Finite Group Databasebeta

← C3×#10604505 #14283931

Subgroups — C3×#10604505 #14283931

Order 1920.

468 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65958C211yesmaximal, characteristic
2C3×C4×#704C211yesmaximal, characteristic, complemented
3C2×C3×#4002C211yesmaximal, characteristic, complemented
4C3×#64927C212yesmaximal, complemented
5C3×#64870C212yesmaximal, complemented
6#10604505C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#704—21no—
8C22×C3×#170—21no—
9C3×C5×C22⋊Q8C411yescharacteristic
10C3×#4093C2211yescharacteristic
11C3×Q8×Dic5C411yescharacteristic
12C3×#4076C411yescharacteristic
13C2×C3×C4×Dic5C2211yescharacteristic
14C3×#4044—21no—
15C3×#4039C411yescharacteristic
16C3×#4038—21no—
17C3×#4036C2211yescharacteristic
18C3×#4032—21no—
19C3×#4027—21no—
20C3×#4002—21no—
21C3×#4002C2212yes—
22C3×#4002C2212yes—
23C3×C4×#170—21no—
24C3×#1524—51nomaximal, Hall
25#65958C2×C311yescharacteristic
26C4×#704C2×C311yescharacteristic, complemented
27C2×#4002C2×C311yescharacteristic, complemented
28#64927C2×C312yescomplemented
29#64870C2×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—
37C3×#704—24no—
38C2×C3×#170—24no—
39C2×C3×#170—22no—
40C2×C3×#170—22no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4C2×C411yescharacteristic
43C3×C5×C22⋊C4—21no—
44C3×#685—21no—
45C3×#685—21no—
46C3×#685—21no—
47C3×#685—21no—
48C3×C4⋊Dic5—21no—
49C3×C4⋊Dic5—21no—
50C3×C4⋊Dic5C2×C411yescharacteristic
51C3×#683—21no—
52C3×C4×Dic5—21no—
53C3×C4×Dic5C2×C411yescharacteristic
54C3×C4×Dic5D411yescharacteristic
55C3×C4×Dic5D411yescharacteristic
56C3×C4×Dic5C2311yescharacteristic
57C3×#529—51no—
58C3×C4×M16—51no—
59C2×C3×#97—51no—
60C3×#330—52no—
61C3×#329—52no—
62C2×#704—21no—
63C22×#170—21no—
64C5×C22⋊Q8C3×C411yescharacteristic
65#4093C22×C311yescharacteristic
66Q8×Dic5C3×C411yescharacteristic
67#4076C3×C411yescharacteristic
68C2×C4×Dic5C22×C311yescharacteristic
69#4044—21no—
70#4039C3×C411yescharacteristic
71#4038—21no—
72#4036C22×C311yescharacteristic
73#4032—21no—
74#4027—21no—
75#4002—21no—
76#4002C22×C312yes—
77#4002C22×C312yes—
78C4×#170—21no—
79#1524—51noSylow
80C23×C3×C5C22⋊C411yescharacteristic
81C22×C3×D5—21no—
82C3×C5×Q8—41no—
83C2×C3×C4×C5—21no—
84C2×C3×C4×C5—21no—
85C2×C3×C4×C5—41no—
86C2×C3×C4×C5C22⋊C411yescharacteristic
87C2×C3×C4×C5C22×C411yescharacteristic
88C3×C22⋊D5—42no—
89C2×C3×Dic5—22no—
90C2×C3×Dic5—22no—
91C2×C3×Dic5—21no—
92C2×C3×Dic5C22⋊C411yescharacteristic
93C2×C3×Dic5C22⋊C411yescharacteristic
94C2×C3×Dic5C2×D411yescharacteristic
95C2×C3×Dic5C2×D411yescharacteristic
96C3×D20—41no—
97C3×C4×D5—42no—
98C3×#170—24no—
99C3×#170—24no—
100C2×C3×M16—101no—
101C22×C3×C8—101no—
102C3×#126—101no—
103C3×#125—51no—
104C3×#124—51no—
105C3×#123—101no—
106C3×C22⋊Q8AGL(1,𝔽5)11yescharacteristic
107C3×#121—51no—
108C3×C4×Q8—51no—
109C3×C4×D4—51no—
110C3×C4×D4—101no—
111C3×#117—101no—
112C2×C3×C42—51no—
113C3×#97—52no—
114C3×#97—52no—
115C3×#97—101no—
116C3×C4×C8—101no—
117C2×C5×Q8C2×C3×C411yescharacteristic
118C22×C4×C5C2×C3×C411yescharacteristic
119C2×C22⋊D5—21no—
120C22×Dic5C3×D411yescharacteristic
121C22×Dic5C3×D411yescharacteristic
122C2×D20C2×C3×C411yescharacteristic
123C2×C4×D5—21no—
124#704—24no—
125C2×#170—24no—
126C2×#170—22no—
127C2×#170—22no—
128C5×C4⋊C4—21no—
129C5×C4⋊C4C2×C3×C411yescharacteristic
130C5×C22⋊C4—21no—
131#685—21no—
132#685—21no—
133#685—21no—
134#685—21no—
135C4⋊Dic5—21no—
136C4⋊Dic5—21no—
137C4⋊Dic5C2×C3×C411yescharacteristic
138#683—21no—
139C4×Dic5—21no—
140C4×Dic5C2×C3×C411yescharacteristic
141C4×Dic5C3×D411yescharacteristic
142C4×Dic5C3×D411yescharacteristic
143C4×Dic5C23×C311yescharacteristic
144#529—51no—
145C4×M16—51no—
146C2×#97—51no—
147#330—52no—
148#329—52no—
149C22×C3×C5—21no—
150C22×C3×C5—21no—
151C22×C3×C5#10412yes—
152C22×C3×C5C2×C22⋊C411yescharacteristic
153C2×C3×D5—41no—
154C2×C3×D5—41no—
155C2×C3×D5—41no—
156C3×C4×C5—21no—
157C3×C4×C5—21no—
158C3×C4×C5—41no—
159C3×C4×C5—41no—
160C3×Dic5—21no—
161C3×Dic5—21no—
162C3×Dic5—21no—
163C3×Dic5—41no—
164C3×Dic5#10412yes—
165C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
166C2×C3×D4—51no—
167C2×C3×D4—101no—
168C22×C3×C4—51no—
169C22×C3×C4—51no—
170C22×C3×C4—101no—
171C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
172C3×M16—104no—
173C2×C3×C8—102no—
174C2×C3×C8—102no—
175C2×C3×C8—104no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—51no—
178C3×C4⋊C4—101no—
179C3×C4⋊C4—101no—
180C3×C4⋊C4—101no—
181C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
182C3×C22⋊C4—21no—
183C3×C22⋊C4—101no—
184C3×C22⋊C4—101no—
185C3×C22⋊C4—101no—
186C3×C22⋊C4—101no—
187C3×C42—51no—
188C3×C42—51no—
189C3×C42—51no—
190C3×C42—51no—
191C3×C42—101no—
192C23×C5C3×C22⋊C411yescharacteristic
193C22×D5—21no—
194C5×Q8—41no—
195C2×C4×C5—21no—
196C2×C4×C5—21no—
197C2×C4×C5—41no—
198C2×C4×C5C3×C22⋊C411yescharacteristic
199C2×C4×C5C22×C3×C411yescharacteristic
200C22⋊D5—42no—
201C2×Dic5—22no—
202C2×Dic5—22no—
203C2×Dic5—21no—
204C2×Dic5C3×C22⋊C411yescharacteristic
205C2×Dic5C3×C22⋊C411yescharacteristic
206C2×Dic5C2×C3×D411yescharacteristic
207C2×Dic5C2×C3×D411yescharacteristic
208D20—41no—
209C4×D5—42no—
210#170—24no—
211#170—24no—
212C2×M16—101no—
213C22×C8—101no—
214#126—101no—
215#125—51no—
216#124—51no—
217#123—101no—
218C22⋊Q8C3×AGL(1,𝔽5)11yescharacteristic
219#121—51no—
220C4×Q8—51no—
221C4×D4—51no—
222C4×D4—101no—
223#117—101no—
224C2×C42—51no—
225#97—52no—
226#97—52no—
227#97—101no—
228C4×C8—101no—
229C2×C3×C5—22no—
230C2×C3×C5#41311yescharacteristic
231C2×C3×C5C2×#10411yescharacteristic
232C2×C3×C5C2×#10411yescharacteristic
233C3×D5—81no—
234C23×C3—101no—
235C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
236C3×Q8—41no—
237C3×D4—201no—
238C3×D4—202no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—41no—
242C2×C3×C4—51no—
243C2×C3×C4—51no—
244C2×C3×C4—51no—
245C2×C3×C4—51no—
246C2×C3×C4—102no—
247C2×C3×C4—102no—
248C2×C3×C4—101no—
249C2×C3×C4—202no—
250C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
251C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
252C3×C8—104no—
253C3×C8—104no—
254C22×C5—21no—
255C22×C5—21no—
256C22×C5C3×#10412yes—
257C22×C5C2×C3×C22⋊C411yescharacteristic
258C2×D5—41no—
259C2×D5—41no—
260C2×D5—41no—
261C4×C5—21no—
262C4×C5—21no—
263C4×C5—41no—
264C4×C5—41no—
265Dic5—21no—
266Dic5—21no—
267Dic5—21no—
268Dic5—41no—
269Dic5C3×#10412yes—
270C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
271C2×D4—51no—
272C2×D4—101no—
273C22×C4—51no—
274C22×C4—51no—
275C22×C4—101no—
276C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
277M16—104no—
278C2×C8—102no—
279C2×C8—102no—
280C2×C8—104no—
281C4⋊C4—21no—
282C4⋊C4—51no—
283C4⋊C4—101no—
284C4⋊C4—101no—
285C4⋊C4—101no—
286C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—101no—
289C22⋊C4—101no—
290C22⋊C4—101no—
291C22⋊C4—101no—
292C42—51no—
293C42—51no—
294C42—51no—
295C42—51no—
296C42—101no—
297C3×C5#152411yescharacteristic, complemented, Hall
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—201no—
301C22×C3—201no—
302C22×C3—201no—
303C22×C3#401012yes—
304C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—41no—
308C3×C4—41no—
309C3×C4—52no—
310C3×C4—101no—
311C3×C4—101no—
312C3×C4—101no—
313C3×C4—201no—
314C2×C5—22no—
315C2×C5C3×#41311yescharacteristic
316C2×C5C2×C3×#10411yescharacteristic
317C2×C5C2×C3×#10411yescharacteristic
318D5—81no—
319C23—101no—
320C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
321Q8—41no—
322D4—201no—
323D4—202no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—51no—
328C2×C4—51no—
329C2×C4—51no—
330C2×C4—51no—
331C2×C4—102no—
332C2×C4—102no—
333C2×C4—101no—
334C2×C4—202no—
335C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
336C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
337C8—104no—
338C8—104no—
339C2×C3—22no—
340C2×C3—401no—
341C2×C3#6577711yescharacteristic
342C2×C3C2×#401011yescharacteristic
343C2×C3D5⋊#10411yescharacteristic
344C5C3×#152411yescharacteristic, complemented, Sylow
345C22—21no—
346C22—21no—
347C22—201no—
348C22—201no—
349C22—201no—
350C22C3×#401012yes—
351C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—52no—
357C4—101no—
358C4—101no—
359C4—101no—
360C4—201no—
361C3#1060450511yescharacteristic, direct product factor, Sylow
362C2—22no—
363C2—401no—
364C2C3×#6577711yescharacteristic
365C2C2×C3×#401011yescharacteristic
366C2C3×D5⋊#10411yescharacteristic
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×#1524
LCSG ▷ C2×C4×C5 ▷ C22×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×C4×#704C2semidirect
3. C2×C3×#4002C2semidirect
4. C3×#64927C2semidirect
5. C3×#64870C2semidirect
6. #10604505C3direct (N × H)
26. C4×#704C2×C3semidirect
27. C2×#4002C2×C3semidirect
28. #64927C2×C3semidirect
29. #64870C2×C3semidirect
297. C3×C5#1524semidirect
344. C5C3×#1524semidirect
361. C3#10604505direct (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×C4×#7042
C2×C3×#40022
C3×#649272
C3×#648702
#106045053
C3×#15245