Group #27196621 — subgroups

Subgroups — #27196621

Order 13184.

230 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 91 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×#15692952	C₂	1	2	yes	maximal
2	#20267689	C₂	1	1	yes	maximal, characteristic
3	#20267553	C₂	1	2	yes	maximal
4	#20267546	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₀₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₀₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₀₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₀₃⋊C₄	—	2	1	no	—
10	#15692952	—	2	4	no	—
11	C₁₀₃⋊#95	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₀₃	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₀₃	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₀₃	—	2	1	no	—
17	C₂²×Dic₁₀₃	—	2	1	no	—
18	C₂×C₁₀₃⋊C₈	—	2	2	no	—
19	C₂×C₁₀₃⋊C₈	—	2	2	no	—
20	C₂×C₁₀₃⋊C₈	Q₈	1	4	yes	—
21	C₄⋊Dic₁₀₃	—	2	1	no	—
22	C₄⋊Dic₁₀₃	—	2	2	no	—
23	C₄⋊Dic₁₀₃	D₄	1	2	yes	—
24	Dic₁₀₃⋊C₄	—	4	2	no	—
25	C₂²×C₄×C₁₀₃	—	2	2	no	—
26	C₂²×C₄×C₁₀₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₀₃	—	2	2	no	—
30	C₄⋊C₄×C₁₀₃	—	2	2	no	—
31	C₄⋊C₄×C₁₀₃	—	4	2	no	—
32	C₄⋊C₄×C₁₀₃	—	4	2	no	—
33	C₄²×C₁₀₃	—	2	2	no	—
34	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₃	—	2	1	no	—
36	C₂×C₄×C₁₀₃	—	2	1	no	—
37	C₂×C₄×C₁₀₃	—	2	1	no	—
38	C₂×C₄×C₁₀₃	—	2	1	no	—
39	C₂×C₄×C₁₀₃	—	4	2	no	—
40	C₂×C₄×C₁₀₃	—	4	2	no	—
41	C₂×C₄×C₁₀₃	—	4	2	no	—
42	C₂×C₄×C₁₀₃	SD₁₆	1	2	yes	—
43	C₂×C₄×C₁₀₃	SD₁₆	1	2	yes	—
44	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₃	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₀₃	#41	1	1	yes	characteristic
47	C₂×C₄×C₁₀₃	#41	1	2	yes	—
48	C₂×Dic₁₀₃	—	2	2	no	—
49	C₂×Dic₁₀₃	—	4	1	no	—
50	C₂×Dic₁₀₃	—	4	1	no	—
51	C₂×Dic₁₀₃	—	4	1	no	—
52	C₂×Dic₁₀₃	—	4	1	no	—
53	C₂×Dic₁₀₃	—	4	1	no	—
54	C₁₀₃⋊C₈	—	2	4	no	—
55	C₂²×C₁₀₃	#121	1	1	yes	characteristic
56	C₂²×C₁₀₃	#123	1	1	yes	characteristic
57	C₂²×C₁₀₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₁₀₃	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₁₀₃	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₁₀₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₁₀₃	#137	1	1	yes	characteristic
62	C₄×C₁₀₃	—	2	2	no	—
63	C₄×C₁₀₃	—	2	2	no	—
64	C₄×C₁₀₃	—	8	2	no	—
65	C₄×C₁₀₃	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₁₀₃	#125	1	2	yes	—
67	Dic₁₀₃	—	4	2	no	—
68	Dic₁₀₃	—	8	1	no	—
69	C₂×C₁₀₃	#398	1	1	yes	characteristic
70	C₂×C₁₀₃	#460	1	1	yes	characteristic
71	C₂×C₁₀₃	#461	1	1	yes	characteristic
72	C₂×C₁₀₃	#481	1	1	yes	characteristic
73	C₂×C₁₀₃	#483	1	1	yes	characteristic
74	C₂×C₁₀₃	#498	1	1	yes	characteristic
75	C₂×C₁₀₃	#501	1	1	yes	characteristic
76	#2084	—	103	1	no	maximal, Sylow
77	C₁₀₃	#2084	1	1	yes	characteristic, complemented, Sylow
78	C₂×#106	—	103	2	no	—
79	#389	—	103	1	no	—
80	#384	D₁₀₃	1	1	yes	characteristic
81	#340	—	103	2	no	—
82	#336	—	103	1	no	—
83	C₂²×C₈	—	103	2	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	103	1	no	—
87	C₂×C₄⋊C₄	—	103	1	no	—
88	C₂×C₄⋊C₄	—	206	1	no	—
89	C₂×C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
90	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
91	#106	—	206	4	no	—
92	#95	—	206	1	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	206	1	no	—
95	C₂²×C₄	—	206	1	no	—
96	C₂²×C₄	C₂²⋊D₁₀₃	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₁₀₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
99	C₂×C₈	—	103	4	no	—
100	C₂×C₈	—	206	2	no	—
101	C₂×C₈	—	206	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	103	2	no	—
107	C₄⋊C₄	—	412	2	no	—
108	C₄⋊C₄	—	206	2	no	—
109	C₄⋊C₄	—	206	1	no	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	412	1	no	—
120	C₂×C₄	—	412	1	no	—
121	C₂×C₄	—	412	1	no	—
122	C₂×C₄	—	412	1	no	—
123	C₂×C₄	—	412	1	no	—
124	C₂×C₄	—	206	2	no	—
125	C₂×C₄	#13110092	1	2	yes	—
126	C₂×C₄	#13110093	1	2	yes	—
127	C₂×C₄	D₄×D₁₀₃	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₁₀₃	1	2	yes	—
129	C₂×C₄	#13110104	1	1	yes	characteristic
130	C₂×C₄	#13110105	1	2	yes	—
131	C₈	—	206	4	no	—
132	C₂²	#15692992	1	1	yes	characteristic
133	C₂²	#15693004	1	1	yes	characteristic
134	C₂²	#15693019	1	1	yes	characteristic
135	C₂²	C₂×#13110092	1	1	yes	characteristic
136	C₂²	C₂×#13110093	1	1	yes	characteristic
137	C₂²	#15693047	1	1	yes	characteristic
138	C₂²	#15693052	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	8	2	no	—
142	C₄	—	824	1	no	—
143	C₄	—	412	2	no	—
144	C₄	#15692999	1	2	yes	—
145	C₄	#15693011	1	2	yes	—
146	C₂	#20267722	1	1	yes	characteristic
147	C₂	#20267988	1	1	yes	characteristic
148	C₂	#20267993	1	1	yes	characteristic
149	C₂	#20268109	1	1	yes	characteristic
150	C₂	#20268121	1	1	yes	characteristic
151	C₂	#20268187	1	1	yes	characteristic
152	C₂	#20268197	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692952

C₂

yes

maximal

#20267689

C₂

yes

maximal, characteristic

#20267553

C₂

yes

maximal

#20267546

C₂

yes

maximal, characteristic

#384×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×C₁₀₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₃

C₂²

yes

characteristic

C₂×Dic₁₀₃⋊C₄

—

#15692952

—

C₁₀₃⋊#95

—

C₂×C₄⋊C₄×C₁₀₃

—

C₂×C₄⋊C₄×C₁₀₃

—

C₂×C₄⋊C₄×C₁₀₃

C₂²

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₂²×Dic₁₀₃

—

C₂²×Dic₁₀₃

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

Q₈

yes

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

D₄

yes

—

Dic₁₀₃⋊C₄

—

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

—

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

SD₁₆

yes

—

C₂×C₄×C₁₀₃

SD₁₆

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×Q₈

yes

—

C₂×C₄×C₁₀₃

#41

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₃

C₂×SD₁₆

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂²⋊Q₈

yes

—

C₄×C₁₀₃

#125

yes

—

Dic₁₀₃

—

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

#2084

yes

characteristic, complemented, Sylow

C₂×#106

—

103

—

#389

—

103

—

#384

D₁₀₃

yes

characteristic

#340

—

103

—

#336

—

103

—

C₂²×C₈

—

103

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

206

—

C₂×C₄⋊C₄

C₂×D₁₀₃

yes

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

#106

—

206

—

#95

—

206

—

C₂²×C₄

—

C₂²×C₄

—

206

—

C₂²×C₄

—

206

—

C₂²×C₄

C₂²⋊D₁₀₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₃

yes

characteristic

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

—

103

—

100

C₂×C₈

—

206

—

101

C₂×C₈

—

206

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

103

—

107

C₄⋊C₄

—

412

—

108

C₄⋊C₄

—

206

—

109

C₄⋊C₄

—

206

—

110

C₄²

—

111

C₂³

C₂×C₂²⋊D₁₀₃

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

412

—

120

C₂×C₄

—

412

—

121

C₂×C₄

—

412

—

122

C₂×C₄

—

412

—

123

C₂×C₄

—

412

—

124

C₂×C₄

—

206

—

125

C₂×C₄

#13110092

yes

—

126

C₂×C₄

#13110093

yes

—

127

C₂×C₄

D₄×D₁₀₃

yes

characteristic

128

C₂×C₄

Q₈×D₁₀₃

yes

—

129

C₂×C₄

#13110104

yes

characteristic

130

C₂×C₄

#13110105

yes

—

131

C₈

—

206

—

132

C₂²

#15692992

yes

characteristic

133

C₂²

#15693004

yes

characteristic

134

C₂²

#15693019

yes

characteristic

135

C₂²

C₂×#13110092

yes

characteristic

136

C₂²

C₂×#13110093

yes

characteristic

137

C₂²

#15693047

yes

characteristic

138

C₂²

#15693052

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

824

—

143

C₄

—

412

—

144

C₄

#15692999

yes

—

145

C₄

#15693011

yes

—

146

C₂

#20267722

yes

characteristic

147

C₂

#20267988

yes

characteristic

148

C₂

#20267993

yes

characteristic

149

C₂

#20268109

yes

characteristic

150

C₂

#20268121

yes

characteristic

151

C₂

#20268187

yes

characteristic

152

C₂

#20268197

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₃
commutator, cosocle	C₂²×C₄×C₁₀₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#384×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂×D₄
hypercenter	#384	D₁₀₃
hypocenter	C₁₀₃	#2084
LCS	G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₃

commutator, cosocle

C₂²×C₄×C₁₀₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#384×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₂×D₄

D₁₀₃

C₁₀₃

G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
77. C₁₀₃	#2084	semidirect

H ≅ G/N

type

77. C₁₀₃

#2084

semidirect

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

subgroup	index
C₂×#15692952	2
#20267689	2
#20267553	2
#20267546	2
#384×C₁₀₃	2
#2084	103

subgroup

index

#384×C₁₀₃

103

Subgroups — #27196621

Conjugacy classes of subgroups

Complements

Maximal subgroups