Finite Group Databasebeta

← #14419518

Subgroups — #14419518

Order 1920.

724 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754415C211yesmaximal, characteristic
2#11752416C211yesmaximal, characteristic, complemented
3C3×C4×#704C211yesmaximal, characteristic, complemented
4#11752037C212yesmaximal, complemented
5#11751987C212yesmaximal, complemented
6#11751894C212yesmaximal, complemented
7#11751872C212yesmaximal, complemented
8#11751800C212yesmaximal, complemented
9#11751774C212yesmaximal, complemented
10#10617266—31nomaximal, Hall
11C2×C3×#704C2212yes—
12C2×#6740C2212yes—
13C2×#6738C2212yes—
14#91819C411yescharacteristic
15C5×#924C411yescharacteristic
16C2×C3×C4×Dic5C2211yescharacteristic
17#91552C2212yes—
18#91546C412yes—
19#91418C2211yescharacteristic
20#91402C2212yes—
21#91388C411yescharacteristic
22#91387C412yes—
23#91375C411yescharacteristic
24Dic5×Dic6C2211yescharacteristic
25#91241C2211yescharacteristic
26#91241C2211yescharacteristic
27#91233C2212yes—
28#91231C2212yes—
29C3×#4002C2212yes—
30C3×C4×#170C2211yescharacteristic
31C3×C4×#170C2211yescharacteristic
32#91177C2212yes—
33Dic3⋊#170C2212yes—
34#91171C2212yes—
35#91171C2212yes—
36#91169C2212yes—
37Dic3×#170C2212yes—
38#78510—51nomaximal, Hall
39#65958—31no—
40#65788—32no—
41#65757—31no—
42C4×#704D311yescharacteristic, complemented
43#65706—32no—
44#65704—32no—
45#65701—32no—
46#65695—32no—
47#65694—32no—
48C22×C3×C4×C5C2×C411yescharacteristic
49C2×#1220C2×C412yes—
50C2×D60C2×C411yescharacteristic
51C2×C5×Dic6C2×C411yescharacteristic
52C22×C3×Dic5C2×C411yescharacteristic
53C22×C3×Dic5C2311yescharacteristic
54C2×#1200C2312yes—
55C2×C3⋊#170C2312yes—
56C2×C3⋊#170C2312yes—
57C3×#704—24no—
58C2×C3×#170C2312yes—
59C2×C3×#170C2312yes—
60#6740—24no—
61#6738—24no—
62#6714C2×C412yes—
63C4⋊Dic15C2×C411yescharacteristic
64C5×#217C2×C412yes—
65C5×C4⋊Dic3C2×C411yescharacteristic
66C5×Dic3⋊C4C2×C412yes—
67C3×C4×Dic5C2×C411yescharacteristic
68C3×C4×Dic5C2×C411yescharacteristic
69C3×C4×Dic5C2311yescharacteristic
70C3×C4×Dic5C2311yescharacteristic
71Dic3⋊Dic5C2×C412yes—
72Dic3⋊Dic5C2×C412yes—
73#6669C2×C412yes—
74#6667C2×C412yes—
75#6667C2×C412yes—
76#6667C2×C412yes—
77Dic3×Dic5C2312yes—
78#5479—51no—
79C3×C4×M16—51no—
80#5128—52no—
81#5127—52no—
82#4926—51no—
83#4706—52no—
84#4701—52no—
85#4690—52no—
86#4689—52no—
87C2×#704C2×D312yes—
88C2×#700—32no—
89C2×D5⋊C8—32no—
90C5×C22⋊Q8—31no—
91#4093—31no—
92Q8×Dic5—31no—
93#4076—31no—
94C2×C4×Dic5C2×D311yescharacteristic
95#4044—32no—
96#4039—31no—
97#4038—32no—
98#4036—31no—
99#4032—32no—
100#4027—32no—
101C22⋊#170—31no—
102C22⋊#170—31no—
103#4004—32no—
104#4004—32no—
105#4003—32no—
106#4003—32no—
107#4002—32no—
108#4002—32no—
109#4002C2×D312yes—
110C4⋊#170—32no—
111C4×#170—32no—
112C4×#170C2×D311yescharacteristic
113C4×#170C2×D311yescharacteristic
114#2967—151noSylow
115C23×C3×C5C22×C411yescharacteristic
116C22×D15C22×C412yes—
117C2×C3×C4×C5—41no—
118C2×C3×C4×C5C22×C411yescharacteristic
119C2×C3×C4×C5C22×C411yescharacteristic
120#1220—42no—
121C2×Dic15C22×C412yes—
122D60—41no—
123C2×C5×Dic3C22×C412yes—
124C2×C5×Dic3C22×C412yes—
125C5×Dic6—41no—
126C2×C3×Dic5—22no—
127C2×C3×Dic5—41no—
128C2×C3×Dic5C22×C411yescharacteristic
129C2×C3×Dic5C22×C411yescharacteristic
130C2×C3×Dic5C22×C411yescharacteristic
131C2×C3×Dic5C2411yescharacteristic
132#1200—22no—
133#1200—22no—
134C3⋊#170—22no—
135C3⋊#170—22no—
136C3×#170—22no—
137C3×#170#4114yes—
138C2×C3×M16—52no—
139C2×C3×C42—51no—
140#930—51no—
141#929—52no—
142C4×#67—52no—
143#924AGL(1,𝔽5)11yescharacteristic
144C2×ΓL(1,𝔽25)—52no—
145C2×D3×C8—52no—
146#876—52no—
147#875—51no—
148C4×D12—51no—
149#872—52no—
150#870—51no—
151C4×Dic6—51no—
152C3×#97—52no—
153C3×C4×C8—51no—
154C3×C4×C8—51no—
155#830—51no—
156#830—51no—
157(C2×D3)⋊C8—52no—
158(C2×D3)⋊C8—52no—
159#815—52no—
160Dic3⋊C8—52no—
161Dic3⋊C8—52no—
162C8×Dic3—52no—
163C4⋊(C3⋊C8)—52no—
164#803—52no—
165C2×C5×Q8—31no—
166C22×C4×C5C4×D311yescharacteristic
167C2×C22⋊D5—32no—
168C22×Dic5C4×D311yescharacteristic
169C22×Dic5C22×D311yescharacteristic
170C2×D20—31no—
171C2×C4×D5—32no—
172#704—24no—
173C2×#170—32no—
174C2×#170—32no—
175C2×#170C22×D312yes—
176C2×#170C22×D312yes—
177#700—64no—
178D5⋊C8—64no—
179C5×C4⋊C4—32no—
180C5×C4⋊C4—31no—
181C5×C22⋊C4—32no—
182#685—32no—
183#685—32no—
184#685—32no—
185#685—32no—
186C4⋊Dic5—31no—
187C4⋊Dic5—32no—
188C4⋊Dic5—32no—
189#683—32no—
190C4×Dic5—32no—
191C4×Dic5C4×D311yescharacteristic
192C4×Dic5C4×D311yescharacteristic
193C4×Dic5C22×D311yescharacteristic
194C4×Dic5C22×D311yescharacteristic
195#529—151no—
196#446—152no—
197#436—152no—
198#435—152no—
199#435—152no—
200#433—152no—
201#432—151no—
202#405—152no—
203C4×M16—51no—
204C22×C3×C5—21no—
205C22×C3×C5—21no—
206C22×C3×C5—21no—
207C22×C3×C5C23×C411yescharacteristic
208C2×D15—22no—
209C2×D15—22no—
210C2×D15—22no—
211C3×C4×C5—21no—
212C3×C4×C5—21no—
213Dic15—22no—
214C3×Dic5—21no—
215C3×Dic5—21no—
216C3×Dic5—21no—
217C3×Dic5C2×#4112yes—
218C5×Dic3—22no—
219C5×Dic3—22no—
220C22×C3×C4—51no—
221C22×C3×C4—51no—
222C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
223C2×#67—52no—
224C2×D12—51no—
225C2×C4×D3—52no—
226C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
227C3×M16—104no—
228C2×C3×C8—52no—
229C2×C3×C8—52no—
230C3×C42—51no—
231C3×C42—51no—
232C3×C42—51no—
233C3×C42—51no—
234#217C2×AGL(1,𝔽5)12yes—
235(C2×D3)⋊C4—52no—
236(C2×D3)⋊C4—52no—
237(C2×D3)⋊C4—52no—
238(C2×D3)⋊C4—52no—
239C4⋊Dic3—51no—
240C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
241Dic3⋊C4—52no—
242Dic3⋊C4—52no—
243Dic3⋊C4—52no—
244Dic3⋊C4C2×AGL(1,𝔽5)12yes—
245C4×Dic3—52no—
246C2×C3⋊C8—52no—
247C2×C3⋊C8—52no—
248ΓL(1,𝔽25)—104no—
249D3×C8—104no—
250C23×C5C2×C4×D311yescharacteristic
251C22×D5—32no—
252C5×Q8—121no—
253C2×C4×C5—32no—
254C2×C4×C5—32no—
255C2×C4×C5—41no—
256C2×C4×C5C2×C4×D311yescharacteristic
257C2×C4×C5C2×C4×D311yescharacteristic
258C22⋊D5—122no—
259C2×Dic5—22no—
260C2×Dic5—32no—
261C2×Dic5—41no—
262C2×Dic5C2×C4×D311yescharacteristic
263C2×Dic5C2×C4×D311yescharacteristic
264C2×Dic5C2×C4×D311yescharacteristic
265C2×Dic5C23×D311yescharacteristic
266D20—121no—
267C4×D5—62no—
268C4×D5—62no—
269#170—22no—
270#170—62no—
271#170—62no—
272#170#23514yes—
273C2×M16—52no—
274C2×M16—152no—
275C22×C8—152no—
276#126—152no—
277#125—151no—
278#124—151no—
279#123—152no—
280C22⋊Q8—31no—
281#121—151no—
282C4×Q8—151no—
283C4×D4—151no—
284C4×D4—152no—
285#117—152no—
286C2×C42—51no—
287C4⋊C8—152no—
288C4⋊C8—152no—
289C4⋊C8—152no—
290C22⋊C8—152no—
291C22⋊C8—152no—
292C22⋊C8—151no—
293C22⋊C8—151no—
294#97—52no—
295#97—152no—
296#97—152no—
297C4×C8—51no—
298C4×C8—51no—
299C4×C8—152no—
300C2×C3×C5—41no—
301C2×C3×C5C4×#4111yescharacteristic
302C2×C3×C5#56811yescharacteristic
303C2×C3×C5#56811yescharacteristic
304D15—42no—
305C23×C3C22×AGL(1,𝔽5)11yescharacteristic
306C22×D3—52no—
307C2×C3×C4—41no—
308C2×C3×C4—51no—
309C2×C3×C4—51no—
310C2×C3×C4—51no—
311C2×C3×C4—51no—
312C2×C3×C4—102no—
313C2×C3×C4—201no—
314C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
315C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
316#67—202no—
317C2×Dic3—52no—
318C2×Dic3C22×AGL(1,𝔽5)12yes—
319C2×Dic3C22×AGL(1,𝔽5)12yes—
320D12—201no—
321C4×D3—102no—
322C4×D3—102no—
323Dic6—41no—
324C3×C8—54no—
325C3×C8—102no—
326C3⋊C8—102no—
327C3⋊C8—102no—
328C22×C5—21no—
329C22×C5—21no—
330C22×C5—21no—
331C22×C5C22×C4×D311yescharacteristic
332C2×D5—62no—
333C2×D5—62no—
334C2×D5—62no—
335C4×C5—21no—
336C4×C5—21no—
337C4×C5—62no—
338C4×C5—62no—
339Dic5—21no—
340Dic5—21no—
341Dic5—21no—
342Dic5—62no—
343Dic5C2×#23512yes—
344C2×Q8—31no—
345C2×D4—152no—
346C2×D4—151no—
347C22×C4—51no—
348C22×C4—51no—
349C22×C4—152no—
350C22×C4D3×AGL(1,𝔽5)11yescharacteristic
351M16—104no—
352M16—304no—
353C2×C8—52no—
354C2×C8—52no—
355C2×C8—152no—
356C2×C8—152no—
357C2×C8—304no—
358C4⋊C4—32no—
359C4⋊C4—31no—
360C4⋊C4—152no—
361C4⋊C4—152no—
362C4⋊C4—152no—
363C4⋊C4—151no—
364C22⋊C4—32no—
365C22⋊C4—152no—
366C22⋊C4—152no—
367C22⋊C4—152no—
368C22⋊C4—152no—
369C42—51no—
370C42—51no—
371C42—51no—
372C42—51no—
373C42—152no—
374C3×C5#296711yescharacteristic, complemented, Hall
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3C23×AGL(1,𝔽5)11yescharacteristic
379C2×D3—102no—
380C2×D3—102no—
381C2×D3—102no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—52no—
385C3×C4—101no—
386C3×C4—101no—
387C3×C4—101no—
388Dic3—22no—
389Dic3—22no—
390Dic3—102no—
391C2×C5—41no—
392C2×C5C4×#23511yescharacteristic
393C2×C5#574511yescharacteristic
394C2×C5#574511yescharacteristic
395D5—122no—
396C23—152no—
397C23C2×D3×AGL(1,𝔽5)11yescharacteristic
398Q8—121no—
399D4—601no—
400D4—602no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—41no—
404C2×C4—51no—
405C2×C4—51no—
406C2×C4—51no—
407C2×C4—51no—
408C2×C4—102no—
409C2×C4—152no—
410C2×C4—201no—
411C2×C4—302no—
412C2×C4—302no—
413C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
414C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
415C8—54no—
416C8—102no—
417C8—302no—
418C8—302no—
419C2×C3—41no—
420C2×C3#6625611yescharacteristic
421C2×C3#6626011yescharacteristic
422C2×C3#41×AGL(1,𝔽5)11yescharacteristic
423D3—202no—
424C5#7851011yescharacteristic, complemented, Sylow
425C22—21no—
426C22—21no—
427C22—21no—
428C22—302no—
429C22—302no—
430C22—302no—
431C22C22×D3×AGL(1,𝔽5)11yescharacteristic
432C4—21no—
433C4—21no—
434C4—52no—
435C4—62no—
436C4—62no—
437C4—101no—
438C4—101no—
439C4—101no—
440C4—302no—
441C3#1061726611yescharacteristic, complemented, Sylow
442C2—41no—
443C2—602no—
444C2#1175682711yescharacteristic
445C2AGL(1,𝔽5)×#23511yescharacteristic
446C2#1175686311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC22×C3×C4×C5C2×C4
prime radical#91819C4
cosocleC2×C3×Dic5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2967
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11752416C2semidirect
3. C3×C4×#704C2semidirect
4. #11752037C2semidirect
5. #11751987C2semidirect
6. #11751894C2semidirect
7. #11751872C2semidirect
8. #11751800C2semidirect
9. #11751774C2semidirect
42. C4×#704D3semidirect
374. C3×C5#2967semidirect
424. C5#78510semidirect
441. C3#10617266semidirect

Maximal subgroups

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

subgroupindex
#117544152
#117524162
C3×C4×#7042
#117520372
#117519872
#117518942
#117518722
#117518002
#117517742
#106172663
#785105