Finite Group Databasebeta

← C3×#10604994 #14284192

Subgroups — C3×#10604994 #14284192

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 408 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×#4002C211yesmaximal, characteristic, complemented
4C3×#64927C212yesmaximal, complemented
5C3×#64921C212yesmaximal, complemented
6#10604994C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×#41C411yescharacteristic
8C2×C3×#709C411yescharacteristic
9C22×C3×#170—21no—
10C3×#4093C2211yescharacteristic
11C3×#4090C411yescharacteristic
12C3×#4086C411yescharacteristic
13C3×#4085—21no—
14C3×#4085—21no—
15C3×#4082C2211yescharacteristic
16C3×C4×C22⋊D5—21no—
17C3×C4×C22⋊D5—21no—
18C2×C3×C4×Dic5C2211yescharacteristic
19C3×C22⋊#170—21no—
20C3×#4004—21no—
21C3×#4002—21no—
22C3×#4002C2212yes—
23C3×#4002C2212yes—
24C3×#1530—51nomaximal, Hall
25#66166C2×C311yescharacteristic
26#65757C2×C311yescharacteristic, complemented
27C2×#4002C2×C311yescharacteristic, complemented
28#64927C2×C312yescomplemented
29#64921C2×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—24no—
46C2×C3×#170—22no—
47C2×C3×#170—22no—
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×#97—51no—
61C3×#329—52no—
62C3×#329—52no—
63C2×C5×#41C3×C411yescharacteristic
64C2×#709C3×C411yescharacteristic
65C22×#170—21no—
66#4093C22×C311yescharacteristic
67#4090C3×C411yescharacteristic
68#4086C3×C411yescharacteristic
69#4085—21no—
70#4085—21no—
71#4082C22×C311yescharacteristic
72C4×C22⋊D5—21no—
73C4×C22⋊D5—21no—
74C2×C4×Dic5C22×C311yescharacteristic
75C22⋊#170—21no—
76#4004—21no—
77#4002—21no—
78#4002C22×C312yes—
79#4002C22×C312yes—
80#1530—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—42no—
95C2×C3×Dic5—22no—
96C2×C3×Dic5—21no—
97C2×C3×Dic5—41no—
98C2×C3×Dic5C22⋊C411yescharacteristic
99C2×C3×Dic5C22⋊C411yescharacteristic
100C2×C3×Dic5C2×D411yescharacteristic
101C2×C3×Dic5C2×D411yescharacteristic
102C3×D20—41no—
103C3×C4×D5—42no—
104C3×Dic10—41no—
105C3×#170—24no—
106C3×#170—42no—
107C2×C3×#41—51no—
108C2×C3×#41AGL(1,𝔽5)11yescharacteristic
109C22×C3×C8—101no—
110C3×C4⋊Q8—51no—
111C3×#127—51no—
112C3×#124—51no—
113C3×#124—51no—
114C3×#121—101no—
115C3×#121—101no—
116C3×C4×D4—101no—
117C3×C4×D4—101no—
118C2×C3×C42—51no—
119C3×C4⋊C8—101no—
120C3×C22⋊C8—101no—
121C3×#97—52no—
122C3×#97—52no—
123C3×#97—101no—
124C5×#41—42no—
125C2×C5×Q8C2×C3×C411yescharacteristic
126C2×C5×D4—21no—
127C2×C5×D4C2×C3×C411yescharacteristic
128C22×C4×C5—21no—
129C22×C4×C5C2×C3×C411yescharacteristic
130C2×C22⋊D5—21no—
131C2×C22⋊D5—21no—
132C2×C22⋊D5—21no—
133C22×Dic5C3×D411yescharacteristic
134C22×Dic5C3×D411yescharacteristic
135#709—42no—
136C2×D20C2×C3×C411yescharacteristic
137C2×C4×D5—21no—
138C2×Dic10C2×C3×C411yescharacteristic
139C2×#170—24no—
140C2×#170—22no—
141C2×#170—22no—
142C22⋊Dic5—21no—
143C22⋊Dic5—21no—
144#685—21no—
145#685—21no—
146#683—21no—
147#683—21no—
148C4×Dic5C2×C3×C411yescharacteristic
149C4×Dic5C3×D411yescharacteristic
150C4×Dic5C3×D411yescharacteristic
151C4×Dic5C23×C311yescharacteristic
152#532—51no—
153#432—51no—
154C2×#97—51no—
155#329—52no—
156#329—52no—
157C22×C3×C5—21no—
158C22×C3×C5—21no—
159C22×C3×C5—41no—
160C22×C3×C5—41no—
161C22×C3×C5—41no—
162C22×C3×C5#10012yes—
163C22×C3×C5C2×C22⋊C411yescharacteristic
164C2×C3×D5—41no—
165C2×C3×D5—41no—
166C2×C3×D5—41no—
167C3×C4×C5—21no—
168C3×C4×C5—21no—
169C3×C4×C5—41no—
170C3×Dic5—21no—
171C3×Dic5—21no—
172C3×Dic5—21no—
173C3×Dic5—41no—
174C3×Dic5#10412yes—
175C3×#41—42no—
176C3×#41—202no—
177C2×C3×Q8—51no—
178C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
179C2×C3×D4—21no—
180C2×C3×D4—51no—
181C2×C3×D4—101no—
182C2×C3×D4—101no—
183C2×C3×D4—101no—
184C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
185C22×C3×C4—21no—
186C22×C3×C4—51no—
187C22×C3×C4—51no—
188C22×C3×C4—101no—
189C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
190C2×C3×C8—102no—
191C2×C3×C8—104no—
192C2×C3×C8—102no—
193C3×C4⋊C4—101no—
194C3×C4⋊C4—101no—
195C3×C22⋊C4—101no—
196C3×C22⋊C4—101no—
197C3×C22⋊C4—101no—
198C3×C22⋊C4—101no—
199C3×C42—51no—
200C3×C42—51no—
201C3×C42—51no—
202C3×C42—51no—
203C23×C5—21no—
204C23×C5C3×C22⋊C411yescharacteristic
205C22×D5—21no—
206C5×Q8—41no—
207C5×D4—41no—
208C5×D4—42no—
209C2×C4×C5—22no—
210C2×C4×C5—21no—
211C2×C4×C5—42no—
212C2×C4×C5C3×C22⋊C411yescharacteristic
213C2×C4×C5C22×C3×C411yescharacteristic
214C22⋊D5—42no—
215C22⋊D5—42no—
216C22⋊D5—42no—
217C2×Dic5—22no—
218C2×Dic5—21no—
219C2×Dic5—41no—
220C2×Dic5C3×C22⋊C411yescharacteristic
221C2×Dic5C3×C22⋊C411yescharacteristic
222C2×Dic5C2×C3×D411yescharacteristic
223C2×Dic5C2×C3×D411yescharacteristic
224D20—41no—
225C4×D5—42no—
226Dic10—41no—
227#170—24no—
228#170—42no—
229C2×#41—51no—
230C2×#41C3×AGL(1,𝔽5)11yescharacteristic
231C22×C8—101no—
232C4⋊Q8—51no—
233#127—51no—
234#124—51no—
235#124—51no—
236#121—101no—
237#121—101no—
238C4×D4—101no—
239C4×D4—101no—
240C2×C42—51no—
241C4⋊C8—101no—
242C22⋊C8—101no—
243#97—52no—
244#97—52no—
245#97—101no—
246C2×C3×C5—22no—
247C2×C3×C5—81no—
248C2×C3×C5C2×#10011yescharacteristic
249C2×C3×C5C2×#10411yescharacteristic
250C2×C3×C5#42111yescharacteristic
251C3×D5—81no—
252C23×C3—21no—
253C23×C3—101no—
254C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
255C3×Q8—41no—
256C3×Q8—201no—
257C3×D4—41no—
258C3×D4—42no—
259C3×D4—201no—
260C3×D4—202no—
261C3×D4—202no—
262C3×D4—202no—
263C2×C3×C4—22no—
264C2×C3×C4—21no—
265C2×C3×C4—42no—
266C2×C3×C4—51no—
267C2×C3×C4—51no—
268C2×C3×C4—51no—
269C2×C3×C4—51no—
270C2×C3×C4—102no—
271C2×C3×C4—101no—
272C2×C3×C4—202no—
273C2×C3×C4—201no—
274C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
275C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
276C3×C8—104no—
277C3×C8—202no—
278C22×C5—21no—
279C22×C5—21no—
280C22×C5—41no—
281C22×C5—41no—
282C22×C5—41no—
283C22×C5C3×#10012yes—
284C22×C5C2×C3×C22⋊C411yescharacteristic
285C2×D5—41no—
286C2×D5—41no—
287C2×D5—41no—
288C4×C5—21no—
289C4×C5—21no—
290C4×C5—41no—
291Dic5—21no—
292Dic5—21no—
293Dic5—21no—
294Dic5—41no—
295Dic5C3×#10412yes—
296#41—42no—
297#41—202no—
298C2×Q8—51no—
299C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
300C2×D4—21no—
301C2×D4—51no—
302C2×D4—101no—
303C2×D4—101no—
304C2×D4—101no—
305C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
306C22×C4—21no—
307C22×C4—51no—
308C22×C4—51no—
309C22×C4—101no—
310C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
311C2×C8—102no—
312C2×C8—104no—
313C2×C8—102no—
314C4⋊C4—101no—
315C4⋊C4—101no—
316C22⋊C4—101no—
317C22⋊C4—101no—
318C22⋊C4—101no—
319C22⋊C4—101no—
320C42—51no—
321C42—51no—
322C42—51no—
323C42—51no—
324C3×C5#153011yescharacteristic, complemented, Hall
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—201no—
331C22×C3—201no—
332C22×C3—201no—
333C22×C3#401312yes—
334C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—41no—
338C3×C4—52no—
339C3×C4—101no—
340C3×C4—101no—
341C3×C4—101no—
342C3×C4—201no—
343C2×C5—22no—
344C2×C5—81no—
345C2×C5C2×C3×#10011yescharacteristic
346C2×C5C2×C3×#10411yescharacteristic
347C2×C5C3×#42111yescharacteristic
348D5—81no—
349C23—21no—
350C23—101no—
351C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
352Q8—41no—
353Q8—201no—
354D4—41no—
355D4—42no—
356D4—201no—
357D4—202no—
358D4—202no—
359D4—202no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—51no—
364C2×C4—51no—
365C2×C4—51no—
366C2×C4—51no—
367C2×C4—102no—
368C2×C4—101no—
369C2×C4—202no—
370C2×C4—201no—
371C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
372C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
373C8—104no—
374C8—202no—
375C2×C3—22no—
376C2×C3—81no—
377C2×C3—401no—
378C2×C3D5⋊#10411yescharacteristic
379C2×C3#6579511yescharacteristic
380C2×C3C2×#401311yescharacteristic
381C5C3×#153011yescharacteristic, complemented, Sylow
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—201no—
388C22—201no—
389C22—201no—
390C22C3×#401312yes—
391C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
392C4—21no—
393C4—21no—
394C4—41no—
395C4—52no—
396C4—101no—
397C4—101no—
398C4—101no—
399C4—201no—
400C3#1060499411yescharacteristic, direct product factor, Sylow
401C2—22no—
402C2—81no—
403C2—401no—
404C2C3×D5⋊#10411yescharacteristic
405C2C3×#6579511yescharacteristic
406C2C2×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×#1530
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×#4002C2semidirect
4. C3×#64927C2semidirect
5. C3×#64921C2semidirect
6. #10604994C3direct (N × H)
26. #65757C2×C3semidirect
27. C2×#4002C2×C3semidirect
28. #64927C2×C3semidirect
29. #64921C2×C3semidirect
324. C3×C5#1530semidirect
381. C5C3×#1530semidirect
400. C3#10604994direct (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×#40022
C3×#649272
C3×#649212
#106049943
C3×#15305