Group #23788247 — subgroups

Subgroups — C₄₁×#12046 #23788247

Order 10496.

216 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₄₁×#2178	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2170	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1373	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1335	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#508	C₄	1	1	yes	characteristic
7	C₄₁×#504	—	2	2	no	—
8	C₄₁×#443	C₂²	1	2	yes	—
9	C₄₁×#368	C₄	1	1	yes	characteristic
10	C₄₁×#364	C₂²	1	2	yes	—
11	C₄₁×#359	C₄	1	2	yes	—
12	C₄₁×#350	—	2	2	no	—
13	C₄×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
15	C₈²×C₄₁	C₂²	1	1	yes	characteristic
16	#135×C₄₁	C₂×C₄	1	2	yes	—
17	#131×C₄₁	—	2	2	no	—
18	Q₃₂×C₄₁	C₈	1	1	yes	characteristic, complemented
19	SD₃₂×C₄₁	C₈	1	2	yes	complemented
20	D₁₆×C₄₁	C₈	1	1	yes	characteristic, complemented
21	M₃₂×C₄₁	—	2	2	no	—
22	M₃₂×C₄₁	—	4	2	no	—
23	C₂×C₁₆×C₄₁	—	4	2	no	—
24	C₂×C₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₄₁	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₄₁	D₄	1	1	yes	characteristic
28	#108×C₄₁	C₂×C₄	1	2	yes	—
29	#104×C₄₁	—	2	2	no	—
30	C₄×C₈×C₄₁	—	2	1	no	—
31	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
32	#41×C₄₁	—	2	2	no	—
33	Q₁₆×C₄₁	C₂×C₈	1	2	yes	—
34	SD₁₆×C₄₁	—	2	2	no	—
35	D₈×C₄₁	C₂×C₈	1	2	yes	—
36	M₁₆×C₄₁	—	2	2	no	—
37	M₁₆×C₄₁	—	4	2	no	—
38	C₂×C₈×C₄₁	—	2	2	no	—
39	C₂×C₈×C₄₁	—	4	2	no	—
40	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₄₁	#41	1	1	yes	characteristic
43	C₁₆×C₄₁	—	2	1	no	—
44	C₁₆×C₄₁	—	2	1	no	—
45	C₁₆×C₄₁	—	4	2	no	—
46	C₁₆×C₄₁	C₂×C₈	1	1	yes	characteristic
47	C₁₆×C₄₁	C₂×C₈	1	1	yes	characteristic
48	C₁₆×C₄₁	D₈	1	1	yes	characteristic
49	C₁₆×C₄₁	D₈	1	1	yes	characteristic
50	Q₈×C₄₁	—	2	2	no	—
51	D₄×C₄₁	—	2	2	no	—
52	D₄×C₄₁	—	4	2	no	—
53	C₂×C₄×C₄₁	—	2	1	no	—
54	C₂×C₄×C₄₁	—	4	2	no	—
55	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
56	C₈×C₄₁	—	2	4	no	—
57	C₈×C₄₁	—	4	2	no	—
58	C₈×C₄₁	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₄₁	#131	1	1	yes	characteristic
60	C₈×C₄₁	C₂×D₈	1	1	yes	characteristic
61	C₈×C₄₁	#135	1	1	yes	characteristic
62	#12046	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂²×C₄₁	—	4	2	no	—
64	C₂²×C₄₁	#443	1	1	yes	characteristic
65	C₄×C₄₁	—	2	2	no	—
66	C₄×C₄₁	—	4	2	no	—
67	C₄×C₄₁	C₈×D₄	1	1	yes	characteristic
68	C₄×C₄₁	C₄×D₈	1	1	yes	characteristic
69	#2178	C₂×C₄₁	1	1	yes	characteristic
70	#2170	C₂×C₄₁	1	2	yes	complemented
71	#1373	C₂×C₄₁	1	1	yes	characteristic, complemented
72	#1335	C₂×C₄₁	1	2	yes	complemented
73	C₈×C₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
74	C₂×C₄₁	—	2	1	no	—
75	C₂×C₄₁	—	8	2	no	—
76	C₂×C₄₁	C₈×D₈	1	1	yes	characteristic
77	#508	C₄×C₄₁	1	1	yes	characteristic
78	#504	—	2	2	no	—
79	#443	C₂²×C₄₁	1	2	yes	—
80	#368	C₄×C₄₁	1	1	yes	characteristic
81	#364	C₂²×C₄₁	1	2	yes	—
82	#359	C₄×C₄₁	1	2	yes	—
83	#350	—	2	2	no	—
84	C₄×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
85	C₄×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
86	C₈²	C₂²×C₄₁	1	1	yes	characteristic
87	C₄₁	#12046	1	1	yes	characteristic, direct product factor, Sylow
88	#135	C₂×C₄×C₄₁	1	2	yes	—
89	#131	—	2	2	no	—
90	Q₃₂	C₈×C₄₁	1	1	yes	characteristic, complemented
91	SD₃₂	C₈×C₄₁	1	2	yes	complemented
92	D₁₆	C₈×C₄₁	1	1	yes	characteristic, complemented
93	M₃₂	—	2	2	no	—
94	M₃₂	—	4	2	no	—
95	C₂×C₁₆	—	4	2	no	—
96	C₂×C₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
97	C₂×C₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
98	C₂×C₁₆	D₄×C₄₁	1	1	yes	characteristic
99	C₂×C₁₆	D₄×C₄₁	1	1	yes	characteristic
100	#108	C₂×C₄×C₄₁	1	2	yes	—
101	#104	—	2	2	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
104	#41	—	2	2	no	—
105	Q₁₆	C₂×C₈×C₄₁	1	2	yes	—
106	SD₁₆	—	2	2	no	—
107	D₈	C₂×C₈×C₄₁	1	2	yes	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	4	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
113	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
114	C₄²	#41×C₄₁	1	1	yes	characteristic
115	C₁₆	—	2	1	no	—
116	C₁₆	—	2	1	no	—
117	C₁₆	—	4	2	no	—
118	C₁₆	C₂×C₈×C₄₁	1	1	yes	characteristic
119	C₁₆	C₂×C₈×C₄₁	1	1	yes	characteristic
120	C₁₆	D₈×C₄₁	1	1	yes	characteristic
121	C₁₆	D₈×C₄₁	1	1	yes	characteristic
122	Q₈	—	2	2	no	—
123	D₄	—	2	2	no	—
124	D₄	—	4	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
128	C₈	—	2	4	no	—
129	C₈	—	4	2	no	—
130	C₈	C₂²×C₈×C₄₁	1	1	yes	characteristic
131	C₈	#131×C₄₁	1	1	yes	characteristic
132	C₈	C₂×D₈×C₄₁	1	1	yes	characteristic
133	C₈	#135×C₄₁	1	1	yes	characteristic
134	C₂²	—	4	2	no	—
135	C₂²	C₄₁×#443	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	4	2	no	—
138	C₄	C₈×D₄×C₄₁	1	1	yes	characteristic
139	C₄	C₄×D₈×C₄₁	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	8	2	no	—
142	C₂	C₈×D₈×C₄₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄₁×#2178

C₂

yes

maximal, characteristic

C₄₁×#2170

C₂

yes

maximal, complemented

C₄₁×#1373

C₂

yes

maximal, characteristic, complemented

C₄₁×#1335

C₂

yes

maximal, complemented

C₈×C₁₆×C₄₁

C₂

yes

maximal, characteristic, complemented

C₄₁×#508

C₄

yes

characteristic

C₄₁×#504

—

C₄₁×#443

C₂²

yes

—

C₄₁×#368

C₄

yes

characteristic

C₄₁×#364

C₂²

yes

—

C₄₁×#359

C₄

yes

—

C₄₁×#350

—

C₄×C₁₆×C₄₁

C₂²

yes

characteristic

C₄×C₁₆×C₄₁

C₂²

yes

characteristic

C₈²×C₄₁

C₂²

yes

characteristic

#135×C₄₁

C₂×C₄

yes

—

#131×C₄₁

—

Q₃₂×C₄₁

C₈

yes

characteristic, complemented

SD₃₂×C₄₁

C₈

yes

complemented

D₁₆×C₄₁

C₈

yes

characteristic, complemented

M₃₂×C₄₁

—

M₃₂×C₄₁

—

C₂×C₁₆×C₄₁

—

C₂×C₁₆×C₄₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₁

D₄

yes

characteristic

C₂×C₁₆×C₄₁

D₄

yes

characteristic

#108×C₄₁

C₂×C₄

yes

—

#104×C₄₁

—

C₄×C₈×C₄₁

—

C₄×C₈×C₄₁

C₂³

yes

characteristic

#41×C₄₁

—

Q₁₆×C₄₁

C₂×C₈

yes

—

SD₁₆×C₄₁

—

D₈×C₄₁

C₂×C₈

yes

—

M₁₆×C₄₁

—

M₁₆×C₄₁

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

—

C₂×C₈×C₄₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₁

C₂×D₄

yes

characteristic

C₄²×C₄₁

#41

yes

characteristic

C₁₆×C₄₁

—

C₁₆×C₄₁

—

C₁₆×C₄₁

—

C₁₆×C₄₁

C₂×C₈

yes

characteristic

C₁₆×C₄₁

C₂×C₈

yes

characteristic

C₁₆×C₄₁

D₈

yes

characteristic

C₁₆×C₄₁

D₈

yes

characteristic

Q₈×C₄₁

—

D₄×C₄₁

—

D₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

C₄×D₄

yes

characteristic

C₈×C₄₁

—

C₈×C₄₁

—

C₈×C₄₁

C₂²×C₈

yes

characteristic

C₈×C₄₁

#131

yes

characteristic

C₈×C₄₁

C₂×D₈

yes

C₈×C₄₁

yes

C₄₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄₁

—

C₂²×C₄₁

#443

yes

characteristic

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

C₈×D₄

yes

characteristic

C₄×C₄₁

C₄×D₈

yes

characteristic

#2178

C₂×C₄₁

yes

characteristic

#2170

C₂×C₄₁

yes

complemented

#1373

C₂×C₄₁

yes

characteristic, complemented

#1335

C₂×C₄₁

yes

complemented

C₈×C₁₆

C₂×C₄₁

yes

characteristic, complemented

C₂×C₄₁

—

C₂×C₄₁

—

C₂×C₄₁

C₈×D₈

yes

characteristic

#508

C₄×C₄₁

yes

characteristic

#504

—

#443

C₂²×C₄₁

yes

—

#368

C₄×C₄₁

yes

characteristic

#364

C₂²×C₄₁

yes

—

#359

C₄×C₄₁

yes

—

#350

—

C₄×C₁₆

C₂²×C₄₁

yes

characteristic

C₄×C₁₆

C₂²×C₄₁

yes

characteristic

C₈²

C₂²×C₄₁

yes

characteristic

C₄₁

#12046

yes

characteristic, direct product factor, Sylow

#135

C₂×C₄×C₄₁

yes

—

#131

—

Q₃₂

C₈×C₄₁

yes

characteristic, complemented

SD₃₂

C₈×C₄₁

yes

complemented

D₁₆

C₈×C₄₁

yes

characteristic, complemented

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₄₁

yes

characteristic

C₂×C₁₆

C₂×C₄×C₄₁

yes

characteristic

C₂×C₁₆

D₄×C₄₁

yes

characteristic

C₂×C₁₆

D₄×C₄₁

yes

characteristic

100

#108

C₂×C₄×C₄₁

yes

—

101

#104

—

102

C₄×C₈

—

103

C₄×C₈

C₂³×C₄₁

yes

characteristic

104

#41

—

105

Q₁₆

C₂×C₈×C₄₁

yes

—

106

SD₁₆

—

107

D₈

C₂×C₈×C₄₁

yes

—

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×C₄×C₄₁

yes

characteristic

113

C₂×C₈

C₂×D₄×C₄₁

yes

characteristic

114

C₄²

#41×C₄₁

yes

characteristic

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

C₁₆

C₂×C₈×C₄₁

yes

characteristic

119

C₁₆

C₂×C₈×C₄₁

yes

characteristic

120

C₁₆

D₈×C₄₁

yes

characteristic

121

C₁₆

D₈×C₄₁

yes

characteristic

122

Q₈

—

123

D₄

—

124

D₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₄×D₄×C₄₁

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

C₂²×C₈×C₄₁

yes

characteristic

131

C₈

#131×C₄₁

yes

characteristic

132

C₈

C₂×D₈×C₄₁

yes

characteristic

133

C₈

#135×C₄₁

yes

characteristic

134

C₂²

—

135

C₂²

C₄₁×#443

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

C₈×D₄×C₄₁

yes

characteristic

139

C₄

C₄×D₈×C₄₁

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

C₈×D₈×C₄₁

yes

characteristic

	subgroup	G/H
center	C₁₆×C₄₁	D₈
commutator	C₈	C₂²×C₈×C₄₁
Frattini, cosocle	C₄×C₈	C₂³×C₄₁
socle	C₂×C₄₁	C₈×D₈
prime radical	C₄₁×#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₄₁ ◁ C₂×C₁₆×C₄₁ ◁ C₄×C₁₆×C₄₁ ◁ G

subgroup

G/H

center

C₁₆×C₄₁

D₈

commutator

C₈

C₂²×C₈×C₄₁

Frattini, cosocle

C₄×C₈

C₂³×C₄₁

socle

C₂×C₄₁

C₈×D₈

prime radical

C₄₁×#508

C₄

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₁₆×C₄₁ ◁ C₂×C₁₆×C₄₁ ◁ C₄×C₁₆×C₄₁ ◁ G

N	H ≅ G/N	type
2. C₄₁×#2170	C₂	semidirect
3. C₄₁×#1373	C₂	semidirect
4. C₄₁×#1335	C₂	semidirect
5. C₈×C₁₆×C₄₁	C₂	semidirect
18. Q₃₂×C₄₁	C₈	semidirect
19. SD₃₂×C₄₁	C₈	semidirect
20. D₁₆×C₄₁	C₈	semidirect
62. #12046	C₄₁	direct (N × H)
70. #2170	C₂×C₄₁	semidirect
71. #1373	C₂×C₄₁	semidirect
72. #1335	C₂×C₄₁	semidirect
73. C₈×C₁₆	C₂×C₄₁	semidirect
87. C₄₁	#12046	direct (N × H)
90. Q₃₂	C₈×C₄₁	semidirect
91. SD₃₂	C₈×C₄₁	semidirect
92. D₁₆	C₈×C₄₁	semidirect

H ≅ G/N

type

2. C₄₁×#2170

C₂

semidirect

3. C₄₁×#1373

C₂

semidirect

4. C₄₁×#1335

C₂

semidirect

5. C₈×C₁₆×C₄₁

C₂

semidirect

18. Q₃₂×C₄₁

C₈

semidirect

19. SD₃₂×C₄₁

C₈

semidirect

20. D₁₆×C₄₁

C₈

semidirect

62. #12046

C₄₁

direct (N × H)

70. #2170

C₂×C₄₁

semidirect

71. #1373

C₂×C₄₁

semidirect

72. #1335

C₂×C₄₁

semidirect

73. C₈×C₁₆

C₂×C₄₁

semidirect

87. C₄₁

#12046

direct (N × H)

90. Q₃₂

C₈×C₄₁

semidirect

91. SD₃₂

C₈×C₄₁

semidirect

92. D₁₆

C₈×C₄₁

semidirect

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

subgroup	index
C₄₁×#2178	2
C₄₁×#2170	2
C₄₁×#1373	2
C₄₁×#1335	2
C₈×C₁₆×C₄₁	2
#12046	41

subgroup

index

C₄₁×#2178

C₄₁×#2170

C₄₁×#1373

C₄₁×#1335

C₈×C₁₆×C₄₁

#12046

Subgroups — C₄₁×#12046 #23788247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C41×#12046 #23788247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#12046 #23788247