Finite Group Databasebeta

← C3×#10605005 #14285030

Subgroups — C3×#10605005 #14285030

Order 1920.

504 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×#66166C211yesmaximal, characteristic
2C3×#65757C211yesmaximal, characteristic, complemented
3C2×C3×#4004C211yesmaximal, characteristic, complemented
4C3×#64931C212yesmaximal, complemented
5C3×#64925C212yesmaximal, complemented
6#10605005C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×#41C411yescharacteristic
8C2×C3×#709C411yescharacteristic
9C22×C3×#170—21no—
10C3×#4093C411yescharacteristic
11C3×#4090C2211yescharacteristic
12C3×#4086C2211yescharacteristic
13C3×#4085—21no—
14C3×#4085—21no—
15C3×#4082C411yescharacteristic
16C3×C4×C22⋊D5—21no—
17C3×C4×C22⋊D5—21no—
18C2×C3×C4×Dic5C2211yescharacteristic
19C3×C22⋊#170—21no—
20C3×#4004—21no—
21C3×#4004C2212yes—
22C3×#4004C2212yes—
23C3×#4002—21no—
24C3×#1551—51nomaximal, Hall
25#66166C2×C311yescharacteristic
26#65757C2×C311yescharacteristic, complemented
27C2×#4004C2×C311yescharacteristic, complemented
28#64931C2×C312yescomplemented
29#64925C2×C312yescomplemented
30C3×C5×#41—42no—
31C2×C3×C5×Q8C2×C411yescharacteristic
32C2×C3×C5×D4—21no—
33C2×C3×C5×D4C2×C411yescharacteristic
34C22×C3×C4×C5—21no—
35C22×C3×C4×C5C2×C411yescharacteristic
36C2×C3×C22⋊D5—21no—
37C2×C3×C22⋊D5—21no—
38C2×C3×C22⋊D5—21no—
39C22×C3×Dic5D411yescharacteristic
40C22×C3×Dic5D411yescharacteristic
41C3×#709—42no—
42C2×C3×D20C2×C411yescharacteristic
43C2×C3×C4×D5—21no—
44C2×C3×Dic10C2×C411yescharacteristic
45C2×C3×#170—22no—
46C2×C3×#170—22no—
47C2×C3×#170—42no—
48C3×C22⋊Dic5—21no—
49C3×C22⋊Dic5—21no—
50C3×#685—21no—
51C3×#685—21no—
52C3×#683—21no—
53C3×#683—21no—
54C3×C4×Dic5C2×C411yescharacteristic
55C3×C4×Dic5D411yescharacteristic
56C3×C4×Dic5D411yescharacteristic
57C3×C4×Dic5C2311yescharacteristic
58C3×#532—51no—
59C3×#432—51no—
60C2×C3×C4⋊C8—51no—
61C3×#333—52no—
62C3×#331—52no—
63C2×C5×#41C3×C411yescharacteristic
64C2×#709C3×C411yescharacteristic
65C22×#170—21no—
66#4093C3×C411yescharacteristic
67#4090C22×C311yescharacteristic
68#4086C22×C311yescharacteristic
69#4085—21no—
70#4085—21no—
71#4082C3×C411yescharacteristic
72C4×C22⋊D5—21no—
73C4×C22⋊D5—21no—
74C2×C4×Dic5C22×C311yescharacteristic
75C22⋊#170—21no—
76#4004—21no—
77#4004C22×C312yes—
78#4004C22×C312yes—
79#4002—21no—
80#1551—51noSylow
81C23×C3×C5—21no—
82C23×C3×C5C22⋊C411yescharacteristic
83C22×C3×D5—21no—
84C3×C5×Q8—41no—
85C3×C5×D4—41no—
86C3×C5×D4—42no—
87C2×C3×C4×C5—22no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—42no—
90C2×C3×C4×C5C22⋊C411yescharacteristic
91C2×C3×C4×C5C22×C411yescharacteristic
92C3×C22⋊D5—42no—
93C3×C22⋊D5—42no—
94C3×C22⋊D5—41no—
95C3×C22⋊D5—41no—
96C2×C3×Dic5—22no—
97C2×C3×Dic5—21no—
98C2×C3×Dic5—21no—
99C2×C3×Dic5—21no—
100C2×C3×Dic5C22⋊C411yescharacteristic
101C2×C3×Dic5C22⋊C411yescharacteristic
102C2×C3×Dic5C2×D411yescharacteristic
103C2×C3×Dic5C2×D411yescharacteristic
104C3×D20—41no—
105C3×C4×D5—42no—
106C3×Dic10—41no—
107C3×#170—42no—
108C3×#170—42no—
109C2×C3×#41—51no—
110C2×C3×#41AGL(1,𝔽5)11yescharacteristic
111C22×C3×C8—101no—
112C3×C4⋊Q8—51no—
113C3×#127—51no—
114C3×#124—51no—
115C3×#124—51no—
116C3×#121—101no—
117C3×#121—101no—
118C3×C4×D4—101no—
119C3×C4×D4—101no—
120C2×C3×C42—51no—
121C3×C4⋊C8—52no—
122C3×C4⋊C8—52no—
123C3×C4⋊C8—101no—
124C3×C22⋊C8—101no—
125C3×#97—101no—
126C5×#41—42no—
127C2×C5×Q8C2×C3×C411yescharacteristic
128C2×C5×D4—21no—
129C2×C5×D4C2×C3×C411yescharacteristic
130C22×C4×C5—21no—
131C22×C4×C5C2×C3×C411yescharacteristic
132C2×C22⋊D5—21no—
133C2×C22⋊D5—21no—
134C2×C22⋊D5—21no—
135C22×Dic5C3×D411yescharacteristic
136C22×Dic5C3×D411yescharacteristic
137#709—42no—
138C2×D20C2×C3×C411yescharacteristic
139C2×C4×D5—21no—
140C2×Dic10C2×C3×C411yescharacteristic
141C2×#170—22no—
142C2×#170—22no—
143C2×#170—42no—
144C22⋊Dic5—21no—
145C22⋊Dic5—21no—
146#685—21no—
147#685—21no—
148#683—21no—
149#683—21no—
150C4×Dic5C2×C3×C411yescharacteristic
151C4×Dic5C3×D411yescharacteristic
152C4×Dic5C3×D411yescharacteristic
153C4×Dic5C23×C311yescharacteristic
154#532—51no—
155#432—51no—
156C2×C4⋊C8—51no—
157#333—52no—
158#331—52no—
159C22×C3×C5—21no—
160C22×C3×C5—21no—
161C22×C3×C5—41no—
162C22×C3×C5—41no—
163C22×C3×C5—41no—
164C22×C3×C5#10012yes—
165C22×C3×C5C2×C22⋊C411yescharacteristic
166C2×C3×D5—41no—
167C2×C3×D5—41no—
168C2×C3×D5—41no—
169C3×C4×C5—21no—
170C3×C4×C5—21no—
171C3×C4×C5—41no—
172C3×Dic5—21no—
173C3×Dic5—21no—
174C3×Dic5—41no—
175C3×Dic5#13512yes—
176C3×Dic5AGL(1,ℤ/8)11yescharacteristic
177C3×Dic5#13711yescharacteristic
178C3×#41—42no—
179C3×#41—202no—
180C2×C3×Q8—51no—
181C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
182C2×C3×D4—21no—
183C2×C3×D4—51no—
184C2×C3×D4—101no—
185C2×C3×D4—101no—
186C2×C3×D4—101no—
187C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
188C22×C3×C4—21no—
189C22×C3×C4—51no—
190C22×C3×C4—51no—
191C22×C3×C4—101no—
192C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
193C2×C3×C8—102no—
194C2×C3×C8—102no—
195C2×C3×C8—202no—
196C3×C4⋊C4—101no—
197C3×C4⋊C4—101no—
198C3×C22⋊C4—101no—
199C3×C22⋊C4—101no—
200C3×C22⋊C4—101no—
201C3×C22⋊C4—101no—
202C3×C42—51no—
203C3×C42—51no—
204C3×C42—51no—
205C3×C42—51no—
206C23×C5—21no—
207C23×C5C3×C22⋊C411yescharacteristic
208C22×D5—21no—
209C5×Q8—41no—
210C5×D4—41no—
211C5×D4—42no—
212C2×C4×C5—22no—
213C2×C4×C5—21no—
214C2×C4×C5—42no—
215C2×C4×C5C3×C22⋊C411yescharacteristic
216C2×C4×C5C22×C3×C411yescharacteristic
217C22⋊D5—42no—
218C22⋊D5—42no—
219C22⋊D5—41no—
220C22⋊D5—41no—
221C2×Dic5—22no—
222C2×Dic5—21no—
223C2×Dic5—21no—
224C2×Dic5—21no—
225C2×Dic5C3×C22⋊C411yescharacteristic
226C2×Dic5C3×C22⋊C411yescharacteristic
227C2×Dic5C2×C3×D411yescharacteristic
228C2×Dic5C2×C3×D411yescharacteristic
229D20—41no—
230C4×D5—42no—
231Dic10—41no—
232#170—42no—
233#170—42no—
234C2×#41—51no—
235C2×#41C3×AGL(1,𝔽5)11yescharacteristic
236C22×C8—101no—
237C4⋊Q8—51no—
238#127—51no—
239#124—51no—
240#124—51no—
241#121—101no—
242#121—101no—
243C4×D4—101no—
244C4×D4—101no—
245C2×C42—51no—
246C4⋊C8—52no—
247C4⋊C8—52no—
248C4⋊C8—101no—
249C22⋊C8—101no—
250#97—101no—
251C2×C3×C5—22no—
252C2×C3×C5—81no—
253C2×C3×C5C2×#10011yescharacteristic
254C2×C3×C5#41611yescharacteristic
255C2×C3×C5#41711yescharacteristic
256C3×D5—81no—
257C23×C3—21no—
258C23×C3—101no—
259C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
260C3×Q8—41no—
261C3×Q8—201no—
262C3×D4—41no—
263C3×D4—42no—
264C3×D4—201no—
265C3×D4—201no—
266C3×D4—201no—
267C3×D4—202no—
268C3×D4—202no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—42no—
272C2×C3×C4—51no—
273C2×C3×C4—51no—
274C2×C3×C4—51no—
275C2×C3×C4—51no—
276C2×C3×C4—101no—
277C2×C3×C4—101no—
278C2×C3×C4—102no—
279C2×C3×C4—101no—
280C2×C3×C4—202no—
281C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
282C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
283C3×C8—202no—
284C3×C8—202no—
285C22×C5—21no—
286C22×C5—21no—
287C22×C5—41no—
288C22×C5—41no—
289C22×C5—41no—
290C22×C5C3×#10012yes—
291C22×C5C2×C3×C22⋊C411yescharacteristic
292C2×D5—41no—
293C2×D5—41no—
294C2×D5—41no—
295C4×C5—21no—
296C4×C5—21no—
297C4×C5—41no—
298Dic5—21no—
299Dic5—21no—
300Dic5—41no—
301Dic5C3×#13512yes—
302Dic5C3×AGL(1,ℤ/8)11yescharacteristic
303Dic5C3×#13711yescharacteristic
304#41—42no—
305#41—202no—
306C2×Q8—51no—
307C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
308C2×D4—21no—
309C2×D4—51no—
310C2×D4—101no—
311C2×D4—101no—
312C2×D4—101no—
313C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
314C22×C4—21no—
315C22×C4—51no—
316C22×C4—51no—
317C22×C4—101no—
318C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
319C2×C8—102no—
320C2×C8—102no—
321C2×C8—202no—
322C4⋊C4—101no—
323C4⋊C4—101no—
324C22⋊C4—101no—
325C22⋊C4—101no—
326C22⋊C4—101no—
327C22⋊C4—101no—
328C42—51no—
329C42—51no—
330C42—51no—
331C42—51no—
332C3×C5#155111yescharacteristic, complemented, Hall
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—201no—
339C22×C3—201no—
340C22×C3—201no—
341C22×C3#401312yes—
342C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—41no—
346C3×C4—51no—
347C3×C4—51no—
348C3×C4—52no—
349C3×C4—101no—
350C3×C4—101no—
351C3×C4—201no—
352C2×C5—22no—
353C2×C5—81no—
354C2×C5C2×C3×#10011yescharacteristic
355C2×C5C3×#41611yescharacteristic
356C2×C5C3×#41711yescharacteristic
357D5—81no—
358C23—21no—
359C23—101no—
360C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
361Q8—41no—
362Q8—201no—
363D4—41no—
364D4—42no—
365D4—201no—
366D4—201no—
367D4—201no—
368D4—202no—
369D4—202no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—51no—
374C2×C4—51no—
375C2×C4—51no—
376C2×C4—51no—
377C2×C4—101no—
378C2×C4—101no—
379C2×C4—102no—
380C2×C4—101no—
381C2×C4—202no—
382C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
383C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
384C8—202no—
385C8—202no—
386C2×C3—22no—
387C2×C3—81no—
388C2×C3—401no—
389C2×C3#6579311yescharacteristic
390C2×C3#6579411yescharacteristic
391C2×C3C2×#401311yescharacteristic
392C5C3×#155111yescharacteristic, complemented, Sylow
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—201no—
399C22—201no—
400C22—201no—
401C22C3×#401312yes—
402C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
403C4—21no—
404C4—21no—
405C4—41no—
406C4—51no—
407C4—51no—
408C4—52no—
409C4—101no—
410C4—101no—
411C4—201no—
412C3#1060500511yescharacteristic, direct product factor, Sylow
413C2—22no—
414C2—81no—
415C2—401no—
416C2C3×#6579311yescharacteristic
417C2C3×#6579411yescharacteristic
418C2C2×C3×#401311yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC2×C3×C5×#41C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#66166C2
cosocleC4×Dic5C23×C3
hypercenterC2×C3×#41AGL(1,𝔽5)
hypocenterC5C3×#1551
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#41

Complements

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

NH ≅ G/Ntype
2. C3×#65757C2semidirect
3. C2×C3×#4004C2semidirect
4. C3×#64931C2semidirect
5. C3×#64925C2semidirect
6. #10605005C3direct (N × H)
26. #65757C2×C3semidirect
27. C2×#4004C2×C3semidirect
28. #64931C2×C3semidirect
29. #64925C2×C3semidirect
332. C3×C5#1551semidirect
392. C5C3×#1551semidirect
412. C3#10605005direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#661662
C3×#657572
C2×C3×#40042
C3×#649312
C3×#649252
#106050053
C3×#15515