Finite Group Databasebeta

← #14283307

Subgroups — #14283307

Order 1920.

548 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×C22⋊Q8C211yesmaximal, characteristic
2#11751312C211yesmaximal, characteristic
3#11751308C211yesmaximal, characteristic
4#11746359C214yesmaximal, complemented
5#10602482—31nomaximal, Hall
6C22×C3×C5×Q8C411yescharacteristic, complemented
7C23×C3×C4×C5C2211yescharacteristic
8C3×C5×C22⋊Q8—21no—
9C3×C5×C22⋊Q8—21no—
10C3×C5×C22⋊Q8C412yescomplemented
11C3×C5×C22⋊Q8C2212yes—
12C2×C3×C5×C4⋊C4—21no—
13C2×C3×C5×C4⋊C4C411yescharacteristic, complemented
14C2×C3×C5×C22⋊C4—21no—
15C2×#6719—21no—
16C2×#6710—21no—
17#91115—21no—
18#91115C2212yes—
19#91114—21no—
20#91114C2212yes—
21#71689—51nomaximal, Hall
22C2×C5×C22⋊Q8D311yescharacteristic
23#65500—31no—
24#65496—31no—
25#64758—34no—
26C24×C3×C5D411yescharacteristic
27C2×C3×C5×Q8—21no—
28C2×C3×C5×Q8—42no—
29C2×C3×C5×Q8C2×C412yes—
30C22×C3×C4×C5—21no—
31C22×C3×C4×C5—21no—
32C22×C3×C4×C5—42no—
33C22×C3×C4×C5C2×C411yescharacteristic
34C22×C3×C4×C5C2×C411yescharacteristic
35C22×C3×C4×C5D411yescharacteristic
36C22×C3×C4×C5D411yescharacteristic
37C22×C3×C4×C5D411yescharacteristic
38C22×C3×C4×C5C2311yescharacteristic
39C22×Dic15—22no—
40C3×C5×C4⋊C4—22no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4—22no—
43C3×C5×C4⋊C4C2×C412yes—
44C3×C5×C22⋊C4—22no—
45C3×C5×C22⋊C4—22no—
46#6719—42no—
47#6710—42no—
48C2×C3⋊#168—22no—
49C2×C3×C22⋊Q8D511yescharacteristic
50#5211—51no—
51#5207—51no—
52#4538—54no—
53C22×C5×Q8Dic311yescharacteristic, complemented
54C23×C4×C5C2×D311yescharacteristic
55C5×C22⋊Q8—21no—
56C5×C22⋊Q8—21no—
57C5×C22⋊Q8Dic312yescomplemented
58C5×C22⋊Q8C2×D312yes—
59C2×C5×C4⋊C4—21no—
60C2×C5×C4⋊C4Dic311yescharacteristic, complemented
61C2×C5×C22⋊C4—21no—
62C2×C22⋊Dic5—61no—
63C2×#681—61no—
64#3963—32no—
65#3963—61no—
66C22⋊#168—32no—
67C22⋊#168—61no—
68#1511—151noSylow
69C23×C3×C5—22no—
70C23×C3×C5—22no—
71C23×C3×C5—41no—
72C23×C3×C5C22⋊C411yescharacteristic
73C23×C3×C5C22⋊C411yescharacteristic
74C23×C3×C5C2×D411yescharacteristic
75C3×C5×Q8—42no—
76C3×C5×Q8—42no—
77C2×C3×C4×C5—22no—
78C2×C3×C4×C5—22no—
79C2×C3×C4×C5—22no—
80C2×C3×C4×C5—22no—
81C2×C3×C4×C5—41no—
82C2×C3×C4×C5—41no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5—41no—
85C2×C3×C4×C5—42no—
86C2×C3×C4×C5—42no—
87C2×C3×C4×C5C22⋊C411yescharacteristic
88C2×C3×C4×C5C22⋊C411yescharacteristic
89C2×C3×C4×C5C22×C411yescharacteristic
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×Dic15—42no—
92C2×Dic15—42no—
93C2×Dic15—42no—
94C3⋊#168—42no—
95C22×C3×Q8Dic511yescharacteristic, complemented
96C23×C3×C4C2×D511yescharacteristic
97C3×C22⋊Q8—21no—
98C3×C22⋊Q8—21no—
99C3×C22⋊Q8Dic512yescomplemented
100C3×C22⋊Q8C2×D512yes—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4Dic511yescharacteristic, complemented
103C2×C3×C22⋊C4—21no—
104C2×#217—101no—
105C2×#208—101no—
106C3⋊#95—52no—
107C3⋊#95—101no—
108#830—52no—
109#830—101no—
110C24×C5#6711yescharacteristic
111C2×C5×Q8—21no—
112C2×C5×Q8—42no—
113C2×C5×Q8C2×Dic312yes—
114C22×C4×C5—21no—
115C22×C4×C5—21no—
116C22×C4×C5—42no—
117C22×C4×C5C2×Dic311yescharacteristic
118C22×C4×C5C2×Dic311yescharacteristic
119C22×C4×C5#6711yescharacteristic
120C22×C4×C5#6711yescharacteristic
121C22×C4×C5#6711yescharacteristic
122C22×C4×C5C22×D311yescharacteristic
123C22×Dic5—62no—
124C5×C4⋊C4—22no—
125C5×C4⋊C4—21no—
126C5×C4⋊C4—22no—
127C5×C4⋊C4C2×Dic312yes—
128C5×C22⋊C4—22no—
129C5×C22⋊C4—22no—
130C22⋊Dic5—122no—
131#681—122no—
132C2×#168—62no—
133C2×C22⋊Q8D1511yescharacteristic
134#407—151no—
135#381—151no—
136#328—154no—
137C22×C3×C5—21no—
138C22×C3×C5—21no—
139C22×C3×C5—21no—
140C22×C3×C5—21no—
141C22×C3×C5—21no—
142C22×C3×C5—21no—
143C22×C3×C5—21no—
144C22×C3×C5—44no—
145C22×C3×C5#9912yes—
146C22×C3×C5C2×C22⋊C411yescharacteristic
147C22×C3×C5#13712yes—
148C3×C4×C5—21no—
149C3×C4×C5—21no—
150C3×C4×C5—21no—
151C3×C4×C5—21no—
152C3×C4×C5—42no—
153C3×C4×C5—42no—
154Dic15—82no—
155C24×C3C22⋊D511yescharacteristic
156C2×C3×Q8—21no—
157C2×C3×Q8—42no—
158C2×C3×Q8C2×Dic512yes—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—42no—
162C22×C3×C4C2×Dic511yescharacteristic
163C22×C3×C4C2×Dic511yescharacteristic
164C22×C3×C4C22⋊D511yescharacteristic
165C22×C3×C4C22⋊D511yescharacteristic
166C22×C3×C4C22⋊D511yescharacteristic
167C22×C3×C4C22×D511yescharacteristic
168C22×Dic3—102no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4C2×Dic512yes—
173C3×C22⋊C4—22no—
174C3×C22⋊C4—22no—
175#217—202no—
176#208—202no—
177C2×C3⋊C8—102no—
178C23×C5—22no—
179C23×C5—22no—
180C23×C5—41no—
181C23×C5#21711yescharacteristic
182C23×C5#21711yescharacteristic
183C23×C5C2×#6711yescharacteristic
184C5×Q8—42no—
185C5×Q8—42no—
186C2×C4×C5—22no—
187C2×C4×C5—22no—
188C2×C4×C5—22no—
189C2×C4×C5—22no—
190C2×C4×C5—41no—
191C2×C4×C5—41no—
192C2×C4×C5—41no—
193C2×C4×C5—41no—
194C2×C4×C5—42no—
195C2×C4×C5—42no—
196C2×C4×C5#21711yescharacteristic
197C2×C4×C5#21711yescharacteristic
198C2×C4×C5C22×Dic311yescharacteristic
199C2×C4×C5C2×#6711yescharacteristic
200C2×Dic5—122no—
201C2×Dic5—122no—
202C2×Dic5—122no—
203#168—122no—
204C22×Q8Dic1511yescharacteristic, complemented
205C23×C4C2×D1511yescharacteristic
206C2×M16—301no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209C22⋊Q8Dic1512yescomplemented
210C22⋊Q8C2×D1512yes—
211C2×C4⋊C4—21no—
212C2×C4⋊C4Dic1511yescharacteristic, complemented
213C2×C22⋊C4—21no—
214C2×C22⋊C4—301no—
215C22⋊C8—152no—
216C22⋊C8—301no—
217#95—152no—
218#95—301no—
219C2×C3×C5—22no—
220C2×C3×C5—22no—
221C2×C3×C5—41no—
222C2×C3×C5C2×#9911yescharacteristic
223C2×C3×C5#41911yescharacteristic
224C2×C3×C5#42111yescharacteristic
225C23×C3—22no—
226C23×C3—22no—
227C23×C3—41no—
228C23×C3C22⋊Dic511yescharacteristic
229C23×C3C22⋊Dic511yescharacteristic
230C23×C3C2×C22⋊D511yescharacteristic
231C3×Q8—42no—
232C3×Q8—42no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4C22⋊Dic511yescharacteristic
244C2×C3×C4C22⋊Dic511yescharacteristic
245C2×C3×C4C22×Dic511yescharacteristic
246C2×C3×C4C2×C22⋊D511yescharacteristic
247C2×Dic3—202no—
248C2×Dic3—202no—
249C2×Dic3—202no—
250C3⋊C8—202no—
251C22×C5—21no—
252C22×C5—21no—
253C22×C5—21no—
254C22×C5—21no—
255C22×C5—21no—
256C22×C5—21no—
257C22×C5—21no—
258C22×C5—44no—
259C22×C5#83412yes—
260C22×C5#94212yes—
261C22×C5C2×#21711yescharacteristic
262C4×C5—21no—
263C4×C5—21no—
264C4×C5—21no—
265C4×C5—21no—
266C4×C5—42no—
267C4×C5—42no—
268Dic5—242no—
269C24#122011yescharacteristic
270C2×Q8—21no—
271C2×Q8—42no—
272C2×Q8C2×Dic1512yes—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—302no—
277C22×C4C2×Dic1511yescharacteristic
278C22×C4C2×Dic1511yescharacteristic
279C22×C4#122011yescharacteristic
280C22×C4#122011yescharacteristic
281C22×C4#122011yescharacteristic
282C22×C4C22×D1511yescharacteristic
283M16—602no—
284C2×C8—302no—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—22no—
288C4⋊C4C2×Dic1512yes—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—602no—
292C3×C5#151111yescharacteristic, complemented, Hall
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—44no—
301C22×C3#396612yes—
302C22×C3#408812yes—
303C22×C3C2×C22⋊Dic511yescharacteristic
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—42no—
309C3×C4—42no—
310Dic3—402no—
311C2×C5—22no—
312C2×C5—22no—
313C2×C5—41no—
314C2×C5C2×#83411yescharacteristic
315C2×C5#522511yescharacteristic
316C2×C5#523611yescharacteristic
317C23—22no—
318C23—22no—
319C23—41no—
320C23#671911yescharacteristic
321C23#671911yescharacteristic
322C23C2×#122011yescharacteristic
323Q8—42no—
324Q8—42no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—602no—
336C2×C4—602no—
337C2×C4—602no—
338C2×C4#671911yescharacteristic
339C2×C4#671911yescharacteristic
340C2×C4C22×Dic1511yescharacteristic
341C2×C4C2×#122011yescharacteristic
342C8—602no—
343C2×C3—22no—
344C2×C3—22no—
345C2×C3—41no—
346C2×C3C2×#396611yescharacteristic
347C2×C3#6551411yescharacteristic
348C2×C3#6552511yescharacteristic
349C5#7168911yescharacteristic, complemented, Sylow
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—44no—
358C22#9111812yes—
359C22#9183112yes—
360C22C2×#671911yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—42no—
367C4—1202no—
368C3#1060248211yescharacteristic, complemented, Sylow
369C2—22no—
370C2—22no—
371C2—41no—
372C2C2×#9111811yescharacteristic
373C2#1175132611yescharacteristic
374C2#1175133711yescharacteristic
subgroupG/H
centerC22C2×#6719
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
FittingC2×C3×C5×C22⋊Q8C2
socleC22×C3×C5C2×C22⋊C4
prime radicalC24×C3×C5D4
cosocleC22×C3×C4×C5C23
hypercenterC2×C22⋊Q8D15
hypocenterC3×C5#1511
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
4. #11746359C2semidirect
6. C22×C3×C5×Q8C4semidirect
10. C3×C5×C22⋊Q8C4semidirect
13. C2×C3×C5×C4⋊C4C4semidirect
53. C22×C5×Q8Dic3semidirect
57. C5×C22⋊Q8Dic3semidirect
60. C2×C5×C4⋊C4Dic3semidirect
95. C22×C3×Q8Dic5semidirect
99. C3×C22⋊Q8Dic5semidirect
102. C2×C3×C4⋊C4Dic5semidirect
204. C22×Q8Dic15semidirect
209. C22⋊Q8Dic15semidirect
212. C2×C4⋊C4Dic15semidirect
292. C3×C5#1511semidirect
349. C5#71689semidirect
368. C3#10602482semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×C22⋊Q82
#117513122
#117513082
#117463592
#106024823
#716895