Finite Group Databasebeta

← C5×#85381 #14511119

Subgroups — C5×#85381 #14511119

Order 1920.

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

normal subgroups only

This group has 452 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, complemented
2C5×#5427C211yesmaximal, characteristic, complemented
3C5×#5424C211yesmaximal, characteristic, complemented
4C5×#5421C211yesmaximal, characteristic, complemented
5C2×C5×#859C211yesmaximal, characteristic, complemented
6C4×C5×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
7C5×#5389C211yesmaximal, characteristic, complemented
8C5×#3301—31nomaximal, Hall
9C2×C5×#41⋊C3C2211yescharacteristic
10C22×C5×SL(2,𝔽3)C2211yescharacteristic
11C2×C5×GL(2,𝔽3)C2211yescharacteristic
12C5×#67⋊C4—41nomaximal
13C4×C5×SL(2,𝔽3)C2211yescharacteristic
14C5×#859C2211yescharacteristic
15C5×#859C2211yescharacteristic
16C5×#859C2211yescharacteristic
17#85381C511yesmaximal, characteristic, direct product factor, Hall
18C5×#553—31no—
19C5×#548D311yescharacteristic, complemented
20C5×#488—31no—
21C5×#485—31no—
22C5×#481—31no—
23C5×#479—31no—
24C5×#466—31no—
25C5×#465—31no—
26C5×#463—31no—
27C5×#449—31no—
28C5×#448—31no—
29C4×C5×SD16—31no—
30C5×C8×D4—31no—
31C5×#416—31no—
32C2×C5×#103—31no—
33C2×C5×#67—41no—
34C22×C5×Dic3—41no—
35C2×C4×C5×D3—41no—
36C5×#41⋊C3—21no—
37C2×C5×SL(2,𝔽3)—21no—
38C2×C5×SL(2,𝔽3)C2311yescharacteristic
39C5×GL(2,𝔽3)—21no—
40C3×C5×C22⋊C4—41no—
41C5×(C2×D3)⋊C4—41no—
42C5×Dic3⋊C4—41no—
43C4×C5×Dic3—41no—
44#548⋊C3C2×C511yescharacteristic, complemented
45#5427C2×C511yescharacteristic, complemented
46#5424C2×C511yescharacteristic, complemented
47#5421C2×C511yescharacteristic, complemented
48C2×#859C2×C511yescharacteristic, complemented
49C4×GL(2,𝔽3)C2×C511yescharacteristic, complemented
50#5389C2×C511yescharacteristic, complemented
51C2×C5×#41C2×D311yescharacteristic
52C22×C5×Q8C2×D311yescharacteristic
53C2×C5×SD16—31no—
54C2×C5×SD16—31no—
55C2×C5×D8—31no—
56C22×C5×C8—31no—
57C22×C5×C8—31no—
58C5×#126—61no—
59C5×#125—31no—
60C5×#124—31no—
61C5×#123—61no—
62C5×C22⋊Q8—31no—
63C5×#121—31no—
64C5×#121—31no—
65C5×#121—31no—
66C5×#121—61no—
67C4×C5×Q8C2×D311yescharacteristic
68C4×C5×D4—31no—
69C4×C5×D4—31no—
70C4×C5×D4—61no—
71C5×#117—31no—
72C2×C5×C4⋊C4—31no—
73C5×#107—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×#102—31no—
84C5×#102—31no—
85C5×#102—31no—
86C5×#102—31no—
87C5×C22⋊C8—31no—
88C5×C22⋊C8—31no—
89C4×C5×C8—31no—
90#3301—31noSylow
91C23×C3×C5—41no—
92C22×C5×D3—41no—
93C2×C3×C4×C5—41no—
94C2×C3×C4×C5—41no—
95C5×#67—44no—
96C2×C5×Dic3—41no—
97C2×C5×Dic3—41no—
98C2×C5×Dic3—41no—
99C2×C5×Dic3—82no—
100C4×C5×D3—82no—
101C5×SL(2,𝔽3)#4111yescharacteristic
102C2×#41⋊C3C22×C511yescharacteristic
103C22×SL(2,𝔽3)C22×C511yescharacteristic
104C2×GL(2,𝔽3)C22×C511yescharacteristic
105#67⋊C4—41no—
106C4×SL(2,𝔽3)C22×C511yescharacteristic
107#859C22×C511yescharacteristic
108#859C22×C511yescharacteristic
109#859C22×C511yescharacteristic
110C5×#41—21no—
111C5×#41—61no—
112C2×C5×Q8—21no—
113C2×C5×Q8—31no—
114C2×C5×Q8—61no—
115C2×C5×Q8C22×D311yescharacteristic
116C2×C5×D4—31no—
117C2×C5×D4—31no—
118C2×C5×D4—31no—
119C2×C5×D4—61no—
120C2×C5×D4—61no—
121C22×C4×C5—31no—
122C22×C4×C5—31no—
123C22×C4×C5—61no—
124C22×C4×C5—61no—
125C5×SD16—61no—
126C5×SD16—61no—
127C5×SD16—121no—
128C5×D8—121no—
129C2×C5×C8—31no—
130C2×C5×C8—31no—
131C2×C5×C8—31no—
132C2×C5×C8—31no—
133C2×C5×C8—121no—
134C2×C5×C8—121no—
135C5×C4⋊C4—31no—
136C5×C4⋊C4—31no—
137C5×C4⋊C4—31no—
138C5×C4⋊C4—31no—
139C5×C4⋊C4—31no—
140C5×C4⋊C4—61no—
141C5×C4⋊C4—61no—
142C5×C4⋊C4—61no—
143C5×C22⋊C4—31no—
144C5×C22⋊C4—61no—
145C5×C22⋊C4—61no—
146C5×C22⋊C4—61no—
147C5×C22⋊C4—61no—
148C5×C22⋊C4—61no—
149C5×C22⋊C4S411yescharacteristic
150C42×C5—31no—
151C42×C5—61no—
152#553—31no—
153#548C5×D311yescharacteristic, complemented
154#488—31no—
155#485—31no—
156#481—31no—
157#479—31no—
158#466—31no—
159#465—31no—
160#463—31no—
161#449—31no—
162#448—31no—
163C4×SD16—31no—
164C8×D4—31no—
165#416—31no—
166C2×#103—31no—
167C22×C3×C5—42no—
168C22×C3×C5—41no—
169C22×C3×C5—81no—
170C22×C3×C5—81no—
171C2×C5×D3—42no—
172C2×C5×D3—81no—
173C2×C5×D3—81no—
174C3×C4×C5—81no—
175C3×C4×C5—81no—
176C5×Dic3—42no—
177C5×Dic3—42no—
178C5×Dic3—81no—
179C2×#67—41no—
180C22×Dic3—41no—
181C2×C4×D3—41no—
182#41⋊C3—21no—
183C2×SL(2,𝔽3)—21no—
184C2×SL(2,𝔽3)C23×C511yescharacteristic
185GL(2,𝔽3)—21no—
186C3×C22⋊C4—41no—
187(C2×D3)⋊C4—41no—
188Dic3⋊C4—41no—
189C4×Dic3—41no—
190C23×C5—31no—
191C23×C5—61no—
192C23×C5C2×S411yescharacteristic
193C5×Q8—31no—
194C5×Q8—61no—
195C5×Q8—61no—
196C5×Q8#23711yescharacteristic
197C5×D4—61no—
198C5×D4—61no—
199C5×D4—61no—
200C5×D4—61no—
201C5×D4—62no—
202C5×D4—122no—
203C5×D4—241no—
204C2×C4×C5—31no—
205C2×C4×C5—31no—
206C2×C4×C5—31no—
207C2×C4×C5—61no—
208C2×C4×C5—61no—
209C2×C4×C5—61no—
210C2×C4×C5—61no—
211C2×C4×C5—61no—
212C2×C4×C5—61no—
213C2×C4×C5—61no—
214C2×C4×C5—241no—
215C2×C4×C5—241no—
216C2×C4×C5C2×S411yescharacteristic
217C2×C4×C5C2×S411yescharacteristic
218C5×C8—61no—
219C5×C8—61no—
220C5×C8—61no—
221C5×C8—61no—
222C2×#41C2×C5×D311yescharacteristic
223C22×Q8C2×C5×D311yescharacteristic
224C2×SD16—31no—
225C2×SD16—31no—
226C2×D8—31no—
227C22×C8—31no—
228C22×C8—31no—
229#126—61no—
230#125—31no—
231#124—31no—
232#123—61no—
233C22⋊Q8—31no—
234#121—31no—
235#121—31no—
236#121—31no—
237#121—61no—
238C4×Q8C2×C5×D311yescharacteristic
239C4×D4—31no—
240C4×D4—31no—
241C4×D4—61no—
242#117—31no—
243C2×C4⋊C4—31no—
244#107—31no—
245#106—31no—
246#106—31no—
247C4⋊C8—31no—
248#103—31no—
249#103—31no—
250#103—31no—
251#103—31no—
252#103—31no—
253#103—31no—
254#102—31no—
255#102—31no—
256#102—31no—
257#102—31no—
258C22⋊C8—31no—
259C22⋊C8—31no—
260C4×C8—31no—
261C2×C3×C5—41no—
262C2×C3×C5—41no—
263C2×C3×C5—41no—
264C2×C3×C5—82no—
265C5×D3—82no—
266C23×C3—41no—
267C22×D3—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270#67—44no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—82no—
275C4×D3—82no—
276SL(2,𝔽3)C5×#4111yescharacteristic
277C22×C5—21no—
278C22×C5—21no—
279C22×C5—61no—
280C22×C5—61no—
281C22×C5—61no—
282C22×C5—121no—
283C22×C5—121no—
284C22×C5—121no—
285C22×C5#98512yes—
286C22×C5C22×S411yescharacteristic
287C4×C5—21no—
288C4×C5—21no—
289C4×C5—31no—
290C4×C5—31no—
291C4×C5—61no—
292C4×C5—61no—
293C4×C5—121no—
294C4×C5—121no—
295C4×C5—121no—
296#41—21no—
297#41—61no—
298C2×Q8—21no—
299C2×Q8—31no—
300C2×Q8—61no—
301C2×Q8C22×C5×D311yescharacteristic
302C2×D4—31no—
303C2×D4—31no—
304C2×D4—31no—
305C2×D4—61no—
306C2×D4—61no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—61no—
310C22×C4—61no—
311SD16—61no—
312SD16—61no—
313SD16—121no—
314D8—121no—
315C2×C8—31no—
316C2×C8—31no—
317C2×C8—31no—
318C2×C8—31no—
319C2×C8—121no—
320C2×C8—121no—
321C4⋊C4—31no—
322C4⋊C4—31no—
323C4⋊C4—31no—
324C4⋊C4—31no—
325C4⋊C4—31no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C22⋊C4—31no—
330C22⋊C4—61no—
331C22⋊C4—61no—
332C22⋊C4—61no—
333C22⋊C4—61no—
334C22⋊C4—61no—
335C22⋊C4C5×S411yescharacteristic
336C42—31no—
337C42—61no—
338C3×C5—41noHall
339C22×C3—42no—
340C22×C3—41no—
341C22×C3—81no—
342C22×C3—81no—
343C2×D3—42no—
344C2×D3—81no—
345C2×D3—81no—
346C3×C4—81no—
347C3×C4—81no—
348Dic3—42no—
349Dic3—42no—
350Dic3—81no—
351C2×C5—22no—
352C2×C5—121no—
353C2×C5—241no—
354C2×C5#591411yescharacteristic
355C2×C5C2×#98511yescharacteristic
356C2×C5#592611yescharacteristic
357C23—31no—
358C23—61no—
359C23C2×C5×S411yescharacteristic
360Q8—31no—
361Q8—61no—
362Q8—61no—
363Q8C5×#23711yescharacteristic
364D4—61no—
365D4—61no—
366D4—61no—
367D4—61no—
368D4—62no—
369D4—122no—
370D4—241no—
371C2×C4—31no—
372C2×C4—31no—
373C2×C4—31no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—241no—
382C2×C4—241no—
383C2×C4C2×C5×S411yescharacteristic
384C2×C4C2×C5×S411yescharacteristic
385C8—61no—
386C8—61no—
387C8—61no—
388C8—61no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—41no—
392C2×C3—82no—
393D3—82no—
394C5#8538111yescharacteristic, direct product factor, Sylow
395C22—21no—
396C22—21no—
397C22—61no—
398C22—61no—
399C22—61no—
400C22—121no—
401C22—121no—
402C22—121no—
403C22C5×#98512yes—
404C22C22×C5×S411yescharacteristic
405C4—21no—
406C4—21no—
407C4—31no—
408C4—31no—
409C4—61no—
410C4—61no—
411C4—121no—
412C4—121no—
413C4—121no—
414C3—41noSylow
415C2—22no—
416C2—121no—
417C2—241no—
418C2C5×#591411yescharacteristic
419C2C2×C5×#98511yescharacteristic
420C2C5×#592611yescharacteristic
subgroupG/H
center, socleC22×C5C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC5×#548D3
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
1. C5×#548⋊C3C2semidirect
2. C5×#5427C2semidirect
3. C5×#5424C2semidirect
4. C5×#5421C2semidirect
5. C2×C5×#859C2semidirect
6. C4×C5×GL(2,𝔽3)C2semidirect
7. C5×#5389C2semidirect
17. #85381C5direct (N × H)
19. C5×#548D3semidirect
44. #548⋊C3C2×C5semidirect
45. #5427C2×C5semidirect
46. #5424C2×C5semidirect
47. #5421C2×C5semidirect
48. C2×#859C2×C5semidirect
49. C4×GL(2,𝔽3)C2×C5semidirect
50. #5389C2×C5semidirect
153. #548C5×D3semidirect
394. C5#85381direct (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×#54272
C5×#54242
C5×#54212
C2×C5×#8592
C4×C5×GL(2,𝔽3)2
C5×#53892
C5×#33013
C5×#67⋊C44
#853815