Finite Group Databasebeta

← C3×#10602162 #14290376

Subgroups — C3×#10602162 #14290376

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66060C211yesmaximal, characteristic
2C3×C5×#500C211yesmaximal, characteristic, complemented
3C3×#65376C211yesmaximal, characteristic, complemented
4C3×#65372C211yesmaximal, characteristic, complemented
5C3×#64767C211yesmaximal, characteristic, complemented
6C3×#64679C211yesmaximal, characteristic, complemented
7C3×#64679C211yesmaximal, characteristic, complemented
8#10602162C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×#713—21no—
10C3×C5×C4⋊Q8C2211yescharacteristic
11C3×C5×C4⋊Q8C2211yescharacteristic
12C3×#4092—21no—
13C2×C3×#688—21no—
14C3×#4041—21no—
15C3×#4040—22no—
16C3×D5×C4⋊C4—21no—
17C3×C4×D20C2211yescharacteristic
18C3×C4×D20C2211yescharacteristic
19C3×C5×#107—22no—
20C3×C4×C5×C8C2211yescharacteristic
21C3×#3967—21no—
22C3×#3941—21no—
23C3×#3939—21no—
24C3×C4⋊#168C2211yescharacteristic
25C3×C4⋊#168C2211yescharacteristic
26C3×#1647—51nomaximal, Hall
27#66060C2×C311yescharacteristic
28C5×#500C2×C311yescharacteristic, complemented
29#65376C2×C311yescharacteristic, complemented
30#65372C2×C311yescharacteristic, complemented
31#64767C2×C311yescharacteristic, complemented
32#64679C2×C311yescharacteristic, complemented
33#64679C2×C311yescharacteristic, complemented
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8—21no—
36C3×#713—42no—
37C3×#713—42no—
38C2×C3×D20—22no—
39C2×C3×D20D411yescharacteristic
40C2×C3×D20D411yescharacteristic
41C2×C3×C4×D5—22no—
42C2×C3×C4×D5—21no—
43C2×C3×C4×D5—21no—
44C2×C3×C5×C8—21no—
45C3×C5×C4⋊C4—21no—
46C3×C5×C4⋊C4—21no—
47C3×C5×C4⋊C4—21no—
48C3×C5×C4⋊C4D412yes—
49C3×C42×C5C2311yescharacteristic
50C3×#688—42no—
51C3×#685—22no—
52C3×#685—21no—
53C3×#685—21no—
54C3×C4⋊Dic5D411yescharacteristic
55C3×C4⋊Dic5D411yescharacteristic
56C3×#683—22no—
57C2×C3×#168—21no—
58C2×C3×#168—21no—
59C3×#547—51no—
60C3×#500D511yescharacteristic, complemented
61C3×#474—51no—
62C3×#461—51no—
63C3×#332—51no—
64C3×D4⋊C8—51no—
65C3×D4⋊C8—51no—
66C2×#713—21no—
67C5×C4⋊Q8C22×C311yescharacteristic
68C5×C4⋊Q8C22×C311yescharacteristic
69#4092—21no—
70C2×#688—21no—
71#4041—21no—
72#4040—22no—
73D5×C4⋊C4—21no—
74C4×D20C22×C311yescharacteristic
75C4×D20C22×C311yescharacteristic
76C5×#107—22no—
77C4×C5×C8C22×C311yescharacteristic
78#3967—21no—
79#3941—21no—
80#3939—21no—
81C4⋊#168C22×C311yescharacteristic
82C4⋊#168C22×C311yescharacteristic
83#1647—51noSylow
84C22×C3×D5—21no—
85C22×C3×D5—21no—
86C3×C5×Q8—41no—
87C3×C5×Q8—41no—
88C3×C5×Q8—41no—
89C3×C5×Q8—41no—
90C2×C3×C4×C5—22no—
91C2×C3×C4×C5—41no—
92C2×C3×C4×C5C2×D411yescharacteristic
93C2×C3×C4×C5C2×D411yescharacteristic
94C2×C3×C4×C5C2×D411yescharacteristic
95C2×C3×Dic5—21no—
96C2×C3×Dic5—21no—
97C3×D20—21no—
98C3×D20—21no—
99C3×D20—21no—
100C3×D20—44no—
101C3×D20D812yes—
102C3×C4×D5—44no—
103C3×C4×D5—41no—
104C3×C4×D5—41no—
105C3×C4×D5—41no—
106C3×C4×D5—41no—
107C3×C5×C8—22no—
108C3×#168—41no—
109C3×#168—41no—
110C2×C3×#41—101no—
111C2×C3×SD16—101no—
112C3×C4⋊Q8C2×D511yescharacteristic
113C3×C4⋊Q8C2×D511yescharacteristic
114C3×#123—102no—
115C3×C22⋊Q8—101no—
116C3×#121—101no—
117C3×C4×D4—51no—
118C3×C4×D4—51no—
119C2×C3×C4⋊C4—101no—
120C3×#107—22no—
121C3×#107—101no—
122C3×C4⋊C8—51no—
123C3×C4⋊C8—51no—
124C3×#103—101no—
125C3×#102—101no—
126C3×C4×C8C2×D511yescharacteristic
127C2×C5×Q8—21no—
128C2×C5×Q8—21no—
129#713—42no—
130#713—42no—
131C2×D20—22no—
132C2×D20C3×D411yescharacteristic
133C2×D20C3×D411yescharacteristic
134C2×C4×D5—22no—
135C2×C4×D5—21no—
136C2×C4×D5—21no—
137C2×C5×C8—21no—
138C5×C4⋊C4—21no—
139C5×C4⋊C4—21no—
140C5×C4⋊C4—21no—
141C5×C4⋊C4C3×D412yes—
142C42×C5C23×C311yescharacteristic
143#688—42no—
144#685—22no—
145#685—21no—
146#685—21no—
147C4⋊Dic5C3×D411yescharacteristic
148C4⋊Dic5C3×D411yescharacteristic
149#683—22no—
150C2×#168—21no—
151C2×#168—21no—
152#547—51no—
153#500C3×D511yescharacteristic, complemented
154#474—51no—
155#461—51no—
156#332—51no—
157D4⋊C8—51no—
158D4⋊C8—51no—
159C22×C3×C5#12011yescharacteristic
160C2×C3×D5—22no—
161C2×C3×D5—41no—
162C2×C3×D5—41no—
163C2×C3×D5—41no—
164C2×C3×D5—41no—
165C2×C3×D5—41no—
166C3×C4×C5—21no—
167C3×C4×C5—42no—
168C3×C4×C5—81no—
169C3×C4×C5C2×D811yescharacteristic
170C3×C4×C5#13511yescharacteristic
171C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
172C3×C4×C5#13711yescharacteristic
173C3×Dic5—41no—
174C3×Dic5—41no—
175C3×#41—202no—
176C3×#41—202no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×D4—51no—
180C2×C3×D4—51no—
181C2×C3×D4—102no—
182C22×C3×C4—102no—
183C22×C3×C4—101no—
184C22×C3×C4—101no—
185C3×SD16—202no—
186C2×C3×C8—21no—
187C2×C3×C8—101no—
188C2×C3×C8—101no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—51no—
193C3×C4⋊C4—51no—
194C3×C4⋊C4—102no—
195C3×C4⋊C4C22⋊D512yes—
196C3×C22⋊C4—102no—
197C3×C22⋊C4—101no—
198C3×C22⋊C4—101no—
199C3×C42C22×D511yescharacteristic
200C22×D5—21no—
201C22×D5—21no—
202C5×Q8—41no—
203C5×Q8—41no—
204C5×Q8—41no—
205C5×Q8—41no—
206C2×C4×C5—22no—
207C2×C4×C5—41no—
208C2×C4×C5C2×C3×D411yescharacteristic
209C2×C4×C5C2×C3×D411yescharacteristic
210C2×C4×C5C2×C3×D411yescharacteristic
211C2×Dic5—21no—
212C2×Dic5—21no—
213D20—21no—
214D20—21no—
215D20—21no—
216D20—44no—
217D20C3×D812yes—
218C4×D5—44no—
219C4×D5—41no—
220C4×D5—41no—
221C4×D5—41no—
222C4×D5—41no—
223C5×C8—22no—
224#168—41no—
225#168—41no—
226C2×#41—101no—
227C2×SD16—101no—
228C4⋊Q8C2×C3×D511yescharacteristic
229C4⋊Q8C2×C3×D511yescharacteristic
230#123—102no—
231C22⋊Q8—101no—
232#121—101no—
233C4×D4—51no—
234C4×D4—51no—
235C2×C4⋊C4—101no—
236#107—22no—
237#107—101no—
238C4⋊C8—51no—
239C4⋊C8—51no—
240#103—101no—
241#102—101no—
242C4×C8C2×C3×D511yescharacteristic
243C2×C3×C5#44711yescharacteristic
244C2×C3×C5#45211yescharacteristic
245C2×C3×C5#45611yescharacteristic
246C3×D5—42no—
247C3×D5—81no—
248C23×C3—101no—
249C23×C3—101no—
250C3×Q8—41no—
251C3×Q8—41no—
252C3×Q8—41no—
253C3×Q8—41no—
254C3×D4—52no—
255C3×D4—101no—
256C3×D4—101no—
257C3×D4—101no—
258C3×D4—204no—
259C2×C3×C4—22no—
260C2×C3×C4—41no—
261C2×C3×C4—101no—
262C2×C3×C4—101no—
263C2×C3×C4—204no—
264C2×C3×C4—201no—
265C2×C3×C4—201no—
266C2×C3×C4—201no—
267C2×C3×C4—201no—
268C2×C3×C4D4×D511yescharacteristic
269C2×C3×C4D4×D511yescharacteristic
270C2×C3×C4C2×C22⋊D511yescharacteristic
271C3×C8—22no—
272C3×C8—201no—
273C3×C8—201no—
274C22×C5C3×#12011yescharacteristic
275C2×D5—22no—
276C2×D5—41no—
277C2×D5—41no—
278C2×D5—41no—
279C2×D5—41no—
280C2×D5—41no—
281C4×C5—21no—
282C4×C5—42no—
283C4×C5—81no—
284C4×C5C2×C3×D811yescharacteristic
285C4×C5C3×#13511yescharacteristic
286C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
287C4×C5C3×#13711yescharacteristic
288Dic5—41no—
289Dic5—41no—
290#41—202no—
291#41—202no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×D4—51no—
295C2×D4—51no—
296C2×D4—102no—
297C22×C4—102no—
298C22×C4—101no—
299C22×C4—101no—
300SD16—202no—
301C2×C8—21no—
302C2×C8—101no—
303C2×C8—101no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—51no—
308C4⋊C4—51no—
309C4⋊C4—102no—
310C4⋊C4C3×C22⋊D512yes—
311C22⋊C4—102no—
312C22⋊C4—101no—
313C22⋊C4—101no—
314C42C22×C3×D511yescharacteristic
315C3×C5#164711yescharacteristic, complemented, Hall
316C22×C3—102no—
317C22×C3—201no—
318C22×C3—201no—
319C22×C3—201no—
320C22×C3—201no—
321C22×C3—201no—
322C22×C3#408311yescharacteristic
323C3×C4—21no—
324C3×C4—42no—
325C3×C4—81no—
326C3×C4—201no—
327C3×C4—201no—
328C3×C4D5×D811yescharacteristic
329C3×C4#405711yescharacteristic
330C3×C4#406411yescharacteristic
331C3×C4#406511yescharacteristic
332C2×C5C3×#44711yescharacteristic
333C2×C5C3×#45211yescharacteristic
334C2×C5C3×#45611yescharacteristic
335D5—42no—
336D5—81no—
337C23—101no—
338C23—101no—
339Q8—41no—
340Q8—41no—
341Q8—41no—
342Q8—41no—
343D4—52no—
344D4—101no—
345D4—101no—
346D4—101no—
347D4—204no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—101no—
351C2×C4—101no—
352C2×C4—204no—
353C2×C4—201no—
354C2×C4—201no—
355C2×C4—201no—
356C2×C4—201no—
357C2×C4C3×D4×D511yescharacteristic
358C2×C4C3×D4×D511yescharacteristic
359C2×C4C2×C3×C22⋊D511yescharacteristic
360C8—22no—
361C8—201no—
362C8—201no—
363C2×C3—202no—
364C2×C3—401no—
365C2×C3#6538411yescharacteristic
366C2×C3#6544511yescharacteristic
367C2×C3#6547611yescharacteristic
368C5C3×#164711yescharacteristic, complemented, Sylow
369C22—102no—
370C22—201no—
371C22—201no—
372C22—201no—
373C22—201no—
374C22—201no—
375C22C3×#408311yescharacteristic
376C4—21no—
377C4—42no—
378C4—81no—
379C4—201no—
380C4—201no—
381C4C3×D5×D811yescharacteristic
382C4C3×#405711yescharacteristic
383C4C3×#406411yescharacteristic
384C4C3×#406511yescharacteristic
385C3#1060216211yescharacteristic, direct product factor, Sylow
386C2—202no—
387C2—401no—
388C2C3×#6538411yescharacteristic
389C2C3×#6544511yescharacteristic
390C2C3×#6547611yescharacteristic
subgroupG/H
centerC22×C3#4083
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
FittingC3×C5×#500C2
socleC22×C3×C5#120
prime radicalC3×#66060C2
hypercenterC3×#500D5
hypocenterC5C3×#1647
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#500

Complements

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

NH ≅ G/Ntype
2. C3×C5×#500C2semidirect
3. C3×#65376C2semidirect
4. C3×#65372C2semidirect
5. C3×#64767C2semidirect
6. C3×#64679C2semidirect
7. C3×#64679C2semidirect
8. #10602162C3direct (N × H)
28. C5×#500C2×C3semidirect
29. #65376C2×C3semidirect
30. #65372C2×C3semidirect
31. #64767C2×C3semidirect
32. #64679C2×C3semidirect
33. #64679C2×C3semidirect
60. C3×#500D5semidirect
153. #500C3×D5semidirect
315. C3×C5#1647semidirect
368. C5C3×#1647semidirect
385. C3#10602162direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660602
C3×C5×#5002
C3×#653762
C3×#653722
C3×#647672
C3×#646792
#106021623
C3×#16475