Finite Group Databasebeta

← #14287824

Subgroups — #14287824

Order 1920.

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

normal subgroups only

This group has 620 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×#537C211yesmaximal, characteristic, complemented
2#11751137C212yesmaximal, complemented
3#11751092C211yesmaximal, characteristic, complemented
4#11746358C212yesmaximal, complemented
5#11746349C211yesmaximal, characteristic, complemented
6#10602671—31nomaximal, Hall
7C2×C3×C5×#41—21no—
8C22×D60C2211yescharacteristic
9C3×C5×C22⋊Q8—22no—
10C3×C5×#121—22no—
11C3×C5×#121—21no—
12C3×C5×#121C2212yes—
13C2×C3×C5×C4⋊C4C2211yescharacteristic
14C2×#6715—22no—
15C2×#6714—21no—
16C2×#6714—21no—
17#91115C2211yescharacteristic
18#91114C2212yes—
19#91093—22no—
20#71878—51nomaximal, Hall
21C5×#537D311yescharacteristic, complemented
22#65325—32no—
23#65280—31no—
24#64757—32no—
25#64748—31no—
26C23×D15—21no—
27C3×C5×#41—42no—
28C3×C5×#41—42no—
29C2×C3×C5×Q8—21no—
30C2×C3×C5×D4—21no—
31C2×C3×C5×D4—22no—
32C2×C3×C5×D4—21no—
33C2×C3×C5×D4D412yes—
34C22×C3×C4×C5—22no—
35C22×C3×C4×C5D411yescharacteristic
36C22×C3×C4×C5D411yescharacteristic
37C22×C3×C4×C5C2311yescharacteristic
38C22×Dic15—21no—
39C2×D60—22no—
40C2×D60—42no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4—22no—
43C3×C5×C4⋊C4D412yes—
44C3×C5×C22⋊C4—22no—
45C3×C5×C22⋊C4—22no—
46#6715—44no—
47#6714—42no—
48#6714—42no—
49C2×C3⋊#168—22no—
50C3×#537D511yescharacteristic, complemented
51#5036—52no—
52#4991—51no—
53#4537—52no—
54#4528—51no—
55C2×C5×#41—21no—
56C22×D20—31no—
57C5×C22⋊Q8—22no—
58C5×#121—22no—
59C5×#121—21no—
60C5×#121C2×D312yes—
61C2×C5×C4⋊C4C2×D311yescharacteristic
62C2×#686—62no—
63C2×#685—61no—
64C2×#685—61no—
65#3963—31no—
66C22⋊#168—32no—
67#3941—62no—
68#1598—151noSylow
69C23×C3×C5—22no—
70C23×C3×C5C2×D411yescharacteristic
71C22×D15—22no—
72C22×D15—41no—
73C22×D15—41no—
74C22×D15—41no—
75C22×D15—41no—
76C22×D15—41no—
77C22×D15—41no—
78C3×C5×Q8—42no—
79C3×C5×D4—42no—
80C3×C5×D4—42no—
81C3×C5×D4—42no—
82C3×C5×D4—42no—
83C3×C5×D4—81no—
84C2×C3×C4×C5—21no—
85C2×C3×C4×C5—21no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—22no—
88C2×C3×C4×C5—41no—
89C2×C3×C4×C5—44no—
90C2×C3×C4×C5—41no—
91C2×C3×C4×C5D812yescomplemented
92C2×C3×C4×C5C2×D412yes—
93C2×Dic15—41no—
94C2×Dic15—41no—
95C2×Dic15—41no—
96D60—24no—
97D60—42no—
98C3⋊#168—42no—
99C2×C3×#41—21no—
100C22×D12—51no—
101C3×C22⋊Q8—22no—
102C3×#121—22no—
103C3×#121—21no—
104C3×#121C2×D512yes—
105C2×C3×C4⋊C4C2×D511yescharacteristic
106C2×#213—102no—
107C2×(C2×D3)⋊C4—101no—
108C2×(C2×D3)⋊C4—101no—
109C3⋊#95—51no—
110#830—52no—
111#809—102no—
112C23×D5—61no—
113C5×#41—42no—
114C5×#41—42no—
115C2×C5×Q8—21no—
116C2×C5×D4—21no—
117C2×C5×D4—22no—
118C2×C5×D4—21no—
119C2×C5×D4#6712yes—
120C22×C4×C5—22no—
121C22×C4×C5#6711yescharacteristic
122C22×C4×C5#6711yescharacteristic
123C22×C4×C5C22×D311yescharacteristic
124C22×Dic5—61no—
125C2×D20—62no—
126C2×D20—122no—
127C5×C4⋊C4—21no—
128C5×C4⋊C4—22no—
129C5×C4⋊C4#6712yes—
130C5×C22⋊C4—22no—
131C5×C22⋊C4—22no—
132#686—124no—
133#685—122no—
134#685—122no—
135C2×#168—62no—
136#537D1511yescharacteristic, complemented
137#447—152no—
138#394—151no—
139#327—152no—
140#324—151no—
141C22×C3×C5—21no—
142C22×C3×C5—21no—
143C22×C3×C5—42no—
144C22×C3×C5—42no—
145C22×C3×C5—42no—
146C22×C3×C5#12011yescharacteristic
147C22×C3×C5C2×D811yescharacteristic
148C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
149C2×D15—42no—
150C2×D15—42no—
151C2×D15—42no—
152C2×D15—81no—
153C2×D15—81no—
154C2×D15—81no—
155C2×D15—81no—
156C3×C4×C5—22no—
157C3×C4×C5—22no—
158C3×C4×C5—42no—
159C3×C4×C5—41no—
160Dic15—81no—
161C23×D3—101no—
162C3×#41—42no—
163C3×#41—42no—
164C2×C3×Q8—21no—
165C2×C3×D4—21no—
166C2×C3×D4—22no—
167C2×C3×D4—21no—
168C2×C3×D4C22⋊D512yes—
169C22×C3×C4—22no—
170C22×C3×C4C22⋊D511yescharacteristic
171C22×C3×C4C22⋊D511yescharacteristic
172C22×C3×C4C22×D511yescharacteristic
173C22×Dic3—101no—
174C2×D12—102no—
175C2×D12—202no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4C22⋊D512yes—
179C3×C22⋊C4—22no—
180C3×C22⋊C4—22no—
181#213—204no—
182(C2×D3)⋊C4—202no—
183(C2×D3)⋊C4—202no—
184C2×C3⋊C8—102no—
185C23×C5—22no—
186C23×C5C2×#6711yescharacteristic
187C22×D5—62no—
188C22×D5—121no—
189C22×D5—121no—
190C22×D5—121no—
191C22×D5—121no—
192C22×D5—121no—
193C22×D5—121no—
194C5×Q8—42no—
195C5×D4—42no—
196C5×D4—42no—
197C5×D4—42no—
198C5×D4—42no—
199C5×D4—81no—
200C2×C4×C5—21no—
201C2×C4×C5—21no—
202C2×C4×C5—21no—
203C2×C4×C5—22no—
204C2×C4×C5—41no—
205C2×C4×C5—44no—
206C2×C4×C5—41no—
207C2×C4×C5#21312yescomplemented
208C2×C4×C5C2×#6712yes—
209C2×Dic5—121no—
210C2×Dic5—121no—
211C2×Dic5—121no—
212D20—64no—
213D20—122no—
214#168—122no—
215C2×#41—21no—
216C22×D4—151no—
217C2×D8—302no—
218C22⋊Q8—22no—
219#121—22no—
220#121—21no—
221#121C2×D1512yes—
222C2×C4⋊C4C2×D1511yescharacteristic
223C2×C22⋊C4—301no—
224C2×C22⋊C4—301no—
225#102—302no—
226C22⋊C8—152no—
227#95—151no—
228C2×C3×C5—21no—
229C2×C3×C5—21no—
230C2×C3×C5—82no—
231C2×C3×C5#44711yescharacteristic
232C2×C3×C5#45411yescharacteristic
233C2×C3×C5B(4,𝔽2)11yescharacteristic
234D15—82no—
235C23×C3—22no—
236C23×C3C2×C22⋊D511yescharacteristic
237C22×D3—102no—
238C22×D3—201no—
239C22×D3—201no—
240C22×D3—201no—
241C22×D3—201no—
242C22×D3—201no—
243C22×D3—201no—
244C3×Q8—42no—
245C3×D4—42no—
246C3×D4—42no—
247C3×D4—42no—
248C3×D4—42no—
249C3×D4—81no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—22no—
254C2×C3×C4—41no—
255C2×C3×C4—44no—
256C2×C3×C4—41no—
257C2×C3×C4#68612yescomplemented
258C2×C3×C4C2×C22⋊D512yes—
259C2×Dic3—201no—
260C2×Dic3—201no—
261C2×Dic3—201no—
262D12—104no—
263D12—202no—
264C3⋊C8—202no—
265C22×C5—21no—
266C22×C5—21no—
267C22×C5—42no—
268C22×C5—42no—
269C22×C5—42no—
270C22×C5C2×#21311yescharacteristic
271C22×C5#93211yescharacteristic
272C22×C5#95311yescharacteristic
273C2×D5—122no—
274C2×D5—122no—
275C2×D5—122no—
276C2×D5—241no—
277C2×D5—241no—
278C2×D5—241no—
279C2×D5—241no—
280C4×C5—22no—
281C4×C5—22no—
282C4×C5—42no—
283C4×C5—41no—
284Dic5—241no—
285C24—301no—
286#41—42no—
287#41—42no—
288C2×Q8—21no—
289C2×D4—21no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—302no—
293C2×D4—602no—
294C2×D4#122012yes—
295C22×C4—22no—
296C22×C4—301no—
297C22×C4#122011yescharacteristic
298C22×C4#122011yescharacteristic
299C22×C4C22×D1511yescharacteristic
300D8—604no—
301C2×C8—302no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4#122012yes—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—602no—
308C22⋊C4—602no—
309C3×C5#159811yescharacteristic, complemented, Hall
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—42no—
313C22×C3—42no—
314C22×C3—42no—
315C22×C3C2×#68611yescharacteristic
316C22×C3#407811yescharacteristic
317C22×C3#409911yescharacteristic
318C2×D3—202no—
319C2×D3—202no—
320C2×D3—202no—
321C2×D3—401no—
322C2×D3—401no—
323C2×D3—401no—
324C2×D3—401no—
325C3×C4—22no—
326C3×C4—22no—
327C3×C4—42no—
328C3×C4—41no—
329Dic3—401no—
330C2×C5—21no—
331C2×C5—21no—
332C2×C5—82no—
333C2×C5#521711yescharacteristic
334C2×C5#524411yescharacteristic
335C2×C5#524511yescharacteristic
336D5—242no—
337C23—22no—
338C23—302no—
339C23—601no—
340C23—601no—
341C23—601no—
342C23—601no—
343C23—601no—
344C23—601no—
345C23C2×#122011yescharacteristic
346Q8—42no—
347D4—42no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4—304no—
353D4—602no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—44no—
360C2×C4—41no—
361C2×C4—601no—
362C2×C4—601no—
363C2×C4—601no—
364C2×C4#671512yescomplemented
365C2×C4C2×#122012yes—
366C8—602no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—82no—
370C2×C3#6550611yescharacteristic
371C2×C3#6553311yescharacteristic
372C2×C3#6553411yescharacteristic
373D3—402no—
374C5#7187811yescharacteristic, complemented, Sylow
375C22—21no—
376C22—21no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—602no—
381C22—602no—
382C22—602no—
383C22—1201no—
384C22—1201no—
385C22—1201no—
386C22—1201no—
387C22C2×#671511yescharacteristic
388C22#9182111yescharacteristic
389C22#9184211yescharacteristic
390C4—22no—
391C4—22no—
392C4—42no—
393C4—41no—
394C4—1201no—
395C3#1060267111yescharacteristic, complemented, Sylow
396C2—21no—
397C2—21no—
398C2—82no—
399C2—1202no—
400C2#1175131811yescharacteristic
401C2#1175134511yescharacteristic
402C2#1175134611yescharacteristic
subgroupG/H
centerC22#91842
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C22×D15
FittingC3×C5×#537C2
socleC22×C3×C5#120
hypercenter#537D15
hypocenterC3×C5#1598
LCSG ▷ C22×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C3×C5×#537C2semidirect
2. #11751137C2semidirect
3. #11751092C2semidirect
4. #11746358C2semidirect
5. #11746349C2semidirect
21. C5×#537D3semidirect
50. C3×#537D5semidirect
91. C2×C3×C4×C5D8semidirect
136. #537D15semidirect
207. C2×C4×C5#213semidirect
257. C2×C3×C4#686semidirect
309. C3×C5#1598semidirect
364. C2×C4#6715semidirect
374. C5#71878semidirect
395. C3#10602671semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5372
#117511372
#117510922
#117463582
#117463492
#106026713
#718785