Finite Group Databasebeta

← #14419555

Subgroups — #14419555

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#11752683C211yesmaximal, characteristic
2#11752343C211yesmaximal, characteristic, complemented
3C3×C4×#700C211yesmaximal, characteristic, complemented
4#11751910C212yesmaximal, complemented
5#11751891C212yesmaximal, complemented
6#11751885C212yesmaximal, complemented
7#11751868C212yesmaximal, complemented
8#11751798C212yesmaximal, complemented
9#11751774C212yesmaximal, complemented
10#10617313—31nomaximal, Hall
11C2×C3×#700C2212yes—
12C2×#6739C2212yes—
13C2×D3×#170C2212yes—
14#91759C411yescharacteristic
15C5×#875C411yescharacteristic
16C3×C42×D5C2211yescharacteristic
17#91449C411yescharacteristic
18#91447C412yes—
19#91440C2212yes—
20Dic5×D12C2211yescharacteristic
21#91398C2212yes—
22#91372C412yes—
23#91359C411yescharacteristic
24Dic5×Dic6C2211yescharacteristic
25#91233C2212yes—
26#91232C2211yescharacteristic
27#91232C2211yescharacteristic
28#91231C2212yes—
29C3×#4002C2212yes—
30C3×C4×#170C2211yescharacteristic
31C3×C4×#170C2211yescharacteristic
32#91177C2212yes—
33Dic3⋊#170C2212yes—
34(C2×D3)⋊#170C2212yes—
35(C2×D3)⋊#170C2212yes—
36#91169C2212yes—
37Dic3×#170C2212yes—
38#78510—51nomaximal, Hall
39#66014—31no—
40#65788—32no—
41#65773—32no—
42#65697—32no—
43#65695—32no—
44#65693—32no—
45#65691—32no—
46#65684—31no—
47C4×#700D311yescharacteristic, complemented
48C2×C5×D12C2×C411yescharacteristic
49C2×C5×Dic6C2×C411yescharacteristic
50C2×C3×C4×D5C2×C411yescharacteristic
51C2×C3×C4×D5C2×C411yescharacteristic
52C2×C3×C4×D5C2311yescharacteristic
53C2×#1201C2×C412yes—
54C2×D3×Dic5C2312yes—
55C2×C3⋊#170C2312yes—
56C2×C3⋊#170C2312yes—
57C2×C3×#170C2312yes—
58C2×C3×#170C2312yes—
59C3×#700—24no—
60#6739—24no—
61D3×#170—24no—
62C3×C42×C5C2×C411yescharacteristic
63C4⋊Dic15C2×C411yescharacteristic
64C4⋊Dic15C2×C411yescharacteristic
65#6712C2×C412yes—
66#6712C2×C412yes—
67C5×(C2×D3)⋊C4C2×C412yes—
68C5×(C2×D3)⋊C4C2×C412yes—
69C3×C4×Dic5C2×C411yescharacteristic
70C3×C4×Dic5C2311yescharacteristic
71C3×C4×Dic5C2311yescharacteristic
72Dic3⋊Dic5C2×C412yes—
73#6669C2×C412yes—
74(C2×D3)⋊Dic5C2×C412yes—
75(C2×D3)⋊Dic5C2×C412yes—
76#6665C2×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×#704—32no—
88C22×#170—32no—
89C2×#700C2×D312yes—
90C5×#124—31no—
91#4092—31no—
92Q8×Dic5—31no—
93#4084—31no—
94D4×Dic5—31no—
95#4029—32no—
96#4027—32no—
97#4025—32no—
98#4023—32no—
99C42×D5C2×D311yescharacteristic
100#4016—31no—
101C22⋊#170—32no—
102C22⋊#170—32no—
103#4004—32no—
104#4004—32no—
105#4003—31no—
106#4003—31no—
107#4002—32no—
108#4002—32no—
109#4002C2×D312yes—
110C4⋊#170—32no—
111C4×#170—32no—
112C4×#170C2×D311yescharacteristic
113C4×#170C2×D311yescharacteristic
114#2967—151noSylow
115C22×C5×D3C22×C412yes—
116C22×C3×D5C22×C411yescharacteristic
117C2×C3×C4×C5C22×C411yescharacteristic
118C2×C3×C4×C5C22×C411yescharacteristic
119C2×C3×C4×C5C22×C411yescharacteristic
120C2×Dic15C22×C412yes—
121C2×Dic15C22×C412yes—
122C2×C5×Dic3C22×C412yes—
123C5×D12—41no—
124C5×Dic6—41no—
125C2×C3×Dic5C22×C411yescharacteristic
126C2×C3×Dic5C22×C411yescharacteristic
127C2×C3×Dic5C2411yescharacteristic
128C3×C4×D5—22no—
129C3×C4×D5—41no—
130C3×C4×D5—41no—
131#1201—42no—
132D3×Dic5—22no—
133D3×Dic5—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#924—51no—
144C2×ΓL(1,𝔽25)—52no—
145C2×D3×C8—52no—
146#876—52no—
147#875AGL(1,𝔽5)11yescharacteristic
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—
166C2×C5×D4—31no—
167C2×C22⋊D5—32no—
168C22×Dic5—32no—
169C2×C4×D5C4×D311yescharacteristic
170C2×C4×D5C4×D311yescharacteristic
171C2×C4×D5C22×D311yescharacteristic
172#704—64no—
173C2×#170—32no—
174C2×#170—32no—
175C2×#170—64no—
176C2×#170C22×D312yes—
177C2×#170C22×D312yes—
178#700—24no—
179C5×C22⋊C4—32no—
180C5×C22⋊C4—32no—
181C42×C5C4×D311yescharacteristic
182C22⋊Dic5—32no—
183C22⋊Dic5—32no—
184#685—32no—
185C4⋊Dic5—31no—
186C4⋊Dic5—31no—
187C4⋊Dic5—32no—
188#683—32no—
189#683—32no—
190#683—32no—
191C4×Dic5—32no—
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×C5C23×C411yescharacteristic
205C2×C5×D3—22no—
206C2×C5×D3—22no—
207C2×C5×D3—22no—
208C2×C3×D5—21no—
209C2×C3×D5—21no—
210C2×C3×D5—21no—
211C3×C4×C5—21no—
212C3×C4×C5—21no—
213C3×C4×C5—21no—
214Dic15—22no—
215Dic15—22no—
216C3×Dic5—21no—
217C3×Dic5—21no—
218C3×Dic5C2×#4112yes—
219C5×Dic3—22no—
220C22×C3×C4—51no—
221C22×C3×C4—51no—
222C22×C3×C4—51no—
223C2×#67—52no—
224C2×D12C2×AGL(1,𝔽5)11yescharacteristic
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×C42C2×AGL(1,𝔽5)11yescharacteristic
234#217—52no—
235(C2×D3)⋊C4—52no—
236(C2×D3)⋊C4—52no—
237(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
238(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
239C4⋊Dic3—51no—
240C4⋊Dic3—51no—
241Dic3⋊C4—52no—
242Dic3⋊C4—52no—
243Dic3⋊C4—52no—
244Dic3⋊C4—52no—
245C4×Dic3—52no—
246C2×C3⋊C8—52no—
247C2×C3⋊C8—52no—
248ΓL(1,𝔽25)—104no—
249D3×C8—104no—
250C23×C5—32no—
251C22×D5C2×C4×D311yescharacteristic
252C5×Q8—121no—
253C5×D4—121no—
254C2×C4×C5—32no—
255C2×C4×C5C2×C4×D311yescharacteristic
256C2×C4×C5C2×C4×D311yescharacteristic
257C2×C4×C5C2×C4×D311yescharacteristic
258C22⋊D5—122no—
259C2×Dic5—32no—
260C2×Dic5—32no—
261C2×Dic5—62no—
262C2×Dic5—62no—
263C2×Dic5C2×C4×D311yescharacteristic
264C2×Dic5C2×C4×D311yescharacteristic
265C2×Dic5C23×D311yescharacteristic
266C4×D5—22no—
267C4×D5—41no—
268C4×D5—41no—
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—31no—
279#123—152no—
280C22⋊Q8—151no—
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×C5C4×#4111yescharacteristic
301C2×C3×C5#56811yescharacteristic
302C2×C3×C5#56811yescharacteristic
303C3×D5—41no—
304C5×D3—42no—
305C23×C3—51no—
306C22×D3C22×AGL(1,𝔽5)12yes—
307C2×C3×C4—51no—
308C2×C3×C4—51no—
309C2×C3×C4—51no—
310C2×C3×C4—102no—
311C2×C3×C4—201no—
312C2×C3×C4—201no—
313C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
314C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
315C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
316#67—202no—
317C2×Dic3—52no—
318C2×Dic3—52no—
319C2×Dic3C22×AGL(1,𝔽5)12yes—
320D12—41no—
321C4×D3—102no—
322C4×D3—102no—
323Dic6—41no—
324C3×C8—54no—
325C3×C8—102no—
326C3⋊C8—102no—
327C3⋊C8—102no—
328C22×C5—62no—
329C22×C5—62no—
330C22×C5—62no—
331C22×C5C22×C4×D311yescharacteristic
332C2×D5—21no—
333C2×D5—21no—
334C2×D5—21no—
335C4×C5—21no—
336C4×C5—21no—
337C4×C5—21no—
338C4×C5—62no—
339Dic5—21no—
340Dic5—21no—
341Dic5—62no—
342Dic5—62no—
343Dic5C2×#23512yes—
344C2×Q8—31no—
345C2×D4—31no—
346C2×D4—152no—
347C22×C4—51no—
348C22×C4—51no—
349C22×C4—51no—
350C22×C4—152no—
351M16—104no—
352M16—304no—
353C2×C8—52no—
354C2×C8—52no—
355C2×C8—152no—
356C2×C8—152no—
357C2×C8—304no—
358C4⋊C4—152no—
359C4⋊C4—152no—
360C4⋊C4—152no—
361C4⋊C4—151no—
362C4⋊C4—152no—
363C4⋊C4—151no—
364C22⋊C4—32no—
365C22⋊C4—32no—
366C22⋊C4—152no—
367C22⋊C4—152no—
368C22⋊C4—152no—
369C42—51no—
370C42—51no—
371C42—51no—
372C42—152no—
373C42D3×AGL(1,𝔽5)11yescharacteristic
374C3×C5#296711yescharacteristic, complemented, Hall
375C22×C3—101no—
376C22×C3—101no—
377C22×C3—101no—
378C22×C3C23×AGL(1,𝔽5)11yescharacteristic
379C2×D3—22no—
380C2×D3—22no—
381C2×D3—22no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—52no—
386C3×C4—101no—
387C3×C4—101no—
388Dic3—22no—
389Dic3—102no—
390Dic3—102no—
391C2×C5—122no—
392C2×C5C4×#23511yescharacteristic
393C2×C5#574511yescharacteristic
394C2×C5#574511yescharacteristic
395D5—41no—
396C23—32no—
397C23—51no—
398Q8—121no—
399D4—121no—
400D4—602no—
401C2×C4—32no—
402C2×C4—51no—
403C2×C4—51no—
404C2×C4—51no—
405C2×C4—102no—
406C2×C4—152no—
407C2×C4—152no—
408C2×C4—201no—
409C2×C4—201no—
410C2×C4—302no—
411C2×C4—302no—
412C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
413C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
414C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
415C8—54no—
416C8—102no—
417C8—302no—
418C8—302no—
419C2×C3—201no—
420C2×C3#6625611yescharacteristic
421C2×C3#6626411yescharacteristic
422C2×C3#41×AGL(1,𝔽5)11yescharacteristic
423D3—42no—
424C5#7851011yescharacteristic, complemented, Sylow
425C22—62no—
426C22—62no—
427C22—62no—
428C22—101no—
429C22—101no—
430C22—101no—
431C22C22×D3×AGL(1,𝔽5)11yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—52no—
436C4—62no—
437C4—101no—
438C4—101no—
439C4—302no—
440C4—302no—
441C3#1061731311yescharacteristic, complemented, Sylow
442C2—122no—
443C2—201no—
444C2#1175682111yescharacteristic
445C2#1175682211yescharacteristic
446C2AGL(1,𝔽5)×#23511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC3×C42×C5C2×C4
prime radical#91449C4
cosocleC2×C3×Dic5C24
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#2967
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11752343C2semidirect
3. C3×C4×#700C2semidirect
4. #11751910C2semidirect
5. #11751891C2semidirect
6. #11751885C2semidirect
7. #11751868C2semidirect
8. #11751798C2semidirect
9. #11751774C2semidirect
47. C4×#700D3semidirect
374. C3×C5#2967semidirect
424. C5#78510semidirect
441. C3#10617313semidirect

Maximal subgroups

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

subgroupindex
#117526832
#117523432
C3×C4×#7002
#117519102
#117518912
#117518852
#117518682
#117517982
#117517742
#106173133
#785105