Finite Group Databasebeta

← C3×#10604408 #14284586

Subgroups — C3×#10604408 #14284586

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65882C211yesmaximal, characteristic
2C3×D5⋊(C4⋊C8)C211yesmaximal, characteristic, complemented
3C3×#65678C211yesmaximal, characteristic, complemented
4C3×#64925C212yesmaximal, complemented
5C3×#64873C212yesmaximal, complemented
6#10604408C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#709—21no—
8C2×C3×D5⋊C8—21no—
9C3×C4×C5×D4C411yescharacteristic
10C3×#4086C2211yescharacteristic
11C3×#4084C411yescharacteristic
12C3×C4×C22⋊D5—21no—
13C3×#4041C411yescharacteristic
14C3×#4034C2211yescharacteristic
15C3×#4031—21no—
16C3×#4030—21no—
17C3×C4×D20C411yescharacteristic
18C3×C42×D5C2211yescharacteristic
19C3×#4004C2212yes—
20C3×#4004C2212yes—
21C3×#4003—21no—
22C3×C4⋊#170—21no—
23C3×C4×#170—21no—
24C3×#1540—51nomaximal, Hall
25#65882C2×C311yescharacteristic
26D5⋊(C4⋊C8)C2×C311yescharacteristic, complemented
27#65678C2×C311yescharacteristic, complemented
28#64925C2×C312yescomplemented
29#64873C2×C312yescomplemented
30C2×C3×C5×D4C2×C411yescharacteristic
31C22×C3×C4×C5—21no—
32C2×C3×C22⋊D5—21no—
33C2×C3×C22⋊D5—21no—
34C2×C3×C22⋊D5—21no—
35C3×#709—42no—
36C3×#709—42no—
37C2×C3×D20—21no—
38C2×C3×D20C2×C411yescharacteristic
39C2×C3×C4×D5—21no—
40C2×C3×C4×D5C2×C411yescharacteristic
41C2×C3×C4×D5D411yescharacteristic
42C2×C3×C4×D5D411yescharacteristic
43C2×C3×Dic10—21no—
44C2×C3×#170—22no—
45C2×C3×#170—22no—
46C3×D5⋊C8—42no—
47C3×C5×C4⋊C4C2×C411yescharacteristic
48C3×C5×C22⋊C4—21no—
49C3×C42×C5C2×C411yescharacteristic
50C3×C22⋊Dic5—21no—
51C3×#685—21no—
52C3×#685—21no—
53C3×C4⋊Dic5C2×C411yescharacteristic
54C3×#683—21no—
55C3×C4×Dic5D411yescharacteristic
56C3×C4×Dic5D411yescharacteristic
57C3×C4×Dic5C2311yescharacteristic
58C3×#532—51no—
59C3×#431—51no—
60C2×C3×C4⋊C8—51no—
61C3×#332—52no—
62C3×#331—52no—
63C2×#709—21no—
64C2×D5⋊C8—21no—
65C4×C5×D4C3×C411yescharacteristic
66#4086C22×C311yescharacteristic
67#4084C3×C411yescharacteristic
68C4×C22⋊D5—21no—
69#4041C3×C411yescharacteristic
70#4034C22×C311yescharacteristic
71#4031—21no—
72#4030—21no—
73C4×D20C3×C411yescharacteristic
74C42×D5C22×C311yescharacteristic
75#4004C22×C312yes—
76#4004C22×C312yes—
77#4003—21no—
78C4⋊#170—21no—
79C4×#170—21no—
80#1540—51noSylow
81C23×C3×C5—21no—
82C22×C3×D5—21no—
83C22×C3×D5C22⋊C411yescharacteristic
84C3×C5×D4—41no—
85C2×C3×C4×C5—21no—
86C2×C3×C4×C5—42no—
87C2×C3×C4×C5C22⋊C411yescharacteristic
88C2×C3×C4×C5C22⋊C411yescharacteristic
89C2×C3×C4×C5C22×C411yescharacteristic
90C3×C22⋊D5—42no—
91C3×C22⋊D5—41no—
92C3×C22⋊D5—41no—
93C3×C22⋊D5—42no—
94C2×C3×Dic5—21no—
95C2×C3×Dic5C22⋊C411yescharacteristic
96C2×C3×Dic5C2×D411yescharacteristic
97C2×C3×Dic5C2×D411yescharacteristic
98C3×D20—41no—
99C3×D20—42no—
100C3×C4×D5—22no—
101C3×C4×D5—41no—
102C3×C4×D5—42no—
103C3×C4×D5D812yes—
104C3×C4×D5SD1612yes—
105C3×Dic10—41no—
106C3×Dic10—41no—
107C3×#170—42no—
108C3×#170—42no—
109C2×C3×#41—101no—
110C22×C3×C8—101no—
111C3×C4⋊Q8—51no—
112C3×#127—51no—
113C3×#124—101no—
114C3×#121—51no—
115C3×#121—51no—
116C3×#121—101no—
117C3×C4×D4—51no—
118C3×C4×D4—101no—
119C3×C4×D4AGL(1,𝔽5)11yescharacteristic
120C2×C3×C42—51no—
121C3×C4⋊C8—52no—
122C3×C4⋊C8—52no—
123C3×C4⋊C8—101no—
124C3×C22⋊C8—101no—
125C3×C4×C8—101no—
126C2×C5×D4C2×C3×C411yescharacteristic
127C22×C4×C5—21no—
128C2×C22⋊D5—21no—
129C2×C22⋊D5—21no—
130C2×C22⋊D5—21no—
131#709—42no—
132#709—42no—
133C2×D20—21no—
134C2×D20C2×C3×C411yescharacteristic
135C2×C4×D5—21no—
136C2×C4×D5C2×C3×C411yescharacteristic
137C2×C4×D5C3×D411yescharacteristic
138C2×C4×D5C3×D411yescharacteristic
139C2×Dic10—21no—
140C2×#170—22no—
141C2×#170—22no—
142D5⋊C8—42no—
143C5×C4⋊C4C2×C3×C411yescharacteristic
144C5×C22⋊C4—21no—
145C42×C5C2×C3×C411yescharacteristic
146C22⋊Dic5—21no—
147#685—21no—
148#685—21no—
149C4⋊Dic5C2×C3×C411yescharacteristic
150#683—21no—
151C4×Dic5C3×D411yescharacteristic
152C4×Dic5C3×D411yescharacteristic
153C4×Dic5C23×C311yescharacteristic
154#532—51no—
155#431—51no—
156C2×C4⋊C8—51no—
157#332—52no—
158#331—52no—
159C22×C3×C5—41no—
160C22×C3×C5—41no—
161C22×C3×C5—41no—
162C22×C3×C5C2×C22⋊C411yescharacteristic
163C2×C3×D5—21no—
164C2×C3×D5—21no—
165C2×C3×D5—41no—
166C2×C3×D5—41no—
167C2×C3×D5—41no—
168C2×C3×D5#10212yes—
169C3×C4×C5—21no—
170C3×C4×C5—21no—
171C3×C4×C5—41no—
172C3×C4×C5#10212yes—
173C3×Dic5—21no—
174C3×Dic5—41no—
175C3×Dic5C2×D811yescharacteristic
176C3×Dic5C2×SD1611yescharacteristic
177C3×Dic5#13512yes—
178C3×#41—202no—
179C3×#41—202no—
180C2×C3×Q8—101no—
181C2×C3×D4—51no—
182C2×C3×D4—101no—
183C2×C3×D4—101no—
184C2×C3×D4—101no—
185C2×C3×D4—101no—
186C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
187C22×C3×C4—21no—
188C22×C3×C4—51no—
189C22×C3×C4—51no—
190C22×C3×C4—51no—
191C22×C3×C4—101no—
192C2×C3×C8—102no—
193C2×C3×C8—102no—
194C2×C3×C8—202no—
195C3×C4⋊C4—51no—
196C3×C4⋊C4—101no—
197C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
198C3×C22⋊C4—21no—
199C3×C22⋊C4—101no—
200C3×C22⋊C4—101no—
201C3×C22⋊C4—101no—
202C3×C42—51no—
203C3×C42—51no—
204C3×C42—51no—
205C3×C42C2×AGL(1,𝔽5)11yescharacteristic
206C23×C5—21no—
207C22×D5—21no—
208C22×D5C3×C22⋊C411yescharacteristic
209C5×D4—41no—
210C2×C4×C5—21no—
211C2×C4×C5—42no—
212C2×C4×C5C3×C22⋊C411yescharacteristic
213C2×C4×C5C3×C22⋊C411yescharacteristic
214C2×C4×C5C22×C3×C411yescharacteristic
215C22⋊D5—42no—
216C22⋊D5—41no—
217C22⋊D5—41no—
218C22⋊D5—42no—
219C2×Dic5—21no—
220C2×Dic5C3×C22⋊C411yescharacteristic
221C2×Dic5C2×C3×D411yescharacteristic
222C2×Dic5C2×C3×D411yescharacteristic
223D20—41no—
224D20—42no—
225C4×D5—22no—
226C4×D5—41no—
227C4×D5—42no—
228C4×D5C3×D812yes—
229C4×D5C3×SD1612yes—
230Dic10—41no—
231Dic10—41no—
232#170—42no—
233#170—42no—
234C2×#41—101no—
235C22×C8—101no—
236C4⋊Q8—51no—
237#127—51no—
238#124—101no—
239#121—51no—
240#121—51no—
241#121—101no—
242C4×D4—51no—
243C4×D4—101no—
244C4×D4C3×AGL(1,𝔽5)11yescharacteristic
245C2×C42—51no—
246C4⋊C8—52no—
247C4⋊C8—52no—
248C4⋊C8—101no—
249C22⋊C8—101no—
250C4×C8—101no—
251C2×C3×C5—81no—
252C2×C3×C5#41311yescharacteristic
253C2×C3×C5C2×#10211yescharacteristic
254C2×C3×C5#41611yescharacteristic
255C3×D5—22no—
256C3×D5—81no—
257C23×C3—21no—
258C23×C3—51no—
259C23×C3—101no—
260C3×Q8—201no—
261C3×Q8—201no—
262C3×D4—41no—
263C3×D4—201no—
264C3×D4—202no—
265C3×D4—202no—
266C3×D4—201no—
267C3×D4—201no—
268C3×D4—202no—
269C2×C3×C4—21no—
270C2×C3×C4—42no—
271C2×C3×C4—51no—
272C2×C3×C4—51no—
273C2×C3×C4—52no—
274C2×C3×C4—52no—
275C2×C3×C4—51no—
276C2×C3×C4—102no—
277C2×C3×C4—101no—
278C2×C3×C4—201no—
279C2×C3×C4—202no—
280C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
281C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
282C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
283C3×C8—202no—
284C3×C8—202no—
285C22×C5—41no—
286C22×C5—41no—
287C22×C5—41no—
288C22×C5C2×C3×C22⋊C411yescharacteristic
289C2×D5—21no—
290C2×D5—21no—
291C2×D5—41no—
292C2×D5—41no—
293C2×D5—41no—
294C2×D5C3×#10212yes—
295C4×C5—21no—
296C4×C5—21no—
297C4×C5—41no—
298C4×C5C3×#10212yes—
299Dic5—21no—
300Dic5—41no—
301Dic5C2×C3×D811yescharacteristic
302Dic5C2×C3×SD1611yescharacteristic
303Dic5C3×#13512yes—
304#41—202no—
305#41—202no—
306C2×Q8—101no—
307C2×D4—51no—
308C2×D4—101no—
309C2×D4—101no—
310C2×D4—101no—
311C2×D4—101no—
312C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
313C22×C4—21no—
314C22×C4—51no—
315C22×C4—51no—
316C22×C4—51no—
317C22×C4—101no—
318C2×C8—102no—
319C2×C8—102no—
320C2×C8—202no—
321C4⋊C4—51no—
322C4⋊C4—101no—
323C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
324C22⋊C4—21no—
325C22⋊C4—101no—
326C22⋊C4—101no—
327C22⋊C4—101no—
328C42—51no—
329C42—51no—
330C42—51no—
331C42C2×C3×AGL(1,𝔽5)11yescharacteristic
332C3×C5#154011yescharacteristic, complemented, Hall
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—52no—
337C22×C3—101no—
338C22×C3—101no—
339C22×C3—201no—
340C22×C3—201no—
341C22×C3—201no—
342C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—41no—
346C3×C4—52no—
347C3×C4—51no—
348C3×C4—51no—
349C3×C4—101no—
350C3×C4—201no—
351C3×C4#400712yes—
352C2×C5—81no—
353C2×C5C3×#41311yescharacteristic
354C2×C5C2×C3×#10211yescharacteristic
355C2×C5C3×#41611yescharacteristic
356D5—22no—
357D5—81no—
358C23—21no—
359C23—51no—
360C23—101no—
361Q8—201no—
362Q8—201no—
363D4—41no—
364D4—201no—
365D4—202no—
366D4—201no—
367D4—201no—
368D4—202no—
369D4—202no—
370C2×C4—21no—
371C2×C4—42no—
372C2×C4—51no—
373C2×C4—51no—
374C2×C4—52no—
375C2×C4—52no—
376C2×C4—51no—
377C2×C4—102no—
378C2×C4—101no—
379C2×C4—201no—
380C2×C4—202no—
381C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
382C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
383C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
384C8—202no—
385C8—202no—
386C2×C3—81no—
387C2×C3—102no—
388C2×C3—401no—
389C2×C3#6576111yescharacteristic
390C2×C3C2×#400711yescharacteristic
391C2×C3#6579411yescharacteristic
392C5C3×#154011yescharacteristic, complemented, Sylow
393C22—41no—
394C22—41no—
395C22—41no—
396C22—52no—
397C22—101no—
398C22—101no—
399C22—201no—
400C22—201no—
401C22—201no—
402C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
403C4—21no—
404C4—21no—
405C4—41no—
406C4—51no—
407C4—51no—
408C4—52no—
409C4—101no—
410C4—201no—
411C4C3×#400712yes—
412C3#1060440811yescharacteristic, direct product factor, Sylow
413C2—81no—
414C2—102no—
415C2—401no—
416C2C3×#6576111yescharacteristic
417C2C2×C3×#400711yescharacteristic
418C2C3×#6579411yescharacteristic
subgroupG/H
centerC2×C3×C4C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C4×C5×D4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#65882C2
cosocleC4×Dic5C23×C3
hypercenterC3×C4×D4AGL(1,𝔽5)
hypocenterC5C3×#1540
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C3×C4 ◁ C3×C42 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
2. C3×D5⋊(C4⋊C8)C2semidirect
3. C3×#65678C2semidirect
4. C3×#64925C2semidirect
5. C3×#64873C2semidirect
6. #10604408C3direct (N × H)
26. D5⋊(C4⋊C8)C2×C3semidirect
27. #65678C2×C3semidirect
28. #64925C2×C3semidirect
29. #64873C2×C3semidirect
332. C3×C5#1540semidirect
392. C5C3×#1540semidirect
412. C3#10604408direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658822
C3×D5⋊(C4⋊C8)2
C3×#656782
C3×#649252
C3×#648732
#106044083
C3×#15405