Finite Group Databasebeta

← C3×#10602713 #14290007

Subgroups — C3×#10602713 #14290007

Order 1920.

500 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#550C211yesmaximal, characteristic
2C3×#65359C211yesmaximal, characteristic, complemented
3C3×#65320C211yesmaximal, characteristic, complemented
4C3×#65296C211yesmaximal, characteristic, complemented
5C3×#64764C211yesmaximal, characteristic, complemented
6C3×Q8⋊#168C211yesmaximal, characteristic, complemented
7C3×D4⋊#168C211yesmaximal, characteristic, complemented
8#10602713C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41—21no—
10C3×C5×#127—21no—
11C3×C5×#127C2211yescharacteristic
12C3×C5×#121—22no—
13C3×C5×#121—21no—
14C3×C4×C5×Q8C2211yescharacteristic
15C3×C4×C5×D4—21no—
16C3×C4×C5×D4C2211yescharacteristic
17C2×C3×#689—21no—
18C3×#4015C2211yescharacteristic
19C3×#3964—21no—
20C3×#3964—22no—
21C3×#3942—21no—
22C3×#3939—21no—
23C3×C4⋊#168C2211yescharacteristic
24C3×C4⋊#168C2211yescharacteristic
25C3×C4×#168C2211yescharacteristic
26C3×#1641—51nomaximal, Hall
27C5×#550C2×C311yescharacteristic
28#65359C2×C311yescharacteristic, complemented
29#65320C2×C311yescharacteristic, complemented
30#65296C2×C311yescharacteristic, complemented
31#64764C2×C311yescharacteristic, complemented
32Q8⋊#168C2×C311yescharacteristic, complemented
33D4⋊#168C2×C311yescharacteristic, complemented
34C3×C5×#41—42no—
35C3×C5×#41—42no—
36C2×C3×C5×Q8D411yescharacteristic
37C2×C3×C5×D4—22no—
38C2×C3×C5×D4—22no—
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4D411yescharacteristic
42C2×C3×C5×D4D412yes—
43C22×C3×C4×C5—22no—
44C22×C3×C4×C5—21no—
45C2×C3×Dic10—21no—
46C3×C5×C4⋊C4—21no—
47C3×C5×C4⋊C4D411yescharacteristic
48C3×C5×C4⋊C4D411yescharacteristic
49C3×C5×C22⋊C4—22no—
50C3×C5×C22⋊C4—21no—
51C3×C42×C5—21no—
52C3×C42×C5C2311yescharacteristic
53C3×#689—42no—
54C3×C4⋊Dic5—21no—
55C3×C4⋊Dic5—21no—
56C2×C3×#168—21no—
57C2×C3×#168—21no—
58C2×C3×#168—21no—
59C3×#550D511yescharacteristic
60C3×#486—51no—
61C3×#474—51no—
62C3×#462—51no—
63C3×#331—51no—
64C3×Q8⋊C8—51no—
65C3×D4⋊C8—51no—
66C2×C5×#41—21no—
67C5×#127—21no—
68C5×#127C22×C311yescharacteristic
69C5×#121—22no—
70C5×#121—21no—
71C4×C5×Q8C22×C311yescharacteristic
72C4×C5×D4—21no—
73C4×C5×D4C22×C311yescharacteristic
74C2×#689—21no—
75#4015C22×C311yescharacteristic
76#3964—21no—
77#3964—22no—
78#3942—21no—
79#3939—21no—
80C4⋊#168C22×C311yescharacteristic
81C4⋊#168C22×C311yescharacteristic
82C4×#168C22×C311yescharacteristic
83#1641—51noSylow
84C23×C3×C5—21no—
85C23×C3×C5—22no—
86C3×C5×Q8—21no—
87C3×C5×Q8D812yes—
88C3×C5×D4—21no—
89C3×C5×D4—21no—
90C3×C5×D4—44no—
91C3×C5×D4—42no—
92C3×C5×D4—41no—
93C3×C5×D4—41no—
94C3×C5×D4—41no—
95C3×C5×D4—41no—
96C3×C5×D4—82no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—44no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5C2×D411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×Dic5—41no—
107C3×Dic10—41no—
108C3×Dic10—41no—
109C3×#168—41no—
110C3×#168—41no—
111C3×#168—41no—
112C2×C3×#41—21no—
113C2×C3×Q16—101no—
114C3×C4⋊Q8—51no—
115C3×#127—21no—
116C3×#127C2×D511yescharacteristic
117C3×#121—22no—
118C3×#121—21no—
119C3×C4×Q8C2×D511yescharacteristic
120C3×C4×D4—21no—
121C3×C4×D4C2×D511yescharacteristic
122C3×#107—101no—
123C3×C4⋊C8—51no—
124C3×C4⋊C8—51no—
125C3×#103—101no—
126C3×#102—101no—
127C3×#102—102no—
128C3×C4×C8—51no—
129C5×#41—42no—
130C5×#41—42no—
131C2×C5×Q8C3×D411yescharacteristic
132C2×C5×D4—22no—
133C2×C5×D4—22no—
134C2×C5×D4—21no—
135C2×C5×D4—21no—
136C2×C5×D4C3×D411yescharacteristic
137C2×C5×D4C3×D412yes—
138C22×C4×C5—22no—
139C22×C4×C5—21no—
140C2×Dic10—21no—
141C5×C4⋊C4—21no—
142C5×C4⋊C4C3×D411yescharacteristic
143C5×C4⋊C4C3×D411yescharacteristic
144C5×C22⋊C4—22no—
145C5×C22⋊C4—21no—
146C42×C5—21no—
147C42×C5C23×C311yescharacteristic
148#689—42no—
149C4⋊Dic5—21no—
150C4⋊Dic5—21no—
151C2×#168—21no—
152C2×#168—21no—
153C2×#168—21no—
154#550C3×D511yescharacteristic
155#486—51no—
156#474—51no—
157#462—51no—
158#331—51no—
159Q8⋊C8—51no—
160D4⋊C8—51no—
161C22×C3×C5—42no—
162C22×C3×C5—41no—
163C22×C3×C5—41no—
164C22×C3×C5—42no—
165C22×C3×C5—42no—
166C22×C3×C5—41no—
167C22×C3×C5#12011yescharacteristic
168C3×C4×C5—22no—
169C3×C4×C5—21no—
170C3×C4×C5—41no—
171C3×C4×C5—41no—
172C3×C4×C5C2×D811yescharacteristic
173C3×C4×C5#13511yescharacteristic
174C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
175C3×C4×C5#13711yescharacteristic
176C3×Dic5—81no—
177C3×#41—42no—
178C3×#41—42no—
179C2×C3×Q8—101no—
180C2×C3×Q8C22⋊D511yescharacteristic
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4C22⋊D511yescharacteristic
186C2×C3×D4C22⋊D512yes—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C3×Q16—202no—
190C2×C3×C8—101no—
191C2×C3×C8—101no—
192C2×C3×C8—101no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—101no—
195C3×C4⋊C4—101no—
196C3×C4⋊C4C22⋊D511yescharacteristic
197C3×C4⋊C4C22⋊D511yescharacteristic
198C3×C22⋊C4—22no—
199C3×C22⋊C4—21no—
200C3×C42—21no—
201C3×C42C22×D511yescharacteristic
202C23×C5—21no—
203C23×C5—22no—
204C5×Q8—21no—
205C5×Q8C3×D812yes—
206C5×D4—21no—
207C5×D4—21no—
208C5×D4—42no—
209C5×D4—44no—
210C5×D4—41no—
211C5×D4—41no—
212C5×D4—41no—
213C5×D4—41no—
214C5×D4—82no—
215C2×C4×C5—21no—
216C2×C4×C5—21no—
217C2×C4×C5—21no—
218C2×C4×C5—44no—
219C2×C4×C5—41no—
220C2×C4×C5—41no—
221C2×C4×C5C2×C3×D411yescharacteristic
222C2×C4×C5C2×C3×D411yescharacteristic
223C2×C4×C5C2×C3×D411yescharacteristic
224C2×Dic5—41no—
225Dic10—41no—
226Dic10—41no—
227#168—41no—
228#168—41no—
229#168—41no—
230C2×#41—21no—
231C2×Q16—101no—
232C4⋊Q8—51no—
233#127—21no—
234#127C2×C3×D511yescharacteristic
235#121—22no—
236#121—21no—
237C4×Q8C2×C3×D511yescharacteristic
238C4×D4—21no—
239C4×D4C2×C3×D511yescharacteristic
240#107—101no—
241C4⋊C8—51no—
242C4⋊C8—51no—
243#103—101no—
244#102—101no—
245#102—102no—
246C4×C8—51no—
247C2×C3×C5—81no—
248C2×C3×C5—82no—
249C2×C3×C5#44711yescharacteristic
250C2×C3×C5#45211yescharacteristic
251C2×C3×C5#45511yescharacteristic
252C23×C3—21no—
253C23×C3—22no—
254C3×Q8—21no—
255C3×Q8—201no—
256C3×Q8—201no—
257C3×Q8#68612yes—
258C3×D4—21no—
259C3×D4—21no—
260C3×D4—42no—
261C3×D4—44no—
262C3×D4—41no—
263C3×D4—41no—
264C3×D4—41no—
265C3×D4—41no—
266C3×D4—82no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—44no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—201no—
274C2×C3×C4C2×C22⋊D511yescharacteristic
275C2×C3×C4C2×C22⋊D511yescharacteristic
276C2×C3×C4C2×C22⋊D511yescharacteristic
277C3×C8—201no—
278C3×C8—201no—
279C3×C8—201no—
280C22×C5—42no—
281C22×C5—41no—
282C22×C5—41no—
283C22×C5—42no—
284C22×C5—42no—
285C22×C5—41no—
286C22×C5C3×#12011yescharacteristic
287C4×C5—22no—
288C4×C5—21no—
289C4×C5—41no—
290C4×C5—41no—
291C4×C5C2×C3×D811yescharacteristic
292C4×C5C3×#13511yescharacteristic
293C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
294C4×C5C3×#13711yescharacteristic
295Dic5—81no—
296#41—42no—
297#41—42no—
298C2×Q8—101no—
299C2×Q8C3×C22⋊D511yescharacteristic
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4C3×C22⋊D511yescharacteristic
305C2×D4C3×C22⋊D512yes—
306C22×C4—22no—
307C22×C4—21no—
308Q16—202no—
309C2×C8—101no—
310C2×C8—101no—
311C2×C8—101no—
312C4⋊C4—21no—
313C4⋊C4—101no—
314C4⋊C4—101no—
315C4⋊C4C3×C22⋊D511yescharacteristic
316C4⋊C4C3×C22⋊D511yescharacteristic
317C22⋊C4—22no—
318C22⋊C4—21no—
319C42—21no—
320C42C22×C3×D511yescharacteristic
321C3×C5#164111yescharacteristic, complemented, Hall
322C22×C3—42no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—41no—
328C22×C3#409911yescharacteristic
329C3×C4—22no—
330C3×C4—21no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—401no—
334C3×C4C2×#68611yescharacteristic
335C3×C4#407811yescharacteristic
336C3×C4#409611yescharacteristic
337C3×C4#409711yescharacteristic
338C2×C5—81no—
339C2×C5—82no—
340C2×C5C3×#44711yescharacteristic
341C2×C5C3×#45211yescharacteristic
342C2×C5C3×#45511yescharacteristic
343C23—21no—
344C23—22no—
345Q8—21no—
346Q8—201no—
347Q8—201no—
348Q8C3×#68612yes—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—44no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—41no—
357D4—82no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—44no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—201no—
365C2×C4C2×C3×C22⋊D511yescharacteristic
366C2×C4C2×C3×C22⋊D511yescharacteristic
367C2×C4C2×C3×C22⋊D511yescharacteristic
368C8—201no—
369C8—201no—
370C8—201no—
371C2×C3—81no—
372C2×C3—82no—
373C2×C3#6550611yescharacteristic
374C2×C3#6552811yescharacteristic
375C2×C3#6553111yescharacteristic
376C5C3×#164111yescharacteristic, complemented, Sylow
377C22—42no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22—42no—
382C22—41no—
383C22C3×#409911yescharacteristic
384C4—22no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—401no—
389C4C2×C3×#68611yescharacteristic
390C4C3×#407811yescharacteristic
391C4C3×#409611yescharacteristic
392C4C3×#409711yescharacteristic
393C3#1060271311yescharacteristic, direct product factor, Sylow
394C2—81no—
395C2—82no—
396C2C3×#6550611yescharacteristic
397C2C3×#6552811yescharacteristic
398C2C3×#6553111yescharacteristic
subgroupG/H
centerC22×C3#4099
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
Fitting, prime radicalC3×C5×#550C2
socleC22×C3×C5#120
hypercenterC3×#550D5
hypocenterC5C3×#1641
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#550

Complements

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

NH ≅ G/Ntype
2. C3×#65359C2semidirect
3. C3×#65320C2semidirect
4. C3×#65296C2semidirect
5. C3×#64764C2semidirect
6. C3×Q8⋊#168C2semidirect
7. C3×D4⋊#168C2semidirect
8. #10602713C3direct (N × H)
28. #65359C2×C3semidirect
29. #65320C2×C3semidirect
30. #65296C2×C3semidirect
31. #64764C2×C3semidirect
32. Q8⋊#168C2×C3semidirect
33. D4⋊#168C2×C3semidirect
321. C3×C5#1641semidirect
376. C5C3×#1641semidirect
393. C3#10602713direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5502
C3×#653592
C3×#653202
C3×#652962
C3×#647642
C3×Q8⋊#1682
C3×D4⋊#1682
#106027133
C3×#16415