Group #41009370 — subgroups

Subgroups — #1750×C₂₁₁ #41009370

Order 27008.

336 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 168 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×#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₈×C₂₁₁	C₂	1	2	yes	maximal
3	C₂×#97×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#336×C₂₁₁	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
8	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
9	#117×C₂₁₁	—	2	2	no	—
10	#117×C₂₁₁	—	2	2	no	—
11	C₂×C₄⋊C₄×C₂₁₁	C₄	1	2	yes	—
12	C₂×C₂²⋊C₄×C₂₁₁	C₄	1	2	yes	—
13	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₂₁₁	C₄	1	8	yes	complemented
16	#97×C₂₁₁	—	2	2	no	—
17	C₄×C₈×C₂₁₁	—	2	2	no	—
18	C₂⁴×C₂₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₁₁	—	2	1	no	—
20	C₂²×C₄×C₂₁₁	—	2	1	no	—
21	C₂²×C₄×C₂₁₁	—	2	2	no	—
22	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₂₁₁	—	2	2	no	—
28	C₂×C₈×C₂₁₁	—	2	2	no	—
29	C₂×C₈×C₂₁₁	—	2	2	no	—
30	C₂×C₈×C₂₁₁	—	2	2	no	—
31	C₂×C₈×C₂₁₁	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₂₁₁	C₂×C₄	1	4	yes	—
33	C₄⋊C₄×C₂₁₁	—	2	4	no	—
34	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
35	C₄²×C₂₁₁	—	2	1	no	—
36	C₄²×C₂₁₁	—	2	1	no	—
37	C₄²×C₂₁₁	D₄	1	2	yes	—
38	C₄²×C₂₁₁	D₄	1	2	yes	—
39	C₂³×C₂₁₁	—	2	1	no	—
40	C₂³×C₂₁₁	—	2	1	no	—
41	C₂³×C₂₁₁	—	2	4	no	—
42	C₂³×C₂₁₁	C₄²	1	2	yes	—
43	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₁₁	—	2	1	no	—
45	C₂×C₄×C₂₁₁	—	2	1	no	—
46	C₂×C₄×C₂₁₁	—	2	4	no	—
47	C₂×C₄×C₂₁₁	—	2	4	no	—
48	C₂×C₄×C₂₁₁	—	2	4	no	—
49	C₂×C₄×C₂₁₁	—	2	2	no	—
50	C₂×C₄×C₂₁₁	—	2	4	no	—
51	C₂×C₄×C₂₁₁	C₄²	1	2	yes	—
52	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₁₁	#41	1	2	yes	—
57	C₈×C₂₁₁	—	2	4	no	—
58	C₈×C₂₁₁	—	2	4	no	—
59	C₂²×C₂₁₁	—	2	2	no	—
60	C₂²×C₂₁₁	—	2	4	no	—
61	C₂²×C₂₁₁	—	4	2	no	—
62	C₂²×C₂₁₁	—	4	2	no	—
63	C₂²×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
64	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₂₁₁	#117	1	1	yes	characteristic
66	C₂²×C₂₁₁	#131	1	4	yes	—
67	C₄×C₂₁₁	—	2	2	no	—
68	C₄×C₂₁₁	—	2	4	no	—
69	C₄×C₂₁₁	—	4	2	no	—
70	C₄×C₂₁₁	C₄×D₄	1	2	yes	—
71	C₄×C₂₁₁	C₄×D₄	1	2	yes	—
72	C₂×C₂₁₁	—	4	2	no	—
73	C₂×C₂₁₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₂₁₁	#405	1	2	yes	—
75	C₂×C₂₁₁	#408	1	2	yes	—
76	C₂×C₂₁₁	#433	1	2	yes	—
77	C₂₁₁	#1750	1	1	yes	characteristic, direct product factor, Sylow
78	#1750	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂×#117	C₂×C₂₁₁	1	1	yes	characteristic
80	C₂×C₂²⋊C₈	C₂×C₂₁₁	1	2	yes	—
81	C₂×#97	C₂×C₂₁₁	1	1	yes	characteristic, complemented
82	C₂×C₄×C₈	C₂×C₂₁₁	1	1	yes	characteristic, complemented
83	#336	C₂×C₂₁₁	1	2	yes	complemented
84	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
85	C₂²×C₈	C₂²×C₂₁₁	1	2	yes	—
86	C₂²×C₈	C₂²×C₂₁₁	1	2	yes	—
87	#117	—	2	2	no	—
88	#117	—	2	2	no	—
89	C₂×C₄⋊C₄	C₄×C₂₁₁	1	2	yes	—
90	C₂×C₂²⋊C₄	C₄×C₂₁₁	1	2	yes	—
91	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
92	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
93	C₂²⋊C₈	C₄×C₂₁₁	1	8	yes	complemented
94	#97	—	2	2	no	—
95	C₄×C₈	—	2	2	no	—
96	C₂⁴	C₂×C₄×C₂₁₁	1	1	yes	characteristic
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
101	C₂²×C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
102	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
103	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
104	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	C₂×C₄×C₂₁₁	1	4	yes	—
110	C₂×C₈	C₂×C₄×C₂₁₁	1	4	yes	—
111	C₄⋊C₄	—	2	4	no	—
112	C₂²⋊C₄	—	2	4	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	2	1	no	—
115	C₄²	D₄×C₂₁₁	1	2	yes	—
116	C₄²	D₄×C₂₁₁	1	2	yes	—
117	C₂³	—	2	1	no	—
118	C₂³	—	2	1	no	—
119	C₂³	—	2	4	no	—
120	C₂³	C₄²×C₂₁₁	1	2	yes	—
121	C₂³	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	C₄²×C₂₁₁	1	2	yes	—
130	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	4	yes	—
131	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
132	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
133	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
134	C₂×C₄	#41×C₂₁₁	1	2	yes	—
135	C₈	—	2	4	no	—
136	C₈	—	2	4	no	—
137	C₂²	—	2	2	no	—
138	C₂²	—	2	4	no	—
139	C₂²	—	4	2	no	—
140	C₂²	—	4	2	no	—
141	C₂²	C₂×C₄²×C₂₁₁	1	1	yes	characteristic
142	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
143	C₂²	#117×C₂₁₁	1	1	yes	characteristic
144	C₂²	#131×C₂₁₁	1	4	yes	—
145	C₄	—	2	2	no	—
146	C₄	—	2	4	no	—
147	C₄	—	4	2	no	—
148	C₄	C₄×D₄×C₂₁₁	1	2	yes	—
149	C₄	C₄×D₄×C₂₁₁	1	2	yes	—
150	C₂	—	4	2	no	—
151	C₂	C₄×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
152	C₂	#405×C₂₁₁	1	2	yes	—
153	C₂	#408×C₂₁₁	1	2	yes	—
154	C₂	#433×C₂₁₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#117×C₂₁₁

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₈×C₂₁₁

C₂

yes

maximal

C₂×#97×C₂₁₁

C₂

yes

maximal, characteristic, complemented

C₂×C₄×C₈×C₂₁₁

C₂

yes

maximal, characteristic, complemented

#336×C₂₁₁

C₂

yes

maximal, complemented

C₂³×C₄×C₂₁₁

C₂²

yes

characteristic

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₂²×C₈×C₂₁₁

C₂²

yes

—

#117×C₂₁₁

—

#117×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

C₄

yes

—

C₂×C₂²⋊C₄×C₂₁₁

C₄

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂²⋊C₈×C₂₁₁

C₄

yes

complemented

#97×C₂₁₁

—

C₄×C₈×C₂₁₁

—

C₂⁴×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

C₂×C₄

yes

—

C₄⋊C₄×C₂₁₁

—

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

—

C₄²×C₂₁₁

—

C₄²×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

D₄

yes

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

C₄²

yes

—

C₂³×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₄²

yes

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄²

yes

characteristic

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₄×D₄

yes

—

C₄×C₂₁₁

C₄×D₄

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₂₁₁

#405

yes

—

C₂×C₂₁₁

#408

yes

—

C₂×C₂₁₁

#433

yes

—

C₂₁₁

#1750

yes

characteristic, direct product factor, Sylow

#1750

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×#117

C₂×C₂₁₁

yes

characteristic

C₂×C₂²⋊C₈

C₂×C₂₁₁

yes

—

C₂×#97

C₂×C₂₁₁

yes

characteristic, complemented

C₂×C₄×C₈

C₂×C₂₁₁

yes

characteristic, complemented

#336

C₂×C₂₁₁

yes

complemented

C₂³×C₄

C₂²×C₂₁₁

yes

characteristic

C₂²×C₈

C₂²×C₂₁₁

yes

—

C₂²×C₈

C₂²×C₂₁₁

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

C₄×C₂₁₁

yes

—

C₂×C₂²⋊C₄

C₄×C₂₁₁

yes

—

C₂×C₄²

C₂²×C₂₁₁

yes

characteristic

C₂×C₄²

C₂²×C₂₁₁

yes

characteristic

C₂²⋊C₈

C₄×C₂₁₁

yes

complemented

#97

—

C₄×C₈

—

C₂⁴

C₂×C₄×C₂₁₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₂×C₄×C₂₁₁

yes

—

101

C₂²×C₄

C₂×C₄×C₂₁₁

yes

characteristic

102

C₂²×C₄

C₂×C₄×C₂₁₁

yes

—

103

C₂²×C₄

C₂×C₄×C₂₁₁

yes

—

104

C₂²×C₄

C₂³×C₂₁₁

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂×C₄×C₂₁₁

yes

—

110

C₂×C₈

C₂×C₄×C₂₁₁

yes

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

D₄×C₂₁₁

yes

—

116

C₄²

D₄×C₂₁₁

yes

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

C₄²×C₂₁₁

yes

—

121

C₂³

C₂²×C₄×C₂₁₁

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₄²×C₂₁₁

yes

—

130

C₂×C₄

C₂²⋊C₄×C₂₁₁

yes

—

131

C₂×C₄

C₂²×C₄×C₂₁₁

yes

—

132

C₂×C₄

C₂×D₄×C₂₁₁

yes

characteristic

133

C₂×C₄

C₂×D₄×C₂₁₁

yes

characteristic

134

C₂×C₄

#41×C₂₁₁

yes

—

135

C₈

—

136

C₈

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₄²×C₂₁₁

yes

characteristic

142

C₂²

C₂×C₂²⋊C₄×C₂₁₁

yes

characteristic

143

C₂²

#117×C₂₁₁

yes

characteristic

144

C₂²

#131×C₂₁₁

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

C₄×D₄×C₂₁₁

yes

—

149

C₄

C₄×D₄×C₂₁₁

yes

—

150

C₂

—

151

C₂

C₄×C₂²⋊C₄×C₂₁₁

yes

characteristic

152

C₂

#405×C₂₁₁

yes

—

153

C₂

#408×C₂₁₁

yes

—

154

C₂

#433×C₂₁₁

yes

—

	subgroup	G/H
center	C₂²×C₄×C₂₁₁	C₂³
commutator	C₂²	C₂×C₄²×C₂₁₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₁₁
socle	C₂³×C₂₁₁	C₂²×C₄
prime radical	C₂⁴×C₂₁₁	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₁₁ ◁ G

subgroup

G/H

center

C₂²×C₄×C₂₁₁

C₂³

commutator

C₂²

C₂×C₄²×C₂₁₁

Frattini, cosocle

C₂²×C₄

C₂³×C₂₁₁

socle

C₂³×C₂₁₁

C₂²×C₄

prime radical

C₂⁴×C₂₁₁

C₂×C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₂₁₁ ◁ G

N	H ≅ G/N	type
3. C₂×#97×C₂₁₁	C₂	semidirect
4. C₂×C₄×C₈×C₂₁₁	C₂	semidirect
5. #336×C₂₁₁	C₂	semidirect
15. C₂²⋊C₈×C₂₁₁	C₄	semidirect
77. C₂₁₁	#1750	direct (N × H)
78. #1750	C₂₁₁	direct (N × H)
81. C₂×#97	C₂×C₂₁₁	semidirect
82. C₂×C₄×C₈	C₂×C₂₁₁	semidirect
83. #336	C₂×C₂₁₁	semidirect
93. C₂²⋊C₈	C₄×C₂₁₁	semidirect

H ≅ G/N

type

3. C₂×#97×C₂₁₁

C₂

semidirect

4. C₂×C₄×C₈×C₂₁₁

C₂

semidirect

5. #336×C₂₁₁

C₂

semidirect

15. C₂²⋊C₈×C₂₁₁

C₄

semidirect

77. C₂₁₁

#1750

direct (N × H)

78. #1750

C₂₁₁

direct (N × H)

81. C₂×#97

C₂×C₂₁₁

semidirect

82. C₂×C₄×C₈

C₂×C₂₁₁

semidirect

83. #336

C₂×C₂₁₁

semidirect

93. C₂²⋊C₈

C₄×C₂₁₁

semidirect

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

subgroup	index
C₂×#117×C₂₁₁	2
C₂×C₂²⋊C₈×C₂₁₁	2
C₂×#97×C₂₁₁	2
C₂×C₄×C₈×C₂₁₁	2
#336×C₂₁₁	2
#1750	211

subgroup

index

C₂×#117×C₂₁₁

C₂×C₂²⋊C₈×C₂₁₁

C₂×#97×C₂₁₁

C₂×C₄×C₈×C₂₁₁

#336×C₂₁₁

#1750

211

Subgroups — #1750×C₂₁₁ #41009370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1750×C211 #41009370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1750×C₂₁₁ #41009370