Finite Group Databasebeta

← #14458156

Subgroups — #14458156

Order 1920.

920 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11753980C212yesmaximal, complemented
2#11753970C212yesmaximal, complemented
3#11750801C211yesmaximal, characteristic, complemented
4C5×#4700C211yesmaximal, characteristic, complemented
5C3×#64982C211yesmaximal, characteristic, complemented
6#11748087C212yesmaximal, complemented
7#11748083C212yesmaximal, complemented
8#11747120C212yesmaximal, complemented
9#11747114C212yesmaximal, complemented
10#10611624—31nomaximal, Hall
11#92017C2212yescomplemented
12#92004C2212yes—
13#91793C2211yescharacteristic
14#91790C2212yescomplemented
15C5×#904C2211yescharacteristic
16C5×#901C2212yescomplemented
17C2×C3×#677C2212yescomplemented
18C2×C3×#677C2212yescomplemented
19C3×#4020C2211yescharacteristic
20C3×#4015C2211yescharacteristic, complemented
21#91315C2212yescomplemented
22#91312C2212yes—
23#91309C2212yescomplemented
24#91298C2212yes—
25#91278—22no—
26#91276—22no—
27#91274—22no—
28#91272—22no—
29#91257—22no—
30#91255—22no—
31#91253—22no—
32D3×#677—22no—
33C3×C4×C5×C8C2211yescharacteristic, complemented
34#91103C2211yescharacteristic, complemented
35#91089C2212yescomplemented
36C5×#819C2211yescharacteristic, complemented
37C5×#805C2212yescomplemented
38#90961C2212yescomplemented
39#90958C2212yescomplemented
40#81774—51nomaximal, Hall
41#66104—32no—
42#66085—32no—
43C5×#443—31no—
44#65430—32no—
45#65187—32no—
46#65115—32no—
47#65111—32no—
48#64989—31no—
49#64982D311yescharacteristic, complemented
50#6817C2312yes—
51C5×#235C2312yes—
52C2×C3×D20—41no—
53C2×C3×D20C2312yes—
54C2×C3×Dic10C2312yes—
55D5×#67—42no—
56#6782—42no—
57#6777—22no—
58D3×D20—22no—
59D15⋊Q8—22no—
60#6769—22no—
61#6768—22no—
62D3×Dic10—22no—
63#6766—22no—
64#6765—22no—
65C2×C3×C5×C8C2311yescharacteristic
66C2×C3×C5×C8C2311yescharacteristic
67C3×C42×C5C2311yescharacteristic
68#6710C2312yes—
69Dic60D411yescharacteristic, complemented
70D120D411yescharacteristic, complemented
71#6706D412yescomplemented
72#6705—22no—
73C8×D15—22no—
74C5×#208C2312yes—
75C5×Dic12D411yescharacteristic, complemented
76C5×D24D411yescharacteristic, complemented
77C5×#204D412yescomplemented
78C5×ΓL(1,𝔽25)—22no—
79C5×D3×C8—22no—
80C3×C4⋊Dic5—41no—
81C3×#677—22no—
82C3×#677—22no—
83C3×#677—22no—
84C3×#677—22no—
85#6662—22no—
86#6661—22no—
87#6658—22no—
88#6657—22no—
89#6656—22no—
90#6655—22no—
91#6653—22no—
92#6652—22no—
93#5762—52no—
94#5762—52no—
95C3×#492—51no—
96#5177—52no—
97#5177—52no—
98#5095—52no—
99#5077—52no—
100#4700—51no—
101#4700D511yescharacteristic, complemented
102#4150—32no—
103#4149—32no—
104C5×#135—31no—
105C5×#131—32no—
106#4097—32no—
107#4095—32no—
108#4064—62no—
109#4062—62no—
110D5×SD16—62no—
111#4057—62no—
112#4055—32no—
113#4055—62no—
114#4054—32no—
115#4054—62no—
116#4050—31no—
117#4050—62no—
118C2×#677—62no—
119C2×#677C2×D312yescomplemented
120C2×#677C2×D312yescomplemented
121#4047—32no—
122#4020C2×D311yescharacteristic
123#4015C2×D311yescharacteristic, complemented
124C5×#108—31no—
125C5×#104—32no—
126C4×C5×C8C2×D311yescharacteristic, complemented
127#3956—32no—
128#3955—32no—
129#3951—31no—
130#3937—32no—
131#3291—151noSylow
132C22×C3×D5—22no—
133C2×D3×D5—42no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5C2411yescharacteristic
136#1220—22no—
137D60C2×D412yes—
138C4×D15—22no—
139Dic30C2×D412yes—
140C5×#67—22no—
141C5×D12C2×D412yes—
142C4×C5×D3—22no—
143C5×Dic6C2×D412yes—
144C2×C3×Dic5—22no—
145C3×D20—22no—
146C3×D20—22no—
147C3×D20—82no—
148C3×Dic10—22no—
149C3×Dic10—22no—
150#1204—42no—
151#1203—42no—
152#1202—42no—
153#1201—42no—
154#1200—42no—
155D3×Dic5—42no—
156D5×Dic3—42no—
157C3×C5×C8C2×D411yescharacteristic
158C3×C5×C8C2×D411yescharacteristic
159C3×C5×C8C2×D411yescharacteristic
160C3×C5×C8C2×D411yescharacteristic
161C3⋊#168—22no—
162C5×C3⋊C8—22no—
163#1007—52no—
164#1004—52no—
165C2×C3×SD16—52no—
166C2×C3×SD16—52no—
167C3×C4⋊Q8—51no—
168C3×#127—51no—
169#942—52no—
170#942—52no—
171#932—52no—
172#932—52no—
173#916—102no—
174#916—102no—
175#915—102no—
176#915—102no—
177#914—102no—
178#914—102no—
179D3×SD16—102no—
180D3×SD16—102no—
181#904—51no—
182#904C2×D511yescharacteristic
183#901—52no—
184#901C2×D512yescomplemented
185C3×C4×C8C2×D511yescharacteristic, complemented
186#836—52no—
187#833—52no—
188#819—51no—
189#819C2×D511yescharacteristic, complemented
190#805—52no—
191#805C2×D512yescomplemented
192C5×#41—32no—
193#713—62no—
194Q8×D5—62no—
195#711—62no—
196#711—122no—
197D4×D5—62no—
198D4×D5—122no—
199#709—32no—
200#709—62no—
201#709—62no—
202C2×D20—41no—
203C2×D20—62no—
204C2×D20C22×D312yes—
205C2×Dic10—62no—
206C2×Dic10C22×D312yes—
207C5×Q16—31no—
208C5×SD16—32no—
209C5×D8—31no—
210C5×M16—32no—
211C5×M16—62no—
212C2×C5×C8—62no—
213C2×C5×C8C22×D311yescharacteristic
214C2×C5×C8C22×D311yescharacteristic
215C42×C5C22×D311yescharacteristic
216#689—62no—
217#688—62no—
218#687—62no—
219#686—62no—
220C4⋊Dic5—41no—
221#681—32no—
222Dic20—31no—
223Dic20—62no—
224D40—31no—
225D40—62no—
226#677—22no—
227#677—22no—
228#677—22no—
229#677—22no—
230#677—32no—
231#677—62no—
232#677—62no—
233#676—62no—
234C8×D5—62no—
235#577—152no—
236#577—152no—
237#492—51no—
238#471—152no—
239#471—152no—
240#456—152no—
241#453—152no—
242#443—31no—
243#443—151no—
244C22×C3×C5H(5,𝔽2)11yescharacteristic
245C2×D15—22no—
246C2×C5×D3—22no—
247C2×C3×D5—42no—
248C2×C3×D5—82no—
249D3×D5—162no—
250C3×C4×C5—22no—
251C3×C4×C5C22×D411yescharacteristic
252C3×C4×C5C22×D411yescharacteristic
253Dic15—22no—
254C3×Dic5—42no—
255C5×Dic3—22no—
256C2×C3×Q8—52no—
257C2×C3×D4—52no—
258C2×C3×D4—201no—
259C2×#67—202no—
260#239—102no—
261#239—102no—
262D3×Q8—102no—
263D3×Q8—102no—
264#237—102no—
265#237—102no—
266D3×D4—102no—
267D3×D4—102no—
268#235—52no—
269#235—202no—
270#235C22×D512yes—
271C3×SD16—102no—
272C3×SD16—102no—
273C3×SD16—102no—
274C3×SD16—102no—
275C2×C3×C8C22×D511yescharacteristic
276C2×C3×C8C22×D511yescharacteristic
277C3×C4⋊C4—201no—
278C3×C42C22×D511yescharacteristic
279#216—102no—
280#216—102no—
281#215—102no—
282#215—102no—
283#214—102no—
284#214—102no—
285#213—102no—
286#213—102no—
287#208—52no—
288#208C22×D512yes—
289Dic12—51no—
290Dic12D2011yescharacteristic, complemented
291D24—51no—
292D24D2011yescharacteristic, complemented
293#204—52no—
294#204D2012yescomplemented
295ΓL(1,𝔽25)—22no—
296ΓL(1,𝔽25)—102no—
297D3×C8—22no—
298D3×C8—102no—
299C22×D5—22no—
300C22×D5—122no—
301C5×Q8—32no—
302C5×D4—32no—
303C5×D4—62no—
304C2×C4×C5—21no—
305C2×C4×C5—62no—
306C2×C4×C5C23×D311yescharacteristic
307C22⋊D5—62no—
308C22⋊D5—122no—
309C22⋊D5—122no—
310C2×Dic5—22no—
311C2×Dic5—122no—
312D20—22no—
313D20—22no—
314D20—32no—
315D20—82no—
316D20—122no—
317C4×D5—62no—
318C4×D5—122no—
319C4×D5—122no—
320Dic10—22no—
321Dic10—22no—
322Dic10—32no—
323Dic10—122no—
324C5×C8—62no—
325C5×C8D3×D411yescharacteristic
326C5×C8D3×D411yescharacteristic
327C5×C8D3×D411yescharacteristic
328C5×C8D3×D411yescharacteristic
329#168—62no—
3302−1+4—152no—
331H(5,𝔽2)—152no—
332#137—152no—
333#137—152no—
334#137—302no—
335#137—302no—
336AGL(1,ℤ/8)—152no—
337AGL(1,ℤ/8)—152no—
338AGL(1,ℤ/8)—302no—
339AGL(1,ℤ/8)—302no—
340#135—31no—
341#135—151no—
342#135—302no—
343#135—302no—
344C2×SD16—52no—
345C2×SD16—52no—
346C2×SD16—302no—
347C2×SD16—302no—
348#131—32no—
349#131—152no—
350C4⋊Q8—51no—
351#127—51no—
352#108—31no—
353#108—151no—
354#104—32no—
355#104—152no—
356#101—152no—
357#100—152no—
358C4×C8D3×D511yescharacteristic, complemented
359C2×C3×C5—21no—
360C2×C3×C5D4211yescharacteristic
361D15—42no—
362C3×D5—82no—
363C5×D3—42no—
364C23×C3—102no—
365C22×D3—202no—
366C3×Q8—102no—
367C3×Q8—102no—
368C3×D4—102no—
369C3×D4—102no—
370C3×D4—402no—
371C2×C3×C4—21no—
372C2×C3×C4—102no—
373C2×C3×C4C23×D511yescharacteristic
374#67—22no—
375#67—102no—
376#67—202no—
377#67—202no—
378C2×Dic3—202no—
379D12—52no—
380D12—202no—
381D12C2×D2012yes—
382C4×D3—22no—
383C4×D3—102no—
384C4×D3—202no—
385C4×D3—202no—
386Dic6—52no—
387Dic6—202no—
388Dic6C2×D2012yes—
389C3×C8C2×D2011yescharacteristic
390C3×C8C2×D2011yescharacteristic
391C3×C8D4×D511yescharacteristic
392C3×C8D4×D511yescharacteristic
393C3⋊C8—22no—
394C3⋊C8—102no—
395C22×C5—62no—
396C22×C5#100411yescharacteristic
397C2×D5—42no—
398C2×D5—62no—
399C2×D5—82no—
400C2×D5—482no—
401C4×C5—22no—
402C4×C5—62no—
403C4×C5C2×D3×D411yescharacteristic
404C4×C5C2×D3×D411yescharacteristic
405Dic5—42no—
406Dic5—62no—
407#41—32no—
408#41—152no—
409#41—302no—
410#41—302no—
411#41—302no—
412#41—302no—
413#41—602no—
414C2×Q8—52no—
415C2×Q8—302no—
416C2×Q8—302no—
417C2×D4—52no—
418C2×D4—201no—
419C2×D4—302no—
420C2×D4—302no—
421C2×D4—602no—
422Q16—31no—
423Q16—151no—
424Q16—302no—
425Q16—302no—
426SD16—32no—
427SD16—102no—
428SD16—102no—
429SD16—102no—
430SD16—102no—
431SD16—152no—
432SD16—302no—
433SD16—302no—
434SD16—302no—
435SD16—302no—
436D8—31no—
437D8—151no—
438D8—302no—
439D8—302no—
440M16—32no—
441M16—62no—
442M16—152no—
443M16—302no—
444C2×C8—62no—
445C2×C8—302no—
446C2×C8C2×D3×D511yescharacteristic
447C2×C8C2×D3×D511yescharacteristic
448C4⋊C4—201no—
449C42C2×D3×D511yescharacteristic
450C3×C5#329111yescharacteristic, complemented, Hall
451C22×C3—202no—
452C22×C3—402no—
453C22×C3#414911yescharacteristic
454C2×D3—22no—
455C2×D3—102no—
456C2×D3—802no—
457C3×C4—22no—
458C3×C4—202no—
459C3×C4C22×D2011yescharacteristic
460C3×C4C2×D4×D511yescharacteristic
461Dic3—22no—
462Dic3—102no—
463C2×C5—21no—
464C2×C5—122no—
465C2×C5#571711yescharacteristic
466D5—82no—
467D5—122no—
468C23—102no—
469C23—602no—
470Q8—32no—
471Q8—102no—
472Q8—102no—
473Q8—152no—
474Q8—602no—
475D4—32no—
476D4—62no—
477D4—102no—
478D4—102no—
479D4—152no—
480D4—302no—
481D4—402no—
482D4—602no—
483D4—602no—
484D4—602no—
485C2×C4—21no—
486C2×C4—62no—
487C2×C4—102no—
488C2×C4—302no—
489C2×C4—602no—
490C2×C4—602no—
491C2×C4—602no—
492C2×C4C22×D3×D511yescharacteristic
493C8—62no—
494C8—302no—
495C8D3×D2011yescharacteristic
496C8D3×D2011yescharacteristic
497C8#677711yescharacteristic
498C8#677711yescharacteristic
499C2×C3—21no—
500C2×C3—402no—
501C2×C3D4×D2011yescharacteristic
502D3—42no—
503D3—202no—
504C5#8177411yescharacteristic, complemented, Sylow
505C22—62no—
506C22—202no—
507C22—302no—
508C22—402no—
509C22—2402no—
510C22#9201711yescharacteristic
511C4—22no—
512C4—62no—
513C4—202no—
514C4—302no—
515C4C2×D3×D2011yescharacteristic
516C4C2×#677711yescharacteristic
517C3#1061162411yescharacteristic, complemented, Sylow
518C2—21no—
519C2—122no—
520C2—402no—
521C2—602no—
522C2#1175274511yescharacteristic
subgroupG/H
centerC2#11752745
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×C8C22
socleC2×C3×C5D42
hypercenterC4×C8D3×D5
hypocenterC3×C5#3291
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
1. #11753980C2semidirect
2. #11753970C2semidirect
3. #11750801C2semidirect
4. C5×#4700C2semidirect
5. C3×#64982C2semidirect
6. #11748087C2semidirect
7. #11748083C2semidirect
8. #11747120C2semidirect
9. #11747114C2semidirect
11. #92017C22semidirect
14. #91790C22semidirect
16. C5×#901C22semidirect
17. C2×C3×#677C22semidirect
18. C2×C3×#677C22semidirect
20. C3×#4015C22semidirect
21. #91315C22semidirect
23. #91309C22semidirect
33. C3×C4×C5×C8C22semidirect
34. #91103C22semidirect
35. #91089C22semidirect
36. C5×#819C22semidirect
37. C5×#805C22semidirect
38. #90961C22semidirect
39. #90958C22semidirect
49. #64982D3semidirect
69. Dic60D4semidirect
70. D120D4semidirect
71. #6706D4semidirect
75. C5×Dic12D4semidirect
76. C5×D24D4semidirect
77. C5×#204D4semidirect
101. #4700D5semidirect
119. C2×#677C2×D3semidirect
120. C2×#677C2×D3semidirect
123. #4015C2×D3semidirect
126. C4×C5×C8C2×D3semidirect
184. #901C2×D5semidirect
185. C3×C4×C8C2×D5semidirect
189. #819C2×D5semidirect
191. #805C2×D5semidirect
290. Dic12D20semidirect
292. D24D20semidirect
294. #204D20semidirect
358. C4×C8D3×D5semidirect
450. C3×C5#3291semidirect
504. C5#81774semidirect
517. C3#10611624semidirect

Maximal subgroups

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

subgroupindex
#117539802
#117539702
#117508012
C5×#47002
C3×#649822
#117480872
#117480832
#117471202
#117471142
#106116243
#817745