Finite Group Databasebeta

← C3×#10602726 #14290574

Subgroups — C3×#10602726 #14290574

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×C5×#556C211yesmaximal, characteristic
2C3×#65367C211yesmaximal, characteristic, complemented
3C3×#65309C211yesmaximal, characteristic, complemented
4C3×#65308C211yesmaximal, characteristic
5C3×#64767C211yesmaximal, characteristic, complemented
6C3×Q8⋊#168C211yesmaximal, characteristic, complemented
7C3×D4⋊#168C211yesmaximal, characteristic
8#10602726C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×C5×#126—22no—
11C3×C5×#124—21no—
12C3×C5×C22⋊Q8—21no—
13C3×C4×C5×Q8—21no—
14C3×C4×C5×Q8C2211yescharacteristic
15C3×C4×C5×D4C2211yescharacteristic
16C3×C5×#117—21no—
17C2×C3×#687—21no—
18C3×#4015C2211yescharacteristic
19C3×#3967—21no—
20C3×#3942—21no—
21C3×#3940—21no—
22C3×#3939—22no—
23C3×C4⋊#168C2211yescharacteristic
24C3×C4⋊#168C2211yescharacteristic
25C3×C4×#168C2211yescharacteristic
26C3×#1650—51nomaximal, Hall
27C5×#556C2×C311yescharacteristic
28#65367C2×C311yescharacteristic, complemented
29#65309C2×C311yescharacteristic, complemented
30#65308C2×C311yescharacteristic
31#64767C2×C311yescharacteristic, complemented
32Q8⋊#168C2×C311yescharacteristic, complemented
33D4⋊#168C2×C311yescharacteristic
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8D411yescharacteristic
36C2×C3×C5×D4D411yescharacteristic
37C22×C3×C4×C5—21no—
38C2×C3×Dic10—21no—
39C3×C5×C4⋊C4—22no—
40C3×C5×C4⋊C4—21no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4D412yes—
43C3×C5×C4⋊C4D411yescharacteristic
44C3×C5×C4⋊C4D411yescharacteristic
45C3×C5×C22⋊C4—22no—
46C3×C5×C22⋊C4—22no—
47C3×C5×C22⋊C4—21no—
48C3×C42×C5—22no—
49C3×C42×C5—21no—
50C3×C42×C5C2311yescharacteristic
51C3×#687—42no—
52C3×C4⋊Dic5—21no—
53C3×C4⋊Dic5—21no—
54C2×C3×#168—21no—
55C2×C3×#168—21no—
56C2×C3×#168—21no—
57C3×#556D511yescharacteristic
58C3×#500—51no—
59C3×#475—51no—
60C3×#461—51no—
61C3×#332—51no—
62C3×Q8⋊C8—51no—
63C3×D4⋊C8—51no—
64C5×C4⋊Q8C22×C311yescharacteristic
65C5×#126—22no—
66C5×#124—21no—
67C5×C22⋊Q8—21no—
68C4×C5×Q8—21no—
69C4×C5×Q8C22×C311yescharacteristic
70C4×C5×D4C22×C311yescharacteristic
71C5×#117—21no—
72C2×#687—21no—
73#4015C22×C311yescharacteristic
74#3967—21no—
75#3942—21no—
76#3940—21no—
77#3939—22no—
78C4⋊#168C22×C311yescharacteristic
79C4⋊#168C22×C311yescharacteristic
80C4×#168C22×C311yescharacteristic
81#1650—51noSylow
82C23×C3×C5—21no—
83C3×C5×Q8—21no—
84C3×C5×Q8—21no—
85C3×C5×Q8—41no—
86C3×C5×Q8—41no—
87C3×C5×D4—21no—
88C3×C5×D4—21no—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5—21no—
92C2×C3×C4×C5—22no—
93C2×C3×C4×C5—41no—
94C2×C3×C4×C5—41no—
95C2×C3×C4×C5C2×D411yescharacteristic
96C2×C3×C4×C5C2×D411yescharacteristic
97C2×C3×C4×C5C2×D411yescharacteristic
98C2×C3×Dic5—41no—
99C3×Dic10—41no—
100C3×Dic10—41no—
101C3×#168—22no—
102C3×#168—41no—
103C3×#168—41no—
104C2×C3×SD16—101no—
105C3×C4⋊Q8—51no—
106C3×C4⋊Q8C2×D511yescharacteristic
107C3×#126—22no—
108C3×#124—21no—
109C3×C22⋊Q8—21no—
110C3×C4×Q8—21no—
111C3×C4×Q8C2×D511yescharacteristic
112C3×C4×D4C2×D511yescharacteristic
113C3×#117—21no—
114C3×#107—102no—
115C3×#106—101no—
116C3×C4⋊C8—51no—
117C3×C4⋊C8—51no—
118C3×#103—101no—
119C3×#103—101no—
120C3×C4×C8—51no—
121C2×C5×Q8—21no—
122C2×C5×Q8C3×D411yescharacteristic
123C2×C5×D4C3×D411yescharacteristic
124C22×C4×C5—21no—
125C2×Dic10—21no—
126C5×C4⋊C4—22no—
127C5×C4⋊C4—21no—
128C5×C4⋊C4—21no—
129C5×C4⋊C4C3×D411yescharacteristic
130C5×C4⋊C4C3×D411yescharacteristic
131C5×C4⋊C4C3×D412yes—
132C5×C22⋊C4—22no—
133C5×C22⋊C4—22no—
134C5×C22⋊C4—21no—
135C42×C5—22no—
136C42×C5—21no—
137C42×C5C23×C311yescharacteristic
138#687—42no—
139C4⋊Dic5—21no—
140C4⋊Dic5—21no—
141C2×#168—21no—
142C2×#168—21no—
143C2×#168—21no—
144#556C3×D511yescharacteristic
145#500—51no—
146#475—51no—
147#461—51no—
148#332—51no—
149Q8⋊C8—51no—
150D4⋊C8—51no—
151C22×C3×C5—41no—
152C22×C3×C5—41no—
153C22×C3×C5—41no—
154C22×C3×C5#12011yescharacteristic
155C3×C4×C5—21no—
156C3×C4×C5—42no—
157C3×C4×C5—41no—
158C3×C4×C5—41no—
159C3×C4×C5—41no—
160C3×C4×C5#13511yescharacteristic
161C3×C4×C5#13511yescharacteristic
162C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
163C3×C4×C5#13711yescharacteristic
164C3×Dic5—81no—
165C2×C3×Q8—21no—
166C2×C3×Q8—101no—
167C2×C3×Q8C22⋊D511yescharacteristic
168C2×C3×D4C22⋊D511yescharacteristic
169C22×C3×C4—21no—
170C3×SD16—202no—
171C2×C3×C8—101no—
172C2×C3×C8—101no—
173C2×C3×C8—101no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—101no—
178C3×C4⋊C4—101no—
179C3×C4⋊C4C22⋊D511yescharacteristic
180C3×C4⋊C4C22⋊D511yescharacteristic
181C3×C4⋊C4C22⋊D512yes—
182C3×C22⋊C4—22no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—21no—
185C3×C42—22no—
186C3×C42—21no—
187C3×C42C22×D511yescharacteristic
188C23×C5—21no—
189C5×Q8—21no—
190C5×Q8—21no—
191C5×Q8—41no—
192C5×Q8—41no—
193C5×D4—21no—
194C5×D4—21no—
195C2×C4×C5—21no—
196C2×C4×C5—21no—
197C2×C4×C5—21no—
198C2×C4×C5—22no—
199C2×C4×C5—41no—
200C2×C4×C5—41no—
201C2×C4×C5C2×C3×D411yescharacteristic
202C2×C4×C5C2×C3×D411yescharacteristic
203C2×C4×C5C2×C3×D411yescharacteristic
204C2×Dic5—41no—
205Dic10—41no—
206Dic10—41no—
207#168—22no—
208#168—41no—
209#168—41no—
210C2×SD16—101no—
211C4⋊Q8—51no—
212C4⋊Q8C2×C3×D511yescharacteristic
213#126—22no—
214#124—21no—
215C22⋊Q8—21no—
216C4×Q8—21no—
217C4×Q8C2×C3×D511yescharacteristic
218C4×D4C2×C3×D511yescharacteristic
219#117—21no—
220#107—102no—
221#106—101no—
222C4⋊C8—51no—
223C4⋊C8—51no—
224#103—101no—
225#103—101no—
226C4×C8—51no—
227C2×C3×C5—81no—
228C2×C3×C5#44911yescharacteristic
229C2×C3×C5#45211yescharacteristic
230C2×C3×C5#45611yescharacteristic
231C23×C3—21no—
232C3×Q8—21no—
233C3×Q8—21no—
234C3×Q8—41no—
235C3×Q8—41no—
236C3×Q8—201no—
237C3×Q8—201no—
238C3×D4—21no—
239C3×D4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—22no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—201no—
247C2×C3×C4C2×C22⋊D511yescharacteristic
248C2×C3×C4C2×C22⋊D511yescharacteristic
249C2×C3×C4C2×C22⋊D511yescharacteristic
250C3×C8—102no—
251C3×C8—201no—
252C3×C8—201no—
253C22×C5—41no—
254C22×C5—41no—
255C22×C5—41no—
256C22×C5C3×#12011yescharacteristic
257C4×C5—21no—
258C4×C5—42no—
259C4×C5—41no—
260C4×C5—41no—
261C4×C5—41no—
262C4×C5C3×#13511yescharacteristic
263C4×C5C3×#13511yescharacteristic
264C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
265C4×C5C3×#13711yescharacteristic
266Dic5—81no—
267C2×Q8—21no—
268C2×Q8—101no—
269C2×Q8C3×C22⋊D511yescharacteristic
270C2×D4C3×C22⋊D511yescharacteristic
271C22×C4—21no—
272SD16—202no—
273C2×C8—101no—
274C2×C8—101no—
275C2×C8—101no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—101no—
280C4⋊C4—101no—
281C4⋊C4C3×C22⋊D511yescharacteristic
282C4⋊C4C3×C22⋊D511yescharacteristic
283C4⋊C4C3×C22⋊D512yes—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—21no—
287C42—22no—
288C42—21no—
289C42C22×C3×D511yescharacteristic
290C3×C5#165011yescharacteristic, complemented, Hall
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3#409911yescharacteristic
295C3×C4—21no—
296C3×C4—42no—
297C3×C4—41no—
298C3×C4—41no—
299C3×C4—41no—
300C3×C4—401no—
301C3×C4#409511yescharacteristic
302C3×C4#409611yescharacteristic
303C3×C4#409611yescharacteristic
304C3×C4#409711yescharacteristic
305C2×C5—81no—
306C2×C5C3×#44911yescharacteristic
307C2×C5C3×#45211yescharacteristic
308C2×C5C3×#45611yescharacteristic
309C23—21no—
310Q8—21no—
311Q8—21no—
312Q8—41no—
313Q8—41no—
314Q8—201no—
315Q8—201no—
316D4—21no—
317D4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—201no—
325C2×C4C2×C3×C22⋊D511yescharacteristic
326C2×C4C2×C3×C22⋊D511yescharacteristic
327C2×C4C2×C3×C22⋊D511yescharacteristic
328C8—102no—
329C8—201no—
330C8—201no—
331C2×C3—81no—
332C2×C3#6552711yescharacteristic
333C2×C3#6552811yescharacteristic
334C2×C3#6553511yescharacteristic
335C5C3×#165011yescharacteristic, complemented, Sylow
336C22—41no—
337C22—41no—
338C22—41no—
339C22C3×#409911yescharacteristic
340C4—21no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—401no—
346C4C3×#409511yescharacteristic
347C4C3×#409611yescharacteristic
348C4C3×#409611yescharacteristic
349C4C3×#409711yescharacteristic
350C3#1060272611yescharacteristic, direct product factor, Sylow
351C2—81no—
352C2C3×#6552711yescharacteristic
353C2C3×#6552811yescharacteristic
354C2C3×#6553511yescharacteristic
subgroupG/H
centerC22×C3#4099
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
FittingC3×C5×#556C2
socleC22×C3×C5#120
prime radicalC2×C3×C5×D4D4
hypercenterC3×#556D5
hypocenterC5C3×#1650
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#556

Complements

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

NH ≅ G/Ntype
2. C3×#65367C2semidirect
3. C3×#65309C2semidirect
5. C3×#64767C2semidirect
6. C3×Q8⋊#168C2semidirect
8. #10602726C3direct (N × H)
28. #65367C2×C3semidirect
29. #65309C2×C3semidirect
31. #64767C2×C3semidirect
32. Q8⋊#168C2×C3semidirect
290. C3×C5#1650semidirect
335. C5C3×#1650semidirect
350. C3#10602726direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5562
C3×#653672
C3×#653092
C3×#653082
C3×#647672
C3×Q8⋊#1682
C3×D4⋊#1682
#106027263
C3×#16505