Finite Group Databasebeta

← #10619191⋊C3 #14514230

Subgroups — #10619191⋊C3 #14514230

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×C5×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C22×C5⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3#66168⋊C3C212yesmaximal, complemented
4#66146⋊C3C211yesmaximal, characteristic, complemented
5#66151⋊C3C212yesmaximal, complemented
6#10619191C311yesmaximal, characteristic, complemented, Hall
7C5×2−1+4⋊C3—22no—
8C2×C5×#41⋊C3C2211yescharacteristic, complemented
9C22×C5×SL(2,𝔽3)C2211yescharacteristic
10C22×C5×SL(2,𝔽3)C2211yescharacteristic, complemented
11C2×C5⋊(#41⋊C3)—22no—
12C2×C5⋊(#41⋊C3)C2212yes—
13C3×#4081—41nomaximal
14Dic5×SL(2,𝔽3)C2212yescomplemented
15#3487⋊C3—51nomaximal, Hall
16C2×C5×2−1+4C2×C311yescharacteristic
17C22×#713C2×C311yescharacteristic, complemented
18#66168—32no—
19#66168C2×C312yescomplemented
20#66164—31no—
21#66164—31no—
22#66162—31no—
23#66162—31no—
24#66151—32no—
25#66151C2×C312yescomplemented
26#66146C2×C311yescharacteristic, complemented
27C2×C3×C5×D4—41no—
28C22×C3×Dic5—41no—
29C5×#41⋊C3—21no—
30C2×C5×SL(2,𝔽3)—21no—
31C2×C5×SL(2,𝔽3)D412yescomplemented
32C2×C5×SL(2,𝔽3)C2311yescharacteristic
33C5⋊(#41⋊C3)—22no—
34C3×C22⋊Dic5—42no—
35C3×C22⋊Dic5—41no—
36C3×#683—42no—
37C2×2−1+4⋊C3D511yescharacteristic, complemented
38C22×#41⋊C3—51no—
39C4⋊(#41⋊C3)—52no—
40#548⋊C3—52no—
41#519⋊C3—51no—
42C5×2−1+4—22no—
43C5×2−1+4—62no—
44C2×C5×#41—31no—
45C2×C5×#41—31no—
46C2×C5×#41—31no—
47C2×C5×#41C22×C311yescharacteristic
48C22×C5×Q8—31no—
49C22×C5×Q8C22×C311yescharacteristic
50C22×C5×Q8C22×C311yescharacteristic
51C2×#713—22no—
52C2×#713—32no—
53C2×#713—62no—
54C2×#713C22×C312yes—
55C22×D20—31no—
56C22×C4×D5—31no—
57#4093—32no—
58#4093—32no—
59#4092—32no—
60#4092—32no—
61Q8×Dic5—32no—
62Q8×Dic5C22×C312yes—
63#4086—32no—
64#4086—32no—
65#4084—32no—
66#4084—32no—
67#4083—32no—
68#4083—31no—
69#4083—31no—
70#4081—31no—
71#4081A411yescharacteristic
72#4076—34no—
73#4076—32no—
74#4076—32no—
75C4×C22⋊D5—34no—
76C4×C22⋊D5—32no—
77C4×C22⋊D5—32no—
78#4072—31no—
79#4072—31no—
80#3487—51noSylow
81C23×C3×C5—41no—
82C23×C3×C5—41no—
83C3×C5×D4—82no—
84C2×C3×C4×C5—41no—
85C2×C3×Dic5—42no—
86C2×C3×Dic5—42no—
87C2×C3×Dic5—82no—
88C5×SL(2,𝔽3)C2×D411yescharacteristic
892−1+4⋊C3—22no—
90C2×#41⋊C3—52no—
91C2×#41⋊C3—102no—
92C2×#41⋊C3C2×D511yescharacteristic, complemented
93C22×SL(2,𝔽3)C2×D511yescharacteristic
94C22×SL(2,𝔽3)C2×D511yescharacteristic, complemented
95C3×#123—201no—
96C4×SL(2,𝔽3)—52no—
97C23×D5—31no—
98C5×#41—21no—
99C5×#41—62no—
100C5×#41—62no—
101C5×#41—62no—
102C5×#41—62no—
103C5×#41—62no—
104C5×#41—62no—
105C5×#41—61no—
106C2×C5×Q8—21no—
107C2×C5×Q8—31no—
108C2×C5×Q8—31no—
109C2×C5×Q8—32no—
110C2×C5×Q8—31no—
111C2×C5×Q8—62no—
112C2×C5×Q8—62no—
113C2×C5×Q8—61no—
114C2×C5×Q8C3×D412yes—
115C2×C5×Q8C23×C311yescharacteristic
116C2×C5×D4—31no—
117C2×C5×D4—31no—
118C2×C5×D4—31no—
119C2×C5×D4C2×A411yescharacteristic
120C22×C4×C5—31no—
121C22×C4×C5—32no—
122C22×C4×C5—31no—
123C22×C4×C5—31no—
124C2×C22⋊D5—34no—
125C2×C22⋊D5—32no—
126C2×C22⋊D5—32no—
127C22×Dic5C2×A411yescharacteristic
128#713—22no—
129#713—64no—
130#713—64no—
131#713—62no—
132C2×D20—32no—
133C2×D20—32no—
134C2×D20—62no—
135C2×D20—62no—
136C2×C4×D5—32no—
137C2×C4×D5—32no—
138C2×C4×D5—62no—
139C2×C4×D5—62no—
140C22⋊Dic5—31no—
141C22⋊Dic5—32no—
142C22⋊Dic5C2×A411yescharacteristic
143C22⋊Dic5C2×A412yes—
144#685—34no—
145#685—32no—
146#685—32no—
147C4⋊Dic5—32no—
148C4⋊Dic5—32no—
149#683—32no—
150#683C2×A412yes—
151C4×Dic5—32no—
152C4×Dic5—32no—
153C2×2−1+4C3×D511yescharacteristic
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#519—51no—
164C22×C3×C5—42no—
165C22×C3×C5—41no—
166C22×C3×C5—82no—
167C22×C3×C5—81no—
168C22×C3×C5—82no—
169C3×C4×C5—81no—
170C3×Dic5—82no—
171C3×Dic5—82no—
172C2×C3×D4—41no—
173C22×C3×C4—201no—
174#41⋊C3—21no—
175#41⋊C3—102no—
176C2×SL(2,𝔽3)—21no—
177C2×SL(2,𝔽3)C22⋊D512yescomplemented
178C2×SL(2,𝔽3)C22×D511yescharacteristic
179C3×C4⋊C4—202no—
180C3×C22⋊C4—201no—
181C3×C22⋊C4—202no—
182C23×C5—31no—
183C23×C5C22×A411yescharacteristic
184C23×C5C22×A411yescharacteristic
185C22×D5—32no—
186C22×D5—62no—
187C22×D5—122no—
188C5×Q8—32no—
189C5×Q8—31no—
190C5×Q8—32no—
191C5×Q8—62no—
192C5×Q8—61no—
193C5×Q8—61no—
194C5×Q8C2×C3×D411yescharacteristic
195C5×D4—22no—
196C5×D4—62no—
197C5×D4—61no—
198C5×D4—61no—
199C5×D4—62no—
200C2×C4×C5—32no—
201C2×C4×C5—32no—
202C2×C4×C5—31no—
203C2×C4×C5—31no—
204C2×C4×C5—31no—
205C2×C4×C5—61no—
206C2×C4×C5—61no—
207C2×C4×C5—61no—
208C2×C4×C5—61no—
209C2×C4×C5—64no—
210C2×C4×C5C22×A411yescharacteristic
211C22⋊D5—122no—
212C22⋊D5—122no—
213C22⋊D5—124no—
214C2×Dic5—22no—
215C2×Dic5—32no—
216C2×Dic5C22×A412yes—
217C2×Dic5C22×A412yes—
218D20—62no—
219D20—62no—
220D20—64no—
221C4×D5—62no—
222C4×D5—62no—
223C4×D5—64no—
2242−1+4—22no—
2252−1+4—62no—
226C2×#41—31no—
227C2×#41—31no—
228C2×#41—31no—
229C2×#41—52no—
230C2×#41—102no—
231C2×#41—152no—
232C2×#41—302no—
233C2×#41C2×C3×D511yescharacteristic
234C22×Q8—31no—
235C22×Q8C2×C3×D511yescharacteristic
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—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—
265C2×C3×C5—161no—
266C23×C3—41no—
267C23×C3—41no—
268C3×D4—82no—
269C2×C3×C4—41no—
270C2×C3×C4—202no—
271C2×C3×C4—202no—
272C2×C3×C4—402no—
273SL(2,𝔽3)C2×C22⋊D511yescharacteristic
274C22×C5—22no—
275C22×C5—21no—
276C22×C5—22no—
277C22×C5—61no—
278C22×C5—62no—
279C22×C5D4×A412yes—
280C22×C5C23×A411yescharacteristic
281C2×D5—62no—
282C2×D5—62no—
283C2×D5—62no—
284C2×D5—124no—
285C4×C5—21no—
286C4×C5—31no—
287C4×C5—32no—
288C4×C5—31no—
289C4×C5—61no—
290Dic5—22no—
291Dic5—22no—
292Dic5—62no—
293C24—151no—
294#41—21no—
295#41—62no—
296#41—62no—
297#41—62no—
298#41—62no—
299#41—62no—
300#41—62no—
301#41—61no—
302#41—102no—
303#41—304no—
304#41—304no—
305#41—302no—
306C2×Q8—21no—
307C2×Q8—31no—
308C2×Q8—31no—
309C2×Q8—32no—
310C2×Q8—31no—
311C2×Q8—62no—
312C2×Q8—62no—
313C2×Q8—61no—
314C2×Q8C3×C22⋊D512yes—
315C2×Q8C22×C3×D511yescharacteristic
316C2×D4—31no—
317C2×D4—31no—
318C2×D4—31no—
319C2×D4—152no—
320C2×D4—152no—
321C2×D4—152no—
322C2×D4—154no—
323C2×D4—152no—
324C2×D4—302no—
325C2×D4—302no—
326C2×D4D5×A411yescharacteristic
327C22×C4—31no—
328C22×C4—32no—
329C22×C4—31no—
330C22×C4—31no—
331C22×C4—51no—
332C22×C4—152no—
333C22×C4—152no—
334C22×C4—302no—
335C22×C4—302no—
336C4⋊C4—52no—
337C4⋊C4—152no—
338C4⋊C4—152no—
339C4⋊C4—152no—
340C22⋊C4—51no—
341C22⋊C4—52no—
342C22⋊C4—152no—
343C22⋊C4—154no—
344C22⋊C4—152no—
345C22⋊C4—151no—
346C22⋊C4—152no—
347C42—152no—
348C42—152no—
349C3×C5—41noHall
350C22×C3—42no—
351C22×C3—41no—
352C22×C3—82no—
353C22×C3—81no—
354C22×C3—82no—
355C3×C4—81no—
356C3×C4—402no—
357C3×C4—402no—
358C2×C5—22no—
359C2×C5—41no—
360C2×C5—121no—
361C2×C5C2×D4×A411yescharacteristic
362C2×C5#594512yes—
363D5—122no—
364C23—31no—
365C23—152no—
366C23—302no—
367C23—602no—
368C23C2×D5×A411yescharacteristic
369C23C2×D5×A411yescharacteristic
370Q8—32no—
371Q8—31no—
372Q8—32no—
373Q8—62no—
374Q8—61no—
375Q8—61no—
376Q8C2×C3×C22⋊D511yescharacteristic
377D4—22no—
378D4—62no—
379D4—61no—
380D4—61no—
381D4—62no—
382D4—302no—
383D4—302no—
384D4—304no—
385D4—602no—
386D4—602no—
387D4—604no—
388C2×C4—32no—
389C2×C4—32no—
390C2×C4—31no—
391C2×C4—31no—
392C2×C4—31no—
393C2×C4—52no—
394C2×C4—52no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—64no—
400C2×C4—102no—
401C2×C4—152no—
402C2×C4—302no—
403C2×C4—302no—
404C2×C4—304no—
405C2×C4C2×D5×A411yescharacteristic
406C2×C3—42no—
407C2×C3—41no—
408C2×C3—82no—
409C2×C3—161no—
410C5#3487⋊C311yescharacteristic, complemented, Sylow
411C22—22no—
412C22—22no—
413C22—21no—
414C22—62no—
415C22—61no—
416C22—302no—
417C22—302no—
418C22—302no—
419C22—604no—
420C22A4×C22⋊D512yes—
421C22C22×D5×A411yescharacteristic
422C4—21no—
423C4—31no—
424C4—32no—
425C4—31no—
426C4—61no—
427C4—102no—
428C4—102no—
429C4—302no—
430C3—41noSylow
431C2—22no—
432C2—41no—
433C2—121no—
434C2—602no—
435C2#66287⋊C312yes—
436C2C2×A4×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, cosocleC2×C5×Q8C23×C3
FittingC2×C5×2−1+4C2×C3
socleC22×C5C23×A4
hypercenterC2×D4D5×A4
hypocenterC5×Q8C2×C3×D4
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×2−1+4⋊C3C2semidirect
2. C22×C5⋊(#41⋊C3)C2semidirect
3. #66168⋊C3C2semidirect
4. #66146⋊C3C2semidirect
5. #66151⋊C3C2semidirect
6. #10619191C3semidirect
8. C2×C5×#41⋊C3C22semidirect
10. C22×C5×SL(2,𝔽3)C22semidirect
14. Dic5×SL(2,𝔽3)C22semidirect
17. C22×#713C2×C3semidirect
19. #66168C2×C3semidirect
25. #66151C2×C3semidirect
26. #66146C2×C3semidirect
31. C2×C5×SL(2,𝔽3)D4semidirect
37. C2×2−1+4⋊C3D5semidirect
92. C2×#41⋊C3C2×D5semidirect
94. C22×SL(2,𝔽3)C2×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
C2×C5×2−1+4⋊C32
C22×C5⋊(#41⋊C3)2
#66168⋊C32
#66146⋊C32
#66151⋊C32
#106191913
C3×#40814
#3487⋊C35