Group #27195358 — subgroups

Subgroups — #27195358

Order 13184.

272 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 89 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	#20268269	C₂	1	1	yes	maximal, characteristic
2	#20267831	C₂	1	2	yes	maximal
3	C₂×#15692940	C₂	1	1	yes	maximal, characteristic
4	#20267551	C₂	1	2	yes	maximal
5	#391×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13110089	C₂²	1	2	yes	—
7	Q₈×Dic₁₀₃	—	2	2	no	—
8	Q₈×Dic₁₀₃	—	2	2	no	—
9	#15692980	—	2	2	no	—
10	#15692980	C₂²	1	2	yes	—
11	#15692954	—	2	2	no	—
12	#15692952	—	2	2	no	—
13	#15692940	C₄	1	4	yes	complemented
14	C₂²×Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₀₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
17	#95×C₁₀₃	—	2	2	no	—
18	#13110089	—	2	2	no	—
19	#13110089	C₂×C₄	1	4	yes	—
20	C₂×C₁₀₃⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₀₃	—	2	2	no	—
22	C₄⋊Dic₁₀₃	—	2	2	no	—
23	C₄⋊Dic₁₀₃	D₄	1	2	yes	—
24	C₄⋊Dic₁₀₃	D₄	1	2	yes	—
25	C₂²⋊Dic₁₀₃	—	2	2	no	—
26	C₄×Dic₁₀₃	—	2	4	no	—
27	C₄×Dic₁₀₃	—	2	2	no	—
28	C₂×Q₈×C₁₀₃	—	2	2	no	—
29	C₂×Q₈×C₁₀₃	—	2	2	no	—
30	C₂×Q₈×C₁₀₃	C₂×C₄	1	2	yes	—
31	C₂×Q₈×C₁₀₃	Q₈	1	2	yes	—
32	C₂²×C₄×C₁₀₃	—	2	1	no	—
33	C₂²×C₄×C₁₀₃	—	2	1	no	—
34	C₂²×C₄×C₁₀₃	D₄	1	2	yes	—
35	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₀₃	—	2	2	no	—
37	C₄²×C₁₀₃	—	2	2	no	—
38	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₁₀₃	—	2	4	no	—
40	Q₈×C₁₀₃	—	2	4	no	—
41	C₂×C₄×C₁₀₃	—	2	2	no	—
42	C₂×C₄×C₁₀₃	—	2	2	no	—
43	C₂×C₄×C₁₀₃	—	2	2	no	—
44	C₂×C₄×C₁₀₃	—	4	1	no	—
45	C₂×C₄×C₁₀₃	—	4	1	no	—
46	C₂×C₄×C₁₀₃	—	4	2	no	—
47	C₂×C₄×C₁₀₃	—	4	1	no	—
48	C₂×C₄×C₁₀₃	C₄⋊C₄	1	4	yes	—
49	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₀₃	#41	1	2	yes	—
53	C₂×Dic₁₀₃	—	2	2	no	—
54	C₂×Dic₁₀₃	—	2	2	no	—
55	C₁₀₃⋊C₈	—	2	4	no	—
56	C₂²×C₁₀₃	—	2	1	no	—
57	C₂²×C₁₀₃	—	2	1	no	—
58	C₂²×C₁₀₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₀₃	#120	1	1	yes	characteristic
60	C₂²×C₁₀₃	#123	1	1	yes	characteristic
61	C₄×C₁₀₃	—	2	4	no	—
62	C₄×C₁₀₃	—	4	2	no	—
63	C₄×C₁₀₃	—	8	1	no	—
64	C₄×C₁₀₃	C₄×D₄	1	2	yes	—
65	C₄×C₁₀₃	C₂²⋊Q₈	1	2	yes	—
66	Dic₁₀₃	—	4	2	no	—
67	Dic₁₀₃	—	4	2	no	—
68	C₂×C₁₀₃	—	2	1	no	—
69	C₂×C₁₀₃	—	2	1	no	—
70	C₂×C₁₀₃	#385	1	1	yes	characteristic
71	C₂×C₁₀₃	#454	1	1	yes	characteristic
72	C₂×C₁₀₃	#456	1	1	yes	characteristic
73	#1857	—	103	1	no	maximal, Sylow
74	C₁₀₃	#1857	1	1	yes	characteristic, complemented, Sylow
75	#519	—	103	1	no	—
76	#428	—	103	2	no	—
77	C₂×#101	—	103	1	no	—
78	#391	D₁₀₃	1	1	yes	characteristic
79	#339	—	103	2	no	—
80	C₂²×Q₈	C₂×D₁₀₃	1	1	yes	characteristic
81	C₂×M₁₆	—	103	2	no	—
82	C₄×Q₈	—	206	2	no	—
83	C₄×Q₈	—	206	2	no	—
84	#117	—	103	2	no	—
85	#117	—	206	2	no	—
86	C₂×C₄⋊C₄	C₂×D₁₀₃	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
88	#107	—	206	2	no	—
89	#106	—	206	2	no	—
90	#101	—	103	4	no	—
91	#95	—	2	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	Dic₂₀₆	1	2	yes	—
95	C₂×Q₈	C₄×D₁₀₃	1	2	yes	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²⋊D₁₀₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
100	M₁₆	—	103	4	no	—
101	M₁₆	—	206	2	no	—
102	C₂×C₈	—	206	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	103	2	no	—
105	C₄⋊C₄	—	103	2	no	—
106	C₄⋊C₄	—	206	2	no	—
107	C₄⋊C₄	—	206	2	no	—
108	C₂²⋊C₄	—	206	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	206	4	no	—
111	C₄²	—	206	2	no	—
112	C₂³	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	206	2	no	—
123	C₂×C₄	—	206	2	no	—
124	C₂×C₄	Dic₁₀₃⋊C₄	1	4	yes	—
125	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
126	C₂×C₄	D₄×D₁₀₃	1	2	yes	—
127	C₂×C₄	C₂×Dic₂₀₆	1	1	yes	characteristic
128	C₂×C₄	#13110104	1	2	yes	—
129	C₈	—	206	4	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×Dic₁₀₃⋊C₄	1	1	yes	characteristic
133	C₂²	#15692989	1	1	yes	characteristic
134	C₂²	#15693002	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	4	2	no	—
137	C₄	—	8	1	no	—
138	C₄	—	412	2	no	—
139	C₄	—	412	2	no	—
140	C₄	(C₂²⋊D₁₀₃)⋊C₄	1	2	yes	—
141	C₄	#15692998	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#20267663	1	1	yes	characteristic
145	C₂	#20267963	1	1	yes	characteristic
146	C₂	#20267973	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20268269

C₂

yes

maximal, characteristic

#20267831

C₂

yes

maximal

C₂×#15692940

C₂

yes

maximal, characteristic

#20267551

C₂

yes

maximal

#391×C₁₀₃

C₂

yes

maximal, characteristic

C₂×#13110089

C₂²

yes

—

Q₈×Dic₁₀₃

—

Q₈×Dic₁₀₃

—

#15692980

—

#15692980

C₂²

yes

—

#15692954

—

#15692952

—

#15692940

C₄

yes

complemented

C₂²×Q₈×C₁₀₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₃

C₂²

yes

characteristic

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

#95×C₁₀₃

—

#13110089

—

#13110089

C₂×C₄

yes

—

C₂×C₁₀₃⋊C₈

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

D₄

yes

—

C₄⋊Dic₁₀₃

D₄

yes

—

C₂²⋊Dic₁₀₃

—

C₄×Dic₁₀₃

—

C₄×Dic₁₀₃

—

C₂×Q₈×C₁₀₃

—

C₂×Q₈×C₁₀₃

—

C₂×Q₈×C₁₀₃

C₂×C₄

yes

—

C₂×Q₈×C₁₀₃

Q₈

yes

—

C₂²×C₄×C₁₀₃

—

C₂²×C₄×C₁₀₃

—

C₂²×C₄×C₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

—

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

Q₈×C₁₀₃

—

Q₈×C₁₀₃

—

C₂×C₄×C₁₀₃

—

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

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

C₂×C₄⋊C₄

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₄×D₄

yes

—

C₄×C₁₀₃

C₂²⋊Q₈

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

103

maximal, Sylow

C₁₀₃

#1857

yes

characteristic, complemented, Sylow

#519

—

103

—

#428

—

103

—

C₂×#101

—

103

—

#391

D₁₀₃

yes

characteristic

#339

—

103

—

C₂²×Q₈

C₂×D₁₀₃

yes

characteristic

C₂×M₁₆

—

103

—

C₄×Q₈

—

206

—

C₄×Q₈

—

206

—

#117

—

103

—

#117

—

206

—

C₂×C₄⋊C₄

C₂×D₁₀₃

yes

characteristic

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

#107

—

206

—

#106

—

206

—

#101

—

103

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₂₀₆

yes

—

C₂×Q₈

C₄×D₁₀₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₀₃

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

100

M₁₆

—

103

—

101

M₁₆

—

206

—

102

C₂×C₈

—

206

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

103

—

105

C₄⋊C₄

—

103

—

106

C₄⋊C₄

—

206

—

107

C₄⋊C₄

—

206

—

108

C₂²⋊C₄

—

206

—

109

C₄²

—

110

C₄²

—

206

—

111

C₄²

—

206

—

112

C₂³

C₂×C₂²⋊D₁₀₃

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

206

—

123

C₂×C₄

—

206

—

124

C₂×C₄

Dic₁₀₃⋊C₄

yes

—

125

C₂×C₄

C₂×C₄×D₁₀₃

yes

characteristic

126

C₂×C₄

D₄×D₁₀₃

yes

—

127

C₂×C₄

C₂×Dic₂₀₆

yes

characteristic

128

C₂×C₄

#13110104

yes

—

129

C₈

—

206

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×Dic₁₀₃⋊C₄

yes

characteristic

133

C₂²

#15692989

yes

characteristic

134

C₂²

#15693002

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

412

—

139

C₄

—

412

—

140

C₄

(C₂²⋊D₁₀₃)⋊C₄

yes

—

141

C₄

#15692998

yes

—

142

C₂

—

143

C₂

—

144

C₂

#20267663

yes

characteristic

145

C₂

#20267963

yes

characteristic

146

C₂

#20267973

yes

characteristic

	subgroup	G/H
center	C₂²	#15692989
commutator	C₂×C₄×C₁₀₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#391×C₁₀₃	C₂
socle	C₂²×C₁₀₃	#120
prime radical	C₂³×C₁₀₃	C₂×D₄
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	#391	D₁₀₃
hypocenter	C₁₀₃	#1857
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

subgroup

G/H

center

C₂²

#15692989

commutator

C₂×C₄×C₁₀₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#391×C₁₀₃

C₂

socle

C₂²×C₁₀₃

#120

prime radical

C₂³×C₁₀₃

C₂×D₄

cosocle

C₂²×C₄×C₁₀₃

C₂³

D₁₀₃

C₁₀₃

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

N	H ≅ G/N	type
13. #15692940	C₄	semidirect
74. C₁₀₃	#1857	semidirect

H ≅ G/N

type

13. #15692940

C₄

semidirect

74. C₁₀₃

#1857

semidirect

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

subgroup	index
#20268269	2
#20267831	2
C₂×#15692940	2
#20267551	2
#391×C₁₀₃	2
#1857	103

subgroup

index

#391×C₁₀₃

103

Subgroups — #27195358

Conjugacy classes of subgroups

Complements

Maximal subgroups