Finite Group Databasebeta

← #72231378

Subgroups — #72231378

Order 3600.

602 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 78 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×C3×D15⋊D5C211yesmaximal, characteristic, complemented
2C2×C3×D3×C5⋊D5C211yesmaximal, characteristic, complemented
3#14268309C212yesmaximal, complemented
4C5×#67150C212yesmaximal, complemented
5#14268268C211yesmaximal, characteristic, complemented
6#11925664—31nomaximal
7#11925419—31nomaximal
8C2×C3×C3⋊(C5⋊D5)C2211yescharacteristic, complemented
9C2×C3×C5×D15C2212yescomplemented
10C2×C3×C52×D3C2211yescharacteristic, complemented
11C2×C32×C5⋊D5C2211yescharacteristic
12C3×D15⋊D5—22no—
13C3×D3×C5⋊D5—22no—
14C5×C3⋊Dic15C2212yescomplemented
15D5×#625—52nomaximal
16C22×C3×C5⋊D5—31no—
17C2×D15⋊D5D311yescharacteristic, complemented
18C2×D3×C5⋊D5D311yescharacteristic, complemented
19C2×C3×D52—31no—
20C5×#1220—32no—
21#10597127—32no—
22#10597126—31no—
23#10597126—31no—
24#10597126—61no—
25#10597123—32no—
26C5×#1201—32no—
27C2×C32×C52C2311yescharacteristic
28C3×C3⋊(C5⋊D5)—21no—
29C3×C5×D15—22no—
30C3×C52×D3—21no—
31C32×C5⋊D5D412yescomplemented
32#87886—91noHall
33C2×C3×D3×D5—52no—
34C2×C3×D3×D5—52no—
35C2×C3×D3×D5—102no—
36#67157—52no—
37#67150—52no—
38#67150D512yescomplemented
39C5×#625—52no—
40D5×C3⋊Dic3—52no—
41C22×C3×C52—31no—
42C2×C3⋊(C5⋊D5)C2×D311yescharacteristic, complemented
43C2×C5×D15C2×D312yescomplemented
44C2×C52×D3C2×D311yescharacteristic, complemented
45C2×C3×C5⋊D5—21no—
46C2×C3×C5⋊D5—31no—
47C2×C3×C5⋊D5—62no—
48C2×C3×C5⋊D5C2×D311yescharacteristic
49C2×C3×C5⋊D5C2×D311yescharacteristic
50C2×C3×C5×D5—32no—
51D15⋊D5—22no—
52D3×C5⋊D5—22no—
53C3×D52—62no—
54C5×Dic15—32no—
55C5×Dic15—32no—
56C5×Dic15—62no—
57D5×#67—152no—
58D5×#67—152no—
59C32×C52C2×D411yescharacteristic, complemented, Hall
60C22×C5⋊D5—31no—
61C2×D52—31no—
62C5×C22⋊D5—92no—
63#6038—92no—
64#6036—91no—
65C2×C3×D15—52no—
66C2×C3×D15—102no—
67C2×C3×D15C2×D512yescomplemented
68C2×C3×C5×D3—22no—
69C2×C3×C5×D3—52no—
70C2×C3×C5×D3C2×D512yescomplemented
71C2×C32×D5—52no—
72C2×C32×D5—102no—
73C3×D3×D5—104no—
74C3×D3×D5—104no—
75C3×D3×D5—104no—
76C3×D3×D5—104no—
77C3⋊Dic15C2×D512yescomplemented
78C5×C3⋊Dic3—52no—
79C2×C3×C52—21no—
80C2×C3×C52—61no—
81C2×C3×C52C22×D311yescharacteristic
82C2×C3×C52C22×D311yescharacteristic
83C3⋊(C5⋊D5)—21no—
84C5×D15—22no—
85C52×D3—21no—
86C3×C5⋊D5—22no—
87C3×C5⋊D5—61no—
88C3×C5⋊D5#6712yescomplemented
89C3×C5⋊D5#6712yescomplemented
90C3×C5×D5—62no—
91C2×#625—251noHall
92C22×C3×D5—152no—
93C22×C3×D5—152no—
94C22×C3×D5—302no—
95C2×D3×D5—52no—
96C2×D3×D5—52no—
97C2×D3×D5—102no—
98#1220—32no—
99#1220—152no—
100C5×#67—152no—
101C5×#67—152no—
102#1202—152no—
103#1202—152no—
104#1201—32no—
105#1201—152no—
106D5×Dic3—152no—
107D5×Dic3—152no—
108D5×Dic3—302no—
109C22×C52—31no—
110C2×C5⋊D5—31no—
111C2×C5⋊D5—62no—
112C2×C5⋊D5D3211yescharacteristic
113C2×C5×D5—32no—
114D52—62no—
115C5×Dic5—92no—
116C2×C32×C5—22no—
117C2×C32×C5C22×D512yes—
118C3×D15—22no—
119C3×D15—102no—
120C3×D15—102no—
121C3×D15—102no—
122C3×C5×D3—22no—
123C3×C5×D3—22no—
124C3×C5×D3—22no—
125C3×C5×D3—102no—
126C32×D5—54no—
127C32×D5—104no—
128D4×D5—452no—
129C3×C52—21no—
130C3×C52C2×#6711yescharacteristic, complemented
131C3×C52C2×#6711yescharacteristic, complemented
132C22×C3×D3—251no—
133C22×C3×D3—251no—
134C2×C3⋊Dic3—251no—
135#625—52no—
136#625—252no—
137C22×C3×C5—32no—
138C22×C3×C5—62no—
139C22×C3×C5—152no—
140C2×D15—52no—
141C2×D15—102no—
142C2×D15D3×D512yescomplemented
143C2×C5×D3—22no—
144C2×C5×D3—52no—
145C2×C5×D3D3×D512yescomplemented
146C2×C3×D5—32no—
147C2×C3×D5—52no—
148C2×C3×D5—52no—
149C2×C3×D5—102no—
150C2×C3×D5—102no—
151C2×C3×D5—102no—
152C2×C3×D5—152no—
153C2×C3×D5—202no—
154C2×C3×D5—302no—
155C2×C3×D5—304no—
156C2×C3×D5—304no—
157C2×C3×D5—304no—
158C2×C3×D5—304no—
159D3×D5—104no—
160D3×D5—104no—
161D3×D5—104no—
162D3×D5—104no—
163Dic15—32no—
164Dic15—32no—
165Dic15—62no—
166C5×Dic3—152no—
167C5×Dic3—152no—
168C5×Dic3—302no—
169C2×C52—61no—
170C2×C52C2×D3211yescharacteristic
171C5⋊D5—61no—
172C5⋊D5#62512yescomplemented
173C5×D5—62no—
174C2×#67—751no—
175C2×#67—751no—
176C32×C5—22no—
177C32×C5D4×D512yescomplemented
178C22×D5—152no—
179C22×D5—152no—
180C22×D5—302no—
181C5×D4—452no—
182C22⋊D5—92no—
183C22⋊D5—452no—
184D20—452no—
185C4×D5—452no—
186C22×C32—251no—
187C2×C3×D3—52no—
188C2×C3×D3—251no—
189C2×C3×D3—502no—
190C2×C3×D3—502no—
191C2×C3×D3D5211yescharacteristic, complemented
192C3⋊Dic3—52no—
193C2×C3×C5—22no—
194C2×C3×C5—22no—
195C2×C3×C5—22no—
196C2×C3×C5—42no—
197C2×C3×C5—62no—
198C2×C3×C5—62no—
199C2×C3×C5—62no—
200C2×C3×C5—302no—
201C2×C3×C5C2×D3×D512yes—
202C2×C3×C5C2×D3×D512yes—
203D15—22no—
204D15—102no—
205D15—102no—
206D15—102no—
207C3×D5—54no—
208C3×D5—54no—
209C3×D5—62no—
210C3×D5—104no—
211C3×D5—104no—
212C3×D5—104no—
213C3×D5—204no—
214C3×D5—302no—
215C3×D5—302no—
216C3×D5—302no—
217C5×D3—22no—
218C5×D3—22no—
219C5×D3—22no—
220C5×D3—102no—
221C52C2×#62511yescharacteristic, complemented, Sylow
222C23×C3—751no—
223C23×C3—751no—
224C22×D3—251no—
225C22×D3—251no—
226#67—152no—
227#67—152no—
228#67—752no—
229#67—752no—
230C2×Dic3—751no—
231C2×Dic3—751no—
232C2×Dic3—1501no—
233C22×C5—32no—
234C22×C5—62no—
235C22×C5—152no—
236C2×D5—32no—
237C2×D5—52no—
238C2×D5—102no—
239C2×D5—152no—
240C2×D5—302no—
241C2×D5—304no—
242C2×D5—304no—
243C2×D5—304no—
244C2×D5—304no—
245C4×C5—452no—
246Dic5—92no—
247C2×C32—252no—
248C2×C32C2×D5211yescharacteristic
249C3×D3—21no—
250C3×D3—102no—
251C3×D3—501no—
252C2×D4—2251noSylow
253C3×C5—22no—
254C3×C5—22no—
255C3×C5—22no—
256C3×C5—42no—
257C3×C5D5×#6712yescomplemented
258C3×C5D5×#6712yescomplemented
259C22×C3—31no—
260C22×C3—152no—
261C22×C3—251no—
262C22×C3—251no—
263C22×C3—501no—
264C22×C3—751no—
265C22×C3—1502no—
266C22×C3—1502no—
267C2×D3—52no—
268C2×D3—251no—
269C2×D3—502no—
270C2×D3—502no—
271C2×D3D15⋊D511yescharacteristic, complemented
272Dic3—152no—
273Dic3—152no—
274Dic3—302no—
275C2×C5—22no—
276C2×C5—62no—
277C2×C5—62no—
278C2×C5—62no—
279C2×C5—302no—
280C2×C5D32×D512yes—
281D5—54no—
282D5—62no—
283D5—104no—
284D5—302no—
285D5—302no—
286D5—302no—
287C32#8788611yescharacteristic, complemented, Sylow
288C23—751no—
289C23—751no—
290D4—452no—
291D4—2252no—
292C2×C4—2251no—
293C2×C3—21no—
294C2×C3—61no—
295C2×C3—252no—
296C2×C3—252no—
297C2×C3—302no—
298C2×C3—502no—
299C2×C3—1501no—
300C2×C3D3×D5211yescharacteristic
301C2×C3C2×D15⋊D511yescharacteristic
302D3—21no—
303D3—102no—
304D3—501no—
305C5—22no—
306C5D5×#62512yescomplemented
307C22—31no—
308C22—152no—
309C22—251no—
310C22—751no—
311C22—1502no—
312C22—1502no—
313C4—452no—
314C3—21no—
315C3#1192541911yescharacteristic, complemented
316C3#1192566411yescharacteristic, complemented
317C2—61no—
318C2—252no—
319C2—302no—
320C2—1501no—
321C2D3×D15⋊D511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×D15⋊D5
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×C3×D15⋊D5C2semidirect
2. C2×C3×D3×C5⋊D5C2semidirect
3. #14268309C2semidirect
4. C5×#67150C2semidirect
5. #14268268C2semidirect
8. C2×C3×C3⋊(C5⋊D5)C22semidirect
9. C2×C3×C5×D15C22semidirect
10. C2×C3×C52×D3C22semidirect
14. C5×C3⋊Dic15C22semidirect
17. C2×D15⋊D5D3semidirect
18. C2×D3×C5⋊D5D3semidirect
31. C32×C5⋊D5D4semidirect
38. #67150D5semidirect
42. C2×C3⋊(C5⋊D5)C2×D3semidirect
43. C2×C5×D15C2×D3semidirect
44. C2×C52×D3C2×D3semidirect
59. C32×C52C2×D4semidirect
67. C2×C3×D15C2×D5semidirect
70. C2×C3×C5×D3C2×D5semidirect
77. C3⋊Dic15C2×D5semidirect
88. C3×C5⋊D5#67semidirect
89. C3×C5⋊D5#67semidirect
130. C3×C52C2×#67semidirect
131. C3×C52C2×#67semidirect
142. C2×D15D3×D5semidirect
145. C2×C5×D3D3×D5semidirect
172. C5⋊D5#625semidirect
177. C32×C5D4×D5semidirect
191. C2×C3×D3D52semidirect
221. C52C2×#625semidirect
257. C3×C5D5×#67semidirect
258. C3×C5D5×#67semidirect
271. C2×D3D15⋊D5semidirect
287. C32#87886semidirect
306. C5D5×#625semidirect
315. C3#11925419semidirect
316. C3#11925664semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D15⋊D52
C2×C3×D3×C5⋊D52
#142683092
C5×#671502
#142682682
#119256643
#119254193
D5×#6255