Group #27195865 — subgroups

Subgroups — #27195865

Order 13184.

248 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 109 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×Dic₁₀₃⋊C₈	C₂	1	2	yes	maximal
2	C₂×#15692932	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₁₀₃	C₂	1	1	yes	maximal, characteristic
4	#20267545	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₀₃	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₀₃	C₂²	1	2	yes	—
10	Dic₁₀₃⋊C₈	—	2	4	no	—
11	#15692932	—	2	2	no	—
12	C₁₀₃⋊#95	C₄	1	1	yes	characteristic
13	C₁₀₃⋊#95	C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₀₃	—	2	2	no	—
17	C₂²×Dic₁₀₃	C₂×C₄	1	2	yes	—
18	C₂²×Dic₁₀₃	C₂×C₄	1	2	yes	—
19	C₂×C₁₀₃⋊C₈	—	2	1	no	—
20	C₂×C₁₀₃⋊C₈	—	2	1	no	—
21	C₂×C₁₀₃⋊C₈	—	2	1	no	—
22	C₂×C₁₀₃⋊C₈	—	2	1	no	—
23	C₂×C₁₀₃⋊C₈	D₄	1	2	yes	—
24	C₂×C₁₀₃⋊C₈	Q₈	1	2	yes	—
25	C₄⋊Dic₁₀₃	—	2	4	no	—
26	C₄×Dic₁₀₃	—	2	2	no	—
27	C₄×Dic₁₀₃	—	2	2	no	—
28	C₂²×C₄×C₁₀₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₃	C₂×C₄	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	2	no	—
33	C₂×C₈×C₁₀₃	—	2	2	no	—
34	C₄²×C₁₀₃	D₄	1	2	yes	—
35	C₄²×C₁₀₃	Q₈	1	2	yes	—
36	C₂³×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₃	—	2	2	no	—
38	C₂×C₄×C₁₀₃	—	2	2	no	—
39	C₂×C₄×C₁₀₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₀₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₀₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₁₀₃	#41	1	1	yes	characteristic
47	C₂×Dic₁₀₃	—	2	2	no	—
48	C₂×Dic₁₀₃	—	2	2	no	—
49	C₂×Dic₁₀₃	—	2	2	no	—
50	C₂×Dic₁₀₃	—	2	2	no	—
51	C₂×Dic₁₀₃	—	2	2	no	—
52	C₂×Dic₁₀₃	M₁₆	1	4	yes	—
53	C₈×C₁₀₃	—	4	2	no	—
54	C₁₀₃⋊C₈	—	2	2	no	—
55	C₁₀₃⋊C₈	—	2	2	no	—
56	C₂²×C₁₀₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₀₃	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₁₀₃	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₁₀₃	C₂×M₁₆	1	2	yes	—
60	C₂²×C₁₀₃	#131	1	2	yes	—
61	C₄×C₁₀₃	—	2	2	no	—
62	C₄×C₁₀₃	—	2	2	no	—
63	C₄×C₁₀₃	#121	1	1	yes	characteristic
64	C₄×C₁₀₃	C₂²⋊Q₈	1	1	yes	characteristic
65	C₄×C₁₀₃	#125	1	1	yes	characteristic
66	C₄×C₁₀₃	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₁₀₃	—	2	4	no	—
68	Dic₁₀₃	—	4	2	no	—
69	C₂×C₁₀₃	#389	1	1	yes	characteristic
70	C₂×C₁₀₃	#423	1	1	yes	characteristic
71	C₂×C₁₀₃	#424	1	1	yes	characteristic
72	C₂×C₁₀₃	#435	1	2	yes	—
73	C₂×C₁₀₃	#446	1	2	yes	—
74	#1940	—	103	1	no	maximal, Sylow
75	C₁₀₃	#1940	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₄⋊C₈	—	103	2	no	—
77	C₂×C₄⋊C₈	D₁₀₃	1	1	yes	characteristic
78	C₂×#97	—	103	1	no	—
79	C₄×C₄⋊C₄	—	103	1	no	—
80	#336	—	103	2	no	—
81	C₂²×C₈	—	103	1	no	—
82	C₂²×C₈	—	103	1	no	—
83	C₂²×C₈	C₂×D₁₀₃	1	2	yes	—
84	C₂×C₄⋊C₄	—	103	2	no	—
85	C₂×C₄²	—	103	2	no	—
86	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
87	C₄⋊C₈	—	2	2	no	—
88	C₄⋊C₈	—	206	4	no	—
89	#97	—	206	2	no	—
90	#95	—	103	1	no	—
91	#95	—	103	1	no	—
92	C₂²×C₄	—	103	2	no	—
93	C₂²×C₄	—	103	2	no	—
94	C₂²×C₄	C₄×D₁₀₃	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₁₀₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	103	2	no	—
101	C₂×C₈	—	103	2	no	—
102	C₂×C₈	—	206	1	no	—
103	C₂×C₈	—	206	1	no	—
104	C₂×C₈	—	206	1	no	—
105	C₂×C₈	—	206	1	no	—
106	C₄⋊C₄	—	206	4	no	—
107	C₄²	—	206	2	no	—
108	C₄²	—	206	2	no	—
109	C₄²	Dic₂₀₆	1	2	yes	—
110	C₄²	C₂²⋊D₁₀₃	1	2	yes	—
111	C₂³	C₂×C₄×D₁₀₃	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	103	4	no	—
115	C₂×C₄	—	206	2	no	—
116	C₂×C₄	—	206	2	no	—
117	C₂×C₄	—	206	2	no	—
118	C₂×C₄	—	206	2	no	—
119	C₂×C₄	—	206	2	no	—
120	C₂×C₄	Dic₁₀₃⋊C₄	1	2	yes	—
121	C₂×C₄	Dic₁₀₃⋊C₄	1	2	yes	—
122	C₂×C₄	D₄×D₁₀₃	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₁₀₃	1	1	yes	characteristic
125	C₂×C₄	C₂×Dic₂₀₆	1	1	yes	characteristic
126	C₂×C₄	#13110104	1	1	yes	characteristic
127	C₂×C₄	#13110106	1	1	yes	characteristic
128	C₈	—	4	2	no	—
129	C₈	—	206	2	no	—
130	C₈	—	206	2	no	—
131	C₂²	C₂×Dic₁₀₃⋊C₄	1	1	yes	characteristic
132	C₂²	#15692983	1	1	yes	characteristic
133	C₂²	#15692986	1	1	yes	characteristic
134	C₂²	M₁₆×D₁₀₃	1	2	yes	—
135	C₂²	#15693027	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	412	2	no	—
139	C₄	—	206	4	no	—
140	C₄	#15692992	1	1	yes	characteristic
141	C₄	#15692997	1	1	yes	characteristic
142	C₄	#15693012	1	1	yes	characteristic
143	C₄	#15693020	1	1	yes	characteristic
144	C₂	#20267687	1	1	yes	characteristic
145	C₂	#20267814	1	1	yes	characteristic
146	C₂	#20267817	1	1	yes	characteristic
147	C₂	#20267861	1	2	yes	—
148	C₂	#20267920	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₀₃⋊C₈

C₂

yes

maximal

C₂×#15692932

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₀₃

C₂

yes

maximal, characteristic

#20267545

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×C₁₀₃⋊C₈

C₂²

yes

characteristic

C₂²×C₁₀₃⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₃

C₄

yes

—

C₂×C₄×Dic₁₀₃

C₂²

yes

—

Dic₁₀₃⋊C₈

—

#15692932

—

C₁₀₃⋊#95

C₄

yes

characteristic

C₁₀₃⋊#95

C₄

yes

characteristic

C₂²×C₈×C₁₀₃

C₂²

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₃

—

C₂²×Dic₁₀₃

C₂×C₄

yes

—

C₂²×Dic₁₀₃

C₂×C₄

yes

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

D₄

yes

—

C₂×C₁₀₃⋊C₈

Q₈

yes

—

C₄⋊Dic₁₀₃

—

C₄×Dic₁₀₃

—

C₄×Dic₁₀₃

—

C₂²×C₄×C₁₀₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₄²×C₁₀₃

D₄

yes

—

C₄²×C₁₀₃

Q₈

yes

—

C₂³×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₃

C₂×Q₈

yes

C₂×C₄×C₁₀₃

yes

C₂×C₄×C₁₀₃

yes

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

M₁₆

yes

—

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₃

C₄×D₄

yes

characteristic

C₂²×C₁₀₃

C₄×Q₈

yes

characteristic

C₂²×C₁₀₃

C₂×M₁₆

yes

—

C₂²×C₁₀₃

#131

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#121

yes

characteristic

C₄×C₁₀₃

C₂²⋊Q₈

yes

characteristic

C₄×C₁₀₃

#125

yes

characteristic

C₄×C₁₀₃

C₄⋊Q₈

yes

characteristic

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

C₂×C₁₀₃

#446

yes

—

#1940

—

103

maximal, Sylow

C₁₀₃

#1940

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

103

—

C₂×C₄⋊C₈

D₁₀₃

yes

characteristic

C₂×#97

—

103

—

C₄×C₄⋊C₄

—

103

—

#336

—

103

—

C₂²×C₈

—

103

—

C₂²×C₈

—

103

—

C₂²×C₈

C₂×D₁₀₃

yes

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄²

—

103

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

206

—

#97

—

206

—

#95

—

103

—

#95

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

C₄×D₁₀₃

yes

characteristic

C₂²×C₄

C₄×D₁₀₃

yes

characteristic

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

103

—

101

C₂×C₈

—

103

—

102

C₂×C₈

—

206

—

103

C₂×C₈

—

206

—

104

C₂×C₈

—

206

—

105

C₂×C₈

—

206

—

106

C₄⋊C₄

—

206

—

107

C₄²

—

206

—

108

C₄²

—

206

—

109

C₄²

Dic₂₀₆

yes

—

110

C₄²

C₂²⋊D₁₀₃

yes

—

111

C₂³

C₂×C₄×D₁₀₃

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

103

—

115

C₂×C₄

—

206

—

116

C₂×C₄

—

206

—

117

C₂×C₄

—

206

—

118

C₂×C₄

—

206

—

119

C₂×C₄

—

206

—

120

C₂×C₄

Dic₁₀₃⋊C₄

yes

—

121

C₂×C₄

Dic₁₀₃⋊C₄

yes

—

122

C₂×C₄

D₄×D₁₀₃

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

124

C₂×C₄

Q₈×D₁₀₃

yes

characteristic

125

C₂×C₄

C₂×Dic₂₀₆

yes

characteristic

126

C₂×C₄

#13110104

yes

characteristic

127

C₂×C₄

#13110106

yes

characteristic

128

C₈

—

129

C₈

—

206

—

130

C₈

—

206

—

131

C₂²

C₂×Dic₁₀₃⋊C₄

yes

characteristic

132

C₂²

#15692983

yes

characteristic

133

C₂²

#15692986

yes

characteristic

134

C₂²

M₁₆×D₁₀₃

yes

—

135

C₂²

#15693027

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

412

—

139

C₄

—

206

—

140

C₄

#15692992

yes

characteristic

141

C₄

#15692997

yes

characteristic

142

C₄

#15693012

yes

characteristic

143

C₄

#15693020

yes

characteristic

144

C₂

#20267687

yes

characteristic

145

C₂

#20267814

yes

characteristic

146

C₂

#20267817

yes

characteristic

147

C₂

#20267861

yes

—

148

C₂

#20267920

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₃
commutator, socle, prime radical	C₂³×C₁₀₃	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₀₃	C₂
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₀₃
hypocenter	C₁₀₃	#1940
LCS	G ▷ C₂³×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₃

commutator, socle, prime radical

C₂³×C₁₀₃

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₀₃

C₂

cosocle

C₂²×C₄×C₁₀₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₀₃

C₁₀₃

G ▷ C₂³×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
75. C₁₀₃	#1940	semidirect

H ≅ G/N

type

75. C₁₀₃

#1940

semidirect

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

subgroup	index
C₂×Dic₁₀₃⋊C₈	2
C₂×#15692932	2
C₄×C₄⋊Dic₁₀₃	2
#20267545	2
C₂×C₄⋊C₈×C₁₀₃	2
#1940	103

subgroup

index

C₂×Dic₁₀₃⋊C₈

C₂×#15692932

C₄×C₄⋊Dic₁₀₃

#20267545

C₂×C₄⋊C₈×C₁₀₃

#1940

103

Subgroups — #27195865

Conjugacy classes of subgroups

Complements

Maximal subgroups