Finite Group Databasebeta

← C3×#10604437 #14281638

Subgroups — C3×#10604437 #14281638

Order 1920.

528 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#713C211yesmaximal, characteristic
2C3×D5⋊(C4⋊C8)C211yesmaximal, characteristic, complemented
3C3×#65678C211yesmaximal, characteristic, complemented
4C3×Q8⋊#170C212yesmaximal, complemented
5C3×#64871C212yesmaximal, complemented
6#10604437C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#713C411yescharacteristic
8C2×C3×D5⋊C8—21no—
9C3×C4×C5×Q8C411yescharacteristic
10C3×Q8×Dic5C2211yescharacteristic
11C3×#4039—21no—
12C3×#4039C2211yescharacteristic
13C3×#4038—21no—
14C3×#4038C411yescharacteristic
15C3×C4×D20—21no—
16C3×C4×D20C411yescharacteristic
17C3×C42×D5—21no—
18C3×C42×D5C2211yescharacteristic
19C3×#4004—21no—
20C3×#4004—21no—
21C3×#4003—21no—
22C3×C4⋊#170C2212yes—
23C3×C4×#170C2212yes—
24C3×#1477—51nomaximal, Hall
25C4×#713C2×C311yescharacteristic
26D5⋊(C4⋊C8)C2×C311yescharacteristic, complemented
27#65678C2×C311yescharacteristic, complemented
28Q8⋊#170C2×C312yescomplemented
29#64871C2×C312yescomplemented
30C2×C3×C5×Q8C2×C411yescharacteristic
31C3×#713—22no—
32C3×#713C814yescomplemented
33C2×C3×D20—21no—
34C2×C3×D20C2×C411yescharacteristic
35C2×C3×C4×D5—21no—
36C2×C3×C4×D5—21no—
37C2×C3×C4×D5—21no—
38C2×C3×C4×D5C2×C411yescharacteristic
39C2×C3×C4×D5D411yescharacteristic
40C2×C3×C4×D5D411yescharacteristic
41C2×C3×#170—22no—
42C2×C3×#170—22no—
43C3×D5⋊C8—42no—
44C3×C5×C4⋊C4—21no—
45C3×C5×C4⋊C4C2×C411yescharacteristic
46C3×C42×C5—21no—
47C3×C42×C5C2×C411yescharacteristic
48C3×#685—21no—
49C3×#685—21no—
50C3×#685—21no—
51C3×C4⋊Dic5—21no—
52C3×C4⋊Dic5C2×C411yescharacteristic
53C3×C4×Dic5—21no—
54C3×C4×Dic5—21no—
55C3×C4×Dic5D411yescharacteristic
56C3×C4×Dic5D411yescharacteristic
57C3×C4×Dic5C2311yescharacteristic
58C3×C4×#41—51no—
59C3×#431—51no—
60C2×C3×C4⋊C8—51no—
61C3×Q8⋊C8—52no—
62C3×D4⋊C8—52no—
63C2×#713C3×C411yescharacteristic
64C2×D5⋊C8—21no—
65C4×C5×Q8C3×C411yescharacteristic
66Q8×Dic5C22×C311yescharacteristic
67#4039—21no—
68#4039C22×C311yescharacteristic
69#4038—21no—
70#4038C3×C411yescharacteristic
71C4×D20—21no—
72C4×D20C3×C411yescharacteristic
73C42×D5—21no—
74C42×D5C22×C311yescharacteristic
75#4004—21no—
76#4004—21no—
77#4003—21no—
78C4⋊#170C22×C312yes—
79C4×#170C22×C312yes—
80#1477—51noSylow
81C22×C3×D5—21no—
82C22×C3×D5C22⋊C411yescharacteristic
83C3×C5×Q8—21no—
84C3×C5×Q8C2×C812yescomplemented
85C2×C3×C4×C5—21no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5C22⋊C411yescharacteristic
88C2×C3×C4×C5C22⋊C411yescharacteristic
89C2×C3×C4×C5C22×C411yescharacteristic
90C2×C3×Dic5—21no—
91C2×C3×Dic5C22⋊C411yescharacteristic
92C2×C3×Dic5C2×D411yescharacteristic
93C2×C3×Dic5C2×D411yescharacteristic
94C3×D20—24no—
95C3×D20—21no—
96C3×D20C2×C812yescomplemented
97C3×C4×D5—24no—
98C3×C4×D5—21no—
99C3×C4×D5—21no—
100C3×C4×D5—41no—
101C3×C4×D5—42no—
102C3×C4×D5—41no—
103C3×C4×D5—41no—
104C3×C4×D5C2×C812yes—
105C3×C4×D5M1612yes—
106C3×#170—42no—
107C3×#170—42no—
108C2×C3×#41—51no—
109C22×C3×C8—101no—
110C3×C4×Q8—51no—
111C3×C4×Q8AGL(1,𝔽5)11yescharacteristic
112C3×C4×D4—51no—
113C3×C4×D4—51no—
114C3×C4×D4—101no—
115C3×C4×D4—101no—
116C3×#117—51no—
117C3×#117—101no—
118C2×C3×C42—51no—
119C2×C3×C42—101no—
120C3×C4⋊C8—52no—
121C3×C4⋊C8—101no—
122C3×C4⋊C8—101no—
123C3×C22⋊C8—101no—
124C3×C4×C8—52no—
125C2×C5×Q8C2×C3×C411yescharacteristic
126#713—22no—
127#713C3×C814yescomplemented
128C2×D20—21no—
129C2×D20C2×C3×C411yescharacteristic
130C2×C4×D5—21no—
131C2×C4×D5—21no—
132C2×C4×D5—21no—
133C2×C4×D5C2×C3×C411yescharacteristic
134C2×C4×D5C3×D411yescharacteristic
135C2×C4×D5C3×D411yescharacteristic
136C2×#170—22no—
137C2×#170—22no—
138D5⋊C8—42no—
139C5×C4⋊C4—21no—
140C5×C4⋊C4C2×C3×C411yescharacteristic
141C42×C5—21no—
142C42×C5C2×C3×C411yescharacteristic
143#685—21no—
144#685—21no—
145#685—21no—
146C4⋊Dic5—21no—
147C4⋊Dic5C2×C3×C411yescharacteristic
148C4×Dic5—21no—
149C4×Dic5—21no—
150C4×Dic5C3×D411yescharacteristic
151C4×Dic5C3×D411yescharacteristic
152C4×Dic5C23×C311yescharacteristic
153C4×#41—51no—
154#431—51no—
155C2×C4⋊C8—51no—
156Q8⋊C8—52no—
157D4⋊C8—52no—
158C22×C3×C5C2×C22⋊C411yescharacteristic
159C2×C3×D5—21no—
160C2×C3×D5—21no—
161C2×C3×D5—22no—
162C2×C3×D5—41no—
163C2×C3×D5—41no—
164C2×C3×D5C22⋊C812yes—
165C3×C4×C5—22no—
166C3×C4×C5—21no—
167C3×C4×C5—21no—
168C3×C4×C5—41no—
169C3×C4×C5C22×C811yescharacteristic
170C3×C4×C5C2×M1611yescharacteristic
171C3×Dic5—21no—
172C3×Dic5—41no—
173C3×Dic5C22⋊C812yes—
174C3×Dic5AGL(1,ℤ/8)11yescharacteristic
175C3×Dic5#13711yescharacteristic
176C3×#41—54no—
177C3×#41—102no—
178C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
179C2×C3×D4—51no—
180C2×C3×D4—101no—
181C22×C3×C4—51no—
182C22×C3×C4—51no—
183C22×C3×C4—51no—
184C22×C3×C4—101no—
185C22×C3×C4—101no—
186C22×C3×C4—101no—
187C2×C3×C8—102no—
188C2×C3×C8—102no—
189C2×C3×C8—202no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—51no—
192C3×C4⋊C4—101no—
193C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
194C3×C22⋊C4—101no—
195C3×C22⋊C4—101no—
196C3×C22⋊C4—101no—
197C3×C42—21no—
198C3×C42—51no—
199C3×C42—51no—
200C3×C42—51no—
201C3×C42—101no—
202C3×C42—101no—
203C3×C42C2×AGL(1,𝔽5)11yescharacteristic
204C22×D5—21no—
205C22×D5C3×C22⋊C411yescharacteristic
206C5×Q8—21no—
207C5×Q8C2×C3×C812yescomplemented
208C2×C4×C5—21no—
209C2×C4×C5—21no—
210C2×C4×C5C3×C22⋊C411yescharacteristic
211C2×C4×C5C3×C22⋊C411yescharacteristic
212C2×C4×C5C22×C3×C411yescharacteristic
213C2×Dic5—21no—
214C2×Dic5C3×C22⋊C411yescharacteristic
215C2×Dic5C2×C3×D411yescharacteristic
216C2×Dic5C2×C3×D411yescharacteristic
217D20—24no—
218D20—21no—
219D20C2×C3×C812yescomplemented
220C4×D5—24no—
221C4×D5—21no—
222C4×D5—21no—
223C4×D5—41no—
224C4×D5—42no—
225C4×D5—41no—
226C4×D5—41no—
227C4×D5C2×C3×C812yes—
228C4×D5C3×M1612yes—
229#170—42no—
230#170—42no—
231C2×#41—51no—
232C22×C8—101no—
233C4×Q8—51no—
234C4×Q8C3×AGL(1,𝔽5)11yescharacteristic
235C4×D4—51no—
236C4×D4—51no—
237C4×D4—101no—
238C4×D4—101no—
239#117—51no—
240#117—101no—
241C2×C42—51no—
242C2×C42—101no—
243C4⋊C8—52no—
244C4⋊C8—101no—
245C4⋊C8—101no—
246C22⋊C8—101no—
247C4×C8—52no—
248C2×C3×C5C2×C22⋊C811yescharacteristic
249C2×C3×C5#41711yescharacteristic
250C2×C3×C5#42111yescharacteristic
251C3×D5—22no—
252C3×D5—42no—
253C23×C3—51no—
254C23×C3—101no—
255C3×Q8—21no—
256C3×Q8D5⋊C812yescomplemented
257C3×D4—52no—
258C3×D4—104no—
259C3×D4—101no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—51no—
263C2×C3×C4—51no—
264C2×C3×C4—52no—
265C2×C3×C4—52no—
266C2×C3×C4—51no—
267C2×C3×C4—101no—
268C2×C3×C4—101no—
269C2×C3×C4—104no—
270C2×C3×C4—101no—
271C2×C3×C4—201no—
272C2×C3×C4—202no—
273C2×C3×C4—201no—
274C2×C3×C4—201no—
275C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
276C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
277C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
278C3×C8—202no—
279C3×C8—202no—
280C22×C5C2×C3×C22⋊C411yescharacteristic
281C2×D5—21no—
282C2×D5—21no—
283C2×D5—22no—
284C2×D5—41no—
285C2×D5—41no—
286C2×D5C3×C22⋊C812yes—
287C4×C5—22no—
288C4×C5—21no—
289C4×C5—21no—
290C4×C5—41no—
291C4×C5C22×C3×C811yescharacteristic
292C4×C5C2×C3×M1611yescharacteristic
293Dic5—21no—
294Dic5—41no—
295Dic5C3×C22⋊C812yes—
296Dic5C3×AGL(1,ℤ/8)11yescharacteristic
297Dic5C3×#13711yescharacteristic
298#41—54no—
299#41—102no—
300C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
301C2×D4—51no—
302C2×D4—101no—
303C22×C4—51no—
304C22×C4—51no—
305C22×C4—51no—
306C22×C4—101no—
307C22×C4—101no—
308C22×C4—101no—
309C2×C8—102no—
310C2×C8—102no—
311C2×C8—202no—
312C4⋊C4—21no—
313C4⋊C4—51no—
314C4⋊C4—101no—
315C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
316C22⋊C4—101no—
317C22⋊C4—101no—
318C22⋊C4—101no—
319C42—21no—
320C42—51no—
321C42—51no—
322C42—51no—
323C42—101no—
324C42—101no—
325C42C2×C3×AGL(1,𝔽5)11yescharacteristic
326C3×C5#147711yescharacteristic, complemented, Hall
327C22×C3—52no—
328C22×C3—101no—
329C22×C3—101no—
330C22×C3—102no—
331C22×C3—201no—
332C22×C3—201no—
333C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
334C3×C4—21no—
335C3×C4—22no—
336C3×C4—21no—
337C3×C4—41no—
338C3×C4—51no—
339C3×C4—51no—
340C3×C4—52no—
341C3×C4—101no—
342C3×C4—201no—
343C3×C4C2×D5⋊C811yescharacteristic
344C3×C4D5⋊M1611yescharacteristic
345C2×C5C2×C3×C22⋊C811yescharacteristic
346C2×C5C3×#41711yescharacteristic
347C2×C5C3×#42111yescharacteristic
348D5—22no—
349D5—42no—
350C23—51no—
351C23—101no—
352Q8—21no—
353Q8C3×D5⋊C812yescomplemented
354D4—52no—
355D4—104no—
356D4—101no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—51no—
360C2×C4—51no—
361C2×C4—52no—
362C2×C4—52no—
363C2×C4—51no—
364C2×C4—101no—
365C2×C4—101no—
366C2×C4—104no—
367C2×C4—101no—
368C2×C4—201no—
369C2×C4—202no—
370C2×C4—201no—
371C2×C4—201no—
372C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
373C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
374C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
375C8—202no—
376C8—202no—
377C2×C3—102no—
378C2×C3—202no—
379C2×C3D5⋊(C22⋊C8)11yescharacteristic
380C2×C3#6578411yescharacteristic
381C2×C3#6579311yescharacteristic
382C5C3×#147711yescharacteristic, complemented, Sylow
383C22—52no—
384C22—101no—
385C22—101no—
386C22—102no—
387C22—201no—
388C22—201no—
389C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
390C4—22no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—52no—
395C4—51no—
396C4—51no—
397C4—101no—
398C4—201no—
399C4C2×C3×D5⋊C811yescharacteristic
400C4C3×D5⋊M1611yescharacteristic
401C3#1060443711yescharacteristic, direct product factor, Sylow
402C2—102no—
403C2—202no—
404C2C3×D5⋊(C22⋊C8)11yescharacteristic
405C2C3×#6578411yescharacteristic
406C2C3×#6579311yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC4×C5C22×C3×C8
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C4×C5×Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×#713C4
cosocleC4×Dic5C23×C3
hypercenterC3×C4×Q8AGL(1,𝔽5)
hypocenterC5C3×#1477
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C3×D5⋊(C4⋊C8)C2semidirect
3. C3×#65678C2semidirect
4. C3×Q8⋊#170C2semidirect
5. C3×#64871C2semidirect
6. #10604437C3direct (N × H)
26. D5⋊(C4⋊C8)C2×C3semidirect
27. #65678C2×C3semidirect
28. Q8⋊#170C2×C3semidirect
29. #64871C2×C3semidirect
32. C3×#713C8semidirect
84. C3×C5×Q8C2×C8semidirect
96. C3×D20C2×C8semidirect
127. #713C3×C8semidirect
207. C5×Q8C2×C3×C8semidirect
219. D20C2×C3×C8semidirect
256. C3×Q8D5⋊C8semidirect
326. C3×C5#1477semidirect
353. Q8C3×D5⋊C8semidirect
382. C5C3×#1477semidirect
401. C3#10604437direct (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×D5⋊(C4⋊C8)2
C3×#656782
C3×Q8⋊#1702
C3×#648712
#106044373
C3×#14775