Group #27194263 — subgroups

Subgroups — #27194263

Order 13184.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#20268527	C₂	1	1	yes	maximal, characteristic
2	#20268083	C₂	1	2	yes	maximal, complemented
3	#20267535	C₂	1	1	yes	maximal, characteristic
4	#20267523	C₂	1	2	yes	maximal, complemented
5	#492×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	#15693008	—	2	2	no	—
7	#15693002	—	2	2	no	—
8	C₄×Dic₂₀₆	C₂²	1	2	yes	—
9	D₄×Dic₁₀₃	—	2	2	no	—
10	#15692954	—	2	2	no	—
11	C₄⋊(C₁₀₃⋊C₈)	C₂²	1	2	yes	—
12	#15692944	—	2	2	no	—
13	C₂×SD₁₆×C₁₀₃	—	2	2	no	—
14	C₄⋊Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
15	#127×C₁₀₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₀₆	D₄	1	2	yes	—
18	C₂²×Dic₁₀₃	—	2	2	no	—
19	C₂×C₁₀₃⋊C₈	—	2	2	no	—
20	C₄⋊Dic₁₀₃	D₄	1	2	yes	—
21	Dic₁₀₃⋊C₄	—	2	2	no	—
22	C₂²⋊Dic₁₀₃	—	2	4	no	—
23	C₂²⋊Dic₁₀₃	—	2	2	no	—
24	C₄×Dic₁₀₃	—	2	2	no	—
25	C₂×Q₈×C₁₀₃	—	2	1	no	—
26	C₂×D₄×C₁₀₃	—	2	2	no	—
27	C₂×D₄×C₁₀₃	D₄	1	2	yes	—
28	SD₁₆×C₁₀₃	—	4	4	no	—
29	C₂×C₈×C₁₀₃	—	2	1	no	—
30	C₄⋊C₄×C₁₀₃	—	2	2	no	—
31	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
32	C₂³×C₁₀₃	—	2	2	no	—
33	Q₈×C₁₀₃	—	4	2	no	—
34	D₄×C₁₀₃	—	4	2	no	—
35	D₄×C₁₀₃	—	4	2	no	—
36	D₄×C₁₀₃	—	4	2	no	—
37	C₂×C₄×C₁₀₃	—	4	1	no	—
38	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₁₀₃	—	2	2	no	—
41	C₂×Dic₁₀₃	—	2	2	no	—
42	C₂×Dic₁₀₃	—	8	2	no	—
43	Dic₂₀₆	—	2	2	no	—
44	Dic₂₀₆	—	2	2	no	—
45	C₈×C₁₀₃	—	2	2	no	—
46	C₁₀₃⋊C₈	—	4	2	no	—
47	C₂²×C₁₀₃	—	4	4	no	—
48	C₂²×C₁₀₃	—	4	2	no	—
49	C₂²×C₁₀₃	#120	1	1	yes	characteristic
50	C₄×C₁₀₃	—	2	1	no	—
51	C₄×C₁₀₃	—	8	1	no	—
52	C₄×C₁₀₃	#135	1	2	yes	—
53	C₄×C₁₀₃	#137	1	2	yes	—
54	Dic₁₀₃	—	4	2	no	—
55	Dic₁₀₃	—	4	2	no	—
56	C₂×C₁₀₃	—	8	2	no	—
57	C₂×C₁₀₃	#452	1	2	yes	—
58	C₂×C₁₀₃	#453	1	1	yes	characteristic
59	#1630	—	103	1	no	maximal, Sylow
60	C₁₀₃	#1630	1	1	yes	characteristic, complemented, Sylow
61	#559	—	103	1	no	—
62	#492	D₁₀₃	1	1	yes	characteristic
63	#477	—	103	2	no	—
64	#331	—	103	1	no	—
65	Q₈⋊C₈	—	103	2	no	—
66	C₂×SD₁₆	—	2	2	no	—
67	C₄⋊Q₈	C₂×D₁₀₃	1	1	yes	characteristic
68	#127	C₂×D₁₀₃	1	1	yes	characteristic
69	#124	—	206	2	no	—
70	#123	—	206	2	no	—
71	C₄×Q₈	—	103	2	no	—
72	C₄×D₄	—	206	2	no	—
73	#107	—	206	2	no	—
74	C₄⋊C₈	—	103	2	no	—
75	#103	—	206	2	no	—
76	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	—	103	2	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	C₂²⋊D₁₀₃	1	2	yes	—
81	C₂²×C₄	—	206	2	no	—
82	SD₁₆	—	4	4	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	206	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	103	2	no	—
87	C₄⋊C₄	—	206	2	no	—
88	C₂²⋊C₄	—	206	4	no	—
89	C₂²⋊C₄	—	206	2	no	—
90	C₄²	—	206	2	no	—
91	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	4	2	no	—
94	Q₈	—	206	2	no	—
95	Q₈	—	206	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	824	2	no	—
101	C₂×C₄	—	206	2	no	—
102	C₂×C₄	—	206	2	no	—
103	C₂×C₄	D₄×D₁₀₃	1	2	yes	—
104	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	412	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#15692989	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	8	1	no	—
112	C₄	—	412	2	no	—
113	C₄	—	412	2	no	—
114	C₄	#15693042	1	2	yes	—
115	C₄	#15693050	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#20267950	1	2	yes	—
118	C₂	#20267958	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20268527

C₂

yes

maximal, characteristic

#20268083

C₂

yes

maximal, complemented

#20267535

C₂

yes

maximal, characteristic

#20267523

C₂

yes

maximal, complemented

#492×C₁₀₃

C₂

yes

maximal, characteristic

#15693008

—

#15693002

—

C₄×Dic₂₀₆

C₂²

yes

—

D₄×Dic₁₀₃

—

#15692954

—

C₄⋊(C₁₀₃⋊C₈)

C₂²

yes

—

#15692944

—

C₂×SD₁₆×C₁₀₃

—

C₄⋊Q₈×C₁₀₃

C₂²

yes

characteristic

#127×C₁₀₃

C₂²

yes

characteristic

C₄×C₈×C₁₀₃

C₂²

yes

characteristic

C₂×Dic₂₀₆

D₄

yes

—

C₂²×Dic₁₀₃

—

C₂×C₁₀₃⋊C₈

—

C₄⋊Dic₁₀₃

D₄

yes

—

Dic₁₀₃⋊C₄

—

C₂²⋊Dic₁₀₃

—

C₂²⋊Dic₁₀₃

—

C₄×Dic₁₀₃

—

C₂×Q₈×C₁₀₃

—

C₂×D₄×C₁₀₃

—

C₂×D₄×C₁₀₃

D₄

yes

—

SD₁₆×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

C₂³

yes

characteristic

C₂³×C₁₀₃

—

Q₈×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

Dic₂₀₆

—

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#120

yes

characteristic

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#135

yes

—

C₄×C₁₀₃

#137

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#452

yes

—

C₂×C₁₀₃

#453

yes

characteristic

#1630

—

103

maximal, Sylow

C₁₀₃

#1630

yes

characteristic, complemented, Sylow

#559

—

103

—

#492

D₁₀₃

yes

characteristic

#477

—

103

—

#331

—

103

—

Q₈⋊C₈

—

103

—

C₂×SD₁₆

—

C₄⋊Q₈

C₂×D₁₀₃

yes

characteristic

#127

C₂×D₁₀₃

yes

characteristic

#124

—

206

—

#123

—

206

—

C₄×Q₈

—

103

—

C₄×D₄

—

206

—

#107

—

206

—

C₄⋊C₈

—

103

—

#103

—

206

—

C₄×C₈

C₂×D₁₀₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

103

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₀₃

yes

—

C₂²×C₄

—

206

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

206

—

C₄⋊C₄

—

C₄⋊C₄

—

103

—

C₄⋊C₄

—

206

—

C₂²⋊C₄

—

206

—

C₂²⋊C₄

—

206

—

C₄²

—

206

—

C₄²

C₂²×D₁₀₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

206

—

Q₈

—

206

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

824

—

101

C₂×C₄

—

206

—

102

C₂×C₄

—

206

—

103

C₂×C₄

D₄×D₁₀₃

yes

—

104

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

105

C₈

—

106

C₈

—

412

—

107

C₂²

—

108

C₂²

—

109

C₂²

#15692989

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

412

—

113

C₄

—

412

—

114

C₄

#15693042

yes

—

115

C₄

#15693050

yes

—

116

C₂

—

117

C₂

#20267950

yes

—

118

C₂

#20267958

yes

characteristic

	subgroup	G/H
center	C₂²	#15692989
commutator, cosocle	C₄²×C₁₀₃	C₂³
Frattini	C₄²	C₂²×D₁₀₃
Fitting	#492×C₁₀₃	C₂
socle	C₂²×C₁₀₃	#120
prime radical	#127×C₁₀₃	C₂²
hypercenter	#492	D₁₀₃
hypocenter	C₁₀₃	#1630
LCS	G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#15692989

commutator, cosocle

C₄²×C₁₀₃

C₂³

Frattini

C₄²

C₂²×D₁₀₃

Fitting

#492×C₁₀₃

C₂

socle

C₂²×C₁₀₃

#120

prime radical

#127×C₁₀₃

C₂²

D₁₀₃

C₁₀₃

G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #20268083	C₂	semidirect
4. #20267523	C₂	semidirect
60. C₁₀₃	#1630	semidirect

H ≅ G/N

type

2. #20268083

C₂

semidirect

4. #20267523

C₂

semidirect

60. C₁₀₃

#1630

semidirect

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

subgroup	index
#20268527	2
#20268083	2
#20267535	2
#20267523	2
#492×C₁₀₃	2
#1630	103

subgroup

index

#492×C₁₀₃

103

Subgroups — #27194263

Conjugacy classes of subgroups

Complements

Maximal subgroups