Finite Group Databasebeta

← #10617719⋊C3 #14514021

Subgroups — #10617719⋊C3 #14514021

Order 1920.

744 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 91 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×#4147⋊C3C212yesmaximal, complemented
2C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#65910⋊C3C212yesmaximal, complemented
4SL(2,𝔽3)×D20C211yesmaximal, characteristic, complemented
5C4⋊(C5⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
6#10617719C311yesmaximal, characteristic, complemented, Hall
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8#4147⋊C3—24no—
9C2×D5×SL(2,𝔽3)C2212yes—
10C2×C5⋊(#41⋊C3)C2212yes—
11C3×#4076—41nomaximal
12C4×C5×SL(2,𝔽3)C2211yescharacteristic
13C4×C5×SL(2,𝔽3)C2211yescharacteristic
14#3497⋊C3—51nomaximal, Hall
15C2×#4147C2×C312yescomplemented
16C2×C4×C5×Q8C2×C311yescharacteristic, complemented
17#65911—31no—
18#65911C2×C311yescharacteristic, complemented
19#65910—32no—
20#65910C2×C312yescomplemented
21Q8×D20—31no—
22Q8×D20C2×C311yescharacteristic, complemented
23#65844—32no—
24#65811—31no—
25#65811—31no—
26C22×C3×C4×C5—41no—
27C2×C3×C22⋊D5—42no—
28C2×C3×D20—41no—
29C2×C5×SL(2,𝔽3)D412yescomplemented
30C2×C5×SL(2,𝔽3)C2311yescharacteristic
31D5×SL(2,𝔽3)—22no—
32C5⋊(#41⋊C3)—22no—
33C3×#685—42no—
34C3×C4⋊Dic5—41no—
35C2×2−1+4⋊C3—52no—
36C4⋊(#41⋊C3)—51no—
37D4×SL(2,𝔽3)—51no—
38#548⋊C3—52no—
39C2×C4×SL(2,𝔽3)D511yescharacteristic, complemented
40C22×C5×Q8C22×C311yescharacteristic
41#4147—24no—
42#4147—64no—
43C2×#713—32no—
44C2×#713C22×C312yes—
45C2×Q8×D5—32no—
46C2×Q8×D5C22×C312yes—
47C2×#709—32no—
48C2×#709—32no—
49C4×C5×Q8—31no—
50C4×C5×Q8—31no—
51C4×C5×Q8C22×C311yescharacteristic
52C4×C5×Q8C22×C311yescharacteristic
53C2×C5×C4⋊C4—31no—
54C2×C42×C5—31no—
55#4076—32no—
56#4076—31no—
57#4076—32no—
58#4076A411yescharacteristic
59#4043—34no—
60#4043—32no—
61#4043—32no—
62#4041—32no—
63#4041—34no—
64#4041—32no—
65#4021—32no—
66#4021—32no—
67#4020—31no—
68#4020—31no—
69C4×D20—32no—
70C4×D20—32no—
71C4×D20—31no—
72C4×D20—32no—
73C4×D20—31no—
74#4015—31no—
75#4015—31no—
76#3497—51noSylow
77C23×C3×C5—41no—
78C22×C3×D5—42no—
79C2×C3×C4×C5—41no—
80C2×C3×C4×C5—41no—
81C2×C3×C4×C5—82no—
82C3×C22⋊D5—84no—
83C2×C3×Dic5—42no—
84C3×D20—82no—
85C5×SL(2,𝔽3)C2×D411yescharacteristic
862−1+4⋊C3—104no—
87C2×#41⋊C3—52no—
88C22×SL(2,𝔽3)—52no—
89C22×SL(2,𝔽3)C2×D511yescharacteristic
90C3×#121—201no—
91C4×SL(2,𝔽3)C2×D511yescharacteristic
92C4×SL(2,𝔽3)C2×D511yescharacteristic
93C2×C5×Q8—31no—
94C2×C5×Q8—32no—
95C2×C5×Q8C3×D412yescomplemented
96C2×C5×Q8C23×C311yescharacteristic
97C22×C4×C5—31no—
98C22×C4×C5—31no—
99C22×C4×C5C2×A411yescharacteristic
100C2×C22⋊D5—32no—
101C2×C22⋊D5C2×A412yes—
102#713—22no—
103#713—64no—
104#713—64no—
105#713—62no—
106Q8×D5—22no—
107Q8×D5—64no—
108Q8×D5—64no—
109Q8×D5—62no—
110#709—64no—
111#709—64no—
112#709—64no—
113#709—64no—
114C2×D20—32no—
115C2×D20—32no—
116C2×D20—31no—
117C2×D20—32no—
118C2×D20—32no—
119C2×D20C2×A411yescharacteristic
120C2×C4×D5—34no—
121C2×C4×D5—32no—
122C2×C4×D5—32no—
123C2×Dic10—32no—
124C2×Dic10—32no—
125C5×C4⋊C4—31no—
126C5×C4⋊C4—31no—
127C5×C4⋊C4—32no—
128C42×C5—31no—
129C42×C5—31no—
130C42×C5—31no—
131C42×C5—31no—
132#685—34no—
133#685—32no—
134#685—32no—
135#685—32no—
136#685C2×A412yes—
137C4⋊Dic5—32no—
138C4⋊Dic5—31no—
139C4⋊Dic5—32no—
140C4⋊Dic5C2×A411yescharacteristic
141C2×2−1+4—52no—
142#550—51no—
143#550—151no—
144D4×Q8—51no—
145D4×Q8—151no—
146#548—52no—
147#548—152no—
148#537—152no—
149#532—151no—
150#532—151no—
151C2×C4×Q8C3×D511yescharacteristic, complemented
152C22×C3×C5—42no—
153C22×C3×C5—41no—
154C22×C3×C5—81no—
155C22×C3×C5—81no—
156C2×C3×D5—82no—
157C2×C3×D5—82no—
158C2×C3×D5—82no—
159C3×C4×C5—42no—
160C3×C4×C5—81no—
161C3×Dic5—82no—
162C2×C3×D4—201no—
163C2×C3×D4—202no—
164C22×C3×C4—41no—
165#41⋊C3—102no—
166C2×SL(2,𝔽3)—102no—
167C2×SL(2,𝔽3)D2012yescomplemented
168C2×SL(2,𝔽3)C22×D511yescharacteristic
169C3×C4⋊C4—201no—
170C3×C22⋊C4—202no—
171C23×C5C22×A411yescharacteristic
172C22×D5—32no—
173C22×D5C22×A412yes—
174C5×Q8—32no—
175C5×Q8—31no—
176C5×Q8—32no—
177C5×Q8C2×C3×D411yescharacteristic
178C2×C4×C5—22no—
179C2×C4×C5—32no—
180C2×C4×C5—32no—
181C2×C4×C5—31no—
182C2×C4×C5—31no—
183C2×C4×C5—31no—
184C2×C4×C5—31no—
185C2×C4×C5—121no—
186C2×C4×C5C22×A411yescharacteristic
187C2×C4×C5C22×A411yescharacteristic
188C22⋊D5—24no—
189C22⋊D5—64no—
190C2×Dic5—32no—
191C2×Dic5C22×A412yes—
192D20—22no—
193D20—64no—
194D20—62no—
195D20—62no—
196D20—62no—
197D20—122no—
198D20—122no—
199C4×D5—62no—
200C4×D5—62no—
201C4×D5—62no—
202C4×D5—62no—
203C4×D5—68no—
204Dic10—64no—
205Dic10—62no—
206Dic10—62no—
2072−1+4—104no—
2082−1+4—304no—
209C2×#41—52no—
210C2×#41—152no—
211C2×#41—152no—
212C2×#41—152no—
213C22×Q8—52no—
214C22×Q8—152no—
215C22×Q8C2×C3×D511yescharacteristic
216C4⋊Q8—151no—
217C4⋊Q8—151no—
218#127—151no—
219#127—151no—
220#124—152no—
221#124—152no—
222C22⋊Q8—154no—
223C22⋊Q8—152no—
224C22⋊Q8—152no—
225#121—51no—
226#121—152no—
227#121—154no—
228#121—152no—
229#121—152no—
230#121—151no—
231#121—152no—
232C4×Q8—31no—
233C4×Q8—31no—
234C4×Q8C2×C3×D511yescharacteristic
235C4×Q8C2×C3×D511yescharacteristic
236C4×D4—152no—
237C4×D4—151no—
238C4×D4—151no—
239C4×D4—152no—
240C4×D4—152no—
241C2×C4⋊C4—31no—
242C2×C42—31no—
243C2×C3×C5—41no—
244C2×C3×C5—41no—
245C2×C3×C5—41no—
246C2×C3×C5—82no—
247C3×D5—162no—
248C23×C3—41no—
249C23×C3—202no—
250C3×D4—402no—
251C3×D4—404no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—82no—
255C2×C3×C4—202no—
256SL(2,𝔽3)C2×D2011yescharacteristic
257C22×C5—21no—
258C22×C5—21no—
259C22×C5D4×A412yes—
260C22×C5C23×A411yescharacteristic
261C2×D5—22no—
262C2×D5—22no—
263C2×D5—22no—
264C2×D5—62no—
265C2×D5—62no—
266C2×D5—62no—
267C4×C5—21no—
268C4×C5—32no—
269C4×C5—31no—
270C4×C5—31no—
271C4×C5—61no—
272C4×C5—61no—
273C4×C52−1+4⋊C312yescomplemented
274Dic5—22no—
275Dic5—62no—
276#41—102no—
277#41—304no—
278#41—304no—
279#41—304no—
280#41—304no—
281#41—304no—
282#41—304no—
283#41—302no—
284C2×Q8—31no—
285C2×Q8—32no—
286C2×Q8—102no—
287C2×Q8—152no—
288C2×Q8—152no—
289C2×Q8—304no—
290C2×Q8—304no—
291C2×Q8—302no—
292C2×Q8C3×D2012yescomplemented
293C2×Q8C22×C3×D511yescharacteristic
294C2×D4—52no—
295C2×D4—51no—
296C2×D4—152no—
297C2×D4—152no—
298C2×D4—152no—
299C2×D4—152no—
300C2×D4—152no—
301C2×D4—151no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—152no—
305C22×C4—152no—
306C22×C4—154no—
307C22×C4D5×A411yescharacteristic
308C4⋊C4—31no—
309C4⋊C4—31no—
310C4⋊C4—32no—
311C4⋊C4—51no—
312C4⋊C4—152no—
313C4⋊C4—152no—
314C4⋊C4—151no—
315C22⋊C4—52no—
316C22⋊C4—152no—
317C22⋊C4—152no—
318C22⋊C4—154no—
319C22⋊C4—152no—
320C42—31no—
321C42—31no—
322C42—31no—
323C42—31no—
324C3×C5—41noHall
325C22×C3—42no—
326C22×C3—41no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—402no—
330C22×C3—402no—
331C22×C3—402no—
332C3×C4—42no—
333C3×C4—81no—
334C3×C4—402no—
335C2×C5—22no—
336C2×C5C2×D4×A411yescharacteristic
337C2×C5C2×2−1+4⋊C311yescharacteristic
338C2×C5#594511yescharacteristic
339D5—42no—
340D5—122no—
341C23—52no—
342C23—152no—
343C23C2×D5×A411yescharacteristic
344Q8—32no—
345Q8—31no—
346Q8—32no—
347Q8—304no—
348Q8—302no—
349Q8—302no—
350Q8C2×C3×D2011yescharacteristic
351D4—104no—
352D4—102no—
353D4—304no—
354D4—302no—
355D4—302no—
356D4—304no—
357D4—302no—
358D4—602no—
359D4—602no—
360C2×C4—22no—
361C2×C4—32no—
362C2×C4—32no—
363C2×C4—31no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—52no—
368C2×C4—121no—
369C2×C4—152no—
370C2×C4—302no—
371C2×C4—302no—
372C2×C4—302no—
373C2×C4—302no—
374C2×C4—308no—
375C2×C4C2×D5×A411yescharacteristic
376C2×C4C2×D5×A411yescharacteristic
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—82no—
381C2×C3—802no—
382C5#3497⋊C311yescharacteristic, complemented, Sylow
383C22—21no—
384C22—21no—
385C22—102no—
386C22—102no—
387C22—102no—
388C22—302no—
389C22—302no—
390C22—302no—
391C22A4×D2012yes—
392C22C22×D5×A411yescharacteristic
393C4—21no—
394C4—31no—
395C4—32no—
396C4—31no—
397C4—61no—
398C4—61no—
399C4—102no—
400C4—302no—
401C4#4147⋊C312yescomplemented
402C3—41noSylow
403C2—22no—
404C2—202no—
405C2—602no—
406C2C2×A4×D2011yescharacteristic
407C2C2×#4147⋊C311yescharacteristic
408C2#66283⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FittingC2×C4×C5×Q8C2×C3
socleC22×C5C23×A4
hypercenterC22×C4D5×A4
hypocenterC5×Q8C2×C3×D4
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#4147⋊C3C2semidirect
2. C2×C4×C5×SL(2,𝔽3)C2semidirect
3. #65910⋊C3C2semidirect
4. SL(2,𝔽3)×D20C2semidirect
5. C4⋊(C5⋊(#41⋊C3))C2semidirect
6. #10617719C3semidirect
15. C2×#4147C2×C3semidirect
16. C2×C4×C5×Q8C2×C3semidirect
18. #65911C2×C3semidirect
20. #65910C2×C3semidirect
22. Q8×D20C2×C3semidirect
29. C2×C5×SL(2,𝔽3)D4semidirect
39. C2×C4×SL(2,𝔽3)D5semidirect
95. C2×C5×Q8C3×D4semidirect
151. C2×C4×Q8C3×D5semidirect
167. C2×SL(2,𝔽3)D20semidirect
273. C4×C52−1+4⋊C3semidirect
292. C2×Q8C3×D20semidirect
382. C5#3497⋊C3semidirect
401. C4#4147⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#4147⋊C32
C2×C4×C5×SL(2,𝔽3)2
#65910⋊C32
SL(2,𝔽3)×D202
C4⋊(C5⋊(#41⋊C3))2
#106177193
C3×#40764
#3497⋊C35