Finite Group Databasebeta

← C3×#10605017 #14297137

Subgroups — C3×#10605017 #14297137

Order 1920.

1340 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1340 conjugacy classes of subgroups — too many to draw. Only its 168 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
1C3×D5×H(5,𝔽2)C211yesmaximal, characteristic
2C3×#65779C213yesmaximal, complemented
3C3×D5⋊#99C213yesmaximal, complemented
4#10605017C311yesmaximal, characteristic, direct product factor, Hall
5C23×C3×AGL(1,𝔽5)C2211yescharacteristic, complemented
6C3×C5×H(5,𝔽2)C411yescharacteristic, complemented
7C3×#4149—23no—
8C3×D5×#41—23no—
9C3×#4144—23no—
10C3×#4144C413yescomplemented
11C2×C3×D4×D5—23no—
12C2×C3×D4×D5C2213yes—
13C2×C3×C22⋊AGL(1,𝔽5)—23no—
14C2×C3×C22⋊AGL(1,𝔽5)C2213yescomplemented
15C3×#4011—23no—
16C3×(C2×AGL(1,𝔽5))⋊C4—23no—
17C3×#3999—23no—
18C3×#1790—51nomaximal, Hall
19D5×H(5,𝔽2)C2×C311yescharacteristic
20#65779C2×C313yescomplemented
21D5⋊#99C2×C313yescomplemented
22C23×C3×D5—21no—
23C23×C3×D5D413yes—
24C23×C3×D5C2311yescharacteristic
25C22×C3×AGL(1,𝔽5)—23no—
26C22×C3×AGL(1,𝔽5)—23no—
27C22×C3×AGL(1,𝔽5)—42no—
28C3×C5×#41—23no—
29C2×C3×C5×D4—23no—
30C2×C3×C5×D4C2×C413yes—
31C2×C3×C22⋊D5—23no—
32C2×C3×C22⋊D5—23no—
33C2×C3×C22⋊D5C2×C413yes—
34C2×C3×C22⋊D5D413yes—
35C3×#713—23no—
36C3×Q8×D5—21no—
37C3×#711—23no—
38C3×#711—26no—
39C3×D4×D5—23no—
40C3×D4×D5—26no—
41C3×D4×D5—26no—
42C3×D4×D5—26no—
43C3×D4×D5—26no—
44C3×#709—26no—
45C3×#709—23no—
46C2×C3×D20—23no—
47C2×C3×D20D413yes—
48C2×C3×C4×D5—23no—
49C2×C3×C4×D5D413yes—
50C3×C22⋊AGL(1,𝔽5)—26no—
51C3×C22⋊AGL(1,𝔽5)—46no—
52C2×C3×H(5,𝔽2)—51no—
53C2×C3×#99—53no—
54C3×#386—53no—
55C23×AGL(1,𝔽5)C22×C311yescharacteristic, complemented
56C5×H(5,𝔽2)C3×C411yescharacteristic, complemented
57#4149—23no—
58D5×#41—23no—
59#4144—23no—
60#4144C3×C413yescomplemented
61C2×D4×D5—23no—
62C2×D4×D5C22×C313yes—
63C2×C22⋊AGL(1,𝔽5)—23no—
64C2×C22⋊AGL(1,𝔽5)C22×C313yescomplemented
65#4011—23no—
66(C2×AGL(1,𝔽5))⋊C4—23no—
67#3999—23no—
68#1790—51noSylow
69C23×C3×C5—21no—
70C23×C3×C5C22⋊C413yescomplemented
71C23×C3×C5C22×C411yescharacteristic
72C22×C3×D5—23no—
73C22×C3×D5—23no—
74C22×C3×D5—23no—
75C22×C3×D5—41no—
76C22×C3×D5—41no—
77C22×C3×D5—43no—
78C22×C3×D5—43no—
79C22×C3×D5—81no—
80C22×C3×D5—81no—
81C22×C3×D5C22⋊C413yescomplemented
82C22×C3×D5C2×D413yes—
83C22×C3×D5C2×D413yes—
84C2×C3×AGL(1,𝔽5)—26no—
85C2×C3×AGL(1,𝔽5)—43no—
86C2×C3×AGL(1,𝔽5)—41no—
87C2×C3×AGL(1,𝔽5)—43no—
88C2×C3×AGL(1,𝔽5)—43no—
89C2×C3×AGL(1,𝔽5)—43no—
90C3×C5×Q8—21no—
91C3×C5×D4—23no—
92C3×C5×D4—26no—
93C2×C3×C4×C5—23no—
94C2×C3×C4×C5C22⋊C413yescomplemented
95C3×C22⋊D5—26no—
96C3×C22⋊D5—26no—
97C3×C22⋊D5—26no—
98C2×C3×Dic5—23no—
99C2×C3×Dic5C22⋊C413yescomplemented
100C3×D20—23no—
101C3×D20—26no—
102C3×C4×D5—26no—
103C3×C4×D5—23no—
104C3×C4×D5—23no—
105C3×Dic10—23no—
106C3×H(5,𝔽2)—53no—
107C3×H(5,𝔽2)—103no—
108C3×H(5,𝔽2)—103no—
109C3×H(5,𝔽2)AGL(1,𝔽5)11yescharacteristic, complemented
110C2×C3×#41—103no—
111C22×C3×D4—53no—
112C22×C3×D4—103no—
113C23×C3×C4—51no—
114C2×C3×C22⋊C4—53no—
115C2×C3×C22⋊C4—103no—
116C3×#99—103no—
117C3×#99—103no—
118C3×#95—103no—
119C23×D5—21no—
120C23×D5C3×D413yes—
121C23×D5C23×C311yescharacteristic
122C22×AGL(1,𝔽5)—23no—
123C22×AGL(1,𝔽5)—23no—
124C22×AGL(1,𝔽5)—42no—
125C5×#41—23no—
126C2×C5×D4—23no—
127C2×C5×D4C2×C3×C413yes—
128C2×C22⋊D5—23no—
129C2×C22⋊D5—23no—
130C2×C22⋊D5C2×C3×C413yes—
131C2×C22⋊D5C3×D413yes—
132#713—23no—
133Q8×D5—21no—
134#711—23no—
135#711—26no—
136D4×D5—23no—
137D4×D5—26no—
138D4×D5—26no—
139D4×D5—26no—
140D4×D5—26no—
141#709—26no—
142#709—23no—
143C2×D20—23no—
144C2×D20C3×D413yes—
145C2×C4×D5—23no—
146C2×C4×D5C3×D413yes—
147C22⋊AGL(1,𝔽5)—26no—
148C22⋊AGL(1,𝔽5)—46no—
149C2×H(5,𝔽2)—51no—
150C2×#99—53no—
151#386—53no—
152C22×C3×C5—23no—
153C22×C3×C5—41no—
154C22×C3×C5—43no—
155C22×C3×C5—81no—
156C22×C3×C5C2×C22⋊C413yes—
157C2×C3×D5—23no—
158C2×C3×D5—23no—
159C2×C3×D5—23no—
160C2×C3×D5—43no—
161C2×C3×D5—43no—
162C2×C3×D5—43no—
163C2×C3×D5—43no—
164C2×C3×D5—83no—
165C2×C3×D5—83no—
166C2×C3×D5#12013yes—
167C2×C3×D5#12011yescharacteristic
168C3×AGL(1,𝔽5)—42no—
169C3×AGL(1,𝔽5)—83no—
170C3×C4×C5—23no—
171C3×Dic5—23no—
172C24×C3—51no—
173C24×C3—53no—
174C24×C3—101no—
175C3×#41—23no—
176C3×#41—106no—
177C3×#41—106no—
178C3×#41—103no—
179C3×#41—103no—
180C3×#41—103no—
181C2×C3×Q8—101no—
182C2×C3×D4—23no—
183C2×C3×D4—53no—
184C2×C3×D4—53no—
185C2×C3×D4—53no—
186C2×C3×D4—103no—
187C2×C3×D4—106no—
188C2×C3×D4—106no—
189C2×C3×D4—106no—
190C2×C3×D4—103no—
191C2×C3×D4—106no—
192C2×C3×D4—103no—
193C2×C3×D4—103no—
194C2×C3×D4C2×AGL(1,𝔽5)13yes—
195C22×C3×C4—53no—
196C22×C3×C4—103no—
197C22×C3×C4—103no—
198C22×C3×C4—103no—
199C22×C3×C4—202no—
200C3×C22⋊C4—106no—
201C3×C22⋊C4—206no—
202C23×C5—21no—
203C23×C5C3×C22⋊C413yescomplemented
204C23×C5C22×C3×C411yescharacteristic
205C22×D5—23no—
206C22×D5—23no—
207C22×D5—23no—
208C22×D5—41no—
209C22×D5—41no—
210C22×D5—43no—
211C22×D5—43no—
212C22×D5—81no—
213C22×D5—81no—
214C22×D5C3×C22⋊C413yescomplemented
215C22×D5C2×C3×D413yes—
216C22×D5C2×C3×D413yes—
217C2×AGL(1,𝔽5)—26no—
218C2×AGL(1,𝔽5)—43no—
219C2×AGL(1,𝔽5)—41no—
220C2×AGL(1,𝔽5)—43no—
221C2×AGL(1,𝔽5)—43no—
222C2×AGL(1,𝔽5)—43no—
223C5×Q8—21no—
224C5×D4—23no—
225C5×D4—26no—
226C2×C4×C5—23no—
227C2×C4×C5C3×C22⋊C413yescomplemented
228C22⋊D5—26no—
229C22⋊D5—26no—
230C22⋊D5—26no—
231C2×Dic5—23no—
232C2×Dic5C3×C22⋊C413yescomplemented
233D20—23no—
234D20—26no—
235C4×D5—26no—
236C4×D5—23no—
237C4×D5—23no—
238Dic10—23no—
239H(5,𝔽2)—53no—
240H(5,𝔽2)—103no—
241H(5,𝔽2)—103no—
242H(5,𝔽2)C3×AGL(1,𝔽5)11yescharacteristic, complemented
243C2×#41—103no—
244C22×D4—53no—
245C22×D4—103no—
246C23×C4—51no—
247C2×C22⋊C4—53no—
248C2×C22⋊C4—103no—
249#99—103no—
250#99—103no—
251#95—103no—
252C2×C3×C5—23no—
253C2×C3×C5—43no—
254C2×C3×C5#37911yescharacteristic
255C3×D5—23no—
256C3×D5—43no—
257C3×D5B(4,𝔽2)11yescharacteristic
258C3×D5#45811yescharacteristic
259C23×C3—21no—
260C23×C3—53no—
261C23×C3—53no—
262C23×C3—53no—
263C23×C3—103no—
264C23×C3—103no—
265C23×C3—103no—
266C23×C3—201no—
267C23×C3—201no—
268C23×C3—203no—
269C23×C3—203no—
270C23×C3—401no—
271C23×C3—401no—
272C23×C3C22⋊AGL(1,𝔽5)13yescomplemented
273C23×C3C22×AGL(1,𝔽5)11yescharacteristic
274C3×Q8—21no—
275C3×Q8—103no—
276C3×D4—23no—
277C3×D4—26no—
278C3×D4—106no—
279C3×D4—103no—
280C3×D4—106no—
281C3×D4—106no—
282C3×D4—106no—
283C2×C3×C4—23no—
284C2×C3×C4—53no—
285C2×C3×C4—103no—
286C2×C3×C4—103no—
287C2×C3×C4—106no—
288C2×C3×C4—103no—
289C2×C3×C4—106no—
290C2×C3×C4—201no—
291C2×C3×C4—203no—
292C2×C3×C4—203no—
293C2×C3×C4—203no—
294C2×C3×C4—203no—
295C2×C3×C4C22⋊AGL(1,𝔽5)13yescomplemented
296C22×C5—23no—
297C22×C5—41no—
298C22×C5—43no—
299C22×C5—81no—
300C22×C5C2×C3×C22⋊C413yes—
301C2×D5—23no—
302C2×D5—23no—
303C2×D5—23no—
304C2×D5—43no—
305C2×D5—43no—
306C2×D5—43no—
307C2×D5—43no—
308C2×D5—83no—
309C2×D5—83no—
310C2×D5C3×#12013yes—
311C2×D5C3×#12011yescharacteristic
312AGL(1,𝔽5)—42no—
313AGL(1,𝔽5)—83no—
314C4×C5—23no—
315Dic5—23no—
316C24—51no—
317C24—53no—
318C24—101no—
319#41—23no—
320#41—106no—
321#41—106no—
322#41—103no—
323#41—103no—
324#41—103no—
325C2×Q8—101no—
326C2×D4—23no—
327C2×D4—53no—
328C2×D4—53no—
329C2×D4—53no—
330C2×D4—103no—
331C2×D4—106no—
332C2×D4—106no—
333C2×D4—106no—
334C2×D4—103no—
335C2×D4—106no—
336C2×D4—103no—
337C2×D4—103no—
338C2×D4C2×C3×AGL(1,𝔽5)13yes—
339C22×C4—53no—
340C22×C4—103no—
341C22×C4—103no—
342C22×C4—103no—
343C22×C4—202no—
344C22⋊C4—106no—
345C22⋊C4—206no—
346C3×C5#179011yescharacteristic, complemented, Hall
347C22×C3—23no—
348C22×C3—41no—
349C22×C3—43no—
350C22×C3—51no—
351C22×C3—53no—
352C22×C3—81no—
353C22×C3—103no—
354C22×C3—103no—
355C22×C3—103no—
356C22×C3—203no—
357C22×C3—203no—
358C22×C3—203no—
359C22×C3—203no—
360C22×C3—403no—
361C22×C3—403no—
362C22×C3C2×C22⋊AGL(1,𝔽5)13yes—
363C3×C4—23no—
364C3×C4—103no—
365C3×C4—202no—
366C3×C4—403no—
367C2×C5—23no—
368C2×C5—43no—
369C2×C5C3×#37911yescharacteristic
370D5—23no—
371D5—43no—
372D5C3×B(4,𝔽2)11yescharacteristic
373D5C3×#45811yescharacteristic
374C23—21no—
375C23—53no—
376C23—53no—
377C23—53no—
378C23—103no—
379C23—103no—
380C23—103no—
381C23—201no—
382C23—201no—
383C23—203no—
384C23—203no—
385C23—401no—
386C23—401no—
387C23C3×C22⋊AGL(1,𝔽5)13yescomplemented
388C23C22×C3×AGL(1,𝔽5)11yescharacteristic
389Q8—21no—
390Q8—103no—
391D4—23no—
392D4—26no—
393D4—106no—
394D4—103no—
395D4—106no—
396D4—106no—
397D4—106no—
398C2×C4—23no—
399C2×C4—53no—
400C2×C4—103no—
401C2×C4—103no—
402C2×C4—106no—
403C2×C4—103no—
404C2×C4—106no—
405C2×C4—201no—
406C2×C4—203no—
407C2×C4—203no—
408C2×C4—203no—
409C2×C4—203no—
410C2×C4C3×C22⋊AGL(1,𝔽5)13yescomplemented
411C2×C3—23no—
412C2×C3—43no—
413C2×C3—51no—
414C2×C3—51no—
415C2×C3—103no—
416C2×C3—203no—
417C2×C3#6580011yescharacteristic
418C5C3×#179011yescharacteristic, complemented, Sylow
419C22—23no—
420C22—41no—
421C22—43no—
422C22—51no—
423C22—53no—
424C22—81no—
425C22—103no—
426C22—103no—
427C22—103no—
428C22—203no—
429C22—203no—
430C22—203no—
431C22—203no—
432C22—403no—
433C22—403no—
434C22C2×C3×C22⋊AGL(1,𝔽5)13yes—
435C4—23no—
436C4—103no—
437C4—202no—
438C4—403no—
439C3#1060501711yescharacteristic, direct product factor, Sylow
440C2—23no—
441C2—43no—
442C2—51no—
443C2—51no—
444C2—103no—
445C2—203no—
446C2C3×#6580011yescharacteristic
subgroupG/H
centerC2×C3#65800
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC3×C5×H(5,𝔽2)C4
socleC2×C3×C5#379
prime radicalC3×D5×H(5,𝔽2)C2
cosocleC23×D5C23×C3
hypercenterC3×H(5,𝔽2)AGL(1,𝔽5)
hypocenterC5C3×#1790
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C3×#65779C2semidirect
3. C3×D5⋊#99C2semidirect
4. #10605017C3direct (N × H)
5. C23×C3×AGL(1,𝔽5)C22semidirect
6. C3×C5×H(5,𝔽2)C4semidirect
10. C3×#4144C4semidirect
14. C2×C3×C22⋊AGL(1,𝔽5)C22semidirect
20. #65779C2×C3semidirect
21. D5⋊#99C2×C3semidirect
55. C23×AGL(1,𝔽5)C22×C3semidirect
56. C5×H(5,𝔽2)C3×C4semidirect
60. #4144C3×C4semidirect
64. C2×C22⋊AGL(1,𝔽5)C22×C3semidirect
70. C23×C3×C5C22⋊C4semidirect
81. C22×C3×D5C22⋊C4semidirect
94. C2×C3×C4×C5C22⋊C4semidirect
99. C2×C3×Dic5C22⋊C4semidirect
109. C3×H(5,𝔽2)AGL(1,𝔽5)semidirect
203. C23×C5C3×C22⋊C4semidirect
214. C22×D5C3×C22⋊C4semidirect
227. C2×C4×C5C3×C22⋊C4semidirect
232. C2×Dic5C3×C22⋊C4semidirect
242. H(5,𝔽2)C3×AGL(1,𝔽5)semidirect
272. C23×C3C22⋊AGL(1,𝔽5)semidirect
295. C2×C3×C4C22⋊AGL(1,𝔽5)semidirect
346. C3×C5#1790semidirect
387. C23C3×C22⋊AGL(1,𝔽5)semidirect
410. C2×C4C3×C22⋊AGL(1,𝔽5)semidirect
418. C5C3×#1790semidirect
439. C3#10605017direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D5×H(5,𝔽2)2
C3×#657792
C3×D5⋊#992
#106050173
C3×#17905