Finite Group Databasebeta

← #14419099

Subgroups — #14419099

Order 1920.

824 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11753406C211yesmaximal, characteristic
2#11752368C211yesmaximal, characteristic, complemented
3C3×#65709C211yesmaximal, characteristic, complemented
4#11751927C211yesmaximal, characteristic, complemented
5#11751913C211yesmaximal, characteristic, complemented
6#11751907C212yesmaximal, complemented
7#11751895C212yesmaximal, complemented
8#11751889C212yesmaximal, complemented
9C4×#6738C211yesmaximal, characteristic, complemented
10#11751777C211yesmaximal, characteristic, complemented
11#11751754C212yesmaximal, complemented
12#10617155—31nomaximal, Hall
13C2×#6768C411yescharacteristic
14C2×#6739C2212yes—
15C2×#6739C2212yes—
16C2×#6738C2211yescharacteristic
17C2×#6738C2211yescharacteristic
18#91781C411yescharacteristic
19C5×#893C411yescharacteristic
20C3×#4033C2211yescharacteristic
21#91446C412yes—
22Dic5×D12C2211yescharacteristic
23#91405C2212yes—
24C4×#1200C2211yescharacteristic
25#91400C412yes—
26C4×D3×Dic5C2212yes—
27Dic3×Dic10C411yescharacteristic
28#91231C2212yes—
29C4⋊(C3⋊#170)C2212yes—
30C4⋊(C3⋊#170)C2211yescharacteristic
31C4×C3⋊#170C2211yescharacteristic
32C3×#4002C2212yes—
33C3×C4⋊#170C2212yes—
34C3×C4⋊#170C2211yescharacteristic
35C3×C4×#170C2211yescharacteristic
36#91171C2212yes—
37(C2×D3)⋊#170C2212yes—
38(C2×D3)⋊#170C2212yes—
39Dic3×#170C2211yescharacteristic
40Dic3×#170C2212yes—
41Dic3×#170C2211yescharacteristic
42#79938—51nomaximal, Hall
43C4×#711—31no—
44#65774—31no—
45#65774—31no—
46C4×#704—32no—
47#65748—32no—
48#65709—31no—
49#65709D311yescharacteristic, complemented
50#65694—32no—
51#65694—32no—
52#65680—31no—
53C4×D5⋊C8—31no—
54C2×C4×D15C2×C411yescharacteristic
55C2×C5×D12C2×C411yescharacteristic
56C2×C4×C5×D3C2×C412yes—
57C2×C3×Dic10C2×C411yescharacteristic
58C2×#1203C2×C412yes—
59C2×#1200C2×C411yescharacteristic
60C2×#1200C2311yescharacteristic
61C2×D3×Dic5C2×C412yes—
62C2×D3×Dic5C2312yes—
63#6768—24no—
64C2×C3⋊#170C2311yescharacteristic
65C2×C3⋊#170C2312yes—
66C2×C3⋊#170C2311yescharacteristic
67C2×C3×#170C2311yescharacteristic
68C2×C3×#170C2312yes—
69C2×C3×#170C2311yescharacteristic
70#6739—24no—
71#6739—24no—
72#6738—22no—
73#6738—22no—
74C3×C5×C4⋊C4C2×C411yescharacteristic
75#6714C2×C412yes—
76C4⋊Dic15C2×C411yescharacteristic
77C4×Dic15C2×C412yes—
78C5×(C2×D3)⋊C4C2×C412yes—
79C4×C5×Dic3C2×C411yescharacteristic
80C3×#683C2×C412yes—
81C3×C4×Dic5C2312yes—
82C3×C4×Dic5C2311yescharacteristic
83#6668C2×C412yes—
84(C2×D3)⋊Dic5C2×C412yes—
85Dic3×Dic5C2×C412yes—
86Dic3×Dic5C2×C411yescharacteristic
87Dic3×Dic5C2311yescharacteristic
88C4×#239—51no—
89C3×#446—51no—
90#5024—51no—
91#4838—51no—
92#4836—52no—
93#4833—52no—
94#4833—51no—
95#4707—52no—
96#4706—52no—
97#4688—51no—
98C4×D3×C8—51no—
99C2×#711—31no—
100C2×#704—32no—
101C2×#704—32no—
102C2×D5⋊C8—31no—
103C2×D5⋊C8—31no—
104C4×C5×D4—31no—
105D4×Dic5—31no—
106C4×C22⋊D5—32no—
107C2×C4×Dic5—32no—
108#4038—31no—
109#4033C2×D311yescharacteristic
110#4027—32no—
111#4023—32no—
112C42×D5—31no—
113C4×Dic10—31no—
114C22⋊#170—32no—
115C22⋊#170—32no—
116#4003—32no—
117#4002—32no—
118#4002C2×D312yes—
119C4⋊#170—31no—
120C4⋊#170—32no—
121C4⋊#170C2×D311yescharacteristic
122C4⋊#170C2×D312yes—
123C4×#170—31no—
124C4×#170—31no—
125C4×#170—32no—
126C4×#170—31no—
127C4×#170C2×D311yescharacteristic
128#2965—151noSylow
129C22×D15C22×C411yescharacteristic
130C22×C5×D3C22×C412yes—
131C2×C3×C4×C5C22×C412yes—
132C2×C3×C4×C5C22×C411yescharacteristic
133C2×Dic15C22×C412yes—
134C2×Dic15C22×C411yescharacteristic
135C4×D15M1614yescomplemented
136C2×C5×Dic3C22×C411yescharacteristic
137C2×C5×Dic3C22×C411yescharacteristic
138C5×D12—22no—
139C4×C5×D3—24no—
140C2×C3×Dic5C22×C411yescharacteristic
141C2×C3×Dic5C22×C412yes—
142C2×C3×Dic5C2411yescharacteristic
143C3×Dic10—22no—
144#1203—24no—
145#1200—21no—
146#1200—21no—
147#1200—22no—
148D3×Dic5—24no—
149D3×Dic5—22no—
150D3×Dic5—22no—
151C3⋊#170—21no—
152C3⋊#170—22no—
153C3⋊#170#4112yescomplemented
154C3×#170—21no—
155C3×#170—22no—
156C3×#170#4112yescomplemented
157C2×#239—51no—
158C3×C4×Q8—51no—
159Q8×Dic3—51no—
160C2×ΓL(1,𝔽25)—52no—
161C2×ΓL(1,𝔽25)—52no—
162C2×D3×C8—51no—
163C2×D3×C8—51no—
164#893—52no—
165#893AGL(1,𝔽5)11yescharacteristic
166#892—51no—
167#892—52no—
168C4×D12—51no—
169C4×D12—52no—
170C42×D3—52no—
171C42×D3—51no—
172C3×C4⋊C8—52no—
173C3×C4⋊C8—51no—
174C3×#97—52no—
175C3×C4×C8—51no—
176(C2×D3)⋊C8—52no—
177(C2×D3)⋊C8—52no—
178(C2×D3)⋊C8—52no—
179C8×Dic3—51no—
180C8×Dic3—51no—
181C8×Dic3—52no—
182C4⋊(C3⋊C8)—51no—
183C4⋊(C3⋊C8)—52no—
184#803—52no—
185C4×C3⋊C8—51no—
186C2×C5×D4—31no—
187C22×C4×C5—32no—
188C2×C22⋊D5—32no—
189C22×Dic5—32no—
190C22×Dic5—32no—
191#711—64no—
192C2×C4×D5—31no—
193C2×C4×D5—31no—
194C2×C4×D5—31no—
195C2×Dic10C4×D311yescharacteristic
196#704—64no—
197#704—64no—
198C2×#170—31no—
199C2×#170—31no—
200C2×#170—32no—
201C2×#170C22×D311yescharacteristic
202C2×#170C22×D312yes—
203C2×#170C22×D311yescharacteristic
204D5⋊C8—62no—
205D5⋊C8—62no—
206C5×C4⋊C4C4×D311yescharacteristic
207C5×C22⋊C4—32no—
208C42×C5—31no—
209C22⋊Dic5—32no—
210#685—32no—
211C4⋊Dic5—31no—
212#683—32no—
213#683C4×D312yes—
214C4×Dic5—32no—
215C4×Dic5—32no—
216C4×Dic5—31no—
217C4×Dic5—31no—
218C4×Dic5C22×D311yescharacteristic
219C4×Dic5C22×D312yes—
220C4×#41—151no—
221#446—51no—
222#446—151no—
223#436—151no—
224#436—151no—
225#435—152no—
226#435—152no—
227#431—151no—
228#431—152no—
229C4×M16—152no—
230C2×C4×C8—151no—
231C22×C3×C5C23×C411yescharacteristic
232C2×D15—21no—
233C2×D15—21no—
234C2×D15C2×M1612yes—
235C2×C5×D3—22no—
236C2×C5×D3—22no—
237C2×C5×D3—22no—
238C3×C4×C5—22no—
239C3×C4×C5C2×M1612yes—
240Dic15—22no—
241Dic15C2×M1612yes—
242C3×Dic5—22no—
243C3×Dic5—21no—
244C3×Dic5C2×#4111yescharacteristic
245C3×Dic5C2×#4111yescharacteristic
246C5×Dic3—21no—
247C5×Dic3#13112yes—
248C2×C3×Q8—51no—
249#239—104no—
250C2×D12—52no—
251C2×D12C2×AGL(1,𝔽5)11yescharacteristic
252C2×C4×D3—51no—
253C2×C4×D3—52no—
254C2×C4×D3—52no—
255C2×C4×D3—51no—
256C2×C4×D3—51no—
257C2×C4×D3C2×AGL(1,𝔽5)12yes—
258C2×C3×C8—51no—
259C2×C3×C8—51no—
260C2×C3×C8—52no—
261C3×C4⋊C4—52no—
262C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
263C3×C42—51no—
264C3×C42—52no—
265(C2×D3)⋊C4—52no—
266(C2×D3)⋊C4—52no—
267(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
268C4⋊Dic3—51no—
269C4⋊Dic3—52no—
270C4×Dic3—52no—
271C4×Dic3—51no—
272C4×Dic3—52no—
273C4×Dic3—51no—
274C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
275C2×C3⋊C8—51no—
276C2×C3⋊C8—51no—
277C2×C3⋊C8—52no—
278ΓL(1,𝔽25)—104no—
279ΓL(1,𝔽25)—104no—
280D3×C8—102no—
281D3×C8—102no—
282C23×C5—32no—
283C22×D5—31no—
284C5×D4—62no—
285C2×C4×C5—31no—
286C2×C4×C5—31no—
287C2×C4×C5—64no—
288C2×C4×C5C2×C4×D312yes—
289C2×C4×C5C2×C4×D311yescharacteristic
290C22⋊D5—64no—
291C2×Dic5—31no—
292C2×Dic5—32no—
293C2×Dic5—64no—
294C2×Dic5—62no—
295C2×Dic5—62no—
296C2×Dic5C2×C4×D312yes—
297C2×Dic5C2×C4×D311yescharacteristic
298C2×Dic5C23×D311yescharacteristic
299C4×D5—34no—
300C4×D5—62no—
301C4×D5—61no—
302C4×D5—61no—
303Dic10—22no—
304#170—21no—
305#170—22no—
306#170—32no—
307#170—61no—
308#170—62no—
309#170#23912yescomplemented
310C2×#41—151no—
311C2×M16—152no—
312C2×M16—152no—
313C22×C8—151no—
314C22×C8—151no—
315C4×Q8—51no—
316C4×Q8—151no—
317C4×D4—31no—
318C4×D4—152no—
319C4×D4—151no—
320C4×D4—152no—
321#117—152no—
322#117—151no—
323C2×C42—151no—
324C2×C42—152no—
325C4⋊C8—52no—
326C4⋊C8—51no—
327C4⋊C8—152no—
328C4⋊C8—151no—
329C22⋊C8—152no—
330C22⋊C8—152no—
331C22⋊C8—152no—
332#97—52no—
333#97—152no—
334C4×C8—51no—
335C4×C8—151no—
336C4×C8—151no—
337C4×C8—151no—
338C4×C8—152no—
339C2×C3×C5C4×#4111yescharacteristic
340C2×C3×C5C22×M1611yescharacteristic
341C2×C3×C5C2×#13111yescharacteristic
342D15—22no—
343C5×D3—42no—
344C22×D3—51no—
345C22×D3C22×AGL(1,𝔽5)12yes—
346C3×Q8—102no—
347C2×C3×C4—51no—
348C2×C3×C4—52no—
349C2×C3×C4—51no—
350C2×C3×C4C22×AGL(1,𝔽5)12yes—
351C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
352C2×Dic3—51no—
353C2×Dic3—52no—
354C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
355C2×Dic3C22×AGL(1,𝔽5)11yescharacteristic
356D12—22no—
357D12—104no—
358C4×D3—24no—
359C4×D3—54no—
360C4×D3—101no—
361C4×D3—101no—
362C4×D3—104no—
363C4×D3—102no—
364C4×D3—102no—
365C4×D3—102no—
366C3×C8—52no—
367C3×C8—101no—
368C3×C8—102no—
369C3⋊C8—52no—
370C3⋊C8—101no—
371C3⋊C8—102no—
372C22×C5—62no—
373C22×C5—62no—
374C22×C5—62no—
375C22×C5C22×C4×D311yescharacteristic
376C2×D5—32no—
377C2×D5—61no—
378C2×D5—61no—
379C4×C5—22no—
380C4×C5—32no—
381C4×C5—61no—
382C4×C5D3×M1612yes—
383Dic5—22no—
384Dic5—21no—
385Dic5—32no—
386Dic5—62no—
387Dic5C2×#23911yescharacteristic
388Dic5D3×#4111yescharacteristic
389#41—304no—
390C2×Q8—51no—
391C2×D4—31no—
392C2×D4—152no—
393C22×C4—32no—
394C22×C4—151no—
395C22×C4—151no—
396C22×C4—152no—
397C22×C4—152no—
398C22×C4—151no—
399M16—304no—
400M16—304no—
401C2×C8—51no—
402C2×C8—51no—
403C2×C8—52no—
404C2×C8—151no—
405C2×C8—151no—
406C2×C8—152no—
407C2×C8—302no—
408C2×C8—302no—
409C4⋊C4—52no—
410C4⋊C4—152no—
411C4⋊C4—151no—
412C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic
413C22⋊C4—32no—
414C22⋊C4—152no—
415C22⋊C4—152no—
416C42—31no—
417C42—51no—
418C42—52no—
419C42—151no—
420C42—151no—
421C42—152no—
422C42—152no—
423C3×C5#296511yescharacteristic, complemented, Hall
424C22×C3C23×AGL(1,𝔽5)11yescharacteristic
425C2×D3—22no—
426C2×D3—22no—
427C2×D3—22no—
428C2×D3—52no—
429C2×D3—101no—
430C2×D3—101no—
431C3×C4—22no—
432C3×C4—51no—
433C3×C4—51no—
434C3×C4—102no—
435C3×C4—101no—
436C3×C4D5⋊M1612yes—
437Dic3—21no—
438Dic3—52no—
439Dic3—102no—
440Dic3#413112yes—
441C2×C5—122no—
442C2×C5C4×#23911yescharacteristic
443C2×C5C2×D3×M1611yescharacteristic
444C2×C5D3×#13111yescharacteristic
445D5—62no—
446C23—32no—
447C23—151no—
448Q8—102no—
449D4—62no—
450D4—304no—
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—51no—
454C2×C4—51no—
455C2×C4—52no—
456C2×C4—64no—
457C2×C4—151no—
458C2×C4—154no—
459C2×C4—152no—
460C2×C4—302no—
461C2×C4—301no—
462C2×C4—301no—
463C2×C4—302no—
464C2×C4—302no—
465C2×C4—304no—
466C2×C4C2×D3×AGL(1,𝔽5)12yes—
467C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
468C8—52no—
469C8—101no—
470C8—102no—
471C8—152no—
472C8—301no—
473C8—302no—
474C2×C3C2×D5⋊M1611yescharacteristic
475C2×C3C2×#413111yescharacteristic
476C2×C3#41×AGL(1,𝔽5)11yescharacteristic
477D3—42no—
478D3—102no—
479C5#7993811yescharacteristic, complemented, Sylow
480C22—62no—
481C22—62no—
482C22—62no—
483C22—152no—
484C22—301no—
485C22—301no—
486C22C22×D3×AGL(1,𝔽5)11yescharacteristic
487C4—22no—
488C4—32no—
489C4—51no—
490C4—51no—
491C4—61no—
492C4—102no—
493C4—101no—
494C4—152no—
495C4—302no—
496C4#9191512yes—
497C3#1061715511yescharacteristic, complemented, Sylow
498C2—122no—
499C2—302no—
500C2C2×#9191511yescharacteristic
501C2D3×#413111yescharacteristic
502C2AGL(1,𝔽5)×#23911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC3×C5×C4⋊C4C2×C4
prime radicalC2×#6768C4
cosocleC2×C3×Dic5C24
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2965
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. #11752368C2semidirect
3. C3×#65709C2semidirect
4. #11751927C2semidirect
5. #11751913C2semidirect
6. #11751907C2semidirect
7. #11751895C2semidirect
8. #11751889C2semidirect
9. C4×#6738C2semidirect
10. #11751777C2semidirect
11. #11751754C2semidirect
49. #65709D3semidirect
135. C4×D15M16semidirect
153. C3⋊#170#41semidirect
156. C3×#170#41semidirect
309. #170#239semidirect
423. C3×C5#2965semidirect
479. C5#79938semidirect
497. C3#10617155semidirect

Maximal subgroups

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

subgroupindex
#117534062
#117523682
C3×#657092
#117519272
#117519132
#117519072
#117518952
#117518892
C4×#67382
#117517772
#117517542
#106171553
#799385