Finite Group Databasebeta

← #72231468

Subgroups — #72231468

Order 3600.

840 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 104 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×D5×C3⋊D15C211yesmaximal, characteristic, complemented
2C2×C3⋊D3×C5⋊D5C211yesmaximal, characteristic, complemented
3#14268269C211yesmaximal, characteristic, complemented
4#14268262C211yesmaximal, characteristic, complemented
5C5×#67140C211yesmaximal, characteristic, complemented
6C5×#67138C211yesmaximal, characteristic, complemented
7C32×#6038C211yesmaximal, characteristic, complemented
8#11925428—34nomaximal
9C2×C3⋊(C3⋊(C5⋊D5))C2211yescharacteristic
10C2×C5×C3⋊D15C2211yescharacteristic, complemented
11C2×C52×C3⋊D3C2211yescharacteristic, complemented
12C2×C32×C5⋊D5C2211yescharacteristic, complemented
13C2×C32×C5×D5C2211yescharacteristic, complemented
14D5×C3⋊D15—22no—
15C3⋊D3×C5⋊D5—22no—
16C5×C3⋊Dic15C2211yescharacteristic, complemented
17C32×C5×Dic5C2211yescharacteristic, complemented
18#10624173—51nomaximal
19#10624160—51nomaximal
20C2×D5×D15—34no—
21C2×D3×C5⋊D5—34no—
22#10597127—34no—
23#10597120—34no—
24C5×#1203—34no—
25C5×#1201—34no—
26C3×#6038D314yescomplemented
27C2×C32×C52C2311yescharacteristic
28C3⋊(C3⋊(C5⋊D5))D412yescomplemented
29C5×C3⋊D15—21no—
30C52×C3⋊D3—21no—
31C32×C5⋊D5—21no—
32C32×C5×D5—21no—
33#87886—91noHall
34C22×C3⋊D15—51no—
35C2×D5×C3⋊D3—51no—
36C2×D5×C3⋊D3—51no—
37C2×D5×C3⋊D3—51no—
38C2×D5×C3⋊D3—102no—
39C4×C3⋊D15—51no—
40C5×#638—51no—
41C5×C4⋊(C3⋊D3)—51no—
42C32×C22⋊D5—51no—
43C32×D20—51no—
44#67140D511yescharacteristic, complemented
45#67139—51no—
46#67138—51no—
47#67138D511yescharacteristic, complemented
48#67137—51no—
49C2×C3⋊(C5⋊D5)—34no—
50C2×C5×D15—34no—
51C2×C52×D3—34no—
52C2×C3×C5⋊D5C2×D314yescomplemented
53C2×C3×C5×D5C2×D314yescomplemented
54D5×D15—68no—
55D3×C5⋊D5—68no—
56C5×Dic15—34no—
57C3×C5×Dic5C2×D314yescomplemented
58#6790—154no—
59#6777—154no—
60C32×C52C2×D411yescharacteristic, complemented, Hall
61C22×C5⋊D5—91no—
62C2×D52—91no—
63C5×C22⋊D5—91no—
64C5×C22⋊D5—91no—
65#6038—91no—
66#6038C3⋊D311yescharacteristic, complemented
67#6036—91no—
68C22×C32×C5—51no—
69C2×C3⋊D15—51no—
70C2×C3⋊D15—51no—
71C2×C3⋊D15—102no—
72C2×C3⋊D15—102no—
73C2×C3⋊D15C2×D511yescharacteristic, complemented
74C2×C5×C3⋊D3—22no—
75C2×C5×C3⋊D3—51no—
76C2×C5×C3⋊D3C2×D511yescharacteristic, complemented
77C2×C5×C3⋊D3C2×D511yescharacteristic, complemented
78C2×C32×D5—51no—
79C2×C32×D5—51no—
80C2×C32×D5—102no—
81C2×C32×D5C2×D511yescharacteristic, complemented
82D5×C3⋊D3—102no—
83D5×C3⋊D3—102no—
84D5×C3⋊D3—102no—
85D5×C3⋊D3—104no—
86D5×C3⋊D3—104no—
87C32×C4×C5—51no—
88C3⋊Dic15C2×D511yescharacteristic, complemented
89C5×C3⋊Dic3—51no—
90C32×Dic5C2×D511yescharacteristic, complemented
91C2×C3×C52C22×D314yes—
92C3⋊(C5⋊D5)—38no—
93C5×D15—64no—
94C52×D3—64no—
95C3×C5⋊D5—24no—
96C3×C5×D5—24no—
97D4×C3⋊D3—251noHall
98C22×D15—154no—
99C2×D3×D5—154no—
100C2×D3×D5—154no—
101C2×D3×D5—154no—
102C2×D3×D5—308no—
103C4×D15—154no—
104C5×#67—154no—
105C5×D12—154no—
106C3×C22⋊D5—54no—
107C3×D20—54no—
108#1203—34no—
109#1202—154no—
110#1201—34no—
111#1201—154no—
112#1200—154no—
113C22×C52—91no—
114C2×C5⋊D5—91no—
115C2×C5⋊D5—182no—
116C2×C5⋊D5C2×C3⋊D311yescharacteristic, complemented
117C2×C5×D5—91no—
118C2×C5×D5C2×C3⋊D311yescharacteristic, complemented
119D52—182no—
120C5×Dic5—91no—
121C5×Dic5C2×C3⋊D311yescharacteristic, complemented
122C2×C32×C5—22no—
123C2×C32×C5—101no—
124C2×C32×C5C22×D511yescharacteristic
125C2×C32×C5C22×D511yescharacteristic
126C3⋊D15—21no—
127C3⋊D15—52no—
128C3⋊D15—52no—
129C3⋊D15—104no—
130C5×C3⋊D3—22no—
131C5×C3⋊D3—22no—
132C5×C3⋊D3—21no—
133C5×C3⋊D3—21no—
134C5×C3⋊D3—101no—
135C32×D5—21no—
136C32×D5—102no—
137C32×D5—102no—
138C32×D5—101no—
139C32×D5—101no—
140D4×D5—451no—
141D4×D5—451no—
142C3×C52D3×D414yescomplemented
143C22×C3⋊D3—251no—
144C22×C3⋊D3—251no—
145C32×D4—251no—
146#638—51no—
147#638—251no—
148C4⋊(C3⋊D3)—51no—
149C4×C3⋊D3—251no—
150C22×C3×C5—54no—
151C2×D15—34no—
152C2×D15—154no—
153C2×D15—154no—
154C2×D15—308no—
155C2×D15—308no—
156C2×C5×D3—34no—
157C2×C5×D3—34no—
158C2×C5×D3—68no—
159C2×C5×D3—154no—
160C2×C3×D5—54no—
161C2×C3×D5—54no—
162C2×C3×D5—108no—
163C2×C3×D5D3×D514yescomplemented
164D3×D5—308no—
165D3×D5—308no—
166D3×D5—3016no—
167D3×D5—3016no—
168D3×D5—308no—
169C3×C4×C5—54no—
170Dic15—34no—
171C3×Dic5D3×D514yescomplemented
172C5×Dic3—154no—
173C2×C52—181no—
174C2×C52C22×C3⋊D311yescharacteristic
175C5⋊D5—21no—
176C5⋊D5—92no—
177C5×D5—21no—
178C5×D5—181no—
179D3×D4—754no—
180C32×C5—22no—
181C32×C5D4×D511yescharacteristic, complemented
182C32×C5D4×D511yescharacteristic, complemented
183C22×D5—451no—
184C22×D5—451no—
185C22×D5—451no—
186C22×D5—451no—
187C22×D5—902no—
188C5×D4—451no—
189C5×D4—451no—
190C22⋊D5—51no—
191C22⋊D5—91no—
192C22⋊D5—91no—
193C22⋊D5—451no—
194D20—51no—
195D20—451no—
196C4×D5—451no—
197C4×D5—451no—
198C22×C32—51no—
199C22×C32—251no—
200C2×C3⋊D3—51no—
201C2×C3⋊D3—251no—
202C2×C3⋊D3—502no—
203C2×C3⋊D3—502no—
204C2×C3⋊D3D5211yescharacteristic, complemented
205C32×C4—51no—
206C3⋊Dic3—51no—
207C2×C3×C5—28no—
208C2×C3×C5—104no—
209C2×C3×C5C2×D3×D514yes—
210C2×C3×C5C2×D3×D514yes—
211D15—64no—
212D15—158no—
213D15—158no—
214D15—3016no—
215C3×D5—24no—
216C3×D5—108no—
217C3×D5—108no—
218C3×D5—104no—
219C3×D5—104no—
220C5×D3—68no—
221C5×D3—68no—
222C5×D3—64no—
223C5×D3—64no—
224C5×D3—304no—
225C52D4×C3⋊D311yescharacteristic, complemented, Sylow
226C22×D3—754no—
227C22×D3—754no—
228C3×D4—254no—
229#67—154no—
230#67—754no—
231D12—154no—
232C4×D3—754no—
233C22×C5—51no—
234C22×C5—91no—
235C22×C5—91no—
236C22×C5—182no—
237C22×C5—451no—
238C2×D5—51no—
239C2×D5—51no—
240C2×D5—91no—
241C2×D5—102no—
242C2×D5—451no—
243C2×D5—451no—
244C2×D5—902no—
245C2×D5—902no—
246C2×D5—902no—
247C2×D5—904no—
248C2×D5—904no—
249C2×D5—902no—
250C2×D5—902no—
251C2×D5D5×C3⋊D311yescharacteristic, complemented
252C4×C5—51no—
253C4×C5—451no—
254Dic5—91no—
255Dic5D5×C3⋊D311yescharacteristic, complemented
256C2×C32—101no—
257C2×C32—501no—
258C2×C32C2×D5211yescharacteristic
259C3⋊D3—21no—
260C3⋊D3—101no—
261C3⋊D3—252no—
262C2×D4—2251noSylow
263C3×C5—28no—
264C3×C5#677714yescomplemented
265C3×C5#679014yescomplemented
266C22×C3—54no—
267C22×C3—254no—
268C2×D3—34no—
269C2×D3—154no—
270C2×D3—754no—
271C2×D3—1508no—
272C2×D3—1508no—
273C3×C4—54no—
274Dic3—154no—
275C2×C5—22no—
276C2×C5—101no—
277C2×C5—182no—
278C2×C5—182no—
279C2×C5—181no—
280C2×C5—181no—
281C2×C5—901no—
282C2×C5C2×D5×C3⋊D311yescharacteristic
283C2×C5C2×D5×C3⋊D311yescharacteristic
284D5—21no—
285D5—102no—
286D5—102no—
287D5—101no—
288D5—101no—
289D5—181no—
290D5—452no—
291D5—452no—
292D5—904no—
293C32#8788611yescharacteristic, complemented, Sylow
294C23—2251no—
295C23—2251no—
296D4—251no—
297D4—451no—
298D4—451no—
299D4—2251no—
300C2×C4—2251no—
301C2×C3—104no—
302C2×C3—504no—
303C2×C3D3×D5214yes—
304D3—64no—
305D3—304no—
306D3—758no—
307C5—22no—
308C5#1062416011yescharacteristic, complemented
309C5#1062417311yescharacteristic, complemented
310C22—51no—
311C22—91no—
312C22—251no—
313C22—451no—
314C22—4502no—
315C22—4502no—
316C22—2251no—
317C4—51no—
318C4—451no—
319C3#1192542814yescomplemented
320C2—101no—
321C2—181no—
322C2—501no—
323C2—901no—
324C2—2252no—
325C2D52×C3⋊D311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D52×C3⋊D3
commutator, Fitting, socle, cosocleC2×C32×C52C23
hypocenterC32×C52C2×D4
LCSG ▷ C2×C32×C52 ▷ C32×C52
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×D5×C3⋊D15C2semidirect
2. C2×C3⋊D3×C5⋊D5C2semidirect
3. #14268269C2semidirect
4. #14268262C2semidirect
5. C5×#67140C2semidirect
6. C5×#67138C2semidirect
7. C32×#6038C2semidirect
10. C2×C5×C3⋊D15C22semidirect
11. C2×C52×C3⋊D3C22semidirect
12. C2×C32×C5⋊D5C22semidirect
13. C2×C32×C5×D5C22semidirect
16. C5×C3⋊Dic15C22semidirect
17. C32×C5×Dic5C22semidirect
26. C3×#6038D3semidirect
28. C3⋊(C3⋊(C5⋊D5))D4semidirect
44. #67140D5semidirect
47. #67138D5semidirect
52. C2×C3×C5⋊D5C2×D3semidirect
53. C2×C3×C5×D5C2×D3semidirect
57. C3×C5×Dic5C2×D3semidirect
60. C32×C52C2×D4semidirect
66. #6038C3⋊D3semidirect
73. C2×C3⋊D15C2×D5semidirect
76. C2×C5×C3⋊D3C2×D5semidirect
77. C2×C5×C3⋊D3C2×D5semidirect
81. C2×C32×D5C2×D5semidirect
88. C3⋊Dic15C2×D5semidirect
90. C32×Dic5C2×D5semidirect
116. C2×C5⋊D5C2×C3⋊D3semidirect
118. C2×C5×D5C2×C3⋊D3semidirect
121. C5×Dic5C2×C3⋊D3semidirect
142. C3×C52D3×D4semidirect
163. C2×C3×D5D3×D5semidirect
171. C3×Dic5D3×D5semidirect
181. C32×C5D4×D5semidirect
182. C32×C5D4×D5semidirect
204. C2×C3⋊D3D52semidirect
225. C52D4×C3⋊D3semidirect
251. C2×D5D5×C3⋊D3semidirect
255. Dic5D5×C3⋊D3semidirect
264. C3×C5#6777semidirect
265. C3×C5#6790semidirect
293. C32#87886semidirect
308. C5#10624160semidirect
309. C5#10624173semidirect
319. C3#11925428semidirect

Maximal subgroups

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

subgroupindex
C2×D5×C3⋊D152
C2×C3⋊D3×C5⋊D52
#142682692
#142682622
C5×#671402
C5×#671382
C32×#60382
#119254283
#106241735
#106241605