Group #27196467 — subgroups

Subgroups — #27196467

Order 13184.

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

normal subgroups only

This group has 288 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	C₂²×Dic₄₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692946	C₂	1	2	yes	maximal
3	#20267697	C₂	1	2	yes	maximal
4	#20267554	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×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×C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
10	#15692946	—	2	4	no	—
11	C₁₀₃⋊#95	—	2	2	no	—
12	C₁₀₃⋊#95	—	2	2	no	—
13	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₀₃	—	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	—	2	2	no	—
22	C₂×Dic₂₀₆	D₄	1	4	yes	—
23	C₂²×Dic₁₀₃	—	2	2	no	—
24	C₂²×Dic₁₀₃	—	2	2	no	—
25	Dic₄₁₂	—	2	8	no	—
26	C₄⋊Dic₁₀₃	—	2	2	no	—
27	C₄⋊Dic₁₀₃	—	2	2	no	—
28	C₂²×C₄×C₁₀₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₀₃	—	2	2	no	—
32	C₂×C₈×C₁₀₃	—	2	1	no	—
33	C₂×C₈×C₁₀₃	—	2	2	no	—
34	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
35	C₄²×C₁₀₃	—	2	1	no	—
36	C₄²×C₁₀₃	—	2	1	no	—
37	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₃	—	2	1	no	—
39	C₂×C₄×C₁₀₃	—	2	1	no	—
40	C₂×C₄×C₁₀₃	—	2	1	no	—
41	C₂×C₄×C₁₀₃	—	2	2	no	—
42	C₂×C₄×C₁₀₃	Q₁₆	1	2	yes	complemented
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	1	yes	characteristic
46	C₂×C₄×C₁₀₃	#41	1	2	yes	—
47	C₂×Dic₁₀₃	—	2	4	no	—
48	C₂×Dic₁₀₃	—	4	2	no	—
49	C₂×Dic₁₀₃	—	4	2	no	—
50	C₂×Dic₁₀₃	—	4	2	no	—
51	C₂×Dic₁₀₃	—	4	2	no	—
52	C₂×Dic₁₀₃	—	4	2	no	—
53	Dic₂₀₆	—	2	4	no	—
54	Dic₂₀₆	—	2	4	no	—
55	Dic₂₀₆	—	4	2	no	—
56	Dic₂₀₆	—	4	2	no	—
57	C₈×C₁₀₃	—	2	2	no	—
58	C₈×C₁₀₃	—	4	1	no	—
59	C₂²×C₁₀₃	#121	1	1	yes	characteristic
60	C₂²×C₁₀₃	#121	1	2	yes	—
61	C₂²×C₁₀₃	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₁₀₃	#135	1	1	yes	characteristic
63	C₂²×C₁₀₃	#137	1	2	yes	—
64	C₄×C₁₀₃	—	2	2	no	—
65	C₄×C₁₀₃	—	4	1	no	—
66	C₄×C₁₀₃	#120	1	1	yes	characteristic
67	C₄×C₁₀₃	#123	1	2	yes	—
68	C₄×C₁₀₃	#124	1	1	yes	characteristic
69	Dic₁₀₃	—	4	4	no	—
70	Dic₁₀₃	—	8	2	no	—
71	C₂×C₁₀₃	#394	1	1	yes	characteristic
72	C₂×C₁₀₃	#462	1	2	yes	—
73	C₂×C₁₀₃	#464	1	2	yes	—
74	C₂×C₁₀₃	#467	1	1	yes	characteristic
75	C₂×C₁₀₃	#470	1	1	yes	characteristic
76	#2056	—	103	1	no	maximal, Sylow
77	C₁₀₃	#2056	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₁₆	—	103	1	no	—
79	C₂×C₄⋊C₈	D₁₀₃	1	1	yes	characteristic
80	C₂×#103	—	103	2	no	—
81	#391	—	103	2	no	—
82	#340	—	103	1	no	—
83	C₂²×Q₈	—	103	2	no	—
84	C₂×Q₁₆	—	206	4	no	—
85	C₂×Q₁₆	—	206	2	no	—
86	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	103	2	no	—
89	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	#103	—	206	4	no	—
92	#95	—	206	2	no	—
93	#95	—	206	2	no	—
94	C₂×Q₈	—	103	4	no	—
95	C₂×Q₈	—	206	2	no	—
96	C₂×Q₈	—	206	2	no	—
97	C₂×Q₈	—	206	2	no	—
98	C₂×Q₈	—	206	2	no	—
99	C₂×Q₈	—	206	2	no	—
100	C₂²×C₄	—	206	2	no	—
101	C₂²×C₄	—	206	2	no	—
102	C₂²×C₄	D₄₁₂	1	1	yes	characteristic
103	C₂²×C₄	D₄₁₂	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
105	Q₁₆	—	206	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₂²⋊D₁₀₃	1	2	yes	—
110	C₄⋊C₄	—	206	2	no	—
111	C₄⋊C₄	—	206	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₂³	C₂×D₄₁₂	1	1	yes	characteristic
115	Q₈	—	412	2	no	—
116	Q₈	—	412	2	no	—
117	Q₈	—	206	4	no	—
118	Q₈	—	206	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	412	2	no	—
124	C₂×C₄	—	412	2	no	—
125	C₂×C₄	—	412	2	no	—
126	C₂×C₄	—	412	2	no	—
127	C₂×C₄	—	412	2	no	—
128	C₂×C₄	—	206	4	no	—
129	C₂×C₄	Dic₄₁₂	1	2	yes	complemented
130	C₂×C₄	D₄×D₁₀₃	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
132	C₂×C₄	#13110105	1	1	yes	characteristic
133	C₂×C₄	#13110106	1	2	yes	—
134	C₈	—	2	2	no	—
135	C₈	—	4	1	no	—
136	C₂²	#15692995	1	1	yes	characteristic
137	C₂²	#15692996	1	2	yes	—
138	C₂²	C₂×Dic₄₁₂	1	1	yes	characteristic
139	C₂²	#15693043	1	1	yes	characteristic
140	C₂²	#15693054	1	2	yes	—
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	824	2	no	—
144	C₄	—	412	4	no	—
145	C₄	#15692989	1	1	yes	characteristic
146	C₄	#15693006	1	2	yes	—
147	C₄	#15693010	1	1	yes	characteristic
148	C₂	#20267708	1	1	yes	characteristic
149	C₂	#20268002	1	2	yes	—
150	C₂	#20268014	1	2	yes	—
151	C₂	#20268032	1	1	yes	characteristic
152	C₂	#20268049	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₄₁₂

C₂

yes

maximal, characteristic

C₂×#15692946

C₂

yes

maximal

#20267697

C₂

yes

maximal

#20267554

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₆

C₂²

yes

—

C₂×Dic₄₁₂

—

C₂×Dic₄₁₂

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

—

#15692946

—

C₁₀₃⋊#95

—

C₁₀₃⋊#95

—

C₂²×C₈×C₁₀₃

C₂²

yes

characteristic

C₂²×C₈×C₁₀₃

C₂²

yes

characteristic

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₃

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

D₄

yes

—

C₂²×Dic₁₀₃

—

C₂²×Dic₁₀₃

—

Dic₄₁₂

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₂²×C₄×C₁₀₃

D₄

yes

characteristic

C₂²×C₄×C₁₀₃

D₄

yes

characteristic

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₄²×C₁₀₃

—

C₄²×C₁₀₃

—

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

Q₁₆

yes

complemented

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

C₂×C₄×C₁₀₃

yes

C₂×C₄×C₁₀₃

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

Dic₂₀₆

—

Dic₂₀₆

—

Dic₂₀₆

—

C₈×C₁₀₃

—

C₈×C₁₀₃

—

C₂²×C₁₀₃

#121

yes

characteristic

C₂²×C₁₀₃

#121

yes

—

C₂²×C₁₀₃

C₂×Q₁₆

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#120

yes

characteristic

C₄×C₁₀₃

#123

yes

—

C₄×C₁₀₃

#124

yes

characteristic

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

#394

yes

characteristic

C₂×C₁₀₃

#462

yes

—

C₂×C₁₀₃

#464

yes

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

103

maximal, Sylow

C₁₀₃

#2056

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

103

—

C₂×C₄⋊C₈

D₁₀₃

yes

characteristic

C₂×#103

—

103

—

#391

—

103

—

#340

—

103

—

C₂²×Q₈

—

103

—

C₂×Q₁₆

—

206

—

C₂×Q₁₆

—

206

—

C₂²×C₈

C₂×D₁₀₃

yes

characteristic

C₂²×C₈

C₂×D₁₀₃

yes

characteristic

C₂×C₄⋊C₄

—

103

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

C₄⋊C₈

—

#103

—

206

—

#95

—

206

—

#95

—

206

—

C₂×Q₈

—

103

—

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

C₂×Q₈

—

206

—

100

C₂²×C₄

—

206

—

101

C₂²×C₄

—

206

—

102

C₂²×C₄

D₄₁₂

yes

characteristic

103

C₂²×C₄

D₄₁₂

yes

characteristic

104

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

105

Q₁₆

—

206

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂²⋊D₁₀₃

yes

—

110

C₄⋊C₄

—

206

—

111

C₄⋊C₄

—

206

—

112

C₄²

—

113

C₄²

—

114

C₂³

C₂×D₄₁₂

yes

characteristic

115

Q₈

—

412

—

116

Q₈

—

412

—

117

Q₈

—

206

—

118

Q₈

—

206

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

412

—

124

C₂×C₄

—

412

—

125

C₂×C₄

—

412

—

126

C₂×C₄

—

412

—

127

C₂×C₄

—

412

—

128

C₂×C₄

—

206

—

129

C₂×C₄

Dic₄₁₂

yes

complemented

130

C₂×C₄

D₄×D₁₀₃

yes

—

131

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

132

C₂×C₄

#13110105

yes

characteristic

133

C₂×C₄

#13110106

yes

—

134

C₈

—

135

C₈

—

136

C₂²

#15692995

yes

characteristic

137

C₂²

#15692996

yes

—

138

C₂²

C₂×Dic₄₁₂

yes

characteristic

139

C₂²

#15693043

yes

characteristic

140

C₂²

#15693054

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

824

—

144

C₄

—

412

—

145

C₄

#15692989

yes

characteristic

146

C₄

#15693006

yes

—

147

C₄

#15693010

yes

characteristic

148

C₂

#20267708

yes

characteristic

149

C₂

#20268002

yes

—

150

C₂

#20268014

yes

—

151

C₂

#20268032

yes

characteristic

152

C₂

#20268049

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₁₂
commutator, cosocle	C₂²×C₄×C₁₀₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	C₂×C₄⋊C₈×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₁₀₃
hypocenter	C₁₀₃	#2056
LCS	G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₄₁₂

commutator, cosocle

C₂²×C₄×C₁₀₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

C₂×C₄⋊C₈×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₁₀₃

hypocenter

C₁₀₃

#2056

LCS

G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
42. C₂×C₄×C₁₀₃	Q₁₆	semidirect
77. C₁₀₃	#2056	semidirect
129. C₂×C₄	Dic₄₁₂	semidirect

H ≅ G/N

type

42. C₂×C₄×C₁₀₃

Q₁₆

semidirect

77. C₁₀₃

#2056

semidirect

129. C₂×C₄

Dic₄₁₂

semidirect

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

subgroup	index
C₂²×Dic₄₁₂	2
C₂×#15692946	2
#20267697	2
#20267554	2
C₂×C₄⋊C₈×C₁₀₃	2
#2056	103

subgroup

index

C₂²×Dic₄₁₂

C₂×C₄⋊C₈×C₁₀₃

103

Subgroups — #27196467

Conjugacy classes of subgroups

Complements

Maximal subgroups