Finite Group Databasebeta

← C3×#10602704 #14289629

Subgroups — C3×#10602704 #14289629

Order 1920.

468 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#547C211yesmaximal, characteristic
2C3×#65382C211yesmaximal, characteristic, complemented
3C3×#65297C211yesmaximal, characteristic, complemented
4C3×#65297C211yesmaximal, characteristic, complemented
5C3×#64767C211yesmaximal, characteristic, complemented
6C3×D4⋊#168C211yesmaximal, characteristic, complemented
7C3×D4⋊#168C211yesmaximal, characteristic, complemented
8#10602704C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41—21no—
10C3×C5×C4⋊Q8C2211yescharacteristic
11C3×C5×#123—22no—
12C3×C5×C22⋊Q8—21no—
13C3×C5×#121—21no—
14C3×C4×C5×D4C2211yescharacteristic
15C3×C4×C5×D4C2211yescharacteristic
16C2×C3×C5×C4⋊C4—21no—
17C3×#4015C2211yescharacteristic
18C3×#3964—21no—
19C3×#3964—21no—
20C3×#3942—22no—
21C3×#3940—21no—
22C3×#3940—21no—
23C3×C4⋊#168C2211yescharacteristic
24C3×C4⋊#168C2211yescharacteristic
25C3×C4×#168C2211yescharacteristic
26C3×#1635—51nomaximal, Hall
27C5×#547C2×C311yescharacteristic
28#65382C2×C311yescharacteristic, complemented
29#65297C2×C311yescharacteristic, complemented
30#65297C2×C311yescharacteristic, complemented
31#64767C2×C311yescharacteristic, complemented
32D4⋊#168C2×C311yescharacteristic, complemented
33D4⋊#168C2×C311yescharacteristic, complemented
34C3×C5×#41—42no—
35C3×C5×#41—42no—
36C2×C3×C5×Q8—21no—
37C2×C3×C5×D4—22no—
38C2×C3×C5×D4D411yescharacteristic
39C2×C3×C5×D4D411yescharacteristic
40C22×C3×C4×C5—22no—
41C22×C3×C4×C5—21no—
42C22×C3×C4×C5—21no—
43C2×C3×Dic10—21no—
44C3×C5×C4⋊C4—22no—
45C3×C5×C4⋊C4—21no—
46C3×C5×C4⋊C4D411yescharacteristic
47C3×C5×C4⋊C4D411yescharacteristic
48C3×C5×C4⋊C4D412yes—
49C3×C5×C22⋊C4—22no—
50C3×C5×C22⋊C4—21no—
51C3×C5×C22⋊C4—21no—
52C3×C42×C5C2311yescharacteristic
53C3×C4⋊Dic5—21no—
54C3×C4⋊Dic5—21no—
55C2×C3×#168—21no—
56C2×C3×#168—21no—
57C2×C3×#168—21no—
58C3×#547D511yescharacteristic
59C3×#487—51no—
60C3×#476—51no—
61C3×#476—51no—
62C3×#332—51no—
63C3×D4⋊C8—51no—
64C3×D4⋊C8—51no—
65C2×C5×#41—21no—
66C5×C4⋊Q8C22×C311yescharacteristic
67C5×#123—22no—
68C5×C22⋊Q8—21no—
69C5×#121—21no—
70C4×C5×D4C22×C311yescharacteristic
71C4×C5×D4C22×C311yescharacteristic
72C2×C5×C4⋊C4—21no—
73#4015C22×C311yescharacteristic
74#3964—21no—
75#3964—21no—
76#3942—22no—
77#3940—21no—
78#3940—21no—
79C4⋊#168C22×C311yescharacteristic
80C4⋊#168C22×C311yescharacteristic
81C4×#168C22×C311yescharacteristic
82#1635—51noSylow
83C23×C3×C5—21no—
84C23×C3×C5—21no—
85C3×C5×Q8—41no—
86C3×C5×Q8—41no—
87C3×C5×D4—21no—
88C3×C5×D4—21no—
89C3×C5×D4—21no—
90C3×C5×D4—44no—
91C3×C5×D4SD1612yes—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—22no—
95C2×C3×C4×C5—44no—
96C2×C3×C4×C5—41no—
97C2×C3×C4×C5—41no—
98C2×C3×C4×C5—41no—
99C2×C3×C4×C5—41no—
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×C4×C5C2×D411yescharacteristic
102C2×C3×C4×C5C2×D411yescharacteristic
103C2×C3×Dic5—41no—
104C3×Dic10—42no—
105C3×#168—41no—
106C3×#168—41no—
107C3×#168—41no—
108C2×C3×#41—21no—
109C3×C4⋊Q8—51no—
110C3×C4⋊Q8C2×D511yescharacteristic
111C3×#123—22no—
112C3×C22⋊Q8—21no—
113C3×#121—21no—
114C3×C4×D4C2×D511yescharacteristic
115C3×C4×D4C2×D511yescharacteristic
116C2×C3×C4⋊C4—21no—
117C3×#106—101no—
118C3×#106—101no—
119C3×C4⋊C8—51no—
120C3×C4⋊C8—51no—
121C3×#103—102no—
122C3×#102—101no—
123C3×#102—101no—
124C3×C4×C8—51no—
125C5×#41—42no—
126C5×#41—42no—
127C2×C5×Q8—21no—
128C2×C5×D4—22no—
129C2×C5×D4C3×D411yescharacteristic
130C2×C5×D4C3×D411yescharacteristic
131C22×C4×C5—22no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C2×Dic10—21no—
135C5×C4⋊C4—22no—
136C5×C4⋊C4—21no—
137C5×C4⋊C4C3×D411yescharacteristic
138C5×C4⋊C4C3×D411yescharacteristic
139C5×C4⋊C4C3×D412yes—
140C5×C22⋊C4—22no—
141C5×C22⋊C4—21no—
142C5×C22⋊C4—21no—
143C42×C5C23×C311yescharacteristic
144C4⋊Dic5—21no—
145C4⋊Dic5—21no—
146C2×#168—21no—
147C2×#168—21no—
148C2×#168—21no—
149#547C3×D511yescharacteristic
150#487—51no—
151#476—51no—
152#476—51no—
153#332—51no—
154D4⋊C8—51no—
155D4⋊C8—51no—
156C22×C3×C5—22no—
157C22×C3×C5—41no—
158C22×C3×C5—41no—
159C22×C3×C5—41no—
160C22×C3×C5—41no—
161C22×C3×C5—41no—
162C22×C3×C5#12011yescharacteristic
163C3×C4×C5—21no—
164C3×C4×C5—41no—
165C3×C4×C5—42no—
166C3×C4×C5—41no—
167C3×C4×C5C2×SD1611yescharacteristic
168C3×C4×C5#13511yescharacteristic
169C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
170C3×C4×C5#13711yescharacteristic
171C3×Dic5—81no—
172C3×#41—42no—
173C3×#41—42no—
174C2×C3×Q8—21no—
175C2×C3×Q8—101no—
176C2×C3×D4—22no—
177C2×C3×D4C22⋊D511yescharacteristic
178C2×C3×D4C22⋊D511yescharacteristic
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C2×C3×C8—101no—
183C2×C3×C8—101no—
184C2×C3×C8—101no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—101no—
188C3×C4⋊C4—101no—
189C3×C4⋊C4C22⋊D512yes—
190C3×C4⋊C4C22⋊D511yescharacteristic
191C3×C4⋊C4C22⋊D511yescharacteristic
192C3×C22⋊C4—22no—
193C3×C22⋊C4—21no—
194C3×C22⋊C4—21no—
195C3×C42C22×D511yescharacteristic
196C23×C5—21no—
197C23×C5—21no—
198C5×Q8—41no—
199C5×Q8—41no—
200C5×D4—21no—
201C5×D4—21no—
202C5×D4—21no—
203C5×D4—44no—
204C5×D4C3×SD1612yes—
205C2×C4×C5—21no—
206C2×C4×C5—21no—
207C2×C4×C5—22no—
208C2×C4×C5—44no—
209C2×C4×C5—41no—
210C2×C4×C5—41no—
211C2×C4×C5—41no—
212C2×C4×C5—41no—
213C2×C4×C5C2×C3×D411yescharacteristic
214C2×C4×C5C2×C3×D411yescharacteristic
215C2×C4×C5C2×C3×D411yescharacteristic
216C2×Dic5—41no—
217Dic10—42no—
218#168—41no—
219#168—41no—
220#168—41no—
221C2×#41—21no—
222C4⋊Q8—51no—
223C4⋊Q8C2×C3×D511yescharacteristic
224#123—22no—
225C22⋊Q8—21no—
226#121—21no—
227C4×D4C2×C3×D511yescharacteristic
228C4×D4C2×C3×D511yescharacteristic
229C2×C4⋊C4—21no—
230#106—101no—
231#106—101no—
232C4⋊C8—51no—
233C4⋊C8—51no—
234#103—102no—
235#102—101no—
236#102—101no—
237C4×C8—51no—
238C2×C3×C5—42no—
239C2×C3×C5—81no—
240C2×C3×C5#45011yescharacteristic
241C2×C3×C5#45211yescharacteristic
242C2×C3×C5#45411yescharacteristic
243C23×C3—21no—
244C23×C3—21no—
245C3×Q8—41no—
246C3×Q8—41no—
247C3×Q8—202no—
248C3×D4—21no—
249C3×D4—21no—
250C3×D4—21no—
251C3×D4—44no—
252C3×D4#68712yes—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—22no—
256C2×C3×C4—44no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—201no—
262C2×C3×C4C2×C22⋊D511yescharacteristic
263C2×C3×C4C2×C22⋊D511yescharacteristic
264C2×C3×C4C2×C22⋊D511yescharacteristic
265C3×C8—201no—
266C3×C8—201no—
267C3×C8—201no—
268C22×C5—22no—
269C22×C5—41no—
270C22×C5—41no—
271C22×C5—41no—
272C22×C5—41no—
273C22×C5—41no—
274C22×C5C3×#12011yescharacteristic
275C4×C5—21no—
276C4×C5—41no—
277C4×C5—42no—
278C4×C5—41no—
279C4×C5C2×C3×SD1611yescharacteristic
280C4×C5C3×#13511yescharacteristic
281C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
282C4×C5C3×#13711yescharacteristic
283Dic5—81no—
284#41—42no—
285#41—42no—
286C2×Q8—21no—
287C2×Q8—101no—
288C2×D4—22no—
289C2×D4C3×C22⋊D511yescharacteristic
290C2×D4C3×C22⋊D511yescharacteristic
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C2×C8—101no—
295C2×C8—101no—
296C2×C8—101no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—101no—
300C4⋊C4—101no—
301C4⋊C4C3×C22⋊D512yes—
302C4⋊C4C3×C22⋊D511yescharacteristic
303C4⋊C4C3×C22⋊D511yescharacteristic
304C22⋊C4—22no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C42C22×C3×D511yescharacteristic
308C3×C5#163511yescharacteristic, complemented, Hall
309C22×C3—22no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3#409911yescharacteristic
316C3×C4—21no—
317C3×C4—41no—
318C3×C4—42no—
319C3×C4—41no—
320C3×C4—401no—
321C3×C4#407811yescharacteristic
322C3×C4C2×#68711yescharacteristic
323C3×C4#409611yescharacteristic
324C3×C4#409711yescharacteristic
325C2×C5—42no—
326C2×C5—81no—
327C2×C5C3×#45011yescharacteristic
328C2×C5C3×#45211yescharacteristic
329C2×C5C3×#45411yescharacteristic
330C23—21no—
331C23—21no—
332Q8—41no—
333Q8—41no—
334Q8—202no—
335D4—21no—
336D4—21no—
337D4—21no—
338D4—44no—
339D4C3×#68712yes—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—201no—
349C2×C4C2×C3×C22⋊D511yescharacteristic
350C2×C4C2×C3×C22⋊D511yescharacteristic
351C2×C4C2×C3×C22⋊D511yescharacteristic
352C8—201no—
353C8—201no—
354C8—201no—
355C2×C3—42no—
356C2×C3—81no—
357C2×C3#6550711yescharacteristic
358C2×C3#6552811yescharacteristic
359C2×C3#6553411yescharacteristic
360C5C3×#163511yescharacteristic, complemented, Sylow
361C22—22no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22C3×#409911yescharacteristic
368C4—21no—
369C4—41no—
370C4—42no—
371C4—41no—
372C4—401no—
373C4C3×#407811yescharacteristic
374C4C2×C3×#68711yescharacteristic
375C4C3×#409611yescharacteristic
376C4C3×#409711yescharacteristic
377C3#1060270411yescharacteristic, direct product factor, Sylow
378C2—42no—
379C2—81no—
380C2C3×#6550711yescharacteristic
381C2C3×#6552811yescharacteristic
382C2C3×#6553411yescharacteristic
subgroupG/H
centerC22×C3#4099
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
Fitting, prime radicalC3×C5×#547C2
socleC22×C3×C5#120
hypercenterC3×#547D5
hypocenterC5C3×#1635
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#547

Complements

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

NH ≅ G/Ntype
2. C3×#65382C2semidirect
3. C3×#65297C2semidirect
4. C3×#65297C2semidirect
5. C3×#64767C2semidirect
6. C3×D4⋊#168C2semidirect
7. C3×D4⋊#168C2semidirect
8. #10602704C3direct (N × H)
28. #65382C2×C3semidirect
29. #65297C2×C3semidirect
30. #65297C2×C3semidirect
31. #64767C2×C3semidirect
32. D4⋊#168C2×C3semidirect
33. D4⋊#168C2×C3semidirect
308. C3×C5#1635semidirect
360. C5C3×#1635semidirect
377. C3#10602704direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5472
C3×#653822
C3×#652972
C3×#647672
C3×D4⋊#1682
#106027043
C3×#16355