Finite Group Databasebeta

← #10617775⋊C3 #14514049

Subgroups — #10617775⋊C3 #14514049

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
1C2×#4147⋊C3C211yesmaximal, characteristic, complemented
2C22×C5⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3C5×#519⋊C3C211yesmaximal, characteristic, complemented
4#65910⋊C3C212yesmaximal, complemented
5C4⋊(C5⋊(#41⋊C3))C212yesmaximal, complemented
6#10617775C311yesmaximal, characteristic, complemented, Hall
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8#4147⋊C3—22no—
9C2×D5×SL(2,𝔽3)C2211yescharacteristic, complemented
10C2×C5⋊(#41⋊C3)—22no—
11C2×C5⋊(#41⋊C3)C2212yes—
12C2×C5⋊(#41⋊C3)C2211yescharacteristic, complemented
13C3×#4032—41nomaximal
14C4×C5×SL(2,𝔽3)C2212yescomplemented
15#3487⋊C3—51nomaximal, Hall
16C2×#4147C2×C311yescharacteristic
17C22×#713C2×C311yescharacteristic, complemented
18C5×#519C2×C311yescharacteristic, complemented
19#65911—32no—
20#65911C2×C312yescomplemented
21#65910—32no—
22#65910C2×C312yescomplemented
23#65859—31no—
24#65859—31no—
25#65858—31no—
26#65858—31no—
27C2×C3×C22⋊D5—41no—
28C22×C3×Dic5—41no—
29C2×C5×SL(2,𝔽3)D412yescomplemented
30C2×C5×SL(2,𝔽3)C2311yescharacteristic
31D5×SL(2,𝔽3)—21no—
32C5⋊(#41⋊C3)—22no—
33C5⋊(#41⋊C3)—21no—
34C3×C5×C22⋊C4—41no—
35C3×#685—42no—
36C3×C4⋊Dic5—42no—
37C2×2−1+4⋊C3—51no—
38C22×#41⋊C3—51no—
39C4⋊(#41⋊C3)—52no—
40#548⋊C3—52no—
41#519⋊C3D511yescharacteristic, complemented
42C22×C5×Q8C22×C311yescharacteristic
43#4147—22no—
44#4147—62no—
45C2×#713—22no—
46C2×#713—31no—
47C2×#713—32no—
48C2×#713—62no—
49C2×#713C22×C312yes—
50C2×#713C22×C311yescharacteristic
51C2×Q8×D5—31no—
52C2×Q8×D5C22×C311yescharacteristic
53C2×#709—31no—
54C2×#709—31no—
55C22×D20—31no—
56C22×C4×D5—31no—
57C4×C5×Q8—32no—
58C4×C5×Q8C22×C312yes—
59C5×#117—31no—
60C5×#117—31no—
61#4043—32no—
62#4043—32no—
63#4041—32no—
64#4041—34no—
65#4041—32no—
66#4041—32no—
67#4041—32no—
68#4032—31no—
69#4032A411yescharacteristic
70#4028—32no—
71#4028—31no—
72#4028—31no—
73#4021—32no—
74#4021—32no—
75#4020—32no—
76#4020—32no—
77C4×D20—34no—
78C4×D20—32no—
79C4×D20—32no—
80#3487—51noSylow
81C23×C3×C5—41no—
82C22×C3×D5—41no—
83C2×C3×C4×C5—42no—
84C3×C22⋊D5—82no—
85C2×C3×Dic5—41no—
86C2×C3×Dic5—42no—
87C2×C3×Dic5—82no—
88C5×SL(2,𝔽3)C2×D411yescharacteristic
892−1+4⋊C3—102no—
90C2×#41⋊C3—52no—
91C2×#41⋊C3—51no—
92C2×#41⋊C3—102no—
93C22×SL(2,𝔽3)—51no—
94C22×SL(2,𝔽3)C2×D511yescharacteristic
95C3×#123—201no—
96C4×SL(2,𝔽3)C2×D512yescomplemented
97C23×D5—31no—
98C2×C5×Q8—31no—
99C2×C5×Q8—32no—
100C2×C5×Q8C3×D412yes—
101C2×C5×Q8C23×C311yescharacteristic
102C22×C4×C5—31no—
103C2×C22⋊D5—31no—
104C2×C22⋊D5C2×A411yescharacteristic
105C22×Dic5C2×A411yescharacteristic
106#713—22no—
107#713—21no—
108#713—64no—
109#713—62no—
110#713—62no—
111#713—64no—
112#713—62no—
113#713—61no—
114Q8×D5—21no—
115Q8×D5—62no—
116Q8×D5—62no—
117Q8×D5—61no—
118#709—62no—
119#709—62no—
120#709—62no—
121#709—62no—
122C2×D20—32no—
123C2×D20—31no—
124C2×D20—32no—
125C2×D20—31no—
126C2×D20—34no—
127C2×D20—32no—
128C2×D20—32no—
129C2×D20—62no—
130C2×D20—62no—
131C2×C4×D5—32no—
132C2×C4×D5—31no—
133C2×C4×D5—32no—
134C2×C4×D5—32no—
135C2×C4×D5—31no—
136C2×C4×D5—62no—
137C2×C4×D5—62no—
138C2×Dic10—31no—
139C2×Dic10—31no—
140C5×C4⋊C4—32no—
141C5×C4⋊C4—32no—
142C5×C22⋊C4—31no—
143C5×C22⋊C4C2×A411yescharacteristic
144C42×C5—32no—
145C42×C5—32no—
146#685—34no—
147#685—32no—
148#685—32no—
149#685—32no—
150#685C2×A412yes—
151C4⋊Dic5—32no—
152C4⋊Dic5C2×A412yes—
153C2×2−1+4—51no—
154C22×#41—51no—
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#519C3×D511yescharacteristic, complemented
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—82no—
171C3×Dic5—81no—
172C2×C3×D4—201no—
173C22×C3×C4—201no—
174#41⋊C3—102no—
175#41⋊C3—101no—
176C2×SL(2,𝔽3)—101no—
177C2×SL(2,𝔽3)D2012yescomplemented
178C2×SL(2,𝔽3)C22×D511yescharacteristic
179C3×C4⋊C4—202no—
180C3×C22⋊C4—41no—
181C3×C22⋊C4—202no—
182C23×C5C22×A411yescharacteristic
183C22×D5—32no—
184C22×D5—31no—
185C22×D5—62no—
186C22×D5—122no—
187C22×D5C22×A411yescharacteristic
188C5×Q8—32no—
189C5×Q8—31no—
190C5×Q8—32no—
191C5×Q8C2×C3×D411yescharacteristic
192C2×C4×C5—32no—
193C2×C4×C5—32no—
194C2×C4×C5—31no—
195C2×C4×C5—31no—
196C2×C4×C5—32no—
197C2×C4×C5C22×A412yes—
198C22⋊D5—22no—
199C22⋊D5—62no—
200C2×Dic5—22no—
201C2×Dic5—31no—
202C2×Dic5C22×A412yes—
203C2×Dic5C22×A411yescharacteristic
204D20—62no—
205D20—62no—
206D20—62no—
207D20—61no—
208D20—61no—
209D20—64no—
210D20—122no—
211D20—122no—
212D20—124no—
213C4×D5—62no—
214C4×D5—62no—
215C4×D5—64no—
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—52no—
227C2×#41—51no—
228C2×#41—102no—
229C2×#41—151no—
230C2×#41—151no—
231C2×#41—151no—
232C2×#41—152no—
233C2×#41—302no—
234C22×Q8—51no—
235C22×Q8—151no—
236C22×Q8C2×C3×D511yescharacteristic
237C22×D4—151no—
238C23×C4—151no—
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—152no—
254#120—151no—
255C4×Q8—32no—
256C4×Q8C2×C3×D512yes—
257C4×D4—152no—
258C4×D4—152no—
259C4×D4—154no—
260#117—31no—
261#117—31no—
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—202no—
271C2×C3×C4—201no—
272C2×C3×C4—402no—
273SL(2,𝔽3)C2×D2011yescharacteristic
274C22×C5—22no—
275C22×C5D4×A412yes—
276C22×C5C23×A411yescharacteristic
277C2×D5—21no—
278C2×D5—22no—
279C2×D5—61no—
280C2×D5—62no—
281C2×D5—62no—
282C2×D5—62no—
283C2×D5—62no—
284C2×D5—124no—
285C4×C5—22no—
286C4×C5—32no—
287C4×C5—31no—
288C4×C5—31no—
289C4×C5—62no—
290Dic5—21no—
291Dic5—22no—
292Dic5—61no—
293C24—151no—
294#41—102no—
295#41—101no—
296#41—302no—
297#41—302no—
298#41—302no—
299#41—302no—
300#41—304no—
301#41—302no—
302#41—302no—
303#41—304no—
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×D2012yes—
315C2×Q8C22×C3×D511yescharacteristic
316C2×D4—51no—
317C2×D4—151no—
318C2×D4—152no—
319C2×D4—151no—
320C2×D4—152no—
321C2×D4—152no—
322C2×D4—154no—
323C2×D4—151no—
324C2×D4—152no—
325C2×D4—302no—
326C2×D4—302no—
327C22×C4—31no—
328C22×C4—51no—
329C22×C4—151no—
330C22×C4—152no—
331C22×C4—152no—
332C22×C4—151no—
333C22×C4—152no—
334C22×C4—302no—
335C22×C4—302no—
336C4⋊C4—32no—
337C4⋊C4—32no—
338C4⋊C4—52no—
339C4⋊C4—152no—
340C22⋊C4—31no—
341C22⋊C4—52no—
342C22⋊C4—152no—
343C22⋊C4—152no—
344C22⋊C4—154no—
345C22⋊C4—152no—
346C22⋊C4D5×A411yescharacteristic
347C42—32no—
348C42—32no—
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×C5C2×D4×A411yescharacteristic
360C2×C5#594512yes—
361D5—41no—
362D5—121no—
363D5—122no—
364C23—51no—
365C23—152no—
366C23—151no—
367C23—302no—
368C23—602no—
369C23C2×D5×A411yescharacteristic
370Q8—32no—
371Q8—31no—
372Q8—32no—
373Q8—302no—
374Q8—301no—
375Q8—301no—
376Q8C2×C3×D2011yescharacteristic
377D4—102no—
378D4—302no—
379D4—302no—
380D4—302no—
381D4—301no—
382D4—301no—
383D4—304no—
384D4—302no—
385D4—602no—
386D4—602no—
387D4—604no—
388C2×C4—32no—
389C2×C4—32no—
390C2×C4—31no—
391C2×C4—31no—
392C2×C4—32no—
393C2×C4—52no—
394C2×C4—51no—
395C2×C4—102no—
396C2×C4—151no—
397C2×C4—302no—
398C2×C4—302no—
399C2×C4—304no—
400C2×C4—301no—
401C2×C4—301no—
402C2×C4—301no—
403C2×C4—301no—
404C2×C4—304no—
405C2×C4C2×D5×A412yes—
406C2×C3—41no—
407C2×C3—42no—
408C2×C3—82no—
409C2×C3—801no—
410C5#3487⋊C311yescharacteristic, complemented, Sylow
411C22—22no—
412C22—101no—
413C22—102no—
414C22—301no—
415C22—302no—
416C22—302no—
417C22—302no—
418C22—302no—
419C22—604no—
420C22A4×D2012yes—
421C22C22×D5×A411yescharacteristic
422C4—22no—
423C4—31no—
424C4—32no—
425C4—31no—
426C4—62no—
427C4—102no—
428C4—101no—
429C4—301no—
430C3—41noSylow
431C2—22no—
432C2—201no—
433C2—602no—
434C2—601no—
435C2C2×A4×D2011yescharacteristic
436C2#66287⋊C312yes—
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FittingC5×#519C2×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. C22×C5⋊(#41⋊C3)C2semidirect
3. C5×#519⋊C3C2semidirect
4. #65910⋊C3C2semidirect
5. C4⋊(C5⋊(#41⋊C3))C2semidirect
6. #10617775C3semidirect
9. C2×D5×SL(2,𝔽3)C22semidirect
12. C2×C5⋊(#41⋊C3)C22semidirect
14. C4×C5×SL(2,𝔽3)C22semidirect
17. C22×#713C2×C3semidirect
18. C5×#519C2×C3semidirect
20. #65911C2×C3semidirect
22. #65910C2×C3semidirect
29. C2×C5×SL(2,𝔽3)D4semidirect
41. #519⋊C3D5semidirect
96. C4×SL(2,𝔽3)C2×D5semidirect
163. #519C3×D5semidirect
177. C2×SL(2,𝔽3)D20semidirect
410. C5#3487⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#4147⋊C32
C22×C5⋊(#41⋊C3)2
C5×#519⋊C32
#65910⋊C32
C4⋊(C5⋊(#41⋊C3))2
#106177753
C3×#40324
#3487⋊C35