Group #27196413 — subgroups

Subgroups — #27196413

Order 13184.

256 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂²×Dic₄₁₂	C₂	1	1	yes	maximal, characteristic
2	#20267817	C₂	1	1	yes	maximal, characteristic
3	#20267801	C₂	1	2	yes	maximal
4	C₂×#15692939	C₂	1	2	yes	maximal
5	#341×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀₆	C₂²	1	2	yes	—
7	C₂×Dic₄₁₂	—	2	4	no	—
8	C₂×Dic₄₁₂	—	2	2	no	—
9	C₂×#13110089	C₂²	1	1	yes	characteristic
10	#15692980	C₂²	1	1	yes	characteristic
11	Dic₁₀₃⋊C₈	—	2	2	no	—
12	#15692946	—	2	4	no	—
13	#15692939	—	2	4	no	—
14	C₂×M₁₆×C₁₀₃	C₂²	1	2	yes	—
15	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₀₆	—	2	2	no	—
17	C₂×Dic₂₀₆	—	2	2	no	—
18	C₂×Dic₂₀₆	—	2	2	no	—
19	C₂×Dic₂₀₆	—	2	2	no	—
20	C₂×Dic₂₀₆	—	2	2	no	—
21	C₂×Dic₂₀₆	D₄	1	4	yes	—
22	C₂²×Dic₁₀₃	—	2	2	no	—
23	Dic₄₁₂	—	2	8	no	—
24	#13110089	—	2	2	no	—
25	C₂×C₁₀₃⋊C₈	—	2	1	no	—
26	C₄⋊Dic₁₀₃	D₄	1	2	yes	—
27	C₂²⋊Dic₁₀₃	—	2	1	no	—
28	C₄×Dic₁₀₃	—	2	1	no	—
29	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
30	M₁₆×C₁₀₃	—	2	2	no	—
31	M₁₆×C₁₀₃	—	2	2	no	—
32	C₂×C₈×C₁₀₃	—	2	2	no	—
33	C₂×C₈×C₁₀₃	—	2	2	no	—
34	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
35	C₂³×C₁₀₃	#41	1	1	yes	characteristic
36	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₃	#41	1	2	yes	—
40	C₂×Dic₁₀₃	—	2	2	no	—
41	C₂×Dic₁₀₃	—	2	4	no	—
42	C₂×Dic₁₀₃	—	4	2	no	—
43	C₂×Dic₁₀₃	—	4	2	no	—
44	Dic₂₀₆	—	2	4	no	—
45	Dic₂₀₆	—	2	4	no	—
46	Dic₂₀₆	—	4	2	no	—
47	Dic₂₀₆	—	4	2	no	—
48	C₈×C₁₀₃	—	2	2	no	—
49	C₈×C₁₀₃	—	4	2	no	—
50	C₁₀₃⋊C₈	—	4	1	no	—
51	C₂²×C₁₀₃	—	2	2	no	—
52	C₂²×C₁₀₃	#121	1	1	yes	characteristic
53	C₂²×C₁₀₃	#123	1	2	yes	—
54	C₄×C₁₀₃	#120	1	1	yes	characteristic
55	C₄×C₁₀₃	#121	1	2	yes	—
56	C₄×C₁₀₃	#124	1	1	yes	characteristic
57	Dic₁₀₃	—	4	4	no	—
58	Dic₁₀₃	—	4	2	no	—
59	C₂×C₁₀₃	—	2	2	no	—
60	C₂×C₁₀₃	#394	1	1	yes	characteristic
61	C₂×C₁₀₃	#473	1	2	yes	—
62	#2047	—	103	1	no	maximal, Sylow
63	C₁₀₃	#2047	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₁₆	—	103	1	no	—
65	#424	—	103	1	no	—
66	#419	—	103	2	no	—
67	C₂×#101	—	103	2	no	—
68	#341	D₁₀₃	1	1	yes	characteristic
69	C₂²×Q₈	—	103	2	no	—
70	C₂×Q₁₆	—	206	4	no	—
71	C₂×Q₁₆	—	206	2	no	—
72	C₂×M₁₆	—	103	1	no	—
73	C₂×M₁₆	C₂×D₁₀₃	1	2	yes	—
74	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
75	#117	—	103	1	no	—
76	C₄⋊C₈	—	206	2	no	—
77	#103	—	206	4	no	—
78	#101	—	206	4	no	—
79	C₂×Q₈	—	103	4	no	—
80	C₂×Q₈	—	206	2	no	—
81	C₂×Q₈	—	206	2	no	—
82	C₂×Q₈	—	206	2	no	—
83	C₂×Q₈	—	206	2	no	—
84	C₂×Q₈	—	206	2	no	—
85	C₂²×C₄	—	206	2	no	—
86	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
87	Q₁₆	—	206	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	206	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	206	1	no	—
94	C₂×C₈	D₄₁₂	1	2	yes	—
95	C₄⋊C₄	—	103	2	no	—
96	C₂²⋊C₄	—	206	1	no	—
97	C₄²	—	206	1	no	—
98	C₂³	#13110106	1	1	yes	characteristic
99	Q₈	—	412	2	no	—
100	Q₈	—	412	2	no	—
101	Q₈	—	206	4	no	—
102	Q₈	—	206	4	no	—
103	C₂×C₄	—	412	2	no	—
104	C₂×C₄	—	412	2	no	—
105	C₂×C₄	—	206	2	no	—
106	C₂×C₄	—	206	4	no	—
107	C₂×C₄	D₄×D₁₀₃	1	2	yes	—
108	C₂×C₄	D₄×D₁₀₃	1	1	yes	characteristic
109	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
110	C₂×C₄	#13110105	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	412	1	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#15692996	1	1	yes	characteristic
116	C₂²	#15693006	1	2	yes	—
117	C₄	—	412	2	no	—
118	C₄	—	412	4	no	—
119	C₄	#15692991	1	1	yes	characteristic
120	C₄	#15692994	1	2	yes	—
121	C₄	#15693009	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#20267709	1	1	yes	characteristic
124	C₂	#20268067	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₄₁₂

C₂

yes

maximal, characteristic

#20267817

C₂

yes

maximal, characteristic

#20267801

C₂

yes

maximal

C₂×#15692939

C₂

yes

maximal

#341×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₆

C₂²

yes

—

C₂×Dic₄₁₂

—

C₂×Dic₄₁₂

—

C₂×#13110089

C₂²

yes

characteristic

#15692980

C₂²

yes

characteristic

Dic₁₀₃⋊C₈

—

#15692946

—

#15692939

—

C₂×M₁₆×C₁₀₃

C₂²

yes

—

C₂²×C₈×C₁₀₃

C₂²

yes

characteristic

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

D₄

yes

—

C₂²×Dic₁₀₃

—

Dic₄₁₂

—

#13110089

—

C₂×C₁₀₃⋊C₈

—

C₄⋊Dic₁₀₃

D₄

yes

—

C₂²⋊Dic₁₀₃

—

C₄×Dic₁₀₃

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

M₁₆×C₁₀₃

—

M₁₆×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₂³×C₁₀₃

#41

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

Dic₂₀₆

—

Dic₂₀₆

—

Dic₂₀₆

—

C₈×C₁₀₃

—

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#121

yes

characteristic

C₂²×C₁₀₃

#123

yes

—

C₄×C₁₀₃

#120

yes

characteristic

C₄×C₁₀₃

#121

yes

—

C₄×C₁₀₃

#124

yes

characteristic

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#394

yes

characteristic

C₂×C₁₀₃

#473

yes

—

#2047

—

103

maximal, Sylow

C₁₀₃

#2047

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

103

—

#424

—

103

—

#419

—

103

—

C₂×#101

—

103

—

#341

D₁₀₃

yes

characteristic

C₂²×Q₈

—

103

—

C₂×Q₁₆

—

206

—

C₂×Q₁₆

—

206

—

C₂×M₁₆

—

103

—

C₂×M₁₆

C₂×D₁₀₃

yes

—

C₂²×C₈

C₂×D₁₀₃

yes

characteristic

#117

—

103

—

C₄⋊C₈

—

206

—

#103

—

206

—

#101

—

206

—

C₂×Q₈

—

103

—

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

C₂²×C₄

—

206

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

Q₁₆

—

206

—

M₁₆

—

M₁₆

—

M₁₆

—

206

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

206

—

C₂×C₈

D₄₁₂

yes

—

C₄⋊C₄

—

103

—

C₂²⋊C₄

—

206

—

C₄²

—

206

—

C₂³

#13110106

yes

characteristic

Q₈

—

412

—

100

Q₈

—

412

—

101

Q₈

—

206

—

102

Q₈

—

206

—

103

C₂×C₄

—

412

—

104

C₂×C₄

—

412

—

105

C₂×C₄

—

206

—

106

C₂×C₄

—

206

—

107

C₂×C₄

D₄×D₁₀₃

yes

—

108

C₂×C₄

D₄×D₁₀₃

yes

characteristic

109

C₂×C₄

C₂×D₄₁₂

yes

characteristic

110

C₂×C₄

#13110105

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

412

—

114

C₂²

—

115

C₂²

#15692996

yes

characteristic

116

C₂²

#15693006

yes

—

117

C₄

—

412

—

118

C₄

—

412

—

119

C₄

#15692991

yes

characteristic

120

C₄

#15692994

yes

—

121

C₄

#15693009

yes

characteristic

122

C₂

—

123

C₂

#20267709

yes

characteristic

124

C₂

#20268067

yes

—

	subgroup	G/H
center	C₂²	#15692996
commutator, cosocle	C₂²×C₄×C₁₀₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#341×C₁₀₃	C₂
socle	C₂²×C₁₀₃	#121
prime radical	C₂³×C₁₀₃	#41
hypercenter	#341	D₁₀₃
hypocenter	C₁₀₃	#2047
LCS	G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#15692996

commutator, cosocle

C₂²×C₄×C₁₀₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#341×C₁₀₃

C₂

C₂²×C₁₀₃

C₂³×C₁₀₃

D₁₀₃

C₁₀₃

G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
63. C₁₀₃	#2047	semidirect

H ≅ G/N

type

63. C₁₀₃

#2047

semidirect

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

subgroup	index
C₂²×Dic₄₁₂	2
#20267817	2
#20267801	2
C₂×#15692939	2
#341×C₁₀₃	2
#2047	103

subgroup

index

C₂²×Dic₄₁₂

#341×C₁₀₃

103

Subgroups — #27196413

Conjugacy classes of subgroups

Complements

Maximal subgroups