Finite Group Databasebeta

← C3×#10604466 #14282828

Subgroups — C3×#10604466 #14282828

Order 1920.

608 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65940C211yesmaximal, characteristic
2C3×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3C2×C3×#4003C211yesmaximal, characteristic, complemented
4C3×#64920C212yesmaximal, complemented
5C3×#64869C212yesmaximal, complemented
6#10604466C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#711C411yescharacteristic, complemented
8C22×C3×C4×D5C2211yescharacteristic
9C2×C3×C22⋊AGL(1,𝔽5)—21no—
10C22×C3×#170—21no—
11C2×C3×C4×AGL(1,𝔽5)—21no—
12C3×C5×#121C411yescharacteristic, complemented
13C3×#4084C2211yescharacteristic
14C3×#4083—21no—
15C3×D4×Dic5—21no—
16C3×#4070C411yescharacteristic, complemented
17C3×#4043C2211yescharacteristic
18C3×#4038C411yescharacteristic, complemented
19C3×#4029—21no—
20C3×#4027—21no—
21C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
22C3×#4003—21no—
23C3×#4003C2212yes—
24C3×#1501—51nomaximal, Hall
25#65940C2×C311yescharacteristic
26C4×C22⋊AGL(1,𝔽5)C2×C311yescharacteristic, complemented
27C2×#4003C2×C311yescharacteristic, complemented
28#64920C2×C312yescomplemented
29#64869C2×C312yescomplemented
30C23×C3×D5D411yescharacteristic
31C22×C3×AGL(1,𝔽5)—22no—
32C2×C3×C5×D4—21no—
33C2×C3×C5×D4C2×C411yescharacteristic
34C22×C3×C4×C5C2×C411yescharacteristic
35C2×C3×C22⋊D5—21no—
36C2×C3×C22⋊D5—21no—
37C22×C3×Dic5—21no—
38C22×C3×Dic5D411yescharacteristic
39C3×#711—42no—
40C2×C3×C4×D5—42no—
41C2×C3×C4×D5C2×C411yescharacteristic
42C2×C3×C4×D5D411yescharacteristic
43C2×C3×C4×D5D411yescharacteristic
44C2×C3×C4×D5C2311yescharacteristic
45C2×C3×Dic10C2×C411yescharacteristic
46C3×C22⋊AGL(1,𝔽5)—24no—
47C2×C3×#170—22no—
48C2×C3×#170—42no—
49C3×C4×AGL(1,𝔽5)—42no—
50C3×C5×C4⋊C4C2×C411yescharacteristic
51C3×C5×C22⋊C4—21no—
52C3×C22⋊Dic5—21no—
53C3×C22⋊Dic5—21no—
54C3×#685—21no—
55C3×#685—21no—
56C3×C4⋊Dic5—21no—
57C3×C4⋊Dic5C2×C411yescharacteristic
58C3×#683—21no—
59C3×C4×Dic5—21no—
60C3×#525—51no—
61C2×C3×C22⋊C8—51no—
62C3×C4×C22⋊C4—51no—
63C3×#328—52no—
64C3×#327—52no—
65C2×#711C3×C411yescharacteristic, complemented
66C22×C4×D5C22×C311yescharacteristic
67C2×C22⋊AGL(1,𝔽5)—21no—
68C22×#170—21no—
69C2×C4×AGL(1,𝔽5)—21no—
70C5×#121C3×C411yescharacteristic, complemented
71#4084C22×C311yescharacteristic
72#4083—21no—
73D4×Dic5—21no—
74#4070C3×C411yescharacteristic, complemented
75#4043C22×C311yescharacteristic
76#4038C3×C411yescharacteristic, complemented
77#4029—21no—
78#4027—21no—
79(C2×AGL(1,𝔽5))⋊C4C22×C312yes—
80#4003—21no—
81#4003C22×C312yes—
82#1501—51noSylow
83C23×C3×C5—21no—
84C23×C3×C5C22⋊C411yescharacteristic
85C22×C3×D5—22no—
86C22×C3×D5—22no—
87C22×C3×D5—41no—
88C22×C3×D5C22⋊C411yescharacteristic
89C22×C3×D5C2×D411yescharacteristic
90C2×C3×AGL(1,𝔽5)—24no—
91C2×C3×AGL(1,𝔽5)—42no—
92C2×C3×AGL(1,𝔽5)—42no—
93C3×C5×D4—41no—
94C3×C5×D4—42no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—41no—
97C2×C3×C4×C5C22⋊C411yescharacteristic
98C2×C3×C4×C5C22×C411yescharacteristic
99C3×C22⋊D5—42no—
100C3×C22⋊D5—42no—
101C2×C3×Dic5—22no—
102C2×C3×Dic5—21no—
103C2×C3×Dic5—21no—
104C2×C3×Dic5—42no—
105C2×C3×Dic5C22⋊C411yescharacteristic
106C2×C3×Dic5C2×D411yescharacteristic
107C3×C4×D5—22no—
108C3×C4×D5—22no—
109C3×C4×D5—41no—
110C3×C4×D5—41no—
111C3×Dic10—41no—
112C3×#170—42no—
113C2×C3×#41—51no—
114C23×C3×C4—51no—
115C22×C3×C8—101no—
116C3×#123—101no—
117C3×C22⋊Q8—51no—
118C3×C22⋊Q8—51no—
119C3×#121—51no—
120C3×#121AGL(1,𝔽5)11yescharacteristic, complemented
121C3×#120—101no—
122C3×C4×D4—101no—
123C3×C4×D4—101no—
124C3×#117—51no—
125C2×C3×C22⋊C4—101no—
126C2×C3×C42—101no—
127C3×C22⋊C8—52no—
128C3×C22⋊C8—101no—
129C3×#95—52no—
130C23×D5C3×D411yescharacteristic
131C22×AGL(1,𝔽5)—22no—
132C2×C5×D4—21no—
133C2×C5×D4C2×C3×C411yescharacteristic
134C22×C4×C5C2×C3×C411yescharacteristic
135C2×C22⋊D5—21no—
136C2×C22⋊D5—21no—
137C22×Dic5—21no—
138C22×Dic5C3×D411yescharacteristic
139#711—42no—
140C2×C4×D5—42no—
141C2×C4×D5C2×C3×C411yescharacteristic
142C2×C4×D5C3×D411yescharacteristic
143C2×C4×D5C3×D411yescharacteristic
144C2×C4×D5C23×C311yescharacteristic
145C2×Dic10C2×C3×C411yescharacteristic
146C22⋊AGL(1,𝔽5)—24no—
147C2×#170—22no—
148C2×#170—42no—
149C4×AGL(1,𝔽5)—42no—
150C5×C4⋊C4C2×C3×C411yescharacteristic
151C5×C22⋊C4—21no—
152C22⋊Dic5—21no—
153C22⋊Dic5—21no—
154#685—21no—
155#685—21no—
156C4⋊Dic5—21no—
157C4⋊Dic5C2×C3×C411yescharacteristic
158#683—21no—
159C4×Dic5—21no—
160#525—51no—
161C2×C22⋊C8—51no—
162C4×C22⋊C4—51no—
163#328—52no—
164#327—52no—
165C22×C3×C5—21no—
166C22×C3×C5—21no—
167C22×C3×C5—41no—
168C22×C3×C5—41no—
169C22×C3×C5—41no—
170C22×C3×C5#10412yes—
171C22×C3×C5C2×C22⋊C411yescharacteristic
172C2×C3×D5—21no—
173C2×C3×D5—21no—
174C2×C3×D5—21no—
175C2×C3×D5—21no—
176C2×C3×D5—21no—
177C2×C3×D5—44no—
178C2×C3×D5#13512yes—
179C3×AGL(1,𝔽5)—44no—
180C3×C4×C5—21no—
181C3×C4×C5—21no—
182C3×C4×C5—41no—
183C3×Dic5—21no—
184C3×Dic5—41no—
185C3×Dic5—41no—
186C3×Dic5#9912yes—
187C24×C3—51no—
188C3×#41—202no—
189C2×C3×Q8—51no—
190C2×C3×D4—21no—
191C2×C3×D4—101no—
192C2×C3×D4—101no—
193C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
194C22×C3×C4—51no—
195C22×C3×C4—51no—
196C22×C3×C4—51no—
197C22×C3×C4—51no—
198C22×C3×C4—51no—
199C22×C3×C4—101no—
200C22×C3×C4—102no—
201C22×C3×C4—202no—
202C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
203C2×C3×C8—102no—
204C2×C3×C8—202no—
205C3×C4⋊C4—51no—
206C3×C4⋊C4—101no—
207C3×C4⋊C4—101no—
208C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
209C3×C22⋊C4—21no—
210C3×C22⋊C4—101no—
211C3×C22⋊C4—101no—
212C3×C22⋊C4—101no—
213C3×C22⋊C4—104no—
214C3×C22⋊C4—101no—
215C3×C42—101no—
216C3×C42—202no—
217C23×C5—21no—
218C23×C5C3×C22⋊C411yescharacteristic
219C22×D5—22no—
220C22×D5—22no—
221C22×D5—41no—
222C22×D5C3×C22⋊C411yescharacteristic
223C22×D5C2×C3×D411yescharacteristic
224C2×AGL(1,𝔽5)—24no—
225C2×AGL(1,𝔽5)—42no—
226C2×AGL(1,𝔽5)—42no—
227C5×D4—41no—
228C5×D4—42no—
229C2×C4×C5—21no—
230C2×C4×C5—41no—
231C2×C4×C5C3×C22⋊C411yescharacteristic
232C2×C4×C5C22×C3×C411yescharacteristic
233C22⋊D5—42no—
234C22⋊D5—42no—
235C2×Dic5—22no—
236C2×Dic5—21no—
237C2×Dic5—21no—
238C2×Dic5—42no—
239C2×Dic5C3×C22⋊C411yescharacteristic
240C2×Dic5C2×C3×D411yescharacteristic
241C4×D5—22no—
242C4×D5—22no—
243C4×D5—41no—
244C4×D5—41no—
245Dic10—41no—
246#170—42no—
247C2×#41—51no—
248C23×C4—51no—
249C22×C8—101no—
250#123—101no—
251C22⋊Q8—51no—
252C22⋊Q8—51no—
253#121—51no—
254#121C3×AGL(1,𝔽5)11yescharacteristic, complemented
255#120—101no—
256C4×D4—101no—
257C4×D4—101no—
258#117—51no—
259C2×C22⋊C4—101no—
260C2×C42—101no—
261C22⋊C8—52no—
262C22⋊C8—101no—
263#95—52no—
264C2×C3×C5—22no—
265C2×C3×C5—81no—
266C2×C3×C5C2×#9911yescharacteristic
267C2×C3×C5#41611yescharacteristic
268C2×C3×C5C2×#10411yescharacteristic
269C3×D5—22no—
270C3×D5—41no—
271C23×C3—21no—
272C23×C3—51no—
273C23×C3—51no—
274C23×C3—102no—
275C23×C3—102no—
276C23×C3—201no—
277C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
278C3×Q8—201no—
279C3×D4—41no—
280C3×D4—42no—
281C3×D4—202no—
282C3×D4—202no—
283C2×C3×C4—21no—
284C2×C3×C4—41no—
285C2×C3×C4—51no—
286C2×C3×C4—51no—
287C2×C3×C4—102no—
288C2×C3×C4—102no—
289C2×C3×C4—102no—
290C2×C3×C4—104no—
291C2×C3×C4—101no—
292C2×C3×C4—101no—
293C2×C3×C4—201no—
294C2×C3×C4—201no—
295C2×C3×C4—202no—
296C2×C3×C4—202no—
297C2×C3×C4—202no—
298C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
299C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
300C3×C8—202no—
301C22×C5—21no—
302C22×C5—21no—
303C22×C5—41no—
304C22×C5—41no—
305C22×C5—41no—
306C22×C5C3×#10412yes—
307C22×C5C2×C3×C22⋊C411yescharacteristic
308C2×D5—21no—
309C2×D5—21no—
310C2×D5—21no—
311C2×D5—21no—
312C2×D5—21no—
313C2×D5—44no—
314C2×D5C3×#13512yes—
315AGL(1,𝔽5)—44no—
316C4×C5—21no—
317C4×C5—21no—
318C4×C5—41no—
319Dic5—21no—
320Dic5—41no—
321Dic5—41no—
322Dic5C3×#9912yes—
323C24—51no—
324#41—202no—
325C2×Q8—51no—
326C2×D4—21no—
327C2×D4—101no—
328C2×D4—101no—
329C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
330C22×C4—51no—
331C22×C4—51no—
332C22×C4—51no—
333C22×C4—51no—
334C22×C4—51no—
335C22×C4—101no—
336C22×C4—102no—
337C22×C4—202no—
338C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
339C2×C8—102no—
340C2×C8—202no—
341C4⋊C4—51no—
342C4⋊C4—101no—
343C4⋊C4—101no—
344C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—101no—
347C22⋊C4—101no—
348C22⋊C4—101no—
349C22⋊C4—104no—
350C22⋊C4—101no—
351C42—101no—
352C42—202no—
353C3×C5#150111yescharacteristic, complemented, Hall
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—52no—
360C22×C3—101no—
361C22×C3—101no—
362C22×C3—101no—
363C22×C3—101no—
364C22×C3—101no—
365C22×C3—204no—
366C22×C3#400812yes—
367C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—41no—
371C3×C4—52no—
372C3×C4—101no—
373C3×C4—204no—
374C3×C4—201no—
375C3×C4—201no—
376C2×C5—22no—
377C2×C5—81no—
378C2×C5C2×C3×#9911yescharacteristic
379C2×C5C3×#41611yescharacteristic
380C2×C5C2×C3×#10411yescharacteristic
381D5—22no—
382D5—41no—
383C23—21no—
384C23—51no—
385C23—51no—
386C23—102no—
387C23—102no—
388C23—201no—
389C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
390Q8—201no—
391D4—41no—
392D4—42no—
393D4—202no—
394D4—202no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—51no—
398C2×C4—51no—
399C2×C4—102no—
400C2×C4—102no—
401C2×C4—102no—
402C2×C4—104no—
403C2×C4—101no—
404C2×C4—101no—
405C2×C4—201no—
406C2×C4—201no—
407C2×C4—202no—
408C2×C4—202no—
409C2×C4—202no—
410C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
411C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
412C8—202no—
413C2×C3—22no—
414C2×C3—81no—
415C2×C3—102no—
416C2×C3—201no—
417C2×C3C2×#400811yescharacteristic
418C2×C3D5⋊#9911yescharacteristic
419C2×C3#6579411yescharacteristic
420C5C3×#150111yescharacteristic, complemented, Sylow
421C22—21no—
422C22—21no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—52no—
427C22—101no—
428C22—101no—
429C22—101no—
430C22—101no—
431C22—101no—
432C22—204no—
433C22C3×#400812yes—
434C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
435C4—21no—
436C4—21no—
437C4—41no—
438C4—52no—
439C4—101no—
440C4—204no—
441C4—201no—
442C4—201no—
443C3#1060446611yescharacteristic, direct product factor, Sylow
444C2—22no—
445C2—81no—
446C2—102no—
447C2—201no—
448C2C2×C3×#400811yescharacteristic
449C2C3×D5⋊#9911yescharacteristic
450C2C3×#6579411yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#121C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#65940C2
cosocleC2×C4×D5C23×C3
hypercenterC3×#121AGL(1,𝔽5)
hypocenterC5C3×#1501
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
2. C3×C4×C22⋊AGL(1,𝔽5)C2semidirect
3. C2×C3×#4003C2semidirect
4. C3×#64920C2semidirect
5. C3×#64869C2semidirect
6. #10604466C3direct (N × H)
7. C2×C3×#711C4semidirect
12. C3×C5×#121C4semidirect
16. C3×#4070C4semidirect
18. C3×#4038C4semidirect
26. C4×C22⋊AGL(1,𝔽5)C2×C3semidirect
27. C2×#4003C2×C3semidirect
28. #64920C2×C3semidirect
29. #64869C2×C3semidirect
65. C2×#711C3×C4semidirect
70. C5×#121C3×C4semidirect
74. #4070C3×C4semidirect
76. #4038C3×C4semidirect
120. C3×#121AGL(1,𝔽5)semidirect
254. #121C3×AGL(1,𝔽5)semidirect
353. C3×C5#1501semidirect
420. C5C3×#1501semidirect
443. C3#10604466direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#659402
C3×C4×C22⋊AGL(1,𝔽5)2
C2×C3×#40032
C3×#649202
C3×#648692
#106044663
C3×#15015