Finite Group Databasebeta

← #10617735⋊C3 #14514031

Subgroups — #10617735⋊C3 #14514031

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
2C4×C5×#41⋊C3C211yesmaximal, characteristic, complemented
3Dic5×#41⋊C3C211yesmaximal, characteristic, complemented
4#65906⋊C3C212yesmaximal, complemented
5#65908⋊C3C212yesmaximal, complemented
6#10617735C311yesmaximal, characteristic, complemented, Hall
7C2×C5×#41⋊C3C2211yescharacteristic
8D5×#41⋊C3C414yescomplemented
9C2×D5×SL(2,𝔽3)C2211yescharacteristic
10C2×C5⋊(#41⋊C3)C2211yescharacteristic
11C3×#4018—41nomaximal
12C4×C5×SL(2,𝔽3)C2212yescomplemented
13Dic5×SL(2,𝔽3)C2212yescomplemented
14#3428⋊C3—51nomaximal, Hall
15C2×D5×#41C2×C311yescharacteristic
16C4×C5×#41C2×C311yescharacteristic, complemented
17#41×Dic5C2×C311yescharacteristic, complemented
18#65908C2×C312yescomplemented
19#65906C2×C312yes—
20#65880—32no—
21#65879—32no—
22#65856—31no—
23D5×#117—31no—
24C4×#709—31no—
25C2×#4018—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×C42×C5—41no—
32C3×#685—42no—
33C3×#683—42no—
34C3×C4×Dic5—41no—
35C22×#41⋊C3—51no—
36#520⋊C3—52no—
37#519⋊C3—52no—
38C4×#41⋊C3—51no—
39C4×#41⋊C3D511yescharacteristic, complemented
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×C42×C5—31no—
53Q8×Dic5C22×C312yes—
54D4×Dic5—32no—
55C4×C22⋊D5—34no—
56C2×#685—32no—
57#4072—31no—
58C2×#683—32no—
59C2×C4×Dic5—31no—
60#4039—32no—
61#4038—32no—
62D5×C4⋊C4—32no—
63#4033—32no—
64#4027—34no—
65D5×C22⋊C4—32no—
66#4023—32no—
67C4×D20—32no—
68#4018—31no—
69#4018—31no—
70#4018—32no—
71#4018—32no—
72#4018—31no—
73#4018A411yescharacteristic
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—34no—
118C2×C4×D5C2×A411yescharacteristic
119C2×Dic10—31no—
120C5×C4⋊C4—32no—
121C5×C22⋊C4—32no—
122C42×C5—31no—
123C42×C5—32no—
124C42×C5C2×A411yescharacteristic
125C22⋊Dic5—32no—
126#685—32no—
127#685—32no—
128#685—32no—
129#685C2×A412yes—
130C4⋊Dic5—32no—
131#683—32no—
132#683—32no—
133#683—32no—
134#683C2×A412yes—
135C4×Dic5—34no—
136C4×Dic5—31no—
137C4×Dic5—32no—
138C4×Dic5C2×A411yescharacteristic
139C22×#41—51no—
140#520—52no—
141#520—152no—
142#519—52no—
143#518—152no—
144C4×#41—51no—
145C4×#41—151no—
146C4×#41—151no—
147C4×#41C3×D511yescharacteristic, complemented
148C2×#117—151no—
149C2×#117—151no—
150C22×C3×C5—41no—
151C2×C3×D5—42no—
152C2×C3×D5—82no—
153C3×C4×C5—42no—
154C3×C4×C5—82no—
155C3×Dic5—42no—
156C3×Dic5—82no—
157C22×C3×C4—201no—
158#41⋊C3—52no—
159#41⋊C3C4×D512yescomplemented
160C2×SL(2,𝔽3)—52no—
161C2×SL(2,𝔽3)C22×D511yescharacteristic
162C3×C4⋊C4—202no—
163C3×C22⋊C4—202no—
164C3×C42—41no—
165C3×C42—201no—
166C23×C5—31no—
167C22×D5—31no—
168C22×D5—34no—
169C22×D5—122no—
170C22×D5C22×A411yescharacteristic
171C5×Q8—31no—
172C5×Q8C22×C3×C411yescharacteristic
173C5×D4—32no—
174C5×D4—32no—
175C2×C4×C5—31no—
176C2×C4×C5—32no—
177C2×C4×C5—32no—
178C2×C4×C5—32no—
179C2×C4×C5—122no—
180C2×C4×C5C22×A412yes—
181C2×C4×C5C22×A411yescharacteristic
182C22⋊D5—38no—
183C2×Dic5—31no—
184C2×Dic5—34no—
185C2×Dic5—32no—
186C2×Dic5—122no—
187C2×Dic5C22×A412yes—
188C2×Dic5C22×A411yescharacteristic
189D20—32no—
190D20—32no—
191C4×D5—32no—
192C4×D5—32no—
193C4×D5—34no—
194C4×D5—32no—
195C4×D5—32no—
196C4×D5—124no—
197C4×D5C4×A414yes—
198Dic10—32no—
199Dic10—32no—
200C2×#41—54no—
201C2×#41—51no—
202C2×#41—151no—
203C2×#41—151no—
204C2×#41—154no—
205C2×#41C2×C3×D511yescharacteristic
206C22×Q8—51no—
207C22×D4—151no—
208C23×C4—151no—
209C4×Q8—52no—
210C4×Q8—152no—
211C4×Q8—152no—
212C4×Q8C2×C3×D512yes—
213C4×D4—32no—
214C4×D4—152no—
215C4×D4—154no—
216C4×D4—154no—
217C4×D4—152no—
218C4×D4—152no—
219#117—31no—
220#117—51no—
221#117—151no—
222#117—152no—
223#117—152no—
224#117—151no—
225#117—152no—
226#117—152no—
227#117—151no—
228#117—151no—
229C2×C4⋊C4—152no—
230C2×C4⋊C4—152no—
231C2×C22⋊C4—152no—
232C2×C22⋊C4—152no—
233C2×C42—31no—
234C2×C42—152no—
235C2×C42—151no—
236C2×C3×C5—41no—
237C2×C3×C5—42no—
238C3×D5—82no—
239C23×C3—201no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—204no—
243C2×C3×C4—201no—
244C2×C3×C4—202no—
245SL(2,𝔽3)C2×C4×D511yescharacteristic
246C22×C5—32no—
247C22×C5—62no—
248C22×C5C23×A411yescharacteristic
249C2×D5—22no—
250C2×D5—32no—
251C2×D5—62no—
252C2×D5—124no—
253C2×D5C2×C4×A412yes—
254C4×C5—22no—
255C4×C5—31no—
256C4×C5—31no—
257C4×C5—62no—
258C4×C5C2×C4×A412yes—
259Dic5—22no—
260Dic5—32no—
261Dic5—62no—
262Dic5C2×C4×A412yes—
263C24—151no—
264#41—32no—
265#41—52no—
266#41—154no—
267#41—154no—
268#41—154no—
269#41—154no—
270#41—152no—
271#41C3×C4×D512yes—
272C2×Q8—52no—
273C2×Q8—151no—
274C2×Q8—152no—
275C2×Q8C22×C3×D511yescharacteristic
276C2×D4—31no—
277C2×D4—151no—
278C2×D4—154no—
279C2×D4—154no—
280C2×D4—152no—
281C22×C4—31no—
282C22×C4—32no—
283C22×C4—51no—
284C22×C4—151no—
285C22×C4—154no—
286C22×C4—154no—
287C22×C4—151no—
288C22×C4—154no—
289C22×C4—151no—
290C22×C4—152no—
291C22×C4—151no—
292C4⋊C4—32no—
293C4⋊C4—52no—
294C4⋊C4—152no—
295C4⋊C4—152no—
296C4⋊C4—152no—
297C4⋊C4—152no—
298C22⋊C4—32no—
299C22⋊C4—52no—
300C22⋊C4—152no—
301C22⋊C4—152no—
302C22⋊C4—152no—
303C22⋊C4—152no—
304C42—32no—
305C42—31no—
306C42—51no—
307C42—152no—
308C42—154no—
309C42—151no—
310C42D5×A411yescharacteristic
311C3×C5—41noHall
312C22×C3—41no—
313C22×C3—202no—
314C22×C3—402no—
315C3×C4—42no—
316C3×C4—82no—
317C3×C4—202no—
318C3×C4—402no—
319C2×C5—62no—
320C2×C5C22×C4×A411yescharacteristic
321C2×C5#594512yes—
322D5—22no—
323D5—62no—
324C23—31no—
325C23—51no—
326C23—154no—
327C23—151no—
328C23—602no—
329Q8—31no—
330Q8—152no—
331Q8—152no—
332Q8C2×C3×C4×D511yescharacteristic
333D4—32no—
334D4—32no—
335D4—152no—
336D4—152no—
337D4—158no—
338C2×C4—31no—
339C2×C4—32no—
340C2×C4—32no—
341C2×C4—32no—
342C2×C4—51no—
343C2×C4—54no—
344C2×C4—52no—
345C2×C4—122no—
346C2×C4—152no—
347C2×C4—152no—
348C2×C4—152no—
349C2×C4—152no—
350C2×C4—154no—
351C2×C4—151no—
352C2×C4—154no—
353C2×C4—152no—
354C2×C4—604no—
355C2×C4—602no—
356C2×C4C2×D5×A412yes—
357C2×C4C2×D5×A411yescharacteristic
358C2×C3—41no—
359C2×C3—42no—
360C2×C3—402no—
361C5#3428⋊C311yescharacteristic, complemented, Sylow
362C22—32no—
363C22—52no—
364C22—62no—
365C22—102no—
366C22—152no—
367C22—302no—
368C22—604no—
369C22C22×D5×A411yescharacteristic
370C4—22no—
371C4—31no—
372C4—31no—
373C4—52no—
374C4—62no—
375C4—102no—
376C4—152no—
377C4—302no—
378C4C4×D5×A412yes—
379C3—41noSylow
380C2—62no—
381C2—102no—
382C2—302no—
383C2C2×C4×D5×A411yescharacteristic
384C2#66283⋊C312yes—
subgroupG/H
centerC2×C4C2×D5×A4
commutator, hypocenterC5×Q8C22×C3×C4
FrattiniC22C22×D5×A4
FittingC4×C5×#41C2×C3
socleC22×C5C23×A4
prime radicalC2×D5×#41⋊C3C2
cosocleC2×C5×Q8C23×C3
hypercenterC42D5×A4
LCSG ▷ C5×Q8
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×C5×#41⋊C3C2semidirect
3. Dic5×#41⋊C3C2semidirect
4. #65906⋊C3C2semidirect
5. #65908⋊C3C2semidirect
6. #10617735C3semidirect
8. D5×#41⋊C3C4semidirect
12. C4×C5×SL(2,𝔽3)C22semidirect
13. Dic5×SL(2,𝔽3)C22semidirect
16. C4×C5×#41C2×C3semidirect
17. #41×Dic5C2×C3semidirect
18. #65908C2×C3semidirect
27. C5×#41⋊C3C2×C4semidirect
29. D5×SL(2,𝔽3)C2×C4semidirect
30. C5⋊(#41⋊C3)C2×C4semidirect
39. C4×#41⋊C3D5semidirect
42. D5×#41C3×C4semidirect
90. C4×SL(2,𝔽3)C2×D5semidirect
147. C4×#41C3×D5semidirect
159. #41⋊C3C4×D5semidirect
361. 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
C4×C5×#41⋊C32
Dic5×#41⋊C32
#65906⋊C32
#65908⋊C32
#106177353
C3×#40184
#3428⋊C35