Finite Group Databasebeta

← C3×#10604458 #14313147

Subgroups — C3×#10604458 #14313147

Order 1920.

528 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65920C211yesmaximal, characteristic
2C3×#65777C211yesmaximal, characteristic, complemented
3C3×#65771C211yesmaximal, characteristic, complemented
4C3×#64934C212yesmaximal, complemented
5C3×#64865C212yesmaximal, complemented
6#10604458C311yesmaximal, characteristic, direct product factor, Hall
7C23×C3×Dic5C411yescharacteristic, complemented
8C2×C3×#711C2211yescharacteristic
9C2×C3×#704—21no—
10C3×#4130—21no—
11C3×C5×#120C411yescharacteristic, complemented
12C3×#4099C2211yescharacteristic
13C3×#4085—21no—
14C3×#4081C411yescharacteristic, complemented
15C3×D4×Dic5—21no—
16C3×#4032C2211yescharacteristic
17C3×#4027—21no—
18C3×#4024—21no—
19C3×#4023C411yescharacteristic, complemented
20C3×#4013C2212yes—
21C3×#4011—21no—
22C3×#4011C2212yes—
23C3×#4005—21no—
24C3×#2120—51nomaximal, Hall
25#65920C2×C311yescharacteristic
26#65777C2×C311yescharacteristic, complemented
27#65771C2×C311yescharacteristic, complemented
28#64934C2×C312yescomplemented
29#64865C2×C312yescomplemented
30C24×C3×C5C2×C411yescharacteristic
31C2×C3×C5×D4—21no—
32C2×C3×C5×D4C2×C411yescharacteristic
33C2×C3×C22⋊D5—21no—
34C2×C3×C22⋊D5D411yescharacteristic
35C2×C3×C22⋊D5C2311yescharacteristic
36C22×C3×Dic5—21no—
37C22×C3×Dic5—21no—
38C22×C3×Dic5—41no—
39C22×C3×Dic5—41no—
40C22×C3×Dic5C2×C411yescharacteristic
41C22×C3×Dic5C2×C411yescharacteristic
42C22×C3×Dic5D411yescharacteristic
43C3×#711—41no—
44C3×#711—41no—
45C2×C3×C4×D5D411yescharacteristic
46C2×C3×Dic10D411yescharacteristic
47C3×C22⋊AGL(1,𝔽5)—22no—
48C3×#704—22no—
49C3×#704—41no—
50C2×C3×#170—41no—
51C3×C4⋊AGL(1,𝔽5)—41no—
52C3×C4×AGL(1,𝔽5)—41no—
53C3×C5×C22⋊C4—21no—
54C3×C5×C22⋊C4C2×C411yescharacteristic
55C3×C22⋊Dic5—21no—
56C3×C22⋊Dic5—21no—
57C3×C22⋊Dic5C2×C411yescharacteristic
58C3×#685—21no—
59C3×C4⋊Dic5—21no—
60C3×#683—21no—
61C3×#683—21no—
62C3×C4×Dic5—21no—
63C3×#525—51no—
64C3×#413—51no—
65C3×#410—51no—
66C3×#352—52no—
67C3×AΣL(1,𝔽16)—52no—
68C23×Dic5C3×C411yescharacteristic, complemented
69C2×#711C22×C311yescharacteristic
70C2×#704—21no—
71#4130—21no—
72C5×#120C3×C411yescharacteristic, complemented
73#4099C22×C311yescharacteristic
74#4085—21no—
75#4081C3×C411yescharacteristic, complemented
76D4×Dic5—21no—
77#4032C22×C311yescharacteristic
78#4027—21no—
79#4024—21no—
80#4023C3×C411yescharacteristic, complemented
81#4013C22×C312yes—
82#4011—21no—
83#4011C22×C312yes—
84#4005—21no—
85#2120—51noSylow
86C23×C3×C5—21no—
87C23×C3×C5—41no—
88C23×C3×C5—42no—
89C23×C3×C5C22⋊C411yescharacteristic
90C23×C3×C5C22×C411yescharacteristic
91C22×C3×D5C2×D411yescharacteristic
92C2×C3×AGL(1,𝔽5)—42no—
93C3×C5×D4—41no—
94C3×C5×D4—42no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5C22⋊C411yescharacteristic
97C3×C22⋊D5—41no—
98C3×C22⋊D5—41no—
99C3×C22⋊D5—42no—
100C2×C3×Dic5—21no—
101C2×C3×Dic5—21no—
102C2×C3×Dic5—21no—
103C2×C3×Dic5—21no—
104C2×C3×Dic5—21no—
105C2×C3×Dic5—21no—
106C2×C3×Dic5—42no—
107C2×C3×Dic5—41no—
108C2×C3×Dic5—41no—
109C2×C3×Dic5—41no—
110C2×C3×Dic5C22⋊C411yescharacteristic
111C2×C3×Dic5C22⋊C411yescharacteristic
112C2×C3×Dic5C2×D411yescharacteristic
113C3×C4×D5—21no—
114C3×C4×D5—21no—
115C3×Dic10—41no—
116C3×#170—42no—
117C2×C3×#41—51no—
118C23×C3×C4—51no—
119C2×C3×M16—101no—
120C3×#123—51no—
121C3×#123—51no—
122C3×C22⋊Q8—101no—
123C3×#121—101no—
124C3×#120—51no—
125C3×#120AGL(1,𝔽5)11yescharacteristic, complemented
126C3×C4×D4—101no—
127C3×C4×D4—101no—
128C3×#117—51no—
129C3×#117—101no—
130C3×#101—101no—
131C3×#100—52no—
132C3×#99—52no—
133C3×#99—101no—
134C24×C5C2×C3×C411yescharacteristic
135C2×C5×D4—21no—
136C2×C5×D4C2×C3×C411yescharacteristic
137C2×C22⋊D5—21no—
138C2×C22⋊D5C3×D411yescharacteristic
139C2×C22⋊D5C23×C311yescharacteristic
140C22×Dic5—21no—
141C22×Dic5—21no—
142C22×Dic5—41no—
143C22×Dic5—41no—
144C22×Dic5C2×C3×C411yescharacteristic
145C22×Dic5C2×C3×C411yescharacteristic
146C22×Dic5C3×D411yescharacteristic
147#711—41no—
148#711—41no—
149C2×C4×D5C3×D411yescharacteristic
150C2×Dic10C3×D411yescharacteristic
151C22⋊AGL(1,𝔽5)—22no—
152#704—22no—
153#704—41no—
154C2×#170—41no—
155C4⋊AGL(1,𝔽5)—41no—
156C4×AGL(1,𝔽5)—41no—
157C5×C22⋊C4—21no—
158C5×C22⋊C4C2×C3×C411yescharacteristic
159C22⋊Dic5—21no—
160C22⋊Dic5—21no—
161C22⋊Dic5C2×C3×C411yescharacteristic
162#685—21no—
163C4⋊Dic5—21no—
164#683—21no—
165#683—21no—
166C4×Dic5—21no—
167#525—51no—
168#413—51no—
169#410—51no—
170#352—52no—
171AΣL(1,𝔽16)—52no—
172C22×C3×C5—22no—
173C22×C3×C5—21no—
174C22×C3×C5—21no—
175C22×C3×C5—41no—
176C22×C3×C5—41no—
177C22×C3×C5—41no—
178C22×C3×C5—42no—
179C22×C3×C5—42no—
180C22×C3×C5—41no—
181C22×C3×C5C2×C22⋊C411yescharacteristic
182C2×C3×D5—21no—
183C2×C3×D5—41no—
184C3×AGL(1,𝔽5)—82no—
185C3×C4×C5—21no—
186C3×C4×C5—41no—
187C3×Dic5—21no—
188C3×Dic5—21no—
189C3×Dic5—41no—
190C3×Dic5—41no—
191C3×Dic5#9911yescharacteristic
192C3×Dic5#9911yescharacteristic
193C24×C3C2×AGL(1,𝔽5)11yescharacteristic
194C3×#41—201no—
195C3×#41—201no—
196C2×C3×Q8—51no—
197C2×C3×D4—21no—
198C2×C3×D4—51no—
199C2×C3×D4—51no—
200C2×C3×D4—101no—
201C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
202C22×C3×C4—51no—
203C22×C3×C4—51no—
204C22×C3×C4—51no—
205C22×C3×C4—51no—
206C22×C3×C4—101no—
207C22×C3×C4—101no—
208C22×C3×C4—201no—
209C22×C3×C4—201no—
210C3×M16—102no—
211C3×M16—201no—
212C2×C3×C8—201no—
213C3×C4⋊C4—101no—
214C3×C4⋊C4—101no—
215C3×C4⋊C4—101no—
216C3×C4⋊C4—201no—
217C3×C22⋊C4—21no—
218C3×C22⋊C4—51no—
219C3×C22⋊C4—101no—
220C3×C22⋊C4—101no—
221C3×C22⋊C4—101no—
222C3×C22⋊C4—102no—
223C3×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
224C3×C42—101no—
225C3×C42—201no—
226C23×C5—21no—
227C23×C5—41no—
228C23×C5—42no—
229C23×C5C3×C22⋊C411yescharacteristic
230C23×C5C22×C3×C411yescharacteristic
231C22×D5C2×C3×D411yescharacteristic
232C2×AGL(1,𝔽5)—42no—
233C5×D4—41no—
234C5×D4—42no—
235C2×C4×C5—21no—
236C2×C4×C5C3×C22⋊C411yescharacteristic
237C22⋊D5—41no—
238C22⋊D5—41no—
239C22⋊D5—42no—
240C2×Dic5—21no—
241C2×Dic5—21no—
242C2×Dic5—21no—
243C2×Dic5—21no—
244C2×Dic5—21no—
245C2×Dic5—21no—
246C2×Dic5—42no—
247C2×Dic5—41no—
248C2×Dic5—41no—
249C2×Dic5—41no—
250C2×Dic5C3×C22⋊C411yescharacteristic
251C2×Dic5C3×C22⋊C411yescharacteristic
252C2×Dic5C2×C3×D411yescharacteristic
253C4×D5—21no—
254C4×D5—21no—
255Dic10—41no—
256#170—42no—
257C2×#41—51no—
258C23×C4—51no—
259C2×M16—101no—
260#123—51no—
261#123—51no—
262C22⋊Q8—101no—
263#121—101no—
264#120—51no—
265#120C3×AGL(1,𝔽5)11yescharacteristic, complemented
266C4×D4—101no—
267C4×D4—101no—
268#117—51no—
269#117—101no—
270#101—101no—
271#100—52no—
272#99—52no—
273#99—101no—
274C2×C3×C5—21no—
275C2×C3×C5—41no—
276C2×C3×C5—42no—
277C2×C3×C5—41no—
278C2×C3×C5C2×#9911yescharacteristic
279C3×D5—41no—
280C23×C3—21no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—51no—
284C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
285C23×C3C22×AGL(1,𝔽5)11yescharacteristic
286C3×Q8—201no—
287C3×D4—41no—
288C3×D4—42no—
289C3×D4—201no—
290C3×D4—201no—
291C3×D4—202no—
292C2×C3×C4—21no—
293C2×C3×C4—51no—
294C2×C3×C4—51no—
295C2×C3×C4—51no—
296C2×C3×C4—101no—
297C2×C3×C4—101no—
298C2×C3×C4—101no—
299C2×C3×C4—101no—
300C2×C3×C4—101no—
301C2×C3×C4—101no—
302C2×C3×C4—101no—
303C2×C3×C4—101no—
304C2×C3×C4—201no—
305C2×C3×C4—201no—
306C2×C3×C4—202no—
307C2×C3×C4—201no—
308C2×C3×C4—202no—
309C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
310C3×C8—202no—
311C22×C5—22no—
312C22×C5—21no—
313C22×C5—21no—
314C22×C5—41no—
315C22×C5—41no—
316C22×C5—41no—
317C22×C5—42no—
318C22×C5—42no—
319C22×C5—41no—
320C22×C5C2×C3×C22⋊C411yescharacteristic
321C2×D5—21no—
322C2×D5—41no—
323AGL(1,𝔽5)—82no—
324C4×C5—21no—
325C4×C5—41no—
326Dic5—21no—
327Dic5—21no—
328Dic5—41no—
329Dic5—41no—
330Dic5C3×#9911yescharacteristic
331Dic5C3×#9911yescharacteristic
332C24C2×C3×AGL(1,𝔽5)11yescharacteristic
333#41—201no—
334#41—201no—
335C2×Q8—51no—
336C2×D4—21no—
337C2×D4—51no—
338C2×D4—51no—
339C2×D4—101no—
340C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
341C22×C4—51no—
342C22×C4—51no—
343C22×C4—51no—
344C22×C4—51no—
345C22×C4—101no—
346C22×C4—101no—
347C22×C4—201no—
348C22×C4—201no—
349M16—102no—
350M16—201no—
351C2×C8—201no—
352C4⋊C4—101no—
353C4⋊C4—101no—
354C4⋊C4—101no—
355C4⋊C4—201no—
356C22⋊C4—21no—
357C22⋊C4—51no—
358C22⋊C4—101no—
359C22⋊C4—101no—
360C22⋊C4—101no—
361C22⋊C4—102no—
362C22⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
363C42—101no—
364C42—201no—
365C3×C5#212011yescharacteristic, complemented, Hall
366C22×C3—22no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—41no—
375C22×C3—101no—
376C22×C3—201no—
377C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
378C3×C4—21no—
379C3×C4—41no—
380C3×C4—51no—
381C3×C4—51no—
382C3×C4—101no—
383C3×C4—101no—
384C3×C4—201no—
385C3×C4—201no—
386C3×C4—402no—
387C2×C5—21no—
388C2×C5—42no—
389C2×C5—41no—
390C2×C5—41no—
391C2×C5C2×C3×#9911yescharacteristic
392D5—41no—
393C23—21no—
394C23—41no—
395C23—42no—
396C23—51no—
397C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
398C23C22×C3×AGL(1,𝔽5)11yescharacteristic
399Q8—201no—
400D4—41no—
401D4—42no—
402D4—201no—
403D4—201no—
404D4—202no—
405C2×C4—21no—
406C2×C4—51no—
407C2×C4—51no—
408C2×C4—51no—
409C2×C4—101no—
410C2×C4—101no—
411C2×C4—101no—
412C2×C4—101no—
413C2×C4—101no—
414C2×C4—101no—
415C2×C4—101no—
416C2×C4—101no—
417C2×C4—201no—
418C2×C4—201no—
419C2×C4—202no—
420C2×C4—201no—
421C2×C4—202no—
422C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
423C8—202no—
424C2×C3—21no—
425C2×C3—42no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—201no—
429C2×C3D5⋊#9911yescharacteristic
430C5C3×#212011yescharacteristic, complemented, Sylow
431C22—22no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—42no—
438C22—42no—
439C22—41no—
440C22—101no—
441C22—201no—
442C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
443C4—21no—
444C4—41no—
445C4—51no—
446C4—51no—
447C4—101no—
448C4—101no—
449C4—201no—
450C4—201no—
451C4—402no—
452C3#1060445811yescharacteristic, direct product factor, Sylow
453C2—21no—
454C2—42no—
455C2—41no—
456C2—41no—
457C2—201no—
458C2C3×D5⋊#9911yescharacteristic
subgroupG/H
centerC2×C3D5⋊#99
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC3×C5×#120C4
socleC2×C3×C5C2×#99
prime radicalC3×#65920C2
cosocleC2×C22⋊D5C23×C3
hypercenterC3×#120AGL(1,𝔽5)
hypocenterC5C3×#2120
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#120

Complements

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

NH ≅ G/Ntype
2. C3×#65777C2semidirect
3. C3×#65771C2semidirect
4. C3×#64934C2semidirect
5. C3×#64865C2semidirect
6. #10604458C3direct (N × H)
7. C23×C3×Dic5C4semidirect
11. C3×C5×#120C4semidirect
14. C3×#4081C4semidirect
19. C3×#4023C4semidirect
26. #65777C2×C3semidirect
27. #65771C2×C3semidirect
28. #64934C2×C3semidirect
29. #64865C2×C3semidirect
68. C23×Dic5C3×C4semidirect
72. C5×#120C3×C4semidirect
75. #4081C3×C4semidirect
80. #4023C3×C4semidirect
125. C3×#120AGL(1,𝔽5)semidirect
265. #120C3×AGL(1,𝔽5)semidirect
365. C3×C5#2120semidirect
430. C5C3×#2120semidirect
452. C3#10604458direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#659202
C3×#657772
C3×#657712
C3×#649342
C3×#648652
#106044583
C3×#21205