₁₀₃	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₁₀₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₀₃	#117	1	1	yes	characteristic
46	C₂²×C₁₀₃	C₂×M₁₆	1	2	yes	—
47	C₂²×C₁₀₃	C₂×M₁₆	1	2	yes	—
48	C₄×C₁₀₃	—	2	2	no	—
49	C₄×C₁₀₃	—	2	4	no	—
50	C₄×C₁₀₃	—	2	4	no	—
51	C₄×C₁₀₃	#97	1	4	yes	complemented
52	C₄×C₁₀₃	C₄×D₄	1	2	yes	—
53	C₄×C₁₀₃	C₄×Q₈	1	2	yes	—
54	C₂×C₁₀₃	C₄×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₁₀₃	C₂×#97	1	1	yes	characteristic
56	C₂×C₁₀₃	C₄×M₁₆	1	1	yes	characteristic
57	C₂×C₁₀₃	#423	1	2	yes	—
58	C₂×C₁₀₃	#432	1	2	yes	—
59	#1767	—	103	1	no	maximal, Sylow
60	C₁₀₃	#1767	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₄⋊C₈	—	103	2	no	—
62	C₂×#97	—	103	2	no	—
63	C₄³	D₁₀₃	1	1	yes	characteristic
64	#336	—	103	2	no	—
65	C₂²×C₈	—	103	4	no	—
66	C₂×C₄²	Dic₁₀₃	1	2	yes	—
67	C₂×C₄²	Dic₁₀₃	1	2	yes	—
68	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
69	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
70	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
71	C₄⋊C₈	—	103	8	no	—
72	#97	—	206	4	no	—
73	C₂²×C₄	C₂×Dic₁₀₃	1	1	yes	characteristic
74	C₂²×C₄	C₂×Dic₁₀₃	1	2	yes	—
75	C₂²×C₄	C₂×Dic₁₀₃	1	2	yes	—
76	C₂²×C₄	C₂×Dic₁₀₃	1	1	yes	characteristic
77	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
78	C₂×C₈	—	103	8	no	—
79	C₂×C₈	—	206	4	no	—
80	C₂×C₈	—	206	4	no	—
81	C₄²	—	2	4	no	—
82	C₄²	—	2	4	no	—
83	C₄²	—	2	2	no	—
84	C₄²	Dic₂₀₆	1	2	yes	—
85	C₄²	C₄×D₁₀₃	1	4	yes	—
86	C₄²	D₄₁₂	1	2	yes	—
87	C₂³	C₂²×Dic₁₀₃	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	C₄×Dic₁₀₃	1	2	yes	—
94	C₂×C₄	C₄×Dic₁₀₃	1	2	yes	—
95	C₂×C₄	C₄⋊Dic₁₀₃	1	4	yes	—
96	C₂×C₄	#13110089	1	4	yes	—
97	C₂×C₄	#13110089	1	4	yes	—
98	C₂×C₄	C₂×C₄×D₁₀₃	1	2	yes	—
99	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₂₀₆	1	1	yes	characteristic
101	C₂×C₄	#13110105	1	2	yes	—
102	C₈	—	206	8	no	—
103	C₂²	C₂×C₄×Dic₁₀₃	1	1	yes	characteristic
104	C₂²	C₂×C₄⋊Dic₁₀₃	1	1	yes	characteristic
105	C₂²	#15692980	1	1	yes	characteristic
106	C₂²	C₂×#13110089	1	2	yes	—
107	C₂²	C₂×#13110089	1	2	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	2	4	no	—
110	C₄	—	2	2	no	—
111	C₄	#15692932	1	4	yes	complemented
112	C₄	C₄×D₄₁₂	1	2	yes	—
113	C₄	C₄×Dic₂₀₆	1	2	yes	—
114	C₂	C₄×C₄⋊Dic₁₀₃	1	1	yes	characteristic
115	C₂	C₂×#15692932	1	1	yes	characteristic
116	C₂	C₄×#13110089	1	1	yes	characteristic
117	C₂	#20267815	1	2	yes	—
118	C₂	#20267848	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₀₃⋊C₈)

C₂

yes

maximal

C₂×#15692932

C₂

yes

maximal

#20267546

C₂

yes

maximal

C₄³×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×C₁₀₃⋊C₈

C₂²

yes

—

C₄⋊(C₁₀₃⋊C₈)

C₄

yes

complemented

#15692932

—

C₂×C₄²×C₁₀₃

C₄

yes

—

C₂×C₄²×C₁₀₃

C₄

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₀₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₃

C₂×C₄

yes

—

C₂²×C₄×C₁₀₃

C₂×C₄

yes

—

C₂²×C₄×C₁₀₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₄²×C₁₀₃

—

C₄²×C₁₀₃

—

C₄²×C₁₀₃

—

C₄²×C₁₀₃

C₂×C₄

yes

—

C₄²×C₁₀₃

D₄

yes

—

C₄²×C₁₀₃

Q₈

yes

—

C₂³×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₄²

yes

—

C₂×C₄×C₁₀₃

C₄²

yes

—

C₂×C₄×C₁₀₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₃

M₁₆

yes

—

C₂×C₄×C₁₀₃

M₁₆

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

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

C₂×C₄²

yes

characteristic

C₂²×C₁₀₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₃

#117

yes

characteristic

C₂²×C₁₀₃

C₂×M₁₆

yes

—

C₂²×C₁₀₃

C₂×M₁₆

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#97

yes

complemented

C₄×C₁₀₃

C₄×D₄

yes

—

C₄×C₁₀₃

C₄×Q₈

yes

—

C₂×C₁₀₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₃

C₂×#97

yes

characteristic

C₂×C₁₀₃

C₄×M₁₆

yes

characteristic

C₂×C₁₀₃

#423

yes

—

C₂×C₁₀₃

#432

yes

—

#1767

—

103

maximal, Sylow

C₁₀₃

#1767

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

103

—

C₂×#97

—

103

—

C₄³

D₁₀₃

yes

characteristic

#336

—

103

—

C₂²×C₈

—

103

—

C₂×C₄²

Dic₁₀₃

yes

—

C₂×C₄²

Dic₁₀₃

yes

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

C₄⋊C₈

—

103

—

#97

—

206

—

C₂²×C₄

C₂×Dic₁₀₃

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₃

yes

—

C₂²×C₄

C₂×Dic₁₀₃

yes

—

C₂²×C₄

C₂×Dic₁₀₃

yes

characteristic

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

—

103

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₂₀₆

yes

—

C₄²

C₄×D₁₀₃

yes

—

C₄²

D₄₁₂

yes

—

C₂³

C₂²×Dic₁₀₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₀₃

yes

—

C₂×C₄

C₄×Dic₁₀₃

yes

—

C₂×C₄

C₄⋊Dic₁₀₃

yes

—

C₂×C₄

#13110089

yes

—

C₂×C₄

#13110089

yes

—

C₂×C₄

C₂×C₄×D₁₀₃

yes

—

C₂×C₄

C₂×D₄₁₂

yes

characteristic

100

C₂×C₄

C₂×Dic₂₀₆

yes

characteristic

101

C₂×C₄

#13110105

yes

—

102

C₈

—

206

—

103

C₂²

C₂×C₄×Dic₁₀₃

yes

characteristic

104

C₂²

C₂×C₄⋊Dic₁₀₃

yes

characteristic

105

C₂²

#15692980

yes

characteristic

106

C₂²

C₂×#13110089

yes

—

107

C₂²

C₂×#13110089

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#15692932

yes

complemented

112

C₄

C₄×D₄₁₂

yes

—

113

C₄

C₄×Dic₂₀₆

yes

—

114

C₂

C₄×C₄⋊Dic₁₀₃

yes

characteristic

115

C₂

C₂×#15692932

yes

characteristic

116

C₂

C₄×#13110089

yes

characteristic

117

C₂

#20267815

yes

—

118

C₂

#20267848

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₃
commutator	C₂²×C₁₀₃	C₂×C₄²
Fitting	C₄³×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	C₄³	D₁₀₃
hypocenter	C₁₀₃	#1767
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₃

commutator

C₂²×C₁₀₃

C₂×C₄²

Fitting

C₄³×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₂²×C₄

cosocle

C₂²×C₄×C₁₀₃

C₂³

hypercenter

C₄³

D₁₀₃

C₁₀₃

G ▷ C₂²×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₂²×C₄ ◁ C₄³

N	H ≅ G/N	type
6. C₄⋊(C₁₀₃⋊C₈)	C₄	semidirect
51. C₄×C₁₀₃	#97	semidirect
60. C₁₀₃	#1767	semidirect
111. C₄	#15692932	semidirect

H ≅ G/N

type

6. C₄⋊(C₁₀₃⋊C₈)

C₄

semidirect

51. C₄×C₁₀₃

#97

semidirect

60. C₁₀₃

#1767

semidirect

111. C₄

#15692932

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₀₃⋊C₈)	2
C₂×#15692932	2
#20267546	2
C₄³×C₁₀₃	2
#1767	103

subgroup

index

C₂×C₄⋊(C₁₀₃⋊C₈)

C₂×#15692932

#20267546

C₄³×C₁₀₃

#1767

103

Subgroups — #27194997

Conjugacy classes of subgroups

Complements

Maximal subgroups