Group #19105747 — subgroups

Subgroups — C₄₁×#1967 #19105747

Order 5248.

312 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 112 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×#124×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#101×C₄₁	C₂	1	2	yes	maximal, complemented
4	C₂×#100×C₄₁	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₄₁	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₄₁	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₄₁	C₂²	1	2	yes	—
9	C₂×M₁₆×C₄₁	C₂²	1	2	yes	—
10	#124×C₄₁	—	2	1	no	—
11	#124×C₄₁	—	2	1	no	—
12	#124×C₄₁	—	2	2	no	—
13	C₂×C₂²⋊C₄×C₄₁	C₄	1	4	yes	—
14	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₄₁	—	2	2	no	—
16	#101×C₄₁	—	2	4	no	—
17	#100×C₄₁	—	2	4	no	—
18	#97×C₄₁	—	2	1	no	—
19	C₄×C₈×C₄₁	—	2	1	no	—
20	C₂⁴×C₄₁	C₂×C₄	1	2	yes	—
21	C₂×Q₈×C₄₁	—	2	2	no	—
22	C₂×Q₈×C₄₁	—	2	1	no	—
23	C₂×Q₈×C₄₁	—	2	1	no	—
24	C₂×Q₈×C₄₁	—	2	2	no	—
25	C₂×D₄×C₄₁	—	2	2	no	—
26	C₂×D₄×C₄₁	—	2	1	no	—
27	C₂×D₄×C₄₁	—	2	1	no	—
28	C₂×D₄×C₄₁	—	2	2	no	—
29	C₂²×C₄×C₄₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₄₁	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
32	M₁₆×C₄₁	—	2	2	no	—
33	M₁₆×C₄₁	—	2	2	no	—
34	M₁₆×C₄₁	—	2	2	no	—
35	M₁₆×C₄₁	D₄	1	4	yes	—
36	C₂×C₈×C₄₁	—	2	2	no	—
37	C₂×C₈×C₄₁	—	2	2	no	—
38	C₂²⋊C₄×C₄₁	—	2	4	no	—
39	C₂²⋊C₄×C₄₁	—	2	4	no	—
40	C₄²×C₄₁	D₄	1	1	yes	characteristic
41	C₄²×C₄₁	D₄	1	1	yes	characteristic
42	C₄²×C₄₁	D₄	1	2	yes	—
43	C₂³×C₄₁	—	2	2	no	—
44	C₂³×C₄₁	—	2	4	no	—
45	C₂³×C₄₁	—	4	4	no	—
46	C₂³×C₄₁	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₄₁	—	4	2	no	—
48	Q₈×C₄₁	—	4	1	no	—
49	Q₈×C₄₁	—	4	1	no	—
50	D₄×C₄₁	—	4	2	no	—
51	D₄×C₄₁	—	4	1	no	—
52	D₄×C₄₁	—	4	1	no	—
53	C₂×C₄×C₄₁	—	2	2	no	—
54	C₂×C₄×C₄₁	—	2	4	no	—
55	C₂×C₄×C₄₁	—	4	2	no	—
56	C₂×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₄₁	#41	1	2	yes	—
62	C₈×C₄₁	—	2	4	no	—
63	C₈×C₄₁	—	4	2	no	—
64	C₂²×C₄₁	—	2	2	no	—
65	C₂²×C₄₁	—	4	2	no	—
66	C₂²×C₄₁	—	4	4	no	—
67	C₂²×C₄₁	—	8	4	no	—
68	C₂²×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄₁	C₄×D₄	1	2	yes	—
70	C₄×C₄₁	—	2	2	no	—
71	C₄×C₄₁	—	2	2	no	—
72	C₄×C₄₁	—	4	2	no	—
73	C₄×C₄₁	#121	1	2	yes	—
74	C₄×C₄₁	#124	1	1	yes	characteristic
75	C₄×C₄₁	#127	1	1	yes	characteristic
76	#1967	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×C₄₁	—	2	2	no	—
78	C₂×C₄₁	—	8	2	no	—
79	C₂×C₄₁	#390	1	1	yes	characteristic
80	C₂×C₄₁	#413	1	2	yes	—
81	C₂×#124	C₂×C₄₁	1	1	yes	characteristic
82	#423	C₂×C₄₁	1	1	yes	characteristic, complemented
83	C₂×#101	C₂×C₄₁	1	2	yes	complemented
84	C₂×#100	C₂×C₄₁	1	2	yes	—
85	C₄×M₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
86	C₄₁	#1967	1	1	yes	characteristic, direct product factor, Sylow
87	C₂²×Q₈	C₂²×C₄₁	1	1	yes	characteristic
88	C₂²×D₄	C₂²×C₄₁	1	1	yes	characteristic
89	C₂×M₁₆	C₂²×C₄₁	1	2	yes	—
90	C₂×M₁₆	C₂²×C₄₁	1	2	yes	—
91	#124	—	2	1	no	—
92	#124	—	2	1	no	—
93	#124	—	2	2	no	—
94	C₂×C₂²⋊C₄	C₄×C₄₁	1	4	yes	—
95	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
96	C₄⋊C₈	—	2	2	no	—
97	#101	—	2	4	no	—
98	#100	—	2	4	no	—
99	#97	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₂⁴	C₂×C₄×C₄₁	1	2	yes	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂²×C₄	C₂×C₄×C₄₁	1	2	yes	—
111	C₂²×C₄	C₂×C₄×C₄₁	1	2	yes	—
112	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	2	2	no	—
116	M₁₆	D₄×C₄₁	1	4	yes	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₄²	D₄×C₄₁	1	1	yes	characteristic
122	C₄²	D₄×C₄₁	1	1	yes	characteristic
123	C₄²	D₄×C₄₁	1	2	yes	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	4	no	—
126	C₂³	—	4	4	no	—
127	C₂³	C₂²×C₄×C₄₁	1	1	yes	characteristic
128	Q₈	—	4	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
138	C₂×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
139	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
140	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
141	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
142	C₂×C₄	#41×C₄₁	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	4	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	2	no	—
147	C₂²	—	4	4	no	—
148	C₂²	—	8	4	no	—
149	C₂²	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
150	C₂²	C₄×D₄×C₄₁	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	#121×C₄₁	1	2	yes	—
155	C₄	#124×C₄₁	1	1	yes	characteristic
156	C₄	#127×C₄₁	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	8	2	no	—
159	C₂	C₄₁×#390	1	1	yes	characteristic
160	C₂	C₄₁×#413	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#124×C₄₁

C₂

yes

maximal, characteristic

C₄₁×#423

C₂

yes

maximal, characteristic, complemented

C₂×#101×C₄₁

C₂

yes

maximal, complemented

C₂×#100×C₄₁

C₂

yes

maximal

C₄×M₁₆×C₄₁

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₄₁

C₂²

yes

characteristic

C₂²×D₄×C₄₁

C₂²

yes

characteristic

C₂×M₁₆×C₄₁

C₂²

yes

—

C₂×M₁₆×C₄₁

C₂²

yes

—

#124×C₄₁

—

#124×C₄₁

—

#124×C₄₁

—

C₂×C₂²⋊C₄×C₄₁

C₄

yes

—

C₂×C₄²×C₄₁

C₂²

yes

characteristic

C₄⋊C₈×C₄₁

—

#101×C₄₁

—

#100×C₄₁

—

#97×C₄₁

—

C₄×C₈×C₄₁

—

C₂⁴×C₄₁

C₂×C₄

yes

—

C₂×Q₈×C₄₁

—

C₂×Q₈×C₄₁

—

C₂×Q₈×C₄₁

—

C₂×Q₈×C₄₁

—

C₂×D₄×C₄₁

—

C₂×D₄×C₄₁

—

C₂×D₄×C₄₁

—

C₂×D₄×C₄₁

—

C₂²×C₄×C₄₁

C₂×C₄

yes

—

C₂²×C₄×C₄₁

C₂×C₄

yes

—

C₂²×C₄×C₄₁

C₂³

yes

characteristic

M₁₆×C₄₁

—

M₁₆×C₄₁

—

M₁₆×C₄₁

—

M₁₆×C₄₁

D₄

yes

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

—

C₂²⋊C₄×C₄₁

—

C₂²⋊C₄×C₄₁

—

C₄²×C₄₁

D₄

yes

characteristic

C₄²×C₄₁

D₄

yes

characteristic

C₄²×C₄₁

D₄

yes

—

C₂³×C₄₁

—

C₂³×C₄₁

—

C₂³×C₄₁

—

C₂³×C₄₁

C₂²×C₄

yes

characteristic

Q₈×C₄₁

—

Q₈×C₄₁

—

Q₈×C₄₁

—

D₄×C₄₁

—

D₄×C₄₁

—

D₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

C₂²⋊C₄

yes

—

C₂×C₄×C₄₁

C₂²⋊C₄

yes

—

C₂×C₄×C₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₄₁

C₂×D₄

yes

—

C₂×C₄×C₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₄₁

#41

yes

—

C₈×C₄₁

—

C₈×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₁

C₄×D₄

yes

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

#121

yes

—

C₄×C₄₁

yes

C₄×C₄₁

yes

C₄₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄₁

—

C₂×C₄₁

—

C₂×C₄₁

#390

yes

characteristic

C₂×C₄₁

#413

yes

—

C₂×#124

C₂×C₄₁

yes

characteristic

#423

C₂×C₄₁

yes

characteristic, complemented

C₂×#101

C₂×C₄₁

yes

complemented

C₂×#100

C₂×C₄₁

yes

—

C₄×M₁₆

C₂×C₄₁

yes

characteristic, complemented

C₄₁

#1967

yes

characteristic, direct product factor, Sylow

C₂²×Q₈

C₂²×C₄₁

yes

characteristic

C₂²×D₄

C₂²×C₄₁

yes

characteristic

C₂×M₁₆

C₂²×C₄₁

yes

—

C₂×M₁₆

C₂²×C₄₁

yes

—

#124

—

#124

—

#124

—

C₂×C₂²⋊C₄

C₄×C₄₁

yes

—

C₂×C₄²

C₂²×C₄₁

yes

characteristic

C₄⋊C₈

—

#101

—

#100

—

#97

—

100

C₄×C₈

—

101

C₂⁴

C₂×C₄×C₄₁

yes

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂²×C₄

C₂×C₄×C₄₁

yes

—

111

C₂²×C₄

C₂×C₄×C₄₁

yes

—

112

C₂²×C₄

C₂³×C₄₁

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

D₄×C₄₁

yes

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

D₄×C₄₁

yes

characteristic

122

C₄²

D₄×C₄₁

yes

characteristic

123

C₄²

D₄×C₄₁

yes

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₂²×C₄×C₄₁

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂²⋊C₄×C₄₁

yes

—

138

C₂×C₄

C₂²⋊C₄×C₄₁

yes

—

139

C₂×C₄

C₂×D₄×C₄₁

yes

characteristic

140

C₂×C₄

C₂×D₄×C₄₁

yes

—

141

C₂×C₄

C₂×D₄×C₄₁

yes

characteristic

142

C₂×C₄

#41×C₄₁

yes

—

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×C₂²⋊C₄×C₄₁

yes

characteristic

150

C₂²

C₄×D₄×C₄₁

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#121×C₄₁

yes

—

155

C₄

#124×C₄₁

yes

characteristic

156

C₄

#127×C₄₁

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

C₄₁×#390

yes

characteristic

160

C₂

C₄₁×#413

yes

—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂³×C₄₁
prime radical	C₂²×D₄×C₄₁	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₄₁

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₄×C₄₁

Frattini, cosocle

C₂²×C₄

C₂³×C₄₁

prime radical

C₂²×D₄×C₄₁

C₂²

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

N	H ≅ G/N	type
2. C₄₁×#423	C₂	semidirect
3. C₂×#101×C₄₁	C₂	semidirect
5. C₄×M₁₆×C₄₁	C₂	semidirect
76. #1967	C₄₁	direct (N × H)
82. #423	C₂×C₄₁	semidirect
83. C₂×#101	C₂×C₄₁	semidirect
85. C₄×M₁₆	C₂×C₄₁	semidirect
86. C₄₁	#1967	direct (N × H)

H ≅ G/N

type

2. C₄₁×#423

C₂

semidirect

3. C₂×#101×C₄₁

C₂

semidirect

5. C₄×M₁₆×C₄₁

C₂

semidirect

76. #1967

C₄₁

direct (N × H)

82. #423

C₂×C₄₁

semidirect

83. C₂×#101

C₂×C₄₁

semidirect

85. C₄×M₁₆

C₂×C₄₁

semidirect

86. C₄₁

#1967

direct (N × H)

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

subgroup	index
C₂×#124×C₄₁	2
C₄₁×#423	2
C₂×#101×C₄₁	2
C₂×#100×C₄₁	2
C₄×M₁₆×C₄₁	2
#1967	41

subgroup

index

C₂×#124×C₄₁

C₄₁×#423

C₂×#101×C₄₁

C₂×#100×C₄₁

C₄×M₁₆×C₄₁

#1967

Subgroups — C₄₁×#1967 #19105747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C41×#1967 #19105747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#1967 #19105747