Finite Group Databasebeta

← #10617718⋊C3 #14514019

Subgroups — #10617718⋊C3 #14514019

Order 1920.

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

normal subgroups only

This group has 588 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×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2SL(2,𝔽3)×C22⋊D5C211yesmaximal, characteristic, complemented
3#66151⋊C3C211yesmaximal, characteristic, complemented
4#65910⋊C3C211yesmaximal, characteristic, complemented
5C4×D5×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#65903⋊C3C211yesmaximal, characteristic, complemented
7C4×C5⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
8#10617718C311yesmaximal, characteristic, complemented, Hall
9C22×C5×SL(2,𝔽3)C2211yescharacteristic
10C2×D5×SL(2,𝔽3)C2211yescharacteristic
11C2×C5⋊(#41⋊C3)C2211yescharacteristic
12C3×C4×C22⋊D5—41nomaximal
13C4×C5×SL(2,𝔽3)C2211yescharacteristic
14C4×C5×SL(2,𝔽3)C2211yescharacteristic
15Dic5×SL(2,𝔽3)C2211yescharacteristic
16Dic5×SL(2,𝔽3)C2211yescharacteristic
17#3482⋊C3—51nomaximal, Hall
18C2×C4×C5×Q8C2×C311yescharacteristic, complemented
19#66151C2×C311yescharacteristic, complemented
20Q8×C22⋊D5C2×C311yescharacteristic
21#65915—31no—
22#65914—31no—
23#65912—31no—
24#65910C2×C311yescharacteristic, complemented
25C4×#713C2×C311yescharacteristic, complemented
26C4×Q8×D5C2×C311yescharacteristic, complemented
27#65903C2×C311yescharacteristic, complemented
28#65902—31no—
29#65847—31no—
30#65841—31no—
31#65813—31no—
32C4×#709—31no—
33C22×C3×C4×C5—41no—
34C2×C3×C22⋊D5—41no—
35C2×C3×C4×D5—41no—
36C2×C5×SL(2,𝔽3)—21no—
37C2×C5×SL(2,𝔽3)C2311yescharacteristic
38D5×SL(2,𝔽3)—21no—
39C5⋊(#41⋊C3)—21no—
40C3×C22⋊Dic5—41no—
41C3×#685—41no—
42C3×#683—41no—
43C3×C4×Dic5—41no—
44#557⋊C3—51no—
45D4×SL(2,𝔽3)—51no—
46#548⋊C3—51no—
47#548⋊C3—51no—
48C4×#41⋊C3—51no—
49C2×C4×SL(2,𝔽3)—51no—
50C2×C4×SL(2,𝔽3)D511yescharacteristic, complemented
51C22×C5×Q8C22×C311yescharacteristic
52C2×#713C22×C311yescharacteristic
53C2×Q8×D5C22×C311yescharacteristic
54C2×#709—31no—
55C4×C5×Q8—31no—
56C4×C5×Q8—31no—
57C4×C5×Q8C22×C311yescharacteristic
58C4×C5×Q8C22×C311yescharacteristic
59C2×C5×C4⋊C4—31no—
60C2×C42×C5—31no—
61#4093—31no—
62#4092—31no—
63#4092—31no—
64Q8×Dic5C22×C311yescharacteristic
65Q8×Dic5C22×C311yescharacteristic
66#4090—31no—
67#4076—31no—
68#4075—32no—
69C4×C22⋊D5—31no—
70C4×C22⋊D5—31no—
71C4×C22⋊D5—31no—
72C4×C22⋊D5A411yescharacteristic
73#4072—31no—
74#4070—31no—
75#4044—32no—
76#4043—31no—
77#4042—32no—
78#4041—31no—
79#4040—32no—
80#4039—31no—
81#4038—31no—
82D5×C4⋊C4—31no—
83#4036—31no—
84#4035—32no—
85#4034—31no—
86#4033—31no—
87#4022—32no—
88#4021—31no—
89C4×D20—31no—
90C4×D20—31no—
91C4×D20—31no—
92#4018—32no—
93#4018—31no—
94#4018—31no—
95C42×D5—31no—
96C42×D5—31no—
97#4016—31no—
98C4×Dic10—31no—
99C4×Dic10—31no—
100C4×Dic10—31no—
101#3482—51noSylow
102C23×C3×C5—41no—
103C22×C3×D5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5—82no—
107C3×C22⋊D5—44no—
108C2×C3×Dic5—41no—
109C2×C3×Dic5—41no—
110C2×C3×Dic5—41no—
111C3×C4×D5—82no—
112C5×SL(2,𝔽3)#4111yescharacteristic, complemented
113C2×#41⋊C3—51no—
114C22×SL(2,𝔽3)—51no—
115C22×SL(2,𝔽3)C2×D511yescharacteristic
116C3×C4×D4—201no—
117C4×SL(2,𝔽3)—51no—
118C4×SL(2,𝔽3)—51no—
119C4×SL(2,𝔽3)C2×D511yescharacteristic
120C4×SL(2,𝔽3)C2×D511yescharacteristic
121C2×C5×Q8—21no—
122C2×C5×Q8—31no—
123C2×C5×Q8—61no—
124C2×C5×Q8C23×C311yescharacteristic
125C22×C4×C5—31no—
126C22×C4×C5—31no—
127C22×C4×C5C2×A411yescharacteristic
128C2×C22⋊D5—31no—
129C2×C22⋊D5C2×A411yescharacteristic
130#713—21no—
131#713—61no—
132Q8×D5—21no—
133Q8×D5—61no—
134#709—64no—
135C2×D20—31no—
136C2×D20—31no—
137C2×C4×D5—31no—
138C2×C4×D5—31no—
139C2×C4×D5—31no—
140C2×C4×D5—31no—
141C2×C4×D5—31no—
142C2×C4×D5C2×A411yescharacteristic
143C2×Dic10—31no—
144C2×Dic10—31no—
145C5×C4⋊C4—31no—
146C5×C4⋊C4—31no—
147C5×C4⋊C4—32no—
148C42×C5—31no—
149C42×C5—31no—
150C42×C5—31no—
151C42×C5—31no—
152C22⋊Dic5—31no—
153C22⋊Dic5C2×A411yescharacteristic
154#685—32no—
155#685—32no—
156#685—32no—
157#685—31no—
158#685—31no—
159#685—31no—
160#685C2×A411yescharacteristic
161C4⋊Dic5—31no—
162C4⋊Dic5—31no—
163C4⋊Dic5—31no—
164C4⋊Dic5—31no—
165#683—32no—
166#683—32no—
167#683—32no—
168#683—31no—
169#683—31no—
170#683—31no—
171#683C2×A411yescharacteristic
172C4×Dic5—31no—
173C4×Dic5—31no—
174C4×Dic5—31no—
175C4×Dic5—31no—
176C4×Dic5—31no—
177C4×Dic5C2×A411yescharacteristic
178#557—51no—
179#556—151no—
180#552—151no—
181#552—151no—
182D4×Q8—51no—
183#548—51no—
184#548—51no—
185#547—151no—
186#533—151no—
187#529—151no—
188#529—151no—
189C4×#41—51no—
190C4×#41—151no—
191C2×C4×Q8—51no—
192C2×C4×Q8C3×D511yescharacteristic, complemented
193C22×C3×C5—42no—
194C22×C3×C5—41no—
195C22×C3×C5—81no—
196C22×C3×C5—81no—
197C2×C3×D5—42no—
198C2×C3×D5—81no—
199C2×C3×D5—81no—
200C3×C4×C5—42no—
201C3×C4×C5—81no—
202C3×Dic5—42no—
203C3×Dic5—81no—
204C3×Dic5—81no—
205C2×C3×D4—201no—
206C22×C3×C4—41no—
207C22×C3×C4—201no—
208#41⋊C3—101no—
209C2×SL(2,𝔽3)—21no—
210C2×SL(2,𝔽3)—101no—
211C2×SL(2,𝔽3)C22×D511yescharacteristic
212C3×C4⋊C4—201no—
213C3×C22⋊C4—201no—
214C3×C22⋊C4—201no—
215C3×C42—201no—
216C23×C5C22×A411yescharacteristic
217C22×D5—31no—
218C22×D5C22×A411yescharacteristic
219C5×Q8—31no—
220C5×Q8—61no—
221C5×Q8—61no—
222C5×Q8C3×#4111yescharacteristic
223C2×C4×C5—22no—
224C2×C4×C5—31no—
225C2×C4×C5—31no—
226C2×C4×C5—31no—
227C2×C4×C5—31no—
228C2×C4×C5—61no—
229C2×C4×C5—61no—
230C2×C4×C5—62no—
231C2×C4×C5C22×A411yescharacteristic
232C2×C4×C5C22×A411yescharacteristic
233C22⋊D5—62no—
234C22⋊D5#41⋊C314yescomplemented
235C2×Dic5—31no—
236C2×Dic5—31no—
237C2×Dic5—31no—
238C2×Dic5C22×A411yescharacteristic
239C2×Dic5C22×A411yescharacteristic
240C2×Dic5C22×A411yescharacteristic
241D20—61no—
242D20—61no—
243D20—62no—
244C4×D5—22no—
245C4×D5—61no—
246C4×D5—61no—
247C4×D5—61no—
248C4×D5—61no—
249C4×D5—62no—
250C4×D5—62no—
251C4×D5—62no—
252Dic10—62no—
253Dic10—61no—
254Dic10—61no—
255C2×#41—51no—
256C2×#41—151no—
257C22×Q8—51no—
258C22×Q8C2×C3×D511yescharacteristic
259C4⋊Q8—151no—
260C4⋊Q8—151no—
261#126—152no—
262#126—152no—
263#125—151no—
264#125—151no—
265#125—152no—
266#124—151no—
267#124—151no—
268#123—152no—
269#123—152no—
270C22⋊Q8—152no—
271C22⋊Q8—151no—
272C22⋊Q8—151no—
273C22⋊Q8—151no—
274C22⋊Q8—151no—
275#121—151no—
276#121—151no—
277C4×Q8—31no—
278C4×Q8—31no—
279C4×Q8—51no—
280C4×Q8—51no—
281C4×Q8—151no—
282C4×Q8—151no—
283C4×Q8—151no—
284C4×Q8—151no—
285C4×Q8C2×C3×D511yescharacteristic
286C4×Q8C2×C3×D511yescharacteristic
287C4×D4—51no—
288C4×D4—151no—
289C4×D4—151no—
290C4×D4—151no—
291C4×D4—151no—
292C4×D4—151no—
293C4×D4—151no—
294C4×D4—151no—
295#117—151no—
296#117—151no—
297#117—152no—
298#117—151no—
299#117—151no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—151no—
302C2×C42—31no—
303C2×C42—151no—
304C2×C42—151no—
305C2×C3×C5—41no—
306C2×C3×C5—41no—
307C2×C3×C5—41no—
308C2×C3×C5—82no—
309C3×D5—82no—
310C23×C3—41no—
311C23×C3—201no—
312C3×D4—204no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—82no—
316C2×C3×C4—201no—
317C2×C3×C4—201no—
318C2×C3×C4—201no—
319C2×C3×C4—402no—
320SL(2,𝔽3)#70911yescharacteristic, complemented
321C22×C5—21no—
322C22×C5—21no—
323C22×C5C2×#41⋊C312yes—
324C22×C5C23×A411yescharacteristic
325C2×D5—21no—
326C2×D5—21no—
327C2×D5—61no—
328C2×D5—61no—
329C2×D5—61no—
330C2×D5C2×#41⋊C312yes—
331C4×C5—21no—
332C4×C5—31no—
333C4×C5—31no—
334C4×C5—61no—
335C4×C5—61no—
336C4×C5—61no—
337C4×C52−1+4⋊C312yes—
338Dic5—21no—
339Dic5—21no—
340Dic5—61no—
341Dic5—61no—
342Dic5—61no—
343Dic5C2×#41⋊C312yes—
344#41—101no—
345#41—304no—
346#41—301no—
347C2×Q8—21no—
348C2×Q8—31no—
349C2×Q8—61no—
350C2×Q8—101no—
351C2×Q8—151no—
352C2×Q8—151no—
353C2×Q8—301no—
354C2×Q8C22×C3×D511yescharacteristic
355C2×D4—51no—
356C2×D4—151no—
357C2×D4—151no—
358C2×D4—151no—
359C22×C4—31no—
360C22×C4—31no—
361C22×C4—51no—
362C22×C4—151no—
363C22×C4—151no—
364C22×C4—151no—
365C22×C4—151no—
366C22×C4—151no—
367C22×C4D5×A411yescharacteristic
368C4⋊C4—31no—
369C4⋊C4—31no—
370C4⋊C4—32no—
371C4⋊C4—51no—
372C4⋊C4—151no—
373C4⋊C4—151no—
374C4⋊C4—151no—
375C4⋊C4—151no—
376C4⋊C4—151no—
377C4⋊C4—151no—
378C4⋊C4—152no—
379C4⋊C4—152no—
380C4⋊C4—152no—
381C4⋊C4—151no—
382C22⋊C4—51no—
383C22⋊C4—51no—
384C22⋊C4—151no—
385C22⋊C4—151no—
386C22⋊C4—152no—
387C22⋊C4—152no—
388C22⋊C4—152no—
389C22⋊C4—151no—
390C22⋊C4—151no—
391C42—31no—
392C42—31no—
393C42—31no—
394C42—31no—
395C42—51no—
396C42—151no—
397C42—151no—
398C42—151no—
399C42—151no—
400C42—151no—
401C3×C5—41noHall
402C22×C3—42no—
403C22×C3—41no—
404C22×C3—81no—
405C22×C3—81no—
406C22×C3—202no—
407C22×C3—401no—
408C22×C3—401no—
409C3×C4—42no—
410C3×C4—81no—
411C3×C4—202no—
412C3×C4—401no—
413C3×C4—401no—
414C2×C5—22no—
415C2×C5C22×#41⋊C311yescharacteristic
416C2×C5A4×#4111yescharacteristic
417C2×C5C2×2−1+4⋊C311yescharacteristic
418D5—22no—
419D5—121no—
420C23—51no—
421C23—151no—
422C23C2×D5×A411yescharacteristic
423Q8—31no—
424Q8—61no—
425Q8—61no—
426Q8—301no—
427Q8—301no—
428Q8—302no—
429Q8C3×#70911yescharacteristic
430D4—54no—
431D4—301no—
432D4—301no—
433D4—302no—
434D4—302no—
435C2×C4—22no—
436C2×C4—31no—
437C2×C4—31no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—51no—
441C2×C4—51no—
442C2×C4—51no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—62no—
446C2×C4—102no—
447C2×C4—151no—
448C2×C4—151no—
449C2×C4—151no—
450C2×C4—301no—
451C2×C4—301no—
452C2×C4—301no—
453C2×C4—301no—
454C2×C4—302no—
455C2×C4—302no—
456C2×C4—302no—
457C2×C4C2×D5×A411yescharacteristic
458C2×C4C2×D5×A411yescharacteristic
459C2×C3—41no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—82no—
463C2×C3—402no—
464C5#3482⋊C311yescharacteristic, complemented, Sylow
465C22—21no—
466C22—21no—
467C22—52no—
468C22—101no—
469C22—101no—
470C22—301no—
471C22—301no—
472C22—301no—
473C22D5×#41⋊C312yes—
474C22C22×D5×A411yescharacteristic
475C4—21no—
476C4—31no—
477C4—31no—
478C4—52no—
479C4—61no—
480C4—61no—
481C4—61no—
482C4—101no—
483C4—101no—
484C4—301no—
485C4—301no—
486C4—301no—
487C4#4147⋊C312yes—
488C3—41noSylow
489C2—22no—
490C2—102no—
491C2—601no—
492C2A4×#70911yescharacteristic
493C2C2×#4147⋊C311yescharacteristic
494C2C2×D5×#41⋊C311yescharacteristic
subgroupG/H
centerC2×C4C2×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FrattiniC22C22×D5×A4
FittingC2×C4×C5×Q8C2×C3
socleC22×C5C23×A4
hypercenterC22×C4D5×A4
hypocenterC5×Q8C3×#41
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C5×SL(2,𝔽3)C2semidirect
2. SL(2,𝔽3)×C22⋊D5C2semidirect
3. #66151⋊C3C2semidirect
4. #65910⋊C3C2semidirect
5. C4×D5×SL(2,𝔽3)C2semidirect
6. #65903⋊C3C2semidirect
7. C4×C5⋊(#41⋊C3)C2semidirect
8. #10617718C3semidirect
18. C2×C4×C5×Q8C2×C3semidirect
19. #66151C2×C3semidirect
24. #65910C2×C3semidirect
25. C4×#713C2×C3semidirect
26. C4×Q8×D5C2×C3semidirect
27. #65903C2×C3semidirect
50. C2×C4×SL(2,𝔽3)D5semidirect
112. C5×SL(2,𝔽3)#41semidirect
192. C2×C4×Q8C3×D5semidirect
234. C22⋊D5#41⋊C3semidirect
320. SL(2,𝔽3)#709semidirect
464. C5#3482⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×SL(2,𝔽3)2
SL(2,𝔽3)×C22⋊D52
#66151⋊C32
#65910⋊C32
C4×D5×SL(2,𝔽3)2
#65903⋊C32
C4×C5⋊(#41⋊C3)2
#106177183
C3×C4×C22⋊D54
#3482⋊C35