Group #27195313 — subgroups

Subgroups — #27195313

Order 13184.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×#15692954	C₂	1	2	yes	maximal
2	C₂×#15692952	C₂	1	2	yes	maximal
3	C₂×#15692932	C₂	1	1	yes	maximal, characteristic
4	#20267661	C₂	1	1	yes	maximal, characteristic
5	#382×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₀₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₀₃	—	2	2	no	—
8	C₂×C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
9	#15692954	—	2	4	no	—
10	#15692952	—	2	4	no	—
11	#15692932	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₄×C₁₀₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₀₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
15	#95×C₁₀₃	—	2	2	no	—
16	C₂²×Dic₁₀₃	—	2	2	no	—
17	C₂×C₁₀₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₁₀₃⋊C₈	D₄	1	4	yes	—
19	C₂×C₁₀₃⋊C₈	Q₈	1	4	yes	—
20	C₄⋊Dic₁₀₃	—	2	2	no	—
21	C₄⋊Dic₁₀₃	—	2	2	no	—
22	C₄⋊Dic₁₀₃	—	4	4	no	—
23	C₂²×C₄×C₁₀₃	—	2	1	no	—
24	C₂²×C₄×C₁₀₃	—	2	1	no	—
25	C₂²×C₄×C₁₀₃	D₄	1	2	yes	—
26	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₀₃	—	2	2	no	—
28	C₄⋊C₄×C₁₀₃	—	2	2	no	—
29	C₄²×C₁₀₃	C₂×C₄	1	2	yes	—
30	C₄²×C₁₀₃	Q₈	1	2	yes	—
31	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₃	—	2	2	no	—
33	C₂×C₄×C₁₀₃	—	2	2	no	—
34	C₂×C₄×C₁₀₃	—	4	1	no	—
35	C₂×C₄×C₁₀₃	—	4	1	no	—
36	C₂×C₄×C₁₀₃	—	4	2	no	—
37	C₂×C₄×C₁₀₃	—	4	2	no	—
38	C₂×C₄×C₁₀₃	C₄⋊C₄	1	4	yes	—
39	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₁₀₃	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×Dic₁₀₃	—	4	2	no	—
44	C₂×Dic₁₀₃	—	4	4	no	—
45	C₁₀₃⋊C₈	C₄⋊C₄	1	8	yes	complemented
46	C₂²×C₁₀₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₀₃	#127	1	1	yes	characteristic
48	C₂²×C₁₀₃	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₁₀₃	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₁₀₃	#137	1	2	yes	—
51	C₄×C₁₀₃	—	2	4	no	—
52	C₄×C₁₀₃	—	8	1	no	—
53	C₄×C₁₀₃	—	8	1	no	—
54	C₄×C₁₀₃	C₂×C₄⋊C₄	1	2	yes	—
55	C₄×C₁₀₃	C₄⋊Q₈	1	2	yes	—
56	Dic₁₀₃	—	8	2	no	—
57	C₂×C₁₀₃	#384	1	1	yes	characteristic
58	C₂×C₁₀₃	#428	1	2	yes	—
59	C₂×C₁₀₃	#496	1	1	yes	characteristic
60	C₂×C₁₀₃	#497	1	1	yes	characteristic
61	C₂×C₁₀₃	#501	1	2	yes	—
62	#1847	—	103	1	no	maximal, Sylow
63	C₁₀₃	#1847	1	1	yes	characteristic, complemented, Sylow
64	C₂×#107	—	103	2	no	—
65	C₂×#106	—	103	2	no	—
66	C₂×#97	—	103	1	no	—
67	#384	—	103	1	no	—
68	#382	D₁₀₃	1	1	yes	characteristic
69	C₂²×C₈	—	103	2	no	—
70	C₂×C₄⋊C₄	—	103	2	no	—
71	C₂×C₄⋊C₄	—	206	2	no	—
72	C₂×C₄⋊C₄	C₂×D₁₀₃	1	1	yes	characteristic
73	C₂×C₄⋊C₄	C₂×D₁₀₃	1	1	yes	characteristic
74	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
75	#107	—	206	4	no	—
76	#106	—	206	4	no	—
77	#97	—	103	4	no	—
78	#95	—	2	2	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	206	2	no	—
82	C₂²×C₄	C₂²⋊D₁₀₃	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
84	C₂×C₈	—	103	4	no	—
85	C₂×C₈	—	103	4	no	—
86	C₂×C₈	—	103	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	412	4	no	—
90	C₄⋊C₄	—	206	2	no	—
91	C₄⋊C₄	—	206	2	no	—
92	C₄²	Dic₂₀₆	1	2	yes	—
93	C₄²	C₄×D₁₀₃	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	412	2	no	—
102	C₂×C₄	—	412	4	no	—
103	C₂×C₄	Dic₁₀₃⋊C₄	1	4	yes	—
104	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
105	C₂×C₄	D₄×D₁₀₃	1	2	yes	—
106	C₂×C₄	Q₈×D₁₀₃	1	2	yes	—
107	C₂×C₄	C₂×Dic₂₀₆	1	1	yes	characteristic
108	C₈	—	103	8	no	—
109	C₂²	C₂×Dic₁₀₃⋊C₄	1	1	yes	characteristic
110	C₂²	#15693017	1	1	yes	characteristic
111	C₂²	#15693019	1	1	yes	characteristic
112	C₂²	#15693046	1	2	yes	—
113	C₂²	#15693051	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	824	2	no	—
118	C₄	C₄⋊C₄×D₁₀₃	1	2	yes	—
119	C₄	#15693020	1	2	yes	—
120	C₂	#20267660	1	1	yes	characteristic
121	C₂	#20267834	1	2	yes	—
122	C₂	#20268181	1	1	yes	characteristic
123	C₂	#20268186	1	1	yes	characteristic
124	C₂	#20268201	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692954

C₂

yes

maximal

C₂×#15692952

C₂

yes

maximal

C₂×#15692932

C₂

yes

maximal, characteristic

#20267661

C₂

yes

maximal, characteristic

#382×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×C₁₀₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₀₃

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

—

#15692954

—

#15692952

—

#15692932

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₀₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₃

C₂²

yes

characteristic

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

#95×C₁₀₃

—

C₂²×Dic₁₀₃

—

C₂×C₁₀₃⋊C₈

C₂×C₄

yes

—

C₂×C₁₀₃⋊C₈

D₄

yes

—

C₂×C₁₀₃⋊C₈

Q₈

yes

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₂²×C₄×C₁₀₃

—

C₂²×C₄×C₁₀₃

—

C₂²×C₄×C₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

C₂×C₄

yes

—

C₄²×C₁₀₃

Q₈

yes

—

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂×Q₈

yes

—

C₂×C₄×C₁₀₃

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₁₀₃⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₁₀₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₃

#127

yes

characteristic

C₂²×C₁₀₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₀₃

#137

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₃

C₄⋊Q₈

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

#384

yes

characteristic

C₂×C₁₀₃

#428

yes

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

#1847

—

103

maximal, Sylow

C₁₀₃

#1847

yes

characteristic, complemented, Sylow

C₂×#107

—

103

—

C₂×#106

—

103

—

C₂×#97

—

103

—

#384

—

103

—

#382

D₁₀₃

yes

characteristic

C₂²×C₈

—

103

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

206

—

C₂×C₄⋊C₄

C₂×D₁₀₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₀₃

yes

characteristic

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

#107

—

206

—

#106

—

206

—

#97

—

103

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

206

—

C₂²×C₄

C₂²⋊D₁₀₃

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

—

103

—

C₂×C₈

—

103

—

C₂×C₈

—

103

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

412

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄²

Dic₂₀₆

yes

—

C₄²

C₄×D₁₀₃

yes

—

C₂³

C₂×C₂²⋊D₁₀₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

412

—

102

C₂×C₄

—

412

—

103

C₂×C₄

Dic₁₀₃⋊C₄

yes

—

104

C₂×C₄

C₂×C₄×D₁₀₃

yes

characteristic

105

C₂×C₄

D₄×D₁₀₃

yes

—

106

C₂×C₄

Q₈×D₁₀₃

yes

—

107

C₂×C₄

C₂×Dic₂₀₆

yes

characteristic

108

C₈

—

103

—

109

C₂²

C₂×Dic₁₀₃⋊C₄

yes

characteristic

110

C₂²

#15693017

yes

characteristic

111

C₂²

#15693019

yes

characteristic

112

C₂²

#15693046

yes

—

113

C₂²

#15693051

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

824

—

118

C₄

C₄⋊C₄×D₁₀₃

yes

—

119

C₄

#15693020

yes

—

120

C₂

#20267660

yes

characteristic

121

C₂

#20267834

yes

—

122

C₂

#20268181

yes

characteristic

123

C₂

#20268186

yes

characteristic

124

C₂

#20268201

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₃
commutator	C₂×C₄×C₁₀₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#382×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂×D₄
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	#382	D₁₀₃
hypocenter	C₁₀₃	#1847
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₃

commutator

C₂×C₄×C₁₀₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#382×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₂×D₄

cosocle

C₂²×C₄×C₁₀₃

C₂³

D₁₀₃

C₁₀₃

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
11. #15692932	C₄	semidirect
45. C₁₀₃⋊C₈	C₄⋊C₄	semidirect
63. C₁₀₃	#1847	semidirect

H ≅ G/N

type

11. #15692932

C₄

semidirect

45. C₁₀₃⋊C₈

C₄⋊C₄

semidirect

63. C₁₀₃

#1847

semidirect

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

subgroup	index
C₂×#15692954	2
C₂×#15692952	2
C₂×#15692932	2
#20267661	2
#382×C₁₀₃	2
#1847	103

subgroup

index

#382×C₁₀₃

103

Subgroups — #27195313

Conjugacy classes of subgroups

Complements

Maximal subgroups