Group #41009713 — subgroups

Subgroups — #2093×C₂₁₁ #41009713

Order 27008.

290 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 108 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×#107×C₂₁₁	C₂	1	2	yes	maximal, complemented
2	C₂×#102×C₂₁₁	C₂	1	2	yes	maximal
3	#390×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#384×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#336×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₂₁₁	—	2	2	no	—
9	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊C₄×C₂₁₁	—	2	2	no	—
12	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
13	#107×C₂₁₁	—	2	4	no	—
14	#102×C₂₁₁	—	2	4	no	—
15	C₂⁴×C₂₁₁	—	2	1	no	—
16	C₂×D₄×C₂₁₁	—	2	2	no	—
17	C₂×D₄×C₂₁₁	—	2	2	no	—
18	C₂×D₄×C₂₁₁	—	2	1	no	—
19	C₂×D₄×C₂₁₁	Q₈	1	2	yes	—
20	C₂²×C₄×C₂₁₁	—	2	1	no	—
21	C₂²×C₄×C₂₁₁	—	2	2	no	—
22	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
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	D₄	1	4	yes	—
27	C₄⋊C₄×C₂₁₁	—	2	2	no	—
28	C₄⋊C₄×C₂₁₁	—	2	2	no	—
29	C₄⋊C₄×C₂₁₁	—	2	2	no	—
30	C₄⋊C₄×C₂₁₁	—	4	4	no	—
31	C₂²⋊C₄×C₂₁₁	—	4	4	no	—
32	C₄²×C₂₁₁	—	2	2	no	—
33	C₂³×C₂₁₁	—	2	2	no	—
34	C₂³×C₂₁₁	—	4	1	no	—
35	C₂³×C₂₁₁	—	4	1	no	—
36	C₂³×C₂₁₁	—	4	2	no	—
37	C₂³×C₂₁₁	—	4	2	no	—
38	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
39	D₄×C₂₁₁	—	2	4	no	—
40	D₄×C₂₁₁	—	4	2	no	—
41	C₂×C₄×C₂₁₁	—	2	2	no	—
42	C₂×C₄×C₂₁₁	—	2	2	no	—
43	C₂×C₄×C₂₁₁	—	4	2	no	—
44	C₂×C₄×C₂₁₁	—	4	1	no	—
45	C₂×C₄×C₂₁₁	—	4	2	no	—
46	C₂×C₄×C₂₁₁	—	4	4	no	—
47	C₂×C₄×C₂₁₁	D₈	1	4	yes	—
48	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₂₁₁	#41	1	2	yes	—
51	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
52	C₈×C₂₁₁	—	2	4	no	—
53	C₂²×C₂₁₁	—	4	2	no	—
54	C₂²×C₂₁₁	—	4	4	no	—
55	C₂²×C₂₁₁	—	8	2	no	—
56	C₂²×C₂₁₁	—	8	1	no	—
57	C₂²×C₂₁₁	—	8	1	no	—
58	C₂²×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂²×C₂₁₁	#123	1	1	yes	characteristic
60	C₂²×C₂₁₁	#127	1	1	yes	characteristic
61	C₂²×C₂₁₁	C₂×D₈	1	2	yes	—
62	C₂²×C₂₁₁	#137	1	2	yes	—
63	C₄×C₂₁₁	—	2	4	no	—
64	C₄×C₂₁₁	—	8	1	no	—
65	C₄×C₂₁₁	—	8	2	no	—
66	C₄×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₂₁₁	#123	1	2	yes	—
68	C₂×C₂₁₁	—	8	2	no	—
69	C₂×C₂₁₁	#399	1	1	yes	characteristic
70	C₂×C₂₁₁	#474	1	2	yes	—
71	C₂×C₂₁₁	#480	1	2	yes	—
72	C₂×C₂₁₁	#493	1	1	yes	characteristic
73	C₂×C₂₁₁	#497	1	1	yes	characteristic
74	C₂₁₁	#2093	1	1	yes	characteristic, direct product factor, Sylow
75	#2093	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂×#107	C₂×C₂₁₁	1	2	yes	complemented
77	C₂×#102	C₂×C₂₁₁	1	2	yes	—
78	#390	C₂×C₂₁₁	1	1	yes	characteristic
79	#384	C₂×C₂₁₁	1	1	yes	characteristic, complemented
80	#336	C₂×C₂₁₁	1	1	yes	characteristic, complemented
81	C₂²×D₄	C₂²×C₂₁₁	1	1	yes	characteristic
82	C₂²×C₈	C₂²×C₂₁₁	1	2	yes	—
83	C₂×C₄⋊C₄	—	2	2	no	—
84	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
85	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
86	C₂×C₂²⋊C₄	—	2	2	no	—
87	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
88	#107	—	2	4	no	—
89	#102	—	2	4	no	—
90	C₂⁴	—	2	1	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	Q₈×C₂₁₁	1	2	yes	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
98	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	D₄×C₂₁₁	1	4	yes	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	4	4	no	—
106	C₂²⋊C₄	—	4	4	no	—
107	C₄²	—	2	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	4	1	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	2	no	—
112	C₂³	—	4	2	no	—
113	C₂³	C₂×D₄×C₂₁₁	1	1	yes	characteristic
114	D₄	—	2	4	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	4	no	—
122	C₂×C₄	D₈×C₂₁₁	1	4	yes	—
123	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
124	C₂×C₄	C₂×Q₈×C₂₁₁	1	1	yes	characteristic
125	C₂×C₄	#41×C₂₁₁	1	2	yes	—
126	C₂×C₄	#41×C₂₁₁	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₂²	—	4	2	no	—
129	C₂²	—	4	4	no	—
130	C₂²	—	8	2	no	—
131	C₂²	—	8	1	no	—
132	C₂²	—	8	1	no	—
133	C₂²	C₂²⋊Q₈×C₂₁₁	1	1	yes	characteristic
134	C₂²	#123×C₂₁₁	1	1	yes	characteristic
135	C₂²	#127×C₂₁₁	1	1	yes	characteristic
136	C₂²	C₂×D₈×C₂₁₁	1	2	yes	—
137	C₂²	#137×C₂₁₁	1	2	yes	—
138	C₄	—	2	4	no	—
139	C₄	—	8	1	no	—
140	C₄	—	8	2	no	—
141	C₄	C₂²⋊Q₈×C₂₁₁	1	2	yes	—
142	C₄	#123×C₂₁₁	1	2	yes	—
143	C₂	—	8	2	no	—
144	C₂	#399×C₂₁₁	1	1	yes	characteristic
145	C₂	#474×C₂₁₁	1	2	yes	—
146	C₂	#480×C₂₁₁	1	2	yes	—
147	C₂	#493×C₂₁₁	1	1	yes	characteristic
148	C₂	#497×C₂₁₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#107×C₂₁₁

C₂

yes

maximal, complemented

C₂×#102×C₂₁₁

C₂

yes

maximal

#390×C₂₁₁

C₂

yes

maximal, characteristic

#384×C₂₁₁

C₂

yes

maximal, characteristic, complemented

#336×C₂₁₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₂₁₁

C₂²

yes

characteristic

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₂×C₄⋊C₄×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₂₁₁

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

#107×C₂₁₁

—

#102×C₂₁₁

—

C₂⁴×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

Q₈

yes

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

D₄×C₂₁₁

—

D₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

D₈

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂²⋊Q₈

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

C₂×D₈

yes

—

C₂²×C₂₁₁

#137

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂²⋊Q₈

yes

—

C₄×C₂₁₁

#123

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#399

yes

characteristic

C₂×C₂₁₁

#474

yes

—

C₂×C₂₁₁

#480

yes

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, direct product factor, Sylow

#2093

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×#107

C₂×C₂₁₁

yes

complemented

C₂×#102

C₂×C₂₁₁

yes

—

#390

C₂×C₂₁₁

yes

characteristic

#384

C₂×C₂₁₁

yes

characteristic, complemented

#336

C₂×C₂₁₁

yes

characteristic, complemented

C₂²×D₄

C₂²×C₂₁₁

yes

characteristic

C₂²×C₈

C₂²×C₂₁₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₂₁₁

yes

—

C₂×C₄⋊C₄

C₂²×C₂₁₁

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂²×C₂₁₁

yes

characteristic

#107

—

#102

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Q₈×C₂₁₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₁₁

yes

—

C₂²×C₄

C₂³×C₂₁₁

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

D₄×C₂₁₁

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂×D₄×C₂₁₁

yes

characteristic

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

D₈×C₂₁₁

yes

—

123

C₂×C₄

C₂×D₄×C₂₁₁

yes

—

124

C₂×C₄

C₂×Q₈×C₂₁₁

yes

characteristic

125

C₂×C₄

#41×C₂₁₁

yes

—

126

C₂×C₄

#41×C₂₁₁

yes

characteristic

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

C₂²⋊Q₈×C₂₁₁

yes

characteristic

134

C₂²

#123×C₂₁₁

yes

characteristic

135

C₂²

#127×C₂₁₁

yes

characteristic

136

C₂²

C₂×D₈×C₂₁₁

yes

—

137

C₂²

#137×C₂₁₁

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

C₂²⋊Q₈×C₂₁₁

yes

—

142

C₄

#123×C₂₁₁

yes

—

143

C₂

—

144

C₂

#399×C₂₁₁

yes

characteristic

145

C₂

#474×C₂₁₁

yes

—

146

C₂

#480×C₂₁₁

yes

—

147

C₂

#493×C₂₁₁

yes

characteristic

148

C₂

#497×C₂₁₁

yes

characteristic

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

subgroup

G/H

center, socle

C₂³×C₂₁₁

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₂₁₁

prime radical

C₂²×D₄×C₂₁₁

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

N	H ≅ G/N	type
1. C₂×#107×C₂₁₁	C₂	semidirect
4. #384×C₂₁₁	C₂	semidirect
5. #336×C₂₁₁	C₂	semidirect
74. C₂₁₁	#2093	direct (N × H)
75. #2093	C₂₁₁	direct (N × H)
76. C₂×#107	C₂×C₂₁₁	semidirect
79. #384	C₂×C₂₁₁	semidirect
80. #336	C₂×C₂₁₁	semidirect

H ≅ G/N

type

1. C₂×#107×C₂₁₁

C₂

semidirect

4. #384×C₂₁₁

C₂

semidirect

5. #336×C₂₁₁

C₂

semidirect

74. C₂₁₁

#2093

direct (N × H)

75. #2093

C₂₁₁

direct (N × H)

76. C₂×#107

C₂×C₂₁₁

semidirect

79. #384

C₂×C₂₁₁

semidirect

80. #336

C₂×C₂₁₁

semidirect

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

subgroup	index
C₂×#107×C₂₁₁	2
C₂×#102×C₂₁₁	2
#390×C₂₁₁	2
#384×C₂₁₁	2
#336×C₂₁₁	2
#2093	211

subgroup

index

C₂×#107×C₂₁₁

C₂×#102×C₂₁₁

#390×C₂₁₁

#384×C₂₁₁

#336×C₂₁₁

#2093

211

Subgroups — #2093×C₂₁₁ #41009713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2093×C211 #41009713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2093×C₂₁₁ #41009713