Finite Group Databasebeta

← #72230951

Subgroups — #72230951

Order 2400.

380 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 56 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#11925624C211yesmaximal, characteristic, complemented
2#11925606C211yesmaximal, characteristic, complemented
3#11925156C211yesmaximal, characteristic, complemented
4#11925129C211yesmaximal, characteristic, complemented
5C5×#6659C211yesmaximal, characteristic, complemented
6C5×#6652C211yesmaximal, characteristic, complemented
7C3×#87771C211yesmaximal, characteristic, complemented
8#11717055—31nomaximal, Hall
9C2×D3×C5⋊D5—21no—
10C4×C3⋊(C5⋊D5)C2211yescharacteristic
11C5×Dic30C2211yescharacteristic, complemented
12C52×D12C2211yescharacteristic
13C3×C4⋊(C5⋊D5)C2211yescharacteristic
14C3×C5×D20C2211yescharacteristic, complemented
15#10597123—21no—
16D5×Dic15—21no—
17#10597114—21no—
18C5×C3⋊#168C2211yescharacteristic, complemented
19C3×C5×#168C2211yescharacteristic, complemented
20#91482—51nomaximal
21#91258—51nomaximal
22D4×C5⋊D5—31no—
23#87871—31no—
24C5×#687—31no—
25C5×#686—31no—
26#87774—31no—
27#87771D311yescharacteristic, complemented
28#87769—31no—
29C2×C3⋊(C5⋊D5)D411yescharacteristic
30C2×C52×D3—21no—
31C2×C3×C5⋊D5—21no—
32C2×C3×C5×D5—21no—
33D3×C5⋊D5—42no—
34C3×C4×C52C2311yescharacteristic
35C3⋊#1039D411yescharacteristic
36C5×Dic15—21no—
37#6819—51no—
38#6777—51no—
39#6777—51no—
40#6777—102no—
41#6766—51no—
42#6705—51no—
43C5×#213—51no—
44C5×#204—51no—
45C3×#686—51no—
46C3×D40—51no—
47#6659D511yescharacteristic, complemented
48#6657—51no—
49#6655—51no—
50#6652D511yescharacteristic, complemented
51#6651—51no—
52C22×C5⋊D5—61no—
53C52×D4—31no—
54C22⋊(C5⋊D5)—61no—
55C4⋊(C5⋊D5)C2×D311yescharacteristic
56C4×C5⋊D5—31no—
57C5×D20C2×D311yescharacteristic, complemented
58C5×Dic10—31no—
59#6038—61no—
60D5×Dic5—61no—
61C5×#168—31no—
62C5×#168C2×D311yescharacteristic, complemented
63#4060—151no—
64#4057—151no—
65C2×C3×C52C2×D411yescharacteristic
66C3⋊(C5⋊D5)—21no—
67C52×D3—41no—
68C3×C5⋊D5—41no—
69C3×C5×D5—41no—
70C2×D3×D5—101no—
71C2×D3×D5—101no—
72C2×D3×D5—102no—
73C2×D3×D5—102no—
74C3×C5×D4—51no—
75#1220—101no—
76C2×Dic15—101no—
77C4×D15—51no—
78C4×D15—51no—
79C4×D15—102no—
80Dic30C2×D511yescharacteristic, complemented
81C5×D12—22no—
82C5×D12C2×D511yescharacteristic
83C5×D12C2×D511yescharacteristic
84C5×Dic6—51no—
85C3×D20—51no—
86C3×D20—51no—
87C3×D20—102no—
88C3×D20C2×D511yescharacteristic, complemented
89#1202—101no—
90#1201—101no—
91#1201—101no—
92#1201—102no—
93#1201—102no—
94D5×Dic3—101no—
95C3×C5×C8—51no—
96C3⋊#168C2×D511yescharacteristic, complemented
97C3×#168C2×D511yescharacteristic, complemented
98C5×C3⋊C8—51no—
99C22×C52—61no—
100C2×C5⋊D5—21no—
101C2×C5⋊D5—31no—
102C2×C5⋊D5—122no—
103C2×C5×D5—21no—
104C4×C52C22×D311yescharacteristic
105#1039—31no—
106C5×Dic5—61no—
107#911—251noHall
108#713—151no—
109#711—151no—
110D4×D5—151no—
111D4×D5—151no—
112D4×D5—302no—
113C5×SD16—151no—
114C5×D8—151no—
115#688—151no—
116#687—31no—
117#686—31no—
118#686—51no—
119D40—51no—
120#677—151no—
121#676—151no—
122#676—151no—
123C3×C52AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
124C22×C3×C5—101no—
125C2×D15—51no—
126C2×D15—51no—
127C2×D15—102no—
128C2×C5×D3—22no—
129C2×C5×D3—22no—
130C2×C5×D3—21no—
131C2×C5×D3—21no—
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—
145Dic15—51no—
146Dic15—51no—
147Dic15—102no—
148C5×Dic3—101no—
149C2×C52—121no—
150C2×C52D3×D411yescharacteristic
151C5⋊D5—41no—
152C5⋊D5—61no—
153C5×D5—41no—
154#237—251no—
155D3×D4—251no—
156C3×D8—251no—
157#214—251no—
158#213—51no—
159#204—51no—
160ΓL(1,𝔽25)—251no—
161C22×D5—301no—
162C22×D5—301no—
163C22×D5—302no—
164C22×D5—302no—
165C5×Q8—151no—
166C5×D4—31no—
167C5×D4—31no—
168C5×D4—51no—
169C5×D4—62no—
170C22⋊D5—301no—
171C22⋊D5—301no—
172C22⋊D5—301no—
173C22⋊D5—302no—
174C22⋊D5—302no—
175C2×Dic5—301no—
176D20—51no—
177D20—51no—
178D20—102no—
179D20—301no—
180D20D3×D511yescharacteristic, complemented
181C4×D5—151no—
182C4×D5—151no—
183C4×D5—302no—
184C4×D5—301no—
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×C5D4×D511yescharacteristic
194C2×C3×C5D4×D511yescharacteristic
195D15—102no—
196D15—102no—
197D15—101no—
198D15—101no—
199C3×D5—41no—
200C3×D5—202no—
201C3×D5—202no—
202C3×D5—201no—
203C3×D5—201no—
204C5×D3—42no—
205C5×D3—42no—
206C5×D3—41no—
207C5×D3—41no—
208C52#91111yescharacteristic, complemented, Sylow
209C22×D3—501no—
210C3×D4—51no—
211C3×D4—251no—
212#67—501no—
213#67—501no—
214C2×Dic3—501no—
215D12D5211yescharacteristic
216C4×D3—251no—
217Dic6—51no—
218C3×C8—51no—
219C3⋊C8—51no—
220C22×C5—62no—
221C22×C5—62no—
222C22×C5—61no—
223C22×C5—61no—
224C22×C5—101no—
225C2×D5—21no—
226C2×D5—102no—
227C2×D5—102no—
228C2×D5—101no—
229C2×D5—101no—
230C2×D5—151no—
231C2×D5—151no—
232C2×D5—302no—
233C2×D5—604no—
234C2×D5—604no—
235C2×D5—602no—
236C2×D5—602no—
237C4×C5—22no—
238C4×C5—301no—
239C4×C5C2×D3×D511yescharacteristic
240C4×C5C2×D3×D511yescharacteristic
241Dic5—61no—
242Dic5—151no—
243Dic5—151no—
244Dic5—302no—
245#41—751no—
246C2×D4—751no—
247SD16—151no—
248SD16—751no—
249D8—151no—
250D8—251no—
251M16—751no—
252C3×C5—22no—
253C3×C5#405711yescharacteristic, complemented
254C3×C5#406011yescharacteristic, complemented
255C22×C3—101no—
256C22×C3—501no—
257C2×D3—21no—
258C2×D3—251no—
259C2×D3—1002no—
260C3×C4C2×D5211yescharacteristic
261Dic3—101no—
262Dic3—251no—
263C2×C5—22no—
264C2×C5—122no—
265C2×C5—122no—
266C2×C5—121no—
267C2×C5—121no—
268C2×C5—201no—
269C2×C5#677711yescharacteristic
270C2×C5#679011yescharacteristic
271D5—41no—
272D5—202no—
273D5—202no—
274D5—201no—
275D5—201no—
276D5—302no—
277D5—302no—
278D5—301no—
279D5—301no—
280C23—1501no—
281Q8—151no—
282D4—31no—
283D4—51no—
284D4—251no—
285D4—1501no—
286D4—1501no—
287C2×C4—751no—
288C2×C4—1501no—
289C8—51no—
290C8—151no—
291C2×C3—201no—
292C2×C3—1001no—
293C2×C3#8788611yescharacteristic
294D3—41no—
295D3—501no—
296C5—22no—
297C5#9125811yescharacteristic, complemented
298C5#9148211yescharacteristic, complemented
299C22—61no—
300C22—101no—
301C22—3002no—
302C22—501no—
303C22—751no—
304C4—301no—
305C4—751no—
306C4D3×D5211yescharacteristic
307C3#1171705511yescharacteristic, complemented, Sylow
308C2—121no—
309C2—201no—
310C2—1001no—
311C2—1501no—
312C2#1192542811yescharacteristic
subgroupG/H
centerC2#11925428
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. #11925624C2semidirect
2. #11925606C2semidirect
3. #11925156C2semidirect
4. #11925129C2semidirect
5. C5×#6659C2semidirect
6. C5×#6652C2semidirect
7. C3×#87771C2semidirect
11. C5×Dic30C22semidirect
14. C3×C5×D20C22semidirect
18. C5×C3⋊#168C22semidirect
19. C3×C5×#168C22semidirect
27. #87771D3semidirect
47. #6659D5semidirect
50. #6652D5semidirect
57. C5×D20C2×D3semidirect
62. C5×#168C2×D3semidirect
80. Dic30C2×D5semidirect
88. C3×D20C2×D5semidirect
96. C3⋊#168C2×D5semidirect
97. C3×#168C2×D5semidirect
123. C3×C52AGL(1,ℤ/8)semidirect
180. D20D3×D5semidirect
189. #168D3×D5semidirect
208. C52#911semidirect
253. C3×C5#4057semidirect
254. C3×C5#4060semidirect
297. C5#91258semidirect
298. C5#91482semidirect
307. C3#11717055semidirect

Maximal subgroups

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

subgroupindex
#119256242
#119256062
#119251562
#119251292
C5×#66592
C5×#66522
C3×#877712
#117170553
#914825
#912585