Finite Group Databasebeta

← #14417818

Subgroups — #14417818

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91801C211yesmaximal, characteristic
2#11754916C212yesmaximal, complemented
3C5×#5152C211yesmaximal, characteristic, complemented
4C3×#65441C211yesmaximal, characteristic, complemented
5#11749011C212yesmaximal, complemented
6#11748998C212yesmaximal, complemented
7#11748959C212yesmaximal, complemented
8#11747876C211yesmaximal, characteristic, complemented
9#11747854C211yesmaximal, characteristic, complemented
10#11747668C211yesmaximal, characteristic, complemented
11#11747632C211yesmaximal, characteristic, complemented
12#10611599—31nomaximal, Hall
13C2×#6819C2212yes—
14C2×C3×C5×D8C2211yescharacteristic
15C2×#6716C2212yes—
16#91801C418yescomplemented
17C2×Dic60C2211yescharacteristic
18C2×C8×D15C2211yescharacteristic
19C5×D4×Dic3C2212yes—
20C3×D4×Dic5C2212yes—
21#91556—22no—
22#91552—22no—
23C2×#6669—22no—
24#91532—22no—
25#91437C2212yes—
26#91386C2211yescharacteristic
27#91340C2212yes—
28#91333C2212yes—
29C2×#6651C2211yescharacteristic
30C5×#832C2212yes—
31C5×#817C2211yescharacteristic
32C5×#815C2211yescharacteristic
33C3×#3964C2212yes—
34C3×#3949C2211yescharacteristic
35C3×#3947C2211yescharacteristic
36#90986C2212yes—
37#90975C2212yes—
38#90974C2212yes—
39Dic3×#168C2211yescharacteristic
40Dic5×C3⋊C8C2211yescharacteristic
41#79868—51nomaximal, Hall
42C2×#4058—31no—
43#65880—32no—
44C5×#442—31no—
45#65441D311yescharacteristic, complemented
46C4×#687—32no—
47D5×#106—31no—
48#65142—31no—
49#65060—32no—
50#65047—32no—
51#65005—31no—
52#64997—31no—
53C2×C3×C5×D4C2312yes—
54C2×#1220—22no—
55C22×Dic15—22no—
56#6819—24no—
57#6819C2×C418yescomplemented
58C2×C4×D15C2311yescharacteristic
59C2×Dic30C2312yes—
60C22×C5×Dic3—22no—
61C22×C3×Dic5—22no—
62C2×#1200—22no—
63C3×C5×D8C2×C414yescomplemented
64C2×C3×C5×C8C2311yescharacteristic
65#6716C2×C418yescomplemented
66C2×C3⋊#168C2311yescharacteristic
67Dic60C2×C414yescomplemented
68C8×D15C2×C414yescomplemented
69C5×#217—22no—
70C5×C4⋊Dic3C2312yes—
71C4×C5×Dic3C2311yescharacteristic
72C2×C5×C3⋊C8C2311yescharacteristic
73C3×C22⋊Dic5—22no—
74C3×C4⋊Dic5C2312yes—
75C3×C4×Dic5C2311yescharacteristic
76C2×C3×#168C2311yescharacteristic
77#6669—22no—
78#6669—22no—
79#6669—22no—
80#6669D412yes—
81#6667D412yes—
82Dic3×Dic5—22no—
83Dic3×Dic5—22no—
84#6651—22no—
85C2×#912—51no—
86#5546—52no—
87C3×#442—51no—
88#5152D511yescharacteristic, complemented
89C4×#214—52no—
90D3×#106—51no—
91#4853—51no—
92#4771—52no—
93#4758—52no—
94#4716—51no—
95#4708—51no—
96C2×#711—32no—
97C2×C5×D8C2×D311yescharacteristic
98C4×C5×D4—32no—
99D4×Dic5C2×D312yes—
100C2×#687—32no—
101C4×C22⋊D5—62no—
102C2×#683—62no—
103#4058—38no—
104C2×Dic20—31no—
105C2×#676—31no—
106C2×C8×D5—31no—
107D5×C4⋊C4—32no—
108#4033—32no—
109#4027—62no—
110#4023—62no—
111#4018—31no—
112C4×Dic10—32no—
113C5×#106—31no—
114C5×#102—32no—
115C5×#97—31no—
116#3964C2×D312yes—
117#3949C2×D311yescharacteristic
118#3948—32no—
119#3947C2×D311yescharacteristic
120C8×Dic5—31no—
121#3942—32no—
122#3940—32no—
123C4×#168—31no—
124#2952—151noSylow
125C23×C3×C5—22no—
126C22×D15C2×D411yescharacteristic
127C3×C5×D4—22no—
128C3×C5×D4C22×C414yes—
129C2×C3×C4×C5C2411yescharacteristic
130#1220—28no—
131C2×Dic15—22no—
132C2×Dic15—28no—
133C2×Dic15C2×D411yescharacteristic
134C4×D15C22×C412yes—
135C4×D15#4112yes—
136Dic30—22no—
137Dic30C22×C414yes—
138C2×C5×Dic3—22no—
139C2×C5×Dic3—82no—
140C2×C5×Dic3C2×D412yes—
141C2×C3×Dic5—22no—
142C2×C3×Dic5—82no—
143C2×C3×Dic5C2×D412yes—
144#1200—82no—
145C3×C5×C8C22×C412yes—
146C3⋊#168C22×C412yes—
147C3×#168—21no—
148C5×C3⋊C8—21no—
149C2×#237—52no—
150C2×C3×D8C2×D511yescharacteristic
151C3×C4×D4—52no—
152D4×Dic3C2×D512yes—
153C2×#214—52no—
154C4×#67—102no—
155C2×Dic3⋊C4—102no—
156#912—58no—
157C2×Dic12—51no—
158C2×ΓL(1,𝔽25)—51no—
159C2×D3×C8—51no—
160D3×C4⋊C4—52no—
161#887—52no—
162#67⋊C4—102no—
163#877—102no—
164#872—51no—
165C4×Dic6—52no—
166C3×#106—51no—
167C3×#102—52no—
168C3×#97—51no—
169#832C2×D512yes—
170#817C2×D511yescharacteristic
171#816—52no—
172#815C2×D511yescharacteristic
173C8×Dic3—51no—
174#810—52no—
175#808—52no—
176C4×C3⋊C8—51no—
177C2×C5×D4C22×D312yes—
178C22×C4×C5—62no—
179C2×C22⋊D5—62no—
180C22×Dic5—22no—
181C22×Dic5—62no—
182#711—38no—
183#711—64no—
184C2×C4×D5—31no—
185C2×C4×D5—62no—
186C2×Dic10—32no—
187C5×D8C4×D314yescomplemented
188C2×C5×C8—31no—
189C2×C5×C8C22×D311yescharacteristic
190C5×C4⋊C4—32no—
191C5×C22⋊C4—62no—
192C42×C5—31no—
193C22⋊Dic5—22no—
194#687—38no—
195#685—32no—
196C4⋊Dic5C22×D312yes—
197#683—32no—
198#683—62no—
199#683—62no—
200#683—62no—
201C4×Dic5—62no—
202C4×Dic5—62no—
203C4×Dic5C22×D311yescharacteristic
204C2×#168—31no—
205C2×#168C22×D311yescharacteristic
206Dic20—34no—
207#676—62no—
208C8×D5—34no—
209C2×#135—151no—
210#520—152no—
211#442—31no—
212#442—51no—
213#441—151no—
214#441—151no—
215C4×SD16—152no—
216C4×SD16—152no—
217C2×#106—151no—
218#417—152no—
219#405—151no—
220C22×C3×C5—24no—
221C22×C3×C5—42no—
222C22×C3×C5—42no—
223C22×C3×C5C22×D411yescharacteristic
224C2×D15—22no—
225C2×D15C4×D412yes—
226C3×C4×C5C23×C411yescharacteristic
227C3×C4×C5C2×#4111yescharacteristic
228Dic15—24no—
229Dic15C4×D412yes—
230C3×Dic5—22no—
231C3×Dic5—42no—
232C5×Dic3—22no—
233C5×Dic3—42no—
234C2×C3×D4C22×D512yes—
235C22×C3×C4—102no—
236C2×#67—102no—
237C22×Dic3—22no—
238C22×Dic3—102no—
239#237—58no—
240#237—104no—
241C2×C4×D3—51no—
242C2×C4×D3—102no—
243C2×Dic6—52no—
244C3×D8C4×D514yescomplemented
245C2×C3×C8—51no—
246C2×C3×C8C22×D511yescharacteristic
247C3×C4⋊C4—52no—
248C3×C22⋊C4—102no—
249C3×C42—51no—
250#217—22no—
251#214—58no—
252(C2×D3)⋊C4—52no—
253C4⋊Dic3C22×D512yes—
254Dic3⋊C4—52no—
255Dic3⋊C4—102no—
256Dic3⋊C4—102no—
257Dic3⋊C4—102no—
258C4×Dic3—102no—
259C4×Dic3—102no—
260C4×Dic3C22×D511yescharacteristic
261C2×C3⋊C8—51no—
262C2×C3⋊C8C22×D511yescharacteristic
263Dic12—54no—
264ΓL(1,𝔽25)—102no—
265D3×C8—54no—
266C23×C5—22no—
267C22×D5—31no—
268C5×D4—22no—
269C5×D4C2×C4×D314yes—
270C2×C4×C5—32no—
271C2×C4×C5—62no—
272C2×C4×C5—242no—
273C2×C4×C5C23×D311yescharacteristic
274C22⋊D5—68no—
275C2×Dic5—22no—
276C2×Dic5—31no—
277C2×Dic5—68no—
278C2×Dic5—62no—
279C2×Dic5—82no—
280C2×Dic5D3×D412yes—
281C4×D5—32no—
282C4×D5—32no—
283C4×D5—242no—
284Dic10—34no—
285Dic10—62no—
286C5×C8—61no—
287C5×C8C2×C4×D312yes—
288#168—21no—
289#168—32no—
290C2×#41—152no—
291#135—158no—
292C2×Q16—151no—
293C2×SD16—152no—
294C2×D8D3×D511yescharacteristic
295C2×M16—151no—
296C22×C8—151no—
297C4×Q8—152no—
298C4×Q8—152no—
299C4×D4—32no—
300C4×D4—52no—
301C4×D4—302no—
302C4×D4—302no—
303#117—151no—
304#117—302no—
305C2×C4⋊C4—152no—
306C2×C4⋊C4—302no—
307#106—31no—
308#106—51no—
309#106—152no—
310#103—152no—
311#103—152no—
312#102—32no—
313#102—52no—
314#97—31no—
315#97—51no—
316C4×C8—151no—
317C4×C8—151no—
318C2×C3×C5—44no—
319C2×C3×C5C2×C4×D411yescharacteristic
320C2×C3×C5#57712yes—
321D15—22no—
322C23×C3—22no—
323C22×D3—51no—
324C3×D4—22no—
325C3×D4C2×C4×D514yes—
326C2×C3×C4—52no—
327C2×C3×C4—102no—
328C2×C3×C4—402no—
329C2×C3×C4C23×D511yescharacteristic
330#67—108no—
331C2×Dic3—22no—
332C2×Dic3—51no—
333C2×Dic3—82no—
334C2×Dic3—108no—
335C2×Dic3—102no—
336C2×Dic3D4×D512yes—
337C4×D3—52no—
338C4×D3—52no—
339C4×D3—402no—
340Dic6—54no—
341Dic6—102no—
342C3×C8—101no—
343C3×C8C2×C4×D512yes—
344C3⋊C8—21no—
345C3⋊C8—52no—
346C22×C5—24no—
347C22×C5—42no—
348C22×C5—42no—
349C22×C5C2×D3×D411yescharacteristic
350C2×D5—32no—
351C2×D5—62no—
352C4×C5—62no—
353C4×C5—122no—
354C4×C5C22×C4×D311yescharacteristic
355C4×C5D3×#4111yescharacteristic
356Dic5—22no—
357Dic5—32no—
358Dic5—42no—
359Dic5—64no—
360#41—158no—
361#41—304no—
362C2×Q8—152no—
363C2×D4—302no—
364C2×D4C2×D3×D512yes—
365C22×C4—62no—
366C22×C4—102no—
367C22×C4—151no—
368C22×C4—302no—
369C22×C4—302no—
370Q16—154no—
371SD16—158no—
372D8#120014yescomplemented
373M16—302no—
374C2×C8—31no—
375C2×C8—51no—
376C2×C8—151no—
377C2×C8—154no—
378C2×C8C2×D3×D511yescharacteristic
379C4⋊C4—32no—
380C4⋊C4—52no—
381C4⋊C4—152no—
382C4⋊C4—302no—
383C4⋊C4—302no—
384C4⋊C4—302no—
385C22⋊C4—62no—
386C22⋊C4—102no—
387C22⋊C4—152no—
388C42—31no—
389C42—51no—
390C42—302no—
391C42—302no—
392C3×C5#295211yescharacteristic, complemented, Hall
393C22×C3—24no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3C2×D4×D511yescharacteristic
397C2×D3—52no—
398C2×D3—102no—
399C3×C4—102no—
400C3×C4—202no—
401C3×C4C22×C4×D511yescharacteristic
402C3×C4D5×#4111yescharacteristic
403Dic3—22no—
404Dic3—42no—
405Dic3—52no—
406Dic3—104no—
407C2×C5—44no—
408C2×C5C4×D3×D411yescharacteristic
409C2×C5#577512yes—
410D5—62no—
411C23—22no—
412C23—151no—
413Q8—154no—
414Q8—302no—
415D4—22no—
416D4—308no—
417D4C2×#120014yes—
418C2×C4—32no—
419C2×C4—52no—
420C2×C4—62no—
421C2×C4—102no—
422C2×C4—151no—
423C2×C4—152no—
424C2×C4—152no—
425C2×C4—242no—
426C2×C4—308no—
427C2×C4—302no—
428C2×C4—402no—
429C2×C4—1202no—
430C2×C4C22×D3×D511yescharacteristic
431C8—61no—
432C8—101no—
433C8—152no—
434C8C2×#120012yes—
435C2×C3—44no—
436C2×C3C4×D4×D511yescharacteristic
437C2×C3#6610912yes—
438D3—102no—
439C5#7986811yescharacteristic, complemented, Sylow
440C22—24no—
441C22—42no—
442C22—42no—
443C22—152no—
444C22—302no—
445C22D3×D4×D511yescharacteristic
446C4—62no—
447C4—102no—
448C4—122no—
449C4—152no—
450C4—202no—
451C4—304no—
452C4#9202411yescharacteristic
453C4C22×#120011yescharacteristic
454C3#1061159911yescharacteristic, complemented, Sylow
455C2—44no—
456C2—302no—
457C2#1175405712yes—
458C2D4×#120011yescharacteristic
subgroupG/H
centerC22D3×D4×D5
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×D8C22
socleC22×C3×C5C22×D4
prime radicalC2×#91801C2
cosocleC2×C3×C4×C5C24
hypercenterC2×D8D3×D5
hypocenterC3×C5#2952
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
2. #11754916C2semidirect
3. C5×#5152C2semidirect
4. C3×#65441C2semidirect
5. #11749011C2semidirect
6. #11748998C2semidirect
7. #11748959C2semidirect
8. #11747876C2semidirect
9. #11747854C2semidirect
10. #11747668C2semidirect
11. #11747632C2semidirect
16. #91801C4semidirect
45. #65441D3semidirect
57. #6819C2×C4semidirect
63. C3×C5×D8C2×C4semidirect
65. #6716C2×C4semidirect
67. Dic60C2×C4semidirect
68. C8×D15C2×C4semidirect
88. #5152D5semidirect
187. C5×D8C4×D3semidirect
244. C3×D8C4×D5semidirect
372. D8#1200semidirect
392. C3×C5#2952semidirect
439. C5#79868semidirect
454. C3#10611599semidirect

Maximal subgroups

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

subgroupindex
C2×#918012
#117549162
C5×#51522
C3×#654412
#117490112
#117489982
#117489592
#117478762
#117478542
#117476682
#117476322
#106115993
#798685