Finite Group Databasebeta

← #72259607

Subgroups — #72259607

Order 4000.

1400 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 1400 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#92562C211yesmaximal, characteristic, complemented
2C2×D5⋊#1043C211yesmaximal, characteristic, complemented
3#14522888C211yesmaximal, characteristic
4C22⋊#92563C211yesmaximal, characteristic, complemented
5#14522680C211yesmaximal, characteristic, complemented
6#14522679C211yesmaximal, characteristic, complemented
7#14522678C211yesmaximal, characteristic, complemented
8C22×C5⋊(C5⋊D5)C2211yescharacteristic
9C2×D5×C5⋊D5C2211yescharacteristic
10C2×#92563C2211yescharacteristic
11C2×#92563C2211yescharacteristic
12C2×#92562—22no—
13C2×#92562C2211yescharacteristic
14C2×#92562C2211yescharacteristic
15D5⋊#1043—22no—
16C5×C22⋊(C5⋊D5)C411yescharacteristic, complemented
17C52×C22⋊D5C411yescharacteristic, complemented
18#11758754C411yescharacteristic, complemented
19#11758753C411yescharacteristic, complemented
20#11758751C2211yescharacteristic
21D4×#1043—51nomaximal
22#11717375—56nomaximal
23C22×C53C2×C411yescharacteristic, complemented
24C2×C5⋊(C5⋊D5)—21no—
25C2×C5⋊(C5⋊D5)—21no—
26C2×C5⋊(C5⋊D5)C2×C411yescharacteristic, complemented
27C2×C5⋊(C5⋊D5)C2311yescharacteristic
28C2×C5×C5⋊D5C2×C411yescharacteristic, complemented
29C2×C52×D5C2×C411yescharacteristic, complemented
30D5×C5⋊D5—21no—
31D5×C5⋊D5—21no—
32#92563—21no—
33#92563—21no—
34#92562—21no—
35#92562D412yescomplemented
36C5×#1039C2×C411yescharacteristic, complemented
37C52×Dic5C2×C411yescharacteristic, complemented
38C22×#1043—51no—
39C22×#1043—51no—
40C22×#1043—56no—
41C22×#1043—1012no—
42C2×D5⋊AGL(1,𝔽5)—56no—
43D4×C5⋊D5—51no—
44#87886—56no—
45C22⋊#1044—56no—
46C22⋊#1043—51no—
47C22⋊#1043—51no—
48C4⋊#1043—51no—
49C4×#1043—51no—
50#87832—56no—
51#87831—56no—
52#87830—56no—
53C2×C53—21no—
54C2×C53C22×C411yescharacteristic
55C5⋊(C5⋊D5)—21no—
56C5⋊(C5⋊D5)C2×D411yescharacteristic
57C5⋊(C5⋊D5)#4111yescharacteristic
58C5×C5⋊D5—21no—
59C52×D5—21no—
60C22×C5⋊D5—51no—
61C22×C5⋊D5—56no—
62C22×C5⋊D5—51no—
63C22×C5⋊D5—1012no—
64C2×D52—56no—
65C2×#1044—56no—
66C2×#1044—56no—
67C2×#1043—51no—
68C2×#1043—56no—
69C2×#1043—51no—
70C2×#1043—51no—
71C2×#1043—51no—
72C2×#1043—56no—
73C2×#1043—102no—
74C2×#1043—102no—
75C2×#1043—1012no—
76C2×#1043—1012no—
77C2×#1043—1012no—
78C2×#1043—1048no—
79D5⋊AGL(1,𝔽5)—1012no—
80C52×D4—51no—
81C22⋊(C5⋊D5)—51no—
82C22⋊(C5⋊D5)AGL(1,𝔽5)11yescharacteristic, complemented
83C4⋊(C5⋊D5)—51no—
84C4×C5⋊D5—51no—
85C5×C22⋊D5—56no—
86C5×C22⋊D5AGL(1,𝔽5)16yescomplemented
87#6038—56no—
88#6038—56no—
89#6036—56no—
90D4×AGL(1,𝔽5)—256no—
91D4×AGL(1,𝔽5)—251no—
92C53C4×D411yescharacteristic, complemented, Sylow
93C22×C52—212no—
94C22×C52—51no—
95C22×C52C2×AGL(1,𝔽5)16yescomplemented
96C22×C52C2×AGL(1,𝔽5)11yescharacteristic, complemented
97C2×C5⋊D5—51no—
98C2×C5⋊D5—56no—
99C2×C5⋊D5—51no—
100C2×C5⋊D5—56no—
101C2×C5⋊D5—101no—
102C2×C5⋊D5—101no—
103C2×C5⋊D5—101no—
104C2×C5⋊D5—106no—
105C2×C5⋊D5—101no—
106C2×C5⋊D5—106no—
107C2×C5⋊D5—1012no—
108C2×C5⋊D5—1012no—
109C2×C5⋊D5—1024no—
110C2×C5⋊D5—1024no—
111C2×C5⋊D5C2×AGL(1,𝔽5)11yescharacteristic, complemented
112C2×C5×D5—56no—
113C2×C5×D5C2×AGL(1,𝔽5)16yescomplemented
114D52—106no—
115D52—106no—
116#1044—106no—
117#1044—106no—
118#1043—52no—
119#1043—512no—
120#1043—101no—
121#1043—106no—
122#1043—101no—
123#1043—101no—
124#1043—1024no—
125#1043—1024no—
126C4×C52—51no—
127#1039C2×AGL(1,𝔽5)11yescharacteristic, complemented
128C5×Dic5—56no—
129C5×Dic5C2×AGL(1,𝔽5)16yescomplemented
130C22×AGL(1,𝔽5)—251no—
131C22×AGL(1,𝔽5)—256no—
132C22×AGL(1,𝔽5)—251no—
133C22×AGL(1,𝔽5)—256no—
134C22×AGL(1,𝔽5)—5012no—
135D4×D5—256no—
136D4×D5—251no—
137C22⋊AGL(1,𝔽5)—251no—
138C22⋊AGL(1,𝔽5)—256no—
139C22⋊AGL(1,𝔽5)—256no—
140C22⋊AGL(1,𝔽5)—251no—
141C4⋊AGL(1,𝔽5)—251no—
142C4⋊AGL(1,𝔽5)—256no—
143C4×AGL(1,𝔽5)—251no—
144C4×AGL(1,𝔽5)—256no—
145C2×C52—224no—
146C2×C52—26no—
147C2×C52—21no—
148C2×C52—212no—
149C2×C52—101no—
150C2×C52C22×AGL(1,𝔽5)16yes—
151C2×C52C22×AGL(1,𝔽5)11yescharacteristic
152C5⋊D5—21no—
153C5⋊D5—51no—
154C5⋊D5—51no—
155C5⋊D5—56no—
156C5⋊D5—56no—
157C5⋊D5—1012no—
158C5⋊D5—1012no—
159C5⋊D5—1024no—
160C5⋊D5—106no—
161C5⋊D5—101no—
162C5×D5—26no—
163C5×D5—106no—
164C22×D5—256no—
165C22×D5—256no—
166C22×D5—251no—
167C22×D5—251no—
168C22×D5—5012no—
169C2×AGL(1,𝔽5)—251no—
170C2×AGL(1,𝔽5)—256no—
171C2×AGL(1,𝔽5)—251no—
172C2×AGL(1,𝔽5)—256no—
173C2×AGL(1,𝔽5)—256no—
174C2×AGL(1,𝔽5)—256no—
175C2×AGL(1,𝔽5)—251no—
176C2×AGL(1,𝔽5)—251no—
177C2×AGL(1,𝔽5)—5012no—
178C2×AGL(1,𝔽5)—5012no—
179C2×AGL(1,𝔽5)—502no—
180C2×AGL(1,𝔽5)—5012no—
181C2×AGL(1,𝔽5)—502no—
182C2×AGL(1,𝔽5)—5012no—
183C2×AGL(1,𝔽5)—5048no—
184C5×D4—56no—
185C5×D4—251no—
186C22⋊D5—56no—
187C22⋊D5—256no—
188C22⋊D5—251no—
189C22⋊D5#104311yescharacteristic, complemented
190D20—256no—
191D20—251no—
192C4×D5—256no—
193C4×D5—251no—
194C4×D4—1251noSylow
195C52—212no—
196C52D4×AGL(1,𝔽5)16yescomplemented
197C52D4×AGL(1,𝔽5)11yescharacteristic, complemented
198C22×C5—212no—
199C22×C5—56no—
200C22×C5—251no—
201C22×C5D5⋊AGL(1,𝔽5)16yescomplemented
202C22×C5C2×#104311yescharacteristic, complemented
203C2×D5—56no—
204C2×D5—256no—
205C2×D5—251no—
206C2×D5—256no—
207C2×D5—251no—
208C2×D5—5012no—
209C2×D5—5012no—
210C2×D5—501no—
211C2×D5—501no—
212C2×D5—506no—
213C2×D5—506no—
214C2×D5—501no—
215C2×D5—506no—
216C2×D5—501no—
217C2×D5—506no—
218C2×D5—5024no—
219C2×D5—5024no—
220C2×D5C2×#104311yescharacteristic, complemented
221AGL(1,𝔽5)—252no—
222AGL(1,𝔽5)—2512no—
223AGL(1,𝔽5)—5024no—
224AGL(1,𝔽5)—501no—
225AGL(1,𝔽5)—506no—
226AGL(1,𝔽5)—506no—
227AGL(1,𝔽5)—506no—
228AGL(1,𝔽5)—501no—
229AGL(1,𝔽5)—501no—
230AGL(1,𝔽5)—5024no—
231C4×C5—56no—
232C4×C5—251no—
233Dic5—56no—
234Dic5C2×#104311yescharacteristic, complemented
235C2×D4—1251no—
236C22×C4—1251no—
237C22×C4—1251no—
238C4⋊C4—1251no—
239C22⋊C4—1251no—
240C22⋊C4—1251no—
241C42—1251no—
242C2×C5—224no—
243C2×C5—21no—
244C2×C5—26no—
245C2×C5—212no—
246C2×C5—106no—
247C2×C5—501no—
248C2×C5C2×D5⋊AGL(1,𝔽5)16yes—
249C2×C5C22×#104311yescharacteristic
250D5—21no—
251D5—106no—
252D5—251no—
253D5—251no—
254D5—256no—
255D5—256no—
256D5—5012no—
257D5—5012no—
258D5—5024no—
259D5—501no—
260D5—506no—
261C23—1251no—
262C23—1251no—
263D4—51no—
264D4—251no—
265D4—1251no—
266D4—1251no—
267C2×C4—1251no—
268C2×C4—1251no—
269C2×C4—1251no—
270C2×C4—1251no—
271C2×C4—1251no—
272C2×C4—2502no—
273C2×C4—2502no—
274C5—212no—
275C5#1171737516yescomplemented
276C5D4×#104311yescharacteristic, complemented
277C22—51no—
278C22—251no—
279C22—1251no—
280C22—1251no—
281C22—2501no—
282C22—2501no—
283C22—2501no—
284C22—2501no—
285C22D5⋊#104311yescharacteristic, complemented
286C4—51no—
287C4—251no—
288C4—1252no—
289C4—2501no—
290C4—2501no—
291C4—2501no—
292C2—21no—
293C2—101no—
294C2—501no—
295C2—1251no—
296C2—1251no—
297C2—2501no—
298C2C2×D5⋊#104311yescharacteristic
subgroupG/H
center, FrattiniC2C2×D5⋊#1043
commutator, socleC2×C53C22×C4
FittingC22×C53C2×C4
prime radical#14522888C2
cosocleC2×C5⋊(C5⋊D5)C23
hypercenterC22D5⋊#1043
hypocenterC53C4×D4
LCSG ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×#92562C2semidirect
2. C2×D5⋊#1043C2semidirect
4. C22⋊#92563C2semidirect
5. #14522680C2semidirect
6. #14522679C2semidirect
7. #14522678C2semidirect
16. C5×C22⋊(C5⋊D5)C4semidirect
17. C52×C22⋊D5C4semidirect
18. #11758754C4semidirect
19. #11758753C4semidirect
23. C22×C53C2×C4semidirect
26. C2×C5⋊(C5⋊D5)C2×C4semidirect
28. C2×C5×C5⋊D5C2×C4semidirect
29. C2×C52×D5C2×C4semidirect
35. #92562D4semidirect
36. C5×#1039C2×C4semidirect
37. C52×Dic5C2×C4semidirect
82. C22⋊(C5⋊D5)AGL(1,𝔽5)semidirect
86. C5×C22⋊D5AGL(1,𝔽5)semidirect
92. C53C4×D4semidirect
95. C22×C52C2×AGL(1,𝔽5)semidirect
96. C22×C52C2×AGL(1,𝔽5)semidirect
111. C2×C5⋊D5C2×AGL(1,𝔽5)semidirect
113. C2×C5×D5C2×AGL(1,𝔽5)semidirect
127. #1039C2×AGL(1,𝔽5)semidirect
129. C5×Dic5C2×AGL(1,𝔽5)semidirect
189. C22⋊D5#1043semidirect
196. C52D4×AGL(1,𝔽5)semidirect
197. C52D4×AGL(1,𝔽5)semidirect
201. C22×C5D5⋊AGL(1,𝔽5)semidirect
202. C22×C5C2×#1043semidirect
220. C2×D5C2×#1043semidirect
234. Dic5C2×#1043semidirect
275. C5#11717375semidirect
276. C5D4×#1043semidirect
285. C22D5⋊#1043semidirect

Maximal subgroups

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

subgroupindex
C22×#925622
C2×D5⋊#10432
#145228882
C22⋊#925632
#145226802
#145226792
#145226782
D4×#10435
#117173755