Group #27196451 — subgroups

Subgroups — #27196451

Order 13184.

264 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 63 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	#20268591	C₂	1	1	yes	maximal, characteristic
2	#20267819	C₂	1	1	yes	maximal, characteristic, complemented
3	#20267785	C₂	1	2	yes	maximal, complemented
4	#20267764	C₂	1	2	yes	maximal, complemented
5	H(3,ℤ/4)×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13110105	C₂²	1	2	yes	—
7	#15693054	—	2	2	no	—
8	#15693049	—	2	2	no	—
9	#15693043	—	2	2	no	—
10	C₂²×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
11	#15692980	C₂²	1	1	yes	characteristic
12	Dic₁₀₃⋊C₈	—	2	2	no	—
13	#15692944	—	2	2	no	—
14	#15692941	—	2	2	no	—
15	#15692935	—	2	4	no	—
16	C₂×M₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
17	#117×C₁₀₃	C₂²	1	2	yes	—
18	#13110105	—	2	2	no	—
19	#13110105	—	2	2	no	—
20	#13110105	—	2	2	no	—
21	#13110105	—	2	2	no	—
22	C₂×Dic₂₀₆	D₄	1	2	yes	—
23	C₂×D₄₁₂	D₄	1	2	yes	—
24	C₂×C₂²⋊D₁₀₃	—	2	2	no	—
25	C₂×C₄×D₁₀₃	—	2	2	no	—
26	Dic₄₁₂	—	2	2	no	—
27	#13110094	—	2	4	no	—
28	D₈₂₄	—	2	2	no	—
29	C₂×C₁₀₃⋊C₈	—	2	2	no	—
30	C₂×C₁₀₃⋊C₈	—	2	1	no	—
31	C₄⋊Dic₁₀₃	—	2	1	no	—
32	C₂²⋊Dic₁₀₃	D₄	1	2	yes	—
33	C₄×Dic₁₀₃	—	2	1	no	—
34	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₀₃	—	2	2	no	—
36	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
37	C₄⋊C₄×C₁₀₃	—	2	2	no	—
38	C₂²⋊C₄×C₁₀₃	—	2	2	no	—
39	C₄²×C₁₀₃	—	2	2	no	—
40	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₁₀₃	—	2	2	no	—
42	C₂×C₄×C₁₀₃	—	4	2	no	—
43	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₃	#41	1	2	yes	—
46	C₂×C₄×C₁₀₃	#41	1	1	yes	characteristic
47	C₂²⋊D₁₀₃	—	4	2	no	—
48	C₂²⋊D₁₀₃	—	4	2	no	—
49	C₂×Dic₁₀₃	—	2	2	no	—
50	C₂×Dic₁₀₃	—	2	2	no	—
51	D₄₁₂	—	2	2	no	—
52	D₄₁₂	—	2	2	no	—
53	C₄×D₁₀₃	—	4	2	no	—
54	C₄×D₁₀₃	—	4	2	no	—
55	Dic₂₀₆	—	2	2	no	—
56	Dic₂₀₆	—	2	2	no	—
57	C₈×C₁₀₃	—	2	2	no	—
58	C₁₀₃⋊C₈	—	4	1	no	—
59	C₂²×C₁₀₃	—	4	1	no	—
60	C₂²×C₁₀₃	#121	1	2	yes	—
61	C₂²×C₁₀₃	#121	1	1	yes	characteristic
62	C₂×D₁₀₃	—	4	2	no	—
63	C₂×D₁₀₃	—	8	2	no	—
64	C₄×C₁₀₃	—	8	2	no	—
65	C₄×C₁₀₃	#120	1	1	yes	characteristic
66	C₄×C₁₀₃	#123	1	2	yes	—
67	C₄×C₁₀₃	#124	1	1	yes	characteristic
68	Dic₁₀₃	—	4	2	no	—
69	Dic₁₀₃	—	4	2	no	—
70	C₂×C₁₀₃	—	2	1	no	—
71	C₂×C₁₀₃	—	2	2	no	—
72	C₂×C₁₀₃	#394	1	1	yes	characteristic
73	D₁₀₃	—	8	2	no	—
74	#2053	—	103	1	no	maximal, Sylow
75	C₁₀₃	#2053	1	1	yes	characteristic, complemented, Sylow
76	#575	—	103	1	no	—
77	#424	—	103	1	no	—
78	#416	—	103	2	no	—
79	#410	—	103	2	no	—
80	H(3,ℤ/4)	D₁₀₃	1	1	yes	characteristic, complemented
81	C₂×#41	—	103	2	no	—
82	#137	—	206	2	no	—
83	AGL(1,ℤ/8)	—	206	2	no	—
84	#135	—	206	2	no	—
85	C₂×M₁₆	C₂×D₁₀₃	1	1	yes	characteristic
86	C₂²×C₈	—	103	1	no	—
87	#117	—	103	1	no	—
88	#117	C₂×D₁₀₃	1	2	yes	—
89	C₄⋊C₈	—	206	2	no	—
90	#103	—	206	2	no	—
91	#102	—	206	2	no	—
92	#99	—	206	4	no	—
93	#41	—	206	2	no	—
94	#41	—	206	2	no	—
95	#41	—	206	2	no	—
96	#41	—	206	2	no	—
97	C₂×Q₈	—	103	2	no	—
98	C₂×D₄	—	103	2	no	—
99	C₂×D₄	—	206	2	no	—
100	C₂²×C₄	—	206	2	no	—
101	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
102	Q₁₆	—	206	2	no	—
103	SD₁₆	—	206	4	no	—
104	D₈	—	206	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	206	2	no	—
107	C₂×C₈	—	206	1	no	—
108	C₂×C₈	D₄₁₂	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	206	1	no	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₂²⋊C₄	—	103	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	206	1	no	—
115	C₂³	—	206	2	no	—
116	C₂³	D₄×D₁₀₃	1	1	yes	characteristic
117	Q₈	—	206	2	no	—
118	Q₈	—	206	2	no	—
119	D₄	—	412	2	no	—
120	D₄	—	412	2	no	—
121	D₄	—	206	2	no	—
122	D₄	—	206	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	412	2	no	—
125	C₂×C₄	—	412	2	no	—
126	C₂×C₄	—	206	2	no	—
127	C₂×C₄	—	206	2	no	—
128	C₂×C₄	D₄×D₁₀₃	1	2	yes	—
129	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
130	C₂×C₄	#13110105	1	2	yes	—
131	C₂×C₄	#13110106	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	412	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	824	2	no	—
136	C₂²	—	412	2	no	—
137	C₂²	#15692994	1	2	yes	—
138	C₂²	#15692996	1	1	yes	characteristic
139	C₄	—	8	2	no	—
140	C₄	—	412	2	no	—
141	C₄	—	412	2	no	—
142	C₄	#15692991	1	1	yes	characteristic
143	C₄	#15693006	1	2	yes	—
144	C₄	#15693009	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	824	2	no	—
148	C₂	#20267709	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20268591

C₂

yes

maximal, characteristic

#20267819

C₂

yes

maximal, characteristic, complemented

#20267785

C₂

yes

maximal, complemented

#20267764

C₂

yes

maximal, complemented

H(3,ℤ/4)×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂×#13110105

C₂²

yes

—

#15693054

—

#15693049

—

#15693043

—

C₂²×C₁₀₃⋊C₈

C₂²

yes

characteristic

#15692980

C₂²

yes

characteristic

Dic₁₀₃⋊C₈

—

#15692944

—

#15692941

—

#15692935

—

C₂×M₁₆×C₁₀₃

C₂²

yes

characteristic

#117×C₁₀₃

C₂²

yes

—

—

—

—

—

C₂×Dic₂₀₆

D₄

yes

—

C₂×D₄₁₂

D₄

yes

—

C₂×C₂²⋊D₁₀₃

—

C₂×C₄×D₁₀₃

—

Dic₄₁₂

—

#13110094

—

D₈₂₄

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₄⋊Dic₁₀₃

—

C₂²⋊Dic₁₀₃

D₄

yes

—

C₄×Dic₁₀₃

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

M₁₆×C₁₀₃

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₄⋊C₄×C₁₀₃

—

C₂²⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

—

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

C₂²×D₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×C₄×C₁₀₃

#41

yes

characteristic

C₂²⋊D₁₀₃

—

C₂²⋊D₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

D₄₁₂

—

D₄₁₂

—

C₄×D₁₀₃

—

C₄×D₁₀₃

—

Dic₂₀₆

—

Dic₂₀₆

—

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#121

yes

—

C₂²×C₁₀₃

#121

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#120

yes

characteristic

C₄×C₁₀₃

#123

yes

—

C₄×C₁₀₃

#124

yes

characteristic

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#394

yes

characteristic

D₁₀₃

—

#2053

—

103

maximal, Sylow

C₁₀₃

#2053

yes

characteristic, complemented, Sylow

#575

—

103

—

#424

—

103

—

#416

—

103

—

#410

—

103

—

H(3,ℤ/4)

D₁₀₃

yes

characteristic, complemented

C₂×#41

—

103

—

#137

—

206

—

AGL(1,ℤ/8)

—

206

—

#135

—

206

—

C₂×M₁₆

C₂×D₁₀₃

yes

characteristic

C₂²×C₈

—

103

—

#117

—

103

—

#117

C₂×D₁₀₃

yes

—

C₄⋊C₈

—

206

—

#103

—

206

—

#102

—

206

—

#99

—

206

—

#41

—

206

—

#41

—

206

—

#41

—

206

—

#41

—

206

—

C₂×Q₈

—

103

—

C₂×D₄

—

103

—

C₂×D₄

—

206

—

100

C₂²×C₄

—

206

—

101

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

102

Q₁₆

—

206

—

103

SD₁₆

—

206

—

104

D₈

—

206

—

105

M₁₆

—

106

C₂×C₈

—

206

—

107

C₂×C₈

—

206

—

108

C₂×C₈

D₄₁₂

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

206

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

103

—

113

C₄²

—

114

C₄²

—

206

—

115

C₂³

—

206

—

116

C₂³

D₄×D₁₀₃

yes

characteristic

117

Q₈

—

206

—

118

Q₈

—

206

—

119

D₄

—

412

—

120

D₄

—

412

—

121

D₄

—

206

—

122

D₄

—

206

—

123

C₂×C₄

—

124

C₂×C₄

—

412

—

125

C₂×C₄

—

412

—

126

C₂×C₄

—

206

—

127

C₂×C₄

—

206

—

128

C₂×C₄

D₄×D₁₀₃

yes

—

129

C₂×C₄

C₂×D₄₁₂

yes

characteristic

130

C₂×C₄

#13110105

yes

—

131

C₂×C₄

#13110106

yes

characteristic

132

C₈

—

133

C₈

—

412

—

134

C₂²

—

135

C₂²

—

824

—

136

C₂²

—

412

—

137

C₂²

#15692994

yes

—

138

C₂²

#15692996

yes

characteristic

139

C₄

—

140

C₄

—

412

—

141

C₄

—

412

—

142

C₄

#15692991

yes

characteristic

143

C₄

#15693006

yes

—

144

C₄

#15693009

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

824

—

148

C₂

#20267709

yes

characteristic

	subgroup	G/H
center	C₄	#15693009
commutator, cosocle	C₂²×C₄×C₁₀₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	H(3,ℤ/4)×C₁₀₃	C₂
socle	C₂×C₁₀₃	#394
prime radical	#20268591	C₂
hypercenter	H(3,ℤ/4)	D₁₀₃
hypocenter	C₁₀₃	#2053
LCS	G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#15693009

commutator, cosocle

C₂²×C₄×C₁₀₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

H(3,ℤ/4)×C₁₀₃

C₂

C₂×C₁₀₃

C₂

D₁₀₃

C₁₀₃

G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #20267819	C₂	semidirect
3. #20267785	C₂	semidirect
4. #20267764	C₂	semidirect
5. H(3,ℤ/4)×C₁₀₃	C₂	semidirect
75. C₁₀₃	#2053	semidirect
80. H(3,ℤ/4)	D₁₀₃	semidirect

H ≅ G/N

type

2. #20267819

C₂

semidirect

3. #20267785

C₂

semidirect

4. #20267764

C₂

semidirect

5. H(3,ℤ/4)×C₁₀₃

C₂

semidirect

75. C₁₀₃

#2053

semidirect

80. H(3,ℤ/4)

D₁₀₃

semidirect

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

subgroup	index
#20268591	2
#20267819	2
#20267785	2
#20267764	2
H(3,ℤ/4)×C₁₀₃	2
#2053	103

subgroup

index

H(3,ℤ/4)×C₁₀₃

103

Subgroups — #27196451

Conjugacy classes of subgroups

Complements

Maximal subgroups