Finite Group Databasebeta

← #14313262

Subgroups — #14313262

Order 1920.

532 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#535C211yesmaximal, characteristic
2#11751341C211yesmaximal, characteristic, complemented
3#11751338C211yesmaximal, characteristic, complemented
4#11746357C214yesmaximal, complemented
5#10602652—31nomaximal, Hall
6C2×C3×C5×#41C2211yescharacteristic
7C22×C3×C5×D4C411yescharacteristic, complemented
8C3×C5×#127—21no—
9C3×C5×#124—21no—
10C3×C5×#121—21no—
11C3×C5×#121—21no—
12C3×C5×#120C412yescomplemented
13C3×C5×#120C2212yes—
14C3×C5×#117C411yescharacteristic, complemented
15#91815—21no—
16C2×#6710—21no—
17#91120—21no—
18#91118—21no—
19#91118C2212yes—
20#91117C2212yes—
21#71859—51nomaximal, Hall
22C5×#535D311yescharacteristic
23#65529—31no—
24#65526—31no—
25#64756—34no—
26C24×C3×C5C2×C412yes—
27C3×C5×#41—41no—
28C3×C5×#41—41no—
29C2×C3×C5×Q8D411yescharacteristic
30C2×C3×C5×D4—22no—
31C2×C3×C5×D4—22no—
32C2×C3×C5×D4—21no—
33C2×C3×C5×D4—21no—
34C2×C3×C5×D4—41no—
35C2×C3×C5×D4—41no—
36C2×C3×C5×D4C2×C411yescharacteristic
37C2×C3×C5×D4D411yescharacteristic
38C2×C3×C5×D4C2311yescharacteristic
39C22×C3×C4×C5C2×C411yescharacteristic
40C22×C3×C4×C5D411yescharacteristic
41C22×C3×C4×C5D411yescharacteristic
42C3×C5×C4⋊C4—21no—
43C3×C5×C22⋊C4—22no—
44C3×C5×C22⋊C4—22no—
45C3×C5×C22⋊C4C2×C412yes—
46C3×C42×C5—21no—
47#6719—22no—
48C4⋊Dic15—41no—
49C4×Dic15—41no—
50#6710—22no—
51#6710—41no—
52C2×C3⋊#168—41no—
53C3×#535D511yescharacteristic
54#5240—51no—
55#5237—51no—
56#4536—54no—
57C2×C5×#41C2×D311yescharacteristic
58C22×C5×D4Dic311yescharacteristic, complemented
59C5×#127—21no—
60C5×#124—21no—
61C5×#121—21no—
62C5×#121—21no—
63C5×#120Dic312yescomplemented
64C5×#120C2×D312yes—
65C5×#117Dic311yescharacteristic, complemented
66#4072—61no—
67C2×#681—61no—
68#3968—61no—
69#3966—32no—
70#3966—61no—
71#3965—32no—
72#2122—151noSylow
73C23×C3×C5—22no—
74C23×C3×C5—42no—
75C23×C3×C5—82no—
76C23×C3×C5C22⋊C411yescharacteristic
77C23×C3×C5C22×C411yescharacteristic
78C23×C3×C5C2×D411yescharacteristic
79C3×C5×Q8—41no—
80C3×C5×D4—41no—
81C3×C5×D4—41no—
82C3×C5×D4—41no—
83C3×C5×D4—42no—
84C3×C5×D4—41no—
85C3×C5×D4—41no—
86C3×C5×D4—82no—
87C3×C5×D4—82no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5—21no—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—22no—
95C2×C3×C4×C5C22⋊C411yescharacteristic
96C2×C3×C4×C5C22⋊C411yescharacteristic
97C2×C3×C4×C5C22⋊C411yescharacteristic
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×Dic15—42no—
100C3⋊#168—42no—
101C2×C3×#41C2×D511yescharacteristic
102C22×C3×D4Dic511yescharacteristic, complemented
103C3×#127—21no—
104C3×#124—21no—
105C3×#121—21no—
106C3×#121—21no—
107C3×#120Dic512yescomplemented
108C3×#120C2×D512yes—
109C3×#117Dic511yescharacteristic, complemented
110#926—101no—
111C2×#208—101no—
112#836—101no—
113#834—52no—
114#834—101no—
115#833—52no—
116C24×C5C2×Dic312yes—
117C5×#41—41no—
118C5×#41—41no—
119C2×C5×Q8#6711yescharacteristic
120C2×C5×D4—22no—
121C2×C5×D4—22no—
122C2×C5×D4—21no—
123C2×C5×D4—21no—
124C2×C5×D4—41no—
125C2×C5×D4—41no—
126C2×C5×D4C2×Dic311yescharacteristic
127C2×C5×D4#6711yescharacteristic
128C2×C5×D4C22×D311yescharacteristic
129C22×C4×C5C2×Dic311yescharacteristic
130C22×C4×C5#6711yescharacteristic
131C22×C4×C5#6711yescharacteristic
132C5×C4⋊C4—21no—
133C5×C22⋊C4—22no—
134C5×C22⋊C4—22no—
135C5×C22⋊C4C2×Dic312yes—
136C42×C5—21no—
137C22⋊Dic5—62no—
138C4⋊Dic5—121no—
139C4×Dic5—121no—
140#681—62no—
141#681—121no—
142C2×#168—121no—
143#535D1511yescharacteristic
144#413—151no—
145#410—151no—
146AΣL(1,𝔽16)—154no—
147C22×C3×C5—21no—
148C22×C3×C5—21no—
149C22×C3×C5—21no—
150C22×C3×C5—41no—
151C22×C3×C5—41no—
152C22×C3×C5—41no—
153C22×C3×C5—42no—
154C22×C3×C5—42no—
155C22×C3×C5—42no—
156C22×C3×C5—84no—
157C22×C3×C5C2×C22⋊C411yescharacteristic
158C3×C4×C5—21no—
159C3×C4×C5—21no—
160C3×C4×C5—21no—
161C3×C4×C5—42no—
162C3×C4×C5#9911yescharacteristic
163C3×C4×C5#9911yescharacteristic
164Dic15—82no—
165C24×C3C2×Dic512yes—
166C3×#41—41no—
167C3×#41—41no—
168C2×C3×Q8C22⋊D511yescharacteristic
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—41no—
174C2×C3×D4—41no—
175C2×C3×D4C2×Dic511yescharacteristic
176C2×C3×D4C22⋊D511yescharacteristic
177C2×C3×D4C22×D511yescharacteristic
178C22×C3×C4C2×Dic511yescharacteristic
179C22×C3×C4C22⋊D511yescharacteristic
180C22×C3×C4C22⋊D511yescharacteristic
181C3×C4⋊C4—21no—
182C3×C22⋊C4—22no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4C2×Dic512yes—
185C3×C42—21no—
186#217—102no—
187C4⋊Dic3—201no—
188C4×Dic3—201no—
189#208—102no—
190#208—201no—
191C2×C3⋊C8—201no—
192C23×C5—22no—
193C23×C5—42no—
194C23×C5—82no—
195C23×C5#21711yescharacteristic
196C23×C5C22×Dic311yescharacteristic
197C23×C5C2×#6711yescharacteristic
198C5×Q8—41no—
199C5×D4—41no—
200C5×D4—41no—
201C5×D4—41no—
202C5×D4—42no—
203C5×D4—41no—
204C5×D4—41no—
205C5×D4—82no—
206C5×D4—82no—
207C2×C4×C5—21no—
208C2×C4×C5—21no—
209C2×C4×C5—21no—
210C2×C4×C5—21no—
211C2×C4×C5—21no—
212C2×C4×C5—21no—
213C2×C4×C5—22no—
214C2×C4×C5#21711yescharacteristic
215C2×C4×C5#21711yescharacteristic
216C2×C4×C5#21711yescharacteristic
217C2×C4×C5C2×#6711yescharacteristic
218C2×Dic5—122no—
219#168—122no—
220C2×#41C2×D1511yescharacteristic
221C22×D4Dic1511yescharacteristic, complemented
222C2×M16—301no—
223#127—21no—
224#124—21no—
225#121—21no—
226#121—21no—
227#120Dic1512yescomplemented
228#120C2×D1512yes—
229#117—301no—
230#117Dic1511yescharacteristic, complemented
231#101—301no—
232#100—152no—
233#99—152no—
234#99—301no—
235C2×C3×C5—21no—
236C2×C3×C5—41no—
237C2×C3×C5—41no—
238C2×C3×C5—41no—
239C2×C3×C5—82no—
240C2×C3×C5C2×#9911yescharacteristic
241C23×C3—22no—
242C23×C3—42no—
243C23×C3—82no—
244C23×C3C22⋊Dic511yescharacteristic
245C23×C3C22×Dic511yescharacteristic
246C23×C3C2×C22⋊D511yescharacteristic
247C3×Q8—41no—
248C3×D4—41no—
249C3×D4—41no—
250C3×D4—41no—
251C3×D4—42no—
252C3×D4—41no—
253C3×D4—41no—
254C3×D4—82no—
255C3×D4—82no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—22no—
263C2×C3×C4C22⋊Dic511yescharacteristic
264C2×C3×C4C22⋊Dic511yescharacteristic
265C2×C3×C4C22⋊Dic511yescharacteristic
266C2×C3×C4C2×C22⋊D511yescharacteristic
267C2×Dic3—202no—
268C3⋊C8—202no—
269C22×C5—21no—
270C22×C5—21no—
271C22×C5—21no—
272C22×C5—41no—
273C22×C5—41no—
274C22×C5—41no—
275C22×C5—42no—
276C22×C5—42no—
277C22×C5—42no—
278C22×C5—84no—
279C22×C5C2×#21711yescharacteristic
280C4×C5—21no—
281C4×C5—21no—
282C4×C5—21no—
283C4×C5—42no—
284C4×C5#83411yescharacteristic
285C4×C5#83411yescharacteristic
286Dic5—242no—
287C24C2×Dic1512yes—
288#41—41no—
289#41—41no—
290C2×Q8#122011yescharacteristic
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4C2×Dic1511yescharacteristic
298C2×D4#122011yescharacteristic
299C2×D4C22×D1511yescharacteristic
300C22×C4C2×Dic1511yescharacteristic
301C22×C4#122011yescharacteristic
302C22×C4#122011yescharacteristic
303M16—302no—
304M16—601no—
305C2×C8—601no—
306C4⋊C4—21no—
307C4⋊C4—601no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—302no—
311C22⋊C4C2×Dic1512yes—
312C42—21no—
313C42—601no—
314C3×C5#212211yescharacteristic, complemented, Hall
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—84no—
325C22×C3C2×C22⋊Dic511yescharacteristic
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—42no—
330C3×C4#396611yescharacteristic
331C3×C4#396611yescharacteristic
332Dic3—402no—
333C2×C5—21no—
334C2×C5—41no—
335C2×C5—41no—
336C2×C5—41no—
337C2×C5—82no—
338C2×C5C2×#83411yescharacteristic
339C23—22no—
340C23—42no—
341C23—82no—
342C23#671911yescharacteristic
343C23C22×Dic1511yescharacteristic
344C23C2×#122011yescharacteristic
345Q8—41no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—42no—
350D4—41no—
351D4—41no—
352D4—82no—
353D4—82no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—602no—
362C2×C4#671911yescharacteristic
363C2×C4#671911yescharacteristic
364C2×C4#671911yescharacteristic
365C2×C4C2×#122011yescharacteristic
366C8—602no—
367C2×C3—21no—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3—82no—
372C2×C3C2×#396611yescharacteristic
373C5#7185911yescharacteristic, complemented, Sylow
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—84no—
384C22C2×#671911yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—1202no—
390C4#9111811yescharacteristic
391C4#9111811yescharacteristic
392C3#1060265211yescharacteristic, complemented, Sylow
393C2—21no—
394C2—41no—
395C2—41no—
396C2—41no—
397C2—82no—
398C2C2×#9111811yescharacteristic
subgroupG/H
centerC2C2×#91118
commutatorC23×C3×C5C22×C4
FrattiniC2×D4C22×D15
Fitting, prime radicalC3×C5×#535C2
socleC2×C3×C5C2×#99
cosocleC2×C3×C5×D4C23
hypercenter#535D15
hypocenterC3×C5#2122
LCSG ▷ C23×C3×C5 ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
2. #11751341C2semidirect
3. #11751338C2semidirect
4. #11746357C2semidirect
7. C22×C3×C5×D4C4semidirect
12. C3×C5×#120C4semidirect
14. C3×C5×#117C4semidirect
58. C22×C5×D4Dic3semidirect
63. C5×#120Dic3semidirect
65. C5×#117Dic3semidirect
102. C22×C3×D4Dic5semidirect
107. C3×#120Dic5semidirect
109. C3×#117Dic5semidirect
221. C22×D4Dic15semidirect
227. #120Dic15semidirect
230. #117Dic15semidirect
314. C3×C5#2122semidirect
373. C5#71859semidirect
392. C3#10602652semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5352
#117513412
#117513382
#117463572
#106026523
#718595