Finite Group Databasebeta

← #10619175⋊C3 #14514218

Subgroups — #10619175⋊C3 #14514218

Order 1920.

732 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C22×C5×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×#4147⋊C3C211yesmaximal, characteristic, complemented
3#66165⋊C3C212yesmaximal, complemented
4#66168⋊C3C212yesmaximal, complemented
5#66146⋊C3C211yesmaximal, characteristic, complemented
6#10619175C311yesmaximal, characteristic, complemented, Hall
7C2×C5×#41⋊C3—22no—
8C2×C5×#41⋊C3C2212yes—
9C22×C5×SL(2,𝔽3)C2211yescharacteristic
10#4147⋊C3—22no—
11C2×D5×SL(2,𝔽3)C2211yescharacteristic, complemented
12C2×C5⋊(#41⋊C3)C2211yescharacteristic, complemented
13C3×#4075—41nomaximal
14Dic5×SL(2,𝔽3)C2212yescomplemented
15#3487⋊C3—51nomaximal, Hall
16C22×C5×#41C2×C311yescharacteristic, complemented
17C2×#4147C2×C311yescharacteristic
18#66168—32no—
19#66168C2×C312yescomplemented
20#66165—32no—
21#66165C2×C312yescomplemented
22#66146C2×C311yescharacteristic, complemented
23#66139—31no—
24#66139—31no—
25#66125—31no—
26#66125—31no—
27C22×C3×C4×C5—41no—
28C2×C3×C22⋊D5—41no—
29C5×#41⋊C3—22no—
30C2×C5×SL(2,𝔽3)D412yescomplemented
31C2×C5×SL(2,𝔽3)C2311yescharacteristic
32D5×SL(2,𝔽3)—21no—
33C5⋊(#41⋊C3)—21no—
34C3×C22⋊Dic5—41no—
35C3×#685—42no—
36C3×#683—42no—
37C2×2−1+4⋊C3—51no—
38C22×#41⋊C3D511yescharacteristic, complemented
39C4⋊(#41⋊C3)—52no—
40#548⋊C3—52no—
41#519⋊C3—51no—
42C2×C5×#41—22no—
43C2×C5×#41—32no—
44C2×C5×#41—62no—
45C2×C5×#41C22×C312yes—
46C22×C5×Q8C22×C311yescharacteristic
47C22×C5×D4—31no—
48C23×C4×C5—31no—
49#4147—22no—
50#4147—62no—
51C2×#713—31no—
52C2×#713C22×C311yescharacteristic
53C2×Q8×D5—31no—
54C2×Q8×D5C22×C311yescharacteristic
55C2×#709—31no—
56C2×#709—31no—
57#4099—32no—
58#4099—31no—
59#4099—31no—
60Q8×Dic5—32no—
61Q8×Dic5C22×C312yes—
62#4086—32no—
63#4086—32no—
64#4084—34no—
65#4084—32no—
66#4084—32no—
67#4082—32no—
68#4082—32no—
69#4076—32no—
70#4076—32no—
71#4075—31no—
72#4075A411yescharacteristic
73C4×C22⋊D5—34no—
74C4×C22⋊D5—32no—
75C4×C22⋊D5—32no—
76#4072—31no—
77#4072—31no—
78#4070—32no—
79#4070—32no—
80#3487—51noSylow
81C23×C3×C5—41no—
82C22×C3×D5—41no—
83C2×C3×C4×C5—42no—
84C2×C3×C4×C5—82no—
85C3×C22⋊D5—82no—
86C2×C3×Dic5—41no—
87C2×C3×Dic5—42no—
88C5×SL(2,𝔽3)C2×D411yescharacteristic
892−1+4⋊C3—102no—
90C2×#41⋊C3—22no—
91C2×#41⋊C3—51no—
92C2×#41⋊C3C2×D512yes—
93C22×SL(2,𝔽3)—51no—
94C22×SL(2,𝔽3)C2×D511yescharacteristic
95C3×#123—201no—
96C4×SL(2,𝔽3)—52no—
97C24×C5—31no—
98C5×#41—22no—
99C5×#41—64no—
100C5×#41—64no—
101C5×#41—62no—
102C2×C5×Q8—31no—
103C2×C5×Q8—32no—
104C2×C5×Q8C3×D412yes—
105C2×C5×Q8C23×C311yescharacteristic
106C2×C5×D4—32no—
107C2×C5×D4—32no—
108C2×C5×D4—62no—
109C2×C5×D4—62no—
110C22×C4×C5—31no—
111C22×C4×C5—32no—
112C22×C4×C5—32no—
113C22×C4×C5—62no—
114C22×C4×C5—62no—
115C22×C4×C5C2×A411yescharacteristic
116C2×C22⋊D5—31no—
117C2×C22⋊D5—34no—
118C2×C22⋊D5—32no—
119C2×C22⋊D5—32no—
120C2×C22⋊D5C2×A411yescharacteristic
121#713—21no—
122#713—62no—
123#713—62no—
124#713—61no—
125Q8×D5—21no—
126Q8×D5—62no—
127Q8×D5—62no—
128Q8×D5—61no—
129#709—62no—
130#709—62no—
131#709—62no—
132#709—62no—
133C2×D20—31no—
134C2×D20—31no—
135C2×C4×D5—32no—
136C2×C4×D5—31no—
137C2×C4×D5—31no—
138C2×Dic10—31no—
139C2×Dic10—31no—
140C22⋊Dic5—34no—
141C22⋊Dic5—31no—
142C22⋊Dic5—32no—
143C22⋊Dic5—32no—
144C22⋊Dic5C2×A411yescharacteristic
145#685—32no—
146#685C2×A412yes—
147C4⋊Dic5—32no—
148C4⋊Dic5—32no—
149#683—32no—
150#683C2×A412yes—
151C4×Dic5—32no—
152C4×Dic5—32no—
153C2×2−1+4—51no—
154C22×#41C3×D511yescharacteristic, complemented
155#550—52no—
156#550—152no—
157#548—52no—
158#548—152no—
159#535—151no—
160#535—151no—
161#525—151no—
162#525—151no—
163#519—51no—
164C22×C3×C5—42no—
165C22×C3×C5—41no—
166C22×C3×C5—82no—
167C2×C3×D5—81no—
168C2×C3×D5—82no—
169C3×C4×C5—82no—
170C3×Dic5—81no—
171C3×Dic5—82no—
172C2×C3×D4—201no—
173C22×C3×C4—41no—
174#41⋊C3—22no—
175#41⋊C3—101no—
176C2×SL(2,𝔽3)—101no—
177C2×SL(2,𝔽3)C22⋊D512yescomplemented
178C2×SL(2,𝔽3)C22×D511yescharacteristic
179C3×C4⋊C4—202no—
180C3×C22⋊C4—202no—
181C3×C22⋊C4—201no—
182C23×C5—32no—
183C23×C5—62no—
184C23×C5—122no—
185C23×C5C22×A411yescharacteristic
186C22×D5—31no—
187C22×D5C22×A411yescharacteristic
188C5×Q8—32no—
189C5×Q8—31no—
190C5×Q8—32no—
191C5×Q8C2×C3×D411yescharacteristic
192C5×D4—62no—
193C5×D4—62no—
194C5×D4—64no—
195C2×C4×C5—22no—
196C2×C4×C5—32no—
197C2×C4×C5—32no—
198C2×C4×C5—31no—
199C2×C4×C5—31no—
200C2×C4×C5—62no—
201C2×C4×C5—62no—
202C2×C4×C5—64no—
203C2×C4×C5C22×A412yes—
204C22⋊D5—22no—
205C22⋊D5—62no—
206C22⋊D5—122no—
207C22⋊D5—122no—
208C22⋊D5—124no—
209C2×Dic5—32no—
210C2×Dic5—31no—
211C2×Dic5C22×A412yes—
212C2×Dic5C22×A411yescharacteristic
213D20—62no—
214D20—61no—
215D20—61no—
216C4×D5—61no—
217C4×D5—61no—
218C4×D5—61no—
219C4×D5—61no—
220C4×D5—64no—
221Dic10—62no—
222Dic10—61no—
223Dic10—61no—
2242−1+4—102no—
2252−1+4—302no—
226C2×#41—22no—
227C2×#41—32no—
228C2×#41—51no—
229C2×#41—62no—
230C2×#41—151no—
231C2×#41—151no—
232C2×#41—151no—
233C2×#41C2×C3×D512yes—
234C22×Q8—51no—
235C22×Q8—151no—
236C22×Q8C2×C3×D511yescharacteristic
237C22×D4—31no—
238C23×C4—31no—
239#127—152no—
240#127—152no—
241#124—152no—
242#124—152no—
243#123—51no—
244#123—151no—
245C22⋊Q8—152no—
246C22⋊Q8—152no—
247#121—152no—
248#121—152no—
249#121—154no—
250#121—152no—
251#121—152no—
252#120—151no—
253#120—151no—
254#120—152no—
255C4×Q8—52no—
256C4×Q8—152no—
257C4×D4—152no—
258C4×D4—152no—
259C4×D4—154no—
260#117—151no—
261#117—151no—
262C2×C3×C5—41no—
263C2×C3×C5—42no—
264C2×C3×C5—82no—
265C3×D5—161no—
266C23×C3—41no—
267C23×C3—201no—
268C3×D4—402no—
269C2×C3×C4—42no—
270C2×C3×C4—82no—
271C2×C3×C4—202no—
272C2×C3×C4—201no—
273SL(2,𝔽3)C2×C22⋊D511yescharacteristic
274C22×C5—22no—
275C22×C5—62no—
276C22×C5—62no—
277C22×C5—62no—
278C22×C5—124no—
279C22×C5D4×A412yes—
280C22×C5C23×A411yescharacteristic
281C2×D5—21no—
282C2×D5—22no—
283C2×D5—61no—
284C2×D5—62no—
285C4×C5—22no—
286C4×C5—32no—
287C4×C5—31no—
288C4×C5—31no—
289Dic5—21no—
290Dic5—22no—
291Dic5—61no—
292Dic5—62no—
293C24—31no—
294#41—22no—
295#41—64no—
296#41—64no—
297#41—62no—
298#41—101no—
299#41—302no—
300#41—302no—
301#41—302no—
302#41—302no—
303#41—302no—
304#41—302no—
305#41—301no—
306C2×Q8—31no—
307C2×Q8—32no—
308C2×Q8—101no—
309C2×Q8—151no—
310C2×Q8—151no—
311C2×Q8—302no—
312C2×Q8—302no—
313C2×Q8—301no—
314C2×Q8C3×C22⋊D512yes—
315C2×Q8C22×C3×D511yescharacteristic
316C2×D4—32no—
317C2×D4—32no—
318C2×D4—51no—
319C2×D4—62no—
320C2×D4—62no—
321C2×D4—151no—
322C2×D4—151no—
323C2×D4—152no—
324C2×D4—151no—
325C2×D4—152no—
326C2×D4—154no—
327C22×C4—31no—
328C22×C4—32no—
329C22×C4—32no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—151no—
333C22×C4—151no—
334C22×C4—152no—
335C22×C4D5×A411yescharacteristic
336C4⋊C4—52no—
337C4⋊C4—152no—
338C4⋊C4—152no—
339C4⋊C4—152no—
340C22⋊C4—51no—
341C22⋊C4—52no—
342C22⋊C4—152no—
343C22⋊C4—152no—
344C22⋊C4—154no—
345C22⋊C4—151no—
346C22⋊C4—152no—
347C42—152no—
348C42—152no—
349C3×C5—41noHall
350C22×C3—42no—
351C22×C3—41no—
352C22×C3—82no—
353C22×C3—401no—
354C22×C3—402no—
355C3×C4—82no—
356C3×C4—402no—
357C3×C4—401no—
358C2×C5—22no—
359C2×C5—122no—
360C2×C5C2×D4×A411yescharacteristic
361C2×C5#594512yes—
362D5—41no—
363D5—121no—
364C23—32no—
365C23—51no—
366C23—62no—
367C23—122no—
368C23—151no—
369C23C2×D5×A411yescharacteristic
370Q8—32no—
371Q8—31no—
372Q8—32no—
373Q8—302no—
374Q8—301no—
375Q8—301no—
376Q8C2×C3×C22⋊D511yescharacteristic
377D4—62no—
378D4—62no—
379D4—64no—
380D4—102no—
381D4—302no—
382D4—301no—
383D4—301no—
384D4—302no—
385D4—602no—
386D4—602no—
387D4—604no—
388C2×C4—22no—
389C2×C4—32no—
390C2×C4—32no—
391C2×C4—31no—
392C2×C4—31no—
393C2×C4—52no—
394C2×C4—51no—
395C2×C4—62no—
396C2×C4—62no—
397C2×C4—64no—
398C2×C4—152no—
399C2×C4—151no—
400C2×C4—301no—
401C2×C4—301no—
402C2×C4—301no—
403C2×C4—301no—
404C2×C4—304no—
405C2×C4C2×D5×A412yes—
406C2×C3—42no—
407C2×C3—41no—
408C2×C3—82no—
409C2×C3—801no—
410C5#3487⋊C311yescharacteristic, complemented, Sylow
411C22—22no—
412C22—62no—
413C22—62no—
414C22—62no—
415C22—102no—
416C22—101no—
417C22—124no—
418C22—302no—
419C22—301no—
420C22A4×C22⋊D512yes—
421C22C22×D5×A411yescharacteristic
422C4—22no—
423C4—31no—
424C4—32no—
425C4—31no—
426C4—102no—
427C4—101no—
428C4—302no—
429C4—301no—
430C3—41noSylow
431C2—22no—
432C2—122no—
433C2—201no—
434C2—601no—
435C2#66283⋊C312yes—
436C2C2×A4×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FittingC22×C5×#41C2×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. C22×C5×#41⋊C3C2semidirect
2. C2×#4147⋊C3C2semidirect
3. #66165⋊C3C2semidirect
4. #66168⋊C3C2semidirect
5. #66146⋊C3C2semidirect
6. #10619175C3semidirect
11. C2×D5×SL(2,𝔽3)C22semidirect
12. C2×C5⋊(#41⋊C3)C22semidirect
14. Dic5×SL(2,𝔽3)C22semidirect
16. C22×C5×#41C2×C3semidirect
19. #66168C2×C3semidirect
21. #66165C2×C3semidirect
22. #66146C2×C3semidirect
30. C2×C5×SL(2,𝔽3)D4semidirect
38. C22×#41⋊C3D5semidirect
154. C22×#41C3×D5semidirect
177. C2×SL(2,𝔽3)C22⋊D5semidirect
410. C5#3487⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C5×#41⋊C32
C2×#4147⋊C32
#66165⋊C32
#66168⋊C32
#66146⋊C32
#106191753
C3×#40754
#3487⋊C35