Finite Group Databasebeta

← #10617786⋊C3 #14514056

Subgroups — #10617786⋊C3 #14514056

Order 1920.

730 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×D5×#41⋊C3C211yesmaximal, characteristic
2C5×#520⋊C3C211yesmaximal, characteristic, complemented
3#66161⋊C3C211yesmaximal, characteristic, complemented
4#65906⋊C3C212yesmaximal, complemented
5C4×C5⋊(#41⋊C3)C212yesmaximal, complemented
6#10617786C311yesmaximal, characteristic, complemented, Hall
7C2×C5×#41⋊C3C2211yescharacteristic
8D5×#41⋊C3C414yescomplemented
9C2×D5×SL(2,𝔽3)C2211yescharacteristic
10C2×C5⋊(#41⋊C3)C2211yescharacteristic
11C3×#4038—41nomaximal
12C4×C5×SL(2,𝔽3)C2212yescomplemented
13Dic5×SL(2,𝔽3)C2212yescomplemented
14#3428⋊C3—51nomaximal, Hall
15C2×D5×#41C2×C311yescharacteristic
16C5×#520C2×C311yescharacteristic, complemented
17#66161C2×C311yescharacteristic, complemented
18C4×#713C2×C312yescomplemented
19#65906C2×C312yes—
20#65879—32no—
21C4×#711—32no—
22#65857—31no—
23D5×#117—31no—
24#65838—31no—
25C2×#4038—31no—
26C2×C3×C4×D5—41no—
27C5×#41⋊C3C2×C412yescomplemented
28C2×C5×SL(2,𝔽3)C2311yescharacteristic
29D5×SL(2,𝔽3)C2×C412yescomplemented
30C5⋊(#41⋊C3)C2×C412yescomplemented
31C3×C5×C4⋊C4—41no—
32C3×#685—42no—
33C3×C4⋊Dic5—41no—
34C3×C4×Dic5—42no—
35C22×#41⋊C3—51no—
36#520⋊C3—51no—
37#520⋊C3D511yescharacteristic, complemented
38#519⋊C3—52no—
39C4×#41⋊C3—52no—
40C2×C5×#41C22×C311yescharacteristic
41D5×#41—34no—
42D5×#41C3×C414yescomplemented
43C2×#713C22×C311yescharacteristic
44C2×Q8×D5C22×C311yescharacteristic
45C2×#711—31no—
46C2×D4×D5—31no—
47C2×#709—31no—
48C22×C4×D5—31no—
49C4×C5×Q8C22×C312yes—
50C4×C5×D4—32no—
51C5×#117—31no—
52C2×C5×C4⋊C4—31no—
53Q8×Dic5C22×C312yes—
54D4×Dic5—32no—
55C4×C22⋊D5—34no—
56C2×#685—32no—
57#4072—31no—
58C2×C4⋊Dic5—31no—
59C2×C4×Dic5—32no—
60#4039—32no—
61#4038—32no—
62#4038—31no—
63#4038—32no—
64#4038—32no—
65#4038A411yescharacteristic
66D5×C4⋊C4—31no—
67D5×C4⋊C4—31no—
68#4033—32no—
69#4027—34no—
70D5×C22⋊C4—32no—
71#4023—32no—
72C4×D20—32no—
73#4018—32no—
74C42×D5—32no—
75C4×Dic10—32no—
76#3428—51noSylow
77C22×C3×D5—41no—
78C2×C3×C4×C5—41no—
79C2×C3×C4×C5—42no—
80C2×C3×Dic5—41no—
81C2×C3×Dic5—42no—
82C3×C4×D5—44no—
83C5×SL(2,𝔽3)C22×C411yescharacteristic
84C2×#41⋊C3—54no—
85C2×#41⋊C3—51no—
86C2×#41⋊C3C2×D511yescharacteristic
87C22×SL(2,𝔽3)—51no—
88C3×#117—201no—
89C4×SL(2,𝔽3)—52no—
90C4×SL(2,𝔽3)C2×D512yescomplemented
91C23×D5—31no—
92C5×#41—32no—
93C5×#41C2×C3×C412yes—
94C2×C5×Q8C23×C311yescharacteristic
95C2×C5×D4—31no—
96C22×C4×C5—31no—
97C22×C4×C5—32no—
98C2×C22⋊D5—32no—
99C22×Dic5—31no—
100C22×Dic5—32no—
101#713—32no—
102#713C2×C3×C412yes—
103Q8×D5—32no—
104Q8×D5C2×C3×C412yes—
105#711—34no—
106#711—34no—
107D4×D5—34no—
108D4×D5—34no—
109#709—34no—
110#709—34no—
111C2×D20—31no—
112C2×C4×D5—31no—
113C2×C4×D5—31no—
114C2×C4×D5—31no—
115C2×C4×D5—34no—
116C2×C4×D5—34no—
117C2×C4×D5—32no—
118C2×C4×D5—32no—
119C2×C4×D5C2×A411yescharacteristic
120C2×Dic10—31no—
121C5×C4⋊C4—32no—
122C5×C4⋊C4—31no—
123C5×C4⋊C4C2×A411yescharacteristic
124C5×C22⋊C4—32no—
125C42×C5—32no—
126C22⋊Dic5—32no—
127#685—34no—
128#685—32no—
129#685C2×A412yes—
130C4⋊Dic5—31no—
131C4⋊Dic5—32no—
132C4⋊Dic5C2×A411yescharacteristic
133#683—32no—
134#683—32no—
135C4×Dic5—34no—
136C4×Dic5—32no—
137C4×Dic5—32no—
138C4×Dic5C2×A412yes—
139C22×#41—51no—
140#520—51no—
141#520—151no—
142#520—151no—
143#520C3×D511yescharacteristic, complemented
144#519—52no—
145#518—152no—
146C4×#41—52no—
147C4×#41—152no—
148C2×#117—151no—
149C2×#117—151no—
150C22×C3×C5—41no—
151C2×C3×D5—42no—
152C2×C3×D5—81no—
153C2×C3×D5—81no—
154C3×C4×C5—42no—
155C3×C4×C5—82no—
156C3×Dic5—42no—
157C3×Dic5—82no—
158C22×C3×C4—201no—
159#41⋊C3—52no—
160#41⋊C3C4×D512yescomplemented
161C2×SL(2,𝔽3)—52no—
162C2×SL(2,𝔽3)C22×D511yescharacteristic
163C3×C4⋊C4—41no—
164C3×C4⋊C4—201no—
165C3×C22⋊C4—202no—
166C3×C42—202no—
167C23×C5—31no—
168C22×D5—31no—
169C22×D5—34no—
170C22×D5—121no—
171C22×D5—121no—
172C22×D5C22×A411yescharacteristic
173C5×Q8—31no—
174C5×Q8C22×C3×C411yescharacteristic
175C5×D4—32no—
176C5×D4—32no—
177C2×C4×C5—31no—
178C2×C4×C5—32no—
179C2×C4×C5—32no—
180C2×C4×C5—32no—
181C2×C4×C5—122no—
182C2×C4×C5C22×A412yes—
183C2×C4×C5C22×A411yescharacteristic
184C22⋊D5—38no—
185C2×Dic5—31no—
186C2×Dic5—34no—
187C2×Dic5—32no—
188C2×Dic5—122no—
189C2×Dic5C22×A412yes—
190C2×Dic5C22×A411yescharacteristic
191D20—32no—
192D20—32no—
193C4×D5—32no—
194C4×D5—32no—
195C4×D5—34no—
196C4×D5—32no—
197C4×D5—32no—
198C4×D5—122no—
199C4×D5—122no—
200C4×D5C4×A414yes—
201Dic10—32no—
202Dic10—32no—
203C2×#41—54no—
204C2×#41—51no—
205C2×#41—151no—
206C2×#41—151no—
207C2×#41—154no—
208C2×#41C2×C3×D511yescharacteristic
209C22×Q8—51no—
210C22×D4—151no—
211C23×C4—151no—
212C4×Q8—52no—
213C4×Q8—152no—
214C4×Q8—152no—
215C4×Q8C2×C3×D512yes—
216C4×D4—32no—
217C4×D4—152no—
218C4×D4—154no—
219C4×D4—152no—
220C4×D4—154no—
221C4×D4—152no—
222#117—31no—
223#117—51no—
224#117—152no—
225#117—152no—
226#117—151no—
227#117—152no—
228#117—151no—
229#117—152no—
230#117—152no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—151no—
233C2×C4⋊C4—151no—
234C2×C4⋊C4—151no—
235C2×C22⋊C4—152no—
236C2×C22⋊C4—152no—
237C2×C42—152no—
238C2×C42—152no—
239C2×C3×C5—41no—
240C2×C3×C5—41no—
241C2×C3×C5—41no—
242C3×D5—82no—
243C23×C3—201no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—204no—
247C2×C3×C4—201no—
248C2×C3×C4—202no—
249SL(2,𝔽3)C2×C4×D511yescharacteristic
250C22×C5—32no—
251C22×C5—61no—
252C22×C5—61no—
253C22×C5C23×A411yescharacteristic
254C2×D5—21no—
255C2×D5—21no—
256C2×D5—32no—
257C2×D5—61no—
258C2×D5—61no—
259C2×D5—124no—
260C2×D5C2×C4×A412yes—
261C4×C5—22no—
262C4×C5—31no—
263C4×C5—31no—
264C4×C5—62no—
265C4×C5C2×C4×A412yes—
266Dic5—22no—
267Dic5—32no—
268Dic5—62no—
269Dic5C2×C4×A412yes—
270C24—151no—
271#41—32no—
272#41—52no—
273#41—154no—
274#41—154no—
275#41—154no—
276#41—154no—
277#41—152no—
278#41C3×C4×D512yes—
279C2×Q8—52no—
280C2×Q8—151no—
281C2×Q8—152no—
282C2×Q8C22×C3×D511yescharacteristic
283C2×D4—31no—
284C2×D4—151no—
285C2×D4—154no—
286C2×D4—154no—
287C2×D4—152no—
288C22×C4—31no—
289C22×C4—32no—
290C22×C4—51no—
291C22×C4—151no—
292C22×C4—154no—
293C22×C4—154no—
294C22×C4—152no—
295C22×C4—151no—
296C22×C4—151no—
297C22×C4—151no—
298C22×C4—152no—
299C22×C4—152no—
300C4⋊C4—32no—
301C4⋊C4—31no—
302C4⋊C4—51no—
303C4⋊C4—152no—
304C4⋊C4—152no—
305C4⋊C4—151no—
306C4⋊C4—152no—
307C4⋊C4D5×A411yescharacteristic
308C22⋊C4—32no—
309C22⋊C4—52no—
310C22⋊C4—152no—
311C22⋊C4—154no—
312C22⋊C4—152no—
313C42—32no—
314C42—52no—
315C42—152no—
316C42—152no—
317C42—154no—
318C3×C5—41noHall
319C22×C3—41no—
320C22×C3—202no—
321C22×C3—401no—
322C22×C3—401no—
323C3×C4—42no—
324C3×C4—82no—
325C3×C4—202no—
326C3×C4—402no—
327C2×C5—62no—
328C2×C5C22×C4×A411yescharacteristic
329C2×C5#594511yescharacteristic
330C2×C5#594511yescharacteristic
331D5—22no—
332D5—62no—
333C23—31no—
334C23—51no—
335C23—154no—
336C23—151no—
337C23—601no—
338C23—601no—
339Q8—31no—
340Q8—152no—
341Q8—152no—
342Q8C2×C3×C4×D511yescharacteristic
343D4—32no—
344D4—32no—
345D4—152no—
346D4—152no—
347D4—158no—
348C2×C4—31no—
349C2×C4—32no—
350C2×C4—32no—
351C2×C4—32no—
352C2×C4—51no—
353C2×C4—54no—
354C2×C4—52no—
355C2×C4—122no—
356C2×C4—152no—
357C2×C4—152no—
358C2×C4—152no—
359C2×C4—152no—
360C2×C4—154no—
361C2×C4—151no—
362C2×C4—154no—
363C2×C4—152no—
364C2×C4—602no—
365C2×C4—602no—
366C2×C4—602no—
367C2×C4C2×D5×A412yes—
368C2×C4C2×D5×A411yescharacteristic
369C2×C3—41no—
370C2×C3—41no—
371C2×C3—41no—
372C2×C3—402no—
373C5#3428⋊C311yescharacteristic, complemented, Sylow
374C22—32no—
375C22—52no—
376C22—61no—
377C22—61no—
378C22—101no—
379C22—101no—
380C22—152no—
381C22—301no—
382C22—301no—
383C22—604no—
384C22C22×D5×A411yescharacteristic
385C4—22no—
386C4—31no—
387C4—31no—
388C4—52no—
389C4—62no—
390C4—102no—
391C4—152no—
392C4—302no—
393C4C4×D5×A412yes—
394C3—41noSylow
395C2—62no—
396C2—102no—
397C2—302no—
398C2C2×C4×D5×A411yescharacteristic
399C2#1175694411yescharacteristic
400C2#66287⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, hypocenterC5×Q8C22×C3×C4
FittingC5×#520C2×C3
socleC22×C5C23×A4
prime radicalC2×D5×#41⋊C3C2
cosocleC2×C5×Q8C23×C3
hypercenterC4⋊C4D5×A4
LCSG ▷ C5×Q8
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#520⋊C3C2semidirect
3. #66161⋊C3C2semidirect
4. #65906⋊C3C2semidirect
5. C4×C5⋊(#41⋊C3)C2semidirect
6. #10617786C3semidirect
8. D5×#41⋊C3C4semidirect
12. C4×C5×SL(2,𝔽3)C22semidirect
13. Dic5×SL(2,𝔽3)C22semidirect
16. C5×#520C2×C3semidirect
17. #66161C2×C3semidirect
18. C4×#713C2×C3semidirect
27. C5×#41⋊C3C2×C4semidirect
29. D5×SL(2,𝔽3)C2×C4semidirect
30. C5⋊(#41⋊C3)C2×C4semidirect
37. #520⋊C3D5semidirect
42. D5×#41C3×C4semidirect
90. C4×SL(2,𝔽3)C2×D5semidirect
143. #520C3×D5semidirect
160. #41⋊C3C4×D5semidirect
373. C5#3428⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D5×#41⋊C32
C5×#520⋊C32
#66161⋊C32
#65906⋊C32
C4×C5⋊(#41⋊C3)2
#106177863
C3×#40384
#3428⋊C35