Finite Group Databasebeta

← #10619165⋊C3 #14514219

Subgroups — #10619165⋊C3 #14514219

Order 1920.

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

normal subgroups only

This group has 754 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
1C22×C5×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×#4150⋊C3C211yesmaximal, characteristic, complemented
3#66165⋊C3C212yesmaximal, complemented
4#66161⋊C3C211yesmaximal, characteristic, complemented
5SL(2,𝔽3)×C22⋊D5C212yesmaximal, complemented
6#10619165C311yesmaximal, characteristic, complemented, Hall
7C2×C5×#41⋊C3—22no—
8C2×C5×#41⋊C3C2211yescharacteristic
9C2×C5×#41⋊C3C2211yescharacteristic
10C22×C5×SL(2,𝔽3)C2211yescharacteristic
11#4150⋊C3—22no—
12C2×D5×SL(2,𝔽3)C2212yescomplemented
13C3×#4076—41nomaximal
14Dic5×SL(2,𝔽3)C2212yescomplemented
15#3489⋊C3—51nomaximal, Hall
16C22×C5×#41C2×C311yescharacteristic, complemented
17C2×#4150C2×C311yescharacteristic, complemented
18#66167—31no—
19#66165C2×C312yescomplemented
20#66161C2×C311yescharacteristic, complemented
21#66159—31no—
22#66158—32no—
23Q8×C22⋊D5C2×C312yescomplemented
24#66140—32no—
25#66125—31no—
26C2×#4070—31no—
27C22×C3×C4×C5—41no—
28C2×C3×C22⋊D5—42no—
29C2×C3×D20—41no—
30C5×#41⋊C3—21no—
31C5×#41⋊C3D412yescomplemented
32C2×C5×SL(2,𝔽3)—21no—
33C2×C5×SL(2,𝔽3)C2311yescharacteristic
34D5×SL(2,𝔽3)—22no—
35C3×#685—42no—
36C3×C4⋊Dic5—41no—
37C2×2−1+4⋊C3—51no—
38C22×#41⋊C3D511yescharacteristic, complemented
39D4×SL(2,𝔽3)—52no—
40#548⋊C3—52no—
41#520⋊C3—51no—
42C2×C5×#41—22no—
43C2×C5×#41—31no—
44C2×C5×#41—31no—
45C2×C5×#41—62no—
46C2×C5×#41C22×C311yescharacteristic
47C2×C5×#41C22×C311yescharacteristic
48C22×C5×Q8C22×C311yescharacteristic
49C22×C5×D4—31no—
50C23×C4×C5—31no—
51#4150—22no—
52#4150—62no—
53C2×Q8×D5C22×C312yes—
54C2×#711—32no—
55C2×#709—31no—
56C22×Dic10—31no—
57#4099—32no—
58C2×C22⋊Dic5—32no—
59#4092—32no—
60Q8×Dic5C22×C312yes—
61#4090—32no—
62#4085—34no—
63#4084—32no—
64#4082—32no—
65#4081—34no—
66D4×Dic5—32no—
67#4076—31no—
68#4076A411yescharacteristic
69#4075—32no—
70C4×C22⋊D5—32no—
71C4×C22⋊D5—32no—
72#4072—31no—
73C2×C4⋊Dic5—31no—
74#4070—32no—
75#4070—31no—
76#4070—32no—
77#4070—31no—
78#4070—32no—
79C2×#683—32no—
80#3489—51noSylow
81C23×C3×C5—41no—
82C22×C3×D5—42no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5—41no—
85C2×C3×C4×C5—82no—
86C3×C22⋊D5—84no—
87C2×C3×Dic5—42no—
88C3×D20—82no—
89C5×SL(2,𝔽3)C2×D411yescharacteristic
902−1+4⋊C3—102no—
91C2×#41⋊C3—22no—
92C2×#41⋊C3C2×D511yescharacteristic
93C2×#41⋊C3C2×D511yescharacteristic
94C22×SL(2,𝔽3)—52no—
95C22×SL(2,𝔽3)C2×D511yescharacteristic
96C3×#121—201no—
97C4×SL(2,𝔽3)—52no—
98C24×C5—31no—
99C5×#41—21no—
100C5×#41—32no—
101C5×#41—62no—
102C5×#41—62no—
103C5×#41—62no—
104C5×#41—62no—
105C5×#41—61no—
106C5×#41C3×D412yescomplemented
107C2×C5×Q8—21no—
108C2×C5×Q8—31no—
109C2×C5×Q8—61no—
110C2×C5×Q8C23×C311yescharacteristic
111C2×C5×D4—31no—
112C2×C5×D4—31no—
113C2×C5×D4—32no—
114C2×C5×D4—62no—
115C2×C5×D4—62no—
116C22×C4×C5—31no—
117C22×C4×C5—31no—
118C22×C4×C5—31no—
119C22×C4×C5—31no—
120C22×C4×C5—31no—
121C22×C4×C5—62no—
122C22×C4×C5—62no—
123C22×C4×C5C2×A411yescharacteristic
124C2×C22⋊D5—32no—
125C2×C22⋊D5—32no—
126C2×C22⋊D5C2×A412yes—
127C22×Dic5—32no—
128C22×Dic5—32no—
129Q8×D5—22no—
130Q8×D5—62no—
131#711—64no—
132#711—64no—
133#709—62no—
134#709—62no—
135C2×D20C2×A411yescharacteristic
136C2×C4×D5—32no—
137C2×Dic10—31no—
138C2×Dic10—32no—
139C2×Dic10—62no—
140C2×Dic10—62no—
141C22⋊Dic5—34no—
142C22⋊Dic5—32no—
143C22⋊Dic5—32no—
144C22⋊Dic5—32no—
145#685—32no—
146#685C2×A412yes—
147C4⋊Dic5—31no—
148C4⋊Dic5—32no—
149C4⋊Dic5C2×A411yescharacteristic
150#683—34no—
151#683—32no—
152#683—32no—
153C4×Dic5—32no—
154C2×2−1+4—51no—
155C22×#41C3×D511yescharacteristic, complemented
156D4×Q8—52no—
157#548—52no—
158#547—152no—
159#546—152no—
160#537—151no—
161#536—151no—
162#525—151no—
163C2×C22⋊Q8—151no—
164#520—51no—
165C22×C3×C5—42no—
166C22×C3×C5—41no—
167C22×C3×C5—81no—
168C22×C3×C5—81no—
169C2×C3×D5—82no—
170C2×C3×D5—82no—
171C2×C3×D5—82no—
172C3×C4×C5—42no—
173C3×C4×C5—81no—
174C3×Dic5—82no—
175C2×C3×D4—201no—
176C2×C3×D4—202no—
177C22×C3×C4—41no—
178#41⋊C3—21no—
179#41⋊C3C22⋊D512yescomplemented
180C2×SL(2,𝔽3)—21no—
181C2×SL(2,𝔽3)—102no—
182C2×SL(2,𝔽3)C22×D511yescharacteristic
183C3×C4⋊C4—201no—
184C3×C22⋊C4—202no—
185C23×C5—31no—
186C23×C5—31no—
187C23×C5—62no—
188C23×C5—62no—
189C23×C5—121no—
190C23×C5C22×A411yescharacteristic
191C22×D5C22×A412yes—
192C5×Q8—31no—
193C5×Q8—61no—
194C5×Q8—61no—
195C5×Q8C2×C3×D411yescharacteristic
196C5×D4—32no—
197C5×D4—32no—
198C5×D4—61no—
199C5×D4—61no—
200C5×D4—64no—
201C2×C4×C5—22no—
202C2×C4×C5—31no—
203C2×C4×C5—32no—
204C2×C4×C5—32no—
205C2×C4×C5—31no—
206C2×C4×C5—61no—
207C2×C4×C5—61no—
208C2×C4×C5—62no—
209C2×C4×C5—61no—
210C2×C4×C5—61no—
211C2×C4×C5—62no—
212C2×C4×C5C22×A411yescharacteristic
213C2×C4×C5C22×A411yescharacteristic
214C22⋊D5—24no—
215C22⋊D5—64no—
216C22⋊D5—122no—
217C2×Dic5—32no—
218C2×Dic5—32no—
219C2×Dic5—64no—
220C2×Dic5—122no—
221C2×Dic5C22×A412yes—
222D20—22no—
223C4×D5—62no—
224C4×D5—62no—
225Dic10—62no—
226Dic10—62no—
227Dic10—62no—
2282−1+4—102no—
2292−1+4—302no—
230C2×#41—22no—
231C2×#41—31no—
232C2×#41—31no—
233C2×#41—62no—
234C2×#41—151no—
235C2×#41—152no—
236C2×#41C2×C3×D511yescharacteristic
237C2×#41C2×C3×D511yescharacteristic
238C22×Q8—52no—
239C22×Q8—151no—
240C22×Q8C2×C3×D511yescharacteristic
241C22×D4—31no—
242C23×C4—31no—
243C4⋊Q8—152no—
244#124—152no—
245#123—152no—
246#123—154no—
247C22⋊Q8—151no—
248C22⋊Q8—152no—
249C22⋊Q8—151no—
250C22⋊Q8—152no—
251C22⋊Q8—152no—
252C22⋊Q8—152no—
253#121—51no—
254#121—151no—
255#121—152no—
256#121—154no—
257#120—152no—
258C4×Q8—52no—
259C4×D4—152no—
260C4×D4—152no—
261C4×D4—152no—
262#117—151no—
263C2×C4⋊C4—152no—
264C2×C4⋊C4—151no—
265C2×C22⋊C4—152no—
266C2×C3×C5—41no—
267C2×C3×C5—41no—
268C2×C3×C5—41no—
269C2×C3×C5—82no—
270C3×D5—162no—
271C23×C3—41no—
272C23×C3—202no—
273C3×D4—402no—
274C3×D4—404no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—82no—
278C2×C3×C4—202no—
279SL(2,𝔽3)C2×C22⋊D511yescharacteristic
280C22×C5—21no—
281C22×C5—21no—
282C22×C5—32no—
283C22×C5—61no—
284C22×C5—61no—
285C22×C5—61no—
286C22×C5—61no—
287C22×C5—61no—
288C22×C5—124no—
289C22×C52−1+4⋊C312yescomplemented
290C22×C5C23×A411yescharacteristic
291C2×D5—22no—
292C2×D5—22no—
293C2×D5—22no—
294C4×C5—21no—
295C4×C5—31no—
296C4×C5—31no—
297C4×C5—61no—
298C4×C5D4×A412yes—
299Dic5—22no—
300Dic5—62no—
301Dic5—62no—
302C24—31no—
303#41—21no—
304#41—32no—
305#41—62no—
306#41—62no—
307#41—62no—
308#41—62no—
309#41—61no—
310#41—302no—
311#41—302no—
312#41—304no—
313#41—304no—
314#41C3×C22⋊D512yescomplemented
315C2×Q8—21no—
316C2×Q8—31no—
317C2×Q8—61no—
318C2×Q8—102no—
319C2×Q8—152no—
320C2×Q8—151no—
321C2×Q8—302no—
322C2×Q8—302no—
323C2×Q8—302no—
324C2×Q8C22×C3×D511yescharacteristic
325C2×D4—31no—
326C2×D4—31no—
327C2×D4—32no—
328C2×D4—52no—
329C2×D4—51no—
330C2×D4—62no—
331C2×D4—62no—
332C2×D4—152no—
333C2×D4—152no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—31no—
338C22×C4—31no—
339C22×C4—62no—
340C22×C4—62no—
341C22×C4—152no—
342C22×C4—152no—
343C22×C4—152no—
344C22×C4D5×A411yescharacteristic
345C4⋊C4—51no—
346C4⋊C4—152no—
347C4⋊C4—152no—
348C4⋊C4—154no—
349C4⋊C4—152no—
350C4⋊C4—151no—
351C22⋊C4—52no—
352C22⋊C4—152no—
353C22⋊C4—152no—
354C22⋊C4—152no—
355C22⋊C4—154no—
356C22⋊C4—152no—
357C42—152no—
358C3×C5—41noHall
359C22×C3—42no—
360C22×C3—41no—
361C22×C3—81no—
362C22×C3—81no—
363C22×C3—402no—
364C22×C3—402no—
365C22×C3—402no—
366C3×C4—42no—
367C3×C4—81no—
368C3×C4—402no—
369C2×C5—22no—
370C2×C5—62no—
371C2×C5—121no—
372C2×C5C2×D4×A411yescharacteristic
373C2×C5C2×2−1+4⋊C311yescharacteristic
374C2×C5#594511yescharacteristic
375D5—42no—
376C23—31no—
377C23—31no—
378C23—52no—
379C23—62no—
380C23—62no—
381C23—121no—
382C23C2×D5×A411yescharacteristic
383Q8—31no—
384Q8—61no—
385Q8—61no—
386Q8—302no—
387Q8—302no—
388Q8—302no—
389Q8C2×C3×C22⋊D511yescharacteristic
390D4—32no—
391D4—32no—
392D4—61no—
393D4—61no—
394D4—64no—
395D4—104no—
396D4—102no—
397D4—304no—
398D4—602no—
399C2×C4—22no—
400C2×C4—31no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—31no—
404C2×C4—52no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—62no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—62no—
411C2×C4—152no—
412C2×C4—152no—
413C2×C4—302no—
414C2×C4—302no—
415C2×C4—304no—
416C2×C4—602no—
417C2×C4C2×D5×A411yescharacteristic
418C2×C4C2×D5×A411yescharacteristic
419C2×C3—41no—
420C2×C3—41no—
421C2×C3—41no—
422C2×C3—82no—
423C2×C3—802no—
424C5#3489⋊C311yescharacteristic, complemented, Sylow
425C22—21no—
426C22—21no—
427C22—32no—
428C22—61no—
429C22—61no—
430C22—61no—
431C22—61no—
432C22—61no—
433C22—102no—
434C22—102no—
435C22—102no—
436C22—124no—
437C22#4150⋊C312yescomplemented
438C22C22×D5×A411yescharacteristic
439C4—21no—
440C4—31no—
441C4—31no—
442C4—61no—
443C4—102no—
444C4—302no—
445C4—302no—
446C4A4×C22⋊D512yes—
447C3—41noSylow
448C2—22no—
449C2—62no—
450C2—121no—
451C2—202no—
452C2C2×#4150⋊C311yescharacteristic
453C2#66283⋊C311yescharacteristic
454C2C2×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×#4150⋊C3C2semidirect
3. #66165⋊C3C2semidirect
4. #66161⋊C3C2semidirect
5. SL(2,𝔽3)×C22⋊D5C2semidirect
6. #10619165C3semidirect
12. C2×D5×SL(2,𝔽3)C22semidirect
14. Dic5×SL(2,𝔽3)C22semidirect
16. C22×C5×#41C2×C3semidirect
17. C2×#4150C2×C3semidirect
19. #66165C2×C3semidirect
20. #66161C2×C3semidirect
23. Q8×C22⋊D5C2×C3semidirect
31. C5×#41⋊C3D4semidirect
38. C22×#41⋊C3D5semidirect
106. C5×#41C3×D4semidirect
155. C22×#41C3×D5semidirect
179. #41⋊C3C22⋊D5semidirect
289. C22×C52−1+4⋊C3semidirect
314. #41C3×C22⋊D5semidirect
424. C5#3489⋊C3semidirect
437. C22#4150⋊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×#4150⋊C32
#66165⋊C32
#66161⋊C32
SL(2,𝔽3)×C22⋊D52
#106191653
C3×#40764
#3489⋊C35