Finite Group Databasebeta

← C3×#10604158 #14298590

Subgroups — C3×#10604158 #14298590

Order 1920.

524 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#709C211yesmaximal, characteristic
2C3×#65761C211yesmaximal, characteristic
3C3×#65745C211yesmaximal, characteristic
4C3×#64863C212yesmaximal, complemented
5C3×#64862C212yesmaximal, complemented
6#10604158C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#709C2211yescharacteristic
8C3×#4130C2212yes—
9C3×D5⋊M16C2212yes—
10C2×C3×C42×C5C411yescharacteristic, complemented
11C3×C4×C22⋊D5—21no—
12C3×C4×C22⋊D5—21no—
13C3×#4072C411yescharacteristic, complemented
14C3×C4×D20—21no—
15C3×#4018C411yescharacteristic, complemented
16C3×#4018C2211yescharacteristic
17C3×C42×D5C411yescharacteristic, complemented
18C3×C42×D5C2211yescharacteristic
19C3×C4×Dic10—21no—
20C3×#4013—21no—
21C3×#4011—21no—
22C3×#4005—21no—
23C3×#3999—21no—
24C3×#1827—51nomaximal, Hall
25C4×#709C2×C311yescharacteristic
26#65761C2×C311yescharacteristic
27#65745C2×C311yescharacteristic
28#64863C2×C312yescomplemented
29#64862C2×C312yescomplemented
30C22×C3×C4×C5—21no—
31C22×C3×C4×C5C2×C411yescharacteristic
32C2×C3×C22⋊D5D411yescharacteristic
33C2×C3×C22⋊D5D411yescharacteristic
34C3×#709—22no—
35C3×#709—22no—
36C2×C3×D20D411yescharacteristic
37C2×C3×C4×D5—21no—
38C2×C3×C4×D5—21no—
39C2×C3×C4×D5C2×C411yescharacteristic
40C2×C3×C4×D5C2311yescharacteristic
41C2×C3×Dic10D411yescharacteristic
42C3×C22⋊AGL(1,𝔽5)—22no—
43C3×#704—22no—
44C3×C4⋊AGL(1,𝔽5)—22no—
45C3×C4×AGL(1,𝔽5)—22no—
46C3×#700—22no—
47C3×D5⋊C8—22no—
48C3×C42×C5—21no—
49C3×C42×C5C2×C411yescharacteristic
50C3×C42×C5C2×C411yescharacteristic
51C3×C22⋊Dic5—21no—
52C3×#685—21no—
53C3×#685—21no—
54C3×C4⋊Dic5—21no—
55C3×#683—21no—
56C3×#683—21no—
57C3×C4×Dic5—21no—
58C3×C4×Dic5—21no—
59C3×C4×Dic5C2×C411yescharacteristic
60C3×C4×Dic5C2×C411yescharacteristic
61C3×C4×#41—51no—
62C3×#413—51no—
63C3×#410—51no—
64C3×#339—52no—
65C3×H(3,ℤ/4)—52no—
66C2×#709C22×C311yescharacteristic
67#4130C22×C312yes—
68D5⋊M16C22×C312yes—
69C2×C42×C5C3×C411yescharacteristic, complemented
70C4×C22⋊D5—21no—
71C4×C22⋊D5—21no—
72#4072C3×C411yescharacteristic, complemented
73C4×D20—21no—
74#4018C3×C411yescharacteristic, complemented
75#4018C22×C311yescharacteristic
76C42×D5C3×C411yescharacteristic, complemented
77C42×D5C22×C311yescharacteristic
78C4×Dic10—21no—
79#4013—21no—
80#4011—21no—
81#4005—21no—
82#3999—21no—
83#1827—51noSylow
84C23×C3×C5C22⋊C411yescharacteristic
85C22×C3×D5C22⋊C411yescharacteristic
86C22×C3×D5C2×D411yescharacteristic
87C2×C3×AGL(1,𝔽5)—42no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5—21no—
92C2×C3×C4×C5—42no—
93C2×C3×C4×C5C22⋊C411yescharacteristic
94C2×C3×C4×C5C22×C411yescharacteristic
95C3×C22⋊D5—22no—
96C3×C22⋊D5—22no—
97C2×C3×Dic5—21no—
98C2×C3×Dic5—21no—
99C2×C3×Dic5C22⋊C411yescharacteristic
100C2×C3×Dic5C2×D411yescharacteristic
101C3×D20—22no—
102C3×C4×D5—21no—
103C3×C4×D5—21no—
104C3×C4×D5—42no—
105C3×C4×D5—42no—
106C3×C4×D5#4112yes—
107C3×C4×D5#4112yes—
108C3×Dic10—22no—
109C3×#170—42no—
110C2×C3×#41—51no—
111C2×C3×M16—52no—
112C3×C4×Q8—101no—
113C3×C4×D4—101no—
114C3×C4×D4—101no—
115C3×C4×D4—101no—
116C3×#117—51no—
117C3×#117—51no—
118C3×#117—51no—
119C3×#117—52no—
120C2×C3×C42—51no—
121C2×C3×C42—51no—
122C2×C3×C42AGL(1,𝔽5)11yescharacteristic, complemented
123C3×#101—101no—
124C3×#100—101no—
125C3×#99—101no—
126C3×#99—101no—
127C22×C4×C5—21no—
128C22×C4×C5C2×C3×C411yescharacteristic
129C2×C22⋊D5C3×D411yescharacteristic
130C2×C22⋊D5C3×D411yescharacteristic
131#709—22no—
132#709—22no—
133C2×D20C3×D411yescharacteristic
134C2×C4×D5—21no—
135C2×C4×D5—21no—
136C2×C4×D5C2×C3×C411yescharacteristic
137C2×C4×D5C23×C311yescharacteristic
138C2×Dic10C3×D411yescharacteristic
139C22⋊AGL(1,𝔽5)—22no—
140#704—22no—
141C4⋊AGL(1,𝔽5)—22no—
142C4×AGL(1,𝔽5)—22no—
143#700—22no—
144D5⋊C8—22no—
145C42×C5—21no—
146C42×C5C2×C3×C411yescharacteristic
147C42×C5C2×C3×C411yescharacteristic
148C22⋊Dic5—21no—
149#685—21no—
150#685—21no—
151C4⋊Dic5—21no—
152#683—21no—
153#683—21no—
154C4×Dic5—21no—
155C4×Dic5—21no—
156C4×Dic5C2×C3×C411yescharacteristic
157C4×Dic5C2×C3×C411yescharacteristic
158C4×#41—51no—
159#413—51no—
160#410—51no—
161#339—52no—
162H(3,ℤ/4)—52no—
163C22×C3×C5—21no—
164C22×C3×C5—41no—
165C22×C3×C5C2×C22⋊C411yescharacteristic
166C2×C3×D5—21no—
167C2×C3×D5—41no—
168C2×C3×D5—41no—
169C2×C3×D5#12312yes—
170C3×AGL(1,𝔽5)—82no—
171C3×C4×C5—22no—
172C3×C4×C5—21no—
173C3×C4×C5—41no—
174C3×C4×C5#11711yescharacteristic
175C3×C4×C5#11711yescharacteristic
176C3×Dic5—21no—
177C3×Dic5—41no—
178C3×Dic5—41no—
179C3×Dic5#12312yes—
180C3×#41—102no—
181C3×#41—102no—
182C2×C3×Q8—51no—
183C2×C3×D4—51no—
184C2×C3×D4—51no—
185C2×C3×D4—51no—
186C22×C3×C4—21no—
187C22×C3×C4—51no—
188C22×C3×C4—51no—
189C22×C3×C4—101no—
190C22×C3×C4—101no—
191C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
192C3×M16—102no—
193C3×M16—102no—
194C2×C3×C8—102no—
195C3×C4⋊C4—101no—
196C3×C4⋊C4—101no—
197C3×C4⋊C4—102no—
198C3×C4⋊C4—101no—
199C3×C22⋊C4—101no—
200C3×C22⋊C4—101no—
201C3×C22⋊C4—101no—
202C3×C22⋊C4—102no—
203C3×C42—21no—
204C3×C42—51no—
205C3×C42—51no—
206C3×C42—101no—
207C3×C42—101no—
208C3×C42—102no—
209C3×C42C2×AGL(1,𝔽5)11yescharacteristic
210C3×C42C2×AGL(1,𝔽5)11yescharacteristic
211C23×C5C3×C22⋊C411yescharacteristic
212C22×D5C3×C22⋊C411yescharacteristic
213C22×D5C2×C3×D411yescharacteristic
214C2×AGL(1,𝔽5)—42no—
215C2×C4×C5—21no—
216C2×C4×C5—21no—
217C2×C4×C5—21no—
218C2×C4×C5—21no—
219C2×C4×C5—42no—
220C2×C4×C5C3×C22⋊C411yescharacteristic
221C2×C4×C5C22×C3×C411yescharacteristic
222C22⋊D5—22no—
223C22⋊D5—22no—
224C2×Dic5—21no—
225C2×Dic5—21no—
226C2×Dic5C3×C22⋊C411yescharacteristic
227C2×Dic5C2×C3×D411yescharacteristic
228D20—22no—
229C4×D5—21no—
230C4×D5—21no—
231C4×D5—42no—
232C4×D5—42no—
233C4×D5C3×#4112yes—
234C4×D5C3×#4112yes—
235Dic10—22no—
236#170—42no—
237C2×#41—51no—
238C2×M16—52no—
239C4×Q8—101no—
240C4×D4—101no—
241C4×D4—101no—
242C4×D4—101no—
243#117—51no—
244#117—51no—
245#117—51no—
246#117—52no—
247C2×C42—51no—
248C2×C42—51no—
249C2×C42C3×AGL(1,𝔽5)11yescharacteristic, complemented
250#101—101no—
251#100—101no—
252#99—101no—
253#99—101no—
254C2×C3×C5—21no—
255C2×C3×C5—41no—
256C2×C3×C5#38111yescharacteristic
257C3×D5—22no—
258C3×D5—41no—
259C23×C3—51no—
260C23×C3—51no—
261C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
262C3×Q8—102no—
263C3×D4—102no—
264C3×D4—102no—
265C3×D4—102no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—42no—
271C2×C3×C4—51no—
272C2×C3×C4—52no—
273C2×C3×C4—52no—
274C2×C3×C4—51no—
275C2×C3×C4—101no—
276C2×C3×C4—101no—
277C2×C3×C4—101no—
278C2×C3×C4—101no—
279C2×C3×C4—202no—
280C2×C3×C4—202no—
281C2×C3×C4—202no—
282C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
283C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
284C3×C8—202no—
285C22×C5—21no—
286C22×C5—41no—
287C22×C5C2×C3×C22⋊C411yescharacteristic
288C2×D5—21no—
289C2×D5—41no—
290C2×D5—41no—
291C2×D5C3×#12312yes—
292AGL(1,𝔽5)—82no—
293C4×C5—22no—
294C4×C5—21no—
295C4×C5—41no—
296C4×C5C3×#11711yescharacteristic
297C4×C5C3×#11711yescharacteristic
298Dic5—21no—
299Dic5—41no—
300Dic5—41no—
301Dic5C3×#12312yes—
302#41—102no—
303#41—102no—
304C2×Q8—51no—
305C2×D4—51no—
306C2×D4—51no—
307C2×D4—51no—
308C22×C4—21no—
309C22×C4—51no—
310C22×C4—51no—
311C22×C4—101no—
312C22×C4—101no—
313C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
314M16—102no—
315M16—102no—
316C2×C8—102no—
317C4⋊C4—101no—
318C4⋊C4—101no—
319C4⋊C4—102no—
320C4⋊C4—101no—
321C22⋊C4—101no—
322C22⋊C4—101no—
323C22⋊C4—101no—
324C22⋊C4—102no—
325C42—21no—
326C42—51no—
327C42—51no—
328C42—101no—
329C42—101no—
330C42—102no—
331C42C2×C3×AGL(1,𝔽5)11yescharacteristic
332C42C2×C3×AGL(1,𝔽5)11yescharacteristic
333C3×C5#182711yescharacteristic, complemented, Hall
334C22×C3—21no—
335C22×C3—41no—
336C22×C3—52no—
337C22×C3—101no—
338C22×C3—201no—
339C22×C3—201no—
340C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
341C3×C4—22no—
342C3×C4—21no—
343C3×C4—41no—
344C3×C4—52no—
345C3×C4—101no—
346C3×C4—201no—
347C3×C4—201no—
348C3×C4—402no—
349C3×C4#413011yescharacteristic
350C3×C4#413011yescharacteristic
351C2×C5—21no—
352C2×C5—41no—
353C2×C5C3×#38111yescharacteristic
354D5—22no—
355D5—41no—
356C23—51no—
357C23—51no—
358C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
359Q8—102no—
360D4—102no—
361D4—102no—
362D4—102no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—51no—
369C2×C4—52no—
370C2×C4—52no—
371C2×C4—51no—
372C2×C4—101no—
373C2×C4—101no—
374C2×C4—101no—
375C2×C4—101no—
376C2×C4—202no—
377C2×C4—202no—
378C2×C4—202no—
379C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
380C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
381C8—202no—
382C2×C3—21no—
383C2×C3—41no—
384C2×C3—102no—
385C2×C3—201no—
386C2×C3#6576411yescharacteristic
387C5C3×#182711yescharacteristic, complemented, Sylow
388C22—21no—
389C22—41no—
390C22—52no—
391C22—101no—
392C22—201no—
393C22—201no—
394C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
395C4—22no—
396C4—21no—
397C4—41no—
398C4—52no—
399C4—101no—
400C4—201no—
401C4—201no—
402C4—402no—
403C4C3×#413011yescharacteristic
404C4C3×#413011yescharacteristic
405C3#1060415811yescharacteristic, direct product factor, Sylow
406C2—21no—
407C2—41no—
408C2—102no—
409C2—201no—
410C2C3×#6576411yescharacteristic
subgroupG/H
centerC3×C4#4130
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC2×C3×C42×C5C4
socleC2×C3×C5#381
prime radicalC2×C3×#709C22
cosocleC2×C4×D5C23×C3
hypercenterC2×C3×C42AGL(1,𝔽5)
hypocenterC5C3×#1827
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C3×C42 ◁ C2×C3×C42

Complements

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

NH ≅ G/Ntype
4. C3×#64863C2semidirect
5. C3×#64862C2semidirect
6. #10604158C3direct (N × H)
10. C2×C3×C42×C5C4semidirect
13. C3×#4072C4semidirect
15. C3×#4018C4semidirect
17. C3×C42×D5C4semidirect
28. #64863C2×C3semidirect
29. #64862C2×C3semidirect
69. C2×C42×C5C3×C4semidirect
72. #4072C3×C4semidirect
74. #4018C3×C4semidirect
76. C42×D5C3×C4semidirect
122. C2×C3×C42AGL(1,𝔽5)semidirect
249. C2×C42C3×AGL(1,𝔽5)semidirect
333. C3×C5#1827semidirect
387. C5C3×#1827semidirect
405. C3#10604158direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#7092
C3×#657612
C3×#657452
C3×#648632
C3×#648622
#106041583
C3×#18275