Finite Group Databasebeta

← C3×#10602486 #14300644

Subgroups — C3×#10602486 #14300644

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#4072C211yesmaximal, characteristic
3C2×C3×#3967C212yesmaximal, complemented
4C2×C3×C22⋊#168C211yesmaximal, characteristic
5C3×#64749C212yesmaximal, complemented
6#10602486C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C5×Q8C2211yescharacteristic
8C23×C3×C4×C5C2211yescharacteristic
9C3×C5×C22⋊Q8—22no—
10C3×C5×C22⋊Q8C414yescomplemented
11C2×C3×C5×C4⋊C4—21no—
12C2×C3×C5×C4⋊C4C2211yescharacteristic
13C2×C3×C5×C22⋊C4—21no—
14C2×C3×C22⋊Dic5—21no—
15C3×#4072—24no—
16C2×C3×C4⋊Dic5C2212yes—
17C2×C3×C4×Dic5—21no—
18C22×C3×#168C2212yes—
19C3×#3967—24no—
20C3×C22⋊#168—22no—
21C3×#1873—51nomaximal, Hall
22C2×C5×C22⋊Q8C2×C311yescharacteristic
23C2×#4072C2×C311yescharacteristic
24C2×#3967C2×C312yescomplemented
25C2×C22⋊#168C2×C311yescharacteristic
26#64749C2×C312yescomplemented
27C24×C3×C5D411yescharacteristic
28C2×C3×C5×Q8—21no—
29C2×C3×C5×Q8—22no—
30C2×C3×C5×Q8—22no—
31C2×C3×C5×Q8C2×C412yes—
32C22×C3×C4×C5—22no—
33C22×C3×C4×C5—22no—
34C22×C3×C4×C5—21no—
35C22×C3×C4×C5—21no—
36C22×C3×C4×C5C2×C412yes—
37C22×C3×C4×C5D411yescharacteristic
38C22×C3×C4×C5D412yes—
39C22×C3×C4×C5C2311yescharacteristic
40C22×C3×Dic5—22no—
41C3×C5×C4⋊C4—24no—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4C2×C412yes—
44C3×C5×C22⋊C4—24no—
45C3×C22⋊Dic5—24no—
46C3×C4⋊Dic5—22no—
47C3×C4⋊Dic5D414yes—
48C3×C4×Dic5—24no—
49C2×C3×#168—22no—
50C2×C3×#168—22no—
51C2×C3×#168—22no—
52C2×C3×C22⋊Q8D511yescharacteristic
53C2×C3×#117—51no—
54C2×C3×#103—52no—
55C2×C3×C22⋊C8—51no—
56C3×#340—52no—
57C22×C5×Q8C22×C311yescharacteristic
58C23×C4×C5C22×C311yescharacteristic
59C5×C22⋊Q8—22no—
60C5×C22⋊Q8C3×C414yescomplemented
61C2×C5×C4⋊C4—21no—
62C2×C5×C4⋊C4C22×C311yescharacteristic
63C2×C5×C22⋊C4—21no—
64C2×C22⋊Dic5—21no—
65#4072—24no—
66C2×C4⋊Dic5C22×C312yes—
67C2×C4×Dic5—21no—
68C22×#168C22×C312yes—
69#3967—24no—
70C22⋊#168—22no—
71#1873—51noSylow
72C23×C3×C5—21no—
73C23×C3×C5—21no—
74C23×C3×C5—22no—
75C23×C3×C5—22no—
76C23×C3×C5C22⋊C412yes—
77C23×C3×C5C2×D411yescharacteristic
78C3×C5×Q8—24no—
79C3×C5×Q8—42no—
80C2×C3×C4×C5—21no—
81C2×C3×C4×C5—21no—
82C2×C3×C4×C5—24no—
83C2×C3×C4×C5—24no—
84C2×C3×C4×C5—22no—
85C2×C3×C4×C5—22no—
86C2×C3×C4×C5—41no—
87C2×C3×C4×C5—41no—
88C2×C3×C4×C5—42no—
89C2×C3×C4×C5C22⋊C412yes—
90C2×C3×C4×C5C22×C411yescharacteristic
91C2×C3×C4×C5C2×D412yes—
92C2×C3×C4×C5C2×D411yescharacteristic
93C2×C3×C4×C5#4112yes—
94C2×C3×Dic5—24no—
95C2×C3×Dic5—42no—
96C2×C3×Dic5—42no—
97C3×#168—42no—
98C22×C3×Q8C2×D511yescharacteristic
99C23×C3×C4C2×D511yescharacteristic
100C22×C3×C8—52no—
101C3×C22⋊Q8—22no—
102C3×C22⋊Q8Dic514yescomplemented
103C3×#117—104no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—52no—
106C2×C3×C4⋊C4C2×D511yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—101no—
109C2×C3×C42—101no—
110C3×#103—104no—
111C3×C22⋊C8—102no—
112C24×C5C3×D411yescharacteristic
113C2×C5×Q8—21no—
114C2×C5×Q8—22no—
115C2×C5×Q8—22no—
116C2×C5×Q8C2×C3×C412yes—
117C22×C4×C5—22no—
118C22×C4×C5—22no—
119C22×C4×C5—21no—
120C22×C4×C5—21no—
121C22×C4×C5C2×C3×C412yes—
122C22×C4×C5C3×D411yescharacteristic
123C22×C4×C5C3×D412yes—
124C22×C4×C5C23×C311yescharacteristic
125C22×Dic5—22no—
126C5×C4⋊C4—24no—
127C5×C4⋊C4—21no—
128C5×C4⋊C4C2×C3×C412yes—
129C5×C22⋊C4—24no—
130C22⋊Dic5—24no—
131C4⋊Dic5—22no—
132C4⋊Dic5C3×D414yes—
133C4×Dic5—24no—
134C2×#168—22no—
135C2×#168—22no—
136C2×#168—22no—
137C2×C22⋊Q8C3×D511yescharacteristic
138C2×#117—51no—
139C2×#103—52no—
140C2×C22⋊C8—51no—
141#340—52no—
142C22×C3×C5—22no—
143C22×C3×C5—22no—
144C22×C3×C5—22no—
145C22×C3×C5—42no—
146C22×C3×C5—42no—
147C22×C3×C5C2×C22⋊C411yescharacteristic
148C22×C3×C5#12011yescharacteristic
149C22×C3×C5#12311yescharacteristic
150C22×C3×C5#13512yes—
151C22×C3×C5#13712yes—
152C3×C4×C5—22no—
153C3×C4×C5—42no—
154C3×C4×C5—42no—
155C3×C4×C5C4×D412yes—
156C3×C4×C5#12112yes—
157C3×Dic5—44no—
158C24×C3C22⋊D511yescharacteristic
159C2×C3×Q8—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8C2×Dic512yes—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—102no—
168C22×C3×C4C2×Dic512yes—
169C22×C3×C4C22⋊D511yescharacteristic
170C22×C3×C4C22⋊D512yes—
171C22×C3×C4C22×D511yescharacteristic
172C2×C3×C8—102no—
173C2×C3×C8—102no—
174C2×C3×C8—102no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—54no—
178C3×C4⋊C4—102no—
179C3×C4⋊C4C2×Dic512yes—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—104no—
182C3×C42—104no—
183C23×C5—21no—
184C23×C5—21no—
185C23×C5—22no—
186C23×C5—22no—
187C23×C5C3×C22⋊C412yes—
188C23×C5C2×C3×D411yescharacteristic
189C5×Q8—24no—
190C5×Q8—42no—
191C2×C4×C5—21no—
192C2×C4×C5—21no—
193C2×C4×C5—24no—
194C2×C4×C5—24no—
195C2×C4×C5—22no—
196C2×C4×C5—22no—
197C2×C4×C5—41no—
198C2×C4×C5—41no—
199C2×C4×C5—42no—
200C2×C4×C5C3×C22⋊C412yes—
201C2×C4×C5C22×C3×C411yescharacteristic
202C2×C4×C5C2×C3×D412yes—
203C2×C4×C5C2×C3×D411yescharacteristic
204C2×C4×C5C3×#4112yes—
205C2×Dic5—24no—
206C2×Dic5—42no—
207C2×Dic5—42no—
208#168—42no—
209C22×Q8C2×C3×D511yescharacteristic
210C23×C4C2×C3×D511yescharacteristic
211C22×C8—52no—
212C22⋊Q8—22no—
213C22⋊Q8C3×Dic514yescomplemented
214#117—104no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—52no—
217C2×C4⋊C4C2×C3×D511yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—101no—
220C2×C42—101no—
221#103—104no—
222C22⋊C8—102no—
223C2×C3×C5—42no—
224C2×C3×C5#38611yescharacteristic
225C2×C3×C5#41611yescharacteristic
226C2×C3×C5#41911yescharacteristic
227C2×C3×C5#45212yes—
228C2×C3×C5#48512yes—
229C23×C3—22no—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—22no—
233C23×C3C22⋊Dic512yes—
234C23×C3C2×C22⋊D511yescharacteristic
235C3×Q8—24no—
236C3×Q8—42no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—104no—
247C2×C3×C4—202no—
248C2×C3×C4—202no—
249C2×C3×C4C22⋊Dic512yes—
250C2×C3×C4D4×D512yes—
251C2×C3×C4#71112yes—
252C2×C3×C4C22×Dic511yescharacteristic
253C2×C3×C4C2×C22⋊D511yescharacteristic
254C3×C8—202no—
255C22×C5—22no—
256C22×C5—22no—
257C22×C5—22no—
258C22×C5—42no—
259C22×C5—42no—
260C22×C5C2×C3×C22⋊C411yescharacteristic
261C22×C5C3×#12011yescharacteristic
262C22×C5C3×#12311yescharacteristic
263C22×C5C3×#13512yes—
264C22×C5C3×#13712yes—
265C4×C5—22no—
266C4×C5—42no—
267C4×C5—42no—
268C4×C5C3×C4×D412yes—
269C4×C5C3×#12112yes—
270Dic5—44no—
271C24C3×C22⋊D511yescharacteristic
272C2×Q8—21no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8C2×C3×Dic512yes—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—102no—
281C22×C4C2×C3×Dic512yes—
282C22×C4C3×C22⋊D511yescharacteristic
283C22×C4C3×C22⋊D512yes—
284C22×C4C22×C3×D511yescharacteristic
285C2×C8—102no—
286C2×C8—102no—
287C2×C8—102no—
288C4⋊C4—24no—
289C4⋊C4—21no—
290C4⋊C4—54no—
291C4⋊C4—102no—
292C4⋊C4C2×C3×Dic512yes—
293C22⋊C4—24no—
294C22⋊C4—104no—
295C42—104no—
296C3×C5#187311yescharacteristic, complemented, Hall
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—22no—
300C22×C3—42no—
301C22×C3—42no—
302C22×C3#408111yescharacteristic
303C22×C3#408311yescharacteristic
304C22×C3#408812yes—
305C22×C3#409612yes—
306C22×C3C2×C22⋊Dic511yescharacteristic
307C3×C4—22no—
308C3×C4—42no—
309C3×C4—42no—
310C3×C4—204no—
311C3×C4D4×Dic512yes—
312C3×C4#408512yes—
313C2×C5—42no—
314C2×C5C3×#38611yescharacteristic
315C2×C5C3×#41611yescharacteristic
316C2×C5C3×#41911yescharacteristic
317C2×C5C3×#45212yes—
318C2×C5C3×#48512yes—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—22no—
323C23C3×C22⋊Dic512yes—
324C23C2×C3×C22⋊D511yescharacteristic
325Q8—24no—
326Q8—42no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—104no—
337C2×C4—202no—
338C2×C4—202no—
339C2×C4C3×C22⋊Dic512yes—
340C2×C4C3×D4×D512yes—
341C2×C4C3×#71112yes—
342C2×C4C22×C3×Dic511yescharacteristic
343C2×C4C2×C3×C22⋊D511yescharacteristic
344C8—202no—
345C2×C3—42no—
346C2×C3#6533412yes—
347C2×C3#6533612yes—
348C2×C3#6550911yescharacteristic
349C2×C3#6551411yescharacteristic
350C2×C3#6552211yescharacteristic
351C5C3×#187311yescharacteristic, complemented, Sylow
352C22—22no—
353C22—22no—
354C22—22no—
355C22—42no—
356C22—42no—
357C22C3×#408111yescharacteristic
358C22C3×#408311yescharacteristic
359C22C3×#408812yes—
360C22C3×#409612yes—
361C22C2×C3×C22⋊Dic511yescharacteristic
362C4—22no—
363C4—42no—
364C4—42no—
365C4—204no—
366C4C3×D4×Dic512yes—
367C4C3×#408512yes—
368C3#1060248611yescharacteristic, direct product factor, Sylow
369C2—42no—
370C2C3×#6533412yes—
371C2C3×#6533612yes—
372C2C3×#6550911yescharacteristic
373C2C3×#6551411yescharacteristic
374C2C3×#6552211yescharacteristic
subgroupG/H
centerC23×C3C2×C22⋊D5
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC2×C3×C5×C22⋊Q8C2
socleC23×C3×C5C2×D4
prime radicalC24×C3×C5D4
cosocleC22×C4×C5C23×C3
hypercenterC2×C3×C22⋊Q8D5
hypocenterC5C3×#1873
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×C3×#3967C2semidirect
5. C3×#64749C2semidirect
6. #10602486C3direct (N × H)
10. C3×C5×C22⋊Q8C4semidirect
24. C2×#3967C2×C3semidirect
26. #64749C2×C3semidirect
60. C5×C22⋊Q8C3×C4semidirect
102. C3×C22⋊Q8Dic5semidirect
213. C22⋊Q8C3×Dic5semidirect
296. C3×C5#1873semidirect
351. C5C3×#1873semidirect
368. C3#10602486direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×C22⋊Q82
C2×C3×#40722
C2×C3×#39672
C2×C3×C22⋊#1682
C3×#647492
#106024863
C3×#18735