Finite Group Databasebeta

← C3×#10599030 #14282157

Subgroups — C3×#10599030 #14282157

Order 1920.

492 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#713C211yesmaximal, characteristic
2C3×#65124C211yesmaximal, characteristic, complemented
3C3×#64974C211yesmaximal, characteristic, complemented
4C3×C5×Q8⋊C8C211yesmaximal, characteristic, complemented
5C3×Q8⋊#168C211yesmaximal, characteristic, complemented
6C3×#64703C211yesmaximal, characteristic, complemented
7C3×#64679C211yesmaximal, characteristic, complemented
8#10599030C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×#713C411yescharacteristic
10C3×C4×C5×Q8C2211yescharacteristic
11C3×Q8×Dic5C411yescharacteristic
12C3×#4039—21no—
13C3×#4039C411yescharacteristic
14C3×#4038—21no—
15C3×#4038C411yescharacteristic
16C3×C4×D20—21no—
17C3×C4×D20C2211yescharacteristic
18C3×C42×D5—21no—
19C3×C42×D5C2211yescharacteristic
20C3×C5×C4⋊C8C2211yescharacteristic
21C3×C4×C5×C8C2211yescharacteristic
22C3×#3952—21no—
23C3×#3952—21no—
24C3×#3947—21no—
25C3×#3946—21no—
26C3×C4⋊#168C2211yescharacteristic
27C3×C4×#168C2211yescharacteristic
28C3×#1488—51nomaximal, Hall
29C4×#713C2×C311yescharacteristic
30#65124C2×C311yescharacteristic, complemented
31#64974C2×C311yescharacteristic, complemented
32C5×Q8⋊C8C2×C311yescharacteristic, complemented
33Q8⋊#168C2×C311yescharacteristic, complemented
34#64703C2×C311yescharacteristic, complemented
35#64679C2×C311yescharacteristic, complemented
36C2×C3×C5×Q8C2×C411yescharacteristic
37C3×#713—22no—
38C3×#713—22no—
39C2×C3×D20—21no—
40C2×C3×D20C2×C411yescharacteristic
41C2×C3×C4×D5—21no—
42C2×C3×C4×D5—21no—
43C2×C3×C4×D5—21no—
44C2×C3×C4×D5C2×C411yescharacteristic
45C2×C3×C4×D5D411yescharacteristic
46C2×C3×C4×D5D411yescharacteristic
47C2×C3×C5×C8—21no—
48C2×C3×C5×C8—21no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4C2×C411yescharacteristic
51C3×C42×C5—21no—
52C3×C42×C5C2311yescharacteristic
53C3×#685—21no—
54C3×#685—21no—
55C3×#685—21no—
56C3×C4⋊Dic5—21no—
57C3×C4⋊Dic5C2×C411yescharacteristic
58C3×C4×Dic5—21no—
59C3×C4×Dic5—21no—
60C3×C4×Dic5C2×C411yescharacteristic
61C3×C4×Dic5D411yescharacteristic
62C3×C4×Dic5D411yescharacteristic
63C2×C3×#168—21no—
64C2×C3×#168—21no—
65C3×C4×#41—51no—
66C3×#432—51no—
67C3×#431—51no—
68C3×Q8⋊C8—51no—
69C3×Q8⋊C8D511yescharacteristic, complemented
70C3×D4⋊C8—51no—
71C3×D4⋊C8—51no—
72C2×#713C3×C411yescharacteristic
73C4×C5×Q8C22×C311yescharacteristic
74Q8×Dic5C3×C411yescharacteristic
75#4039—21no—
76#4039C3×C411yescharacteristic
77#4038—21no—
78#4038C3×C411yescharacteristic
79C4×D20—21no—
80C4×D20C22×C311yescharacteristic
81C42×D5—21no—
82C42×D5C22×C311yescharacteristic
83C5×C4⋊C8C22×C311yescharacteristic
84C4×C5×C8C22×C311yescharacteristic
85#3952—21no—
86#3952—21no—
87#3947—21no—
88#3946—21no—
89C4⋊#168C22×C311yescharacteristic
90C4×#168C22×C311yescharacteristic
91#1488—51noSylow
92C22×C3×D5—21no—
93C22×C3×D5C22⋊C411yescharacteristic
94C3×C5×Q8—21no—
95C3×C5×Q8M1612yes—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5C22×C411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×Dic5—21no—
102C2×C3×Dic5C22⋊C411yescharacteristic
103C2×C3×Dic5C22⋊C411yescharacteristic
104C2×C3×Dic5C22⋊C411yescharacteristic
105C3×D20—24no—
106C3×D20—21no—
107C3×D20M1612yes—
108C3×C4×D5—24no—
109C3×C4×D5—21no—
110C3×C4×D5—21no—
111C3×C4×D5—21no—
112C3×C4×D5—21no—
113C3×C4×D5—41no—
114C3×C4×D5—42no—
115C3×C4×D5—41no—
116C3×C4×D5—41no—
117C3×C5×C8—41no—
118C3×C5×C8—41no—
119C3×#168—41no—
120C3×#168—41no—
121C2×C3×#41—51no—
122C3×C4×Q8—51no—
123C3×C4×Q8C2×D511yescharacteristic
124C3×C4×D4—51no—
125C3×C4×D4—51no—
126C3×C4×D4—101no—
127C3×C4×D4—101no—
128C3×#117—51no—
129C3×#117—101no—
130C2×C3×C42—51no—
131C2×C3×C42—101no—
132C3×C4⋊C8—51no—
133C3×C4⋊C8—101no—
134C3×C4⋊C8C2×D511yescharacteristic
135C3×C22⋊C8—101no—
136C3×C22⋊C8—101no—
137C3×#97—101no—
138C3×C4×C8—51no—
139C3×C4×C8C2×D511yescharacteristic
140C2×C5×Q8C2×C3×C411yescharacteristic
141#713—22no—
142#713—22no—
143C2×D20—21no—
144C2×D20C2×C3×C411yescharacteristic
145C2×C4×D5—21no—
146C2×C4×D5—21no—
147C2×C4×D5—21no—
148C2×C4×D5C2×C3×C411yescharacteristic
149C2×C4×D5C3×D411yescharacteristic
150C2×C4×D5C3×D411yescharacteristic
151C2×C5×C8—21no—
152C2×C5×C8—21no—
153C5×C4⋊C4—21no—
154C5×C4⋊C4C2×C3×C411yescharacteristic
155C42×C5—21no—
156C42×C5C23×C311yescharacteristic
157#685—21no—
158#685—21no—
159#685—21no—
160C4⋊Dic5—21no—
161C4⋊Dic5C2×C3×C411yescharacteristic
162C4×Dic5—21no—
163C4×Dic5—21no—
164C4×Dic5C2×C3×C411yescharacteristic
165C4×Dic5C3×D411yescharacteristic
166C4×Dic5C3×D411yescharacteristic
167C2×#168—21no—
168C2×#168—21no—
169C4×#41—51no—
170#432—51no—
171#431—51no—
172Q8⋊C8—51no—
173Q8⋊C8C3×D511yescharacteristic, complemented
174D4⋊C8—51no—
175D4⋊C8—51no—
176C22×C3×C5C2×C22⋊C411yescharacteristic
177C2×C3×D5—21no—
178C2×C3×D5—22no—
179C2×C3×D5—21no—
180C2×C3×D5—21no—
181C2×C3×D5—41no—
182C2×C3×D5—41no—
183C3×C4×C5—22no—
184C3×C4×C5—21no—
185C3×C4×C5—41no—
186C3×C4×C5C2×M1611yescharacteristic
187C3×C4×C5C2×M1611yescharacteristic
188C3×C4×C5#13511yescharacteristic
189C3×C4×C5#13511yescharacteristic
190C3×Dic5—21no—
191C3×Dic5—21no—
192C3×Dic5—21no—
193C3×Dic5—41no—
194C3×#41—102no—
195C3×#41—102no—
196C2×C3×Q8C4×D511yescharacteristic
197C2×C3×D4—51no—
198C2×C3×D4—101no—
199C22×C3×C4—51no—
200C22×C3×C4—51no—
201C22×C3×C4—51no—
202C22×C3×C4—101no—
203C22×C3×C4—101no—
204C22×C3×C4—101no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—101no—
208C2×C3×C8—101no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—51no—
211C3×C4⋊C4—101no—
212C3×C4⋊C4C4×D511yescharacteristic
213C3×C22⋊C4—101no—
214C3×C22⋊C4—101no—
215C3×C22⋊C4—101no—
216C3×C42—21no—
217C3×C42—51no—
218C3×C42—51no—
219C3×C42—51no—
220C3×C42—101no—
221C3×C42—101no—
222C3×C42C22×D511yescharacteristic
223C22×D5—21no—
224C22×D5C3×C22⋊C411yescharacteristic
225C5×Q8—21no—
226C5×Q8C3×M1612yes—
227C2×C4×C5—21no—
228C2×C4×C5—21no—
229C2×C4×C5C22×C3×C411yescharacteristic
230C2×C4×C5C2×C3×D411yescharacteristic
231C2×C4×C5C2×C3×D411yescharacteristic
232C2×Dic5—21no—
233C2×Dic5C3×C22⋊C411yescharacteristic
234C2×Dic5C3×C22⋊C411yescharacteristic
235C2×Dic5C3×C22⋊C411yescharacteristic
236D20—24no—
237D20—21no—
238D20C3×M1612yes—
239C4×D5—24no—
240C4×D5—21no—
241C4×D5—21no—
242C4×D5—21no—
243C4×D5—21no—
244C4×D5—41no—
245C4×D5—42no—
246C4×D5—41no—
247C4×D5—41no—
248C5×C8—41no—
249C5×C8—41no—
250#168—41no—
251#168—41no—
252C2×#41—51no—
253C4×Q8—51no—
254C4×Q8C2×C3×D511yescharacteristic
255C4×D4—51no—
256C4×D4—51no—
257C4×D4—101no—
258C4×D4—101no—
259#117—51no—
260#117—101no—
261C2×C42—51no—
262C2×C42—101no—
263C4⋊C8—51no—
264C4⋊C8—101no—
265C4⋊C8C2×C3×D511yescharacteristic
266C22⋊C8—101no—
267C22⋊C8—101no—
268#97—101no—
269C4×C8—51no—
270C4×C8C2×C3×D511yescharacteristic
271C2×C3×C5#40711yescharacteristic
272C2×C3×C5#41611yescharacteristic
273C2×C3×C5#42111yescharacteristic
274C3×D5—41no—
275C3×D5—42no—
276C23×C3—51no—
277C23×C3—101no—
278C3×Q8—21no—
279C3×Q8#67612yes—
280C3×D4—52no—
281C3×D4—104no—
282C3×D4—101no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—51no—
286C2×C3×C4—51no—
287C2×C3×C4—51no—
288C2×C3×C4—104no—
289C2×C3×C4—101no—
290C2×C3×C4—101no—
291C2×C3×C4—101no—
292C2×C3×C4—101no—
293C2×C3×C4—101no—
294C2×C3×C4—201no—
295C2×C3×C4—202no—
296C2×C3×C4—201no—
297C2×C3×C4—201no—
298C2×C3×C4C2×C4×D511yescharacteristic
299C2×C3×C4D4×D511yescharacteristic
300C2×C3×C4D4×D511yescharacteristic
301C3×C8—41no—
302C3×C8—41no—
303C3×C8—201no—
304C3×C8—201no—
305C22×C5C2×C3×C22⋊C411yescharacteristic
306C2×D5—21no—
307C2×D5—22no—
308C2×D5—21no—
309C2×D5—21no—
310C2×D5—41no—
311C2×D5—41no—
312C4×C5—22no—
313C4×C5—21no—
314C4×C5—41no—
315C4×C5C2×C3×M1611yescharacteristic
316C4×C5C2×C3×M1611yescharacteristic
317C4×C5C3×#13511yescharacteristic
318C4×C5C3×#13511yescharacteristic
319Dic5—21no—
320Dic5—21no—
321Dic5—21no—
322Dic5—41no—
323#41—102no—
324#41—102no—
325C2×Q8C3×C4×D511yescharacteristic
326C2×D4—51no—
327C2×D4—101no—
328C22×C4—51no—
329C22×C4—51no—
330C22×C4—51no—
331C22×C4—101no—
332C22×C4—101no—
333C22×C4—101no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—101no—
337C2×C8—101no—
338C4⋊C4—21no—
339C4⋊C4—51no—
340C4⋊C4—101no—
341C4⋊C4C3×C4×D511yescharacteristic
342C22⋊C4—101no—
343C22⋊C4—101no—
344C22⋊C4—101no—
345C42—21no—
346C42—51no—
347C42—51no—
348C42—51no—
349C42—101no—
350C42—101no—
351C42C22×C3×D511yescharacteristic
352C3×C5#148811yescharacteristic, complemented, Hall
353C22×C3—101no—
354C22×C3—101no—
355C22×C3—102no—
356C22×C3—101no—
357C22×C3—201no—
358C22×C3—201no—
359C22×C3D5×C22⋊C411yescharacteristic
360C3×C4—21no—
361C3×C4—22no—
362C3×C4—41no—
363C3×C4—101no—
364C3×C4—101no—
365C3×C4—101no—
366C3×C4—201no—
367C3×C4C2×#67611yescharacteristic
368C3×C4D5×M1611yescharacteristic
369C3×C4#406211yescharacteristic
370C3×C4#406511yescharacteristic
371C2×C5C3×#40711yescharacteristic
372C2×C5C3×#41611yescharacteristic
373C2×C5C3×#42111yescharacteristic
374D5—41no—
375D5—42no—
376C23—51no—
377C23—101no—
378Q8—21no—
379Q8C3×#67612yes—
380D4—52no—
381D4—104no—
382D4—101no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—51no—
386C2×C4—51no—
387C2×C4—51no—
388C2×C4—104no—
389C2×C4—101no—
390C2×C4—101no—
391C2×C4—101no—
392C2×C4—101no—
393C2×C4—101no—
394C2×C4—201no—
395C2×C4—202no—
396C2×C4—201no—
397C2×C4—201no—
398C2×C4C2×C3×C4×D511yescharacteristic
399C2×C4C3×D4×D511yescharacteristic
400C2×C4C3×D4×D511yescharacteristic
401C8—41no—
402C8—41no—
403C8—201no—
404C8—201no—
405C2×C3—201no—
406C2×C3—202no—
407C2×C3#6501511yescharacteristic
408C2×C3#6509311yescharacteristic
409C2×C3#6511011yescharacteristic
410C5C3×#148811yescharacteristic, complemented, Sylow
411C22—101no—
412C22—101no—
413C22—102no—
414C22—101no—
415C22—201no—
416C22—201no—
417C22C3×D5×C22⋊C411yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—101no—
422C4—101no—
423C4—101no—
424C4—201no—
425C4C2×C3×#67611yescharacteristic
426C4C3×D5×M1611yescharacteristic
427C4C3×#406211yescharacteristic
428C4C3×#406511yescharacteristic
429C3#1059903011yescharacteristic, direct product factor, Sylow
430C2—201no—
431C2—202no—
432C2C3×#6501511yescharacteristic
433C2C3×#6509311yescharacteristic
434C2C3×#6511011yescharacteristic
subgroupG/H
centerC2×C3×C4D4×D5
commutatorC2×C4×C5C22×C3×C4
FrattiniC42C22×C3×D5
FittingC3×C5×Q8⋊C8C2
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×#713C4
cosocleC42×C5C23×C3
hypercenterC3×Q8⋊C8D5
hypocenterC5C3×#1488
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3×C4 ◁ C3×C42 ◁ C3×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. C3×#65124C2semidirect
3. C3×#64974C2semidirect
4. C3×C5×Q8⋊C8C2semidirect
5. C3×Q8⋊#168C2semidirect
6. C3×#64703C2semidirect
7. C3×#64679C2semidirect
8. #10599030C3direct (N × H)
30. #65124C2×C3semidirect
31. #64974C2×C3semidirect
32. C5×Q8⋊C8C2×C3semidirect
33. Q8⋊#168C2×C3semidirect
34. #64703C2×C3semidirect
35. #64679C2×C3semidirect
69. C3×Q8⋊C8D5semidirect
173. Q8⋊C8C3×D5semidirect
352. C3×C5#1488semidirect
410. C5C3×#1488semidirect
429. C3#10599030direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#7132
C3×#651242
C3×#649742
C3×C5×Q8⋊C82
C3×Q8⋊#1682
C3×#647032
C3×#646792
#105990303
C3×#14885