Finite Group Databasebeta

← C5×#69153 #14316685

Subgroups — C5×#69153 #14316685

Order 1920.

516 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C5×#5679C211yesmaximal, characteristic, complemented
2C5×#5061C211yesmaximal, characteristic, complemented
3C5×#4742C211yesmaximal, characteristic, complemented
4C5×#4741C211yesmaximal, characteristic
5C3×C5×#354C211yesmaximal, characteristic, complemented
6C5×#4472C211yesmaximal, characteristic, complemented
7C5×#4471C211yesmaximal, characteristic, complemented
8C5×#2197—31nomaximal, Hall
9C5×#1004C2211yescharacteristic
10C3×C5×#124C2211yescharacteristic
11C5×#946—21no—
12C5×#942—21no—
13C5×#937C2211yescharacteristic
14C5×#935C2211yescharacteristic
15C5×#886—21no—
16C5×#883—21no—
17C5×#882—21no—
18C5×#67⋊C4—21no—
19C5×D3×C22⋊C4—21no—
20C5×#879—21no—
21C5×#872—21no—
22C3×C5×#99C2211yescharacteristic
23C3×C5×#99C2211yescharacteristic
24C5×#833C2211yescharacteristic
25C5×#806—21no—
26C5×#806—21no—
27C5×#805—21no—
28#69153C511yesmaximal, characteristic, direct product factor, Hall
29C5×#551—31no—
30C5×#458—31no—
31C5×B(4,𝔽2)—31no—
32C5×#455—31no—
33C5×#354D311yescharacteristic, complemented
34C5×#352—31no—
35C5×AΣL(1,𝔽16)—31no—
36C23×C5×D3D411yescharacteristic
37C2×C3×C5×Q8D411yescharacteristic, complemented
38C2×C3×C5×D4C2311yescharacteristic
39C2×C5×#67—21no—
40C2×C5×#67—21no—
41C2×C5×#67—21no—
42C22×C5×Dic3D411yescharacteristic, complemented
43C5×#237—41no—
44C5×D3×D4—41no—
45C5×#235—21no—
46C2×C5×D12—41no—
47C2×C4×C5×D3—21no—
48C2×C4×C5×D3—21no—
49C3×C5×C22⋊C4—21no—
50C3×C5×C22⋊C4—21no—
51C3×C5×C22⋊C4—21no—
52C3×C5×C22⋊C4—21no—
53C3×C42×C5D411yescharacteristic, complemented
54C5×#217—21no—
55C5×#217D411yescharacteristic
56C5×#217D411yescharacteristic, complemented
57C5×#216—41no—
58C5×#215—41no—
59C5×(C2×D3)⋊C4—21no—
60C5×(C2×D3)⋊C4—21no—
61C5×(C2×D3)⋊C4—21no—
62C5×(C2×D3)⋊C4—41no—
63C5×(C2×D3)⋊C4—41no—
64C5×C4⋊Dic3—21no—
65C5×C4⋊Dic3—41no—
66C5×Dic3⋊C4—21no—
67C5×Dic3⋊C4—21no—
68C5×Dic3⋊C4—21no—
69C4×C5×Dic3—21no—
70C5×#208—21no—
71#5679C2×C511yescharacteristic, complemented
72#5061C2×C511yescharacteristic, complemented
73#4742C2×C511yescharacteristic, complemented
74#4741C2×C511yescharacteristic
75C3×#354C2×C511yescharacteristic, complemented
76#4472C2×C511yescharacteristic, complemented
77#4471C2×C511yescharacteristic, complemented
78C5×H(5,𝔽2)—31no—
79C5×#137—61no—
80C5×#126—61no—
81C5×#124C2×D311yescharacteristic
82C5×#123—31no—
83C5×#123—61no—
84C5×#123—61no—
85C5×C22⋊Q8—61no—
86C5×#120—31no—
87C5×#120—61no—
88C4×C5×D4—61no—
89C5×#117—61no—
90C2×C5×C22⋊C4—61no—
91C5×#104—61no—
92C5×#100—31no—
93C5×#99—61no—
94C5×#99—61no—
95C5×#99C2×D311yescharacteristic
96C5×#99C2×D311yescharacteristic
97#2197—31noSylow
98C23×C3×C5C2×D411yescharacteristic
99C23×C3×C5C2×D411yescharacteristic
100C22×C5×D3—21no—
101C22×C5×D3—21no—
102C22×C5×D3—21no—
103C22×C5×D3—42no—
104C22×C5×D3—41no—
105C3×C5×Q8—41no—
106C3×C5×D4—41no—
107C2×C3×C4×C5—21no—
108C2×C3×C4×C5—21no—
109C2×C3×C4×C5—41no—
110C2×C3×C4×C5—41no—
111C2×C3×C4×C5C2×D411yescharacteristic
112C5×#67—42no—
113C5×#67—41no—
114C5×#67—41no—
115C5×#67—41no—
116C2×C5×Dic3—21no—
117C2×C5×Dic3—21no—
118C2×C5×Dic3—21no—
119C2×C5×Dic3—21no—
120C2×C5×Dic3—21no—
121C2×C5×Dic3—41no—
122C5×D12—21no—
123C5×D12—82no—
124C4×C5×D3—41no—
125C4×C5×D3—42no—
126C4×C5×D3—81no—
127C5×Dic6—21no—
128C5×C3⋊C8—41no—
129#1004C22×C511yescharacteristic
130C3×#124C22×C511yescharacteristic
131#946—21no—
132#942—21no—
133#937C22×C511yescharacteristic
134#935C22×C511yescharacteristic
135#886—21no—
136#883—21no—
137#882—21no—
138#67⋊C4—21no—
139D3×C22⋊C4—21no—
140#879—21no—
141#872—21no—
142C3×#99C22×C511yescharacteristic
143C3×#99C22×C511yescharacteristic
144#833C22×C511yescharacteristic
145#806—21no—
146#806—21no—
147#805—21no—
148C24×C5—31no—
149C5×#41—61no—
150C5×#41—121no—
151C2×C5×Q8D1211yescharacteristic, complemented
152C2×C5×D4—61no—
153C2×C5×D4—61no—
154C2×C5×D4—61no—
155C2×C5×D4—121no—
156C2×C5×D4—121no—
157C2×C5×D4C22×D311yescharacteristic
158C22×C4×C5—31no—
159C22×C4×C5—61no—
160C22×C4×C5—61no—
161C5×Q16—121no—
162C5×SD16—121no—
163C5×M16—61no—
164C5×C4⋊C4—61no—
165C5×C4⋊C4—61no—
166C5×C4⋊C4—61no—
167C5×C4⋊C4—61no—
168C5×C4⋊C4—121no—
169C5×C22⋊C4—21no—
170C5×C22⋊C4—21no—
171C5×C22⋊C4—21no—
172C5×C22⋊C4—21no—
173C5×C22⋊C4—31no—
174C5×C22⋊C4—31no—
175C5×C22⋊C4—61no—
176C5×C22⋊C4—61no—
177C5×C22⋊C4—61no—
178C5×C22⋊C4—61no—
179C5×C22⋊C4—121no—
180C5×C22⋊C4—121no—
181C42×C5—61no—
182C42×C5D1211yescharacteristic, complemented
183#551—31no—
184#458—31no—
185B(4,𝔽2)—31no—
186#455—31no—
187#354C5×D311yescharacteristic, complemented
188#352—31no—
189AΣL(1,𝔽16)—31no—
190C22×C3×C5—21no—
191C22×C3×C5—21no—
192C22×C3×C5—41no—
193C22×C3×C5—41no—
194C22×C3×C5#12011yescharacteristic
195C2×C5×D3—22no—
196C2×C5×D3—41no—
197C2×C5×D3—41no—
198C2×C5×D3—42no—
199C2×C5×D3—41no—
200C2×C5×D3—81no—
201C2×C5×D3—81no—
202C2×C5×D3—81no—
203C3×C4×C5—21no—
204C3×C4×C5—41no—
205C3×C4×C5—41no—
206C3×C4×C5—81no—
207C3×C4×C5—81no—
208C5×Dic3—41no—
209C5×Dic3—41no—
210C5×Dic3—41no—
211C5×Dic3—41no—
212C23×D3C5×D411yescharacteristic
213C2×C3×Q8C5×D411yescharacteristic, complemented
214C2×C3×D4C23×C511yescharacteristic
215C2×#67—21no—
216C2×#67—21no—
217C2×#67—21no—
218C22×Dic3C5×D411yescharacteristic, complemented
219#237—41no—
220D3×D4—41no—
221#235—21no—
222C2×D12—41no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C3×C22⋊C4—21no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—21no—
228C3×C22⋊C4—21no—
229C3×C42C5×D411yescharacteristic, complemented
230#217—21no—
231#217C5×D411yescharacteristic
232#217C5×D411yescharacteristic, complemented
233#216—41no—
234#215—41no—
235(C2×D3)⋊C4—21no—
236(C2×D3)⋊C4—21no—
237(C2×D3)⋊C4—21no—
238(C2×D3)⋊C4—41no—
239(C2×D3)⋊C4—41no—
240C4⋊Dic3—21no—
241C4⋊Dic3—41no—
242Dic3⋊C4—21no—
243Dic3⋊C4—21no—
244Dic3⋊C4—21no—
245C4×Dic3—21no—
246#208—21no—
247C23×C5—61no—
248C23×C5—61no—
249C23×C5—61no—
250C23×C5—121no—
251C23×C5—122no—
252C23×C5D3×D411yescharacteristic
253C23×C5D3×D411yescharacteristic
254C5×Q8—41no—
255C5×Q8—61no—
256C5×D4—41no—
257C5×D4—61no—
258C5×D4—121no—
259C5×D4—121no—
260C5×D4—122no—
261C5×D4—121no—
262C5×D4—242no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—41no—
266C2×C4×C5—41no—
267C2×C4×C5—61no—
268C2×C4×C5—61no—
269C2×C4×C5—61no—
270C2×C4×C5—61no—
271C2×C4×C5—61no—
272C2×C4×C5—121no—
273C2×C4×C5—122no—
274C2×C4×C5—121no—
275C2×C4×C5—241no—
276C2×C4×C5C2×D1211yescharacteristic
277C5×C8—121no—
278H(5,𝔽2)—31no—
279#137—61no—
280#126—61no—
281#124C2×C5×D311yescharacteristic
282#123—31no—
283#123—61no—
284#123—61no—
285C22⋊Q8—61no—
286#120—31no—
287#120—61no—
288C4×D4—61no—
289#117—61no—
290C2×C22⋊C4—61no—
291#104—61no—
292#100—31no—
293#99—61no—
294#99—61no—
295#99C2×C5×D311yescharacteristic
296#99C2×C5×D311yescharacteristic
297C2×C3×C5—21no—
298C2×C3×C5—41no—
299C2×C3×C5—41no—
300C2×C3×C5B(4,𝔽2)11yescharacteristic
301C5×D3—42no—
302C5×D3—81no—
303C23×C3C2×C5×D411yescharacteristic
304C23×C3C2×C5×D411yescharacteristic
305C22×D3—21no—
306C22×D3—21no—
307C22×D3—21no—
308C22×D3—42no—
309C22×D3—41no—
310C3×Q8—41no—
311C3×D4—41no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4C2×C5×D411yescharacteristic
317#67—42no—
318#67—41no—
319#67—41no—
320#67—41no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—41no—
327D12—21no—
328D12—82no—
329C4×D3—41no—
330C4×D3—42no—
331C4×D3—81no—
332Dic6—21no—
333C3⋊C8—41no—
334C22×C5—21no—
335C22×C5—21no—
336C22×C5—41no—
337C22×C5—41no—
338C22×C5—62no—
339C22×C5—121no—
340C22×C5—122no—
341C22×C5—121no—
342C22×C5—121no—
343C22×C5—241no—
344C22×C5—241no—
345C22×C5—241no—
346C22×C5#88211yescharacteristic
347C4×C5—21no—
348C4×C5—41no—
349C4×C5—41no—
350C4×C5—81no—
351C4×C5—81no—
352C4×C5—121no—
353C4×C5—121no—
354C4×C5—121no—
355C4×C5—121no—
356C24—31no—
357#41—61no—
358#41—121no—
359C2×Q8C5×D1211yescharacteristic, complemented
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—121no—
364C2×D4—121no—
365C2×D4C22×C5×D311yescharacteristic
366C22×C4—31no—
367C22×C4—61no—
368C22×C4—61no—
369Q16—121no—
370SD16—121no—
371M16—61no—
372C4⋊C4—61no—
373C4⋊C4—61no—
374C4⋊C4—61no—
375C4⋊C4—61no—
376C4⋊C4—121no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—21no—
380C22⋊C4—21no—
381C22⋊C4—31no—
382C22⋊C4—31no—
383C22⋊C4—61no—
384C22⋊C4—61no—
385C22⋊C4—61no—
386C22⋊C4—61no—
387C22⋊C4—121no—
388C22⋊C4—121no—
389C42—61no—
390C42C5×D1211yescharacteristic, complemented
391C3×C5#219711yescharacteristic, complemented, Hall
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3C5×#12011yescharacteristic
397C2×D3—22no—
398C2×D3—41no—
399C2×D3—41no—
400C2×D3—42no—
401C2×D3—41no—
402C2×D3—81no—
403C2×D3—81no—
404C2×D3—81no—
405C3×C4—21no—
406C3×C4—41no—
407C3×C4—41no—
408C3×C4—81no—
409C3×C4—81no—
410Dic3—41no—
411Dic3—41no—
412Dic3—41no—
413Dic3—41no—
414C2×C5—21no—
415C2×C5—41no—
416C2×C5—41no—
417C2×C5—122no—
418C2×C5—241no—
419C2×C5#474111yescharacteristic
420C23—61no—
421C23—61no—
422C23—61no—
423C23—121no—
424C23—122no—
425C23C5×D3×D411yescharacteristic
426C23C5×D3×D411yescharacteristic
427Q8—41no—
428Q8—61no—
429D4—41no—
430D4—61no—
431D4—121no—
432D4—121no—
433D4—122no—
434D4—121no—
435D4—242no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—121no—
446C2×C4—122no—
447C2×C4—121no—
448C2×C4—241no—
449C2×C4C2×C5×D1211yescharacteristic
450C8—121no—
451C2×C3—21no—
452C2×C3—41no—
453C2×C3—41no—
454C2×C3C5×B(4,𝔽2)11yescharacteristic
455D3—42no—
456D3—81no—
457C5#6915311yescharacteristic, direct product factor, Sylow
458C22—21no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—62no—
463C22—121no—
464C22—122no—
465C22—121no—
466C22—121no—
467C22—241no—
468C22—241no—
469C22—241no—
470C22C5×#88211yescharacteristic
471C4—21no—
472C4—41no—
473C4—41no—
474C4—81no—
475C4—81no—
476C4—121no—
477C4—121no—
478C4—121no—
479C4—121no—
480C3C5×#219711yescharacteristic, complemented, Sylow
481C2—21no—
482C2—41no—
483C2—41no—
484C2—122no—
485C2—241no—
486C2C5×#474111yescharacteristic
subgroupG/H
centerC2×C5#4741
commutator, cosocleC2×C3×D4C23×C5
FrattiniC2×D4C22×C5×D3
FittingC3×C5×#354C2
socleC2×C3×C5B(4,𝔽2)
prime radicalC5×#4741C2
hypercenterC5×#354D3
hypocenterC3C5×#2197
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C5×D4 ◁ C5×#354

Complements

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

NH ≅ G/Ntype
1. C5×#5679C2semidirect
2. C5×#5061C2semidirect
3. C5×#4742C2semidirect
5. C3×C5×#354C2semidirect
6. C5×#4472C2semidirect
7. C5×#4471C2semidirect
28. #69153C5direct (N × H)
33. C5×#354D3semidirect
37. C2×C3×C5×Q8D4semidirect
42. C22×C5×Dic3D4semidirect
53. C3×C42×C5D4semidirect
56. C5×#217D4semidirect
71. #5679C2×C5semidirect
72. #5061C2×C5semidirect
73. #4742C2×C5semidirect
75. C3×#354C2×C5semidirect
76. #4472C2×C5semidirect
77. #4471C2×C5semidirect
151. C2×C5×Q8D12semidirect
182. C42×C5D12semidirect
187. #354C5×D3semidirect
213. C2×C3×Q8C5×D4semidirect
218. C22×Dic3C5×D4semidirect
229. C3×C42C5×D4semidirect
232. #217C5×D4semidirect
359. C2×Q8C5×D12semidirect
390. C42C5×D12semidirect
391. C3×C5#2197semidirect
457. C5#69153direct (N × H)
480. C3C5×#2197semidirect

Maximal subgroups

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

subgroupindex
C5×#56792
C5×#50612
C5×#47422
C5×#47412
C3×C5×#3542
C5×#44722
C5×#44712
C5×#21973
#691535