Finite Group Databasebeta

← C3×#10604403 #14281696

Subgroups — C3×#10604403 #14281696

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×D4×D5C211yesmaximal, characteristic
2C3×#65680C211yesmaximal, characteristic, complemented
3C3×#65678C211yesmaximal, characteristic, complemented
4C3×D4⋊#170C212yesmaximal, complemented
5C3×#64871C212yesmaximal, complemented
6#10604403C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×D4×D5C411yescharacteristic
8C22×C3×C4×D5—21no—
9C3×C4×C5×D4C411yescharacteristic
10C3×D4×Dic5C2211yescharacteristic
11C3×C4×C22⋊D5—21no—
12C3×#4039C2211yescharacteristic
13C3×D5×C4⋊C4C411yescharacteristic
14C3×#4027—21no—
15C3×D5×C22⋊C4—21no—
16C3×C4×D20C411yescharacteristic
17C3×C42×D5C2211yescharacteristic
18C3×#4004—21no—
19C3×#4003—21no—
20C3×#4003—21no—
21C3×C4⋊#170C2212yescomplemented
22C3×C4×#170—21no—
23C3×C4×#170C2212yescomplemented
24C3×#1478—51nomaximal, Hall
25C4×D4×D5C2×C311yescharacteristic
26#65680C2×C311yescharacteristic, complemented
27#65678C2×C311yescharacteristic, complemented
28D4⋊#170C2×C312yescomplemented
29#64871C2×C312yescomplemented
30C23×C3×D5—21no—
31C2×C3×C5×D4C2×C411yescharacteristic
32C22×C3×C4×C5—21no—
33C2×C3×C22⋊D5—21no—
34C22×C3×Dic5—21no—
35C3×D4×D5—22no—
36C3×D4×D5C814yescomplemented
37C2×C3×D20C2×C411yescharacteristic
38C2×C3×C4×D5—21no—
39C2×C3×C4×D5—21no—
40C2×C3×C4×D5—21no—
41C2×C3×C4×D5—44no—
42C2×C3×C4×D5C2×C411yescharacteristic
43C2×C3×C4×D5D411yescharacteristic
44C2×C3×C4×D5D411yescharacteristic
45C2×C3×#170—22no—
46C2×C3×#170—22no—
47C3×C5×C4⋊C4C2×C411yescharacteristic
48C3×C5×C22⋊C4—21no—
49C3×C42×C5C2×C411yescharacteristic
50C3×C22⋊Dic5—21no—
51C3×#685—21no—
52C3×#685—21no—
53C3×C4⋊Dic5C2×C411yescharacteristic
54C3×#683—21no—
55C3×C4×Dic5D411yescharacteristic
56C3×C4×Dic5D411yescharacteristic
57C3×C4×Dic5C2311yescharacteristic
58C2×C3×C4×D4—51no—
59C3×#431—51no—
60C3×#431—51no—
61C3×D4⋊C8—52no—
62C3×D4⋊C8—52no—
63C2×D4×D5C3×C411yescharacteristic
64C22×C4×D5—21no—
65C4×C5×D4C3×C411yescharacteristic
66D4×Dic5C22×C311yescharacteristic
67C4×C22⋊D5—21no—
68#4039C22×C311yescharacteristic
69D5×C4⋊C4C3×C411yescharacteristic
70#4027—21no—
71D5×C22⋊C4—21no—
72C4×D20C3×C411yescharacteristic
73C42×D5C22×C311yescharacteristic
74#4004—21no—
75#4003—21no—
76#4003—21no—
77C4⋊#170C22×C312yescomplemented
78C4×#170—21no—
79C4×#170C22×C312yescomplemented
80#1478—51noSylow
81C23×C3×C5—21no—
82C22×C3×D5—21no—
83C22×C3×D5—24no—
84C22×C3×D5—42no—
85C22×C3×D5—42no—
86C22×C3×D5C22⋊C411yescharacteristic
87C3×C5×D4—21no—
88C3×C5×D4C2×C812yes—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—42no—
91C2×C3×C4×C5C22⋊C411yescharacteristic
92C2×C3×C4×C5C22⋊C411yescharacteristic
93C2×C3×C4×C5C22×C411yescharacteristic
94C3×C22⋊D5—24no—
95C2×C3×Dic5—21no—
96C2×C3×Dic5—42no—
97C2×C3×Dic5C22⋊C411yescharacteristic
98C2×C3×Dic5C2×D411yescharacteristic
99C2×C3×Dic5C2×D411yescharacteristic
100C3×D20—21no—
101C3×D20C2×C812yes—
102C3×C4×D5—22no—
103C3×C4×D5—41no—
104C3×C4×D5—42no—
105C3×C4×D5—42no—
106C3×C4×D5—42no—
107C3×C4×D5C2×C812yes—
108C3×C4×D5M1612yes—
109C3×#170—42no—
110C3×#170—42no—
111C22×C3×D4—51no—
112C23×C3×C4—101no—
113C3×C4×D4—51no—
114C3×C4×D4—51no—
115C3×C4×D4—51no—
116C3×C4×D4—101no—
117C3×C4×D4—101no—
118C3×C4×D4AGL(1,𝔽5)11yescharacteristic
119C2×C3×C4⋊C4—51no—
120C2×C3×C22⋊C4—101no—
121C2×C3×C42—51no—
122C3×C4⋊C8—52no—
123C3×C4⋊C8—101no—
124C3×C22⋊C8—101no—
125C3×C22⋊C8—101no—
126C3×C4×C8—52no—
127C3×C4×C8—101no—
128C23×D5—21no—
129C2×C5×D4C2×C3×C411yescharacteristic
130C22×C4×C5—21no—
131C2×C22⋊D5—21no—
132C22×Dic5—21no—
133D4×D5—22no—
134D4×D5C3×C814yescomplemented
135C2×D20C2×C3×C411yescharacteristic
136C2×C4×D5—21no—
137C2×C4×D5—21no—
138C2×C4×D5—21no—
139C2×C4×D5—44no—
140C2×C4×D5C2×C3×C411yescharacteristic
141C2×C4×D5C3×D411yescharacteristic
142C2×C4×D5C3×D411yescharacteristic
143C2×#170—22no—
144C2×#170—22no—
145C5×C4⋊C4C2×C3×C411yescharacteristic
146C5×C22⋊C4—21no—
147C42×C5C2×C3×C411yescharacteristic
148C22⋊Dic5—21no—
149#685—21no—
150#685—21no—
151C4⋊Dic5C2×C3×C411yescharacteristic
152#683—21no—
153C4×Dic5C3×D411yescharacteristic
154C4×Dic5C3×D411yescharacteristic
155C4×Dic5C23×C311yescharacteristic
156C2×C4×D4—51no—
157#431—51no—
158#431—51no—
159D4⋊C8—52no—
160D4⋊C8—52no—
161C22×C3×C5—22no—
162C22×C3×C5—41no—
163C22×C3×C5—41no—
164C22×C3×C5C2×C22⋊C411yescharacteristic
165C2×C3×D5—21no—
166C2×C3×D5—21no—
167C2×C3×D5—22no—
168C2×C3×D5—48no—
169C2×C3×D5—41no—
170C2×C3×D5—41no—
171C2×C3×D5C22⋊C812yes—
172C3×C4×C5—21no—
173C3×C4×C5—21no—
174C3×C4×C5—41no—
175C3×C4×C5C22×C811yescharacteristic
176C3×C4×C5C2×M1611yescharacteristic
177C3×Dic5—21no—
178C3×Dic5—41no—
179C3×Dic5C22⋊C812yes—
180C3×Dic5AGL(1,ℤ/8)12yes—
181C24×C3—101no—
182C2×C3×D4—54no—
183C2×C3×D4—51no—
184C2×C3×D4—102no—
185C2×C3×D4—101no—
186C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
187C22×C3×C4—21no—
188C22×C3×C4—51no—
189C22×C3×C4—51no—
190C22×C3×C4—51no—
191C22×C3×C4—101no—
192C22×C3×C4—101no—
193C22×C3×C4—101no—
194C22×C3×C4—101no—
195C22×C3×C4—204no—
196C2×C3×C8—102no—
197C2×C3×C8—102no—
198C3×C4⋊C4—51no—
199C3×C4⋊C4—101no—
200C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
201C3×C22⋊C4—21no—
202C3×C22⋊C4—101no—
203C3×C22⋊C4—101no—
204C3×C22⋊C4—101no—
205C3×C42—51no—
206C3×C42—51no—
207C3×C42—51no—
208C3×C42C2×AGL(1,𝔽5)11yescharacteristic
209C23×C5—21no—
210C22×D5—21no—
211C22×D5—24no—
212C22×D5—42no—
213C22×D5—42no—
214C22×D5C3×C22⋊C411yescharacteristic
215C5×D4—21no—
216C5×D4C2×C3×C812yes—
217C2×C4×C5—21no—
218C2×C4×C5—42no—
219C2×C4×C5C3×C22⋊C411yescharacteristic
220C2×C4×C5C3×C22⋊C411yescharacteristic
221C2×C4×C5C22×C3×C411yescharacteristic
222C22⋊D5—24no—
223C2×Dic5—21no—
224C2×Dic5—42no—
225C2×Dic5C3×C22⋊C411yescharacteristic
226C2×Dic5C2×C3×D411yescharacteristic
227C2×Dic5C2×C3×D411yescharacteristic
228D20—21no—
229D20C2×C3×C812yes—
230C4×D5—22no—
231C4×D5—41no—
232C4×D5—42no—
233C4×D5—42no—
234C4×D5—42no—
235C4×D5C2×C3×C812yes—
236C4×D5C3×M1612yes—
237#170—42no—
238#170—42no—
239C22×D4—51no—
240C23×C4—101no—
241C4×D4—51no—
242C4×D4—51no—
243C4×D4—51no—
244C4×D4—101no—
245C4×D4—101no—
246C4×D4C3×AGL(1,𝔽5)11yescharacteristic
247C2×C4⋊C4—51no—
248C2×C22⋊C4—101no—
249C2×C42—51no—
250C4⋊C8—52no—
251C4⋊C8—101no—
252C22⋊C8—101no—
253C22⋊C8—101no—
254C4×C8—52no—
255C4×C8—101no—
256C2×C3×C5—42no—
257C2×C3×C5C2×C22⋊C811yescharacteristic
258C2×C3×C5#41811yescharacteristic
259C2×C3×C5#42111yescharacteristic
260C3×D5—22no—
261C3×D5—42no—
262C23×C3—21no—
263C23×C3—51no—
264C23×C3—104no—
265C23×C3—101no—
266C23×C3—202no—
267C23×C3—202no—
268C3×D4—21no—
269C3×D4—52no—
270C3×D4—104no—
271C3×D4—101no—
272C3×D4D5⋊C812yes—
273C2×C3×C4—21no—
274C2×C3×C4—42no—
275C2×C3×C4—51no—
276C2×C3×C4—51no—
277C2×C3×C4—52no—
278C2×C3×C4—52no—
279C2×C3×C4—51no—
280C2×C3×C4—102no—
281C2×C3×C4—101no—
282C2×C3×C4—201no—
283C2×C3×C4—202no—
284C2×C3×C4—202no—
285C2×C3×C4—202no—
286C2×C3×C4—202no—
287C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
288C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
289C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
290C3×C8—202no—
291C3×C8—202no—
292C22×C5—22no—
293C22×C5—41no—
294C22×C5—41no—
295C22×C5C2×C3×C22⋊C411yescharacteristic
296C2×D5—21no—
297C2×D5—21no—
298C2×D5—22no—
299C2×D5—48no—
300C2×D5—41no—
301C2×D5—41no—
302C2×D5C3×C22⋊C812yes—
303C4×C5—21no—
304C4×C5—21no—
305C4×C5—41no—
306C4×C5C22×C3×C811yescharacteristic
307C4×C5C2×C3×M1611yescharacteristic
308Dic5—21no—
309Dic5—41no—
310Dic5C3×C22⋊C812yes—
311Dic5C3×AGL(1,ℤ/8)12yes—
312C24—101no—
313C2×D4—54no—
314C2×D4—51no—
315C2×D4—102no—
316C2×D4—101no—
317C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
318C22×C4—21no—
319C22×C4—51no—
320C22×C4—51no—
321C22×C4—51no—
322C22×C4—101no—
323C22×C4—101no—
324C22×C4—101no—
325C22×C4—101no—
326C22×C4—204no—
327C2×C8—102no—
328C2×C8—102no—
329C4⋊C4—51no—
330C4⋊C4—101no—
331C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—101no—
334C22⋊C4—101no—
335C22⋊C4—101no—
336C42—51no—
337C42—51no—
338C42—51no—
339C42C2×C3×AGL(1,𝔽5)11yescharacteristic
340C3×C5#147811yescharacteristic, complemented, Hall
341C22×C3—22no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—52no—
345C22×C3—101no—
346C22×C3—101no—
347C22×C3—102no—
348C22×C3—208no—
349C22×C3—201no—
350C22×C3—201no—
351C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—41no—
355C3×C4—52no—
356C3×C4—52no—
357C3×C4—101no—
358C3×C4—201no—
359C3×C4C2×D5⋊C811yescharacteristic
360C3×C4D5⋊M1611yescharacteristic
361C2×C5—42no—
362C2×C5C2×C3×C22⋊C811yescharacteristic
363C2×C5C3×#41811yescharacteristic
364C2×C5C3×#42111yescharacteristic
365D5—22no—
366D5—42no—
367C23—21no—
368C23—51no—
369C23—104no—
370C23—101no—
371C23—202no—
372C23—202no—
373D4—21no—
374D4—52no—
375D4—104no—
376D4—101no—
377D4C3×D5⋊C812yes—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—51no—
381C2×C4—51no—
382C2×C4—52no—
383C2×C4—52no—
384C2×C4—51no—
385C2×C4—102no—
386C2×C4—101no—
387C2×C4—201no—
388C2×C4—202no—
389C2×C4—202no—
390C2×C4—202no—
391C2×C4—202no—
392C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
393C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
394C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
395C8—202no—
396C8—202no—
397C2×C3—42no—
398C2×C3—102no—
399C2×C3—202no—
400C2×C3D5⋊(C22⋊C8)11yescharacteristic
401C2×C3#6576711yescharacteristic
402C2×C3#6579511yescharacteristic
403C5C3×#147811yescharacteristic, complemented, Sylow
404C22—22no—
405C22—41no—
406C22—41no—
407C22—52no—
408C22—101no—
409C22—101no—
410C22—102no—
411C22—208no—
412C22—201no—
413C22—201no—
414C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
415C4—21no—
416C4—21no—
417C4—41no—
418C4—52no—
419C4—52no—
420C4—101no—
421C4—201no—
422C4C2×C3×D5⋊C811yescharacteristic
423C4C3×D5⋊M1611yescharacteristic
424C3#1060440311yescharacteristic, direct product factor, Sylow
425C2—42no—
426C2—102no—
427C2—202no—
428C2C3×D5⋊(C22⋊C8)11yescharacteristic
429C2C3×#6576711yescharacteristic
430C2C3×#6579511yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC4×C5C22×C3×C8
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C4×C5×D4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×D4×D5C4
cosocleC4×Dic5C23×C3
hypercenterC3×C4×D4AGL(1,𝔽5)
hypocenterC5C3×#1478
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
2. C3×#65680C2semidirect
3. C3×#65678C2semidirect
4. C3×D4⋊#170C2semidirect
5. C3×#64871C2semidirect
6. #10604403C3direct (N × H)
21. C3×C4⋊#170C22semidirect
23. C3×C4×#170C22semidirect
26. #65680C2×C3semidirect
27. #65678C2×C3semidirect
28. D4⋊#170C2×C3semidirect
29. #64871C2×C3semidirect
36. C3×D4×D5C8semidirect
77. C4⋊#170C22×C3semidirect
79. C4×#170C22×C3semidirect
134. D4×D5C3×C8semidirect
340. C3×C5#1478semidirect
403. C5C3×#1478semidirect
424. C3#10604403direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×D4×D52
C3×#656802
C3×#656782
C3×D4⋊#1702
C3×#648712
#106044033
C3×#14785