Finite Group Databasebeta

← C5×#85377 #14511115

Subgroups — C5×#85377 #14511115

Order 1920.

422 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 68 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×#548⋊C3C211yesmaximal, characteristic
2C5×#5426C211yesmaximal, characteristic, complemented
3C5×#5425C211yesmaximal, characteristic, complemented
4C5×#5419C211yesmaximal, characteristic, complemented
5C2×C5×#859C211yesmaximal, characteristic, complemented
6C5×#5388C211yesmaximal, characteristic, complemented
7C5×#5386C211yesmaximal, characteristic, complemented
8C5×#3316—31nomaximal, Hall
9C2×C5×#41⋊C3C2211yescharacteristic
10C22×C5×SL(2,𝔽3)C2211yescharacteristic
11C2×C5×#226C2211yescharacteristic
12C5×#878—41nomaximal
13C4×C5×SL(2,𝔽3)C2211yescharacteristic
14C5×#859C2211yescharacteristic
15C5×#859C2211yescharacteristic
16C5×#859C2211yescharacteristic
17#85377C511yesmaximal, characteristic, direct product factor, Hall
18C5×#554—31no—
19C5×#548D311yescharacteristic
20C5×#489—31no—
21C5×#483—31no—
22C5×#481—31no—
23C5×#475—31no—
24C5×#470—31no—
25C5×#465—31no—
26C5×#464—31no—
27C5×#452—31no—
28C5×#451—31no—
29C5×#441—31no—
30C5×#435—31no—
31C5×#417—31no—
32C2×C5×#103—31no—
33C22×C5×Dic3—41no—
34C2×C5×Dic6—41no—
35C5×#41⋊C3—21no—
36C2×C5×SL(2,𝔽3)—21no—
37C2×C5×SL(2,𝔽3)C2311yescharacteristic
38C5×#226—21no—
39C3×C5×C22⋊C4—41no—
40C5×#217—41no—
41C5×C4⋊Dic3—41no—
42C5×Dic3⋊C4—41no—
43C5×Dic3⋊C4—41no—
44#548⋊C3C2×C511yescharacteristic
45#5426C2×C511yescharacteristic, complemented
46#5425C2×C511yescharacteristic, complemented
47#5419C2×C511yescharacteristic, complemented
48C2×#859C2×C511yescharacteristic, complemented
49#5388C2×C511yescharacteristic, complemented
50#5386C2×C511yescharacteristic, complemented
51C2×C5×#41C2×D311yescharacteristic
52C22×C5×Q8C2×D311yescharacteristic
53C2×C5×Q16—31no—
54C2×C5×Q16—31no—
55C2×C5×SD16—31no—
56C2×C5×M16—31no—
57C22×C5×C8—31no—
58C5×C4⋊Q8—31no—
59C5×#125—61no—
60C5×#124—31no—
61C5×C22⋊Q8—31no—
62C5×C22⋊Q8—31no—
63C5×C22⋊Q8—31no—
64C5×C22⋊Q8—61no—
65C5×C22⋊Q8—61no—
66C5×#121—31no—
67C4×C5×Q8—31no—
68C4×C5×Q8C2×D311yescharacteristic
69C4×C5×D4—31no—
70C4×C5×D4—61no—
71C2×C5×C4⋊C4—31no—
72C2×C5×C4⋊C4—31no—
73C5×#106—31no—
74C5×#106—31no—
75C5×#106—31no—
76C5×C4⋊C8—31no—
77C5×#103—31no—
78C5×#103—31no—
79C5×#103—31no—
80C5×#103—31no—
81C5×#103—31no—
82C5×#103—31no—
83C5×#103—31no—
84C5×#102—31no—
85C5×#102—31no—
86C5×#102—31no—
87C5×C22⋊C8—31no—
88C5×C22⋊C8—31no—
89C5×#97—31no—
90#3316—31noSylow
91C23×C3×C5—41no—
92C2×C3×C4×C5—41no—
93C2×C3×C4×C5—41no—
94C2×C5×Dic3—41no—
95C2×C5×Dic3—41no—
96C2×C5×Dic3—41no—
97C2×C5×Dic3—41no—
98C2×C5×Dic3—82no—
99C5×Dic6—82no—
100C5×SL(2,𝔽3)#4111yescharacteristic
101C2×#41⋊C3C22×C511yescharacteristic
102C22×SL(2,𝔽3)C22×C511yescharacteristic
103C2×#226C22×C511yescharacteristic
104#878—41no—
105C4×SL(2,𝔽3)C22×C511yescharacteristic
106#859C22×C511yescharacteristic
107#859C22×C511yescharacteristic
108#859C22×C511yescharacteristic
109C5×#41—21no—
110C5×#41—61no—
111C2×C5×Q8—21no—
112C2×C5×Q8—31no—
113C2×C5×Q8—31no—
114C2×C5×Q8—61no—
115C2×C5×Q8—61no—
116C2×C5×Q8C22×D311yescharacteristic
117C2×C5×D4—31no—
118C2×C5×D4—31no—
119C22×C4×C5—31no—
120C22×C4×C5—31no—
121C22×C4×C5—61no—
122C22×C4×C5—61no—
123C5×Q16—61no—
124C5×Q16—61no—
125C5×Q16—121no—
126C5×SD16—121no—
127C5×M16—121no—
128C2×C5×C8—31no—
129C2×C5×C8—31no—
130C2×C5×C8—31no—
131C2×C5×C8—31no—
132C2×C5×C8—121no—
133C5×C4⋊C4—31no—
134C5×C4⋊C4—31no—
135C5×C4⋊C4—31no—
136C5×C4⋊C4—31no—
137C5×C4⋊C4—31no—
138C5×C4⋊C4—61no—
139C5×C4⋊C4—61no—
140C5×C4⋊C4—61no—
141C5×C4⋊C4—61no—
142C5×C4⋊C4—61no—
143C5×C4⋊C4—61no—
144C5×C22⋊C4—31no—
145C5×C22⋊C4—61no—
146C5×C22⋊C4—61no—
147C5×C22⋊C4—61no—
148C5×C22⋊C4S411yescharacteristic
149C42×C5—31no—
150C42×C5—61no—
151#554—31no—
152#548C5×D311yescharacteristic
153#489—31no—
154#483—31no—
155#481—31no—
156#475—31no—
157#470—31no—
158#465—31no—
159#464—31no—
160#452—31no—
161#451—31no—
162#441—31no—
163#435—31no—
164#417—31no—
165C2×#103—31no—
166C22×C3×C5—42no—
167C22×C3×C5—41no—
168C22×C3×C5—81no—
169C22×C3×C5—81no—
170C3×C4×C5—81no—
171C3×C4×C5—81no—
172C5×Dic3—42no—
173C5×Dic3—81no—
174C5×Dic3—81no—
175C5×Dic3—81no—
176C22×Dic3—41no—
177C2×Dic6—41no—
178#41⋊C3—21no—
179C2×SL(2,𝔽3)—21no—
180C2×SL(2,𝔽3)C23×C511yescharacteristic
181#226—21no—
182C3×C22⋊C4—41no—
183#217—41no—
184C4⋊Dic3—41no—
185Dic3⋊C4—41no—
186Dic3⋊C4—41no—
187C23×C5—31no—
188C23×C5C2×S411yescharacteristic
189C5×Q8—31no—
190C5×Q8—61no—
191C5×Q8—61no—
192C5×Q8—61no—
193C5×Q8—61no—
194C5×Q8—241no—
195C5×Q8#23711yescharacteristic
196C5×D4—61no—
197C5×D4—61no—
198C5×D4—62no—
199C2×C4×C5—31no—
200C2×C4×C5—31no—
201C2×C4×C5—31no—
202C2×C4×C5—61no—
203C2×C4×C5—61no—
204C2×C4×C5—61no—
205C2×C4×C5—61no—
206C2×C4×C5—61no—
207C2×C4×C5—61no—
208C2×C4×C5—61no—
209C2×C4×C5—61no—
210C2×C4×C5—241no—
211C2×C4×C5—241no—
212C2×C4×C5C2×S411yescharacteristic
213C2×C4×C5C2×S411yescharacteristic
214C5×C8—61no—
215C5×C8—61no—
216C5×C8—61no—
217C5×C8—61no—
218C2×#41C2×C5×D311yescharacteristic
219C22×Q8C2×C5×D311yescharacteristic
220C2×Q16—31no—
221C2×Q16—31no—
222C2×SD16—31no—
223C2×M16—31no—
224C22×C8—31no—
225C4⋊Q8—31no—
226#125—61no—
227#124—31no—
228C22⋊Q8—31no—
229C22⋊Q8—31no—
230C22⋊Q8—31no—
231C22⋊Q8—61no—
232C22⋊Q8—61no—
233#121—31no—
234C4×Q8—31no—
235C4×Q8C2×C5×D311yescharacteristic
236C4×D4—31no—
237C4×D4—61no—
238C2×C4⋊C4—31no—
239C2×C4⋊C4—31no—
240#106—31no—
241#106—31no—
242#106—31no—
243C4⋊C8—31no—
244#103—31no—
245#103—31no—
246#103—31no—
247#103—31no—
248#103—31no—
249#103—31no—
250#103—31no—
251#102—31no—
252#102—31no—
253#102—31no—
254C22⋊C8—31no—
255C22⋊C8—31no—
256#97—31no—
257C2×C3×C5—41no—
258C2×C3×C5—41no—
259C2×C3×C5—41no—
260C2×C3×C5—82no—
261C23×C3—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—82no—
269Dic6—82no—
270SL(2,𝔽3)C5×#4111yescharacteristic
271C22×C5—21no—
272C22×C5—21no—
273C22×C5—61no—
274C22×C5—61no—
275C22×C5—61no—
276C22×C5#98412yes—
277C22×C5C22×S411yescharacteristic
278C4×C5—21no—
279C4×C5—21no—
280C4×C5—31no—
281C4×C5—31no—
282C4×C5—61no—
283C4×C5—61no—
284C4×C5—121no—
285C4×C5—121no—
286C4×C5—121no—
287C4×C5—121no—
288#41—21no—
289#41—61no—
290C2×Q8—21no—
291C2×Q8—31no—
292C2×Q8—31no—
293C2×Q8—61no—
294C2×Q8—61no—
295C2×Q8C22×C5×D311yescharacteristic
296C2×D4—31no—
297C2×D4—31no—
298C22×C4—31no—
299C22×C4—31no—
300C22×C4—61no—
301C22×C4—61no—
302Q16—61no—
303Q16—61no—
304Q16—121no—
305SD16—121no—
306M16—121no—
307C2×C8—31no—
308C2×C8—31no—
309C2×C8—31no—
310C2×C8—31no—
311C2×C8—121no—
312C4⋊C4—31no—
313C4⋊C4—31no—
314C4⋊C4—31no—
315C4⋊C4—31no—
316C4⋊C4—31no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C22⋊C4—31no—
324C22⋊C4—61no—
325C22⋊C4—61no—
326C22⋊C4—61no—
327C22⋊C4C5×S411yescharacteristic
328C42—31no—
329C42—61no—
330C3×C5—41noHall
331C22×C3—42no—
332C22×C3—41no—
333C22×C3—81no—
334C22×C3—81no—
335C3×C4—81no—
336C3×C4—81no—
337Dic3—42no—
338Dic3—81no—
339Dic3—81no—
340Dic3—81no—
341C2×C5—22no—
342C2×C5—121no—
343C2×C5#591411yescharacteristic
344C2×C5C2×#98411yescharacteristic
345C2×C5#592611yescharacteristic
346C23—31no—
347C23C2×C5×S411yescharacteristic
348Q8—31no—
349Q8—61no—
350Q8—61no—
351Q8—61no—
352Q8—61no—
353Q8—241no—
354Q8C5×#23711yescharacteristic
355D4—61no—
356D4—61no—
357D4—62no—
358C2×C4—31no—
359C2×C4—31no—
360C2×C4—31no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—241no—
370C2×C4—241no—
371C2×C4C2×C5×S411yescharacteristic
372C2×C4C2×C5×S411yescharacteristic
373C8—61no—
374C8—61no—
375C8—61no—
376C8—61no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—82no—
381C5#8537711yescharacteristic, direct product factor, Sylow
382C22—21no—
383C22—21no—
384C22—61no—
385C22—61no—
386C22—61no—
387C22C5×#98412yes—
388C22C22×C5×S411yescharacteristic
389C4—21no—
390C4—21no—
391C4—31no—
392C4—31no—
393C4—61no—
394C4—61no—
395C4—121no—
396C4—121no—
397C4—121no—
398C4—121no—
399C3—41noSylow
400C2—22no—
401C2—121no—
402C2C5×#591411yescharacteristic
403C2C2×C5×#98411yescharacteristic
404C2C5×#592611yescharacteristic
subgroupG/H
center, socleC22×C5C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC5×#548D3
prime radicalC5×#548⋊C3C2
hypercenterC5×C22⋊C4S4
hypocenterSL(2,𝔽3)C5×#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C5 ◁ C5×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#5426C2semidirect
3. C5×#5425C2semidirect
4. C5×#5419C2semidirect
5. C2×C5×#859C2semidirect
6. C5×#5388C2semidirect
7. C5×#5386C2semidirect
17. #85377C5direct (N × H)
45. #5426C2×C5semidirect
46. #5425C2×C5semidirect
47. #5419C2×C5semidirect
48. C2×#859C2×C5semidirect
49. #5388C2×C5semidirect
50. #5386C2×C5semidirect
381. C5#85377direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#548⋊C32
C5×#54262
C5×#54252
C5×#54192
C2×C5×#8592
C5×#53882
C5×#53862
C5×#33163
C5×#8784
#853775