Finite Group Databasebeta

← C5×#70124 #14303073

Subgroups — C5×#70124 #14303073

Order 1920.

616 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5816C211yesmaximal, characteristic
2C5×#5745C211yesmaximal, characteristic
3C2×C3×C5×#101C211yesmaximal, characteristic, complemented
4C2×C5×#836C211yesmaximal, characteristic, complemented
5C5×#5132C211yesmaximal, characteristic
6C5×#4559C211yesmaximal, characteristic, complemented
7C5×#4556C211yesmaximal, characteristic, complemented
8C5×#1915—31nomaximal, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C2×C5×#235C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C2×C3×C5×M16C2211yescharacteristic
13C5×#947—21no—
14C5×#946—22no—
15C5×#944—21no—
16C5×#929C414yes—
17C5×#926C2211yescharacteristic
18C2×C5×#208C2211yescharacteristic
19C2×C5×#208C2211yescharacteristic
20C5×#907—22no—
21C5×D3×M16—22no—
22C5×#901—22no—
23C3×C5×#101—22no—
24C5×#836—22no—
25C5×#824—21no—
26C5×#823—21no—
27#70124C511yesmaximal, characteristic, direct product factor, Hall
28C5×#568—31no—
29C5×#536—31no—
30C5×#413—31no—
31C2×C5×#101—31no—
32C2×C5×#101D311yescharacteristic, complemented
33C5×#344—31no—
34C5×#344—31no—
35C2×C3×C5×Q8—21no—
36C2×C3×C5×Q8—21no—
37C2×C3×C5×Q8—21no—
38C2×C3×C5×Q8—21no—
39C2×C3×C5×Q8—21no—
40C2×C3×C5×Q8—21no—
41C22×C3×C4×C5C2×C412yes—
42C22×C3×C4×C5C2311yescharacteristic
43C2×C5×#67C2×C412yes—
44C5×#235—21no—
45C5×#235—21no—
46C5×#235—21no—
47C5×#235—21no—
48C2×C5×D12D411yescharacteristic
49C2×C4×C5×D3D412yes—
50C2×C5×Dic6D411yescharacteristic
51C3×C5×M16—21no—
52C3×C5×M16—21no—
53C3×C5×M16—21no—
54C3×C5×M16D412yes—
55C2×C3×C5×C8—21no—
56C2×C3×C5×C8—21no—
57C5×#217C2×C412yes—
58C5×(C2×D3)⋊C4—24no—
59C5×C4⋊Dic3—21no—
60C5×Dic3⋊C4—24no—
61C4×C5×Dic3—21no—
62C5×#208—21no—
63C5×#208—21no—
64C5×#208—21no—
65C5×#208D412yes—
66C2×C5×C3⋊C8—21no—
67C2×C5×C3⋊C8—21no—
68C5×ΓL(1,𝔽25)—24no—
69C5×D3×C8—24no—
70#5816C2×C511yescharacteristic
71#5745C2×C511yescharacteristic
72C2×C3×#101C2×C511yescharacteristic, complemented
73C2×#836C2×C511yescharacteristic, complemented
74#5132C2×C511yescharacteristic
75#4559C2×C511yescharacteristic, complemented
76#4556C2×C511yescharacteristic, complemented
77C2×C5×#41—31no—
78C22×C5×Q8C2×D311yescharacteristic
79C5×#131—62no—
80C5×#131—62no—
81C2×C5×M16—31no—
82C2×C5×M16—31no—
83C2×C5×M16—62no—
84C2×C5×M16C2×D311yescharacteristic
85C2×C5×M16C2×D311yescharacteristic
86C5×C4⋊Q8—61no—
87C5×#124—61no—
88C5×#123—34no—
89C5×C22⋊Q8—62no—
90C5×#117—31no—
91C5×#101—22no—
92C5×#101—61no—
93C5×#101—62no—
94C5×#100—61no—
95#1915—31noSylow
96C23×C3×C5C22×C411yescharacteristic
97C22×C5×D3C22⋊C412yes—
98C3×C5×Q8—41no—
99C3×C5×Q8—41no—
100C3×C5×Q8—41no—
101C3×C5×Q8—41no—
102C2×C3×C4×C5—22no—
103C2×C3×C4×C5—22no—
104C2×C3×C4×C5—22no—
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5#4111yescharacteristic
110C2×C3×C4×C5#4111yescharacteristic
111C5×#67—22no—
112C5×#67—22no—
113C2×C5×Dic3—22no—
114C2×C5×Dic3C22⋊C412yes—
115C5×D12—21no—
116C5×D12—21no—
117C4×C5×D3—22no—
118C4×C5×D3—22no—
119C5×Dic6—21no—
120C5×Dic6—21no—
121C3×C5×C8—22no—
122C3×C5×C8—41no—
123C5×C3⋊C8—22no—
124C5×C3⋊C8—41no—
125C22×C3×Q8C22×C511yescharacteristic
126C2×#235C22×C511yescharacteristic
127C2×C3×M16C22×C511yescharacteristic
128C2×C3×M16C22×C511yescharacteristic
129#947—21no—
130#946—22no—
131#944—21no—
132#929C4×C514yes—
133#926C22×C511yescharacteristic
134C2×#208C22×C511yescharacteristic
135C2×#208C22×C511yescharacteristic
136#907—22no—
137D3×M16—22no—
138#901—22no—
139C3×#101—22no—
140#836—22no—
141#824—21no—
142#823—21no—
143C5×#41—61no—
144C5×#41—61no—
145C5×#41—61no—
146C5×#41—61no—
147C2×C5×Q8—21no—
148C2×C5×Q8—21no—
149C2×C5×Q8—21no—
150C2×C5×Q8—21no—
151C2×C5×Q8—21no—
152C2×C5×Q8—21no—
153C2×C5×Q8—31no—
154C2×C5×D4—32no—
155C2×C5×D4—31no—
156C22×C4×C5—32no—
157C22×C4×C5C4×D312yes—
158C22×C4×C5C22×D311yescharacteristic
159C5×M16—21no—
160C5×M16—21no—
161C5×M16—21no—
162C5×M16—32no—
163C5×M16—64no—
164C5×M16—61no—
165C5×M16—61no—
166C5×M16—61no—
167C5×M16#6712yes—
168C2×C5×C8—21no—
169C2×C5×C8—21no—
170C2×C5×C8—64no—
171C2×C5×C8—61no—
172C2×C5×C8—61no—
173C5×C4⋊C4—64no—
174C5×C4⋊C4—61no—
175C5×C22⋊C4—32no—
176C5×C22⋊C4—64no—
177C42×C5—61no—
178#568—31no—
179#536—31no—
180#413—31no—
181C2×#101—31no—
182C2×#101C5×D311yescharacteristic, complemented
183#344—31no—
184#344—31no—
185C22×C3×C5—41no—
186C22×C3×C5C2×C22⋊C411yescharacteristic
187C22×C3×C5C4×D411yescharacteristic
188C22×C3×C5C4×D411yescharacteristic
189C2×C5×D3—22no—
190C2×C5×D3—42no—
191C3×C4×C5—42no—
192C3×C4×C5#12011yescharacteristic
193C3×C4×C5#12111yescharacteristic
194C3×C4×C5#12111yescharacteristic
195C3×C4×C5#12311yescharacteristic
196C5×Dic3—22no—
197C5×Dic3—42no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C22×C3×C4C2×C4×C512yes—
205C22×C3×C4C23×C511yescharacteristic
206C2×#67C2×C4×C512yes—
207#235—21no—
208#235—21no—
209#235—21no—
210#235—21no—
211C2×D12C5×D411yescharacteristic
212C2×C4×D3C5×D412yes—
213C2×Dic6C5×D411yescharacteristic
214C3×M16—21no—
215C3×M16—21no—
216C3×M16—21no—
217C3×M16C5×D412yes—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220#217C2×C4×C512yes—
221(C2×D3)⋊C4—24no—
222C4⋊Dic3—21no—
223Dic3⋊C4—24no—
224C4×Dic3—21no—
225#208—21no—
226#208—21no—
227#208—21no—
228#208C5×D412yes—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—21no—
231ΓL(1,𝔽25)—24no—
232D3×C8—24no—
233C23×C5—32no—
234C23×C5C2×C4×D311yescharacteristic
235C5×Q8—41no—
236C5×Q8—41no—
237C5×Q8—41no—
238C5×Q8—41no—
239C5×Q8—61no—
240C5×Q8—61no—
241C5×D4—62no—
242C5×D4—62no—
243C5×D4—61no—
244C5×D4—61no—
245C2×C4×C5—22no—
246C2×C4×C5—22no—
247C2×C4×C5—22no—
248C2×C4×C5—32no—
249C2×C4×C5—62no—
250C2×C4×C5—62no—
251C2×C4×C5—62no—
252C2×C4×C5#23511yescharacteristic
253C2×C4×C5D3×D411yescharacteristic
254C2×C4×C5D3×D411yescharacteristic
255C2×C4×C5D3×D411yescharacteristic
256C2×C4×C5#23711yescharacteristic
257C2×C4×C5C2×#6711yescharacteristic
258C5×C8—22no—
259C5×C8—41no—
260C5×C8—62no—
261C5×C8—121no—
262C2×#41—31no—
263C22×Q8C2×C5×D311yescharacteristic
264#131—62no—
265#131—62no—
266C2×M16—31no—
267C2×M16—31no—
268C2×M16—62no—
269C2×M16C2×C5×D311yescharacteristic
270C2×M16C2×C5×D311yescharacteristic
271C4⋊Q8—61no—
272#124—61no—
273#123—34no—
274C22⋊Q8—62no—
275#117—31no—
276#101—22no—
277#101—61no—
278#101—62no—
279#100—61no—
280C2×C3×C5—21no—
281C2×C3×C5—21no—
282C2×C3×C5—21no—
283C2×C3×C5#38611yescharacteristic
284C5×D3—42no—
285C23×C3C22×C4×C511yescharacteristic
286C22×D3C5×C22⋊C412yes—
287C3×Q8—41no—
288C3×Q8—41no—
289C3×Q8—41no—
290C3×Q8—41no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4C2×C5×D411yescharacteristic
295C2×C3×C4C2×C5×D411yescharacteristic
296C2×C3×C4C2×C5×D411yescharacteristic
297C2×C3×C4C2×C5×D411yescharacteristic
298C2×C3×C4C5×#4111yescharacteristic
299C2×C3×C4C5×#4111yescharacteristic
300#67—22no—
301#67—22no—
302C2×Dic3—22no—
303C2×Dic3C5×C22⋊C412yes—
304D12—21no—
305D12—21no—
306C4×D3—22no—
307C4×D3—22no—
308Dic6—21no—
309Dic6—21no—
310C3×C8—22no—
311C3×C8—41no—
312C3⋊C8—22no—
313C3⋊C8—41no—
314C22×C5—41no—
315C22×C5—62no—
316C22×C5—122no—
317C22×C5D3×C22⋊C411yescharacteristic
318C22×C5#67⋊C411yescharacteristic
319C22×C5C4×#6711yescharacteristic
320C4×C5—42no—
321C4×C5—62no—
322C4×C5—122no—
323C4×C5#88311yescharacteristic
324C4×C5#88411yescharacteristic
325C4×C5#93711yescharacteristic
326C4×C5#93811yescharacteristic
327#41—61no—
328#41—61no—
329#41—61no—
330#41—61no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—31no—
338C2×D4—32no—
339C2×D4—31no—
340C22×C4—32no—
341C22×C4C4×C5×D312yes—
342C22×C4C22×C5×D311yescharacteristic
343M16—21no—
344M16—21no—
345M16—21no—
346M16—32no—
347M16—64no—
348M16—61no—
349M16—61no—
350M16—61no—
351M16C5×#6712yes—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—64no—
355C2×C8—61no—
356C2×C8—61no—
357C4⋊C4—64no—
358C4⋊C4—61no—
359C22⋊C4—32no—
360C22⋊C4—64no—
361C42—61no—
362C3×C5#191511yescharacteristic, complemented, Hall
363C22×C3—41no—
364C22×C3C2×C5×C22⋊C411yescharacteristic
365C22×C3C4×C5×D411yescharacteristic
366C22×C3C4×C5×D411yescharacteristic
367C2×D3—22no—
368C2×D3—42no—
369C3×C4—42no—
370C3×C4C5×#12011yescharacteristic
371C3×C4C5×#12111yescharacteristic
372C3×C4C5×#12111yescharacteristic
373C3×C4C5×#12311yescharacteristic
374Dic3—22no—
375Dic3—42no—
376C2×C5—21no—
377C2×C5—21no—
378C2×C5—21no—
379C2×C5—122no—
380C2×C5#495611yescharacteristic
381C23—32no—
382C23C2×C4×C5×D311yescharacteristic
383Q8—41no—
384Q8—41no—
385Q8—41no—
386Q8—41no—
387Q8—61no—
388Q8—61no—
389D4—62no—
390D4—62no—
391D4—61no—
392D4—61no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—32no—
397C2×C4—62no—
398C2×C4—62no—
399C2×C4—62no—
400C2×C4C5×#23511yescharacteristic
401C2×C4C5×D3×D411yescharacteristic
402C2×C4C5×D3×D411yescharacteristic
403C2×C4C5×D3×D411yescharacteristic
404C2×C4C5×#23711yescharacteristic
405C2×C4C2×C5×#6711yescharacteristic
406C8—22no—
407C8—41no—
408C8—62no—
409C8—121no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3C5×#38611yescharacteristic
414D3—42no—
415C5#7012411yescharacteristic, direct product factor, Sylow
416C22—41no—
417C22—62no—
418C22—122no—
419C22C5×D3×C22⋊C411yescharacteristic
420C22C5×#67⋊C411yescharacteristic
421C22C4×C5×#6711yescharacteristic
422C4—42no—
423C4—62no—
424C4—122no—
425C4C5×#88311yescharacteristic
426C4C5×#88411yescharacteristic
427C4C5×#93711yescharacteristic
428C4C5×#93811yescharacteristic
429C3C5×#191511yescharacteristic, complemented, Sylow
430C2—21no—
431C2—21no—
432C2—21no—
433C2—122no—
434C2C5×#495611yescharacteristic
subgroupG/H
centerC2×C5#4956
commutatorC23×C3C22×C4×C5
FrattiniC22×C4C22×C5×D3
FittingC2×C3×C5×#101C2
socleC2×C3×C5#386
prime radicalC2×C5×#235C22
cosocleC22×C3×C4C23×C5
hypercenterC2×C5×#101D3
hypocenterC3C5×#1915
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#101

Complements

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

NH ≅ G/Ntype
3. C2×C3×C5×#101C2semidirect
4. C2×C5×#836C2semidirect
6. C5×#4559C2semidirect
7. C5×#4556C2semidirect
27. #70124C5direct (N × H)
32. C2×C5×#101D3semidirect
72. C2×C3×#101C2×C5semidirect
73. C2×#836C2×C5semidirect
75. #4559C2×C5semidirect
76. #4556C2×C5semidirect
182. C2×#101C5×D3semidirect
362. C3×C5#1915semidirect
415. C5#70124direct (N × H)
429. C3C5×#1915semidirect

Maximal subgroups

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

subgroupindex
C5×#58162
C5×#57452
C2×C3×C5×#1012
C2×C5×#8362
C5×#51322
C5×#45592
C5×#45562
C5×#19153
#701245