Finite Group Databasebeta

← #72230953

Subgroups — #72230953

Order 2400.

380 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 62 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
1D3×C4⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
2C5×#6765C211yesmaximal, characteristic, complemented
3#11925156C211yesmaximal, characteristic, complemented
4#11925136C211yesmaximal, characteristic, complemented
5#11925132C211yesmaximal, characteristic, complemented
6C5×#6650C211yesmaximal, characteristic, complemented
7C3×#87771C211yesmaximal, characteristic, complemented
8#11716943—31nomaximal, Hall
9C2×D3×C5⋊D5—21no—
10C4⋊(C3⋊(C5⋊D5))C2211yescharacteristic
11C5×Dic30C2211yescharacteristic, complemented
12C4×C52×D3C2211yescharacteristic
13C3×C4⋊(C5⋊D5)C2211yescharacteristic
14C3×C5×D20C2211yescharacteristic, complemented
15#10597115—21no—
16C5×#1201—21no—
17C5×D5×Dic3—21no—
18C5×C3⋊#168C2211yescharacteristic, complemented
19C3×C5×#168C2211yescharacteristic, complemented
20#91480—51nomaximal
21#91254—51nomaximal
22C2×C4⋊(C5⋊D5)—31no—
23C5×#709—31no—
24C5×#681—31no—
25#87774—31no—
26#87774—31no—
27#87771—31no—
28#87771D311yescharacteristic, complemented
29C2×C3⋊(C5⋊D5)—21no—
30C2×C52×D3D411yescharacteristic
31C2×C3×C5⋊D5—21no—
32C2×C3×C5×D5—21no—
33D3×C5⋊D5—42no—
34C3×C4×C52C2311yescharacteristic
35C5×Dic15—21no—
36C52×Dic3D411yescharacteristic
37C5×#237—51no—
38D3×D20—51no—
39D3×D20—51no—
40D3×D20—102no—
41#6765D511yescharacteristic, complemented
42#6715—51no—
43#6706—51no—
44C5×ΓL(1,𝔽25)—51no—
45C3×#686—51no—
46C3×D40—51no—
47#6660—51no—
48#6657—51no—
49#6655—51no—
50#6653—51no—
51#6650D511yescharacteristic, complemented
52C22×C5⋊D5—61no—
53C2×C4×C52—31no—
54C4⋊(C5⋊D5)—31no—
55C4⋊(C5⋊D5)—61no—
56C4⋊(C5⋊D5)C2×D311yescharacteristic
57C5×C22⋊D5—61no—
58C5×D20C2×D311yescharacteristic, complemented
59C4×C5×D5—61no—
60C5×Dic10—31no—
61C5×#168—31no—
62C5×#168C2×D311yescharacteristic, complemented
63#4095—151no—
64#4054—151no—
65C2×C3×C52C2×D411yescharacteristic
66C3⋊(C5⋊D5)—41no—
67C52×D3—21no—
68C3×C5⋊D5—41no—
69C3×C5×D5—41no—
70C2×D3×D5—102no—
71C2×D3×D5—102no—
72C2×D3×D5—101no—
73C2×D3×D5—101no—
74C3×C5×D4—51no—
75D60—51no—
76D60—51no—
77D60—102no—
78Dic30C2×D511yescharacteristic, complemented
79C5×#67—101no—
80C2×C5×Dic3—101no—
81C4×C5×D3—22no—
82C4×C5×D3C2×D511yescharacteristic
83C4×C5×D3C2×D511yescharacteristic
84C5×Dic6—51no—
85C3×D20—51no—
86C3×D20—51no—
87C3×D20—102no—
88C3×D20C2×D511yescharacteristic, complemented
89#1202—102no—
90#1202—102no—
91#1202—101no—
92#1202—101no—
93#1201—21no—
94D5×Dic3—21no—
95C3×C5×C8—51no—
96C3⋊#168C2×D511yescharacteristic, complemented
97C3×#168C2×D511yescharacteristic, complemented
98C5×C3⋊C8—51no—
99C22×C52—31no—
100C2×C5⋊D5—21no—
101C2×C5⋊D5—61no—
102C2×C5⋊D5—122no—
103C2×C5×D5—21no—
104C4×C52—31no—
105C4×C52C22×D311yescharacteristic
106C5×Dic5—61no—
107#911—251noHall
108C5×#41—151no—
109#709—31no—
110C2×D20—151no—
111C2×D20—151no—
112C2×D20—302no—
113C5×M16—151no—
114#688—151no—
115#688—151no—
116#686—51no—
117#686—151no—
118#681—31no—
119D40—51no—
120D40—151no—
121#677—151no—
122#677—151no—
123C3×C52AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
124C22×C3×C5—101no—
125C2×D15—102no—
126C2×D15—102no—
127C2×D15—101no—
128C2×D15—101no—
129C2×C5×D3—22no—
130C2×C5×D3D2011yescharacteristic
131C2×C5×D3C22⋊D511yescharacteristic
132C2×C3×D5—21no—
133C2×C3×D5—102no—
134C2×C3×D5—102no—
135C2×C3×D5—101no—
136C2×C3×D5—101no—
137D3×D5—204no—
138D3×D5—204no—
139D3×D5—202no—
140D3×D5—202no—
141C3×C4×C5—22no—
142C3×C4×C5C22×D511yescharacteristic
143C3×C4×C5C22×D511yescharacteristic
144Dic15—21no—
145C5×Dic3—22no—
146C5×Dic3—101no—
147C5×Dic3D2011yescharacteristic
148C5×Dic3C22⋊D511yescharacteristic
149C2×C52—61no—
150C2×C52D3×D411yescharacteristic
151C5⋊D5—41no—
152C5⋊D5—121no—
153C5×D5—41no—
154#237—51no—
155D3×D4—251no—
156C3×D8—251no—
157#214—251no—
158#213—251no—
159#204—251no—
160ΓL(1,𝔽25)—51no—
161C22×D5—301no—
162C22×D5—301no—
163C22×D5—302no—
164C22×D5—302no—
165C5×Q8—151no—
166C5×D4—51no—
167C5×D4—301no—
168C2×C4×C5—31no—
169C2×C4×C5—31no—
170C2×C4×C5—62no—
171C2×C4×C5—301no—
172C22⋊D5—61no—
173D20—51no—
174D20—51no—
175D20—102no—
176D20—151no—
177D20—151no—
178D20—302no—
179D20—301no—
180D20—301no—
181D20—302no—
182D20—302no—
183D20D3×D511yescharacteristic, complemented
184C4×D5—61no—
185Dic10—31no—
186C5×C8—51no—
187C5×C8—151no—
188#168—31no—
189#168D3×D511yescharacteristic, complemented
190AGL(1,ℤ/8)—751noSylow
191C2×C3×C5—22no—
192C2×C3×C5—201no—
193C2×C3×C5C2×D2011yescharacteristic
194C2×C3×C5C2×C22⋊D511yescharacteristic
195D15—202no—
196D15—202no—
197D15—201no—
198D15—201no—
199C3×D5—41no—
200C3×D5—202no—
201C3×D5—202no—
202C3×D5—201no—
203C3×D5—201no—
204C5×D3—22no—
205C5×D3—22no—
206C5×D3—21no—
207C5×D3—21no—
208C52#91111yescharacteristic, complemented, Sylow
209C22×D3—501no—
210C3×D4—51no—
211C3×D4—251no—
212#67—101no—
213#67—501no—
214C2×Dic3—101no—
215D12—251no—
216C4×D3D5211yescharacteristic
217Dic6—51no—
218C3×C8—51no—
219C3⋊C8—51no—
220C22×C5—31no—
221C22×C5—31no—
222C22×C5—62no—
223C22×C5—101no—
224C2×D5—21no—
225C2×D5—102no—
226C2×D5—102no—
227C2×D5—101no—
228C2×D5—101no—
229C2×D5—302no—
230C2×D5—302no—
231C2×D5—301no—
232C2×D5—301no—
233C2×D5—604no—
234C2×D5—604no—
235C2×D5—602no—
236C2×D5—602no—
237C4×C5—22no—
238C4×C5—31no—
239C4×C5—31no—
240C4×C5—62no—
241C4×C5—301no—
242C4×C5C2×D3×D511yescharacteristic
243C4×C5C2×D3×D511yescharacteristic
244Dic5—61no—
245#41—151no—
246C2×D4—751no—
247SD16—751no—
248SD16—751no—
249D8—251no—
250D8—751no—
251M16—151no—
252C3×C5—22no—
253C3×C5#405411yescharacteristic, complemented
254C3×C5#409511yescharacteristic, complemented
255C22×C3—101no—
256C22×C3—501no—
257C2×D3—501no—
258C2×D3—1002no—
259C2×D3#603811yescharacteristic
260C3×C4C2×D5211yescharacteristic
261Dic3—101no—
262Dic3#603811yescharacteristic
263C2×C5—22no—
264C2×C5—62no—
265C2×C5—62no—
266C2×C5—61no—
267C2×C5—61no—
268C2×C5—201no—
269C2×C5D3×D2011yescharacteristic
270C2×C5D3×C22⋊D511yescharacteristic
271D5—41no—
272D5—202no—
273D5—202no—
274D5—201no—
275D5—201no—
276D5—602no—
277D5—602no—
278D5—601no—
279D5—601no—
280C23—1501no—
281Q8—151no—
282D4—51no—
283D4—251no—
284D4—301no—
285D4—751no—
286D4—1501no—
287C2×C4—31no—
288C2×C4—301no—
289C8—51no—
290C8—151no—
291C2×C3—201no—
292C2×C3—1001no—
293C2×C3C2×#603811yescharacteristic
294D3—21no—
295D3—1001no—
296C5—22no—
297C5#9125411yescharacteristic, complemented
298C5#9148011yescharacteristic, complemented
299C22—31no—
300C22—101no—
301C22—3002no—
302C22—501no—
303C22—1501no—
304C4—31no—
305C4—301no—
306C4D3×D5211yescharacteristic
307C3#1171694311yescharacteristic, complemented, Sylow
308C2—61no—
309C2—201no—
310C2—3001no—
311C2—1001no—
312C2D3×#603811yescharacteristic
subgroupG/H
centerC2D3×#6038
commutator, Fitting, cosocleC3×C4×C52C23
Frattini, hypercenterC4D3×D52
socleC2×C3×C52C2×D4
hypocenterC3×C52AGL(1,ℤ/8)
LCSG ▷ C3×C4×C52 ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. D3×C4⋊(C5⋊D5)C2semidirect
2. C5×#6765C2semidirect
3. #11925156C2semidirect
4. #11925136C2semidirect
5. #11925132C2semidirect
6. C5×#6650C2semidirect
7. C3×#87771C2semidirect
11. C5×Dic30C22semidirect
14. C3×C5×D20C22semidirect
18. C5×C3⋊#168C22semidirect
19. C3×C5×#168C22semidirect
28. #87771D3semidirect
41. #6765D5semidirect
51. #6650D5semidirect
58. C5×D20C2×D3semidirect
62. C5×#168C2×D3semidirect
78. Dic30C2×D5semidirect
88. C3×D20C2×D5semidirect
96. C3⋊#168C2×D5semidirect
97. C3×#168C2×D5semidirect
123. C3×C52AGL(1,ℤ/8)semidirect
183. D20D3×D5semidirect
189. #168D3×D5semidirect
208. C52#911semidirect
253. C3×C5#4054semidirect
254. C3×C5#4095semidirect
297. C5#91254semidirect
298. C5#91480semidirect
307. C3#11716943semidirect

Maximal subgroups

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

subgroupindex
D3×C4⋊(C5⋊D5)2
C5×#67652
#119251562
#119251362
#119251322
C5×#66502
C3×#877712
#117169433
#914805
#912545