Finite Group Databasebeta

← C5×#71915 #14289754

Subgroups — C5×#71915 #14289754

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×D4×Q8C211yesmaximal, characteristic
2C5×#5093C211yesmaximal, characteristic, complemented
3C5×#5021C211yesmaximal, characteristic, complemented
4C5×#5008C211yesmaximal, characteristic
5C5×#4547C211yesmaximal, characteristic, complemented
6C5×#4535C211yesmaximal, characteristic, complemented
7C5×D4⋊(C3⋊C8)C211yesmaximal, characteristic
8C5×#1637—31nomaximal, Hall
9C22×C3×C5×Q8—21no—
10C3×C5×C4⋊Q8—21no—
11C3×C5×C4⋊Q8C2211yescharacteristic
12C3×C5×C22⋊Q8—21no—
13C3×C5×C22⋊Q8—22no—
14C3×C4×C5×Q8C2211yescharacteristic
15C3×C4×C5×D4—21no—
16C3×C4×C5×D4C2211yescharacteristic
17C5×#869C2211yescharacteristic
18C5×#835—21no—
19C5×#832—21no—
20C5×#810—22no—
21C5×#808—21no—
22C5×#807—21no—
23C5×C4⋊(C3⋊C8)C2211yescharacteristic
24C5×C4⋊(C3⋊C8)C2211yescharacteristic
25C4×C5×C3⋊C8C2211yescharacteristic
26#71915C511yesmaximal, characteristic, direct product factor, Hall
27C5×D4×Q8D311yescharacteristic
28C5×#487—31no—
29C5×#477—31no—
30C5×#476—31no—
31C5×#332—31no—
32C5×Q8⋊C8—31no—
33C5×D4⋊C8—31no—
34C2×C3×C5×Q8—22no—
35C2×C3×C5×Q8—21no—
36C2×C3×C5×Q8—42no—
37C2×C3×C5×Q8—42no—
38C2×C3×C5×Q8D411yescharacteristic
39C2×C3×C5×D4D411yescharacteristic
40C22×C3×C4×C5—22no—
41C22×C3×C4×C5—21no—
42C2×C5×Dic6—21no—
43C3×C5×C4⋊C4—22no—
44C3×C5×C4⋊C4—21no—
45C3×C5×C4⋊C4—21no—
46C3×C5×C4⋊C4D411yescharacteristic
47C3×C5×C4⋊C4D411yescharacteristic
48C3×C5×C4⋊C4D412yes—
49C3×C5×C22⋊C4—22no—
50C3×C5×C22⋊C4—21no—
51C3×C42×C5—21no—
52C3×C42×C5C2311yescharacteristic
53C5×C4⋊Dic3—21no—
54C5×C4⋊Dic3—21no—
55C2×C5×C3⋊C8—21no—
56C2×C5×C3⋊C8—21no—
57C2×C5×C3⋊C8—21no—
58C3×D4×Q8C2×C511yescharacteristic
59#5093C2×C511yescharacteristic, complemented
60#5021C2×C511yescharacteristic, complemented
61#5008C2×C511yescharacteristic
62#4547C2×C511yescharacteristic, complemented
63#4535C2×C511yescharacteristic, complemented
64D4⋊(C3⋊C8)C2×C511yescharacteristic
65C22×C5×Q8—21no—
66C5×C4⋊Q8—21no—
67C5×C4⋊Q8—31no—
68C5×C4⋊Q8C2×D311yescharacteristic
69C5×C22⋊Q8—21no—
70C5×C22⋊Q8—22no—
71C4×C5×Q8C2×D311yescharacteristic
72C4×C5×D4—21no—
73C4×C5×D4C2×D311yescharacteristic
74C5×#107—61no—
75C5×#106—61no—
76C5×C4⋊C8—31no—
77C5×C4⋊C8—31no—
78C5×#103—61no—
79C5×#103—62no—
80C5×#102—61no—
81C4×C5×C8—31no—
82#1637—31noSylow
83C23×C3×C5—21no—
84C3×C5×Q8—21no—
85C3×C5×Q8—44no—
86C3×C5×Q8—41no—
87C3×C5×Q8—41no—
88C3×C5×Q8SD1612yes—
89C3×C5×D4—21no—
90C3×C5×D4Q1612yes—
91C2×C3×C4×C5—21no—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—22no—
94C2×C3×C4×C5—21no—
95C2×C3×C4×C5—44no—
96C2×C3×C4×C5—41no—
97C2×C3×C4×C5—41no—
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C5×Dic3—41no—
102C5×Dic6—42no—
103C5×C3⋊C8—41no—
104C5×C3⋊C8—41no—
105C5×C3⋊C8—41no—
106C22×C3×Q8—21no—
107C3×C4⋊Q8—21no—
108C3×C4⋊Q8C22×C511yescharacteristic
109C3×C22⋊Q8—21no—
110C3×C22⋊Q8—22no—
111C3×C4×Q8C22×C511yescharacteristic
112C3×C4×D4—21no—
113C3×C4×D4C22×C511yescharacteristic
114#869C22×C511yescharacteristic
115#835—21no—
116#832—21no—
117#810—22no—
118#808—21no—
119#807—21no—
120C4⋊(C3⋊C8)C22×C511yescharacteristic
121C4⋊(C3⋊C8)C22×C511yescharacteristic
122C4×C3⋊C8C22×C511yescharacteristic
123C2×C5×Q8—22no—
124C2×C5×Q8—21no—
125C2×C5×Q8—42no—
126C2×C5×Q8—42no—
127C2×C5×Q8—61no—
128C2×C5×Q8#6711yescharacteristic
129C2×C5×D4#6711yescharacteristic
130C22×C4×C5—22no—
131C22×C4×C5—21no—
132C2×C5×C8—61no—
133C2×C5×C8—61no—
134C2×C5×C8—61no—
135C5×C4⋊C4—22no—
136C5×C4⋊C4—21no—
137C5×C4⋊C4—21no—
138C5×C4⋊C4—61no—
139C5×C4⋊C4—61no—
140C5×C4⋊C4#6711yescharacteristic
141C5×C4⋊C4#6711yescharacteristic
142C5×C4⋊C4#6712yes—
143C5×C22⋊C4—22no—
144C5×C22⋊C4—21no—
145C42×C5—21no—
146C42×C5C22×D311yescharacteristic
147D4×Q8C5×D311yescharacteristic
148#487—31no—
149#477—31no—
150#476—31no—
151#332—31no—
152Q8⋊C8—31no—
153D4⋊C8—31no—
154C22×C3×C5—22no—
155C22×C3×C5—41no—
156C22×C3×C5—41no—
157C22×C3×C5#12011yescharacteristic
158C3×C4×C5—22no—
159C3×C4×C5—21no—
160C3×C4×C5—42no—
161C3×C4×C5—41no—
162C3×C4×C5—41no—
163C3×C4×C5C2×SD1611yescharacteristic
164C3×C4×C5C2×Q1611yescharacteristic
165C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
166C3×C4×C5#13711yescharacteristic
167C5×Dic3—81no—
168C2×C3×Q8—22no—
169C2×C3×Q8—21no—
170C2×C3×Q8—42no—
171C2×C3×Q8—42no—
172C2×C3×Q8C5×D411yescharacteristic
173C2×C3×D4C5×D411yescharacteristic
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C2×Dic6—21no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4C5×D411yescharacteristic
181C3×C4⋊C4C5×D411yescharacteristic
182C3×C4⋊C4C5×D412yes—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—21no—
185C3×C42—21no—
186C3×C42C23×C511yescharacteristic
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189C2×C3⋊C8—21no—
190C2×C3⋊C8—21no—
191C2×C3⋊C8—21no—
192C23×C5—21no—
193C5×Q8—21no—
194C5×Q8—44no—
195C5×Q8—41no—
196C5×Q8—41no—
197C5×Q8—122no—
198C5×Q8#21412yes—
199C5×D4—21no—
200C5×D4#21612yes—
201C2×C4×C5—21no—
202C2×C4×C5—21no—
203C2×C4×C5—22no—
204C2×C4×C5—21no—
205C2×C4×C5—44no—
206C2×C4×C5—41no—
207C2×C4×C5—41no—
208C2×C4×C5—121no—
209C2×C4×C5C2×#6711yescharacteristic
210C2×C4×C5C2×#6711yescharacteristic
211C2×C4×C5C2×#6711yescharacteristic
212C5×C8—121no—
213C5×C8—121no—
214C5×C8—121no—
215C22×Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—31no—
218C4⋊Q8C2×C5×D311yescharacteristic
219C22⋊Q8—21no—
220C22⋊Q8—22no—
221C4×Q8C2×C5×D311yescharacteristic
222C4×D4—21no—
223C4×D4C2×C5×D311yescharacteristic
224#107—61no—
225#106—61no—
226C4⋊C8—31no—
227C4⋊C8—31no—
228#103—61no—
229#103—62no—
230#102—61no—
231C4×C8—31no—
232C2×C3×C5—42no—
233C2×C3×C5#45011yescharacteristic
234C2×C3×C5#45111yescharacteristic
235C2×C3×C5#45411yescharacteristic
236C23×C3—21no—
237C3×Q8—21no—
238C3×Q8—44no—
239C3×Q8—41no—
240C3×Q8—41no—
241C3×Q8C5×SD1612yes—
242C3×D4—21no—
243C3×D4C5×Q1612yes—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—22no—
247C2×C3×C4—21no—
248C2×C3×C4—44no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4C2×C5×D411yescharacteristic
252C2×C3×C4C2×C5×D411yescharacteristic
253C2×C3×C4C2×C5×D411yescharacteristic
254C2×Dic3—41no—
255Dic6—42no—
256C3⋊C8—41no—
257C3⋊C8—41no—
258C3⋊C8—41no—
259C22×C5—22no—
260C22×C5—41no—
261C22×C5—41no—
262C22×C5#95311yescharacteristic
263C4×C5—22no—
264C4×C5—21no—
265C4×C5—42no—
266C4×C5—41no—
267C4×C5—41no—
268C4×C5—241no—
269C4×C5#93211yescharacteristic
270C4×C5C2×#21411yescharacteristic
271C4×C5#94211yescharacteristic
272C4×C5C2×#21611yescharacteristic
273C2×Q8—22no—
274C2×Q8—21no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—61no—
278C2×Q8C5×#6711yescharacteristic
279C2×D4C5×#6711yescharacteristic
280C22×C4—22no—
281C22×C4—21no—
282C2×C8—61no—
283C2×C8—61no—
284C2×C8—61no—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4C5×#6712yes—
291C4⋊C4C5×#6711yescharacteristic
292C4⋊C4C5×#6711yescharacteristic
293C22⋊C4—22no—
294C22⋊C4—21no—
295C42—21no—
296C42C22×C5×D311yescharacteristic
297C3×C5#163711yescharacteristic, complemented, Hall
298C22×C3—22no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3C5×#12011yescharacteristic
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—42no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4C2×C5×SD1611yescharacteristic
308C3×C4C2×C5×Q1611yescharacteristic
309C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
310C3×C4C5×#13711yescharacteristic
311Dic3—81no—
312C2×C5—42no—
313C2×C5#521811yescharacteristic
314C2×C5#522811yescharacteristic
315C2×C5#524511yescharacteristic
316C23—21no—
317Q8—21no—
318Q8—44no—
319Q8—41no—
320Q8—41no—
321Q8—122no—
322Q8C5×#21412yes—
323D4—21no—
324D4C5×#21612yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—121no—
333C2×C4C2×C5×#6711yescharacteristic
334C2×C4C2×C5×#6711yescharacteristic
335C2×C4C2×C5×#6711yescharacteristic
336C8—121no—
337C8—121no—
338C8—121no—
339C2×C3—42no—
340C2×C3C5×#45011yescharacteristic
341C2×C3C5×#45111yescharacteristic
342C2×C3C5×#45411yescharacteristic
343C5#7191511yescharacteristic, direct product factor, Sylow
344C22—22no—
345C22—41no—
346C22—41no—
347C22C5×#95311yescharacteristic
348C4—22no—
349C4—21no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—241no—
354C4C5×#93211yescharacteristic
355C4C2×C5×#21411yescharacteristic
356C4C5×#94211yescharacteristic
357C4C2×C5×#21611yescharacteristic
358C3C5×#163711yescharacteristic, complemented, Sylow
359C2—42no—
360C2C5×#521811yescharacteristic
361C2C5×#522811yescharacteristic
362C2C5×#524511yescharacteristic
subgroupG/H
centerC22×C5#953
commutator, cosocleC3×C42C23×C5
FrattiniC42C22×C5×D3
FittingC3×C5×D4×Q8C2
socleC22×C3×C5#120
prime radicalC2×C3×C5×D4D4
hypercenterC5×D4×Q8D3
hypocenterC3C5×#1637
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C42×C5 ◁ C5×D4×Q8

Complements

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

NH ≅ G/Ntype
2. C5×#5093C2semidirect
3. C5×#5021C2semidirect
5. C5×#4547C2semidirect
6. C5×#4535C2semidirect
26. #71915C5direct (N × H)
59. #5093C2×C5semidirect
60. #5021C2×C5semidirect
62. #4547C2×C5semidirect
63. #4535C2×C5semidirect
297. C3×C5#1637semidirect
343. C5#71915direct (N × H)
358. C3C5×#1637semidirect

Maximal subgroups

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

subgroupindex
C3×C5×D4×Q82
C5×#50932
C5×#50212
C5×#50082
C5×#45472
C5×#45352
C5×D4⋊(C3⋊C8)2
C5×#16373
#719155