Finite Group Databasebeta

← C3×#10602717 #14289488

Subgroups — C3×#10602717 #14289488

Order 1920.

428 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#554C211yesmaximal, characteristic
2C3×#65381C211yesmaximal, characteristic, complemented
3C3×#65310C211yesmaximal, characteristic, complemented
4C3×#65297C211yesmaximal, characteristic
5C3×#64766C211yesmaximal, characteristic, complemented
6C3×Q8⋊#168C211yesmaximal, characteristic, complemented
7C3×D4⋊#168C211yesmaximal, characteristic
8#10602717C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×C5×#125—21no—
11C3×C5×C22⋊Q8—21no—
12C3×C5×C22⋊Q8—22no—
13C3×C4×C5×Q8C2211yescharacteristic
14C3×C4×C5×D4—21no—
15C3×C4×C5×D4C2211yescharacteristic
16C2×C3×C5×C4⋊C4—21no—
17C2×C3×#689—22no—
18C3×#4015C2211yescharacteristic
19C3×#3967—21no—
20C3×#3964—21no—
21C3×#3940—21no—
22C3×#3939—21no—
23C3×C4⋊#168C2211yescharacteristic
24C3×C4⋊#168C2211yescharacteristic
25C3×C4×#168C2211yescharacteristic
26C3×#1632—51nomaximal, Hall
27C5×#554C2×C311yescharacteristic
28#65381C2×C311yescharacteristic, complemented
29#65310C2×C311yescharacteristic, complemented
30#65297C2×C311yescharacteristic
31#64766C2×C311yescharacteristic, complemented
32Q8⋊#168C2×C311yescharacteristic, complemented
33D4⋊#168C2×C311yescharacteristic
34C2×C3×C5×Q8D411yescharacteristic
35C2×C3×C5×Q8D412yes—
36C2×C3×C5×D4D411yescharacteristic
37C22×C3×C4×C5—22no—
38C22×C3×C4×C5—21no—
39C2×C3×Dic10—21no—
40C3×C5×C4⋊C4—22no—
41C3×C5×C4⋊C4—22no—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4—21no—
44C3×C5×C4⋊C4—21no—
45C3×C5×C4⋊C4D411yescharacteristic
46C3×C5×C4⋊C4D411yescharacteristic
47C3×C5×C22⋊C4—22no—
48C3×C5×C22⋊C4—21no—
49C3×C42×C5—21no—
50C3×C42×C5C2311yescharacteristic
51C3×#689—44no—
52C3×C4⋊Dic5—21no—
53C3×C4⋊Dic5—21no—
54C2×C3×#168—21no—
55C2×C3×#168—21no—
56C2×C3×#168—21no—
57C3×#554D511yescharacteristic
58C3×#494—51no—
59C3×#477—51no—
60C3×#476—51no—
61C3×#333—51no—
62C3×Q8⋊C8—51no—
63C3×D4⋊C8—51no—
64C5×C4⋊Q8C22×C311yescharacteristic
65C5×#125—21no—
66C5×C22⋊Q8—21no—
67C5×C22⋊Q8—22no—
68C4×C5×Q8C22×C311yescharacteristic
69C4×C5×D4—21no—
70C4×C5×D4C22×C311yescharacteristic
71C2×C5×C4⋊C4—21no—
72C2×#689—22no—
73#4015C22×C311yescharacteristic
74#3967—21no—
75#3964—21no—
76#3940—21no—
77#3939—21no—
78C4⋊#168C22×C311yescharacteristic
79C4⋊#168C22×C311yescharacteristic
80C4×#168C22×C311yescharacteristic
81#1632—51noSylow
82C23×C3×C5—21no—
83C3×C5×Q8—21no—
84C3×C5×Q8—21no—
85C3×C5×Q8—42no—
86C3×C5×D4—21no—
87C3×C5×D4Q1612yescomplemented
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5—22no—
92C2×C3×C4×C5—41no—
93C2×C3×C4×C5—41no—
94C2×C3×C4×C5—82no—
95C2×C3×C4×C5C2×D411yescharacteristic
96C2×C3×C4×C5C2×D411yescharacteristic
97C2×C3×C4×C5C2×D411yescharacteristic
98C2×C3×Dic5—41no—
99C3×Dic10—42no—
100C3×#168—22no—
101C3×#168—41no—
102C3×#168—41no—
103C2×C3×Q16—102no—
104C3×C4⋊Q8—51no—
105C3×C4⋊Q8C2×D511yescharacteristic
106C3×#125—21no—
107C3×C22⋊Q8—21no—
108C3×C22⋊Q8—22no—
109C3×C4×Q8C2×D511yescharacteristic
110C3×C4×D4—21no—
111C3×C4×D4C2×D511yescharacteristic
112C2×C3×C4⋊C4—21no—
113C3×#107—101no—
114C3×#106—101no—
115C3×C4⋊C8—51no—
116C3×C4⋊C8—51no—
117C3×#103—101no—
118C3×#102—101no—
119C3×C4×C8—51no—
120C2×C5×Q8C3×D411yescharacteristic
121C2×C5×Q8C3×D412yes—
122C2×C5×D4C3×D411yescharacteristic
123C22×C4×C5—22no—
124C22×C4×C5—21no—
125C2×Dic10—21no—
126C5×C4⋊C4—22no—
127C5×C4⋊C4—22no—
128C5×C4⋊C4—21no—
129C5×C4⋊C4—21no—
130C5×C4⋊C4—21no—
131C5×C4⋊C4C3×D411yescharacteristic
132C5×C4⋊C4C3×D411yescharacteristic
133C5×C22⋊C4—22no—
134C5×C22⋊C4—21no—
135C42×C5—21no—
136C42×C5C23×C311yescharacteristic
137#689—44no—
138C4⋊Dic5—21no—
139C4⋊Dic5—21no—
140C2×#168—21no—
141C2×#168—21no—
142C2×#168—21no—
143#554C3×D511yescharacteristic
144#494—51no—
145#477—51no—
146#476—51no—
147#333—51no—
148Q8⋊C8—51no—
149D4⋊C8—51no—
150C22×C3×C5—22no—
151C22×C3×C5—41no—
152C22×C3×C5—41no—
153C22×C3×C5#12011yescharacteristic
154C3×C4×C5—21no—
155C3×C4×C5—41no—
156C3×C4×C5—41no—
157C3×C4×C5—41no—
158C3×C4×C5—42no—
159C3×C4×C5C2×Q1611yescharacteristic
160C3×C4×C5#13511yescharacteristic
161C3×C4×C5#13711yescharacteristic
162C3×C4×C5#13711yescharacteristic
163C3×Dic5—81no—
164C2×C3×Q8—101no—
165C2×C3×Q8C22⋊D511yescharacteristic
166C2×C3×Q8C22⋊D512yes—
167C2×C3×D4C22⋊D511yescharacteristic
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C3×Q16—204no—
171C2×C3×C8—101no—
172C2×C3×C8—101no—
173C2×C3×C8—101no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—101no—
180C3×C4⋊C4—101no—
181C3×C4⋊C4C22⋊D511yescharacteristic
182C3×C4⋊C4C22⋊D511yescharacteristic
183C3×C22⋊C4—22no—
184C3×C22⋊C4—21no—
185C3×C42—21no—
186C3×C42C22×D511yescharacteristic
187C23×C5—21no—
188C5×Q8—21no—
189C5×Q8—21no—
190C5×Q8—42no—
191C5×D4—21no—
192C5×D4C3×Q1612yescomplemented
193C2×C4×C5—21no—
194C2×C4×C5—21no—
195C2×C4×C5—21no—
196C2×C4×C5—22no—
197C2×C4×C5—41no—
198C2×C4×C5—41no—
199C2×C4×C5—82no—
200C2×C4×C5C2×C3×D411yescharacteristic
201C2×C4×C5C2×C3×D411yescharacteristic
202C2×C4×C5C2×C3×D411yescharacteristic
203C2×Dic5—41no—
204Dic10—42no—
205#168—22no—
206#168—41no—
207#168—41no—
208C2×Q16—102no—
209C4⋊Q8—51no—
210C4⋊Q8C2×C3×D511yescharacteristic
211#125—21no—
212C22⋊Q8—21no—
213C22⋊Q8—22no—
214C4×Q8C2×C3×D511yescharacteristic
215C4×D4—21no—
216C4×D4C2×C3×D511yescharacteristic
217C2×C4⋊C4—21no—
218#107—101no—
219#106—101no—
220C4⋊C8—51no—
221C4⋊C8—51no—
222#103—101no—
223#102—101no—
224C4×C8—51no—
225C2×C3×C5—42no—
226C2×C3×C5#45111yescharacteristic
227C2×C3×C5#45211yescharacteristic
228C2×C3×C5#45311yescharacteristic
229C23×C3—21no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—42no—
233C3×Q8—202no—
234C3×D4—21no—
235C3×D4#68912yescomplemented
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—82no—
243C2×C3×C4—201no—
244C2×C3×C4C2×C22⋊D511yescharacteristic
245C2×C3×C4C2×C22⋊D511yescharacteristic
246C2×C3×C4C2×C22⋊D511yescharacteristic
247C3×C8—102no—
248C3×C8—201no—
249C3×C8—201no—
250C22×C5—22no—
251C22×C5—41no—
252C22×C5—41no—
253C22×C5C3×#12011yescharacteristic
254C4×C5—21no—
255C4×C5—41no—
256C4×C5—41no—
257C4×C5—41no—
258C4×C5—42no—
259C4×C5C2×C3×Q1611yescharacteristic
260C4×C5C3×#13511yescharacteristic
261C4×C5C3×#13711yescharacteristic
262C4×C5C3×#13711yescharacteristic
263Dic5—81no—
264C2×Q8—101no—
265C2×Q8C3×C22⋊D511yescharacteristic
266C2×Q8C3×C22⋊D512yes—
267C2×D4C3×C22⋊D511yescharacteristic
268C22×C4—22no—
269C22×C4—21no—
270Q16—204no—
271C2×C8—101no—
272C2×C8—101no—
273C2×C8—101no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—101no—
280C4⋊C4—101no—
281C4⋊C4C3×C22⋊D511yescharacteristic
282C4⋊C4C3×C22⋊D511yescharacteristic
283C22⋊C4—22no—
284C22⋊C4—21no—
285C42—21no—
286C42C22×C3×D511yescharacteristic
287C3×C5#163211yescharacteristic, complemented, Hall
288C22×C3—22no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3#409911yescharacteristic
292C3×C4—21no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—42no—
297C3×C4—401no—
298C3×C4#408811yescharacteristic
299C3×C4C2×#68911yescharacteristic
300C3×C4#409611yescharacteristic
301C3×C4#409711yescharacteristic
302C2×C5—42no—
303C2×C5C3×#45111yescharacteristic
304C2×C5C3×#45211yescharacteristic
305C2×C5C3×#45311yescharacteristic
306C23—21no—
307Q8—21no—
308Q8—21no—
309Q8—42no—
310Q8—202no—
311D4—21no—
312D4C3×#68912yescomplemented
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—82no—
320C2×C4—201no—
321C2×C4C2×C3×C22⋊D511yescharacteristic
322C2×C4C2×C3×C22⋊D511yescharacteristic
323C2×C4C2×C3×C22⋊D511yescharacteristic
324C8—102no—
325C8—201no—
326C8—201no—
327C2×C3—42no—
328C2×C3#6551711yescharacteristic
329C2×C3#6552811yescharacteristic
330C2×C3#6553011yescharacteristic
331C5C3×#163211yescharacteristic, complemented, Sylow
332C22—22no—
333C22—41no—
334C22—41no—
335C22C3×#409911yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—42no—
341C4—401no—
342C4C3×#408811yescharacteristic
343C4C2×C3×#68911yescharacteristic
344C4C3×#409611yescharacteristic
345C4C3×#409711yescharacteristic
346C3#1060271711yescharacteristic, direct product factor, Sylow
347C2—42no—
348C2C3×#6551711yescharacteristic
349C2C3×#6552811yescharacteristic
350C2C3×#6553011yescharacteristic
subgroupG/H
centerC22×C3#4099
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
FittingC3×C5×#554C2
socleC22×C3×C5#120
prime radicalC2×C3×C5×D4D4
hypercenterC3×#554D5
hypocenterC5C3×#1632
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#554

Complements

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

NH ≅ G/Ntype
2. C3×#65381C2semidirect
3. C3×#65310C2semidirect
5. C3×#64766C2semidirect
6. C3×Q8⋊#168C2semidirect
8. #10602717C3direct (N × H)
28. #65381C2×C3semidirect
29. #65310C2×C3semidirect
31. #64766C2×C3semidirect
32. Q8⋊#168C2×C3semidirect
87. C3×C5×D4Q16semidirect
192. C5×D4C3×Q16semidirect
235. C3×D4#689semidirect
287. C3×C5#1632semidirect
312. D4C3×#689semidirect
331. C5C3×#1632semidirect
346. C3#10602717direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5542
C3×#653812
C3×#653102
C3×#652972
C3×#647662
C3×Q8⋊#1682
C3×D4⋊#1682
#106027173
C3×#16325