Finite Group Databasebeta

← #14423161

Subgroups — #14423161

Order 1920.

692 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11752685C211yesmaximal, characteristic
2#11752342C211yesmaximal, characteristic, complemented
3C3×#65682C211yesmaximal, characteristic, complemented
4#11751905C212yesmaximal, complemented
5#11751902C212yesmaximal, complemented
6#11751886C212yesmaximal, complemented
7#11751781C212yesmaximal, complemented
8#11751771C212yesmaximal, complemented
9#11751763C212yesmaximal, complemented
10#10617342—31nomaximal, Hall
11C2×C3×#700C2211yescharacteristic
12C2×C3×#700C2211yescharacteristic
13C2×#6740C2212yes—
14C2×#6738C2212yes—
15#91764C411yescharacteristic
16C5×#870C411yescharacteristic
17C3×C42×D5C2211yescharacteristic
18#91450C411yescharacteristic
19#91444C2212yes—
20#91418C2211yescharacteristic
21#91402C2212yes—
22#91387C412yes—
23#91360C411yescharacteristic
24#91348C412yes—
25#91333C2211yescharacteristic
26#91232C2212yes—
27#91231C2212yes—
28C4⋊(C3⋊#170)C2211yescharacteristic
29C4⋊(C3⋊#170)C2211yescharacteristic
30C3×#4004C2212yes—
31C3×#4004C2212yes—
32#91177C2212yes—
33Dic3⋊#170C2212yes—
34#91171C2212yes—
35#91171C2212yes—
36#91169C2212yes—
37Dic3×#170C2212yes—
38#79054—51nomaximal, Hall
39#66025—31no—
40#65708—32no—
41#65707—32no—
42#65706—32no—
43#65705—32no—
44#65704—32no—
45#65704—32no—
46#65683—31no—
47#65682D311yescharacteristic, complemented
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—
57C2×C3×#170C2312yes—
58C2×C3×#170C2312yes—
59C3×#700—41no—
60C3×#700—41no—
61#6740—42no—
62#6738—42no—
63C3×C42×C5C2×C411yescharacteristic
64#6714C2×C412yes—
65#6714C2×C412yes—
66C5×C4⋊Dic3C2×C411yescharacteristic
67C5×C4⋊Dic3C2×C411yescharacteristic
68C5×Dic3⋊C4C2×C412yes—
69C5×Dic3⋊C4C2×C412yes—
70C3×C4×Dic5C2×C411yescharacteristic
71C3×C4×Dic5C2311yescharacteristic
72C3×C4×Dic5C2311yescharacteristic
73Dic3⋊Dic5C2×C412yes—
74#6669C2×C412yes—
75#6667C2×C412yes—
76#6667C2×C412yes—
77#6665C2×C412yes—
78Dic3×Dic5C2312yes—
79#5479—51no—
80C3×#423—51no—
81#5128—52no—
82#5127—52no—
83#4926—51no—
84#4840—52no—
85#4839—52no—
86#4836—52no—
87#4828—52no—
88C2×#700—32no—
89C2×#700C2×D311yescharacteristic
90C2×#700C2×D311yescharacteristic
91C2×D5⋊C8—32no—
92C5×#125—31no—
93#4044—32no—
94#4043—31no—
95#4041—31no—
96#4040—32no—
97#4039—32no—
98#4039—31no—
99#4038—32no—
100#4033—31no—
101#4021—31no—
102C42×D5C2×D311yescharacteristic
103#4004—32no—
104#4004C2×D312yes—
105#4004C2×D312yes—
106#4003—32no—
107#4003—32no—
108#4003—32no—
109#4002—32no—
110#4002—32no—
111C4⋊#170—32no—
112C4⋊#170—31no—
113C4⋊#170—31no—
114C4×#170—32no—
115#2995—151noSylow
116C22×D15C22×C412yes—
117C22×C3×D5C22×C411yescharacteristic
118C2×C3×C4×C5C22×C411yescharacteristic
119C2×C3×C4×C5C22×C411yescharacteristic
120C2×C3×C4×C5C22×C411yescharacteristic
121C2×Dic15C22×C412yes—
122D60—41no—
123Dic30—41no—
124C2×C5×Dic3C22×C412yes—
125C2×C5×Dic3C22×C412yes—
126C2×C3×Dic5C22×C411yescharacteristic
127C2×C3×Dic5C22×C411yescharacteristic
128C2×C3×Dic5C2411yescharacteristic
129C3×C4×D5—21no—
130C3×C4×D5—21no—
131C3×C4×D5—41no—
132C3×C4×D5—41no—
133#1202—42no—
134#1200—22no—
135#1200—22no—
136C3⋊#170—22no—
137C3⋊#170—22no—
138C3×#170—22no—
139C3×#170—22no—
140C2×C3×M16—51no—
141C2×C3×M16—51no—
142C2×C3×C42—51no—
143#930—51no—
144#929—52no—
145C4×#67—52no—
146#924—51no—
147C2×ΓL(1,𝔽25)—52no—
148C2×D3×C8—52no—
149#876—52no—
150#875—51no—
151C4×D12—51no—
152#872—52no—
153#870AGL(1,𝔽5)11yescharacteristic
154C4×Dic6—51no—
155C3×C4⋊C8—52no—
156C3×C4⋊C8—52no—
157#830—52no—
158(C2×D3)⋊C8—52no—
159(C2×D3)⋊C8—52no—
160#815—52no—
161Dic3⋊C8—52no—
162Dic3⋊C8—52no—
163C8×Dic3—52no—
164C4⋊(C3⋊C8)—51no—
165C4⋊(C3⋊C8)—51no—
166#803—52no—
167C2×D20—32no—
168C2×D20—31no—
169C2×C4×D5—32no—
170C2×C4×D5C4×D311yescharacteristic
171C2×C4×D5C4×D311yescharacteristic
172C2×C4×D5C22×D311yescharacteristic
173C2×Dic10—31no—
174C2×#170—32no—
175C2×#170—32no—
176C2×#170C22×D312yes—
177C2×#170C22×D312yes—
178#700—41no—
179#700—41no—
180#700—122no—
181D5⋊C8—122no—
182C5×C4⋊C4—32no—
183C5×C4⋊C4—32no—
184C5×C4⋊C4—31no—
185C5×C4⋊C4—31no—
186C42×C5C4×D311yescharacteristic
187#685—32no—
188#685—32no—
189#685—32no—
190#685—32no—
191#685—32no—
192C4⋊Dic5—32no—
193#683—32no—
194C4×Dic5—32no—
195C4×Dic5C4×D311yescharacteristic
196C4×Dic5C22×D311yescharacteristic
197C4×Dic5C22×D311yescharacteristic
198#529—151no—
199#446—152no—
200#436—152no—
201#435—152no—
202#435—152no—
203#433—152no—
204#432—151no—
205#424—152no—
206#423—51no—
207C22×C3×C5C23×C411yescharacteristic
208C2×D15—22no—
209C2×D15—22no—
210C2×D15—22no—
211C2×C3×D5—21no—
212C2×C3×D5—21no—
213C2×C3×D5—21no—
214C3×C4×C5—21no—
215C3×C4×C5—21no—
216C3×C4×C5—21no—
217Dic15—22no—
218C3×Dic5—21no—
219C3×Dic5—21no—
220C3×Dic5H(5,𝔽2)11yescharacteristic
221C3×Dic52−1+411yescharacteristic
222C5×Dic3—22no—
223C5×Dic3—22no—
224C22×C3×C4—51no—
225C22×C3×C4—51no—
226C22×C3×C4—51no—
227C2×#67—52no—
228C2×D12—51no—
229C2×C4×D3—52no—
230C2×Dic6—51no—
231C3×M16—201no—
232C3×M16—201no—
233C2×C3×C8—52no—
234C2×C3×C8—52no—
235C3×C42—51no—
236C3×C42—51no—
237C3×C42—51no—
238C3×C42C2×AGL(1,𝔽5)11yescharacteristic
239#217—52no—
240(C2×D3)⋊C4—52no—
241(C2×D3)⋊C4—52no—
242(C2×D3)⋊C4—52no—
243(C2×D3)⋊C4—52no—
244C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
245C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
246Dic3⋊C4—52no—
247Dic3⋊C4—52no—
248Dic3⋊C4C2×AGL(1,𝔽5)12yes—
249Dic3⋊C4C2×AGL(1,𝔽5)12yes—
250C4×Dic3—52no—
251C2×C3⋊C8—52no—
252C2×C3⋊C8—52no—
253ΓL(1,𝔽25)—202no—
254D3×C8—202no—
255C22×D5—32no—
256C22×D5C2×C4×D311yescharacteristic
257C2×C4×C5—32no—
258C2×C4×C5—32no—
259C2×C4×C5C2×C4×D311yescharacteristic
260C2×C4×C5C2×C4×D311yescharacteristic
261C2×C4×C5C2×C4×D311yescharacteristic
262C2×Dic5—32no—
263C2×Dic5C2×C4×D311yescharacteristic
264C2×Dic5C2×C4×D311yescharacteristic
265C2×Dic5C23×D311yescharacteristic
266D20—121no—
267D20—122no—
268C4×D5—21no—
269C4×D5—21no—
270C4×D5—41no—
271C4×D5—41no—
272C4×D5—62no—
273C4×D5—62no—
274Dic10—121no—
275#170—22no—
276#170—22no—
277#170—62no—
278#170—62no—
279C2×M16—51no—
280C2×M16—51no—
281C2×M16—152no—
282C22×C8—152no—
283#126—152no—
284#125—31no—
285#124—151no—
286#123—152no—
287C22⋊Q8—151no—
288#121—151no—
289C4×Q8—151no—
290C4×D4—151no—
291C4×D4—152no—
292#117—152no—
293C2×C42—51no—
294C4⋊C8—52no—
295C4⋊C8—52no—
296C4⋊C8—151no—
297C4⋊C8—151no—
298C4⋊C8—152no—
299C4⋊C8—152no—
300C22⋊C8—152no—
301C22⋊C8—152no—
302C22⋊C8—152no—
303#97—152no—
304#97—152no—
305C4×C8—152no—
306C2×C3×C5#52011yescharacteristic
307C2×C3×C5#56811yescharacteristic
308C2×C3×C5#56811yescharacteristic
309D15—42no—
310C3×D5—41no—
311C23×C3—51no—
312C22×D3—52no—
313C2×C3×C4—51no—
314C2×C3×C4—51no—
315C2×C3×C4—51no—
316C2×C3×C4—101no—
317C2×C3×C4—101no—
318C2×C3×C4—201no—
319C2×C3×C4—201no—
320C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
321C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
322C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
323#67—202no—
324C2×Dic3—52no—
325C2×Dic3C22×AGL(1,𝔽5)12yes—
326C2×Dic3C22×AGL(1,𝔽5)12yes—
327D12—201no—
328C4×D3—102no—
329C4×D3—102no—
330Dic6—201no—
331C3×C8—102no—
332C3×C8—102no—
333C3⋊C8—102no—
334C3⋊C8—102no—
335C22×C5C22×C4×D311yescharacteristic
336C2×D5—21no—
337C2×D5—21no—
338C2×D5—21no—
339C2×D5—62no—
340C2×D5—62no—
341C2×D5—62no—
342C4×C5—21no—
343C4×C5—21no—
344C4×C5—21no—
345C4×C5—62no—
346C4×C5—62no—
347Dic5—21no—
348Dic5—21no—
349Dic5—62no—
350Dic5#100411yescharacteristic
351Dic5#100711yescharacteristic
352C2×Q8—151no—
353C2×D4—152no—
354C2×D4—151no—
355C22×C4—51no—
356C22×C4—51no—
357C22×C4—51no—
358C22×C4—152no—
359M16—201no—
360M16—201no—
361M16—602no—
362C2×C8—52no—
363C2×C8—52no—
364C2×C8—152no—
365C2×C8—152no—
366C2×C8—602no—
367C4⋊C4—32no—
368C4⋊C4—31no—
369C4⋊C4—32no—
370C4⋊C4—31no—
371C4⋊C4—152no—
372C4⋊C4—152no—
373C22⋊C4—152no—
374C22⋊C4—152no—
375C22⋊C4—152no—
376C22⋊C4—152no—
377C22⋊C4—152no—
378C42—51no—
379C42—51no—
380C42—51no—
381C42—152no—
382C42D3×AGL(1,𝔽5)11yescharacteristic
383C3×C5#299511yescharacteristic, complemented, Hall
384C22×C3—101no—
385C22×C3—101no—
386C22×C3—101no—
387C22×C3C23×AGL(1,𝔽5)11yescharacteristic
388C2×D3—102no—
389C2×D3—102no—
390C2×D3—102no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—51no—
395C3×C4—51no—
396C3×C4—101no—
397C3×C4—101no—
398Dic3—22no—
399Dic3—22no—
400Dic3—102no—
401C2×C5#550411yescharacteristic
402C2×C5#574511yescharacteristic
403C2×C5#574511yescharacteristic
404D5—41no—
405D5—122no—
406C23—51no—
407C23—152no—
408Q8—601no—
409D4—601no—
410D4—602no—
411C2×C4—32no—
412C2×C4—32no—
413C2×C4—51no—
414C2×C4—51no—
415C2×C4—51no—
416C2×C4—101no—
417C2×C4—101no—
418C2×C4—152no—
419C2×C4—201no—
420C2×C4—201no—
421C2×C4—302no—
422C2×C4—302no—
423C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
424C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
425C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
426C8—102no—
427C8—102no—
428C8—302no—
429C8—302no—
430C2×C3—201no—
431C2×C3#6626011yescharacteristic
432C2×C3#6626411yescharacteristic
433C2×C3#6626611yescharacteristic
434D3—202no—
435C5#7905411yescharacteristic, complemented, Sylow
436C22—101no—
437C22—101no—
438C22—101no—
439C22—302no—
440C22—302no—
441C22—302no—
442C22C22×D3×AGL(1,𝔽5)11yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—51no—
447C4—51no—
448C4—62no—
449C4—62no—
450C4—101no—
451C4—101no—
452C4—302no—
453C3#1061734211yescharacteristic, complemented, Sylow
454C2—201no—
455C2—602no—
456C2#1175682311yescharacteristic
457C2#1175682711yescharacteristic
458C2#1175683611yescharacteristic
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#2995
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11752342C2semidirect
3. C3×#65682C2semidirect
4. #11751905C2semidirect
5. #11751902C2semidirect
6. #11751886C2semidirect
7. #11751781C2semidirect
8. #11751771C2semidirect
9. #11751763C2semidirect
47. #65682D3semidirect
383. C3×C5#2995semidirect
435. C5#79054semidirect
453. C3#10617342semidirect

Maximal subgroups

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

subgroupindex
#117526852
#117523422
C3×#656822
#117519052
#117519022
#117518862
#117517812
#117517712
#117517632
#106173423
#790545