Group #39913402 — subgroups

Subgroups — C₂×C₁₀₃×#1309 #39913402

Order 26368.

658 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 286 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×#423×C₁₀₃	C₂	1	3	yes	maximal
2	C₁₀₃×#1309	C₂	1	4	yes	maximal, direct product factor
3	C₂²×M₁₆×C₁₀₃	C₄	1	6	yes	—
4	C₂²×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
5	#423×C₁₀₃	—	2	6	no	—
6	#423×C₁₀₃	C₄	1	6	yes	—
7	#423×C₁₀₃	C₂²	1	6	yes	—
8	C₂×C₄⋊C₈×C₁₀₃	—	2	6	no	—
9	C₂³×C₄×C₁₀₃	C₂×C₄	1	3	yes	—
10	C₂×M₁₆×C₁₀₃	—	2	6	no	—
11	C₂×M₁₆×C₁₀₃	—	4	12	no	—
12	C₂×M₁₆×C₁₀₃	C₂×C₄	1	12	yes	—
13	C₂²×C₈×C₁₀₃	—	2	6	no	—
14	C₂×C₄²×C₁₀₃	C₂×C₄	1	3	yes	—
15	C₂×C₄²×C₁₀₃	D₄	1	3	yes	—
16	C₂×C₄²×C₁₀₃	D₄	1	3	yes	—
17	C₂×C₄²×C₁₀₃	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₁₀₃	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
20	C₄⋊C₈×C₁₀₃	—	2	12	no	—
21	C₄⋊C₈×C₁₀₃	—	2	12	no	—
22	C₂⁴×C₁₀₃	C₄²	1	1	yes	characteristic
23	C₂²×C₄×C₁₀₃	—	2	6	no	—
24	C₂²×C₄×C₁₀₃	—	2	6	no	—
25	C₂²×C₄×C₁₀₃	C₄²	1	3	yes	—
26	C₂²×C₄×C₁₀₃	C₂²⋊C₄	1	3	yes	—
27	C₂²×C₄×C₁₀₃	C₂²⋊C₄	1	3	yes	—
28	C₂²×C₄×C₁₀₃	C₄⋊C₄	1	3	yes	—
29	C₂²×C₄×C₁₀₃	C₄⋊C₄	1	3	yes	—
30	C₂²×C₄×C₁₀₃	C₂²×C₄	1	3	yes	—
31	M₁₆×C₁₀₃	—	4	12	no	—
32	M₁₆×C₁₀₃	—	4	12	no	—
33	C₂×C₈×C₁₀₃	—	2	12	no	—
34	C₂×C₈×C₁₀₃	—	4	12	no	—
35	C₄²×C₁₀₃	C₂²⋊C₄	1	6	yes	—
36	C₄²×C₁₀₃	C₄⋊C₄	1	3	yes	—
37	C₄²×C₁₀₃	C₄⋊C₄	1	3	yes	—
38	C₄²×C₁₀₃	C₂×D₄	1	3	yes	—
39	C₄²×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
40	C₂³×C₁₀₃	—	2	6	no	—
41	C₂³×C₁₀₃	#95	1	1	yes	characteristic
42	C₂³×C₁₀₃	#95	1	1	yes	characteristic
43	C₂³×C₁₀₃	C₂×C₄²	1	1	yes	characteristic
44	C₂×C₄×C₁₀₃	—	2	6	no	—
45	C₂×C₄×C₁₀₃	—	2	6	no	—
46	C₂×C₄×C₁₀₃	—	2	6	no	—
47	C₂×C₄×C₁₀₃	—	2	3	no	—
48	C₂×C₄×C₁₀₃	—	2	3	no	—
49	C₂×C₄×C₁₀₃	—	2	3	no	—
50	C₂×C₄×C₁₀₃	—	2	3	no	—
51	C₂×C₄×C₁₀₃	#95	1	3	yes	—
52	C₂×C₄×C₁₀₃	#95	1	3	yes	—
53	C₂×C₄×C₁₀₃	#100	1	6	yes	—
54	C₂×C₄×C₁₀₃	#101	1	6	yes	—
55	C₂×C₄×C₁₀₃	C₂×C₂²⋊C₄	1	3	yes	—
56	C₂×C₄×C₁₀₃	C₂×C₄⋊C₄	1	3	yes	—
57	C₈×C₁₀₃	—	4	12	no	—
58	C₂²×C₁₀₃	—	2	3	no	—
59	C₂²×C₁₀₃	—	2	3	no	—
60	C₂²×C₁₀₃	—	4	4	no	—
61	C₂²×C₁₀₃	#343	1	6	yes	—
62	C₂²×C₁₀₃	C₂×#95	1	1	yes	characteristic
63	C₄×C₁₀₃	—	2	3	no	—
64	C₄×C₁₀₃	—	2	3	no	—
65	C₄×C₁₀₃	#343	1	6	yes	—
66	C₄×C₁₀₃	C₂×#100	1	3	yes	—
67	C₄×C₁₀₃	C₂×#101	1	3	yes	—
68	C₂×#1309	C₁₀₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×C₁₀₃	—	4	1	no	—
70	C₂×C₁₀₃	—	4	1	no	—
71	C₂×C₁₀₃	#1309	1	4	yes	direct product factor
72	C₂×C₁₀₃	C₂×#343	1	3	yes	—
73	C₂×#423	C₂×C₁₀₃	1	3	yes	—
74	#1309	C₂×C₁₀₃	1	4	yes	direct product factor
75	C₁₀₃	C₂×#1309	1	1	yes	characteristic, direct product factor, Sylow
76	C₂²×M₁₆	C₄×C₁₀₃	1	6	yes	—
77	C₂²×C₄²	C₂²×C₁₀₃	1	1	yes	characteristic
78	#423	—	2	6	no	—
79	#423	C₄×C₁₀₃	1	6	yes	—
80	#423	C₂²×C₁₀₃	1	6	yes	—
81	C₂×C₄⋊C₈	—	2	6	no	—
82	C₂³×C₄	C₂×C₄×C₁₀₃	1	3	yes	—
83	C₂×M₁₆	—	2	6	no	—
84	C₂×M₁₆	—	4	12	no	—
85	C₂×M₁₆	C₂×C₄×C₁₀₃	1	12	yes	—
86	C₂²×C₈	—	2	6	no	—
87	C₂×C₄²	C₂×C₄×C₁₀₃	1	3	yes	—
88	C₂×C₄²	D₄×C₁₀₃	1	3	yes	—
89	C₂×C₄²	D₄×C₁₀₃	1	3	yes	—
90	C₂×C₄²	Q₈×C₁₀₃	1	1	yes	characteristic
91	C₂×C₄²	Q₈×C₁₀₃	1	1	yes	characteristic
92	C₂×C₄²	C₂³×C₁₀₃	1	1	yes	characteristic
93	C₄⋊C₈	—	2	12	no	—
94	C₄⋊C₈	—	2	12	no	—
95	C₂⁴	C₄²×C₁₀₃	1	1	yes	characteristic
96	C₂²×C₄	—	2	6	no	—
97	C₂²×C₄	—	2	6	no	—
98	C₂²×C₄	C₄²×C₁₀₃	1	3	yes	—
99	C₂²×C₄	C₂²⋊C₄×C₁₀₃	1	3	yes	—
100	C₂²×C₄	C₂²⋊C₄×C₁₀₃	1	3	yes	—
101	C₂²×C₄	C₄⋊C₄×C₁₀₃	1	3	yes	—
102	C₂²×C₄	C₄⋊C₄×C₁₀₃	1	3	yes	—
103	C₂²×C₄	C₂²×C₄×C₁₀₃	1	3	yes	—
104	M₁₆	—	4	12	no	—
105	M₁₆	—	4	12	no	—
106	C₂×C₈	—	2	12	no	—
107	C₂×C₈	—	4	12	no	—
108	C₄²	C₂²⋊C₄×C₁₀₃	1	6	yes	—
109	C₄²	C₄⋊C₄×C₁₀₃	1	3	yes	—
110	C₄²	C₄⋊C₄×C₁₀₃	1	3	yes	—
111	C₄²	C₂×D₄×C₁₀₃	1	3	yes	—
112	C₄²	C₂×Q₈×C₁₀₃	1	1	yes	characteristic
113	C₂³	—	2	6	no	—
114	C₂³	#95×C₁₀₃	1	1	yes	characteristic
115	C₂³	#95×C₁₀₃	1	1	yes	characteristic
116	C₂³	C₂×C₄²×C₁₀₃	1	1	yes	characteristic
117	C₂×C₄	—	2	6	no	—
118	C₂×C₄	—	2	6	no	—
119	C₂×C₄	—	2	6	no	—
120	C₂×C₄	—	2	3	no	—
121	C₂×C₄	—	2	3	no	—
122	C₂×C₄	—	2	3	no	—
123	C₂×C₄	—	2	3	no	—
124	C₂×C₄	#95×C₁₀₃	1	3	yes	—
125	C₂×C₄	#95×C₁₀₃	1	3	yes	—
126	C₂×C₄	#100×C₁₀₃	1	6	yes	—
127	C₂×C₄	#101×C₁₀₃	1	6	yes	—
128	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₃	1	3	yes	—
129	C₂×C₄	C₂×C₄⋊C₄×C₁₀₃	1	3	yes	—
130	C₈	—	4	12	no	—
131	C₂²	—	2	3	no	—
132	C₂²	—	2	3	no	—
133	C₂²	—	4	4	no	—
134	C₂²	#343×C₁₀₃	1	6	yes	—
135	C₂²	C₂×#95×C₁₀₃	1	1	yes	characteristic
136	C₄	—	2	3	no	—
137	C₄	—	2	3	no	—
138	C₄	#343×C₁₀₃	1	6	yes	—
139	C₄	C₂×#100×C₁₀₃	1	3	yes	—
140	C₄	C₂×#101×C₁₀₃	1	3	yes	—
141	C₂	—	4	1	no	—
142	C₂	—	4	1	no	—
143	C₂	C₁₀₃×#1309	1	4	yes	direct product factor
144	C₂	C₂×#343×C₁₀₃	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#423×C₁₀₃

C₂

yes

maximal

C₁₀₃×#1309

C₂

yes

maximal, direct product factor

C₂²×M₁₆×C₁₀₃

C₄

yes

—

C₂²×C₄²×C₁₀₃

C₂²

yes

characteristic

#423×C₁₀₃

—

#423×C₁₀₃

C₄

yes

—

#423×C₁₀₃

C₂²

yes

—

C₂×C₄⋊C₈×C₁₀₃

—

C₂³×C₄×C₁₀₃

C₂×C₄

yes

—

C₂×M₁₆×C₁₀₃

—

C₂×M₁₆×C₁₀₃

—

C₂×M₁₆×C₁₀₃

C₂×C₄

yes

—

C₂²×C₈×C₁₀₃

—

C₂×C₄²×C₁₀₃

C₂×C₄

yes

—

C₂×C₄²×C₁₀₃

D₄

yes

—

C₂×C₄²×C₁₀₃

D₄

yes

—

C₂×C₄²×C₁₀₃

Q₈

yes

characteristic

C₂×C₄²×C₁₀₃

Q₈

yes

characteristic

C₂×C₄²×C₁₀₃

C₂³

yes

characteristic

C₄⋊C₈×C₁₀₃

—

C₄⋊C₈×C₁₀₃

—

C₂⁴×C₁₀₃

C₄²

yes

characteristic

C₂²×C₄×C₁₀₃

—

C₂²×C₄×C₁₀₃

—

C₂²×C₄×C₁₀₃

C₄²

yes

—

C₂²×C₄×C₁₀₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₀₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₀₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₀₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₀₃

C₂²×C₄

yes

—

M₁₆×C₁₀₃

—

M₁₆×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₄²×C₁₀₃

C₂²⋊C₄

yes

—

C₄²×C₁₀₃

C₄⋊C₄

yes

—

C₄²×C₁₀₃

C₄⋊C₄

yes

—

C₄²×C₁₀₃

C₂×D₄

yes

—

C₄²×C₁₀₃

C₂×Q₈

yes

characteristic

C₂³×C₁₀₃

—

C₂³×C₁₀₃

yes

C₂³×C₁₀₃

yes

C₂³×C₁₀₃

C₂×C₄²

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

#95

yes

—

C₂×C₄×C₁₀₃

#95

yes

—

C₂×C₄×C₁₀₃

#100

yes

—

C₂×C₄×C₁₀₃

#101

yes

—

C₂×C₄×C₁₀₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂×C₄⋊C₄

yes

—

C₈×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#343

yes

—

C₂²×C₁₀₃

C₂×#95

yes

characteristic

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#343

yes

—

C₄×C₁₀₃

C₂×#100

yes

—

C₄×C₁₀₃

C₂×#101

yes

—

C₂×#1309

C₁₀₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#1309

yes

direct product factor

C₂×C₁₀₃

C₂×#343

yes

—

C₂×#423

C₂×C₁₀₃

yes

—

#1309

C₂×C₁₀₃

yes

direct product factor

C₁₀₃

C₂×#1309

yes

characteristic, direct product factor, Sylow

C₂²×M₁₆

C₄×C₁₀₃

yes

—

C₂²×C₄²

C₂²×C₁₀₃

yes

characteristic

#423

—

#423

C₄×C₁₀₃

yes

—

#423

C₂²×C₁₀₃

yes

—

C₂×C₄⋊C₈

—

C₂³×C₄

C₂×C₄×C₁₀₃

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄×C₁₀₃

yes

—

C₂²×C₈

—

C₂×C₄²

C₂×C₄×C₁₀₃

yes

—

C₂×C₄²

D₄×C₁₀₃

yes

—

C₂×C₄²

D₄×C₁₀₃

yes

—

C₂×C₄²

Q₈×C₁₀₃

yes

characteristic

C₂×C₄²

Q₈×C₁₀₃

yes

characteristic

C₂×C₄²

C₂³×C₁₀₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂⁴

C₄²×C₁₀₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄²×C₁₀₃

yes

—

C₂²×C₄

C₂²⋊C₄×C₁₀₃

yes

—

100

C₂²×C₄

C₂²⋊C₄×C₁₀₃

yes

—

101

C₂²×C₄

C₄⋊C₄×C₁₀₃

yes

—

102

C₂²×C₄

C₄⋊C₄×C₁₀₃

yes

—

103

C₂²×C₄

C₂²×C₄×C₁₀₃

yes

—

104

M₁₆

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₄²

C₂²⋊C₄×C₁₀₃

yes

—

109

C₄²

C₄⋊C₄×C₁₀₃

yes

—

110

C₄²

C₄⋊C₄×C₁₀₃

yes

—

111

C₄²

C₂×D₄×C₁₀₃

yes

—

112

C₄²

C₂×Q₈×C₁₀₃

yes

characteristic

113

C₂³

—

114

C₂³

#95×C₁₀₃

yes

characteristic

115

C₂³

#95×C₁₀₃

yes

characteristic

116

C₂³

C₂×C₄²×C₁₀₃

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#95×C₁₀₃

yes

—

125

C₂×C₄

#95×C₁₀₃

yes

—

126

C₂×C₄

#100×C₁₀₃

yes

—

127

C₂×C₄

#101×C₁₀₃

yes

—

128

C₂×C₄

C₂×C₂²⋊C₄×C₁₀₃

yes

—

129

C₂×C₄

C₂×C₄⋊C₄×C₁₀₃

yes

—

130

C₈

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

#343×C₁₀₃

yes

—

135

C₂²

C₂×#95×C₁₀₃

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

#343×C₁₀₃

yes

—

139

C₄

C₂×#100×C₁₀₃

yes

—

140

C₄

C₂×#101×C₁₀₃

yes

—

141

C₂

—

142

C₂

—

143

C₂

C₁₀₃×#1309

yes

direct product factor

144

C₂

C₂×#343×C₁₀₃

yes

—

	subgroup	G/H
center, socle	C₂³×C₁₀₃	#95
commutator	C₂³	C₂×C₄²×C₁₀₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₃
prime radical	C₂⁴×C₁₀₃	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₃ ◁ C₂²×C₄²×C₁₀₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₀₃

#95

commutator

C₂³

C₂×C₄²×C₁₀₃

Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₃

prime radical

C₂⁴×C₁₀₃

C₄²

LCS

G ▷ C₂³ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₃ ◁ C₂²×C₄²×C₁₀₃ ◁ G

N	H ≅ G/N	type
2. C₁₀₃×#1309	C₂	direct (N × H)
68. C₂×#1309	C₁₀₃	direct (N × H)
71. C₂×C₁₀₃	#1309	direct (N × H)
74. #1309	C₂×C₁₀₃	direct (N × H)
75. C₁₀₃	C₂×#1309	direct (N × H)
143. C₂	C₁₀₃×#1309	direct (N × H)

H ≅ G/N

type

2. C₁₀₃×#1309

C₂

direct (N × H)

68. C₂×#1309

C₁₀₃

direct (N × H)

71. C₂×C₁₀₃

#1309

direct (N × H)

74. #1309

C₂×C₁₀₃

direct (N × H)

75. C₁₀₃

C₂×#1309

direct (N × H)

143. C₂

C₁₀₃×#1309

direct (N × H)

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

subgroup	index
C₂×#423×C₁₀₃	2
C₁₀₃×#1309	2
C₂×#1309	103

subgroup

index

C₂×#423×C₁₀₃

C₁₀₃×#1309

C₂×#1309

103

Subgroups — C₂×C₁₀₃×#1309 #39913402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C103×#1309 #39913402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₃×#1309 #39913402