Finite Group Databasebeta

← C5×#70038 #14309030

Subgroups — C5×#70038 #14309030

Order 1920.

488 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5517C211yesmaximal, characteristic, complemented
2C3×C5×#410C211yesmaximal, characteristic
3C5×#5236C211yesmaximal, characteristic
4C2×C5×#837C211yesmaximal, characteristic
5C5×#4533C211yesmaximal, characteristic, complemented
6C5×#4532C211yesmaximal, characteristic, complemented
7C5×#4531C211yesmaximal, characteristic, complemented
8C5×#2031—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C3×C5×#117C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12C5×#926C2211yescharacteristic
13C2×C4×C5×Dic3C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C2×C5×#208C2211yescharacteristic
16C5×#890—21no—
17C5×#888—21no—
18C5×#878—22no—
19C4×C5×Dic6—22no—
20C3×C5×#99—22no—
21C5×#837—21no—
22C5×#837—21no—
23C5×#837—22no—
24#70038C511yesmaximal, characteristic, direct product factor, Hall
25C5×#533—31no—
26C5×#421—31no—
27C2×C5×#104—31no—
28C5×#410D311yescharacteristic
29C5×#344—31no—
30C5×#339—31no—
31C5×H(3,ℤ/4)—31no—
32C3×C5×#41—21no—
33C3×C5×#41—21no—
34C3×C5×#41—21no—
35C3×C5×#41—21no—
36C2×C3×C5×Q8Q811yescharacteristic, complemented
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4Q811yescharacteristic, complemented
39C22×C3×C4×C5—21no—
40C22×C3×C4×C5C2311yescharacteristic
41C22×C5×Dic3—21no—
42C2×C5×Dic6—22no—
43C3×C5×C4⋊C4—21no—
44C3×C5×C4⋊C4—21no—
45C3×C5×C22⋊C4—21no—
46C3×C5×C22⋊C4D412yes—
47C3×C42×C5—21no—
48C3×C42×C5—21no—
49C5×#217—21no—
50C5×C4⋊Dic3—22no—
51C5×C4⋊Dic3—21no—
52C5×Dic3⋊C4—22no—
53C5×Dic3⋊C4—22no—
54C4×C5×Dic3—21no—
55C4×C5×Dic3D412yes—
56C4×C5×Dic3D411yescharacteristic
57C4×C5×Dic3D411yescharacteristic
58C5×#208—21no—
59C5×#208—21no—
60C5×#208—21no—
61C5×#208—21no—
62C2×C5×C3⋊C8—21no—
63C2×C5×C3⋊C8—21no—
64#5517C2×C511yescharacteristic, complemented
65C3×#410C2×C511yescharacteristic
66#5236C2×C511yescharacteristic
67C2×#837C2×C511yescharacteristic
68#4533C2×C511yescharacteristic, complemented
69#4532C2×C511yescharacteristic, complemented
70#4531C2×C511yescharacteristic, complemented
71C2×C5×#41C2×D311yescharacteristic
72C2×C5×M16—31no—
73C2×C5×M16—31no—
74C5×C4⋊Q8—61no—
75C5×#125—61no—
76C5×C22⋊Q8—62no—
77C4×C5×Q8—62no—
78C5×#117—31no—
79C5×#117C2×D311yescharacteristic
80C5×#117C2×D311yescharacteristic
81C2×C42×C5—31no—
82C5×#104—61no—
83C5×#104—61no—
84C5×#104—62no—
85C5×#99—22no—
86#2031—31noSylow
87C23×C3×C5—21no—
88C23×C3×C5C2×D411yescharacteristic
89C3×C5×Q8—21no—
90C3×C5×Q8—21no—
91C3×C5×D4—21no—
92C3×C5×D4—21no—
93C3×C5×D4—41no—
94C3×C5×D4—41no—
95C2×C3×C4×C5—22no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—41no—
98C2×C3×C4×C5—41no—
99C2×C3×C4×C5—41no—
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×C4×C5C2×D411yescharacteristic
102C2×C3×C4×C5C2×Q811yescharacteristic
103C2×C3×C4×C5#4111yescharacteristic
104C2×C3×C4×C5#4111yescharacteristic
105C2×C3×C4×C5#4111yescharacteristic
106C2×C5×Dic3—21no—
107C2×C5×Dic3—21no—
108C2×C5×Dic3—22no—
109C2×C5×Dic3—42no—
110C5×Dic6—44no—
111C5×C3⋊C8—41no—
112C5×C3⋊C8—41no—
113C2×C3×#41C22×C511yescharacteristic
114C3×#117C22×C511yescharacteristic
115C3×#117C22×C511yescharacteristic
116#926C22×C511yescharacteristic
117C2×C4×Dic3C22×C511yescharacteristic
118C2×#208C22×C511yescharacteristic
119C2×#208C22×C511yescharacteristic
120#890—21no—
121#888—21no—
122#878—22no—
123C4×Dic6—22no—
124C3×#99—22no—
125#837—21no—
126#837—21no—
127#837—22no—
128C5×#41—21no—
129C5×#41—21no—
130C5×#41—21no—
131C5×#41—21no—
132C2×C5×Q8—62no—
133C2×C5×Q8Dic611yescharacteristic, complemented
134C2×C5×D4—21no—
135C2×C5×D4Dic611yescharacteristic, complemented
136C22×C4×C5—21no—
137C22×C4×C5—61no—
138C22×C4×C5C22×D311yescharacteristic
139C5×M16—61no—
140C5×M16—61no—
141C5×M16—61no—
142C5×M16—61no—
143C2×C5×C8—61no—
144C2×C5×C8—61no—
145C5×C4⋊C4—21no—
146C5×C4⋊C4—21no—
147C5×C4⋊C4—62no—
148C5×C4⋊C4—62no—
149C5×C4⋊C4—62no—
150C5×C4⋊C4—61no—
151C5×C22⋊C4—21no—
152C5×C22⋊C4—61no—
153C5×C22⋊C4#6712yes—
154C42×C5—21no—
155C42×C5—21no—
156C42×C5—31no—
157C42×C5—31no—
158C42×C5—32no—
159C42×C5—61no—
160#533—31no—
161#421—31no—
162C2×#104—31no—
163#410C5×D311yescharacteristic
164#344—31no—
165#339—31no—
166H(3,ℤ/4)—31no—
167C22×C3×C5—41no—
168C22×C3×C5—41no—
169C22×C3×C5—81no—
170C22×C3×C5#12011yescharacteristic
171C22×C3×C5C22⋊Q811yescharacteristic
172C22×C3×C5#12411yescharacteristic
173C3×C4×C5—42no—
174C3×C4×C5—41no—
175C3×C4×C5—81no—
176C3×C4×C5C22⋊Q811yescharacteristic
177C3×C4×C5C22⋊Q811yescharacteristic
178C3×C4×C5#12411yescharacteristic
179C3×C4×C5#12411yescharacteristic
180C5×Dic3—22no—
181C5×Dic3—42no—
182C5×Dic3—41no—
183C3×#41—21no—
184C3×#41—21no—
185C3×#41—21no—
186C3×#41—21no—
187C2×C3×Q8C5×Q811yescharacteristic, complemented
188C2×C3×D4—21no—
189C2×C3×D4C5×Q811yescharacteristic, complemented
190C22×C3×C4—21no—
191C22×C3×C4C23×C511yescharacteristic
192C22×Dic3—21no—
193C2×Dic6—22no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C22⋊C4—21no—
197C3×C22⋊C4C5×D412yes—
198C3×C42—21no—
199C3×C42—21no—
200#217—21no—
201C4⋊Dic3—22no—
202C4⋊Dic3—21no—
203Dic3⋊C4—22no—
204Dic3⋊C4—22no—
205C4×Dic3—21no—
206C4×Dic3C5×D412yes—
207C4×Dic3C5×D411yescharacteristic
208C4×Dic3C5×D411yescharacteristic
209#208—21no—
210#208—21no—
211#208—21no—
212#208—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C23×C5—21no—
216C23×C5C2×#6711yescharacteristic
217C5×Q8—21no—
218C5×Q8—21no—
219C5×Q8—124no—
220C5×D4—21no—
221C5×D4—21no—
222C5×D4—41no—
223C5×D4—41no—
224C2×C4×C5—21no—
225C2×C4×C5—22no—
226C2×C4×C5—41no—
227C2×C4×C5—41no—
228C2×C4×C5—41no—
229C2×C4×C5—61no—
230C2×C4×C5—61no—
231C2×C4×C5—62no—
232C2×C4×C5—122no—
233C2×C4×C5C2×Dic611yescharacteristic
234C2×C4×C5#23511yescharacteristic
235C2×C4×C5D3×D411yescharacteristic
236C2×C4×C5D3×D411yescharacteristic
237C2×C4×C5#23711yescharacteristic
238C2×C4×C5#23711yescharacteristic
239C5×C8—121no—
240C5×C8—121no—
241C2×#41C2×C5×D311yescharacteristic
242C2×M16—31no—
243C2×M16—31no—
244C4⋊Q8—61no—
245#125—61no—
246C22⋊Q8—62no—
247C4×Q8—62no—
248#117—31no—
249#117C2×C5×D311yescharacteristic
250#117C2×C5×D311yescharacteristic
251C2×C42—31no—
252#104—61no—
253#104—61no—
254#104—62no—
255#99—22no—
256C2×C3×C5—21no—
257C2×C3×C5—21no—
258C2×C3×C5—21no—
259C2×C3×C5—81no—
260C2×C3×C5#39311yescharacteristic
261C23×C3—21no—
262C23×C3C2×C5×D411yescharacteristic
263C3×Q8—21no—
264C3×Q8—21no—
265C3×D4—21no—
266C3×D4—21no—
267C3×D4—41no—
268C3×D4—41no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4C2×C5×D411yescharacteristic
275C2×C3×C4C2×C5×D411yescharacteristic
276C2×C3×C4C2×C5×Q811yescharacteristic
277C2×C3×C4C5×#4111yescharacteristic
278C2×C3×C4C5×#4111yescharacteristic
279C2×C3×C4C5×#4111yescharacteristic
280C2×Dic3—21no—
281C2×Dic3—21no—
282C2×Dic3—22no—
283C2×Dic3—42no—
284Dic6—44no—
285C3⋊C8—41no—
286C3⋊C8—41no—
287C22×C5—41no—
288C22×C5—41no—
289C22×C5—81no—
290C22×C5#92411yescharacteristic
291C22×C5#93611yescharacteristic
292C22×C5#93711yescharacteristic
293C4×C5—42no—
294C4×C5—41no—
295C4×C5—62no—
296C4×C5—81no—
297C4×C5—121no—
298C4×C5—122no—
299C4×C5#87811yescharacteristic
300C4×C5#87811yescharacteristic
301C4×C5#88511yescharacteristic
302C4×C5#88511yescharacteristic
303#41—21no—
304#41—21no—
305#41—21no—
306#41—21no—
307C2×Q8—62no—
308C2×Q8C5×Dic611yescharacteristic, complemented
309C2×D4—21no—
310C2×D4C5×Dic611yescharacteristic, complemented
311C22×C4—21no—
312C22×C4—61no—
313C22×C4C22×C5×D311yescharacteristic
314M16—61no—
315M16—61no—
316M16—61no—
317M16—61no—
318C2×C8—61no—
319C2×C8—61no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—62no—
323C4⋊C4—62no—
324C4⋊C4—62no—
325C4⋊C4—61no—
326C22⋊C4—21no—
327C22⋊C4—61no—
328C22⋊C4C5×#6712yes—
329C42—21no—
330C42—21no—
331C42—31no—
332C42—31no—
333C42—32no—
334C42—61no—
335C3×C5#203111yescharacteristic, complemented, Hall
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—81no—
339C22×C3C5×#12011yescharacteristic
340C22×C3C5×C22⋊Q811yescharacteristic
341C22×C3C5×#12411yescharacteristic
342C3×C4—42no—
343C3×C4—41no—
344C3×C4—81no—
345C3×C4C5×C22⋊Q811yescharacteristic
346C3×C4C5×C22⋊Q811yescharacteristic
347C3×C4C5×#12411yescharacteristic
348C3×C4C5×#12411yescharacteristic
349Dic3—22no—
350Dic3—42no—
351Dic3—41no—
352C2×C5—21no—
353C2×C5—21no—
354C2×C5—21no—
355C2×C5—81no—
356C2×C5#494711yescharacteristic
357C23—21no—
358C23C2×C5×#6711yescharacteristic
359Q8—21no—
360Q8—21no—
361Q8—124no—
362D4—21no—
363D4—21no—
364D4—41no—
365D4—41no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—62no—
374C2×C4—122no—
375C2×C4C2×C5×Dic611yescharacteristic
376C2×C4C5×#23511yescharacteristic
377C2×C4C5×D3×D411yescharacteristic
378C2×C4C5×D3×D411yescharacteristic
379C2×C4C5×#23711yescharacteristic
380C2×C4C5×#23711yescharacteristic
381C8—121no—
382C8—121no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—81no—
387C2×C3C5×#39311yescharacteristic
388C5#7003811yescharacteristic, direct product factor, Sylow
389C22—41no—
390C22—41no—
391C22—81no—
392C22C5×#92411yescharacteristic
393C22C5×#93611yescharacteristic
394C22C5×#93711yescharacteristic
395C4—42no—
396C4—41no—
397C4—62no—
398C4—81no—
399C4—121no—
400C4—122no—
401C4C5×#87811yescharacteristic
402C4C5×#87811yescharacteristic
403C4C5×#88511yescharacteristic
404C4C5×#88511yescharacteristic
405C3C5×#203111yescharacteristic, complemented, Sylow
406C2—21no—
407C2—21no—
408C2—21no—
409C2—81no—
410C2C5×#494711yescharacteristic
subgroupG/H
centerC4×C5#885
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#410C2
socleC2×C3×C5#393
prime radicalC2×C3×C5×#41C22
hypercenterC5×#410D3
hypocenterC3C5×#2031
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#410

Complements

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

NH ≅ G/Ntype
1. C5×#5517C2semidirect
5. C5×#4533C2semidirect
6. C5×#4532C2semidirect
7. C5×#4531C2semidirect
24. #70038C5direct (N × H)
36. C2×C3×C5×Q8Q8semidirect
38. C2×C3×C5×D4Q8semidirect
64. #5517C2×C5semidirect
68. #4533C2×C5semidirect
69. #4532C2×C5semidirect
70. #4531C2×C5semidirect
133. C2×C5×Q8Dic6semidirect
135. C2×C5×D4Dic6semidirect
187. C2×C3×Q8C5×Q8semidirect
189. C2×C3×D4C5×Q8semidirect
308. C2×Q8C5×Dic6semidirect
310. C2×D4C5×Dic6semidirect
335. C3×C5#2031semidirect
388. C5#70038direct (N × H)
405. C3C5×#2031semidirect

Maximal subgroups

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

subgroupindex
C5×#55172
C3×C5×#4102
C5×#52362
C2×C5×#8372
C5×#45332
C5×#45322
C5×#45312
C5×#20313
#700385