Finite Group Databasebeta

← C2×#26042320 #37550474

Subgroups — C2×#26042320 #37550474

Order 23168.

608 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×#10624382C212yesmaximal, complemented
2#26042320C218yesmaximal, direct product factor
3C2×#19153884C211yesmaximal, characteristic, complemented
4C2×#19153878C212yesmaximal, complemented
5C2×#19153870C211yesmaximal, characteristic, complemented
6C2×D8×D181C211yesmaximal, characteristic
7C23×#10624382—22no—
8C2×D181⋊C8C2211yescharacteristic
9D4×#10624382—24no—
10D4×#10624382C2218yescomplemented
11C2×C4⋊#10624382C2212yes—
12C2×C22⋊#10624382—22no—
13C2×C4×#10624382C2211yescharacteristic
14#19153884C2214yescomplemented
15#19153878C2218yescomplemented
16#19153870C2214yescomplemented
17C2×D4×D181C2212yes—
18C2×D1448C411yescharacteristic, complemented
19C2×#15668434C412yescomplemented
20D8×D181C414yescomplemented
21D8×D181C2214yes—
22C2×C8×D181C2211yescharacteristic
23C2×D8×C181C411yescharacteristic, complemented
24C22×#10624382—28no—
25C22×#10624382—22no—
26C22×#10624382—48no—
27C22×#10624382D412yes—
28D181⋊C8—21no—
29D181⋊C8C2312yes—
30C2×#13089685—21no—
31C4⋊#10624382—22no—
32C4⋊#10624382C2314yes—
33C22⋊#10624382—28no—
34C4×#10624382D412yes—
35C4×#10624382C2312yes—
36C23×D181—22no—
37C2×D724C2×C412yescomplemented
38C2×C22⋊D181—22no—
39D4×D181—22no—
40D4×D181—22no—
41D4×D181C2×C414yescomplemented
42D4×D181C2314yes—
43C2×C4×D181C2311yescharacteristic
44D1448C2×C414yescomplemented
45#15668434C2×C418yescomplemented
46C2×#13089678C2×C411yescharacteristic, complemented
47C8×D181C2×C412yescomplemented
48C8×D181C2312yes—
49C2×D4×C181C2×C412yescomplemented
50D8×C181C2×C414yescomplemented
51C2×C8×C181C2×C411yescharacteristic, complemented
52C2×#10624382—24no—
53C2×#10624382—24no—
54C2×#10624382—416no—
55C2×#10624382—44no—
56C2×#10624382C2×D414yes—
57#13089685—22no—
58C22×D181—22no—
59C22×D181—24no—
60C22×D181—24no—
61C22×D181—44no—
62C22×D181—44no—
63C22×D181C2×D411yescharacteristic
64D724—22no—
65D724C22×C414yescomplemented
66C22⋊D181—24no—
67C22⋊D181—24no—
68C2×Dic181#4111yescharacteristic
69C4×D181C22×C411yescharacteristic
70C4×D181C2×D411yescharacteristic
71C4×D181#4111yescharacteristic
72C4×D181C2411yescharacteristic
73#13089678C22×C412yescomplemented
74C23×C181—22no—
75D4×C181—22no—
76D4×C181C22×C414yescomplemented
77C2×C4×C181C22×C411yescharacteristic
78C8×C181C22×C412yescomplemented
79C22×C181—24no—
80C22×C181—44no—
81C22×C181C4×D411yescharacteristic
82C2×D181—24no—
83C2×D181—44no—
84C2×D181—44no—
85C2×D181—44no—
86C2×D181—48no—
87C2×D181C4×D411yescharacteristic
88C2×D181AGL(1,ℤ/8)14yes—
89C2×D181C22×D411yescharacteristic
90#10624382—24no—
91#10624382—44no—
92C4×C181C4×D411yescharacteristic
93C4×C181C23×C411yescharacteristic
94Dic181C4×D411yescharacteristic
95Dic181C2×#4111yescharacteristic
96C2×C181—44no—
97C2×C181#44212yescomplemented
98C2×C181C2×C4×D411yescharacteristic
99D181—44no—
100D181#44212yescomplemented
101D181C2×AGL(1,ℤ/8)12yes—
102C181C2×#44211yescharacteristic, complemented, Sylow
103C2×#442—1811nomaximal, Sylow
104C22×D8—1811no—
105C2×C4×D4—1812no—
106#442—1818no—
107C2×#106—1811no—
108C2×#102—1812no—
109C2×#97—1811no—
110C22×D4—1812no—
111C23×C4—3622no—
112C2×D8—1812no—
113C2×D8—1814no—
114C2×D8—1814no—
115C2×D8—1811no—
116C2×D8#1062438211yescharacteristic, complemented
117C22×C8—1811no—
118C22×C8—1811no—
119C4×D4—3624no—
120C4×D4—1818no—
121C2×C4⋊C4—1812no—
122C2×C22⋊C4—3622no—
123C2×C42—1811no—
124#106—1814no—
125#102—1818no—
126#97—1814no—
127C24—3622no—
128C2×D4—3622no—
129C2×D4—3622no—
130C2×D4—3622no—
131C2×D4—1814no—
132C2×D4—1814no—
133C2×D4—1812no—
134C2×D4C2×#1062438212yescomplemented
135C22×C4—3622no—
136C22×C4—3628no—
137C22×C4—1812no—
138C22×C4—1811no—
139C22×C4—7248no—
140D8—1818no—
141D8—1814no—
142D8C2×#1062438214yescomplemented
143C2×C8—3621no—
144C2×C8—3621no—
145C2×C8—1812no—
146C2×C8—1811no—
147C2×C8—1812no—
148C2×C8—1812no—
149C2×C8C2×#1062438211yescharacteristic, complemented
150C4⋊C4—3622no—
151C4⋊C4—1814no—
152C22⋊C4—3628no—
153C42—1812no—
154C42—1812no—
155C23—22no—
156C23—3624no—
157C23—3624no—
158C23—3622no—
159C23—1811no—
160C23—7244no—
161C23—7244no—
162D4—22no—
163D4—3624no—
164D4—3624no—
165D4—3622no—
166D4—1814no—
167D4C22×#1062438214yescomplemented
168C2×C4—3624no—
169C2×C4—3624no—
170C2×C4—1814no—
171C2×C4—1811no—
172C2×C4—1811no—
173C2×C4—1811no—
174C2×C4—1811no—
175C2×C4—1811no—
176C2×C4—72416no—
177C2×C4—7244no—
178C2×C4C22×#1062438211yescharacteristic
179C8—3622no—
180C8—1812no—
181C8C22×#1062438212yescomplemented
182C22—24no—
183C22—44no—
184C22—3624no—
185C22—1814no—
186C22—1811no—
187C22—1811no—
188C22—7244no—
189C22—7244no—
190C22—7248no—
191C22—7244no—
192C22D4×#1062438211yescharacteristic
193C4—3624no—
194C4—1811no—
195C4—1811no—
196C4—7244no—
197C4D4×#1062438211yescharacteristic
198C4C23×#1062438211yescharacteristic
199C2—44no—
200C2—1812no—
201C2—1812no—
202C2—7244no—
203C2#2604232012yesdirect product factor
204C2C2×D4×#1062438211yescharacteristic
subgroupG/H
centerC22D4×#10624382
commutatorC4×C181C23×C4
FrattiniC4C23×#10624382
FittingC2×D8×C181C4
socleC22×C181C4×D4
prime radicalC2×D8×D181C2
cosocleC4×D181C24
hypercenterC2×D8#10624382
hypocenterC181C2×#442
LCSG ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×D4×#10624382C2semidirect
2. #26042320C2direct (N × H)
3. C2×#19153884C2semidirect
4. C2×#19153878C2semidirect
5. C2×#19153870C2semidirect
10. D4×#10624382C22semidirect
14. #19153884C22semidirect
15. #19153878C22semidirect
16. #19153870C22semidirect
18. C2×D1448C4semidirect
19. C2×#15668434C4semidirect
20. D8×D181C4semidirect
23. C2×D8×C181C4semidirect
37. C2×D724C2×C4semidirect
41. D4×D181C2×C4semidirect
44. D1448C2×C4semidirect
45. #15668434C2×C4semidirect
46. C2×#13089678C2×C4semidirect
47. C8×D181C2×C4semidirect
49. C2×D4×C181C2×C4semidirect
50. D8×C181C2×C4semidirect
51. C2×C8×C181C2×C4semidirect
65. D724C22×C4semidirect
73. #13089678C22×C4semidirect
76. D4×C181C22×C4semidirect
78. C8×C181C22×C4semidirect
97. C2×C181#442semidirect
100. D181#442semidirect
102. C181C2×#442semidirect
116. C2×D8#10624382semidirect
134. C2×D4C2×#10624382semidirect
142. D8C2×#10624382semidirect
149. C2×C8C2×#10624382semidirect
167. D4C22×#10624382semidirect
181. C8C22×#10624382semidirect
203. C2#26042320direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×#106243822
#260423202
C2×#191538842
C2×#191538782
C2×#191538702
C2×D8×D1812
C2×#442181