Finite Group Databasebeta

← #14423162

Subgroups — #14423162

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#11752683C211yesmaximal, characteristic
2#11752341C211yesmaximal, characteristic, complemented
3C3×#65683C211yesmaximal, characteristic, complemented
4#11751899C212yesmaximal, complemented
5#11751891C212yesmaximal, complemented
6#11751887C212yesmaximal, complemented
7#11751799C212yesmaximal, complemented
8#11751774C212yesmaximal, complemented
9#11751773C212yesmaximal, complemented
10#10617322—31nomaximal, Hall
11C2×#6758C2211yescharacteristic
12C2×#6758C2211yescharacteristic
13C2×#6739C2212yes—
14C2×D3×#170C2212yes—
15#91759C411yescharacteristic
16C5×#875C411yescharacteristic
17C3×C42×D5C2211yescharacteristic
18#91449C411yescharacteristic
19#91447C412yes—
20#91440C2212yes—
21Dic5×D12C2211yescharacteristic
22#91398C2212yes—
23#91372C412yes—
24#91359C411yescharacteristic
25Dic5×Dic6C2211yescharacteristic
26#91233C2212yes—
27#91233C2212yes—
28C3×#4003C2212yes—
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#79342—51nomaximal, Hall
39#66014—31no—
40#65788—32no—
41#65773—32no—
42#65697—32no—
43#65696—32no—
44#65695—32no—
45#65693—32no—
46#65683D311yescharacteristic, complemented
47#65682—31no—
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—
57#6758—41no—
58#6758—41no—
59C2×C3×#170C2312yes—
60C2×C3×#170C2312yes—
61#6739—42no—
62D3×#170—42no—
63C3×C42×C5C2×C411yescharacteristic
64C4⋊Dic15C2×C411yescharacteristic
65C4⋊Dic15C2×C411yescharacteristic
66#6712C2×C412yes—
67#6712C2×C412yes—
68C5×(C2×D3)⋊C4C2×C412yes—
69C5×(C2×D3)⋊C4C2×C412yes—
70C3×C4×Dic5C2×C411yescharacteristic
71C3×C4×Dic5C2311yescharacteristic
72C3×C4×Dic5C2311yescharacteristic
73Dic3⋊Dic5C2×C412yes—
74#6669C2×C412yes—
75(C2×D3)⋊Dic5C2×C412yes—
76(C2×D3)⋊Dic5C2×C412yes—
77#6665C2×C412yes—
78Dic3×Dic5C2312yes—
79#5479—51no—
80C3×#432—51no—
81#4924—51no—
82#4840—52no—
83#4839—52no—
84#4729—52no—
85#4728—52no—
86#4706—52no—
87#4701—52no—
88C2×#704—32no—
89C22×#170—32no—
90C2×#700—31no—
91C2×#700—31no—
92C5×#124—31no—
93#4092—31no—
94Q8×Dic5—31no—
95#4084—31no—
96D4×Dic5—31no—
97#4029—32no—
98#4027—32no—
99#4025—32no—
100#4023—32no—
101C42×D5C2×D311yescharacteristic
102#4016—31no—
103C22⋊#170—32no—
104C22⋊#170—32no—
105#4004—32no—
106#4004—32no—
107#4004—32no—
108#4003C2×D312yes—
109#4002—32no—
110#4002C2×D312yes—
111C4⋊#170—32no—
112C4⋊#170C2×D311yescharacteristic
113C4⋊#170C2×D311yescharacteristic
114C4×#170—32no—
115#2995—151noSylow
116C22×C5×D3C22×C412yes—
117C22×C3×D5C22×C411yescharacteristic
118C2×C3×C4×C5C22×C411yescharacteristic
119C2×C3×C4×C5C22×C411yescharacteristic
120C2×C3×C4×C5C22×C411yescharacteristic
121C2×Dic15C22×C412yes—
122C2×Dic15C22×C412yes—
123C2×C5×Dic3C22×C412yes—
124C5×D12—41no—
125C5×Dic6—41no—
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#1201—42no—
134D3×Dic5—22no—
135D3×Dic5—22no—
136C3⋊#170—22no—
137C3⋊#170—22no—
138C3×#170—22no—
139C3×#170—22no—
140C2×C3×C42—51no—
141#930—51no—
142#929—52no—
143C4×#67—52no—
144#924—51no—
145C2×#208—51no—
146C2×#208—51no—
147C2×ΓL(1,𝔽25)—52no—
148C2×D3×C8—52no—
149#876—52no—
150#875AGL(1,𝔽5)11yescharacteristic
151C4×D12—51no—
152#872—52no—
153#870—51no—
154C4×Dic6—51no—
155C3×C4⋊C8—51no—
156C3×C4⋊C8—51no—
157C3×C22⋊C8—52no—
158C3×#97—52no—
159(C2×D3)⋊C8—52no—
160(C2×D3)⋊C8—52no—
161#815—52no—
162Dic3⋊C8—52no—
163Dic3⋊C8—52no—
164C8×Dic3—52no—
165C4⋊(C3⋊C8)—52no—
166C4⋊(C3⋊C8)—52no—
167C2×C5×Q8—31no—
168C2×C5×D4—31no—
169C2×C22⋊D5—32no—
170C22×Dic5—32no—
171C2×C4×D5C4×D311yescharacteristic
172C2×C4×D5C4×D311yescharacteristic
173C2×C4×D5C22×D311yescharacteristic
174#704—122no—
175C2×#170—32no—
176C2×#170—32no—
177C2×#170—122no—
178C2×#170C22×D312yes—
179C2×#170C22×D312yes—
180#700—121no—
181#700—121no—
182C5×C22⋊C4—32no—
183C5×C22⋊C4—32no—
184C42×C5C4×D311yescharacteristic
185C22⋊Dic5—32no—
186C22⋊Dic5—32no—
187#685—32no—
188C4⋊Dic5—31no—
189C4⋊Dic5—31no—
190C4⋊Dic5—32no—
191#683—32no—
192#683—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—51no—
205#424—152no—
206#423—151no—
207C22×C3×C5C23×C411yescharacteristic
208C2×C5×D3—22no—
209C2×C5×D3—22no—
210C2×C5×D3—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—
218Dic15—22no—
219C3×Dic5—21no—
220C3×Dic5—21no—
221C3×Dic5H(5,𝔽2)11yescharacteristic
222C3×Dic52−1+411yescharacteristic
223C5×Dic3—22no—
224C22×C3×C4—51no—
225C22×C3×C4—51no—
226C22×C3×C4—51no—
227C2×#67—52no—
228C2×D12C2×AGL(1,𝔽5)11yescharacteristic
229C2×C4×D3—52no—
230C2×Dic6C2×AGL(1,𝔽5)11yescharacteristic
231C2×C3×C8—52no—
232C2×C3×C8—52no—
233C3×C42—51no—
234C3×C42—51no—
235C3×C42—51no—
236C3×C42C2×AGL(1,𝔽5)11yescharacteristic
237#217—52no—
238(C2×D3)⋊C4—52no—
239(C2×D3)⋊C4—52no—
240(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
241(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
242C4⋊Dic3—51no—
243C4⋊Dic3—51no—
244Dic3⋊C4—52no—
245Dic3⋊C4—52no—
246Dic3⋊C4—52no—
247Dic3⋊C4—52no—
248C4×Dic3—52no—
249#208—201no—
250#208—201no—
251C2×C3⋊C8—52no—
252C2×C3⋊C8—52no—
253ΓL(1,𝔽25)—202no—
254D3×C8—202no—
255C23×C5—32no—
256C22×D5C2×C4×D311yescharacteristic
257C5×Q8—121no—
258C5×D4—121no—
259C2×C4×C5—32no—
260C2×C4×C5C2×C4×D311yescharacteristic
261C2×C4×C5C2×C4×D311yescharacteristic
262C2×C4×C5C2×C4×D311yescharacteristic
263C22⋊D5—122no—
264C2×Dic5—32no—
265C2×Dic5—32no—
266C2×Dic5—62no—
267C2×Dic5—62no—
268C2×Dic5C2×C4×D311yescharacteristic
269C2×Dic5C2×C4×D311yescharacteristic
270C2×Dic5C23×D311yescharacteristic
271C4×D5—21no—
272C4×D5—21no—
273C4×D5—41no—
274C4×D5—41no—
275#170—22no—
276#170—22no—
277#170—62no—
278#170—62no—
279C2×M16—152no—
280C2×M16—151no—
281C2×M16—151no—
282C22×C8—152no—
283#126—152no—
284#125—151no—
285#124—31no—
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—51no—
295C4⋊C8—51no—
296C4⋊C8—152no—
297C4⋊C8—152no—
298C4⋊C8—152no—
299C4⋊C8—152no—
300C22⋊C8—52no—
301C22⋊C8—152no—
302C22⋊C8—152no—
303#97—52no—
304#97—152no—
305C4×C8—152no—
306C2×C3×C5#52011yescharacteristic
307C2×C3×C5#56811yescharacteristic
308C2×C3×C5#56811yescharacteristic
309C3×D5—41no—
310C5×D3—42no—
311C23×C3—51no—
312C22×D3C22×AGL(1,𝔽5)12yes—
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×Dic3—52no—
326C2×Dic3C22×AGL(1,𝔽5)12yes—
327D12—41no—
328C4×D3—102no—
329C4×D3—102no—
330Dic6—41no—
331C3×C8—102no—
332C3×C8—102no—
333C3⋊C8—102no—
334C3⋊C8—102no—
335C22×C5—62no—
336C22×C5—62no—
337C22×C5—62no—
338C22×C5C22×C4×D311yescharacteristic
339C2×D5—21no—
340C2×D5—21no—
341C2×D5—21no—
342C4×C5—21no—
343C4×C5—21no—
344C4×C5—21no—
345C4×C5—62no—
346Dic5—21no—
347Dic5—21no—
348Dic5—62no—
349Dic5—62no—
350Dic5#100911yescharacteristic
351Dic5#101011yescharacteristic
352C2×Q8—31no—
353C2×D4—31no—
354C2×D4—152no—
355C22×C4—51no—
356C22×C4—51no—
357C22×C4—51no—
358C22×C4—152no—
359M16—602no—
360M16—601no—
361M16—601no—
362C2×C8—52no—
363C2×C8—52no—
364C2×C8—152no—
365C2×C8—152no—
366C2×C8—602no—
367C4⋊C4—152no—
368C4⋊C4—152no—
369C4⋊C4—152no—
370C4⋊C4—151no—
371C4⋊C4—152no—
372C4⋊C4—151no—
373C22⋊C4—32no—
374C22⋊C4—32no—
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—22no—
389C2×D3—22no—
390C2×D3—22no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—51no—
395C3×C4—51no—
396C3×C4—101no—
397C3×C4—101no—
398Dic3—22no—
399Dic3—102no—
400Dic3—102no—
401C2×C5—122no—
402C2×C5#552311yescharacteristic
403C2×C5#574511yescharacteristic
404C2×C5#574511yescharacteristic
405D5—41no—
406C23—32no—
407C23—51no—
408Q8—121no—
409D4—121no—
410D4—602no—
411C2×C4—32no—
412C2×C4—51no—
413C2×C4—51no—
414C2×C4—51no—
415C2×C4—101no—
416C2×C4—101no—
417C2×C4—152no—
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#6625611yescharacteristic
432C2×C3#6626411yescharacteristic
433C2×C3#6626611yescharacteristic
434D3—42no—
435C5#7934211yescharacteristic, complemented, Sylow
436C22—62no—
437C22—62no—
438C22—62no—
439C22—101no—
440C22—101no—
441C22—101no—
442C22C22×D3×AGL(1,𝔽5)11yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—51no—
447C4—51no—
448C4—62no—
449C4—101no—
450C4—101no—
451C4—302no—
452C4—302no—
453C3#1061732211yescharacteristic, complemented, Sylow
454C2—122no—
455C2—201no—
456C2#1175682111yescharacteristic
457C2#1175682211yescharacteristic
458C2#1175683811yescharacteristic
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#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. #11752341C2semidirect
3. C3×#65683C2semidirect
4. #11751899C2semidirect
5. #11751891C2semidirect
6. #11751887C2semidirect
7. #11751799C2semidirect
8. #11751774C2semidirect
9. #11751773C2semidirect
46. #65683D3semidirect
383. C3×C5#2995semidirect
435. C5#79342semidirect
453. C3#10617322semidirect

Maximal subgroups

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

subgroupindex
#117526832
#117523412
C3×#656832
#117518992
#117518912
#117518872
#117517992
#117517742
#117517732
#106173223
#793425