Finite Group Databasebeta

← C5×#71207 #14289997

Subgroups — C5×#71207 #14289997

Order 1920.

500 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5718C211yesmaximal, characteristic
2C3×C5×#486C211yesmaximal, characteristic, complemented
3C5×#5090C211yesmaximal, characteristic, complemented
4C5×#5087C211yesmaximal, characteristic, complemented
5C5×#4544C211yesmaximal, characteristic, complemented
6C5×#4459C211yesmaximal, characteristic, complemented
7C5×#4456C211yesmaximal, characteristic, complemented
8C5×#1641—31nomaximal, Hall
9C2×C5×#237—21no—
10C3×C5×C4⋊Q8C2211yescharacteristic
11C3×C5×#127C2211yescharacteristic
12C2×C5×#216—21no—
13C5×#940—21no—
14C5×#939—22no—
15C5×#938—21no—
16C5×D4×Dic3—21no—
17C4×C5×D12C2211yescharacteristic
18C4×C5×Dic6C2211yescharacteristic
19C3×C5×#102—22no—
20C3×C4×C5×C8C2211yescharacteristic
21C5×#835—21no—
22C5×#809—21no—
23C5×#807—21no—
24C5×C4⋊(C3⋊C8)C2211yescharacteristic
25C5×C4⋊(C3⋊C8)C2211yescharacteristic
26#71207C511yesmaximal, characteristic, direct product factor, Hall
27C5×#550—31no—
28C5×#486D311yescharacteristic, complemented
29C5×#474—31no—
30C5×#462—31no—
31C5×#331—31no—
32C5×Q8⋊C8—31no—
33C5×D4⋊C8—31no—
34C2×C3×C5×Q8—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4D412yes—
38C2×C5×#67—22no—
39C2×C5×#67—22no—
40C22×C5×Dic3—22no—
41C5×#237—42no—
42C5×#237—42no—
43C2×C5×D12D411yescharacteristic
44C2×C4×C5×D3—21no—
45C2×C5×Dic6D411yescharacteristic
46C2×C3×C5×C8—21no—
47C3×C5×C4⋊C4—21no—
48C3×C5×C4⋊C4—21no—
49C3×C42×C5C2311yescharacteristic
50C5×#217—22no—
51C5×#216—42no—
52C5×(C2×D3)⋊C4—21no—
53C5×C4⋊Dic3D411yescharacteristic
54C5×C4⋊Dic3D411yescharacteristic
55C5×Dic3⋊C4—21no—
56C4×C5×Dic3—21no—
57C2×C5×C3⋊C8—21no—
58C2×C5×C3⋊C8—21no—
59#5718C2×C511yescharacteristic
60C3×#486C2×C511yescharacteristic, complemented
61#5090C2×C511yescharacteristic, complemented
62#5087C2×C511yescharacteristic, complemented
63#4544C2×C511yescharacteristic, complemented
64#4459C2×C511yescharacteristic, complemented
65#4456C2×C511yescharacteristic, complemented
66C2×C5×#41—61no—
67C2×C5×Q16—61no—
68C5×C4⋊Q8C2×D311yescharacteristic
69C5×#127—61no—
70C5×#127C2×D311yescharacteristic
71C5×#121—61no—
72C5×#121—62no—
73C4×C5×Q8—31no—
74C4×C5×D4—31no—
75C4×C5×D4—61no—
76C5×#107—61no—
77C5×C4⋊C8—31no—
78C5×C4⋊C8—31no—
79C5×#103—61no—
80C5×#102—22no—
81C5×#102—61no—
82C4×C5×C8C2×D311yescharacteristic
83#1641—31noSylow
84C23×C3×C5—22no—
85C22×C5×D3—21no—
86C3×C5×Q8—41no—
87C3×C5×Q8—41no—
88C3×C5×D4—42no—
89C3×C5×D4—41no—
90C3×C5×D4—41no—
91C3×C5×D4—41no—
92C3×C5×D4—41no—
93C2×C3×C4×C5—41no—
94C2×C3×C4×C5C2×D411yescharacteristic
95C2×C3×C4×C5C2×D411yescharacteristic
96C2×C3×C4×C5C2×D411yescharacteristic
97C5×#67—44no—
98C5×#67—82no—
99C2×C5×Dic3—21no—
100C2×C5×Dic3—21no—
101C2×C5×Dic3—21no—
102C2×C5×Dic3—44no—
103C5×D12—21no—
104C5×D12—21no—
105C4×C5×D3—41no—
106C4×C5×D3—41no—
107C5×Dic6—21no—
108C5×Dic6D812yes—
109C3×C5×C8—41no—
110C5×C3⋊C8—41no—
111C5×C3⋊C8—41no—
112C2×#237—21no—
113C3×C4⋊Q8C22×C511yescharacteristic
114C3×#127C22×C511yescharacteristic
115C2×#216—21no—
116#940—21no—
117#939—22no—
118#938—21no—
119D4×Dic3—21no—
120C4×D12C22×C511yescharacteristic
121C4×Dic6C22×C511yescharacteristic
122C3×#102—22no—
123C3×C4×C8C22×C511yescharacteristic
124#835—21no—
125#809—21no—
126#807—21no—
127C4⋊(C3⋊C8)C22×C511yescharacteristic
128C4⋊(C3⋊C8)C22×C511yescharacteristic
129C5×#41—122no—
130C5×#41—122no—
131C2×C5×Q8—21no—
132C2×C5×Q8—31no—
133C2×C5×D4—21no—
134C2×C5×D4—21no—
135C2×C5×D4—31no—
136C2×C5×D4—62no—
137C2×C5×D4—62no—
138C2×C5×D4#6712yes—
139C22×C4×C5—62no—
140C22×C4×C5—61no—
141C5×Q16—122no—
142C2×C5×C8—21no—
143C2×C5×C8—61no—
144C2×C5×C8—61no—
145C5×C4⋊C4—21no—
146C5×C4⋊C4—21no—
147C5×C4⋊C4—31no—
148C5×C4⋊C4—31no—
149C5×C4⋊C4—61no—
150C5×C22⋊C4—62no—
151C5×C22⋊C4—61no—
152C42×C5—61no—
153C42×C5C22×D311yescharacteristic
154#550—31no—
155#486C5×D311yescharacteristic, complemented
156#474—31no—
157#462—31no—
158#331—31no—
159Q8⋊C8—31no—
160D4⋊C8—31no—
161C22×C3×C5—42no—
162C22×C3×C5—42no—
163C22×C3×C5—42no—
164C22×C3×C5#12011yescharacteristic
165C2×C5×D3—41no—
166C2×C5×D3—41no—
167C2×C5×D3—41no—
168C3×C4×C5—21no—
169C3×C4×C5—81no—
170C3×C4×C5C2×D811yescharacteristic
171C3×C4×C5#13511yescharacteristic
172C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
173C3×C4×C5#13711yescharacteristic
174C5×Dic3—22no—
175C5×Dic3—41no—
176C5×Dic3—41no—
177C2×C3×Q8—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4C5×D412yes—
181C2×#67—22no—
182C2×#67—22no—
183C22×Dic3—22no—
184#237—42no—
185#237—42no—
186C2×D12C5×D411yescharacteristic
187C2×C4×D3—21no—
188C2×Dic6C5×D411yescharacteristic
189C2×C3×C8—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C42C23×C511yescharacteristic
193#217—22no—
194#216—42no—
195(C2×D3)⋊C4—21no—
196C4⋊Dic3C5×D411yescharacteristic
197C4⋊Dic3C5×D411yescharacteristic
198Dic3⋊C4—21no—
199C4×Dic3—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C23×C5—22no—
203C23×C5—61no—
204C5×Q8—32no—
205C5×Q8—41no—
206C5×Q8—41no—
207C5×Q8—61no—
208C5×D4—42no—
209C5×D4—41no—
210C5×D4—41no—
211C5×D4—41no—
212C5×D4—41no—
213C5×D4—61no—
214C5×D4—61no—
215C5×D4—124no—
216C5×D4—242no—
217C2×C4×C5—41no—
218C2×C4×C5—61no—
219C2×C4×C5—61no—
220C2×C4×C5—61no—
221C2×C4×C5—124no—
222C2×C4×C5—121no—
223C2×C4×C5—121no—
224C2×C4×C5D3×D411yescharacteristic
225C2×C4×C5D3×D411yescharacteristic
226C2×C4×C5C2×#6711yescharacteristic
227C5×C8—41no—
228C5×C8—121no—
229C5×C8—121no—
230C2×#41—61no—
231C2×Q16—61no—
232C4⋊Q8C2×C5×D311yescharacteristic
233#127—61no—
234#127C2×C5×D311yescharacteristic
235#121—61no—
236#121—62no—
237C4×Q8—31no—
238C4×D4—31no—
239C4×D4—61no—
240#107—61no—
241C4⋊C8—31no—
242C4⋊C8—31no—
243#103—61no—
244#102—22no—
245#102—61no—
246C4×C8C2×C5×D311yescharacteristic
247C2×C3×C5—82no—
248C2×C3×C5#44711yescharacteristic
249C2×C3×C5#45211yescharacteristic
250C2×C3×C5#45511yescharacteristic
251C5×D3—81no—
252C23×C3—22no—
253C22×D3—21no—
254C3×Q8—41no—
255C3×Q8—41no—
256C3×D4—42no—
257C3×D4—41no—
258C3×D4—41no—
259C3×D4—41no—
260C3×D4—41no—
261C2×C3×C4—41no—
262C2×C3×C4C2×C5×D411yescharacteristic
263C2×C3×C4C2×C5×D411yescharacteristic
264C2×C3×C4C2×C5×D411yescharacteristic
265#67—44no—
266#67—82no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—44no—
271D12—21no—
272D12—21no—
273C4×D3—41no—
274C4×D3—41no—
275Dic6—21no—
276Dic6C5×D812yes—
277C3×C8—41no—
278C3⋊C8—41no—
279C3⋊C8—41no—
280C22×C5—42no—
281C22×C5—42no—
282C22×C5—42no—
283C22×C5—121no—
284C22×C5—121no—
285C22×C5—121no—
286C22×C5#93711yescharacteristic
287C4×C5—21no—
288C4×C5—62no—
289C4×C5—81no—
290C4×C5—121no—
291C4×C5—121no—
292C4×C5D3×D811yescharacteristic
293C4×C5#91111yescharacteristic
294C4×C5#91511yescharacteristic
295C4×C5#91611yescharacteristic
296#41—122no—
297#41—122no—
298C2×Q8—21no—
299C2×Q8—31no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—31no—
303C2×D4—62no—
304C2×D4—62no—
305C2×D4C5×#6712yes—
306C22×C4—62no—
307C22×C4—61no—
308Q16—122no—
309C2×C8—21no—
310C2×C8—61no—
311C2×C8—61no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—31no—
315C4⋊C4—31no—
316C4⋊C4—61no—
317C22⋊C4—62no—
318C22⋊C4—61no—
319C42—61no—
320C42C22×C5×D311yescharacteristic
321C3×C5#164111yescharacteristic, complemented, Hall
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3C5×#12011yescharacteristic
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—41no—
329C3×C4—21no—
330C3×C4—81no—
331C3×C4C2×C5×D811yescharacteristic
332C3×C4C5×#13511yescharacteristic
333C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
334C3×C4C5×#13711yescharacteristic
335Dic3—22no—
336Dic3—41no—
337Dic3—41no—
338C2×C5—82no—
339C2×C5—241no—
340C2×C5#506111yescharacteristic
341C2×C5#515611yescharacteristic
342C2×C5#517311yescharacteristic
343C23—22no—
344C23—61no—
345Q8—32no—
346Q8—41no—
347Q8—41no—
348Q8—61no—
349D4—42no—
350D4—41no—
351D4—41no—
352D4—41no—
353D4—41no—
354D4—61no—
355D4—61no—
356D4—124no—
357D4—242no—
358C2×C4—41no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—124no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4C5×D3×D411yescharacteristic
366C2×C4C5×D3×D411yescharacteristic
367C2×C4C2×C5×#6711yescharacteristic
368C8—41no—
369C8—121no—
370C8—121no—
371C2×C3—82no—
372C2×C3C5×#44711yescharacteristic
373C2×C3C5×#45211yescharacteristic
374C2×C3C5×#45511yescharacteristic
375D3—81no—
376C5#7120711yescharacteristic, direct product factor, Sylow
377C22—42no—
378C22—42no—
379C22—42no—
380C22—121no—
381C22—121no—
382C22—121no—
383C22C5×#93711yescharacteristic
384C4—21no—
385C4—62no—
386C4—81no—
387C4—121no—
388C4—121no—
389C4C5×D3×D811yescharacteristic
390C4C5×#91111yescharacteristic
391C4C5×#91511yescharacteristic
392C4C5×#91611yescharacteristic
393C3C5×#164111yescharacteristic, complemented, Sylow
394C2—82no—
395C2—241no—
396C2C5×#506111yescharacteristic
397C2C5×#515611yescharacteristic
398C2C5×#517311yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC3×C42C23×C5
FrattiniC42C22×C5×D3
FittingC3×C5×#486C2
socleC22×C3×C5#120
prime radicalC5×#5718C2
hypercenterC5×#486D3
hypocenterC3C5×#1641
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C42×C5 ◁ C5×#486

Complements

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

NH ≅ G/Ntype
2. C3×C5×#486C2semidirect
3. C5×#5090C2semidirect
4. C5×#5087C2semidirect
5. C5×#4544C2semidirect
6. C5×#4459C2semidirect
7. C5×#4456C2semidirect
26. #71207C5direct (N × H)
28. C5×#486D3semidirect
60. C3×#486C2×C5semidirect
61. #5090C2×C5semidirect
62. #5087C2×C5semidirect
63. #4544C2×C5semidirect
64. #4459C2×C5semidirect
65. #4456C2×C5semidirect
155. #486C5×D3semidirect
321. C3×C5#1641semidirect
376. C5#71207direct (N × H)
393. C3C5×#1641semidirect

Maximal subgroups

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

subgroupindex
C5×#57182
C3×C5×#4862
C5×#50902
C5×#50872
C5×#45442
C5×#44592
C5×#44562
C5×#16413
#712075