Group #41009610 — subgroups

Subgroups — #1990×C₂₁₁ #41009610

Order 27008.

300 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 160 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×C₄⋊Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#384×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#340×C₂₁₁	C₂	1	4	yes	maximal
5	C₂²×Q₈×C₂₁₁	—	2	1	no	—
6	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
7	C₄⋊Q₈×C₂₁₁	—	2	2	no	—
8	C₄⋊Q₈×C₂₁₁	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄×C₂₁₁	—	2	1	no	—
10	C₂×C₄⋊C₄×C₂₁₁	—	2	1	no	—
11	C₂×C₄⋊C₄×C₂₁₁	—	2	1	no	—
12	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₂₁₁	—	2	2	no	—
16	C₂×Q₈×C₂₁₁	—	2	4	no	—
17	C₂×Q₈×C₂₁₁	—	4	2	no	—
18	C₂×Q₈×C₂₁₁	—	4	2	no	—
19	C₂²×C₄×C₂₁₁	—	2	2	no	—
20	C₂²×C₄×C₂₁₁	—	2	2	no	—
21	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₂₁₁	—	2	2	no	—
25	C₂×C₈×C₂₁₁	—	2	2	no	—
26	C₂×C₈×C₂₁₁	Q₈	1	4	yes	complemented
27	C₄⋊C₄×C₂₁₁	—	2	4	no	—
28	C₄⋊C₄×C₂₁₁	—	2	2	no	—
29	C₄⋊C₄×C₂₁₁	—	2	4	no	—
30	C₄⋊C₄×C₂₁₁	—	4	2	no	—
31	C₄⋊C₄×C₂₁₁	—	4	2	no	—
32	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
33	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
34	C₄²×C₂₁₁	D₄	1	2	yes	—
35	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
36	Q₈×C₂₁₁	—	4	8	no	—
37	C₂×C₄×C₂₁₁	—	2	4	no	—
38	C₂×C₄×C₂₁₁	—	4	2	no	—
39	C₂×C₄×C₂₁₁	—	4	2	no	—
40	C₂×C₄×C₂₁₁	—	4	2	no	—
41	C₂×C₄×C₂₁₁	—	4	2	no	—
42	C₂×C₄×C₂₁₁	—	4	2	no	—
43	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₂₁₁	D₈	1	2	yes	—
46	C₂×C₄×C₂₁₁	SD₁₆	1	2	yes	—
47	C₂×C₄×C₂₁₁	SD₁₆	1	2	yes	—
48	C₂×C₄×C₂₁₁	Q₁₆	1	2	yes	—
49	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₂₁₁	#41	1	2	yes	—
53	C₈×C₂₁₁	—	2	4	no	—
54	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₁₁	#124	1	1	yes	characteristic
56	C₂²×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₂₁₁	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₂₁₁	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₂₁₁	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₂₁₁	—	4	4	no	—
62	C₄×C₂₁₁	—	8	2	no	—
63	C₄×C₂₁₁	#102	1	4	yes	—
64	C₄×C₂₁₁	#103	1	4	yes	—
65	C₄×C₂₁₁	C₄×Q₈	1	2	yes	—
66	C₄×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₂₁₁	#391	1	1	yes	characteristic
68	C₂×C₂₁₁	C₂×#102	1	1	yes	characteristic
69	C₂×C₂₁₁	C₂×#103	1	1	yes	characteristic
70	C₂×C₂₁₁	#486	1	1	yes	characteristic
71	C₂×C₂₁₁	#487	1	1	yes	characteristic
72	C₂×C₂₁₁	#498	1	1	yes	characteristic
73	C₂×C₂₁₁	#500	1	1	yes	characteristic
74	C₂₁₁	#1990	1	1	yes	characteristic, direct product factor, Sylow
75	#1990	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂×C₄⋊Q₈	C₂×C₂₁₁	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂×C₂₁₁	1	1	yes	characteristic
78	#384	C₂×C₂₁₁	1	1	yes	characteristic
79	#340	C₂×C₂₁₁	1	4	yes	—
80	C₂²×Q₈	—	2	1	no	—
81	C₂²×C₈	C₂²×C₂₁₁	1	2	yes	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	C₄×C₂₁₁	1	4	yes	complemented
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	2	1	no	—
87	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
88	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
89	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
90	C₄×C₈	—	2	2	no	—
91	C₂×Q₈	—	2	4	no	—
92	C₂×Q₈	—	4	2	no	—
93	C₂×Q₈	—	4	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	D₄×C₂₁₁	1	1	yes	characteristic
97	C₂²×C₄	D₄×C₂₁₁	1	1	yes	characteristic
98	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	Q₈×C₂₁₁	1	4	yes	complemented
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	4	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	—
108	C₄²	C₂×C₄×C₂₁₁	1	2	yes	—
109	C₄²	D₄×C₂₁₁	1	2	yes	—
110	C₂³	C₂×D₄×C₂₁₁	1	1	yes	characteristic
111	Q₈	—	4	8	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	2	yes	—
119	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	2	yes	—
120	C₂×C₄	D₈×C₂₁₁	1	2	yes	—
121	C₂×C₄	SD₁₆×C₂₁₁	1	2	yes	—
122	C₂×C₄	SD₁₆×C₂₁₁	1	2	yes	—
123	C₂×C₄	Q₁₆×C₂₁₁	1	2	yes	—
124	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
125	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
126	C₂×C₄	C₂×Q₈×C₂₁₁	1	2	yes	—
127	C₂×C₄	#41×C₂₁₁	1	2	yes	—
128	C₈	—	2	4	no	—
129	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
130	C₂²	#124×C₂₁₁	1	1	yes	characteristic
131	C₂²	C₄⋊Q₈×C₂₁₁	1	1	yes	characteristic
132	C₂²	C₂×D₈×C₂₁₁	1	1	yes	characteristic
133	C₂²	C₂×SD₁₆×C₂₁₁	1	1	yes	characteristic
134	C₂²	C₂×SD₁₆×C₂₁₁	1	1	yes	characteristic
135	C₂²	C₂×Q₁₆×C₂₁₁	1	1	yes	characteristic
136	C₄	—	4	4	no	—
137	C₄	—	8	2	no	—
138	C₄	#102×C₂₁₁	1	4	yes	—
139	C₄	#103×C₂₁₁	1	4	yes	—
140	C₄	C₄×Q₈×C₂₁₁	1	2	yes	—
141	C₄	C₂²⋊Q₈×C₂₁₁	1	2	yes	—
142	C₂	#391×C₂₁₁	1	1	yes	characteristic
143	C₂	C₂×#102×C₂₁₁	1	1	yes	characteristic
144	C₂	C₂×#103×C₂₁₁	1	1	yes	characteristic
145	C₂	#486×C₂₁₁	1	1	yes	characteristic
146	C₂	#487×C₂₁₁	1	1	yes	characteristic
147	C₂	#498×C₂₁₁	1	1	yes	characteristic
148	C₂	#500×C₂₁₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₂₁₁

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₂₁₁

C₂

yes

maximal, characteristic

#384×C₂₁₁

C₂

yes

maximal, characteristic

#340×C₂₁₁

C₂

yes

maximal

C₂²×Q₈×C₂₁₁

—

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₄⋊Q₈×C₂₁₁

—

C₄⋊Q₈×C₂₁₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₄×C₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

Q₈

yes

complemented

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

D₄

yes

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

D₈

yes

—

C₂×C₄×C₂₁₁

SD₁₆

yes

—

C₂×C₄×C₂₁₁

SD₁₆

yes

—

C₂×C₄×C₂₁₁

Q₁₆

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₁₁

#124

yes

characteristic

C₂²×C₂₁₁

C₄⋊Q₈

yes

characteristic

C₂²×C₂₁₁

C₂×D₈

yes

characteristic

C₂²×C₂₁₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₁₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₁₁

C₂×Q₁₆

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#102

yes

—

C₄×C₂₁₁

#103

yes

—

C₄×C₂₁₁

C₄×Q₈

yes

—

C₄×C₂₁₁

C₂²⋊Q₈

yes

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, direct product factor, Sylow

#1990

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂×C₂₁₁

yes

characteristic

C₂×C₄×C₈

C₂×C₂₁₁

yes

characteristic

#384

C₂×C₂₁₁

yes

characteristic

#340

C₂×C₂₁₁

yes

—

C₂²×Q₈

—

C₂²×C₈

C₂²×C₂₁₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₄×C₂₁₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₂₁₁

yes

—

C₂×C₄⋊C₄

C₂²×C₂₁₁

yes

—

C₂×C₄²

C₂²×C₂₁₁

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₁₁

yes

characteristic

C₂²×C₄

D₄×C₂₁₁

yes

characteristic

C₂²×C₄

C₂³×C₂₁₁

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

Q₈×C₂₁₁

yes

complemented

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×C₄×C₂₁₁

yes

—

108

C₄²

C₂×C₄×C₂₁₁

yes

—

109

C₄²

D₄×C₂₁₁

yes

—

110

C₂³

C₂×D₄×C₂₁₁

yes

characteristic

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂²⋊C₄×C₂₁₁

yes

—

119

C₂×C₄

C₂²⋊C₄×C₂₁₁

yes

—

120

C₂×C₄

D₈×C₂₁₁

yes

—

121

C₂×C₄

SD₁₆×C₂₁₁

yes

—

122

C₂×C₄

SD₁₆×C₂₁₁

yes

—

123

C₂×C₄

Q₁₆×C₂₁₁

yes

—

124

C₂×C₄

C₂²×C₄×C₂₁₁

yes

characteristic

125

C₂×C₄

C₂×D₄×C₂₁₁

yes

characteristic

126

C₂×C₄

C₂×Q₈×C₂₁₁

yes

—

127

C₂×C₄

#41×C₂₁₁

yes

—

128

C₈

—

129

C₂²

C₂×C₂²⋊C₄×C₂₁₁

yes

characteristic

130

C₂²

#124×C₂₁₁

yes

characteristic

131

C₂²

C₄⋊Q₈×C₂₁₁

yes

characteristic

132

C₂²

C₂×D₈×C₂₁₁

yes

characteristic

133

C₂²

C₂×SD₁₆×C₂₁₁

yes

characteristic

134

C₂²

C₂×SD₁₆×C₂₁₁

yes

characteristic

135

C₂²

C₂×Q₁₆×C₂₁₁

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

#102×C₂₁₁

yes

—

139

C₄

#103×C₂₁₁

yes

—

140

C₄

C₄×Q₈×C₂₁₁

yes

—

141

C₄

C₂²⋊Q₈×C₂₁₁

yes

—

142

C₂

#391×C₂₁₁

yes

characteristic

143

C₂

C₂×#102×C₂₁₁

yes

characteristic

144

C₂

C₂×#103×C₂₁₁

yes

characteristic

145

C₂

#486×C₂₁₁

yes

characteristic

146

C₂

#487×C₂₁₁

yes

characteristic

147

C₂

#498×C₂₁₁

yes

characteristic

148

C₂

#500×C₂₁₁

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₁₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₁₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₁₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₁₁

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₂₁₁

Frattini, cosocle

C₂²×C₄

C₂³×C₂₁₁

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

N	H ≅ G/N	type
8. C₄⋊Q₈×C₂₁₁	C₄	semidirect
26. C₂×C₈×C₂₁₁	Q₈	semidirect
74. C₂₁₁	#1990	direct (N × H)
75. #1990	C₂₁₁	direct (N × H)
83. C₄⋊Q₈	C₄×C₂₁₁	semidirect
101. C₂×C₈	Q₈×C₂₁₁	semidirect

H ≅ G/N

type

8. C₄⋊Q₈×C₂₁₁

C₄

semidirect

26. C₂×C₈×C₂₁₁

Q₈

semidirect

74. C₂₁₁

#1990

direct (N × H)

75. #1990

C₂₁₁

direct (N × H)

83. C₄⋊Q₈

C₄×C₂₁₁

semidirect

101. C₂×C₈

Q₈×C₂₁₁

semidirect

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

subgroup	index
C₂×C₄⋊Q₈×C₂₁₁	2
C₂×C₄×C₈×C₂₁₁	2
#384×C₂₁₁	2
#340×C₂₁₁	2
#1990	211

subgroup

index

C₂×C₄⋊Q₈×C₂₁₁

C₂×C₄×C₈×C₂₁₁

#384×C₂₁₁

#340×C₂₁₁

#1990

211

Subgroups — #1990×C₂₁₁ #41009610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1990×C211 #41009610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1990×C₂₁₁ #41009610