Finite Group Databasebeta

← C2×#10624033 #13089011

Subgroups — C2×#10624033 #13089011

Order 1440.

998 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 172 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⋊D3)⋊D5C211yesmaximal, characteristic, complemented
2C22×D3×D15C211yesmaximal, characteristic, complemented
3C2×#67151C211yesmaximal, characteristic, complemented
4C2×C5×#626C211yesmaximal, characteristic, complemented
5C2×#67140C211yesmaximal, characteristic, complemented
6C2×C3×#1203C211yesmaximal, characteristic, complemented
7C2×C3×#1200C211yesmaximal, characteristic, complemented
8#10624033C218yesmaximal, direct product factor
9C2×#6790—31nomaximal
10C2×D5×D12—31nomaximal
11C22×C3⋊D15C2211yescharacteristic, complemented
12C22×C5×C3⋊D3C2211yescharacteristic, complemented
13C22×C3×D15C2211yescharacteristic
14C22×C3×D15C2211yescharacteristic, complemented
15C22×C3×C5×D3C2211yescharacteristic, complemented
16C2×(C3⋊D3)⋊D5—24no—
17C2×(C3⋊D3)⋊D5C2214yescomplemented
18C2×D3×D15—24no—
19C2×D3×D15C2214yescomplemented
20C2×C3×C5×Dic3C2211yescharacteristic, complemented
21C2×C32×Dic5C2211yescharacteristic, complemented
22#67151C2214yescomplemented
23C5×#626C2214yescomplemented
24#67140C2214yescomplemented
25C3×#1203C2214yescomplemented
26C3×#1200C2214yescomplemented
27C2×D3×D12—51nomaximal, Hall
28C23×D15—31no—
29C22×D3×D5—31no—
30C22×D3×D5—31no—
31C22×D3×D5—31no—
32C2×D60—31no—
33C2×C5×#67—31no—
34C2×C5×D12—31no—
35C2×C3×C22⋊D5—31no—
36C2×C3×C4×D5—31no—
37#6790—38no—
38C2×#1203—31no—
39C2×#1203—31no—
40C2×#1203—61no—
41C2×#1203D311yescharacteristic, complemented
42C2×#1202—31no—
43C2×#1200D311yescharacteristic, complemented
44D5×D12—38no—
45C22×C32×C5C2311yescharacteristic
46C2×C3⋊D15—22no—
47C2×C3⋊D15C2312yescomplemented
48C2×C5×C3⋊D3—22no—
49C2×C5×C3⋊D3C2312yescomplemented
50C2×C3×D15—22no—
51C2×C3×D15D414yescomplemented
52C2×C3×D15C2312yes—
53C2×C3×D15C2312yescomplemented
54C2×C3×C5×D3—22no—
55C2×C3×C5×D3C2312yescomplemented
56(C3⋊D3)⋊D5—28no—
57D3×D15—28no—
58C3×C5×Dic3C2312yescomplemented
59C32×Dic5C2312yescomplemented
60C2×D4×D5—91noHall
61C22×D32—51no—
62C22×D32—51no—
63C2×C4⋊(C3⋊D3)—51no—
64C2×C3×D12—51no—
65C2×C3×C4×D3—51no—
66C2×#626—51no—
67C2×#626D511yescharacteristic, complemented
68D3×D12—58no—
69C23×C3×C5—31no—
70C22×D15—31no—
71C22×D15—31no—
72C22×D15—34no—
73C22×D15—64no—
74C22×D15—61no—
75C22×D15C2×D311yescharacteristic
76C22×D15C2×D311yescharacteristic, complemented
77C22×C5×D3—31no—
78C22×C5×D3—31no—
79C22×C5×D3—61no—
80C22×C5×D3C2×D311yescharacteristic, complemented
81C22×C3×D5—31no—
82C22×C3×D5—31no—
83C2×D3×D5—34no—
84C2×D3×D5—34no—
85C2×D3×D5—34no—
86C2×D3×D5—64no—
87C2×D3×D5—64no—
88C2×D3×D5—64no—
89C2×C3×C4×C5—31no—
90D60—34no—
91C5×#67—34no—
92C2×C5×Dic3C2×D311yescharacteristic, complemented
93C5×D12—34no—
94C3×C22⋊D5—34no—
95C2×C3×Dic5—21no—
96C2×C3×Dic5C2×D311yescharacteristic, complemented
97C2×C3×Dic5C2×D311yescharacteristic, complemented
98C3×C4×D5—34no—
99#1203—34no—
100#1203—34no—
101#1203—64no—
102#1203C2×D314yescomplemented
103#1202—34no—
104#1200C2×D314yescomplemented
105C2×D3×D4—151no—
106C22×D12—151no—
107C2×C32×C5C2×D412yescomplemented
108C2×C32×C5C2411yescharacteristic
109C3⋊D15—22no—
110C5×C3⋊D3—22no—
111C3×D15—22no—
112C3×D15C2×D414yescomplemented
113C3×C5×D3—22no—
114C23×D5—91no—
115C23×D5—91no—
116C2×C5×D4—91no—
117C2×C22⋊D5—31no—
118C2×C22⋊D5—91no—
119D4×D5—98no—
120C2×D20—91no—
121C2×C4×D5—31no—
122C22×C3⋊D3—51no—
123C22×C3⋊D3C2×D511yescharacteristic, complemented
124C22×C3×D3—51no—
125C22×C3×D3—51no—
126C22×C3×D3C2×D511yescharacteristic, complemented
127C2×D32—54no—
128C2×D32—54no—
129C2×D32—104no—
130C2×D32—104no—
131C2×C32×C4—51no—
132C4⋊(C3⋊D3)—54no—
133C2×C3×Dic3C2×D511yescharacteristic, complemented
134C3×D12—54no—
135C3×C4×D3—54no—
136#626—54no—
137#626C2×D514yescomplemented
138C22×C3×C5—21no—
139C22×C3×C5—32no—
140C22×C3×C5—62no—
141C22×C3×C5C22×D311yescharacteristic
142C22×C3×C5C22×D311yescharacteristic
143C2×D15—22no—
144C2×D15—32no—
145C2×D15—32no—
146C2×D15—62no—
147C2×D15—62no—
148C2×D15—62no—
149C2×D15—68no—
150C2×D15—122no—
151C2×D15D1214yescomplemented
152C2×D15C22×D312yes—
153C2×D15C22×D312yescomplemented
154C2×C5×D3—22no—
155C2×C5×D3—32no—
156C2×C5×D3—32no—
157C2×C5×D3—62no—
158C2×C5×D3—62no—
159C2×C5×D3—62no—
160C2×C5×D3—122no—
161C2×C5×D3C22×D312yescomplemented
162C2×C3×D5—34no—
163C2×C3×D5—32no—
164C2×C3×D5—32no—
165C2×C3×D5—62no—
166D3×D5—68no—
167D3×D5—68no—
168D3×D5—68no—
169C3×C4×C5—32no—
170C3×Dic5—22no—
171C3×Dic5C22×D312yescomplemented
172C3×Dic5C22×D312yescomplemented
173C5×Dic3C22×D312yescomplemented
174C23×D3—151no—
175C23×D3—151no—
176C23×D3—151no—
177C23×D3—151no—
178C2×C3×D4—151no—
179C22×C3×C4—151no—
180C2×#67—31no—
181C2×#67—151no—
182D3×D4—158no—
183C2×D12—31no—
184C2×D12—51no—
185C2×D12—158no—
186C2×D12—151no—
187C2×D12—151no—
188C2×D12—151no—
189C2×D12—301no—
190C2×C4×D3—51no—
191C32×C5C22×D411yescharacteristic, complemented, Hall
192C23×C5—31no—
193C23×C5—91no—
194C22×D5—31no—
195C22×D5—31no—
196C22×D5—94no—
197C22×D5—94no—
198C22×D5—91no—
199C22×D5—184no—
200C22×D5—184no—
201C5×D4—94no—
202C2×C4×C5—31no—
203C22⋊D5—34no—
204C22⋊D5—94no—
205C2×Dic5D3211yescharacteristic, complemented
206D20—94no—
207C4×D5—34no—
208C22×C32C22×D511yescharacteristic
209C2×C3⋊D3—22no—
210C2×C3⋊D3—52no—
211C2×C3⋊D3—102no—
212C2×C3⋊D3C22×D512yescomplemented
213C2×C3×D3—22no—
214C2×C3×D3—54no—
215C2×C3×D3—52no—
216C2×C3×D3—52no—
217C2×C3×D3—102no—
218C2×C3×D3C22×D512yescomplemented
219D32—108no—
220D32—108no—
221C32×C4—52no—
222C3×Dic3C22×D512yescomplemented
223C22×D4—451noSylow
224C2×C3×C5—21no—
225C2×C3×C5—22no—
226C2×C3×C5—62no—
227C2×C3×C5C2×D1212yescomplemented
228C2×C3×C5D3×D412yescomplemented
229C2×C3×C5C23×D311yescharacteristic
230C2×C3×C5C23×D311yescharacteristic
231D15—22no—
232D15—62no—
233D15—62no—
234D15—122no—
235D15C2×D1214yescomplemented
236C3×D5—34no—
237C3×D5—62no—
238C5×D3—22no—
239C5×D3—62no—
240C5×D3—62no—
241C5×D3—122no—
242C23×C3—31no—
243C23×C3—151no—
244C23×C3—151no—
245C22×D3—31no—
246C22×D3—31no—
247C22×D3—51no—
248C22×D3—51no—
249C22×D3—61no—
250C22×D3—154no—
251C22×D3—154no—
252C22×D3—154no—
253C22×D3—151no—
254C22×D3—151no—
255C22×D3—154no—
256C22×D3—301no—
257C22×D3—304no—
258C22×D3—304no—
259C22×D3—304no—
260C22×D3—304no—
261C22×D3D3×D511yescharacteristic, complemented
262C3×D4—154no—
263C2×C3×C4—31no—
264C2×C3×C4—51no—
265C2×C3×C4—51no—
266C2×C3×C4—101no—
267C2×C3×C4—154no—
268#67—34no—
269#67—154no—
270C2×Dic3D3×D511yescharacteristic, complemented
271D12—34no—
272D12—54no—
273D12—154no—
274D12—154no—
275D12—154no—
276D12—304no—
277C4×D3—54no—
278C22×C5—32no—
279C22×C5—62no—
280C22×C5—92no—
281C22×C5—182no—
282C22×C5C2×D3211yescharacteristic
283C2×D5—32no—
284C2×D5—34no—
285C2×D5—32no—
286C2×D5—62no—
287C2×D5—92no—
288C2×D5—188no—
289C2×D5—188no—
290C2×D5—182no—
291C4×C5—32no—
292Dic5C2×D3212yescomplemented
293C2×C32D4×D512yescomplemented
294C2×C32C23×D511yescharacteristic
295C3⋊D3—22no—
296C3⋊D3—102no—
297C3×D3—22no—
298C3×D3—54no—
299C3×D3—102no—
300C24—451no—
301C24—451no—
302C2×D4—91no—
303C2×D4—151no—
304C2×D4—451no—
305C2×D4—451no—
306C2×D4—458no—
307C22×C4—151no—
308C3×C5—21no—
309C3×C5C22×D1211yescharacteristic, complemented
310C3×C5C2×D3×D411yescharacteristic, complemented
311C22×C3—21no—
312C22×C3—32no—
313C22×C3—62no—
314C22×C3—154no—
315C22×C3—152no—
316C22×C3—152no—
317C22×C3—302no—
318C22×C3C2×D3×D511yescharacteristic
319C22×C3C2×D3×D511yescharacteristic
320C2×D3—22no—
321C2×D3—32no—
322C2×D3—32no—
323C2×D3—54no—
324C2×D3—52no—
325C2×D3—52no—
326C2×D3—62no—
327C2×D3—62no—
328C2×D3—62no—
329C2×D3—102no—
330C2×D3—122no—
331C2×D3—152no—
332C2×D3—152no—
333C2×D3—302no—
334C2×D3—308no—
335C2×D3—308no—
336C2×D3—308no—
337C2×D3—308no—
338C2×D3—302no—
339C2×D3—302no—
340C2×D3—602no—
341C2×D3C2×D3×D512yescomplemented
342C3×C4—32no—
343C3×C4—52no—
344C3×C4—52no—
345C3×C4—102no—
346Dic3C2×D3×D512yescomplemented
347C2×C5—62no—
348C2×C5—182no—
349C2×C5D3×D1212yescomplemented
350C2×C5C22×D3211yescharacteristic
351D5—34no—
352D5—62no—
353D5—182no—
354C32C2×D4×D511yescharacteristic, complemented, Sylow
355C23—31no—
356C23—91no—
357C23—151no—
358C23—151no—
359C23—454no—
360C23—454no—
361C23—451no—
362C23—904no—
363C23—904no—
364D4—94no—
365D4—154no—
366D4—454no—
367D4—454no—
368C2×C4—31no—
369C2×C4—51no—
370C2×C4—154no—
371C2×C3—22no—
372C2×C3—21no—
373C2×C3—62no—
374C2×C3—154no—
375C2×C3—302no—
376C2×C3D5×D1212yescomplemented
377C2×C3#679012yescomplemented
378C2×C3C22×D3×D511yescharacteristic
379C2×C3C22×D3×D511yescharacteristic
380D3—22no—
381D3—54no—
382D3—62no—
383D3—62no—
384D3—102no—
385D3—122no—
386D3—302no—
387D3—302no—
388D3—602no—
389C5C2×D3×D1211yescharacteristic, complemented, Sylow
390C22—32no—
391C22—62no—
392C22—92no—
393C22—154no—
394C22—152no—
395C22—152no—
396C22—182no—
397C22—302no—
398C22—452no—
399C22—908no—
400C22—908no—
401C22—902no—
402C22D32×D511yescharacteristic
403C4—32no—
404C4—52no—
405C3—21no—
406C3C2×D5×D1211yescharacteristic, complemented
407C3C2×#679011yescharacteristic, complemented
408C2—62no—
409C2—154no—
410C2—182no—
411C2—302no—
412C2—902no—
413C2#1062403312yesdirect product factor
414C2C2×D32×D511yescharacteristic
subgroupG/H
center, hypercenterC22D32×D5
commutator, cosocleC2×C32×C5C24
FrattiniC2C2×D32×D5
Fitting, socleC22×C32×C5C23
hypocenterC32×C5C22×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×(C3⋊D3)⋊D5C2semidirect
2. C22×D3×D15C2semidirect
3. C2×#67151C2semidirect
4. C2×C5×#626C2semidirect
5. C2×#67140C2semidirect
6. C2×C3×#1203C2semidirect
7. C2×C3×#1200C2semidirect
8. #10624033C2direct (N × H)
11. C22×C3⋊D15C22semidirect
12. C22×C5×C3⋊D3C22semidirect
14. C22×C3×D15C22semidirect
15. C22×C3×C5×D3C22semidirect
17. C2×(C3⋊D3)⋊D5C22semidirect
19. C2×D3×D15C22semidirect
20. C2×C3×C5×Dic3C22semidirect
21. C2×C32×Dic5C22semidirect
22. #67151C22semidirect
23. C5×#626C22semidirect
24. #67140C22semidirect
25. C3×#1203C22semidirect
26. C3×#1200C22semidirect
41. C2×#1203D3semidirect
43. C2×#1200D3semidirect
47. C2×C3⋊D15C23semidirect
49. C2×C5×C3⋊D3C23semidirect
51. C2×C3×D15D4semidirect
53. C2×C3×D15C23semidirect
55. C2×C3×C5×D3C23semidirect
58. C3×C5×Dic3C23semidirect
59. C32×Dic5C23semidirect
67. C2×#626D5semidirect
76. C22×D15C2×D3semidirect
80. C22×C5×D3C2×D3semidirect
92. C2×C5×Dic3C2×D3semidirect
96. C2×C3×Dic5C2×D3semidirect
97. C2×C3×Dic5C2×D3semidirect
102. #1203C2×D3semidirect
104. #1200C2×D3semidirect
107. C2×C32×C5C2×D4semidirect
112. C3×D15C2×D4semidirect
123. C22×C3⋊D3C2×D5semidirect
126. C22×C3×D3C2×D5semidirect
133. C2×C3×Dic3C2×D5semidirect
137. #626C2×D5semidirect
151. C2×D15D12semidirect
153. C2×D15C22×D3semidirect
161. C2×C5×D3C22×D3semidirect
171. C3×Dic5C22×D3semidirect
172. C3×Dic5C22×D3semidirect
173. C5×Dic3C22×D3semidirect
191. C32×C5C22×D4semidirect
205. C2×Dic5D32semidirect
212. C2×C3⋊D3C22×D5semidirect
218. C2×C3×D3C22×D5semidirect
222. C3×Dic3C22×D5semidirect
227. C2×C3×C5C2×D12semidirect
228. C2×C3×C5D3×D4semidirect
235. D15C2×D12semidirect
261. C22×D3D3×D5semidirect
270. C2×Dic3D3×D5semidirect
292. Dic5C2×D32semidirect
293. C2×C32D4×D5semidirect
309. C3×C5C22×D12semidirect
310. C3×C5C2×D3×D4semidirect
341. C2×D3C2×D3×D5semidirect
346. Dic3C2×D3×D5semidirect
349. C2×C5D3×D12semidirect
354. C32C2×D4×D5semidirect
376. C2×C3D5×D12semidirect
377. C2×C3#6790semidirect
389. C5C2×D3×D12semidirect
406. C3C2×D5×D12semidirect
407. C3C2×#6790semidirect
413. C2#10624033direct (N × H)

Maximal subgroups

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

subgroupindex
C22×(C3⋊D3)⋊D52
C22×D3×D152
C2×#671512
C2×C5×#6262
C2×#671402
C2×C3×#12032
C2×C3×#12002
#106240332
C2×#67903
C2×D5×D123
C2×D3×D125