Finite Group Databasebeta

← C2×#14268304 #72230721

Subgroups — C2×#14268304 #72230721

Order 3600.

688 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 112 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×C3×C5×D15C211yesmaximal, characteristic, complemented
2C22×C3⋊(C3⋊(C5⋊D5))C211yesmaximal, characteristic, complemented
3C2×C3×C5×Dic15C211yesmaximal, characteristic, complemented
4#14268304C214yesmaximal, direct product factor
5C2×#10597123—31nomaximal
6C2×#10597122—31nomaximal
7C22×C32×C52C2211yescharacteristic
8C2×C3⋊(C3⋊(C5⋊D5))—22no—
9C2×C3⋊(C3⋊(C5⋊D5))C2212yescomplemented
10C2×C3×C5×D15—22no—
11C2×C3×C5×D15C2212yescomplemented
12C3×C5×Dic15C2212yescomplemented
13C2×#67152—51nomaximal
14C2×#67151—51nomaximal
15C22×C3⋊(C5⋊D5)—31no—
16C22×C3⋊(C5⋊D5)—31no—
17C22×C3⋊(C5⋊D5)—61no—
18C22×C5×D15D311yescharacteristic, complemented
19C22×C3×C5×D5—31no—
20C2×C5×Dic15D311yescharacteristic, complemented
21C2×C3×C5×Dic5—31no—
22#10597123—34no—
23#10597122—34no—
24C2×C32×C52D412yescomplemented
25C2×C32×C52C2311yescharacteristic
26C3⋊(C3⋊(C5⋊D5))—22no—
27C3×C5×D15—22no—
28C2×#6038—91noHall
29C22×C3⋊D15—51no—
30C22×C3⋊D15—51no—
31C22×C3⋊D15—102no—
32C22×C3×D15D511yescharacteristic, complemented
33C22×C3×C5×D3—51no—
34C2×C3×Dic15D511yescharacteristic, complemented
35C2×C3×C5×Dic3—51no—
36#67152—54no—
37#67151—54no—
38C22×C3×C52—21no—
39C22×C3×C52C2×D311yescharacteristic
40C22×C3×C52C2×D311yescharacteristic
41C2×C3⋊(C5⋊D5)—32no—
42C2×C3⋊(C5⋊D5)—32no—
43C2×C3⋊(C5⋊D5)—62no—
44C2×C3⋊(C5⋊D5)—62no—
45C2×C3⋊(C5⋊D5)—62no—
46C2×C3⋊(C5⋊D5)—62no—
47C2×C3⋊(C5⋊D5)—62no—
48C2×C5×D15—22no—
49C2×C5×D15C2×D312yescomplemented
50C2×C3×C5×D5—32no—
51C2×C3×C5×D5—62no—
52C5×Dic15C2×D312yescomplemented
53C3×C5×Dic5—32no—
54C2×#1220—151no—
55C2×D60—151no—
56C2×#1203—151no—
57C2×#1202—151no—
58C32×C52C2×D411yescharacteristic, complemented, Hall
59C22×C5⋊D5—91no—
60C22×C5×D5—31no—
61C2×C5×Dic5—31no—
62#6038—94no—
63C22×C32×C5—22no—
64C22×C32×C5C2×D511yescharacteristic
65C22×C32×C5C2×D511yescharacteristic
66C2×C3⋊D15—52no—
67C2×C3⋊D15—52no—
68C2×C3⋊D15—104no—
69C2×C3⋊D15—104no—
70C2×C3⋊D15—102no—
71C2×C3⋊D15—102no—
72C2×C3⋊D15—104no—
73C2×C3×D15—22no—
74C2×C3×D15C2×D512yescomplemented
75C2×C3×C5×D3—52no—
76C2×C3×C5×D3—102no—
77C3×Dic15C2×D512yescomplemented
78C3×C5×Dic3—52no—
79C2×C3×C52—22no—
80C2×C3×C52—21no—
81C2×C3×C52D1212yescomplemented
82C2×C3×C52#6712yescomplemented
83C2×C3×C52C22×D311yescharacteristic
84C2×C3×C52C22×D311yescharacteristic
85C3⋊(C5⋊D5)—62no—
86C3⋊(C5⋊D5)—62no—
87C3⋊(C5⋊D5)—62no—
88C3⋊(C5⋊D5)—62no—
89C5×D15—22no—
90C3×C5×D5—62no—
91C2×#626—251noHall
92C23×C3×C5—151no—
93C22×D15—151no—
94C22×D15—151no—
95C22×D15—151no—
96C22×D15—151no—
97C22×D15—302no—
98C22×D15—302no—
99C22×D15—301no—
100C22×D15—301no—
101C22×D15—302no—
102C22×D15—302no—
103C22×D15D1511yescharacteristic, complemented
104C22×C5×D3—51no—
105C22×C3×D5—31no—
106C2×C3×C4×C5—151no—
107#1220—154no—
108C2×Dic15D1511yescharacteristic, complemented
109D60—154no—
110C2×C5×Dic3—51no—
111C2×C3×Dic5—31no—
112#1203—154no—
113#1202—154no—
114C22×C52D3211yescharacteristic
115C2×C5⋊D5—92no—
116C2×C5⋊D5—182no—
117C2×C5×D5—32no—
118C2×C5×D5—62no—
119C5×Dic5—32no—
120C2×C32×C5—24no—
121C2×C32×C5—22no—
122C2×C32×C5D2012yescomplemented
123C2×C32×C5C22⋊D512yescomplemented
124C2×C32×C5C22×D511yescharacteristic
125C2×C32×C5C22×D511yescharacteristic
126C3⋊D15—104no—
127C3⋊D15—104no—
128C3⋊D15—102no—
129C3⋊D15—102no—
130C3×D15—22no—
131C3×C5×D3—102no—
132C2×C22⋊D5—451no—
133C2×D20—451no—
134C3×C52—21no—
135C3×C52C2×D1211yescharacteristic, complemented
136C3×C52C2×#6711yescharacteristic, complemented
137C22×C3⋊D3—251no—
138C22×C3×D3—51no—
139C2×C3×Dic3—51no—
140#626—254no—
141C22×C3×C5—22no—
142C22×C3×C5—22no—
143C22×C3×C5—21no—
144C22×C3×C5—21no—
145C22×C3×C5—22no—
146C22×C3×C5—22no—
147C22×C3×C5—152no—
148C22×C3×C5—302no—
149C22×C3×C5D3×D511yescharacteristic
150C22×C3×C5D3×D511yescharacteristic
151C22×C3×C5C2×D1511yescharacteristic
152C22×C3×C5C2×D1511yescharacteristic
153C2×D15—22no—
154C2×D15—152no—
155C2×D15—152no—
156C2×D15—152no—
157C2×D15—152no—
158C2×D15—304no—
159C2×D15—304no—
160C2×D15—302no—
161C2×D15—302no—
162C2×D15—304no—
163C2×D15—304no—
164C2×D15—302no—
165C2×D15—302no—
166C2×D15—304no—
167C2×D15—304no—
168C2×D15—304no—
169C2×D15—304no—
170C2×D15—302no—
171C2×D15—302no—
172C2×D15—304no—
173C2×D15—304no—
174C2×D15—302no—
175C2×D15—302no—
176C2×D15—302no—
177C2×D15—302no—
178C2×D15—304no—
179C2×D15—304no—
180C2×D15C2×D1512yescomplemented
181C2×C5×D3—52no—
182C2×C5×D3—102no—
183C2×C3×D5—32no—
184C2×C3×D5—62no—
185C3×C4×C5—152no—
186Dic15C2×D1512yescomplemented
187C3×Dic5—32no—
188C5×Dic3—52no—
189C2×C52#62612yescomplemented
190C2×C52C2×D3211yescharacteristic
191C5⋊D5—182no—
192C5×D5—62no—
193C2×#67—751no—
194C2×D12—751no—
195C32×C5—22no—
196C32×C5C2×D2011yescharacteristic, complemented
197C32×C5C2×C22⋊D511yescharacteristic, complemented
198C23×C5—151no—
199C22×D5—31no—
200C22×D5—451no—
201C22×D5—451no—
202C22×D5—902no—
203C2×C4×C5—151no—
204C22⋊D5—454no—
205C2×Dic5—31no—
206D20—454no—
207C22×C32D5211yescharacteristic
208C2×C3⋊D3—252no—
209C2×C3⋊D3—502no—
210C2×C3×D3—52no—
211C2×C3×D3—102no—
212C3×Dic3—52no—
213C2×C3×C5—22no—
214C2×C3×C5—22no—
215C2×C3×C5—24no—
216C2×C3×C5—24no—
217C2×C3×C5—22no—
218C2×C3×C5—22no—
219C2×C3×C5—21no—
220C2×C3×C5—21no—
221C2×C3×C5—24no—
222C2×C3×C5—24no—
223C2×C3×C5—22no—
224C2×C3×C5—22no—
225C2×C3×C5—302no—
226C2×C3×C5#120212yescomplemented
227C2×C3×C5#120312yescomplemented
228C2×C3×C5D6012yescomplemented
229C2×C3×C5#122012yescomplemented
230C2×C3×C5C2×D3×D511yescharacteristic
231C2×C3×C5C2×D3×D511yescharacteristic
232C2×C3×C5C22×D1511yescharacteristic
233C2×C3×C5C22×D1511yescharacteristic
234D15—22no—
235D15—304no—
236D15—304no—
237D15—302no—
238D15—302no—
239D15—304no—
240D15—304no—
241D15—304no—
242D15—304no—
243D15—302no—
244D15—302no—
245D15—304no—
246D15—304no—
247D15—302no—
248D15—302no—
249D15—302no—
250D15—302no—
251C3×D5—62no—
252C5×D3—102no—
253C52C2×#62611yescharacteristic, complemented, Sylow
254C23×C3—151no—
255C22×D3—51no—
256C22×D3—751no—
257C22×D3—751no—
258C22×D3—1501no—
259C2×C3×C4—151no—
260#67—754no—
261C2×Dic3—51no—
262D12—754no—
263C22×C5—22no—
264C22×C5—152no—
265C22×C5—302no—
266C22×C5D3×D1511yescharacteristic
267C22×C5D3×D1511yescharacteristic
268C2×D5—32no—
269C2×D5—62no—
270C2×D5—452no—
271C2×D5—452no—
272C2×D5—904no—
273C2×D5—904no—
274C2×D5—902no—
275C2×D5—902no—
276C2×D5—904no—
277C4×C5—152no—
278Dic5—32no—
279C2×C32#603812yescomplemented
280C2×C32C2×D5211yescharacteristic
281C3⋊D3—502no—
282C3×D3—102no—
283C2×D4—2251noSylow
284C3×C5—22no—
285C3×C5—22no—
286C3×C5—21no—
287C3×C5—21no—
288C3×C5—22no—
289C3×C5—22no—
290C3×C5C2×#120211yescharacteristic, complemented
291C3×C5C2×#120311yescharacteristic, complemented
292C3×C5C2×D6011yescharacteristic, complemented
293C3×C5C2×#122011yescharacteristic, complemented
294C22×C3—21no—
295C22×C3—152no—
296C22×C3—302no—
297C22×C3D5×D1511yescharacteristic
298C22×C3D5×D1511yescharacteristic
299C2×D3—52no—
300C2×D3—102no—
301C2×D3—752no—
302C2×D3—752no—
303C2×D3—1502no—
304C2×D3—1502no—
305C2×D3—1502no—
306C2×D3—1502no—
307C2×D3—1502no—
308C3×C4—152no—
309Dic3—52no—
310C2×C5—22no—
311C2×C5—24no—
312C2×C5—302no—
313C2×C5#6715112yescomplemented
314C2×C5#6715212yescomplemented
315C2×C5C2×D3×D1511yescharacteristic
316C2×C5C2×D3×D1511yescharacteristic
317D5—62no—
318D5—904no—
319D5—904no—
320D5—902no—
321D5—902no—
322C32C2×#603811yescharacteristic, complemented, Sylow
323C23—151no—
324C23—2251no—
325D4—2254no—
326C2×C4—151no—
327C2×C3—21no—
328C2×C3—22no—
329C2×C3—302no—
330C2×C3#1059712212yescomplemented
331C2×C3#1059712312yescomplemented
332C2×C3C2×D5×D1511yescharacteristic
333C2×C3C2×D5×D1511yescharacteristic
334D3—102no—
335D3—1502no—
336D3—1502no—
337D3—1502no—
338D3—1502no—
339C5—22no—
340C5C2×#6715111yescharacteristic, complemented
341C5C2×#6715211yescharacteristic, complemented
342C22—152no—
343C22—302no—
344C22—4502no—
345C22—2252no—
346C22D15211yescharacteristic
347C4—152no—
348C3—21no—
349C3C2×#1059712211yescharacteristic, complemented
350C3C2×#1059712311yescharacteristic, complemented
351C2—302no—
352C2—4502no—
353C2#1426830412yesdirect product factor
354C2C2×D15211yescharacteristic
subgroupG/H
center, hypercenterC22D152
commutator, cosocleC2×C32×C52C23
FrattiniC2C2×D152
Fitting, socleC22×C32×C52C22
hypocenterC32×C52C2×D4
LCSG ▷ C2×C32×C52 ▷ C32×C52
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×C5×D15C2semidirect
2. C22×C3⋊(C3⋊(C5⋊D5))C2semidirect
3. C2×C3×C5×Dic15C2semidirect
4. #14268304C2direct (N × H)
9. C2×C3⋊(C3⋊(C5⋊D5))C22semidirect
11. C2×C3×C5×D15C22semidirect
12. C3×C5×Dic15C22semidirect
18. C22×C5×D15D3semidirect
20. C2×C5×Dic15D3semidirect
24. C2×C32×C52D4semidirect
32. C22×C3×D15D5semidirect
34. C2×C3×Dic15D5semidirect
49. C2×C5×D15C2×D3semidirect
52. C5×Dic15C2×D3semidirect
58. C32×C52C2×D4semidirect
74. C2×C3×D15C2×D5semidirect
77. C3×Dic15C2×D5semidirect
81. C2×C3×C52D12semidirect
82. C2×C3×C52#67semidirect
103. C22×D15D15semidirect
108. C2×Dic15D15semidirect
122. C2×C32×C5D20semidirect
123. C2×C32×C5C22⋊D5semidirect
135. C3×C52C2×D12semidirect
136. C3×C52C2×#67semidirect
180. C2×D15C2×D15semidirect
186. Dic15C2×D15semidirect
189. C2×C52#626semidirect
196. C32×C5C2×D20semidirect
197. C32×C5C2×C22⋊D5semidirect
226. C2×C3×C5#1202semidirect
227. C2×C3×C5#1203semidirect
228. C2×C3×C5D60semidirect
229. C2×C3×C5#1220semidirect
253. C52C2×#626semidirect
279. C2×C32#6038semidirect
290. C3×C5C2×#1202semidirect
291. C3×C5C2×#1203semidirect
292. C3×C5C2×D60semidirect
293. C3×C5C2×#1220semidirect
313. C2×C5#67151semidirect
314. C2×C5#67152semidirect
322. C32C2×#6038semidirect
330. C2×C3#10597122semidirect
331. C2×C3#10597123semidirect
340. C5C2×#67151semidirect
341. C5C2×#67152semidirect
349. C3C2×#10597122semidirect
350. C3C2×#10597123semidirect
353. C2#14268304direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C5×D152
C22×C3⋊(C3⋊(C5⋊D5))2
C2×C3×C5×Dic152
#142683042
C2×#105971233
C2×#105971223
C2×#671525
C2×#671515