Finite Group Databasebeta

← C3×#10604532 #14313313

Subgroups — C3×#10604532 #14313313

Order 1920.

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

normal subgroups only

This group has 468 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×#65982C211yesmaximal, characteristic
2C3×#65777C211yesmaximal, characteristic
3C3×#65771C211yesmaximal, characteristic
4C3×#64916C212yesmaximal, complemented
5C3×#64865C212yesmaximal, complemented
6#10604532C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#711C2211yescharacteristic
8C22×C3×Dic10C411yescharacteristic, complemented
9C2×C3×#704—21no—
10C3×#4130—21no—
11C3×C5×#123C411yescharacteristic, complemented
12C3×#4081C411yescharacteristic, complemented
13C3×#4075C2211yescharacteristic
14C3×#4042—21no—
15C3×#4034—21no—
16C3×#4032C2211yescharacteristic
17C3×#4031—21no—
18C3×#4024—21no—
19C3×#4023C411yescharacteristic, complemented
20C3×#4013C2212yes—
21C3×#4011—21no—
22C3×#4011C2212yes—
23C3×#4005—21no—
24C3×#2123—51nomaximal, Hall
25#65982C2×C311yescharacteristic
26#65777C2×C311yescharacteristic
27#65771C2×C311yescharacteristic
28#64916C2×C312yescomplemented
29#64865C2×C312yescomplemented
30C2×C3×C5×D4C2×C411yescharacteristic
31C22×C3×C4×C5C2×C411yescharacteristic
32C2×C3×C22⋊D5D411yescharacteristic
33C2×C3×C22⋊D5C2311yescharacteristic
34C22×C3×Dic5C2×C411yescharacteristic
35C22×C3×Dic5C2×C411yescharacteristic
36C22×C3×Dic5D411yescharacteristic
37C3×#711—41no—
38C3×#711—41no—
39C2×C3×C4×D5D411yescharacteristic
40C2×C3×Dic10—21no—
41C2×C3×Dic10—21no—
42C2×C3×Dic10—41no—
43C2×C3×Dic10—41no—
44C2×C3×Dic10D411yescharacteristic
45C3×C22⋊AGL(1,𝔽5)—22no—
46C3×#704—22no—
47C3×#704—41no—
48C2×C3×#170—41no—
49C3×C4⋊AGL(1,𝔽5)—41no—
50C3×C4×AGL(1,𝔽5)—41no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C22⋊C4—21no—
53C3×C5×C22⋊C4C2×C411yescharacteristic
54C3×C22⋊Dic5—21no—
55C3×C22⋊Dic5C2×C411yescharacteristic
56C3×#685—21no—
57C3×#685—21no—
58C3×C4⋊Dic5—21no—
59C3×#683—21no—
60C3×#683—21no—
61C3×#683—21no—
62C3×C4×Dic5—21no—
63C3×#536—51no—
64C3×#413—51no—
65C3×#410—51no—
66C3×#352—52no—
67C3×#352—52no—
68C2×#711C22×C311yescharacteristic
69C22×Dic10C3×C411yescharacteristic, complemented
70C2×#704—21no—
71#4130—21no—
72C5×#123C3×C411yescharacteristic, complemented
73#4081C3×C411yescharacteristic, complemented
74#4075C22×C311yescharacteristic
75#4042—21no—
76#4034—21no—
77#4032C22×C311yescharacteristic
78#4031—21no—
79#4024—21no—
80#4023C3×C411yescharacteristic, complemented
81#4013C22×C312yes—
82#4011—21no—
83#4011C22×C312yes—
84#4005—21no—
85#2123—51noSylow
86C23×C3×C5C22⋊C411yescharacteristic
87C23×C3×C5C22×C411yescharacteristic
88C22×C3×D5C2×D411yescharacteristic
89C2×C3×AGL(1,𝔽5)—42no—
90C3×C5×D4—41no—
91C2×C3×C4×C5—21no—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—41no—
94C2×C3×C4×C5C22⋊C411yescharacteristic
95C3×C22⋊D5—41no—
96C3×C22⋊D5—41no—
97C2×C3×Dic5—21no—
98C2×C3×Dic5—21no—
99C2×C3×Dic5—21no—
100C2×C3×Dic5—21no—
101C2×C3×Dic5—21no—
102C2×C3×Dic5—21no—
103C2×C3×Dic5—41no—
104C2×C3×Dic5C22⋊C411yescharacteristic
105C2×C3×Dic5C22⋊C411yescharacteristic
106C2×C3×Dic5C2×D411yescharacteristic
107C3×C4×D5—21no—
108C3×C4×D5—21no—
109C3×Dic10—41no—
110C3×Dic10—42no—
111C3×Dic10—41no—
112C3×Dic10—41no—
113C3×#170—42no—
114C2×C3×#41—51no—
115C22×C3×Q8—51no—
116C2×C3×M16—101no—
117C3×C4⋊Q8—101no—
118C3×#124—101no—
119C3×#123—51no—
120C3×#123—51no—
121C3×#123—51no—
122C3×#123AGL(1,𝔽5)11yescharacteristic, complemented
123C3×C22⋊Q8—101no—
124C3×C22⋊Q8—101no—
125C3×#117—51no—
126C3×#117—101no—
127C3×#101—101no—
128C3×#100—52no—
129C3×#99—52no—
130C3×#99—101no—
131C2×C5×D4C2×C3×C411yescharacteristic
132C22×C4×C5C2×C3×C411yescharacteristic
133C2×C22⋊D5C3×D411yescharacteristic
134C2×C22⋊D5C23×C311yescharacteristic
135C22×Dic5C2×C3×C411yescharacteristic
136C22×Dic5C2×C3×C411yescharacteristic
137C22×Dic5C3×D411yescharacteristic
138#711—41no—
139#711—41no—
140C2×C4×D5C3×D411yescharacteristic
141C2×Dic10—21no—
142C2×Dic10—21no—
143C2×Dic10—41no—
144C2×Dic10—41no—
145C2×Dic10C3×D411yescharacteristic
146C22⋊AGL(1,𝔽5)—22no—
147#704—22no—
148#704—41no—
149C2×#170—41no—
150C4⋊AGL(1,𝔽5)—41no—
151C4×AGL(1,𝔽5)—41no—
152C5×C4⋊C4—21no—
153C5×C22⋊C4—21no—
154C5×C22⋊C4C2×C3×C411yescharacteristic
155C22⋊Dic5—21no—
156C22⋊Dic5C2×C3×C411yescharacteristic
157#685—21no—
158#685—21no—
159C4⋊Dic5—21no—
160#683—21no—
161#683—21no—
162#683—21no—
163C4×Dic5—21no—
164#536—51no—
165#413—51no—
166#410—51no—
167#352—52no—
168#352—52no—
169C22×C3×C5—21no—
170C22×C3×C5—21no—
171C22×C3×C5—41no—
172C22×C3×C5—41no—
173C22×C3×C5C2×C22⋊C411yescharacteristic
174C2×C3×D5—21no—
175C2×C3×D5—41no—
176C3×AGL(1,𝔽5)—82no—
177C3×C4×C5—21no—
178C3×C4×C5—41no—
179C3×C4×C5—41no—
180C3×Dic5—21no—
181C3×Dic5—21no—
182C3×Dic5—41no—
183C3×Dic5—41no—
184C3×Dic5#9911yescharacteristic
185C3×Dic5#9911yescharacteristic
186C3×#41—201no—
187C3×#41—201no—
188C2×C3×Q8—51no—
189C2×C3×Q8—101no—
190C2×C3×Q8—101no—
191C2×C3×Q8—201no—
192C2×C3×Q8—201no—
193C2×C3×D4—51no—
194C2×C3×D4—51no—
195C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
196C22×C3×C4—51no—
197C22×C3×C4—51no—
198C22×C3×C4—51no—
199C22×C3×C4—51no—
200C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
201C3×M16—102no—
202C3×M16—201no—
203C2×C3×C8—201no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—101no—
206C3×C4⋊C4—101no—
207C3×C4⋊C4—101no—
208C3×C4⋊C4—101no—
209C3×C4⋊C4—201no—
210C3×C22⋊C4—21no—
211C3×C22⋊C4—51no—
212C3×C22⋊C4—101no—
213C3×C22⋊C4—101no—
214C3×C22⋊C4—101no—
215C3×C22⋊C4—102no—
216C3×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
217C3×C42—101no—
218C3×C42—201no—
219C23×C5C3×C22⋊C411yescharacteristic
220C23×C5C22×C3×C411yescharacteristic
221C22×D5C2×C3×D411yescharacteristic
222C2×AGL(1,𝔽5)—42no—
223C5×D4—41no—
224C2×C4×C5—21no—
225C2×C4×C5—21no—
226C2×C4×C5—41no—
227C2×C4×C5C3×C22⋊C411yescharacteristic
228C22⋊D5—41no—
229C22⋊D5—41no—
230C2×Dic5—21no—
231C2×Dic5—21no—
232C2×Dic5—21no—
233C2×Dic5—21no—
234C2×Dic5—21no—
235C2×Dic5—21no—
236C2×Dic5—41no—
237C2×Dic5C3×C22⋊C411yescharacteristic
238C2×Dic5C3×C22⋊C411yescharacteristic
239C2×Dic5C2×C3×D411yescharacteristic
240C4×D5—21no—
241C4×D5—21no—
242Dic10—41no—
243Dic10—42no—
244Dic10—41no—
245Dic10—41no—
246#170—42no—
247C2×#41—51no—
248C22×Q8—51no—
249C2×M16—101no—
250C4⋊Q8—101no—
251#124—101no—
252#123—51no—
253#123—51no—
254#123—51no—
255#123C3×AGL(1,𝔽5)11yescharacteristic, complemented
256C22⋊Q8—101no—
257C22⋊Q8—101no—
258#117—51no—
259#117—101no—
260#101—101no—
261#100—52no—
262#99—52no—
263#99—101no—
264C2×C3×C5—21no—
265C2×C3×C5—41no—
266C2×C3×C5—41no—
267C2×C3×C5C2×#9911yescharacteristic
268C3×D5—41no—
269C23×C3—51no—
270C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
271C23×C3C22×AGL(1,𝔽5)11yescharacteristic
272C3×Q8—201no—
273C3×Q8—201no—
274C3×Q8—201no—
275C3×Q8—202no—
276C3×D4—41no—
277C3×D4—201no—
278C3×D4—201no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—41no—
282C2×C3×C4—51no—
283C2×C3×C4—51no—
284C2×C3×C4—51no—
285C2×C3×C4—101no—
286C2×C3×C4—101no—
287C2×C3×C4—101no—
288C2×C3×C4—101no—
289C2×C3×C4—101no—
290C2×C3×C4—101no—
291C2×C3×C4—101no—
292C2×C3×C4—101no—
293C2×C3×C4—201no—
294C2×C3×C4—202no—
295C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
296C3×C8—202no—
297C22×C5—21no—
298C22×C5—21no—
299C22×C5—41no—
300C22×C5—41no—
301C22×C5C2×C3×C22⋊C411yescharacteristic
302C2×D5—21no—
303C2×D5—41no—
304AGL(1,𝔽5)—82no—
305C4×C5—21no—
306C4×C5—41no—
307C4×C5—41no—
308Dic5—21no—
309Dic5—21no—
310Dic5—41no—
311Dic5—41no—
312Dic5C3×#9911yescharacteristic
313Dic5C3×#9911yescharacteristic
314#41—201no—
315#41—201no—
316C2×Q8—51no—
317C2×Q8—101no—
318C2×Q8—101no—
319C2×Q8—201no—
320C2×Q8—201no—
321C2×D4—51no—
322C2×D4—51no—
323C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
324C22×C4—51no—
325C22×C4—51no—
326C22×C4—51no—
327C22×C4—51no—
328C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
329M16—102no—
330M16—201no—
331C2×C8—201no—
332C4⋊C4—21no—
333C4⋊C4—101no—
334C4⋊C4—101no—
335C4⋊C4—101no—
336C4⋊C4—101no—
337C4⋊C4—201no—
338C22⋊C4—21no—
339C22⋊C4—51no—
340C22⋊C4—101no—
341C22⋊C4—101no—
342C22⋊C4—101no—
343C22⋊C4—102no—
344C22⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
345C42—101no—
346C42—201no—
347C3×C5#212311yescharacteristic, complemented, Hall
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—101no—
353C22×C3—201no—
354C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
355C3×C4—21no—
356C3×C4—41no—
357C3×C4—41no—
358C3×C4—51no—
359C3×C4—51no—
360C3×C4—101no—
361C3×C4—101no—
362C3×C4—201no—
363C3×C4—201no—
364C3×C4—402no—
365C2×C5—21no—
366C2×C5—41no—
367C2×C5—41no—
368C2×C5C2×C3×#9911yescharacteristic
369D5—41no—
370C23—51no—
371C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
372C23C22×C3×AGL(1,𝔽5)11yescharacteristic
373Q8—201no—
374Q8—201no—
375Q8—201no—
376Q8—202no—
377D4—41no—
378D4—201no—
379D4—201no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—51no—
384C2×C4—51no—
385C2×C4—51no—
386C2×C4—101no—
387C2×C4—101no—
388C2×C4—101no—
389C2×C4—101no—
390C2×C4—101no—
391C2×C4—101no—
392C2×C4—101no—
393C2×C4—101no—
394C2×C4—201no—
395C2×C4—202no—
396C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
397C8—202no—
398C2×C3—21no—
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—201no—
402C2×C3D5⋊#9911yescharacteristic
403C5C3×#212311yescharacteristic, complemented, Sylow
404C22—21no—
405C22—21no—
406C22—41no—
407C22—41no—
408C22—101no—
409C22—201no—
410C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
411C4—21no—
412C4—41no—
413C4—41no—
414C4—51no—
415C4—51no—
416C4—101no—
417C4—101no—
418C4—201no—
419C4—201no—
420C4—402no—
421C3#1060453211yescharacteristic, direct product factor, Sylow
422C2—21no—
423C2—41no—
424C2—41no—
425C2—201no—
426C2C3×D5⋊#9911yescharacteristic
subgroupG/H
centerC2×C3D5⋊#99
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC3×C5×#123C4
socleC2×C3×C5C2×#99
prime radicalC2×C3×#711C22
cosocleC2×C22⋊D5C23×C3
hypercenterC3×#123AGL(1,𝔽5)
hypocenterC5C3×#2123
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#123

Complements

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

NH ≅ G/Ntype
4. C3×#64916C2semidirect
5. C3×#64865C2semidirect
6. #10604532C3direct (N × H)
8. C22×C3×Dic10C4semidirect
11. C3×C5×#123C4semidirect
12. C3×#4081C4semidirect
19. C3×#4023C4semidirect
28. #64916C2×C3semidirect
29. #64865C2×C3semidirect
69. C22×Dic10C3×C4semidirect
72. C5×#123C3×C4semidirect
73. #4081C3×C4semidirect
80. #4023C3×C4semidirect
122. C3×#123AGL(1,𝔽5)semidirect
255. #123C3×AGL(1,𝔽5)semidirect
347. C3×C5#2123semidirect
403. C5C3×#2123semidirect
421. C3#10604532direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#659822
C3×#657772
C3×#657712
C3×#649162
C3×#648652
#106045323
C3×#21235