Finite Group Databasebeta

← #14419552

Subgroups — #14419552

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#11752685C211yesmaximal, characteristic
2C4×#6758C211yesmaximal, characteristic, complemented
3C3×#65684C211yesmaximal, characteristic, complemented
4#11751905C212yesmaximal, complemented
5#11751903C212yesmaximal, complemented
6#11751874C212yesmaximal, complemented
7#11751810C212yesmaximal, complemented
8#11751779C212yesmaximal, complemented
9#11751771C212yesmaximal, complemented
10#10617333—31nomaximal, Hall
11C2×#6758C2212yes—
12C2×#6740C2212yes—
13C2×#6738C2212yes—
14#91764C411yescharacteristic
15C5×#870C411yescharacteristic
16C3×C42×D5C2211yescharacteristic
17#91450C411yescharacteristic
18#91444C2212yes—
19#91418C2211yescharacteristic
20#91402C2212yes—
21#91387C412yes—
22#91360C411yescharacteristic
23#91348C412yes—
24#91333C2211yescharacteristic
25#91231C2212yes—
26C4×C3⋊#170C2211yescharacteristic
27C4×C3⋊#170C2211yescharacteristic
28C3×#4004C2212yes—
29C3×#4003C2211yescharacteristic
30C3×#4003C2211yescharacteristic
31C3×#4002C2212yes—
32#91177C2212yes—
33Dic3⋊#170C2212yes—
34#91171C2212yes—
35#91171C2212yes—
36#91169C2212yes—
37Dic3×#170C2212yes—
38#79326—51nomaximal, Hall
39#66025—31no—
40#65707—32no—
41#65706—32no—
42#65705—32no—
43#65704—32no—
44#65704—32no—
45#65701—32no—
46#65684D311yescharacteristic, complemented
47C4×#700—31no—
48C2×D60C2×C411yescharacteristic
49C2×Dic30C2×C411yescharacteristic
50C2×C3×C4×D5C2×C411yescharacteristic
51C2×C3×C4×D5C2×C411yescharacteristic
52C2×C3×C4×D5C2311yescharacteristic
53C2×#1202C2×C412yes—
54C2×#1200C2312yes—
55C2×C3⋊#170C2312yes—
56C2×C3⋊#170C2312yes—
57#6758—24no—
58C2×C3×#170C2312yes—
59C2×C3×#170C2312yes—
60#6740—24no—
61#6738—24no—
62C3×C42×C5C2×C411yescharacteristic
63#6714C2×C412yes—
64#6714C2×C412yes—
65C5×C4⋊Dic3C2×C411yescharacteristic
66C5×C4⋊Dic3C2×C411yescharacteristic
67C5×Dic3⋊C4C2×C412yes—
68C5×Dic3⋊C4C2×C412yes—
69C3×C4×Dic5C2×C411yescharacteristic
70C3×C4×Dic5C2311yescharacteristic
71C3×C4×Dic5C2311yescharacteristic
72Dic3⋊Dic5C2×C412yes—
73#6669C2×C412yes—
74#6667C2×C412yes—
75#6667C2×C412yes—
76#6665C2×C412yes—
77Dic3×Dic5C2312yes—
78#5479—51no—
79C3×#432—51no—
80C4×#208—51no—
81#4836—52no—
82#4828—52no—
83#4729—52no—
84#4728—52no—
85#4709—52no—
86#4708—52no—
87C2×#700—32no—
88C2×#700—32no—
89C2×D5⋊C8—32no—
90C5×#125—31no—
91#4044—32no—
92#4043—31no—
93#4041—31no—
94#4040—32no—
95#4039—32no—
96#4039—31no—
97#4038—32no—
98#4033—31no—
99#4021—31no—
100C42×D5C2×D311yescharacteristic
101#4004—32no—
102#4004C2×D312yes—
103#4003—32no—
104#4003—32no—
105#4003C2×D311yescharacteristic
106#4003C2×D311yescharacteristic
107#4002—32no—
108#4002—32no—
109#4002C2×D312yes—
110C4⋊#170—32no—
111C4×#170—32no—
112C4×#170—31no—
113C4×#170—31no—
114#2967—151noSylow
115C22×D15C22×C412yes—
116C22×C3×D5C22×C411yescharacteristic
117C2×C3×C4×C5C22×C411yescharacteristic
118C2×C3×C4×C5C22×C411yescharacteristic
119C2×C3×C4×C5C22×C411yescharacteristic
120C2×Dic15C22×C412yes—
121D60—41no—
122Dic30—41no—
123C2×C5×Dic3C22×C412yes—
124C2×C5×Dic3C22×C412yes—
125C2×C3×Dic5C22×C411yescharacteristic
126C2×C3×Dic5C22×C411yescharacteristic
127C2×C3×Dic5C2411yescharacteristic
128C3×C4×D5—22no—
129C3×C4×D5—41no—
130C3×C4×D5—41no—
131#1202—42no—
132#1200—22no—
133#1200—22no—
134C3⋊#170—22no—
135C3⋊#170#4114yes—
136C3×#170—22no—
137C3×#170—22no—
138C2×C3×C42—51no—
139#930—51no—
140#929—52no—
141C4×#67—52no—
142#924—51no—
143C2×#208—52no—
144C2×ΓL(1,𝔽25)—52no—
145C2×D3×C8—52no—
146#876—52no—
147#875—51no—
148C4×D12—51no—
149#872—52no—
150#870AGL(1,𝔽5)11yescharacteristic
151C4×Dic6—51no—
152C3×C4⋊C8—52no—
153C3×C22⋊C8—51no—
154C3×C22⋊C8—51no—
155C3×#97—52no—
156(C2×D3)⋊C8—52no—
157(C2×D3)⋊C8—52no—
158#815—52no—
159Dic3⋊C8—52no—
160Dic3⋊C8—52no—
161C8×Dic3—52no—
162#803—52no—
163C4×C3⋊C8—51no—
164C4×C3⋊C8—51no—
165C2×D20—32no—
166C2×D20—31no—
167C2×C4×D5—32no—
168C2×C4×D5C4×D311yescharacteristic
169C2×C4×D5C4×D311yescharacteristic
170C2×C4×D5C22×D311yescharacteristic
171C2×Dic10—31no—
172C2×#170—32no—
173C2×#170—32no—
174C2×#170C22×D312yes—
175C2×#170C22×D312yes—
176#700—64no—
177#700—64no—
178D5⋊C8—64no—
179C5×C4⋊C4—32no—
180C5×C4⋊C4—32no—
181C5×C4⋊C4—31no—
182C5×C4⋊C4—31no—
183C42×C5C4×D311yescharacteristic
184#685—32no—
185#685—32no—
186#685—32no—
187#685—32no—
188#685—32no—
189C4⋊Dic5—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—51no—
202#405—152no—
203C4×M16—151no—
204C22×C3×C5C23×C411yescharacteristic
205C2×D15—22no—
206C2×D15—22no—
207C2×D15—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—
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×C4—51no—
223C2×#67—52no—
224C2×D12—51no—
225C2×C4×D3—52no—
226C2×Dic6—51no—
227C2×C3×C8—52no—
228C2×C3×C8—52no—
229C3×C42—51no—
230C3×C42—51no—
231C3×C42—51no—
232C3×C42C2×AGL(1,𝔽5)11yescharacteristic
233#217—52no—
234(C2×D3)⋊C4—52no—
235(C2×D3)⋊C4—52no—
236(C2×D3)⋊C4—52no—
237(C2×D3)⋊C4—52no—
238C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
239C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
240Dic3⋊C4—52no—
241Dic3⋊C4—52no—
242Dic3⋊C4C2×AGL(1,𝔽5)12yes—
243Dic3⋊C4C2×AGL(1,𝔽5)12yes—
244C4×Dic3—52no—
245#208—104no—
246C2×C3⋊C8—52no—
247C2×C3⋊C8—52no—
248ΓL(1,𝔽25)—104no—
249D3×C8—104no—
250C22×D5—32no—
251C22×D5C2×C4×D311yescharacteristic
252C2×C4×C5—32no—
253C2×C4×C5—32no—
254C2×C4×C5C2×C4×D311yescharacteristic
255C2×C4×C5C2×C4×D311yescharacteristic
256C2×C4×C5C2×C4×D311yescharacteristic
257C2×Dic5—32no—
258C2×Dic5C2×C4×D311yescharacteristic
259C2×Dic5C2×C4×D311yescharacteristic
260C2×Dic5C23×D311yescharacteristic
261D20—121no—
262D20—122no—
263C4×D5—22no—
264C4×D5—41no—
265C4×D5—41no—
266C4×D5—62no—
267C4×D5—62no—
268Dic10—121no—
269#170—22no—
270#170—22no—
271#170—34no—
272#170—62no—
273C2×M16—152no—
274C2×M16—152no—
275C22×C8—152no—
276#126—152no—
277#125—31no—
278#124—151no—
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—52no—
288C4⋊C8—152no—
289C4⋊C8—152no—
290C22⋊C8—51no—
291C22⋊C8—51no—
292C22⋊C8—152no—
293C22⋊C8—152no—
294#97—52no—
295#97—152no—
296#97—152no—
297C4×C8—151no—
298C4×C8—151no—
299C4×C8—152no—
300C2×C3×C5C4×#4111yescharacteristic
301C2×C3×C5#56811yescharacteristic
302C2×C3×C5#56811yescharacteristic
303D15—42no—
304C3×D5—41no—
305C23×C3—51no—
306C22×D3—52no—
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×Dic3C22×AGL(1,𝔽5)12yes—
319C2×Dic3C22×AGL(1,𝔽5)12yes—
320D12—201no—
321C4×D3—102no—
322C4×D3—102no—
323Dic6—201no—
324C3×C8—102no—
325C3×C8—102no—
326C3⋊C8—54no—
327C3⋊C8—102no—
328C22×C5C22×C4×D311yescharacteristic
329C2×D5—21no—
330C2×D5—21no—
331C2×D5—21no—
332C2×D5—62no—
333C2×D5—62no—
334C2×D5—62no—
335C4×C5—21no—
336C4×C5—21no—
337C4×C5—21no—
338C4×C5—62no—
339C4×C5—62no—
340Dic5—21no—
341Dic5—21no—
342Dic5—62no—
343Dic5D3×#4112yes—
344C2×Q8—151no—
345C2×D4—152no—
346C2×D4—151no—
347C22×C4—51no—
348C22×C4—51no—
349C22×C4—51no—
350C22×C4—152no—
351M16—304no—
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—32no—
361C4⋊C4—31no—
362C4⋊C4—152no—
363C4⋊C4—152no—
364C22⋊C4—152no—
365C22⋊C4—152no—
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—102no—
380C2×D3—102no—
381C2×D3—102no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—52no—
386C3×C4—101no—
387C3×C4—101no—
388Dic3—22no—
389Dic3—22no—
390Dic3—102no—
391C2×C5#552211yescharacteristic
392C2×C5#574511yescharacteristic
393C2×C5#574511yescharacteristic
394D5—41no—
395D5—122no—
396C23—51no—
397C23—152no—
398Q8—601no—
399D4—601no—
400D4—602no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—51no—
404C2×C4—51no—
405C2×C4—51no—
406C2×C4—102no—
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—102no—
416C8—102no—
417C8—154no—
418C8—302no—
419C2×C3—201no—
420C2×C3#6626011yescharacteristic
421C2×C3#6626411yescharacteristic
422C2×C3#41×AGL(1,𝔽5)11yescharacteristic
423D3—202no—
424C5#7932611yescharacteristic, complemented, Sylow
425C22—101no—
426C22—101no—
427C22—101no—
428C22—302no—
429C22—302no—
430C22—302no—
431C22C22×D3×AGL(1,𝔽5)11yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—52no—
436C4—62no—
437C4—62no—
438C4—101no—
439C4—101no—
440C4—302no—
441C3#1061733311yescharacteristic, complemented, Sylow
442C2—201no—
443C2—602no—
444C2#1175682311yescharacteristic
445C2#1175682711yescharacteristic
446C2#1175683511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC3×C42×C5C2×C4
prime radical#91450C4
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. C4×#6758C2semidirect
3. C3×#65684C2semidirect
4. #11751905C2semidirect
5. #11751903C2semidirect
6. #11751874C2semidirect
7. #11751810C2semidirect
8. #11751779C2semidirect
9. #11751771C2semidirect
46. #65684D3semidirect
374. C3×C5#2967semidirect
424. C5#79326semidirect
441. C3#10617333semidirect

Maximal subgroups

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

subgroupindex
#117526852
C4×#67582
C3×#656842
#117519052
#117519032
#117518742
#117518102
#117517792
#117517712
#106173333
#793265