Finite Group Databasebeta

← C2×#10624030 #13089006

Subgroups — C2×#10624030 #13089006

Order 1440.

1078 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 164 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×D3×D15C211yesmaximal, characteristic, complemented
2C22×D5×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×#67156C211yesmaximal, characteristic, complemented
4C2×#67149C211yesmaximal, characteristic, complemented
5C2×C5×#626C211yesmaximal, characteristic, complemented
6C2×C3×#1202C211yesmaximal, characteristic, complemented
7C2×C3×#1201C211yesmaximal, characteristic, complemented
8#10624030C218yesmaximal, direct product factor
9C2×#6790—31nomaximal
10C2×#6777—31nomaximal
11C22×C3⋊D15C2211yescharacteristic
12C22×C5×C3⋊D3C2211yescharacteristic, complemented
13C22×C3×D15C2211yescharacteristic, complemented
14C22×C3×C5×D3C2211yescharacteristic, complemented
15C22×C32×D5C2211yescharacteristic, complemented
16C2×D3×D15—24no—
17C2×D3×D15C2214yescomplemented
18C2×D5×C3⋊D3—24no—
19C2×D5×C3⋊D3C2214yescomplemented
20C2×C3×Dic15C2211yescharacteristic, complemented
21C2×C3×C5×Dic3C2211yescharacteristic, complemented
22#67156C2214yescomplemented
23#67149C2214yescomplemented
24C5×#626C2214yescomplemented
25C3×#1202C2214yescomplemented
26C3×#1201C2214yescomplemented
27C2×#3811—51nomaximal, Hall
28C23×D15—31no—
29C22×D3×D5—31no—
30C22×D3×D5—31no—
31C22×D3×D5—31no—
32C22×D3×D5—61no—
33C2×C4×D15—31no—
34C2×C5×#67—31no—
35C2×C5×D12—31no—
36C2×C3×C22⋊D5—31no—
37C2×C3×D20—31no—
38#6790—38no—
39C2×#1203—31no—
40C2×#1202D311yescharacteristic, complemented
41C2×#1201—31no—
42C2×#1201D311yescharacteristic, complemented
43C2×#1200—31no—
44#6777—38no—
45C22×C32×C5C2311yescharacteristic
46C2×C3⋊D15D414yescomplemented
47C2×C3⋊D15C2312yes—
48C2×C5×C3⋊D3—22no—
49C2×C5×C3⋊D3C2312yescomplemented
50C2×C3×D15—22no—
51C2×C3×D15C2312yescomplemented
52C2×C3×C5×D3—22no—
53C2×C3×C5×D3C2312yescomplemented
54C2×C32×D5—22no—
55C2×C32×D5C2312yescomplemented
56D3×D15—28no—
57D5×C3⋊D3—28no—
58C3×Dic15C2312yescomplemented
59C3×C5×Dic3C2312yescomplemented
60C2×D4×D5—91noHall
61C23×C3⋊D3—51no—
62C22×D32—51no—
63C2×C3×#67—51no—
64C2×C3×#67—51no—
65#3811—58no—
66C2×#626—51no—
67C2×#626D511yescharacteristic, complemented
68C2×#624—51no—
69C23×C3×C5—31no—
70C22×D15—31no—
71C22×D15—31no—
72C22×D15—34no—
73C22×D15—64no—
74C22×D15—61no—
75C22×D15C2×D311yescharacteristic, complemented
76C22×C5×D3—31no—
77C22×C5×D3—31no—
78C22×C5×D3—61no—
79C22×C5×D3C2×D311yescharacteristic, complemented
80C22×C3×D5—21no—
81C22×C3×D5—31no—
82C22×C3×D5C2×D311yescharacteristic, complemented
83C22×C3×D5C2×D311yescharacteristic, complemented
84C2×D3×D5—34no—
85C2×D3×D5—34no—
86C2×D3×D5—34no—
87C2×D3×D5—64no—
88C2×D3×D5—64no—
89C2×D3×D5—64no—
90C2×D3×D5—68no—
91C2×D3×D5—64no—
92C2×C3×C4×C5—31no—
93C2×Dic15C2×D311yescharacteristic, complemented
94C4×D15—34no—
95C5×#67—34no—
96C2×C5×Dic3C2×D311yescharacteristic, complemented
97C5×D12—34no—
98C3×C22⋊D5—34no—
99C2×C3×Dic5—31no—
100C3×D20—34no—
101#1203—34no—
102#1202C2×D314yescomplemented
103#1201—34no—
104#1201C2×D314yescomplemented
105#1200—34no—
106C2×D3×D4—151no—
107C2×D3×D4—151no—
108C2×C32×C5C2×D412yescomplemented
109C2×C32×C5C2411yescharacteristic
110C3⋊D15C2×D414yescomplemented
111C5×C3⋊D3—22no—
112C3×D15—22no—
113C3×C5×D3—22no—
114C32×D5—22no—
115C23×D5—91no—
116C23×D5—91no—
117C2×C5×D4—91no—
118C2×C22⋊D5—31no—
119C2×C22⋊D5—91no—
120D4×D5—98no—
121C2×D20—31no—
122C2×C4×D5—91no—
123C23×C32—51no—
124C22×C3⋊D3—51no—
125C22×C3⋊D3—54no—
126C22×C3⋊D3—104no—
127C22×C3⋊D3C2×D511yescharacteristic, complemented
128C22×C3×D3—51no—
129C22×C3×D3C2×D511yescharacteristic, complemented
130C2×D32—54no—
131C2×D32—104no—
132C3×#67—54no—
133C3×#67—54no—
134C2×C3×Dic3—51no—
135C2×C3×Dic3C2×D511yescharacteristic, complemented
136#626—54no—
137#626C2×D514yescomplemented
138#624—54no—
139C22×C3×C5—21no—
140C22×C3×C5—32no—
141C22×C3×C5—62no—
142C22×C3×C5C22×D311yescharacteristic
143C22×C3×C5C22×D311yescharacteristic
144C2×D15—22no—
145C2×D15—34no—
146C2×D15—34no—
147C2×D15—32no—
148C2×D15—32no—
149C2×D15—64no—
150C2×D15—62no—
151C2×D15—68no—
152C2×D15C22×D312yescomplemented
153C2×C5×D3—22no—
154C2×C5×D3—32no—
155C2×C5×D3—32no—
156C2×C5×D3—62no—
157C2×C5×D3—64no—
158C2×C5×D3—62no—
159C2×C5×D3—62no—
160C2×C5×D3C22×D312yescomplemented
161C2×C3×D5—24no—
162C2×C3×D5—22no—
163C2×C3×D5—22no—
164C2×C3×D5—22no—
165C2×C3×D5—32no—
166C2×C3×D5—62no—
167C2×C3×D5C22×D312yescomplemented
168C2×C3×D5C22×D312yescomplemented
169D3×D5—68no—
170D3×D5—68no—
171D3×D5—68no—
172D3×D5—616no—
173C3×C4×C5—32no—
174Dic15C22×D312yescomplemented
175C3×Dic5—32no—
176C5×Dic3C22×D312yescomplemented
177C23×D3—151no—
178C23×D3—151no—
179C23×D3—151no—
180C23×D3—151no—
181C23×D3—301no—
182C2×C3×D4—151no—
183C2×C3×D4—151no—
184C2×#67—31no—
185C2×#67—51no—
186C2×#67—51no—
187C2×#67—151no—
188D3×D4—158no—
189D3×D4—158no—
190C2×D12—31no—
191C2×D12—151no—
192C2×C4×D3—151no—
193C2×C4×D3—151no—
194C32×C5C22×D411yescharacteristic, complemented, Hall
195C23×C5—31no—
196C23×C5—91no—
197C22×D5—31no—
198C22×D5—94no—
199C22×D5—91no—
200C22×D5—94no—
201C22×D5—184no—
202C22×D5—184no—
203C22×D5D3211yescharacteristic, complemented
204C5×D4—94no—
205C2×C4×C5—31no—
206C22⋊D5—34no—
207C22⋊D5—94no—
208C2×Dic5—31no—
209D20—34no—
210C4×D5—94no—
211C22×C32—52no—
212C22×C32—102no—
213C22×C32C22×D511yescharacteristic
214C2×C3⋊D3—22no—
215C2×C3⋊D3—54no—
216C2×C3⋊D3—52no—
217C2×C3⋊D3—108no—
218C2×C3⋊D3C22×D512yescomplemented
219C2×C3×D3—22no—
220C2×C3×D3—52no—
221C2×C3×D3—102no—
222C2×C3×D3C22×D512yescomplemented
223D32—108no—
224C3×Dic3—52no—
225C3×Dic3C22×D512yescomplemented
226C22×D4—451noSylow
227C2×C3×C5—21no—
228C2×C3×C5—22no—
229C2×C3×C5—62no—
230C2×C3×C5D3×D412yescomplemented
231C2×C3×C5D3×D412yescomplemented
232C2×C3×C5C23×D311yescharacteristic
233C2×C3×C5C23×D311yescharacteristic
234D15—22no—
235D15—34no—
236D15—34no—
237D15—64no—
238C3×D5—24no—
239C3×D5—22no—
240C3×D5—22no—
241C3×D5—62no—
242C5×D3—22no—
243C5×D3—64no—
244C5×D3—62no—
245C5×D3—62no—
246C23×C3—31no—
247C23×C3—51no—
248C23×C3—51no—
249C23×C3—101no—
250C23×C3—151no—
251C22×D3—31no—
252C22×D3—31no—
253C22×D3—51no—
254C22×D3—61no—
255C22×D3—154no—
256C22×D3—154no—
257C22×D3—154no—
258C22×D3—151no—
259C22×D3—151no—
260C22×D3—154no—
261C22×D3—304no—
262C22×D3—301no—
263C22×D3—304no—
264C22×D3—304no—
265C22×D3—304no—
266C22×D3—308no—
267C22×D3—304no—
268C22×D3D3×D511yescharacteristic, complemented
269C3×D4—154no—
270C3×D4—154no—
271C2×C3×C4—31no—
272C2×C3×C4—151no—
273#67—34no—
274#67—54no—
275#67—54no—
276#67—154no—
277C2×Dic3—51no—
278C2×Dic3D3×D511yescharacteristic, complemented
279D12—34no—
280D12—154no—
281C4×D3—154no—
282C4×D3—154no—
283C22×C5—32no—
284C22×C5—62no—
285C22×C5—92no—
286C22×C5—182no—
287C22×C5C2×D3211yescharacteristic
288C2×D5—22no—
289C2×D5—32no—
290C2×D5—62no—
291C2×D5—92no—
292C2×D5—94no—
293C2×D5—188no—
294C2×D5—188no—
295C2×D5C2×D3212yescomplemented
296C4×C5—32no—
297Dic5—32no—
298C2×C32—102no—
299C2×C32D4×D512yescomplemented
300C2×C32C23×D511yescharacteristic
301C3⋊D3—22no—
302C3⋊D3—54no—
303C3×D3—22no—
304C3×D3—102no—
305C24—451no—
306C24—451no—
307C2×D4—91no—
308C2×D4—151no—
309C2×D4—151no—
310C2×D4—451no—
311C2×D4—458no—
312C22×C4—451no—
313C3×C5—21no—
314C3×C5C2×D3×D411yescharacteristic, complemented
315C3×C5C2×D3×D411yescharacteristic, complemented
316C22×C3—21no—
317C22×C3—32no—
318C22×C3—52no—
319C22×C3—52no—
320C22×C3—62no—
321C22×C3—102no—
322C22×C3—104no—
323C22×C3—102no—
324C22×C3—102no—
325C22×C3—152no—
326C22×C3—302no—
327C22×C3C2×D3×D511yescharacteristic
328C22×C3C2×D3×D511yescharacteristic
329C2×D3—22no—
330C2×D3—32no—
331C2×D3—32no—
332C2×D3—52no—
333C2×D3—62no—
334C2×D3—64no—
335C2×D3—62no—
336C2×D3—62no—
337C2×D3—102no—
338C2×D3—154no—
339C2×D3—154no—
340C2×D3—152no—
341C2×D3—152no—
342C2×D3—304no—
343C2×D3—302no—
344C2×D3—308no—
345C2×D3—308no—
346C2×D3—308no—
347C2×D3—3016no—
348C2×D3—308no—
349C2×D3C2×D3×D512yescomplemented
350C3×C4—32no—
351C3×C4—152no—
352Dic3—52no—
353Dic3C2×D3×D512yescomplemented
354C2×C5—62no—
355C2×C5—182no—
356C2×C5#381112yescomplemented
357C2×C5C22×D3211yescharacteristic
358D5—22no—
359D5—62no—
360D5—94no—
361C32C2×D4×D511yescharacteristic, complemented, Sylow
362C23—31no—
363C23—51no—
364C23—91no—
365C23—151no—
366C23—454no—
367C23—454no—
368C23—451no—
369C23—904no—
370C23—904no—
371D4—94no—
372D4—154no—
373D4—154no—
374D4—454no—
375C2×C4—31no—
376C2×C4—151no—
377C2×C4—454no—
378C2×C3—22no—
379C2×C3—21no—
380C2×C3—62no—
381C2×C3—104no—
382C2×C3—102no—
383C2×C3—102no—
384C2×C3—302no—
385C2×C3#677712yescomplemented
386C2×C3#679012yescomplemented
387C2×C3C22×D3×D511yescharacteristic
388C2×C3C22×D3×D511yescharacteristic
389D3—22no—
390D3—64no—
391D3—62no—
392D3—62no—
393D3—102no—
394D3—154no—
395D3—154no—
396D3—304no—
397C5C2×#381111yescharacteristic, complemented, Sylow
398C22—32no—
399C22—52no—
400C22—62no—
401C22—92no—
402C22—102no—
403C22—152no—
404C22—182no—
405C22—302no—
406C22—454no—
407C22—452no—
408C22—908no—
409C22—908no—
410C22D32×D511yescharacteristic
411C4—32no—
412C4—152no—
413C3—21no—
414C3C2×#677711yescharacteristic, complemented
415C3C2×#679011yescharacteristic, complemented
416C2—62no—
417C2—102no—
418C2—182no—
419C2—302no—
420C2—454no—
421C2#1062403012yesdirect product factor
422C2C2×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×D3×D15C2semidirect
2. C22×D5×C3⋊D3C2semidirect
3. C2×#67156C2semidirect
4. C2×#67149C2semidirect
5. C2×C5×#626C2semidirect
6. C2×C3×#1202C2semidirect
7. C2×C3×#1201C2semidirect
8. #10624030C2direct (N × H)
12. C22×C5×C3⋊D3C22semidirect
13. C22×C3×D15C22semidirect
14. C22×C3×C5×D3C22semidirect
15. C22×C32×D5C22semidirect
17. C2×D3×D15C22semidirect
19. C2×D5×C3⋊D3C22semidirect
20. C2×C3×Dic15C22semidirect
21. C2×C3×C5×Dic3C22semidirect
22. #67156C22semidirect
23. #67149C22semidirect
24. C5×#626C22semidirect
25. C3×#1202C22semidirect
26. C3×#1201C22semidirect
40. C2×#1202D3semidirect
42. C2×#1201D3semidirect
46. C2×C3⋊D15D4semidirect
49. C2×C5×C3⋊D3C23semidirect
51. C2×C3×D15C23semidirect
53. C2×C3×C5×D3C23semidirect
55. C2×C32×D5C23semidirect
58. C3×Dic15C23semidirect
59. C3×C5×Dic3C23semidirect
67. C2×#626D5semidirect
75. C22×D15C2×D3semidirect
79. C22×C5×D3C2×D3semidirect
82. C22×C3×D5C2×D3semidirect
83. C22×C3×D5C2×D3semidirect
93. C2×Dic15C2×D3semidirect
96. C2×C5×Dic3C2×D3semidirect
102. #1202C2×D3semidirect
104. #1201C2×D3semidirect
108. C2×C32×C5C2×D4semidirect
110. C3⋊D15C2×D4semidirect
127. C22×C3⋊D3C2×D5semidirect
129. C22×C3×D3C2×D5semidirect
135. C2×C3×Dic3C2×D5semidirect
137. #626C2×D5semidirect
152. C2×D15C22×D3semidirect
160. C2×C5×D3C22×D3semidirect
167. C2×C3×D5C22×D3semidirect
168. C2×C3×D5C22×D3semidirect
174. Dic15C22×D3semidirect
176. C5×Dic3C22×D3semidirect
194. C32×C5C22×D4semidirect
203. C22×D5D32semidirect
218. C2×C3⋊D3C22×D5semidirect
222. C2×C3×D3C22×D5semidirect
225. C3×Dic3C22×D5semidirect
230. C2×C3×C5D3×D4semidirect
231. C2×C3×C5D3×D4semidirect
268. C22×D3D3×D5semidirect
278. C2×Dic3D3×D5semidirect
295. C2×D5C2×D32semidirect
299. C2×C32D4×D5semidirect
314. C3×C5C2×D3×D4semidirect
315. C3×C5C2×D3×D4semidirect
349. C2×D3C2×D3×D5semidirect
353. Dic3C2×D3×D5semidirect
356. C2×C5#3811semidirect
361. C32C2×D4×D5semidirect
385. C2×C3#6777semidirect
386. C2×C3#6790semidirect
397. C5C2×#3811semidirect
414. C3C2×#6777semidirect
415. C3C2×#6790semidirect
421. C2#10624030direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×D152
C22×D5×C3⋊D32
C2×#671562
C2×#671492
C2×C5×#6262
C2×C3×#12022
C2×C3×#12012
#106240302
C2×#67903
C2×#67773
C2×#38115