Finite Group Databasebeta

← C3×#10604481 #14282374

Subgroups — C3×#10604481 #14282374

Order 1920.

552 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65930C211yesmaximal, characteristic
2C3×#65758C211yesmaximal, characteristic, complemented
3C2×C3×C4×#170C211yesmaximal, characteristic, complemented
4C3×D4⋊#170C212yesmaximal, complemented
5C3×#64871C212yesmaximal, complemented
6#10604481C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#711—21no—
8C2×C3×#704—21no—
9C22×C3×#170—21no—
10C3×C5×#121C411yescharacteristic
11C3×#4086C411yescharacteristic
12C3×#4085—21no—
13C3×D4×Dic5C2211yescharacteristic
14C3×#4076C411yescharacteristic
15C2×C3×C4×Dic5C2211yescharacteristic
16C3×#4039C2211yescharacteristic
17C3×#4034C411yescharacteristic
18C3×#4031—21no—
19C3×#4027—21no—
20C3×#4004—21no—
21C3×C4⋊#170C2212yes—
22C3×C4×#170—21no—
23C3×C4×#170C2212yes—
24C3×#1492—51nomaximal, Hall
25#65930C2×C311yescharacteristic
26#65758C2×C311yescharacteristic, complemented
27C2×C4×#170C2×C311yescharacteristic, complemented
28D4⋊#170C2×C312yescomplemented
29#64871C2×C312yescomplemented
30C2×C3×C5×D4—21no—
31C2×C3×C5×D4C2×C411yescharacteristic
32C22×C3×C4×C5C2×C411yescharacteristic
33C2×C3×C22⋊D5—21no—
34C2×C3×C22⋊D5—21no—
35C2×C3×C22⋊D5—21no—
36C22×C3×Dic5—21no—
37C22×C3×Dic5D411yescharacteristic
38C22×C3×Dic5D411yescharacteristic
39C3×#711—42no—
40C3×#711—42no—
41C2×C3×D20C2×C411yescharacteristic
42C2×C3×C4×D5—21no—
43C2×C3×Dic10—21no—
44C3×#704—42no—
45C2×C3×#170—24no—
46C2×C3×#170—22no—
47C2×C3×#170—22no—
48C3×C5×C4⋊C4C2×C411yescharacteristic
49C3×C5×C22⋊C4—21no—
50C3×C22⋊Dic5—21no—
51C3×#685—21no—
52C3×#685—21no—
53C3×C4⋊Dic5C2×C411yescharacteristic
54C3×#683—21no—
55C3×C4×Dic5C2×C411yescharacteristic
56C3×C4×Dic5D411yescharacteristic
57C3×C4×Dic5D411yescharacteristic
58C3×C4×Dic5C2311yescharacteristic
59C3×#532—51no—
60C3×#423—51no—
61C2×C3×C4×C8—51no—
62C3×D4⋊C8—52no—
63C3×D4⋊C8—52no—
64C2×#711—21no—
65C2×#704—21no—
66C22×#170—21no—
67C5×#121C3×C411yescharacteristic
68#4086C3×C411yescharacteristic
69#4085—21no—
70D4×Dic5C22×C311yescharacteristic
71#4076C3×C411yescharacteristic
72C2×C4×Dic5C22×C311yescharacteristic
73#4039C22×C311yescharacteristic
74#4034C3×C411yescharacteristic
75#4031—21no—
76#4027—21no—
77#4004—21no—
78C4⋊#170C22×C312yes—
79C4×#170—21no—
80C4×#170C22×C312yes—
81#1492—51noSylow
82C23×C3×C5—21no—
83C23×C3×C5C22⋊C411yescharacteristic
84C22×C3×D5—21no—
85C3×C5×D4—22no—
86C3×C5×D4—42no—
87C2×C3×C4×C5—22no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5C22⋊C411yescharacteristic
90C2×C3×C4×C5C22×C411yescharacteristic
91C3×C22⋊D5—42no—
92C3×C22⋊D5—42no—
93C3×C22⋊D5—42no—
94C2×C3×Dic5—22no—
95C2×C3×Dic5—21no—
96C2×C3×Dic5—41no—
97C2×C3×Dic5—41no—
98C2×C3×Dic5C22⋊C411yescharacteristic
99C2×C3×Dic5C22⋊C411yescharacteristic
100C2×C3×Dic5D812yes—
101C2×C3×Dic5SD1612yes—
102C2×C3×Dic5C2×D411yescharacteristic
103C2×C3×Dic5C2×D411yescharacteristic
104C3×D20—22no—
105C3×C4×D5—41no—
106C3×C4×D5—41no—
107C3×Dic10—42no—
108C3×#170—24no—
109C3×#170—42no—
110C2×C3×#41—101no—
111C2×C3×M16—101no—
112C22×C3×C8—101no—
113C3×C4⋊Q8—51no—
114C3×#127—51no—
115C3×#124—101no—
116C3×#121—51no—
117C3×#121—101no—
118C3×#121AGL(1,𝔽5)11yescharacteristic
119C3×C4×D4—51no—
120C3×C4×D4—51no—
121C3×C4×D4—101no—
122C2×C3×C42—51no—
123C3×C4⋊C8—52no—
124C3×C4⋊C8—101no—
125C3×C4×C8—52no—
126C3×C4×C8—101no—
127C2×C5×D4—21no—
128C2×C5×D4C2×C3×C411yescharacteristic
129C22×C4×C5C2×C3×C411yescharacteristic
130C2×C22⋊D5—21no—
131C2×C22⋊D5—21no—
132C2×C22⋊D5—21no—
133C22×Dic5—21no—
134C22×Dic5C3×D411yescharacteristic
135C22×Dic5C3×D411yescharacteristic
136#711—42no—
137#711—42no—
138C2×D20C2×C3×C411yescharacteristic
139C2×C4×D5—21no—
140C2×Dic10—21no—
141#704—42no—
142C2×#170—24no—
143C2×#170—22no—
144C2×#170—22no—
145C5×C4⋊C4C2×C3×C411yescharacteristic
146C5×C22⋊C4—21no—
147C22⋊Dic5—21no—
148#685—21no—
149#685—21no—
150C4⋊Dic5C2×C3×C411yescharacteristic
151#683—21no—
152C4×Dic5C2×C3×C411yescharacteristic
153C4×Dic5C3×D411yescharacteristic
154C4×Dic5C3×D411yescharacteristic
155C4×Dic5C23×C311yescharacteristic
156#532—51no—
157#423—51no—
158C2×C4×C8—51no—
159D4⋊C8—52no—
160D4⋊C8—52no—
161C22×C3×C5—21no—
162C22×C3×C5—21no—
163C22×C3×C5—41no—
164C22×C3×C5—41no—
165C22×C3×C5—41no—
166C22×C3×C5#10212yes—
167C22×C3×C5C2×C22⋊C411yescharacteristic
168C2×C3×D5—41no—
169C2×C3×D5—41no—
170C2×C3×D5—41no—
171C3×C4×C5—21no—
172C3×C4×C5—41no—
173C3×C4×C5#13112yes—
174C3×Dic5—21no—
175C3×Dic5—41no—
176C3×Dic5#10212yes—
177C3×Dic5#10412yes—
178C3×Dic5C2×D811yescharacteristic
179C3×Dic5C2×SD1611yescharacteristic
180C3×#41—202no—
181C3×#41—202no—
182C2×C3×Q8—101no—
183C2×C3×D4—21no—
184C2×C3×D4—51no—
185C2×C3×D4—101no—
186C2×C3×D4—101no—
187C2×C3×D4—101no—
188C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
189C22×C3×C4—51no—
190C22×C3×C4—51no—
191C22×C3×C4—101no—
192C22×C3×C4—101no—
193C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
194C3×M16—202no—
195C2×C3×C8—102no—
196C2×C3×C8—104no—
197C2×C3×C8—102no—
198C3×C4⋊C4—51no—
199C3×C4⋊C4—101no—
200C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
201C3×C22⋊C4—21no—
202C3×C22⋊C4—101no—
203C3×C22⋊C4—101no—
204C3×C22⋊C4—101no—
205C3×C42—51no—
206C3×C42—51no—
207C3×C42—51no—
208C3×C42—51no—
209C23×C5—21no—
210C23×C5C3×C22⋊C411yescharacteristic
211C22×D5—21no—
212C5×D4—22no—
213C5×D4—42no—
214C2×C4×C5—22no—
215C2×C4×C5—21no—
216C2×C4×C5C3×C22⋊C411yescharacteristic
217C2×C4×C5C22×C3×C411yescharacteristic
218C22⋊D5—42no—
219C22⋊D5—42no—
220C22⋊D5—42no—
221C2×Dic5—22no—
222C2×Dic5—21no—
223C2×Dic5—41no—
224C2×Dic5—41no—
225C2×Dic5C3×C22⋊C411yescharacteristic
226C2×Dic5C3×C22⋊C411yescharacteristic
227C2×Dic5C3×D812yes—
228C2×Dic5C3×SD1612yes—
229C2×Dic5C2×C3×D411yescharacteristic
230C2×Dic5C2×C3×D411yescharacteristic
231D20—22no—
232C4×D5—41no—
233C4×D5—41no—
234Dic10—42no—
235#170—24no—
236#170—42no—
237C2×#41—101no—
238C2×M16—101no—
239C22×C8—101no—
240C4⋊Q8—51no—
241#127—51no—
242#124—101no—
243#121—51no—
244#121—101no—
245#121C3×AGL(1,𝔽5)11yescharacteristic
246C4×D4—51no—
247C4×D4—51no—
248C4×D4—101no—
249C2×C42—51no—
250C4⋊C8—52no—
251C4⋊C8—101no—
252C4×C8—52no—
253C4×C8—101no—
254C2×C3×C5—22no—
255C2×C3×C5—81no—
256C2×C3×C5#40811yescharacteristic
257C2×C3×C5C2×#10211yescharacteristic
258C2×C3×C5C2×#10411yescharacteristic
259C3×D5—81no—
260C23×C3—21no—
261C23×C3—101no—
262C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
263C3×Q8—202no—
264C3×D4—22no—
265C3×D4—42no—
266C3×D4—102no—
267C3×D4—202no—
268C3×D4—202no—
269C3×D4—202no—
270C2×C3×C4—22no—
271C2×C3×C4—21no—
272C2×C3×C4—51no—
273C2×C3×C4—51no—
274C2×C3×C4—51no—
275C2×C3×C4—52no—
276C2×C3×C4—52no—
277C2×C3×C4—51no—
278C2×C3×C4—102no—
279C2×C3×C4—101no—
280C2×C3×C4—201no—
281C2×C3×C4—201no—
282C2×C3×C4—201no—
283C2×C3×C4—201no—
284C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
285C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
286C3×C8—104no—
287C3×C8—202no—
288C22×C5—21no—
289C22×C5—21no—
290C22×C5—41no—
291C22×C5—41no—
292C22×C5—41no—
293C22×C5C3×#10212yes—
294C22×C5C2×C3×C22⋊C411yescharacteristic
295C2×D5—41no—
296C2×D5—41no—
297C2×D5—41no—
298C4×C5—21no—
299C4×C5—41no—
300C4×C5C3×#13112yes—
301Dic5—21no—
302Dic5—41no—
303Dic5C3×#10212yes—
304Dic5C3×#10412yes—
305Dic5C2×C3×D811yescharacteristic
306Dic5C2×C3×SD1611yescharacteristic
307#41—202no—
308#41—202no—
309C2×Q8—101no—
310C2×D4—21no—
311C2×D4—51no—
312C2×D4—101no—
313C2×D4—101no—
314C2×D4—101no—
315C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
316C22×C4—51no—
317C22×C4—51no—
318C22×C4—101no—
319C22×C4—101no—
320C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
321M16—202no—
322C2×C8—102no—
323C2×C8—104no—
324C2×C8—102no—
325C4⋊C4—51no—
326C4⋊C4—101no—
327C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
328C22⋊C4—21no—
329C22⋊C4—101no—
330C22⋊C4—101no—
331C22⋊C4—101no—
332C42—51no—
333C42—51no—
334C42—51no—
335C42—51no—
336C3×C5#149211yescharacteristic, complemented, Hall
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—201no—
343C22×C3—201no—
344C22×C3—201no—
345C22×C3#400712yes—
346C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
347C3×C4—21no—
348C3×C4—41no—
349C3×C4—52no—
350C3×C4—52no—
351C3×C4—51no—
352C3×C4—51no—
353C3×C4—101no—
354C3×C4—201no—
355C3×C4#413112yes—
356C2×C5—22no—
357C2×C5—81no—
358C2×C5C3×#40811yescharacteristic
359C2×C5C2×C3×#10211yescharacteristic
360C2×C5C2×C3×#10411yescharacteristic
361D5—81no—
362C23—21no—
363C23—101no—
364C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
365Q8—202no—
366D4—22no—
367D4—42no—
368D4—102no—
369D4—202no—
370D4—202no—
371D4—202no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—52no—
375C2×C4—52no—
376C2×C4—51no—
377C2×C4—51no—
378C2×C4—51no—
379C2×C4—51no—
380C2×C4—102no—
381C2×C4—101no—
382C2×C4—201no—
383C2×C4—201no—
384C2×C4—201no—
385C2×C4—201no—
386C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
387C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
388C8—104no—
389C8—202no—
390C2×C3—22no—
391C2×C3—81no—
392C2×C3—401no—
393C2×C3C2×#400711yescharacteristic
394C2×C3#6577511yescharacteristic
395C2×C3D5⋊#10411yescharacteristic
396C5C3×#149211yescharacteristic, complemented, Sylow
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—201no—
403C22—201no—
404C22—201no—
405C22C3×#400712yes—
406C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
407C4—21no—
408C4—41no—
409C4—51no—
410C4—51no—
411C4—52no—
412C4—52no—
413C4—101no—
414C4—201no—
415C4C3×#413112yes—
416C3#1060448111yescharacteristic, direct product factor, Sylow
417C2—22no—
418C2—81no—
419C2—401no—
420C2C2×C3×#400711yescharacteristic
421C2C3×#6577511yescharacteristic
422C2C3×D5⋊#10411yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#121C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#65930C2
cosocleC4×Dic5C23×C3
hypercenterC3×#121AGL(1,𝔽5)
hypocenterC5C3×#1492
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
2. C3×#65758C2semidirect
3. C2×C3×C4×#170C2semidirect
4. C3×D4⋊#170C2semidirect
5. C3×#64871C2semidirect
6. #10604481C3direct (N × H)
26. #65758C2×C3semidirect
27. C2×C4×#170C2×C3semidirect
28. D4⋊#170C2×C3semidirect
29. #64871C2×C3semidirect
336. C3×C5#1492semidirect
396. C5C3×#1492semidirect
416. C3#10604481direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#659302
C3×#657582
C2×C3×C4×#1702
C3×D4⋊#1702
C3×#648712
#106044813
C3×#14925