Finite Group Databasebeta

← C3×#10603987 #14304784

Subgroups — C3×#10603987 #14304784

Order 1920.

460 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65787C211yesmaximal, characteristic
2C2×C3×#4010C211yesmaximal, characteristic
3C2×C3×C8×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4C3×#65708C211yesmaximal, characteristic, complemented
5C3×#65093C211yesmaximal, characteristic
6C3×#64896C211yesmaximal, characteristic, complemented
7C3×#64875C211yesmaximal, characteristic, complemented
8#10603987C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×#713C2211yescharacteristic
10C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
11C2×C3×#700C2211yescharacteristic
12C2×C3×#700C2211yescharacteristic
13C2×C3×D5⋊C8C2211yescharacteristic
14C2×C3×C8×D5C2211yescharacteristic
15C3×#4038C2211yescharacteristic
16C3×#4010—22no—
17C3×#4004—21no—
18C3×#4003—21no—
19C3×#4003—21no—
20C3×C4⋊#170—21no—
21C3×C8×AGL(1,𝔽5)—22no—
22C3×C5×#103C411yescharacteristic, complemented
23C3×#3967C411yescharacteristic, complemented
24C3×#3953C411yescharacteristic, complemented
25C3×#3941C411yescharacteristic, complemented
26C3×#1937—51nomaximal, Hall
27#65787C2×C311yescharacteristic
28C2×#4010C2×C311yescharacteristic
29C2×C8×AGL(1,𝔽5)C2×C311yescharacteristic, complemented
30#65708C2×C311yescharacteristic, complemented
31#65093C2×C311yescharacteristic
32#64896C2×C311yescharacteristic, complemented
33#64875C2×C311yescharacteristic, complemented
34C22×C3×AGL(1,𝔽5)—21no—
35C2×C3×C5×Q8C2×C411yescharacteristic
36C3×#713—21no—
37C3×#713—21no—
38C3×#713—21no—
39C3×#713—21no—
40C2×C3×D20—21no—
41C2×C3×D20C2×C411yescharacteristic
42C2×C3×C4×D5—21no—
43C2×C3×C4×D5C2311yescharacteristic
44C2×C3×#170—21no—
45C2×C3×#170—21no—
46C2×C3×#170D411yescharacteristic
47C2×C3×#170D411yescharacteristic
48C3×C4×AGL(1,𝔽5)—21no—
49C3×C4×AGL(1,𝔽5)D412yes—
50C3×#700—21no—
51C3×#700—21no—
52C3×#700—21no—
53C3×#700—21no—
54C3×D5⋊C8—21no—
55C3×D5⋊C8—21no—
56C2×C3×C5×C8C2×C411yescharacteristic
57C3×C5×C4⋊C4C2×C411yescharacteristic
58C3×#685—21no—
59C3×C4⋊Dic5C2×C411yescharacteristic
60C3×C4×Dic5—21no—
61C2×C3×#168C2×C411yescharacteristic
62C3×C8×D5—21no—
63C3×C8×D5—21no—
64C3×#424—51no—
65C2×C3×#104—51no—
66C3×#416—51no—
67C3×#408—51no—
68C2×C3×C4×C8—51no—
69C3×#341—51no—
70C3×#339—51no—
71C2×#713C22×C311yescharacteristic
72C2×C4×AGL(1,𝔽5)C22×C311yescharacteristic
73C2×#700C22×C311yescharacteristic
74C2×#700C22×C311yescharacteristic
75C2×D5⋊C8C22×C311yescharacteristic
76C2×C8×D5C22×C311yescharacteristic
77#4038C22×C311yescharacteristic
78#4010—22no—
79#4004—21no—
80#4003—21no—
81#4003—21no—
82C4⋊#170—21no—
83C8×AGL(1,𝔽5)—22no—
84C5×#103C3×C411yescharacteristic, complemented
85#3967C3×C411yescharacteristic, complemented
86#3953C3×C411yescharacteristic, complemented
87#3941C3×C411yescharacteristic, complemented
88#1937—51noSylow
89C22×C3×D5—21no—
90C22×C3×D5#4111yescharacteristic
91C2×C3×AGL(1,𝔽5)—22no—
92C2×C3×AGL(1,𝔽5)—22no—
93C2×C3×AGL(1,𝔽5)—22no—
94C3×C5×Q8—21no—
95C3×C5×Q8—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5C22×C411yescharacteristic
99C2×C3×Dic5—21no—
100C2×C3×Dic5C2×D411yescharacteristic
101C3×D20—21no—
102C3×D20—21no—
103C3×D20—41no—
104C3×D20—41no—
105C3×C4×D5—41no—
106C3×C4×D5—41no—
107C3×C4×D5C2×D411yescharacteristic
108C3×C4×D5#4111yescharacteristic
109C3×C4×D5#4111yescharacteristic
110C3×C4×D5#4111yescharacteristic
111C3×#170—21no—
112C3×#170—21no—
113C3×#170—41no—
114C3×#170—41no—
115C3×C5×C8—21no—
116C3×#168—21no—
117C2×C3×#41—51no—
118C2×C3×M16—51no—
119C2×C3×M16—51no—
120C22×C3×C8—51no—
121C22×C3×C8—51no—
122C3×#117—51no—
123C2×C3×C42—51no—
124C3×C4⋊C8—101no—
125C3×C4⋊C8—101no—
126C3×#104—102no—
127C3×#103—51no—
128C3×#103AGL(1,𝔽5)11yescharacteristic, complemented
129C3×#102—51no—
130C3×#102—51no—
131C3×C22⋊C8—101no—
132C3×C22⋊C8—101no—
133C3×C4×C8—102no—
134C22×AGL(1,𝔽5)—21no—
135C2×C5×Q8C2×C3×C411yescharacteristic
136#713—21no—
137#713—21no—
138#713—21no—
139#713—21no—
140C2×D20—21no—
141C2×D20C2×C3×C411yescharacteristic
142C2×C4×D5—21no—
143C2×C4×D5C23×C311yescharacteristic
144C2×#170—21no—
145C2×#170—21no—
146C2×#170C3×D411yescharacteristic
147C2×#170C3×D411yescharacteristic
148C4×AGL(1,𝔽5)—21no—
149C4×AGL(1,𝔽5)C3×D412yes—
150#700—21no—
151#700—21no—
152#700—21no—
153#700—21no—
154D5⋊C8—21no—
155D5⋊C8—21no—
156C2×C5×C8C2×C3×C411yescharacteristic
157C5×C4⋊C4C2×C3×C411yescharacteristic
158#685—21no—
159C4⋊Dic5C2×C3×C411yescharacteristic
160C4×Dic5—21no—
161C2×#168C2×C3×C411yescharacteristic
162C8×D5—21no—
163C8×D5—21no—
164#424—51no—
165C2×#104—51no—
166#416—51no—
167#408—51no—
168C2×C4×C8—51no—
169#341—51no—
170#339—51no—
171C22×C3×C5C4×D411yescharacteristic
172C2×C3×D5—21no—
173C2×C3×D5—21no—
174C2×C3×D5—41no—
175C2×C3×D5—41no—
176C2×C3×D5—41no—
177C2×C3×D5#12311yescharacteristic
178C2×C3×D5#12611yescharacteristic
179C3×AGL(1,𝔽5)—24no—
180C3×C4×C5—41no—
181C3×C4×C5—41no—
182C3×C4×C5#11711yescharacteristic
183C3×C4×C5C4×D411yescharacteristic
184C3×Dic5—41no—
185C3×Dic5#12111yescharacteristic
186C3×Dic5#12411yescharacteristic
187C3×#41—101no—
188C3×#41—101no—
189C3×#41—101no—
190C3×#41—101no—
191C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
192C2×C3×D4—51no—
193C2×C3×D4—101no—
194C22×C3×C4—51no—
195C22×C3×C4—101no—
196C22×C3×C4—101no—
197C3×M16—101no—
198C3×M16—101no—
199C3×M16—101no—
200C3×M16—101no—
201C2×C3×C8—51no—
202C2×C3×C8—51no—
203C2×C3×C8—51no—
204C2×C3×C8—101no—
205C2×C3×C8—101no—
206C2×C3×C8—101no—
207C2×C3×C8—101no—
208C2×C3×C8—101no—
209C2×C3×C8—101no—
210C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
211C3×C4⋊C4—51no—
212C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
213C3×C22⋊C4—101no—
214C3×C42—52no—
215C3×C42—101no—
216C3×C42—101no—
217C22×D5—21no—
218C22×D5C3×#4111yescharacteristic
219C2×AGL(1,𝔽5)—22no—
220C2×AGL(1,𝔽5)—22no—
221C2×AGL(1,𝔽5)—22no—
222C5×Q8—21no—
223C5×Q8—21no—
224C2×C4×C5—21no—
225C2×C4×C5—21no—
226C2×C4×C5C22×C3×C411yescharacteristic
227C2×Dic5—21no—
228C2×Dic5C2×C3×D411yescharacteristic
229D20—21no—
230D20—21no—
231D20—41no—
232D20—41no—
233C4×D5—41no—
234C4×D5—41no—
235C4×D5C2×C3×D411yescharacteristic
236C4×D5C3×#4111yescharacteristic
237C4×D5C3×#4111yescharacteristic
238C4×D5C3×#4111yescharacteristic
239#170—21no—
240#170—21no—
241#170—41no—
242#170—41no—
243C5×C8—21no—
244#168—21no—
245C2×#41—51no—
246C2×M16—51no—
247C2×M16—51no—
248C22×C8—51no—
249C22×C8—51no—
250#117—51no—
251C2×C42—51no—
252C4⋊C8—101no—
253C4⋊C8—101no—
254#104—102no—
255#103—51no—
256#103C3×AGL(1,𝔽5)11yescharacteristic, complemented
257#102—51no—
258#102—51no—
259C22⋊C8—101no—
260C22⋊C8—101no—
261C4×C8—102no—
262C2×C3×C5#38811yescharacteristic
263C2×C3×C5#44311yescharacteristic
264C2×C3×C5#44311yescharacteristic
265C3×D5—21no—
266C3×D5—21no—
267C3×D5—81no—
268C23×C3—51no—
269C23×C3—101no—
270C3×Q8—21no—
271C3×Q8—21no—
272C3×D4—101no—
273C3×D4—101no—
274C3×D4—201no—
275C3×D4—201no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—51no—
279C2×C3×C4—51no—
280C2×C3×C4—51no—
281C2×C3×C4—51no—
282C2×C3×C4—51no—
283C2×C3×C4—102no—
284C2×C3×C4—102no—
285C2×C3×C4—102no—
286C2×C3×C4—101no—
287C2×C3×C4—201no—
288C2×C3×C4—201no—
289C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
290C3×C8—21no—
291C3×C8—101no—
292C3×C8—101no—
293C3×C8—101no—
294C3×C8—201no—
295C3×C8—201no—
296C22×C5C3×C4×D411yescharacteristic
297C2×D5—21no—
298C2×D5—21no—
299C2×D5—41no—
300C2×D5—41no—
301C2×D5—41no—
302C2×D5C3×#12311yescharacteristic
303C2×D5C3×#12611yescharacteristic
304AGL(1,𝔽5)—24no—
305C4×C5—41no—
306C4×C5—41no—
307C4×C5C3×#11711yescharacteristic
308C4×C5C3×C4×D411yescharacteristic
309Dic5—41no—
310Dic5C3×#12111yescharacteristic
311Dic5C3×#12411yescharacteristic
312#41—101no—
313#41—101no—
314#41—101no—
315#41—101no—
316C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
317C2×D4—51no—
318C2×D4—101no—
319C22×C4—51no—
320C22×C4—101no—
321C22×C4—101no—
322M16—101no—
323M16—101no—
324M16—101no—
325M16—101no—
326C2×C8—51no—
327C2×C8—51no—
328C2×C8—51no—
329C2×C8—101no—
330C2×C8—101no—
331C2×C8—101no—
332C2×C8—101no—
333C2×C8—101no—
334C2×C8—101no—
335C2×C8C2×C3×AGL(1,𝔽5)11yescharacteristic
336C4⋊C4—51no—
337C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
338C22⋊C4—101no—
339C42—52no—
340C42—101no—
341C42—101no—
342C3×C5#193711yescharacteristic, complemented, Hall
343C22×C3—51no—
344C22×C3—51no—
345C22×C3—101no—
346C22×C3—101no—
347C22×C3—201no—
348C22×C3—201no—
349C22×C3—201no—
350C22×C3D4×AGL(1,𝔽5)11yescharacteristic
351C3×C4—41no—
352C3×C4—41no—
353C3×C4—51no—
354C3×C4—51no—
355C3×C4—104no—
356C3×C4—201no—
357C3×C4#413011yescharacteristic
358C3×C4D4×AGL(1,𝔽5)11yescharacteristic
359C2×C5C3×#38811yescharacteristic
360C2×C5C3×#44311yescharacteristic
361C2×C5C3×#44311yescharacteristic
362D5—21no—
363D5—21no—
364D5—81no—
365C23—51no—
366C23—101no—
367Q8—21no—
368Q8—21no—
369D4—101no—
370D4—101no—
371D4—201no—
372D4—201no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—51no—
376C2×C4—51no—
377C2×C4—51no—
378C2×C4—51no—
379C2×C4—51no—
380C2×C4—102no—
381C2×C4—102no—
382C2×C4—102no—
383C2×C4—101no—
384C2×C4—201no—
385C2×C4—201no—
386C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
387C8—21no—
388C8—101no—
389C8—101no—
390C8—101no—
391C8—201no—
392C8—201no—
393C2×C3—101no—
394C2×C3—101no—
395C2×C3—401no—
396C2×C3#6570011yescharacteristic
397C2×C3#6573611yescharacteristic
398C2×C3#6574011yescharacteristic
399C5C3×#193711yescharacteristic, complemented, Sylow
400C22—51no—
401C22—51no—
402C22—101no—
403C22—101no—
404C22—201no—
405C22—201no—
406C22—201no—
407C22C3×D4×AGL(1,𝔽5)11yescharacteristic
408C4—41no—
409C4—41no—
410C4—51no—
411C4—51no—
412C4—104no—
413C4—201no—
414C4C3×#413011yescharacteristic
415C4C3×D4×AGL(1,𝔽5)11yescharacteristic
416C3#1060398711yescharacteristic, direct product factor, Sylow
417C2—101no—
418C2—101no—
419C2—401no—
420C2C3×#6570011yescharacteristic
421C2C3×#6573611yescharacteristic
422C2C3×#6574011yescharacteristic
subgroupG/H
centerC22×C3D4×AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#103C4
socleC22×C3×C5C4×D4
prime radicalC2×C3×#713C22
cosocleC2×C4×D5C23×C3
hypercenterC3×#103AGL(1,𝔽5)
hypocenterC5C3×#1937
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
3. C2×C3×C8×AGL(1,𝔽5)C2semidirect
4. C3×#65708C2semidirect
6. C3×#64896C2semidirect
7. C3×#64875C2semidirect
8. #10603987C3direct (N × H)
22. C3×C5×#103C4semidirect
23. C3×#3967C4semidirect
24. C3×#3953C4semidirect
25. C3×#3941C4semidirect
29. C2×C8×AGL(1,𝔽5)C2×C3semidirect
30. #65708C2×C3semidirect
32. #64896C2×C3semidirect
33. #64875C2×C3semidirect
84. C5×#103C3×C4semidirect
85. #3967C3×C4semidirect
86. #3953C3×C4semidirect
87. #3941C3×C4semidirect
128. C3×#103AGL(1,𝔽5)semidirect
256. #103C3×AGL(1,𝔽5)semidirect
342. C3×C5#1937semidirect
399. C5C3×#1937semidirect
416. C3#10603987direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#657872
C2×C3×#40102
C2×C3×C8×AGL(1,𝔽5)2
C3×#657082
C3×#650932
C3×#648962
C3×#648752
#106039873
C3×#19375