Finite Group Databasebeta

← #10617787⋊C3 #14514050

Subgroups — #10617787⋊C3 #14514050

Order 1920.

760 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 113 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×C5⋊#994C211yesmaximal, characteristic
2C5×#520⋊C3C211yesmaximal, characteristic, complemented
3Dic5×#41⋊C3C211yesmaximal, characteristic, complemented
4#65908⋊C3C212yesmaximal, complemented
5C4×C5⋊(#41⋊C3)C212yesmaximal, complemented
6#10617787C311yesmaximal, characteristic, complemented, Hall
7C2×C5×#41⋊C3C2211yescharacteristic
8C5⋊#994C414yescomplemented
9C2×C5⋊(#41⋊C3)C2212yes—
10C3×#4033—41nomaximal
11C4×C5×SL(2,𝔽3)C2212yescomplemented
12Dic5×SL(2,𝔽3)C2211yescharacteristic, complemented
13Dic5×SL(2,𝔽3)C2211yescharacteristic, complemented
14C4×#994—51nomaximal, Hall
15C2×#4149C2×C311yescharacteristic
16C5×#520C2×C311yescharacteristic
17#41×Dic5C2×C311yescharacteristic
18#65908C2×C312yes—
19C4×#713C2×C312yes—
20#65879—32no—
21C4×D4×D5—32no—
22#65856—31no—
23#65855—31no—
24#65838—31no—
25C2×#4039—31no—
26C2×C3×Dic10—41no—
27C5×#41⋊C3C2×C412yescomplemented
28C2×C5×SL(2,𝔽3)C2311yescharacteristic
29C5⋊(#41⋊C3)C2×C414yescomplemented
30C3×C5×C4⋊C4—41no—
31C3×#683—42no—
32C3×C4×Dic5—41no—
33C3×C4×Dic5—42no—
34C2×#994—51no—
35#520⋊C3—52no—
36#520⋊C3D511yescharacteristic, complemented
37C4×#41⋊C3—51no—
38C4×#41⋊C3—52no—
39C2×C5×#41C22×C311yescharacteristic
40#4149—34no—
41#4149C3×C414yescomplemented
42C2×#713C22×C312yes—
43C2×D4×D5—32no—
44C2×#709—31no—
45C22×D20—31no—
46C22×C4×D5—32no—
47C4×C5×Q8C22×C312yes—
48C4×C5×D4—32no—
49C5×#117—31no—
50C2×C5×C4⋊C4—31no—
51Q8×Dic5C22×C311yescharacteristic
52Q8×Dic5C22×C311yescharacteristic
53D4×Dic5—31no—
54D4×Dic5—31no—
55C4×C22⋊D5—34no—
56C2×#685—32no—
57#4072—31no—
58C2×C4×Dic5—31no—
59#4039—32no—
60#4039—32no—
61#4039—31no—
62#4039—32no—
63#4039—31no—
64#4038—32no—
65D5×C4⋊C4—32no—
66#4033—31no—
67#4033A411yescharacteristic
68#4027—32no—
69#4027—32no—
70D5×C22⋊C4—34no—
71C4×D20—32no—
72C4×D20—32no—
73#4018—32no—
74C42×D5—32no—
75C4×H(5,𝔽2)—51noSylow
76C2×C3×C4×C5—41no—
77C2×C3×C4×C5—42no—
78C2×C3×Dic5—41no—
79C2×C3×Dic5—42no—
80C2×C3×Dic5—41no—
81C3×Dic10—44no—
82C5×SL(2,𝔽3)C22×C411yescharacteristic, complemented
83#994—54no—
84C2×#41⋊C3—52no—
85C2×#41⋊C3C2×D511yescharacteristic
86C3×C4×Q8—201no—
87C4×SL(2,𝔽3)—51no—
88C4×SL(2,𝔽3)—51no—
89C4×SL(2,𝔽3)C2×D512yescomplemented
90C23×D5—32no—
91C5×#41—32no—
92C5×#41C2×C3×C412yes—
93C2×C5×Q8C23×C311yescharacteristic
94C2×C5×D4—31no—
95C22×C4×C5—31no—
96C22×C4×C5—32no—
97C2×C22⋊D5—32no—
98C22×Dic5—31no—
99C22×Dic5—31no—
100#713—34no—
101#713C2×C3×C414yes—
102D4×D5—38no—
103D4×D5—38no—
104#709—34no—
105#709—34no—
106C2×D20—32no—
107C2×D20—34no—
108C2×D20—31no—
109C2×D20—34no—
110C2×C4×D5—32no—
111C2×C4×D5—32no—
112C2×C4×D5—34no—
113C2×C4×D5—32no—
114C2×C4×D5—124no—
115C2×Dic10C2×A411yescharacteristic
116C5×C4⋊C4—32no—
117C5×C4⋊C4—31no—
118C5×C4⋊C4C2×A411yescharacteristic
119C5×C22⋊C4—32no—
120C42×C5—32no—
121C22⋊Dic5—32no—
122#685—34no—
123#685—32no—
124#685—32no—
125C4⋊Dic5—31no—
126C4⋊Dic5—31no—
127#683—32no—
128#683C2×A412yes—
129C4×Dic5—31no—
130C4×Dic5—31no—
131C4×Dic5—32no—
132C4×Dic5—31no—
133C4×Dic5C2×A412yes—
134C4×Dic5C2×A411yescharacteristic
135C2×H(5,𝔽2)—51no—
136#520—52no—
137#520—151no—
138#520C3×D511yescharacteristic
139#518—151no—
140#518—152no—
141C4×#41—51no—
142C4×#41—52no—
143C4×#41—151no—
144C2×C4×D4—151no—
145C2×C4×D4—152no—
146C22×C3×C5—41no—
147C3×C4×C5—42no—
148C3×C4×C5—82no—
149C3×Dic5—44no—
150C3×Dic5—42no—
151C3×Dic5—81no—
152C2×C3×Q8—201no—
153#41⋊C3—54no—
154#41⋊C3C4×D512yescomplemented
155C2×SL(2,𝔽3)C22×D511yescharacteristic
156C3×C4⋊C4—41no—
157C3×C4⋊C4—202no—
158C3×C42—201no—
159C3×C42—202no—
160C23×C5—31no—
161C22×D5—32no—
162C22×D5—38no—
163C22×D5—122no—
164C22×D5—122no—
165C5×Q8—31no—
166C5×Q8C22×C3×C411yescharacteristic
167C5×D4—32no—
168C5×D4—32no—
169C2×C4×C5—31no—
170C2×C4×C5—32no—
171C2×C4×C5—32no—
172C2×C4×C5—32no—
173C2×C4×C5—122no—
174C2×C4×C5C22×A412yes—
175C2×C4×C5C22×A411yescharacteristic
176C22⋊D5—38no—
177C2×Dic5—31no—
178C2×Dic5—31no—
179C2×Dic5—62no—
180C2×Dic5—121no—
181C2×Dic5C22×A411yescharacteristic
182C2×Dic5C22×A412yes—
183C2×Dic5C22×A411yescharacteristic
184D20—34no—
185D20—34no—
186D20—34no—
187C4×D5—34no—
188C4×D5—34no—
189C4×D5—64no—
190C4×D5—122no—
191C4×D5—124no—
192Dic10C4×A414yes—
193H(5,𝔽2)—54no—
194H(5,𝔽2)—154no—
195C2×#41—52no—
196C2×#41—151no—
197C2×#41C2×C3×D511yescharacteristic
198C22×D4—151no—
199C22×D4—152no—
200C23×C4—152no—
201C4×Q8—51no—
202C4×Q8—51no—
203C4×Q8—51no—
204C4×Q8—151no—
205C4×Q8C2×C3×D512yes—
206C4×D4—32no—
207C4×D4—152no—
208C4×D4—152no—
209C4×D4—152no—
210C4×D4—151no—
211C4×D4—152no—
212C4×D4—152no—
213C4×D4—152no—
214C4×D4—154no—
215C4×D4—151no—
216C4×D4—151no—
217C4×D4—151no—
218C4×D4—152no—
219#117—31no—
220#117—152no—
221#117—152no—
222#117—151no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—152no—
225C2×C22⋊C4—154no—
226C2×C22⋊C4—152no—
227C2×C42—152no—
228C2×C42—151no—
229C2×C3×C5—41no—
230C2×C3×C5—41no—
231C2×C3×C5—41no—
232C3×Q8—204no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—201no—
236C2×C3×C4—202no—
237C2×C3×C4—201no—
238SL(2,𝔽3)C2×C4×D511yescharacteristic, complemented
239C22×C5—32no—
240C22×C5—61no—
241C22×C5—61no—
242C22×C5C23×A411yescharacteristic
243C2×D5—34no—
244C2×D5—62no—
245C2×D5—62no—
246C2×D5—128no—
247C4×C5—22no—
248C4×C5—31no—
249C4×C5—31no—
250C4×C5—62no—
251C4×C5C2×C4×A412yes—
252Dic5—21no—
253Dic5—61no—
254Dic5—61no—
255Dic5C2×C4×A414yes—
256Dic5#99412yescomplemented
257C24—152no—
258#41—32no—
259#41—54no—
260#41—154no—
261#41—154no—
262#41—154no—
263#41C3×C4×D512yes—
264C2×Q8—51no—
265C2×Q8C22×C3×D511yescharacteristic
266C2×D4—31no—
267C2×D4—152no—
268C2×D4—154no—
269C2×D4—154no—
270C2×D4—152no—
271C2×D4—158no—
272C2×D4—158no—
273C2×D4—151no—
274C22×C4—31no—
275C22×C4—32no—
276C22×C4—152no—
277C22×C4—152no—
278C22×C4—154no—
279C22×C4—151no—
280C22×C4—151no—
281C22×C4—152no—
282C22×C4—604no—
283C4⋊C4—32no—
284C4⋊C4—31no—
285C4⋊C4—52no—
286C4⋊C4—152no—
287C4⋊C4—151no—
288C4⋊C4—151no—
289C4⋊C4D5×A411yescharacteristic
290C22⋊C4—32no—
291C22⋊C4—152no—
292C22⋊C4—154no—
293C22⋊C4—152no—
294C22⋊C4—152no—
295C42—32no—
296C42—52no—
297C42—51no—
298C42—151no—
299C42—151no—
300C42—152no—
301C42—151no—
302C3×C5—41noHall
303C22×C3—41no—
304C3×C4—42no—
305C3×C4—82no—
306C3×C4—202no—
307C3×C4—204no—
308C3×C4—401no—
309C2×C5—62no—
310C2×C5C22×C4×A411yescharacteristic
311C2×C5C2×#99411yescharacteristic
312C2×C5#594511yescharacteristic
313D5—64no—
314C23—31no—
315C23—158no—
316C23—152no—
317C23—602no—
318C23—602no—
319Q8—31no—
320Q8—54no—
321Q8C2×C3×C4×D511yescharacteristic
322D4—32no—
323D4—32no—
324D4—154no—
325D4—154no—
326D4—158no—
327D4—154no—
328C2×C4—31no—
329C2×C4—32no—
330C2×C4—32no—
331C2×C4—32no—
332C2×C4—51no—
333C2×C4—52no—
334C2×C4—51no—
335C2×C4—122no—
336C2×C4—154no—
337C2×C4—154no—
338C2×C4—151no—
339C2×C4—151no—
340C2×C4—304no—
341C2×C4—302no—
342C2×C4—602no—
343C2×C4—604no—
344C2×C4—601no—
345C2×C4C2×D5×A412yes—
346C2×C4C2×D5×A411yescharacteristic
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—41no—
350C5C4×#99411yescharacteristic, complemented, Sylow
351C22—32no—
352C22—61no—
353C22—61no—
354C22—154no—
355C22—302no—
356C22—302no—
357C22—608no—
358C22C22×D5×A411yescharacteristic
359C4—22no—
360C4—31no—
361C4—31no—
362C4—54no—
363C4—52no—
364C4—62no—
365C4—101no—
366C4—301no—
367C4—301no—
368C4C4×D5×A412yes—
369C3—41noSylow
370C2—62no—
371C2—304no—
372C2C2×C4×D5×A411yescharacteristic
373C2D5×#99411yescharacteristic
374C2#66287⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, hypocenterC5×Q8C22×C3×C4
FittingC5×#520C2×C3
socleC22×C5C23×A4
prime radicalC2×C5⋊#994C2
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. Dic5×#41⋊C3C2semidirect
4. #65908⋊C3C2semidirect
5. C4×C5⋊(#41⋊C3)C2semidirect
6. #10617787C3semidirect
8. C5⋊#994C4semidirect
11. C4×C5×SL(2,𝔽3)C22semidirect
12. Dic5×SL(2,𝔽3)C22semidirect
13. Dic5×SL(2,𝔽3)C22semidirect
27. C5×#41⋊C3C2×C4semidirect
29. C5⋊(#41⋊C3)C2×C4semidirect
36. #520⋊C3D5semidirect
41. #4149C3×C4semidirect
82. C5×SL(2,𝔽3)C22×C4semidirect
89. C4×SL(2,𝔽3)C2×D5semidirect
154. #41⋊C3C4×D5semidirect
238. SL(2,𝔽3)C2×C4×D5semidirect
256. Dic5#994semidirect
350. C5C4×#994semidirect

Maximal subgroups

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

subgroupindex
C2×C5⋊#9942
C5×#520⋊C32
Dic5×#41⋊C32
#65908⋊C32
C4×C5⋊(#41⋊C3)2
#106177873
C3×#40334
C4×#9945