Group #27196472 — subgroups

Subgroups — #27196472

Order 13184.

282 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 97 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×#15693021	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692954	C₂	1	2	yes	maximal
3	#20267553	C₂	1	2	yes	maximal
4	#20267546	C₂	1	1	yes	maximal, characteristic
5	#389×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀₆	—	2	1	no	—
7	C₂²×C₁₀₃⋊C₈	C₂²	1	2	yes	—
8	#15693021	—	2	4	no	—
9	C₂×C₄⋊Dic₁₀₃	—	2	1	no	—
10	C₂×C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
11	#15692954	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₀₃	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₀₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
16	#95×C₁₀₃	—	2	1	no	—
17	C₂×Dic₂₀₆	—	2	4	no	—
18	C₂×Dic₂₀₆	—	4	2	no	—
19	C₂×Dic₂₀₆	—	4	2	no	—
20	C₂²×Dic₁₀₃	—	2	2	no	—
21	C₂×C₁₀₃⋊C₈	—	2	2	no	—
22	C₂×C₁₀₃⋊C₈	—	2	2	no	—
23	C₂×C₁₀₃⋊C₈	Q₈	1	4	yes	complemented
24	C₄⋊Dic₁₀₃	—	2	4	no	—
25	C₄⋊Dic₁₀₃	—	2	2	no	—
26	C₄⋊Dic₁₀₃	D₄	1	4	yes	—
27	C₂²×C₄×C₁₀₃	—	2	1	no	—
28	C₂²×C₄×C₁₀₃	—	2	1	no	—
29	C₂²×C₄×C₁₀₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₁₀₃	—	2	1	no	—
33	C₄⋊C₄×C₁₀₃	—	2	2	no	—
34	C₄⋊C₄×C₁₀₃	—	4	2	no	—
35	C₄⋊C₄×C₁₀₃	Q₈	1	2	yes	complemented
36	C₄²×C₁₀₃	—	2	2	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	1	no	—
42	C₂×C₄×C₁₀₃	—	2	2	no	—
43	C₂×C₄×C₁₀₃	—	4	1	no	—
44	C₂×C₄×C₁₀₃	—	4	1	no	—
45	C₂×C₄×C₁₀₃	—	4	1	no	—
46	C₂×C₄×C₁₀₃	—	4	1	no	—
47	C₂×C₄×C₁₀₃	—	4	1	no	—
48	C₂×C₄×C₁₀₃	D₈	1	2	yes	—
49	C₂×C₄×C₁₀₃	Q₁₆	1	2	yes	—
50	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₀₃	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₀₃	#41	1	1	yes	characteristic
54	C₂×Dic₁₀₃	—	2	4	no	—
55	C₂×Dic₁₀₃	—	4	2	no	—
56	C₂×Dic₁₀₃	—	4	2	no	—
57	Dic₂₀₆	—	4	8	no	—
58	C₁₀₃⋊C₈	—	2	4	no	—
59	C₂²×C₁₀₃	#120	1	1	yes	characteristic
60	C₂²×C₁₀₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₁₀₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₁₀₃	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₁₀₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₂²×C₁₀₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₁₀₃	#137	1	1	yes	characteristic
66	C₄×C₁₀₃	—	2	2	no	—
67	C₄×C₁₀₃	—	2	2	no	—
68	C₄×C₁₀₃	—	4	2	no	—
69	C₄×C₁₀₃	—	8	1	no	—
70	C₄×C₁₀₃	C₂²⋊Q₈	1	2	yes	—
71	C₄×C₁₀₃	C₄⋊Q₈	1	2	yes	—
72	Dic₁₀₃	—	4	4	no	—
73	C₂×C₁₀₃	#395	1	1	yes	characteristic
74	C₂×C₁₀₃	#447	1	1	yes	characteristic
75	C₂×C₁₀₃	#451	1	1	yes	characteristic
76	C₂×C₁₀₃	#474	1	1	yes	characteristic
77	C₂×C₁₀₃	#477	1	1	yes	characteristic
78	C₂×C₁₀₃	#500	1	1	yes	characteristic
79	C₂×C₁₀₃	#501	1	1	yes	characteristic
80	#2057	—	103	1	no	maximal, Sylow
81	C₁₀₃	#2057	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄⋊Q₈	—	103	1	no	—
83	C₂×#107	—	103	2	no	—
84	#389	D₁₀₃	1	1	yes	characteristic
85	#340	—	103	2	no	—
86	#336	—	103	1	no	—
87	C₂²×Q₈	—	206	1	no	—
88	C₂²×C₈	—	103	2	no	—
89	C₄⋊Q₈	—	206	4	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	103	2	no	—
92	C₂×C₄⋊C₄	—	206	1	no	—
93	C₂×C₄⋊C₄	C₂×D₁₀₃	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂×D₁₀₃	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
96	#107	—	206	4	no	—
97	#95	—	2	1	no	—
98	C₂×Q₈	—	412	2	no	—
99	C₂×Q₈	—	412	2	no	—
100	C₂×Q₈	—	206	4	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	206	2	no	—
104	C₂²×C₄	C₂²⋊D₁₀₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₁₀₃	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
107	C₂×C₈	—	103	4	no	—
108	C₂×C₈	—	206	2	no	—
109	C₂×C₈	—	206	2	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	103	4	no	—
114	C₄⋊C₄	—	206	4	no	—
115	C₄⋊C₄	—	206	2	no	—
116	C₄⋊C₄	Dic₂₀₆	1	2	yes	complemented
117	C₄²	—	2	2	no	—
118	C₂³	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
119	Q₈	—	412	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	412	2	no	—
131	C₂×C₄	—	412	2	no	—
132	C₂×C₄	—	206	4	no	—
133	C₂×C₄	#13110090	1	2	yes	—
134	C₂×C₄	#13110095	1	2	yes	—
135	C₂×C₄	D₄×D₁₀₃	1	2	yes	—
136	C₂×C₄	Q₈×D₁₀₃	1	2	yes	—
137	C₂×C₄	C₂×Dic₂₀₆	1	1	yes	characteristic
138	C₂×C₄	#13110105	1	1	yes	characteristic
139	C₈	—	206	4	no	—
140	C₂²	#15692989	1	1	yes	characteristic
141	C₂²	#15693000	1	1	yes	characteristic
142	C₂²	#15693019	1	1	yes	characteristic
143	C₂²	C₂×#13110090	1	1	yes	characteristic
144	C₂²	C₂×#13110095	1	1	yes	characteristic
145	C₂²	#15693046	1	1	yes	characteristic
146	C₂²	#15693051	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	8	1	no	—
151	C₄	—	412	4	no	—
152	C₄	#15692999	1	2	yes	—
153	C₄	#15693020	1	2	yes	—
154	C₂	#20267711	1	1	yes	characteristic
155	C₂	#20267924	1	1	yes	characteristic
156	C₂	#20267946	1	1	yes	characteristic
157	C₂	#20268071	1	1	yes	characteristic
158	C₂	#20268086	1	1	yes	characteristic
159	C₂	#20268194	1	1	yes	characteristic
160	C₂	#20268201	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15693021

C₂

yes

maximal, characteristic

C₂×#15692954

C₂

yes

maximal

#20267553

C₂

yes

maximal

#20267546

C₂

yes

maximal, characteristic

#389×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₆

—

C₂²×C₁₀₃⋊C₈

C₂²

yes

—

#15693021

—

C₂×C₄⋊Dic₁₀₃

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

—

#15692954

—

C₂×C₄⋊C₄×C₁₀₃

—

C₂×C₄⋊C₄×C₁₀₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₃

C₂²

yes

characteristic

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

#95×C₁₀₃

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂×Dic₂₀₆

—

C₂²×Dic₁₀₃

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

Q₈

yes

complemented

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

—

C₂²×C₄×C₁₀₃

—

C₂²×C₄×C₁₀₃

D₄

yes

characteristic

C₂²×C₄×C₁₀₃

D₄

yes

characteristic

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

Q₈

yes

complemented

C₄²×C₁₀₃

—

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

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

D₈

yes

—

C₂×C₄×C₁₀₃

Q₁₆

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂×Q₈

yes

—

C₂×C₄×C₁₀₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

characteristic

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

#120

yes

characteristic

C₂²×C₁₀₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₀₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₃

C₂×D₈

yes

characteristic

C₂²×C₁₀₃

C₂×Q₁₆

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂²⋊Q₈

yes

—

C₄×C₁₀₃

C₄⋊Q₈

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

103

maximal, Sylow

C₁₀₃

#2057

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

103

—

C₂×#107

—

103

—

#389

D₁₀₃

yes

characteristic

#340

—

103

—

#336

—

103

—

C₂²×Q₈

—

206

—

C₂²×C₈

—

103

—

C₄⋊Q₈

—

206

—

C₂×C₄⋊C₄

—

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

—

#95

—

C₂×Q₈

—

412

—

C₂×Q₈

—

412

—

100

C₂×Q₈

—

206

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

206

—

104

C₂²×C₄

C₂²⋊D₁₀₃

yes

characteristic

105

C₂²×C₄

C₂²⋊D₁₀₃

yes

characteristic

106

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

107

C₂×C₈

—

103

—

108

C₂×C₈

—

206

—

109

C₂×C₈

—

206

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

103

—

114

C₄⋊C₄

—

206

—

115

C₄⋊C₄

—

206

—

116

C₄⋊C₄

Dic₂₀₆

yes

complemented

117

C₄²

—

118

C₂³

C₂×C₂²⋊D₁₀₃

yes

characteristic

119

Q₈

—

412

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

412

—

131

C₂×C₄

—

412

—

132

C₂×C₄

—

206

—

133

C₂×C₄

#13110090

yes

—

134

C₂×C₄

#13110095

yes

—

135

C₂×C₄

D₄×D₁₀₃

yes

—

136

C₂×C₄

Q₈×D₁₀₃

yes

—

137

C₂×C₄

C₂×Dic₂₀₆

yes

characteristic

138

C₂×C₄

#13110105

yes

characteristic

139

C₈

—

206

—

140

C₂²

#15692989

yes

characteristic

141

C₂²

#15693000

yes

characteristic

142

C₂²

#15693019

yes

characteristic

143

C₂²

C₂×#13110090

yes

characteristic

144

C₂²

C₂×#13110095

yes

characteristic

145

C₂²

#15693046

yes

characteristic

146

C₂²

#15693051

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

412

—

152

C₄

#15692999

yes

—

153

C₄

#15693020

yes

—

154

C₂

#20267711

yes

characteristic

155

C₂

#20267924

yes

characteristic

156

C₂

#20267946

yes

characteristic

157

C₂

#20268071

yes

characteristic

158

C₂

#20268086

yes

characteristic

159

C₂

#20268194

yes

characteristic

160

C₂

#20268201

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₃
commutator, cosocle	C₂²×C₄×C₁₀₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#389×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂×D₄
hypercenter	#389	D₁₀₃
hypocenter	C₁₀₃	#2057
LCS	G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₃

commutator, cosocle

C₂²×C₄×C₁₀₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#389×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₂×D₄

D₁₀₃

C₁₀₃

G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
23. C₂×C₁₀₃⋊C₈	Q₈	semidirect
35. C₄⋊C₄×C₁₀₃	Q₈	semidirect
81. C₁₀₃	#2057	semidirect
116. C₄⋊C₄	Dic₂₀₆	semidirect

H ≅ G/N

type

23. C₂×C₁₀₃⋊C₈

Q₈

semidirect

35. C₄⋊C₄×C₁₀₃

Q₈

semidirect

81. C₁₀₃

#2057

semidirect

116. C₄⋊C₄

Dic₂₀₆

semidirect

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

subgroup	index
C₂×#15693021	2
C₂×#15692954	2
#20267553	2
#20267546	2
#389×C₁₀₃	2
#2057	103

subgroup

index

#389×C₁₀₃

103

Subgroups — #27196472

Conjugacy classes of subgroups

Complements

Maximal subgroups