Finite Group Databasebeta

← C3×#10604510 #14281869

Subgroups — C3×#10604510 #14281869

Order 1920.

424 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65958C211yesmaximal, characteristic
2C3×#65754C211yesmaximal, characteristic, complemented
3C3×#65748C211yesmaximal, characteristic, complemented
4C3×Q8⋊#170C212yesmaximal, complemented
5C3×#64871C212yesmaximal, complemented
6#10604510C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×C22⋊Q8C411yescharacteristic
8C3×#4093C411yescharacteristic
9C3×Q8×Dic5C2211yescharacteristic
10C3×#4076C411yescharacteristic
11C2×C3×C4×Dic5C2211yescharacteristic
12C3×#4044—21no—
13C3×#4039C2211yescharacteristic
14C3×#4038—21no—
15C3×#4036C411yescharacteristic
16C3×#4032—21no—
17C3×#4027—21no—
18C3×C22⋊#170—21no—
19C3×C22⋊#170—21no—
20C3×#4004—21no—
21C3×C4⋊#170C2212yes—
22C3×C4×#170—21no—
23C3×C4×#170C2212yes—
24C3×#1482—51nomaximal, Hall
25#65958C2×C311yescharacteristic
26#65754C2×C311yescharacteristic, complemented
27#65748C2×C311yescharacteristic, complemented
28Q8⋊#170C2×C312yescomplemented
29#64871C2×C312yescomplemented
30C2×C3×C5×Q8C2×C411yescharacteristic
31C22×C3×C4×C5C2×C411yescharacteristic
32C2×C3×C22⋊D5—21no—
33C22×C3×Dic5D411yescharacteristic
34C22×C3×Dic5D411yescharacteristic
35C2×C3×D20C2×C411yescharacteristic
36C2×C3×C4×D5—21no—
37C2×C3×#170—22no—
38C2×C3×#170—22no—
39C3×C5×C4⋊C4—21no—
40C3×C5×C4⋊C4C2×C411yescharacteristic
41C3×C5×C22⋊C4—21no—
42C3×#685—21no—
43C3×#685—21no—
44C3×#685—21no—
45C3×#685—21no—
46C3×C4⋊Dic5—21no—
47C3×C4⋊Dic5—21no—
48C3×C4⋊Dic5C2×C411yescharacteristic
49C3×#683—21no—
50C3×C4×Dic5—21no—
51C3×C4×Dic5C2×C411yescharacteristic
52C3×C4×Dic5D411yescharacteristic
53C3×C4×Dic5D411yescharacteristic
54C3×C4×Dic5C2311yescharacteristic
55C3×#529—51no—
56C3×#431—51no—
57C3×#431—51no—
58C3×Q8⋊C8—52no—
59C3×D4⋊C8—52no—
60C5×C22⋊Q8C3×C411yescharacteristic
61#4093C3×C411yescharacteristic
62Q8×Dic5C22×C311yescharacteristic
63#4076C3×C411yescharacteristic
64C2×C4×Dic5C22×C311yescharacteristic
65#4044—21no—
66#4039C22×C311yescharacteristic
67#4038—21no—
68#4036C3×C411yescharacteristic
69#4032—21no—
70#4027—21no—
71C22⋊#170—21no—
72C22⋊#170—21no—
73#4004—21no—
74C4⋊#170C22×C312yes—
75C4×#170—21no—
76C4×#170C22×C312yes—
77#1482—51noSylow
78C23×C3×C5C22⋊C411yescharacteristic
79C22×C3×D5—21no—
80C3×C5×Q8—22no—
81C2×C3×C4×C5—22no—
82C2×C3×C4×C5—21no—
83C2×C3×C4×C5—21no—
84C2×C3×C4×C5C22⋊C411yescharacteristic
85C2×C3×C4×C5C22×C411yescharacteristic
86C3×C22⋊D5—42no—
87C2×C3×Dic5—22no—
88C2×C3×Dic5—21no—
89C2×C3×Dic5—41no—
90C2×C3×Dic5C22⋊C411yescharacteristic
91C2×C3×Dic5C22⋊C411yescharacteristic
92C2×C3×Dic5C2×D411yescharacteristic
93C2×C3×Dic5C2×D411yescharacteristic
94C3×D20—22no—
95C3×C4×D5—42no—
96C3×#170—42no—
97C3×#170—42no—
98C3×#126—101no—
99C3×#125—51no—
100C3×#124—51no—
101C3×#123—101no—
102C3×C22⋊Q8AGL(1,𝔽5)11yescharacteristic
103C3×#121—51no—
104C3×C4×Q8—51no—
105C3×C4×D4—51no—
106C3×C4×D4—101no—
107C3×#117—101no—
108C2×C3×C42—51no—
109C3×C4⋊C8—52no—
110C3×C4⋊C8—101no—
111C3×C22⋊C8—101no—
112C3×C22⋊C8—101no—
113C3×C4×C8—52no—
114C3×C4×C8—101no—
115C2×C5×Q8C2×C3×C411yescharacteristic
116C22×C4×C5C2×C3×C411yescharacteristic
117C2×C22⋊D5—21no—
118C22×Dic5C3×D411yescharacteristic
119C22×Dic5C3×D411yescharacteristic
120C2×D20C2×C3×C411yescharacteristic
121C2×C4×D5—21no—
122C2×#170—22no—
123C2×#170—22no—
124C5×C4⋊C4—21no—
125C5×C4⋊C4C2×C3×C411yescharacteristic
126C5×C22⋊C4—21no—
127#685—21no—
128#685—21no—
129#685—21no—
130#685—21no—
131C4⋊Dic5—21no—
132C4⋊Dic5—21no—
133C4⋊Dic5C2×C3×C411yescharacteristic
134#683—21no—
135C4×Dic5—21no—
136C4×Dic5C2×C3×C411yescharacteristic
137C4×Dic5C3×D411yescharacteristic
138C4×Dic5C3×D411yescharacteristic
139C4×Dic5C23×C311yescharacteristic
140#529—51no—
141#431—51no—
142#431—51no—
143Q8⋊C8—52no—
144D4⋊C8—52no—
145C22×C3×C5—21no—
146C22×C3×C5—21no—
147C22×C3×C5—21no—
148C22×C3×C5C2×C22⋊C411yescharacteristic
149C2×C3×D5—41no—
150C2×C3×D5—41no—
151C2×C3×D5—41no—
152C3×C4×C5—21no—
153C3×C4×C5—41no—
154C3×C4×C5—41no—
155C3×C4×C5#13112yes—
156C3×Dic5—21no—
157C3×Dic5—21no—
158C3×Dic5—21no—
159C3×Dic5—41no—
160C3×Dic5#13512yes—
161C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
162C2×C3×D4—51no—
163C2×C3×D4—101no—
164C22×C3×C4—51no—
165C22×C3×C4—51no—
166C22×C3×C4—101no—
167C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
168C2×C3×C8—102no—
169C2×C3×C8—102no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—51no—
172C3×C4⋊C4—101no—
173C3×C4⋊C4—101no—
174C3×C4⋊C4—101no—
175C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
176C3×C22⋊C4—21no—
177C3×C22⋊C4—101no—
178C3×C22⋊C4—101no—
179C3×C22⋊C4—101no—
180C3×C22⋊C4—101no—
181C3×C42—51no—
182C3×C42—51no—
183C3×C42—51no—
184C3×C42—51no—
185C3×C42—101no—
186C23×C5C3×C22⋊C411yescharacteristic
187C22×D5—21no—
188C5×Q8—22no—
189C2×C4×C5—22no—
190C2×C4×C5—21no—
191C2×C4×C5—21no—
192C2×C4×C5C3×C22⋊C411yescharacteristic
193C2×C4×C5C22×C3×C411yescharacteristic
194C22⋊D5—42no—
195C2×Dic5—22no—
196C2×Dic5—21no—
197C2×Dic5—41no—
198C2×Dic5C3×C22⋊C411yescharacteristic
199C2×Dic5C3×C22⋊C411yescharacteristic
200C2×Dic5C2×C3×D411yescharacteristic
201C2×Dic5C2×C3×D411yescharacteristic
202D20—22no—
203C4×D5—42no—
204#170—42no—
205#170—42no—
206#126—101no—
207#125—51no—
208#124—51no—
209#123—101no—
210C22⋊Q8C3×AGL(1,𝔽5)11yescharacteristic
211#121—51no—
212C4×Q8—51no—
213C4×D4—51no—
214C4×D4—101no—
215#117—101no—
216C2×C42—51no—
217C4⋊C8—52no—
218C4⋊C8—101no—
219C22⋊C8—101no—
220C22⋊C8—101no—
221C4×C8—52no—
222C4×C8—101no—
223C2×C3×C5—41no—
224C2×C3×C5#40811yescharacteristic
225C2×C3×C5#41611yescharacteristic
226C2×C3×C5#42111yescharacteristic
227C3×D5—81no—
228C23×C3—101no—
229C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
230C3×Q8—22no—
231C3×D4—102no—
232C3×D4—202no—
233C2×C3×C4—22no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—51no—
237C2×C3×C4—51no—
238C2×C3×C4—51no—
239C2×C3×C4—51no—
240C2×C3×C4—102no—
241C2×C3×C4—101no—
242C2×C3×C4—202no—
243C2×C3×C4—201no—
244C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
245C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
246C3×C8—202no—
247C3×C8—202no—
248C22×C5—21no—
249C22×C5—21no—
250C22×C5—21no—
251C22×C5C2×C3×C22⋊C411yescharacteristic
252C2×D5—41no—
253C2×D5—41no—
254C2×D5—41no—
255C4×C5—21no—
256C4×C5—41no—
257C4×C5—41no—
258C4×C5C3×#13112yes—
259Dic5—21no—
260Dic5—21no—
261Dic5—21no—
262Dic5—41no—
263Dic5C3×#13512yes—
264C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
265C2×D4—51no—
266C2×D4—101no—
267C22×C4—51no—
268C22×C4—51no—
269C22×C4—101no—
270C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
271C2×C8—102no—
272C2×C8—102no—
273C4⋊C4—21no—
274C4⋊C4—51no—
275C4⋊C4—101no—
276C4⋊C4—101no—
277C4⋊C4—101no—
278C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—101no—
281C22⋊C4—101no—
282C22⋊C4—101no—
283C22⋊C4—101no—
284C42—51no—
285C42—51no—
286C42—51no—
287C42—51no—
288C42—101no—
289C3×C5#148211yescharacteristic, complemented, Hall
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—201no—
294C22×C3—201no—
295C22×C3—201no—
296C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
297C3×C4—21no—
298C3×C4—41no—
299C3×C4—41no—
300C3×C4—52no—
301C3×C4—101no—
302C3×C4—101no—
303C3×C4—101no—
304C3×C4—201no—
305C3×C4#413112yes—
306C2×C5—41no—
307C2×C5C3×#40811yescharacteristic
308C2×C5C3×#41611yescharacteristic
309C2×C5C3×#42111yescharacteristic
310D5—81no—
311C23—101no—
312C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
313Q8—22no—
314D4—102no—
315D4—202no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—51no—
320C2×C4—51no—
321C2×C4—51no—
322C2×C4—51no—
323C2×C4—102no—
324C2×C4—101no—
325C2×C4—202no—
326C2×C4—201no—
327C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
328C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
329C8—202no—
330C8—202no—
331C2×C3—41no—
332C2×C3—401no—
333C2×C3#6577511yescharacteristic
334C2×C3#6578411yescharacteristic
335C2×C3#6579411yescharacteristic
336C5C3×#148211yescharacteristic, complemented, Sylow
337C22—21no—
338C22—21no—
339C22—21no—
340C22—201no—
341C22—201no—
342C22—201no—
343C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—52no—
348C4—101no—
349C4—101no—
350C4—101no—
351C4—201no—
352C4C3×#413112yes—
353C3#1060451011yescharacteristic, direct product factor, Sylow
354C2—41no—
355C2—401no—
356C2C3×#6577511yescharacteristic
357C2C3×#6578411yescharacteristic
358C2C3×#6579411yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×C22⋊Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#4076C4
cosocleC4×Dic5C23×C3
hypercenterC3×C22⋊Q8AGL(1,𝔽5)
hypocenterC5C3×#1482
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×#65754C2semidirect
3. C3×#65748C2semidirect
4. C3×Q8⋊#170C2semidirect
5. C3×#64871C2semidirect
6. #10604510C3direct (N × H)
26. #65754C2×C3semidirect
27. #65748C2×C3semidirect
28. Q8⋊#170C2×C3semidirect
29. #64871C2×C3semidirect
289. C3×C5#1482semidirect
336. C5C3×#1482semidirect
353. C3#10604510direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#659582
C3×#657542
C3×#657482
C3×Q8⋊#1702
C3×#648712
#106045103
C3×#14825