Finite Group Databasebeta

← #37548125

Subgroups — #37548125

Order 23168.

1258 conjugacy classes of subgroups; only the 455 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 455 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
1#26042021C213yesmaximal, complemented
2#26041991C216yesmaximal, complemented
3#26041911C216yesmaximal, complemented
4#26041847C216yesmaximal, complemented
5#26041844C216yesmaximal, complemented
6#26041828C213yesmaximal, complemented
7H(3,𝔽4)×C181C211yesmaximal, characteristic, complemented
8C2×#15668450C2213yescomplemented
9C2×#15668448C2213yescomplemented
10C22×C22⋊D181C2216yescomplemented
11C2×D4×D181C2216yescomplemented
12#19153814C2213yescomplemented
13#19153813C2212yescomplemented
14#19153812C2213yes—
15#19153812C2216yescomplemented
16#19153805C2216yescomplemented
17#19153804C2213yescomplemented
18#19153802C22112yescomplemented
19#19153789C22112yescomplemented
20#19153789C22112yescomplemented
21#19153788C22112yescomplemented
22#19153786C22112yescomplemented
23#19153786C2216yescomplemented
24#19153785C2212yescomplemented
25#19153784C2216yescomplemented
26#19153776C22112yescomplemented
27#19153773C2216yescomplemented
28#19153772C2216yescomplemented
29C42×D181C2211yescharacteristic, complemented
30#124×C181C2216yescomplemented
31#124×C181C2213yescomplemented
32#120×C181C2216yescomplemented
33C23×D181C2316yes—
34#15668450—46no—
35#15668448—46no—
36C2×Dic362C2313yes—
37C2×D724C2313yes—
38C2×D724C2316yes—
39C2×D724C2316yes—
40C2×C22⋊D181—412no—
41C2×C22⋊D181C23112yes—
42C2×C22⋊D181C23112yes—
43C2×C22⋊D181C2316yes—
44D4×D181—412no—
45C22×Dic181C2316yes—
46C2×C4×D181C2313yes—
47C2×C4×D181C2316yes—
48#15668428C23112yes—
49C22⋊Dic181C2316yes—
50#15668426C23112yes—
51#15668426C23112yes—
52C4×Dic181C2313yes—
53C4×Dic181C2316yes—
54C24×C181C2312yes—
55C2×Q8×C181C2313yes—
56C2×D4×C181C2313yes—
57C2×D4×C181C2316yes—
58C22⋊C4×C181C2316yes—
59C22⋊C4×C181C23112yes—
60C42×C181C2311yescharacteristic
61C42×C181C2312yes—
62C22×D181—412no—
63C22×D181—46no—
64C22×D181C2416yes—
65C22×D181C2411yescharacteristic
66Dic362—43no—
67D724—46no—
68D724—46no—
69D724—43no—
70C22⋊D181—224no—
71C22⋊D181—412no—
72C22⋊D181—46no—
73C2×Dic181—212no—
74C2×Dic181C2416yes—
75C2×Dic181C2413yes—
76C4×D181—212no—
77C4×D181—26no—
78C23×C181—46no—
79C23×C181C2416yes—
80Q8×C181—43no—
81D4×C181—46no—
82D4×C181—43no—
83C2×C4×C181C2413yes—
84C2×C4×C181C2416yes—
85C22×C181—26no—
86C22×C181—212no—
87C22×C181—84no—
88C22×C181C2511yescharacteristic
89C2×D181—23no—
90C2×D181—26no—
91C2×D181—212no—
92C2×D181—812no—
93C4×C181—26no—
94C4×C181—23no—
95Dic181—26no—
96Dic181H(5,𝔽2)16yes—
97C2×C181—46no—
98C2×C181C2×H(5,𝔽2)13yes—
99D181—41no—
100D181—46no—
101C181#354911yescharacteristic, complemented, Sylow
102#3549—1811nomaximal, Sylow
103H(3,𝔽4)—1813no—
104H(3,𝔽4)D18111yescharacteristic, complemented
105#555—1816no—
106D42—1816no—
107#535—1816no—
108#535—1816no—
109#532—1813no—
110C2×#41—1813no—
111C2×#41—1813no—
112C22×D4—1816no—
113C22×D4—1816no—
114C4⋊Q8—1813no—
115#127—1813no—
116#127—1816no—
117#127—1812no—
118#124—1816no—
119#124—18112no—
120#124—1813no—
121#124C2×D18113yescomplemented
122#124C2×D18116yescomplemented
123#121—18112no—
124#121—18112no—
125#121—18112no—
126#120—1812no—
127#120—1816no—
128#120—1816no—
129#120—18112no—
130#120C2×D18116yescomplemented
131C4×D4—18112no—
132#117—1816no—
133#117—1816no—
134C2×C42—1811no—
135C24—1816no—
136C24C22×D18112yes—
137#41—7246no—
138#41—7246no—
139C2×Q8—1813no—
140C2×Q8C22×D18113yes—
141C2×D4—18112no—
142C2×D4—18112no—
143C2×D4—1816no—
144C2×D4—1813no—
145C2×D4—1816no—
146C2×D4—1816no—
147C2×D4—72412no—
148C2×D4—72412no—
149C2×D4C22×D18113yes—
150C2×D4C22×D18116yes—
151C22×C4—1816no—
152C22×C4—1813no—
153C22×C4—1816no—
154C4⋊C4—18112no—
155C22⋊C4—1816no—
156C22⋊C4—18112no—
157C22⋊C4—18112no—
158C22⋊C4C22×D18116yes—
159C22⋊C4C22×D181112yes—
160C42—1813no—
161C42—1816no—
162C42C22×D18111yescharacteristic
163C42C22×D18112yes—
164C23—46no—
165C23—1816no—
166C23—1811no—
167C23—7246no—
168C23—72412no—
169C23C23×D18116yes—
170Q8—43no—
171Q8—7243no—
172D4—46no—
173D4—43no—
174D4—36224no—
175D4—7246no—
176D4—7246no—
177D4—7243no—
178D4—72412no—
179D4—7246no—
180C2×C4—36212no—
181C2×C4—3626no—
182C2×C4—36212no—
183C2×C4—1816no—
184C2×C4—1813no—
185C2×C4C23×D18113yes—
186C2×C4C23×D18116yes—
187C22—26no—
188C22—212no—
189C22—84no—
190C22—3623no—
191C22—3626no—
192C22—36212no—
193C22—144812no—
194C22C24×D18111yescharacteristic
195C4—26no—
196C4—23no—
197C4—3626no—
198C4—1816no—
199C2—46no—
200C2—7246no—
201C2—7241no—
202C2H(5,𝔽2)×D18113yes—
subgroupG/H
center, FrattiniC22C24×D181
commutator, socle, cosocleC22×C181C25
FittingH(3,𝔽4)×C181C2
hypercenterH(3,𝔽4)D181
hypocenterC181#3549
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #26042021C2semidirect
2. #26041991C2semidirect
3. #26041911C2semidirect
4. #26041847C2semidirect
5. #26041844C2semidirect
6. #26041828C2semidirect
7. H(3,𝔽4)×C181C2semidirect
8. C2×#15668450C22semidirect
9. C2×#15668448C22semidirect
10. C22×C22⋊D181C22semidirect
11. C2×D4×D181C22semidirect
12. #19153814C22semidirect
13. #19153813C22semidirect
15. #19153812C22semidirect
16. #19153805C22semidirect
17. #19153804C22semidirect
18. #19153802C22semidirect
19. #19153789C22semidirect
20. #19153789C22semidirect
21. #19153788C22semidirect
22. #19153786C22semidirect
23. #19153786C22semidirect
24. #19153785C22semidirect
25. #19153784C22semidirect
26. #19153776C22semidirect
27. #19153773C22semidirect
28. #19153772C22semidirect
29. C42×D181C22semidirect
30. #124×C181C22semidirect
31. #124×C181C22semidirect
32. #120×C181C22semidirect
101. C181#3549semidirect
104. H(3,𝔽4)D181semidirect
121. #124C2×D181semidirect
122. #124C2×D181semidirect
130. #120C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420212
#260419912
#260419112
#260418472
#260418442
#260418282
H(3,𝔽4)×C1812
#3549181