Finite Group Databasebeta

← #14287965

Subgroups — #14287965

Order 1920.

616 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#534C211yesmaximal, characteristic
2#11751141C212yesmaximal, complemented
3#11751048C211yesmaximal, characteristic, complemented
4#11746358C212yesmaximal, complemented
5#11746346C211yesmaximal, characteristic, complemented
6#10602639—31nomaximal, Hall
7C22×C3×C5×D4—21no—
8C22×Dic30C2211yescharacteristic, complemented
9C3×C5×#123—22no—
10C3×C5×#121—21no—
11C3×C5×#121C2212yes—
12C3×C5×#120—22no—
13C2×C3×C5×C22⋊C4C2211yescharacteristic
14C2×#6719—21no—
15C2×#6716—22no—
16#91115—21no—
17#91115C2211yescharacteristic, complemented
18#91114C2212yescomplemented
19#91094—22no—
20#71846—51nomaximal, Hall
21C5×#534D311yescharacteristic
22#65329—32no—
23#65236—31no—
24#64757—32no—
25#64745—31no—
26C24×C3×C5—21no—
27C24×C3×C5D411yescharacteristic
28C2×C3×C5×D4—22no—
29C2×C3×C5×D4—22no—
30C2×C3×C5×D4—21no—
31C2×C3×C5×D4—42no—
32C2×C3×C5×D4—42no—
33C2×C3×C5×D4D412yes—
34C22×C3×C4×C5—21no—
35C22×C3×C4×C5D411yescharacteristic
36C22×C3×C4×C5C2311yescharacteristic
37C22×Dic15—21no—
38C22×Dic15—21no—
39C2×Dic30—22no—
40C2×Dic30—42no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4D412yes—
43C3×C5×C22⋊C4—22no—
44C3×C5×C22⋊C4—22no—
45C3×C5×C22⋊C4—22no—
46#6719—42no—
47#6716—44no—
48C2×C3⋊#168—22no—
49C3×#534D511yescharacteristic
50#5040—52no—
51#4947—51no—
52#4537—52no—
53#4525—51no—
54C22×C5×D4—21no—
55C22×Dic10—31no—
56C5×#123—22no—
57C5×#121—21no—
58C5×#121C2×D312yes—
59C5×#120—22no—
60C2×C5×C22⋊C4C2×D311yescharacteristic
61C2×C22⋊Dic5—61no—
62C2×#687—62no—
63#3963—31no—
64#3963—61no—
65C22⋊#168—32no—
66#3942—62no—
67#1601—151noSylow
68C23×C3×C5—21no—
69C23×C3×C5—21no—
70C23×C3×C5—22no—
71C23×C3×C5—41no—
72C23×C3×C5—41no—
73C23×C3×C5—42no—
74C23×C3×C5—81no—
75C23×C3×C5—81no—
76C23×C3×C5SD1612yescomplemented
77C23×C3×C5C2×D411yescharacteristic
78C3×C5×D4—42no—
79C3×C5×D4—44no—
80C3×C5×D4—42no—
81C3×C5×D4—42no—
82C3×C5×D4—81no—
83C2×C3×C4×C5—21no—
84C2×C3×C4×C5—22no—
85C2×C3×C4×C5—41no—
86C2×C3×C4×C5—42no—
87C2×C3×C4×C5—41no—
88C2×C3×C4×C5C2×D412yes—
89C2×Dic15—22no—
90C2×Dic15—41no—
91C2×Dic15—41no—
92C2×Dic15—41no—
93C2×Dic15—41no—
94C2×Dic15—41no—
95Dic30—24no—
96Dic30—42no—
97C3⋊#168—42no—
98C22×C3×D4—21no—
99C22×Dic6—51no—
100C3×#123—22no—
101C3×#121—21no—
102C3×#121C2×D512yes—
103C3×#120—22no—
104C2×C3×C22⋊C4C2×D511yescharacteristic
105C2×#217—101no—
106C2×#214—102no—
107C3⋊#95—51no—
108C3⋊#95—101no—
109#830—52no—
110#810—102no—
111C24×C5—21no—
112C24×C5#6711yescharacteristic
113C2×C5×D4—22no—
114C2×C5×D4—22no—
115C2×C5×D4—21no—
116C2×C5×D4—42no—
117C2×C5×D4—42no—
118C2×C5×D4#6712yes—
119C22×C4×C5—21no—
120C22×C4×C5#6711yescharacteristic
121C22×C4×C5C22×D311yescharacteristic
122C22×Dic5—61no—
123C22×Dic5—61no—
124C2×Dic10—62no—
125C2×Dic10—122no—
126C5×C4⋊C4—21no—
127C5×C4⋊C4#6712yes—
128C5×C22⋊C4—22no—
129C5×C22⋊C4—22no—
130C5×C22⋊C4—22no—
131C22⋊Dic5—122no—
132#687—124no—
133C2×#168—62no—
134#534D1511yescharacteristic
135#448—152no—
136#393—151no—
137#327—152no—
138#323—151no—
139C22×C3×C5—21no—
140C22×C3×C5—21no—
141C22×C3×C5—22no—
142C22×C3×C5—42no—
143C22×C3×C5—41no—
144C22×C3×C5—41no—
145C22×C3×C5—42no—
146C22×C3×C5—42no—
147C22×C3×C5—42no—
148C22×C3×C5—81no—
149C22×C3×C5—84no—
150C22×C3×C5#12011yescharacteristic
151C22×C3×C5C2×SD1611yescharacteristic
152C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
153C3×C4×C5—22no—
154C3×C4×C5—42no—
155C3×C4×C5—41no—
156Dic15—42no—
157Dic15—81no—
158C24×C3—21no—
159C24×C3C22⋊D511yescharacteristic
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—21no—
163C2×C3×D4—42no—
164C2×C3×D4—42no—
165C2×C3×D4C22⋊D512yes—
166C22×C3×C4—21no—
167C22×C3×C4C22⋊D511yescharacteristic
168C22×C3×C4C22×D511yescharacteristic
169C22×Dic3—101no—
170C22×Dic3—101no—
171C2×Dic6—102no—
172C2×Dic6—202no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4C22⋊D512yes—
175C3×C22⋊C4—22no—
176C3×C22⋊C4—22no—
177C3×C22⋊C4—22no—
178#217—202no—
179#214—204no—
180C2×C3⋊C8—102no—
181C23×C5—21no—
182C23×C5—21no—
183C23×C5—22no—
184C23×C5—41no—
185C23×C5—41no—
186C23×C5—42no—
187C23×C5—81no—
188C23×C5—81no—
189C23×C5#21412yescomplemented
190C23×C5C2×#6711yescharacteristic
191C5×D4—42no—
192C5×D4—44no—
193C5×D4—42no—
194C5×D4—42no—
195C5×D4—81no—
196C2×C4×C5—21no—
197C2×C4×C5—22no—
198C2×C4×C5—41no—
199C2×C4×C5—42no—
200C2×C4×C5—41no—
201C2×C4×C5C2×#6712yes—
202C2×Dic5—62no—
203C2×Dic5—121no—
204C2×Dic5—121no—
205C2×Dic5—121no—
206C2×Dic5—121no—
207C2×Dic5—121no—
208Dic10—64no—
209Dic10—122no—
210#168—122no—
211C22×Q8—151no—
212C22×D4—21no—
213C2×SD16—302no—
214#123—22no—
215#121—21no—
216#121C2×D1512yes—
217#120—22no—
218C2×C22⋊C4—301no—
219C2×C22⋊C4C2×D1511yescharacteristic
220#103—302no—
221C22⋊C8—152no—
222#95—151no—
223#95—301no—
224C2×C3×C5—21no—
225C2×C3×C5—21no—
226C2×C3×C5—42no—
227C2×C3×C5—82no—
228C2×C3×C5#45011yescharacteristic
229C2×C3×C5#45511yescharacteristic
230C2×C3×C5B(4,𝔽2)11yescharacteristic
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—22no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—42no—
237C23×C3—81no—
238C23×C3—81no—
239C23×C3#68712yescomplemented
240C23×C3C2×C22⋊D511yescharacteristic
241C3×D4—42no—
242C3×D4—44no—
243C3×D4—42no—
244C3×D4—42no—
245C3×D4—81no—
246C2×C3×C4—21no—
247C2×C3×C4—22no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—41no—
251C2×C3×C4C2×C22⋊D512yes—
252C2×Dic3—102no—
253C2×Dic3—201no—
254C2×Dic3—201no—
255C2×Dic3—201no—
256C2×Dic3—201no—
257C2×Dic3—201no—
258Dic6—104no—
259Dic6—202no—
260C3⋊C8—202no—
261C22×C5—21no—
262C22×C5—21no—
263C22×C5—22no—
264C22×C5—42no—
265C22×C5—41no—
266C22×C5—41no—
267C22×C5—42no—
268C22×C5—42no—
269C22×C5—42no—
270C22×C5—81no—
271C22×C5—84no—
272C22×C5#93211yescharacteristic
273C22×C5C2×#21411yescharacteristic
274C22×C5#95311yescharacteristic
275C4×C5—22no—
276C4×C5—42no—
277C4×C5—41no—
278Dic5—122no—
279Dic5—241no—
280C24—21no—
281C24#122011yescharacteristic
282C2×Q8—302no—
283C2×Q8—602no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4#122012yes—
290C22×C4—21no—
291C22×C4—301no—
292C22×C4—301no—
293C22×C4#122011yescharacteristic
294C22×C4C22×D1511yescharacteristic
295SD16—604no—
296C2×C8—302no—
297C4⋊C4—21no—
298C4⋊C4#122012yes—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—602no—
303C3×C5#160111yescharacteristic, complemented, Hall
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—22no—
307C22×C3—42no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—42no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—81no—
314C22×C3—84no—
315C22×C3#407811yescharacteristic
316C22×C3C2×#68711yescharacteristic
317C22×C3#409911yescharacteristic
318C3×C4—22no—
319C3×C4—42no—
320C3×C4—41no—
321Dic3—202no—
322Dic3—401no—
323C2×C5—21no—
324C2×C5—21no—
325C2×C5—42no—
326C2×C5—82no—
327C2×C5#521811yescharacteristic
328C2×C5#524211yescharacteristic
329C2×C5#524411yescharacteristic
330C23—21no—
331C23—21no—
332C23—22no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—81no—
337C23—81no—
338C23#671612yescomplemented
339C23C2×#122011yescharacteristic
340Q8—304no—
341Q8—602no—
342D4—42no—
343D4—44no—
344D4—42no—
345D4—42no—
346D4—81no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—302no—
353C2×C4—601no—
354C2×C4—601no—
355C2×C4—601no—
356C2×C4—601no—
357C2×C4—601no—
358C2×C4C2×#122012yes—
359C8—602no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—42no—
363C2×C3—82no—
364C2×C3#6550711yescharacteristic
365C2×C3#6553111yescharacteristic
366C2×C3#6553311yescharacteristic
367C5#7184611yescharacteristic, complemented, Sylow
368C22—21no—
369C22—21no—
370C22—22no—
371C22—42no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—81no—
378C22—84no—
379C22#9182111yescharacteristic
380C22C2×#671611yescharacteristic
381C22#9184211yescharacteristic
382C4—22no—
383C4—42no—
384C4—41no—
385C4—602no—
386C4—1201no—
387C3#1060263911yescharacteristic, complemented, Sylow
388C2—21no—
389C2—21no—
390C2—42no—
391C2—82no—
392C2#1175131911yescharacteristic
393C2#1175134311yescharacteristic
394C2#1175134511yescharacteristic
subgroupG/H
centerC22#91842
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C22×D15
Fitting, prime radicalC3×C5×#534C2
socleC22×C3×C5#120
hypercenter#534D15
hypocenterC3×C5#1601
LCSG ▷ C22×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
2. #11751141C2semidirect
3. #11751048C2semidirect
4. #11746358C2semidirect
5. #11746346C2semidirect
8. C22×Dic30C22semidirect
17. #91115C22semidirect
18. #91114C22semidirect
76. C23×C3×C5SD16semidirect
189. C23×C5#214semidirect
239. C23×C3#687semidirect
303. C3×C5#1601semidirect
338. C23#6716semidirect
367. C5#71846semidirect
387. C3#10602639semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5342
#117511412
#117510482
#117463582
#117463462
#106026393
#718465