Finite Group Databasebeta

← C3×#10601919 #14290571

Subgroups — C3×#10601919 #14290571

Order 1920.

400 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65918C211yesmaximal, characteristic
2C3×C5×#475C211yesmaximal, characteristic, complemented
3C3×#65372C211yesmaximal, characteristic
4C3×#65367C211yesmaximal, characteristic, complemented
5C3×Q8⋊#168C211yesmaximal, characteristic, complemented
6C3×#64706C211yesmaximal, characteristic, complemented
7C3×#64703C211yesmaximal, characteristic
8#10601919C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×C4×C5×Q8C2211yescharacteristic
11C3×#4092—21no—
12C2×C3×#688—21no—
13C3×#4044—22no—
14C3×#4038—21no—
15C3×#4021—21no—
16C3×C4×D20C2211yescharacteristic
17C3×#4015C2211yescharacteristic
18C3×C4×Dic10—21no—
19C3×C5×#106—21no—
20C3×C5×C4⋊C8C2211yescharacteristic
21C3×C5×#103—21no—
22C3×#3967—21no—
23C3×#3939—22no—
24C3×C4⋊#168C2211yescharacteristic
25C3×C4×#168C2211yescharacteristic
26C3×#1650—51nomaximal, Hall
27#65918C2×C311yescharacteristic
28C5×#475C2×C311yescharacteristic, complemented
29#65372C2×C311yescharacteristic
30#65367C2×C311yescharacteristic, complemented
31Q8⋊#168C2×C311yescharacteristic, complemented
32#64706C2×C311yescharacteristic, complemented
33#64703C2×C311yescharacteristic
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8D411yescharacteristic
36C2×C3×D20D411yescharacteristic
37C2×C3×C4×D5—21no—
38C2×C3×Dic10—21no—
39C2×C3×C5×C8—21no—
40C3×C5×C4⋊C4—21no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4D411yescharacteristic
44C3×C42×C5—21no—
45C3×C42×C5C2311yescharacteristic
46C3×#688—42no—
47C3×#685—22no—
48C3×#685—22no—
49C3×#685—21no—
50C3×C4⋊Dic5—21no—
51C3×C4⋊Dic5D412yes—
52C3×C4⋊Dic5D411yescharacteristic
53C3×#683—22no—
54C3×C4×Dic5—22no—
55C2×C3×#168—21no—
56C2×C3×#168—21no—
57C3×#556—51no—
58C3×#500—51no—
59C3×#475D511yescharacteristic, complemented
60C3×#461—51no—
61C3×#332—51no—
62C3×Q8⋊C8—51no—
63C3×D4⋊C8—51no—
64C5×C4⋊Q8C22×C311yescharacteristic
65C4×C5×Q8C22×C311yescharacteristic
66#4092—21no—
67C2×#688—21no—
68#4044—22no—
69#4038—21no—
70#4021—21no—
71C4×D20C22×C311yescharacteristic
72#4015C22×C311yescharacteristic
73C4×Dic10—21no—
74C5×#106—21no—
75C5×C4⋊C8C22×C311yescharacteristic
76C5×#103—21no—
77#3967—21no—
78#3939—22no—
79C4⋊#168C22×C311yescharacteristic
80C4×#168C22×C311yescharacteristic
81#1650—51noSylow
82C22×C3×D5—21no—
83C3×C5×Q8—21no—
84C3×C5×Q8—21no—
85C3×C5×Q8—41no—
86C3×C5×Q8—41no—
87C2×C3×C4×C5—21no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—41no—
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×C3×C4×C5C2×D411yescharacteristic
93C2×C3×Dic5—21no—
94C2×C3×Dic5—22no—
95C3×D20—21no—
96C3×D20—21no—
97C3×C4×D5—41no—
98C3×C4×D5—41no—
99C3×Dic10—41no—
100C3×Dic10—41no—
101C3×C5×C8—41no—
102C3×#168—22no—
103C3×#168—41no—
104C2×C3×SD16—101no—
105C3×C4⋊Q8—51no—
106C3×C4⋊Q8C2×D511yescharacteristic
107C3×#126—102no—
108C3×#124—101no—
109C3×C22⋊Q8—101no—
110C3×C4×Q8—101no—
111C3×C4×Q8C2×D511yescharacteristic
112C3×C4×D4—51no—
113C3×#117—101no—
114C3×#107—102no—
115C3×#106—21no—
116C3×C4⋊C8—51no—
117C3×C4⋊C8C2×D511yescharacteristic
118C3×#103—21no—
119C3×#103—101no—
120C3×C4×C8—51no—
121C2×C5×Q8—21no—
122C2×C5×Q8C3×D411yescharacteristic
123C2×D20C3×D411yescharacteristic
124C2×C4×D5—21no—
125C2×Dic10—21no—
126C2×C5×C8—21no—
127C5×C4⋊C4—21no—
128C5×C4⋊C4—21no—
129C5×C4⋊C4—21no—
130C5×C4⋊C4C3×D411yescharacteristic
131C42×C5—21no—
132C42×C5C23×C311yescharacteristic
133#688—42no—
134#685—22no—
135#685—22no—
136#685—21no—
137C4⋊Dic5—21no—
138C4⋊Dic5C3×D412yes—
139C4⋊Dic5C3×D411yescharacteristic
140#683—22no—
141C4×Dic5—22no—
142C2×#168—21no—
143C2×#168—21no—
144#556—51no—
145#500—51no—
146#475C3×D511yescharacteristic, complemented
147#461—51no—
148#332—51no—
149Q8⋊C8—51no—
150D4⋊C8—51no—
151C22×C3×C5#12011yescharacteristic
152C2×C3×D5—41no—
153C2×C3×D5—41no—
154C2×C3×D5—41no—
155C3×C4×C5—21no—
156C3×C4×C5—41no—
157C3×C4×C5—41no—
158C3×C4×C5—81no—
159C3×C4×C5#13511yescharacteristic
160C3×C4×C5#13511yescharacteristic
161C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
162C3×C4×C5#13711yescharacteristic
163C3×Dic5—42no—
164C3×Dic5—41no—
165C2×C3×Q8—21no—
166C2×C3×Q8—101no—
167C2×C3×Q8C22⋊D511yescharacteristic
168C2×C3×D4—51no—
169C22×C3×C4—101no—
170C3×SD16—202no—
171C2×C3×C8—21no—
172C2×C3×C8—101no—
173C2×C3×C8—101no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—51no—
178C3×C4⋊C4—52no—
179C3×C4⋊C4—102no—
180C3×C4⋊C4—101no—
181C3×C4⋊C4C22⋊D511yescharacteristic
182C3×C22⋊C4—102no—
183C3×C22⋊C4—102no—
184C3×C22⋊C4—101no—
185C3×C42—21no—
186C3×C42—102no—
187C3×C42C22×D511yescharacteristic
188C22×D5—21no—
189C5×Q8—21no—
190C5×Q8—21no—
191C5×Q8—41no—
192C5×Q8—41no—
193C2×C4×C5—21no—
194C2×C4×C5—21no—
195C2×C4×C5—41no—
196C2×C4×C5C2×C3×D411yescharacteristic
197C2×C4×C5C2×C3×D411yescharacteristic
198C2×C4×C5C2×C3×D411yescharacteristic
199C2×Dic5—21no—
200C2×Dic5—22no—
201D20—21no—
202D20—21no—
203C4×D5—41no—
204C4×D5—41no—
205Dic10—41no—
206Dic10—41no—
207C5×C8—41no—
208#168—22no—
209#168—41no—
210C2×SD16—101no—
211C4⋊Q8—51no—
212C4⋊Q8C2×C3×D511yescharacteristic
213#126—102no—
214#124—101no—
215C22⋊Q8—101no—
216C4×Q8—101no—
217C4×Q8C2×C3×D511yescharacteristic
218C4×D4—51no—
219#117—101no—
220#107—102no—
221#106—21no—
222C4⋊C8—51no—
223C4⋊C8C2×C3×D511yescharacteristic
224#103—21no—
225#103—101no—
226C4×C8—51no—
227C2×C3×C5#44911yescharacteristic
228C2×C3×C5#45211yescharacteristic
229C2×C3×C5#45611yescharacteristic
230C3×D5—81no—
231C23×C3—101no—
232C3×Q8—21no—
233C3×Q8—21no—
234C3×Q8—41no—
235C3×Q8—41no—
236C3×Q8—201no—
237C3×Q8—201no—
238C3×D4—101no—
239C3×D4—101no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—41no—
243C2×C3×C4—101no—
244C2×C3×C4—102no—
245C2×C3×C4—201no—
246C2×C3×C4—201no—
247C2×C3×C4D4×D511yescharacteristic
248C2×C3×C4D4×D511yescharacteristic
249C2×C3×C4C2×C22⋊D511yescharacteristic
250C3×C8—41no—
251C3×C8—102no—
252C3×C8—201no—
253C22×C5C3×#12011yescharacteristic
254C2×D5—41no—
255C2×D5—41no—
256C2×D5—41no—
257C4×C5—21no—
258C4×C5—41no—
259C4×C5—41no—
260C4×C5—81no—
261C4×C5C3×#13511yescharacteristic
262C4×C5C3×#13511yescharacteristic
263C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
264C4×C5C3×#13711yescharacteristic
265Dic5—42no—
266Dic5—41no—
267C2×Q8—21no—
268C2×Q8—101no—
269C2×Q8C3×C22⋊D511yescharacteristic
270C2×D4—51no—
271C22×C4—101no—
272SD16—202no—
273C2×C8—21no—
274C2×C8—101no—
275C2×C8—101no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—51no—
280C4⋊C4—52no—
281C4⋊C4—102no—
282C4⋊C4—101no—
283C4⋊C4C3×C22⋊D511yescharacteristic
284C22⋊C4—102no—
285C22⋊C4—102no—
286C22⋊C4—101no—
287C42—21no—
288C42—102no—
289C42C22×C3×D511yescharacteristic
290C3×C5#165011yescharacteristic, complemented, Hall
291C22×C3—201no—
292C22×C3—201no—
293C22×C3—201no—
294C22×C3#408311yescharacteristic
295C3×C4—21no—
296C3×C4—41no—
297C3×C4—41no—
298C3×C4—81no—
299C3×C4—201no—
300C3×C4—202no—
301C3×C4#406011yescharacteristic
302C3×C4#406211yescharacteristic
303C3×C4#408811yescharacteristic
304C3×C4#409611yescharacteristic
305C2×C5C3×#44911yescharacteristic
306C2×C5C3×#45211yescharacteristic
307C2×C5C3×#45611yescharacteristic
308D5—81no—
309C23—101no—
310Q8—21no—
311Q8—21no—
312Q8—41no—
313Q8—41no—
314Q8—201no—
315Q8—201no—
316D4—101no—
317D4—101no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—101no—
322C2×C4—102no—
323C2×C4—201no—
324C2×C4—201no—
325C2×C4C3×D4×D511yescharacteristic
326C2×C4C3×D4×D511yescharacteristic
327C2×C4C2×C3×C22⋊D511yescharacteristic
328C8—41no—
329C8—102no—
330C8—201no—
331C2×C3—401no—
332C2×C3#6533611yescharacteristic
333C2×C3#6546111yescharacteristic
334C2×C3#6549411yescharacteristic
335C5C3×#165011yescharacteristic, complemented, Sylow
336C22—201no—
337C22—201no—
338C22—201no—
339C22C3×#408311yescharacteristic
340C4—21no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—201no—
345C4—202no—
346C4C3×#406011yescharacteristic
347C4C3×#406211yescharacteristic
348C4C3×#408811yescharacteristic
349C4C3×#409611yescharacteristic
350C3#1060191911yescharacteristic, direct product factor, Sylow
351C2—401no—
352C2C3×#6533611yescharacteristic
353C2C3×#6546111yescharacteristic
354C2C3×#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×#1650
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×#65367C2semidirect
5. C3×Q8⋊#168C2semidirect
6. C3×#64706C2semidirect
8. #10601919C3direct (N × H)
28. C5×#475C2×C3semidirect
30. #65367C2×C3semidirect
31. Q8⋊#168C2×C3semidirect
32. #64706C2×C3semidirect
59. C3×#475D5semidirect
146. #475C3×D5semidirect
290. C3×C5#1650semidirect
335. C5C3×#1650semidirect
350. C3#10601919direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#659182
C3×C5×#4752
C3×#653722
C3×#653672
C3×Q8⋊#1682
C3×#647062
C3×#647032
#106019193
C3×#16505