Finite Group Databasebeta

← #14419352

Subgroups — #14419352

Order 1920.

760 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11753535C211yesmaximal, characteristic
2#11752362C211yesmaximal, characteristic, complemented
3C3×#65709C211yesmaximal, characteristic, complemented
4#11751913C211yesmaximal, characteristic, complemented
5#11751891C212yesmaximal, complemented
6#11751870C212yesmaximal, complemented
7#11751867C212yesmaximal, complemented
8#11751867C211yesmaximal, characteristic, complemented
9C4×D3×#170C211yesmaximal, characteristic, complemented
10#11751807C211yesmaximal, characteristic, complemented
11#11751799C212yesmaximal, complemented
12#10617233—31nomaximal, Hall
13C2×#6739C2212yes—
14C2×D3×#170C2211yescharacteristic
15C2×D3×#170C2211yescharacteristic
16C2×D3×#170C2212yes—
17#91779C411yescharacteristic
18C5×#895C411yescharacteristic
19C3×#4033C2211yescharacteristic
20#91423C412yes—
21Dic5×D12C2212yes—
22Dic5×D12C2211yescharacteristic
23#91398C2212yes—
24C4×D3×Dic5C2211yescharacteristic
25#91370C411yescharacteristic
26#91369C411yescharacteristic
27#91342C412yes—
28#91233C2212yes—
29C4⋊(C3⋊#170)C2211yescharacteristic
30C4×C3⋊#170C2211yescharacteristic
31C4×C3⋊#170C2212yes—
32C3×#4002C2212yes—
33C3×C4⋊#170C2212yes—
34C3×C4⋊#170C2211yescharacteristic
35C3×C4×#170C2211yescharacteristic
36(C2×D3)⋊#170C2212yes—
37(C2×D3)⋊#170C2212yes—
38(C2×D3)⋊#170C2212yes—
39#91169C2212yes—
40Dic3×#170C2211yescharacteristic
41Dic3×#170C2211yescharacteristic
42#79939—51nomaximal, Hall
43#65934—31no—
44#65774—31no—
45#65773—32no—
46D4×#170—31no—
47D4×#170—32no—
48#65759—31no—
49C2×C4×#170—31no—
50#65709D311yescharacteristic, complemented
51#65703—31no—
52#65697—32no—
53#65691—32no—
54C2×C5×D12C2×C412yes—
55C2×C5×D12C2×C411yescharacteristic
56C2×C4×C5×D3C2×C411yescharacteristic
57C2×C3×Dic10C2×C411yescharacteristic
58C2×D3×Dic5C2×C411yescharacteristic
59C2×D3×Dic5C2311yescharacteristic
60C2×D3×Dic5C2312yes—
61C2×C3⋊#170C2311yescharacteristic
62C2×C3⋊#170C2312yes—
63C2×C3⋊#170C2311yescharacteristic
64C2×C3×#170C2311yescharacteristic
65C2×C3×#170C2312yes—
66C2×C3×#170C2311yescharacteristic
67#6739—24no—
68D3×#170—22no—
69D3×#170—24no—
70D3×#170—22no—
71C3×C5×C4⋊C4C2×C411yescharacteristic
72C4⋊Dic15C2×C412yes—
73C4⋊Dic15C2×C411yescharacteristic
74#6712C2×C412yes—
75C4×Dic15C2×C411yescharacteristic
76C5×(C2×D3)⋊C4C2×C412yes—
77C3×#683C2×C412yes—
78C3×C4×Dic5C2311yescharacteristic
79C3×C4×Dic5C2312yes—
80#6669C2×C412yes—
81(C2×D3)⋊Dic5C2×C412yes—
82(C2×D3)⋊Dic5C2×C412yes—
83(C2×D3)⋊Dic5C2×C412yes—
84(C2×D3)⋊Dic5C2×C412yes—
85Dic3×Dic5C2×C411yescharacteristic
86Dic3×Dic5C2311yescharacteristic
87#5580—51no—
88C3×#446—51no—
89Q8×C3⋊C8—51no—
90#4840—52no—
91#4835—51no—
92#4834—51no—
93#4834—52no—
94#4708—52no—
95#4706—52no—
96#4688—51no—
97C4×D3×C8—51no—
98C2×#704—32no—
99C22×#170—32no—
100C22×#170—31no—
101C22×#170—31no—
102C5×#121—31no—
103#4082—31no—
104#4081—32no—
105D4×Dic5—32no—
106D4×Dic5—31no—
107#4070—31no—
108C2×C4×Dic5—31no—
109#4036—31no—
110#4033C2×D311yescharacteristic
111#4025—32no—
112#4023—32no—
113C22⋊#170—32no—
114C22⋊#170—32no—
115C22⋊#170—32no—
116#4004—32no—
117#4002—32no—
118#4002C2×D312yes—
119C4⋊#170—31no—
120C4⋊#170C2×D312yes—
121C4⋊#170C2×D311yescharacteristic
122C4×#170—31no—
123C4×#170—31no—
124C4×#170—32no—
125C4×#170—31no—
126C4×#170C2×D311yescharacteristic
127#2966—151noSylow
128C22×C5×D3C22×C412yes—
129C22×C5×D3C22×C411yescharacteristic
130C2×C3×C4×C5C22×C412yes—
131C2×C3×C4×C5C22×C411yescharacteristic
132C2×Dic15C22×C411yescharacteristic
133C2×Dic15C22×C411yescharacteristic
134C2×Dic15C22×C412yes—
135C2×C5×Dic3C22×C411yescharacteristic
136C5×D12—24no—
137C5×D12—22no—
138C4×C5×D3—22no—
139C2×C3×Dic5C22×C411yescharacteristic
140C2×C3×Dic5C22×C412yes—
141C2×C3×Dic5C2411yescharacteristic
142C3×Dic10—22no—
143D3×Dic5—22no—
144D3×Dic5—21no—
145D3×Dic5—21no—
146D3×Dic5—22no—
147D3×Dic5—22no—
148C3⋊#170—21no—
149C3⋊#170—22no—
150C3⋊#170#4112yes—
151C3×#170—21no—
152C3×#170—22no—
153C3×#170#4112yes—
154C3×C4×Q8—51no—
155#947—51no—
156#946—51no—
157C2×ΓL(1,𝔽25)—52no—
158C2×D3×C8—51no—
159C2×D3×C8—51no—
160C2×D3×C8—52no—
161#898—52no—
162#895AGL(1,𝔽5)11yescharacteristic
163#894—52no—
164#890—51no—
165C4×D12—51no—
166C4×D12—52no—
167#872—52no—
168C42×D3—51no—
169C3×C4⋊C8—51no—
170C3×C4⋊C8—52no—
171C3×#97—52no—
172C3×C4×C8—51no—
173(C2×D3)⋊C8—52no—
174(C2×D3)⋊C8—52no—
175(C2×D3)⋊C8—52no—
176#815—52no—
177C8×Dic3—51no—
178C8×Dic3—51no—
179C4⋊(C3⋊C8)—51no—
180C4⋊(C3⋊C8)—52no—
181C4×C3⋊C8—51no—
182C4×C3⋊C8—52no—
183C2×C5×D4—32no—
184C2×C5×D4—31no—
185C22×C4×C5—31no—
186C22×Dic5—31no—
187C22×Dic5—31no—
188C22×Dic5—32no—
189C2×Dic10C4×D311yescharacteristic
190#704—64no—
191C2×#170—31no—
192C2×#170—31no—
193C2×#170—32no—
194C2×#170—64no—
195C2×#170—62no—
196C2×#170—62no—
197C2×#170C22×D311yescharacteristic
198C2×#170C22×D312yes—
199C2×#170C22×D311yescharacteristic
200C5×C4⋊C4C4×D311yescharacteristic
201C5×C22⋊C4—32no—
202C22⋊Dic5—32no—
203C22⋊Dic5—32no—
204C22⋊Dic5—32no—
205C22⋊Dic5—32no—
206C4⋊Dic5—32no—
207C4⋊Dic5—31no—
208#683—32no—
209#683—32no—
210#683C4×D312yes—
211C4×Dic5—31no—
212C4×Dic5—31no—
213C4×Dic5—31no—
214C4×Dic5C22×D311yescharacteristic
215C4×Dic5C22×D312yes—
216#529—151no—
217#446—51no—
218C8×Q8—151no—
219#436—151no—
220#435—152no—
221C8×D4—151no—
222C8×D4—152no—
223#433—152no—
224#432—151no—
225#405—152no—
226C2×C4×C8—151no—
227C22×C3×C5C23×C411yescharacteristic
228C2×C5×D3—21no—
229C2×C5×D3—21no—
230C2×C5×D3—22no—
231C2×C5×D3—22no—
232C2×C5×D3—22no—
233C2×C5×D3#13112yes—
234C3×C4×C5—22no—
235C3×C4×C5#13112yes—
236Dic15—21no—
237Dic15—22no—
238Dic15—21no—
239C3×Dic5—21no—
240C3×Dic5—22no—
241C3×Dic5C2×#4111yescharacteristic
242C3×Dic5C2×#4111yescharacteristic
243C5×Dic3—21no—
244C2×C3×Q8—51no—
245C2×D12C2×AGL(1,𝔽5)12yes—
246C2×D12C2×AGL(1,𝔽5)11yescharacteristic
247C2×C4×D3—51no—
248C2×C4×D3—52no—
249C2×C4×D3—51no—
250C2×C4×D3C2×AGL(1,𝔽5)11yescharacteristic
251C2×C3×C8—51no—
252C2×C3×C8—51no—
253C2×C3×C8—52no—
254C3×C4⋊C4—52no—
255C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
256C3×C42—51no—
257C3×C42—52no—
258(C2×D3)⋊C4—52no—
259(C2×D3)⋊C4—52no—
260(C2×D3)⋊C4—52no—
261(C2×D3)⋊C4—52no—
262(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
263C4⋊Dic3—52no—
264C4⋊Dic3—51no—
265Dic3⋊C4—52no—
266Dic3⋊C4—52no—
267C4×Dic3—51no—
268C4×Dic3—51no—
269C4×Dic3—51no—
270C2×C3⋊C8—51no—
271C2×C3⋊C8—51no—
272C2×C3⋊C8—52no—
273ΓL(1,𝔽25)—104no—
274D3×C8—102no—
275D3×C8—104no—
276D3×C8—102no—
277C23×C5—31no—
278C23×C5—32no—
279C5×D4—64no—
280C5×D4—62no—
281C2×C4×C5—31no—
282C2×C4×C5—62no—
283C2×C4×C5C2×C4×D312yes—
284C2×C4×C5C2×C4×D311yescharacteristic
285C2×Dic5—31no—
286C2×Dic5—31no—
287C2×Dic5—32no—
288C2×Dic5—62no—
289C2×Dic5—61no—
290C2×Dic5—62no—
291C2×Dic5—61no—
292C2×Dic5—62no—
293C2×Dic5C2×C4×D311yescharacteristic
294C2×Dic5C2×C4×D312yes—
295C2×Dic5C23×D311yescharacteristic
296Dic10—22no—
297#170—21no—
298#170—22no—
299#170—32no—
300#170—61no—
301#170—62no—
302#170#23912yes—
303C2×M16—152no—
304C22×C8—151no—
305C22×C8—151no—
306C22×C8—152no—
307#126—152no—
308#125—151no—
309#124—151no—
310#123—152no—
311C22⋊Q8—151no—
312#121—31no—
313C4×Q8—51no—
314C4×D4—152no—
315C4×D4—151no—
316#117—152no—
317C2×C42—151no—
318C4⋊C8—51no—
319C4⋊C8—52no—
320C4⋊C8—151no—
321C4⋊C8—152no—
322C22⋊C8—152no—
323C22⋊C8—152no—
324C22⋊C8—152no—
325#97—52no—
326#97—152no—
327C4×C8—51no—
328C4×C8—151no—
329C4×C8—152no—
330C4×C8—151no—
331C4×C8—151no—
332C2×C3×C5C4×#4111yescharacteristic
333C2×C3×C5C2×#13111yescharacteristic
334C2×C3×C5C2×#13111yescharacteristic
335C5×D3—22no—
336C5×D3—42no—
337C22×D3C22×AGL(1,𝔽5)12yes—
338C22×D3C22×AGL(1,𝔽5)11yescharacteristic
339C3×Q8—102no—
340C2×C3×C4—51no—
341C2×C3×C4—51no—
342C2×C3×C4—52no—
343C2×C3×C4C22×AGL(1,𝔽5)12yes—
344C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
345C2×Dic3—51no—
346C2×Dic3—51no—
347C2×Dic3—52no—
348C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
349D12—24no—
350D12—22no—
351C4×D3—22no—
352C4×D3—101no—
353C4×D3—102no—
354C4×D3—101no—
355C4×D3—102no—
356C4×D3—102no—
357C3×C8—52no—
358C3×C8—101no—
359C3×C8—102no—
360C3⋊C8—52no—
361C3⋊C8—101no—
362C3⋊C8—102no—
363C22×C5—32no—
364C22×C5—61no—
365C22×C5—61no—
366C22×C5—62no—
367C22×C5—62no—
368C22×C5—62no—
369C22×C5C22×C4×D311yescharacteristic
370C4×C5—22no—
371C4×C5—61no—
372C4×C5#90712yes—
373Dic5—22no—
374Dic5—21no—
375Dic5—61no—
376Dic5—61no—
377Dic5—62no—
378Dic5C2×#23911yescharacteristic
379Dic5D3×#4111yescharacteristic
380C2×Q8—51no—
381C2×D4—32no—
382C2×D4—31no—
383C22×C4—31no—
384C22×C4—151no—
385C22×C4—152no—
386C22×C4—151no—
387M16—304no—
388C2×C8—51no—
389C2×C8—51no—
390C2×C8—52no—
391C2×C8—151no—
392C2×C8—151no—
393C2×C8—152no—
394C2×C8—302no—
395C2×C8—302no—
396C2×C8—304no—
397C4⋊C4—52no—
398C4⋊C4—152no—
399C4⋊C4—152no—
400C4⋊C4—151no—
401C4⋊C4—152no—
402C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
403C22⋊C4—32no—
404C22⋊C4—152no—
405C22⋊C4—152no—
406C22⋊C4—152no—
407C22⋊C4—152no—
408C42—51no—
409C42—52no—
410C42—151no—
411C42—151no—
412C42—151no—
413C3×C5#296611yescharacteristic, complemented, Hall
414C22×C3C23×AGL(1,𝔽5)11yescharacteristic
415C2×D3—21no—
416C2×D3—21no—
417C2×D3—22no—
418C2×D3—22no—
419C2×D3—22no—
420C2×D3#413112yes—
421C3×C4—22no—
422C3×C4—51no—
423C3×C4—51no—
424C3×C4—101no—
425C3×C4—102no—
426C3×C4#413112yes—
427Dic3—21no—
428Dic3—101no—
429Dic3—101no—
430Dic3—102no—
431C2×C5—62no—
432C2×C5—122no—
433C2×C5C4×#23911yescharacteristic
434C2×C5C2×#90711yescharacteristic
435C2×C5D3×#13111yescharacteristic
436C23—31no—
437C23—32no—
438Q8—102no—
439D4—64no—
440D4—62no—
441C2×C4—31no—
442C2×C4—51no—
443C2×C4—51no—
444C2×C4—52no—
445C2×C4—62no—
446C2×C4—151no—
447C2×C4—151no—
448C2×C4—152no—
449C2×C4—301no—
450C2×C4—301no—
451C2×C4—302no—
452C2×C4—302no—
453C2×C4—302no—
454C2×C4C2×D3×AGL(1,𝔽5)12yes—
455C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
456C8—52no—
457C8—101no—
458C8—102no—
459C8—152no—
460C8—301no—
461C8—302no—
462C2×C3C2×#413111yescharacteristic
463C2×C3C2×#413111yescharacteristic
464C2×C3#41×AGL(1,𝔽5)11yescharacteristic
465D3—22no—
466D3—42no—
467C5#7993911yescharacteristic, complemented, Sylow
468C22—32no—
469C22—61no—
470C22—61no—
471C22—62no—
472C22—62no—
473C22—62no—
474C22C22×D3×AGL(1,𝔽5)11yescharacteristic
475C4—22no—
476C4—51no—
477C4—51no—
478C4—61no—
479C4—102no—
480C4—101no—
481C4—301no—
482C4—301no—
483C4—302no—
484C4#9191312yes—
485C3#1061723311yescharacteristic, complemented, Sylow
486C2—62no—
487C2—122no—
488C2C2×#9191311yescharacteristic
489C2D3×#413111yescharacteristic
490C2AGL(1,𝔽5)×#23911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC3×C5×C4⋊C4C2×C4
prime radicalC5×#895C4
cosocleC2×C3×Dic5C24
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2966
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11752362C2semidirect
3. C3×#65709C2semidirect
4. #11751913C2semidirect
5. #11751891C2semidirect
6. #11751870C2semidirect
7. #11751867C2semidirect
8. #11751867C2semidirect
9. C4×D3×#170C2semidirect
10. #11751807C2semidirect
11. #11751799C2semidirect
50. #65709D3semidirect
413. C3×C5#2966semidirect
467. C5#79939semidirect
485. C3#10617233semidirect

Maximal subgroups

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

subgroupindex
#117535352
#117523622
C3×#657092
#117519132
#117518912
#117518702
#117518672
C4×D3×#1702
#117518072
#117517992
#106172333
#799395