Finite Group Databasebeta

← C3×#10601920 #14290634

Subgroups — C3×#10601920 #14290634

Order 1920.

468 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×D20C211yesmaximal, characteristic
2C3×C5×#475C211yesmaximal, characteristic, complemented
3C3×#65372C211yesmaximal, characteristic
4C3×#65368C211yesmaximal, characteristic, complemented
5C3×Q8⋊#168C211yesmaximal, characteristic, complemented
6C3×#64703C211yesmaximal, characteristic
7C3×#64702C211yesmaximal, characteristic, complemented
8#10601920C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×Q8×D5—21no—
10C3×C5×C4⋊Q8C2211yescharacteristic
11C3×C4×C5×Q8C2211yescharacteristic
12C2×C3×#688—21no—
13C3×#4043—21no—
14C3×#4043—22no—
15C3×C4×D20—21no—
16C3×C4×D20C2211yescharacteristic
17C3×#4015—21no—
18C3×#4015C2211yescharacteristic
19C3×C5×#106—21no—
20C3×C5×C4⋊C8C2211yescharacteristic
21C3×C5×#103—21no—
22C3×#3967—21no—
23C3×#3940—22no—
24C3×C4⋊#168C2211yescharacteristic
25C3×C4×#168C2211yescharacteristic
26C3×#1651—51nomaximal, Hall
27Q8×D20C2×C311yescharacteristic
28C5×#475C2×C311yescharacteristic, complemented
29#65372C2×C311yescharacteristic
30#65368C2×C311yescharacteristic, complemented
31Q8⋊#168C2×C311yescharacteristic, complemented
32#64703C2×C311yescharacteristic
33#64702C2×C311yescharacteristic, complemented
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8D411yescharacteristic
36C3×Q8×D5—42no—
37C3×Q8×D5—42no—
38C2×C3×D20D411yescharacteristic
39C2×C3×C4×D5—22no—
40C2×C3×C4×D5—21no—
41C2×C3×Dic10—22no—
42C2×C3×Dic10—21no—
43C2×C3×C5×C8—21no—
44C3×C5×C4⋊C4—21no—
45C3×C5×C4⋊C4—21no—
46C3×C5×C4⋊C4—21no—
47C3×C5×C4⋊C4D411yescharacteristic
48C3×C42×C5—21no—
49C3×C42×C5C2311yescharacteristic
50C3×#688—42no—
51C3×#685—22no—
52C3×#685—21no—
53C3×C4⋊Dic5—22no—
54C3×C4⋊Dic5—21no—
55C3×C4⋊Dic5D412yes—
56C3×C4⋊Dic5D411yescharacteristic
57C2×C3×#168—21no—
58C2×C3×#168—21no—
59C3×D4×Q8—51no—
60C3×#498—51no—
61C3×#475D511yescharacteristic, complemented
62C3×#461—51no—
63C3×#332—51no—
64C3×Q8⋊C8—51no—
65C3×D4⋊C8—51no—
66C2×Q8×D5—21no—
67C5×C4⋊Q8C22×C311yescharacteristic
68C4×C5×Q8C22×C311yescharacteristic
69C2×#688—21no—
70#4043—21no—
71#4043—22no—
72C4×D20—21no—
73C4×D20C22×C311yescharacteristic
74#4015—21no—
75#4015C22×C311yescharacteristic
76C5×#106—21no—
77C5×C4⋊C8C22×C311yescharacteristic
78C5×#103—21no—
79#3967—21no—
80#3940—22no—
81C4⋊#168C22×C311yescharacteristic
82C4×#168C22×C311yescharacteristic
83#1651—51noSylow
84C22×C3×D5—21no—
85C3×C5×Q8—21no—
86C3×C5×Q8—41no—
87C3×C5×Q8—41no—
88C3×C5×Q8SD1612yescomplemented
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—21no—
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—22no—
97C3×D20—21no—
98C3×D20SD1612yes—
99C3×C4×D5—44no—
100C3×C4×D5—41no—
101C3×C4×D5—41no—
102C3×Dic10—44no—
103C3×Dic10—41no—
104C3×Dic10—41no—
105C3×C5×C8—41no—
106C3×#168—22no—
107C3×#168—41no—
108C22×C3×Q8—101no—
109C2×C3×SD16—101no—
110C3×C4⋊Q8—51no—
111C3×C4⋊Q8—101no—
112C3×C4⋊Q8C2×D511yescharacteristic
113C3×C22⋊Q8—102no—
114C3×C22⋊Q8—101no—
115C3×C4×Q8C2×D511yescharacteristic
116C3×C4×D4—51no—
117C3×C4×D4—101no—
118C3×#106—21no—
119C3×#106—102no—
120C3×C4⋊C8—51no—
121C3×C4⋊C8C2×D511yescharacteristic
122C3×#103—21no—
123C3×#103—101no—
124C3×C4×C8—51no—
125C2×C5×Q8—21no—
126C2×C5×Q8C3×D411yescharacteristic
127Q8×D5—42no—
128Q8×D5—42no—
129C2×D20C3×D411yescharacteristic
130C2×C4×D5—22no—
131C2×C4×D5—21no—
132C2×Dic10—22no—
133C2×Dic10—21no—
134C2×C5×C8—21no—
135C5×C4⋊C4—21no—
136C5×C4⋊C4—21no—
137C5×C4⋊C4—21no—
138C5×C4⋊C4C3×D411yescharacteristic
139C42×C5—21no—
140C42×C5C23×C311yescharacteristic
141#688—42no—
142#685—22no—
143#685—21no—
144C4⋊Dic5—22no—
145C4⋊Dic5—21no—
146C4⋊Dic5C3×D412yes—
147C4⋊Dic5C3×D411yescharacteristic
148C2×#168—21no—
149C2×#168—21no—
150D4×Q8—51no—
151#498—51no—
152#475C3×D511yescharacteristic, complemented
153#461—51no—
154#332—51no—
155Q8⋊C8—51no—
156D4⋊C8—51no—
157C22×C3×C5#12011yescharacteristic
158C2×C3×D5—22no—
159C2×C3×D5—41no—
160C2×C3×D5—41no—
161C3×C4×C5—22no—
162C3×C4×C5—21no—
163C3×C4×C5—41no—
164C3×C4×C5—81no—
165C3×C4×C5C2×SD1611yescharacteristic
166C3×C4×C5C2×SD1611yescharacteristic
167C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
168C3×C4×C5#13711yescharacteristic
169C3×Dic5—41no—
170C3×Dic5—42no—
171C2×C3×Q8—21no—
172C2×C3×Q8—102no—
173C2×C3×Q8—101no—
174C2×C3×Q8—202no—
175C2×C3×Q8—202no—
176C2×C3×Q8C22⋊D511yescharacteristic
177C2×C3×D4—51no—
178C22×C3×C4—102no—
179C22×C3×C4—101no—
180C3×SD16—202no—
181C2×C3×C8—21no—
182C2×C3×C8—101no—
183C2×C3×C8—101no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—51no—
188C3×C4⋊C4—52no—
189C3×C4⋊C4—102no—
190C3×C4⋊C4—101no—
191C3×C4⋊C4C22⋊D511yescharacteristic
192C3×C22⋊C4—102no—
193C3×C22⋊C4—101no—
194C3×C42—21no—
195C3×C42C22×D511yescharacteristic
196C22×D5—21no—
197C5×Q8—21no—
198C5×Q8—41no—
199C5×Q8—41no—
200C5×Q8C3×SD1612yescomplemented
201C2×C4×C5—21no—
202C2×C4×C5—21no—
203C2×C4×C5—41no—
204C2×C4×C5C2×C3×D411yescharacteristic
205C2×C4×C5C2×C3×D411yescharacteristic
206C2×C4×C5C2×C3×D411yescharacteristic
207C2×Dic5—21no—
208C2×Dic5—22no—
209D20—21no—
210D20C3×SD1612yes—
211C4×D5—44no—
212C4×D5—41no—
213C4×D5—41no—
214Dic10—44no—
215Dic10—41no—
216Dic10—41no—
217C5×C8—41no—
218#168—22no—
219#168—41no—
220C22×Q8—101no—
221C2×SD16—101no—
222C4⋊Q8—51no—
223C4⋊Q8—101no—
224C4⋊Q8C2×C3×D511yescharacteristic
225C22⋊Q8—102no—
226C22⋊Q8—101no—
227C4×Q8C2×C3×D511yescharacteristic
228C4×D4—51no—
229C4×D4—101no—
230#106—21no—
231#106—102no—
232C4⋊C8—51no—
233C4⋊C8C2×C3×D511yescharacteristic
234#103—21no—
235#103—101no—
236C4×C8—51no—
237C2×C3×C5#44811yescharacteristic
238C2×C3×C5#45011yescharacteristic
239C2×C3×C5#45611yescharacteristic
240C3×D5—42no—
241C23×C3—101no—
242C3×Q8—21no—
243C3×Q8—41no—
244C3×Q8—41no—
245C3×Q8—204no—
246C3×Q8—201no—
247C3×Q8—201no—
248C3×Q8#68812yescomplemented
249C3×D4—52no—
250C3×D4—101no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—41no—
254C2×C3×C4—102no—
255C2×C3×C4—101no—
256C2×C3×C4—204no—
257C2×C3×C4—201no—
258C2×C3×C4—201no—
259C2×C3×C4D4×D511yescharacteristic
260C2×C3×C4D4×D511yescharacteristic
261C2×C3×C4C2×C22⋊D511yescharacteristic
262C3×C8—41no—
263C3×C8—102no—
264C3×C8—201no—
265C22×C5C3×#12011yescharacteristic
266C2×D5—22no—
267C2×D5—41no—
268C2×D5—41no—
269C4×C5—22no—
270C4×C5—21no—
271C4×C5—41no—
272C4×C5—81no—
273C4×C5C2×C3×SD1611yescharacteristic
274C4×C5C2×C3×SD1611yescharacteristic
275C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
276C4×C5C3×#13711yescharacteristic
277Dic5—41no—
278Dic5—42no—
279C2×Q8—21no—
280C2×Q8—102no—
281C2×Q8—101no—
282C2×Q8—202no—
283C2×Q8—202no—
284C2×Q8C3×C22⋊D511yescharacteristic
285C2×D4—51no—
286C22×C4—102no—
287C22×C4—101no—
288SD16—202no—
289C2×C8—21no—
290C2×C8—101no—
291C2×C8—101no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—51no—
296C4⋊C4—52no—
297C4⋊C4—102no—
298C4⋊C4—101no—
299C4⋊C4C3×C22⋊D511yescharacteristic
300C22⋊C4—102no—
301C22⋊C4—101no—
302C42—21no—
303C42C22×C3×D511yescharacteristic
304C3×C5#165111yescharacteristic, complemented, Hall
305C22×C3—102no—
306C22×C3—201no—
307C22×C3—201no—
308C22×C3#408311yescharacteristic
309C3×C4—22no—
310C3×C4—21no—
311C3×C4—41no—
312C3×C4—81no—
313C3×C4—202no—
314C3×C4—201no—
315C3×C4D5×SD1611yescharacteristic
316C3×C4#406111yescharacteristic
317C3×C4C2×#68811yescharacteristic
318C3×C4#409511yescharacteristic
319C2×C5C3×#44811yescharacteristic
320C2×C5C3×#45011yescharacteristic
321C2×C5C3×#45611yescharacteristic
322D5—42no—
323C23—101no—
324Q8—21no—
325Q8—41no—
326Q8—41no—
327Q8—204no—
328Q8—201no—
329Q8—201no—
330Q8C3×#68812yescomplemented
331D4—52no—
332D4—101no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—102no—
337C2×C4—101no—
338C2×C4—204no—
339C2×C4—201no—
340C2×C4—201no—
341C2×C4C3×D4×D511yescharacteristic
342C2×C4C3×D4×D511yescharacteristic
343C2×C4C2×C3×C22⋊D511yescharacteristic
344C8—41no—
345C8—102no—
346C8—201no—
347C2×C3—202no—
348C2×C3#6533511yescharacteristic
349C2×C3#6545911yescharacteristic
350C2×C3#6549411yescharacteristic
351C5C3×#165111yescharacteristic, complemented, Sylow
352C22—102no—
353C22—201no—
354C22—201no—
355C22C3×#408311yescharacteristic
356C4—22no—
357C4—21no—
358C4—41no—
359C4—81no—
360C4—202no—
361C4—201no—
362C4C3×D5×SD1611yescharacteristic
363C4C3×#406111yescharacteristic
364C4C2×C3×#68811yescharacteristic
365C4C3×#409511yescharacteristic
366C3#1060192011yescharacteristic, direct product factor, Sylow
367C2—202no—
368C2C3×#6533511yescharacteristic
369C2C3×#6545911yescharacteristic
370C2C3×#6549411yescharacteristic
subgroupG/H
centerC22×C3#4083
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
FittingC3×C5×#475C2
socleC22×C3×C5#120
prime radicalC2×C3×D20D4
hypercenterC3×#475D5
hypocenterC5C3×#1651
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#475

Complements

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

NH ≅ G/Ntype
2. C3×C5×#475C2semidirect
4. C3×#65368C2semidirect
5. C3×Q8⋊#168C2semidirect
7. C3×#64702C2semidirect
8. #10601920C3direct (N × H)
28. C5×#475C2×C3semidirect
30. #65368C2×C3semidirect
31. Q8⋊#168C2×C3semidirect
33. #64702C2×C3semidirect
61. C3×#475D5semidirect
88. C3×C5×Q8SD16semidirect
152. #475C3×D5semidirect
200. C5×Q8C3×SD16semidirect
248. C3×Q8#688semidirect
304. C3×C5#1651semidirect
330. Q8C3×#688semidirect
351. C5C3×#1651semidirect
366. C3#10601920direct (N × H)

Maximal subgroups

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

subgroupindex
C3×Q8×D202
C3×C5×#4752
C3×#653722
C3×#653682
C3×Q8⋊#1682
C3×#647032
C3×#647022
#106019203
C3×#16515