Finite Group Databasebeta

← #14285056

Subgroups — #14285056

Order 1920.

504 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 102 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
1#11756484C211yesmaximal, characteristic
2#11747000C211yesmaximal, characteristic, complemented
3#11746867C211yesmaximal, characteristic, complemented
4C5×#4546C211yesmaximal, characteristic, complemented
5C3×#64766C211yesmaximal, characteristic, complemented
6#11745854C211yesmaximal, characteristic, complemented
7#11745854C211yesmaximal, characteristic, complemented
8#10599278—31nomaximal, Hall
9C2×#6821C412yes—
10C3×C5×C4⋊Q8C2211yescharacteristic
11#91836C412yes—
12#91784—21no—
13#91784—21no—
14#91782—21no—
15#91782—21no—
16#91763C2211yescharacteristic
17C42×D15C2211yescharacteristic
18C2×#6648—21no—
19C5×C4⋊(C3⋊C8)C2211yescharacteristic
20C5×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4⋊#168C2211yescharacteristic
22C3×C4⋊#168C2211yescharacteristic
23#90983—21no—
24#90957—21no—
25#90954—21no—
26#68485—51nomaximal, Hall
27#66059—31no—
28#65124—31no—
29D5×C4⋊C8—31no—
30C5×#333—31no—
31#64766D311yescharacteristic, complemented
32#64705—31no—
33#64705—31no—
34C2×C3×C5×Q8C2×C412yes—
35#6821—44no—
36C2×D60—22no—
37C2×D60—21no—
38C2×D60—21no—
39C2×D60C2×C412yes—
40C2×C4×D15—22no—
41C2×C4×D15C2×C411yescharacteristic
42C2×C4×D15D411yescharacteristic
43C2×C4×D15D411yescharacteristic
44C3×C5×C4⋊C4—21no—
45C3×C5×C4⋊C4—21no—
46C3×C42×C5C2311yescharacteristic
47#6714—22no—
48#6714—22no—
49C4×Dic15C2×C411yescharacteristic
50C4×Dic15D411yescharacteristic
51C4×Dic15D411yescharacteristic
52C2×C5×C3⋊C8—21no—
53C2×C5×C3⋊C8—21no—
54C2×C3×#168—21no—
55C2×C3×#168—21no—
56#6648—42no—
57#5725—51no—
58#4835—51no—
59D3×C4⋊C8—51no—
60C3×#333—51no—
61#4546D511yescharacteristic, complemented
62#4485—51no—
63#4485—51no—
64C2×#713—32no—
65C5×C4⋊Q8C2×D311yescharacteristic
66#4093—32no—
67C2×C8×D5—61no—
68#4041—61no—
69#4041—61no—
70#4039—61no—
71#4039—61no—
72#4020—31no—
73C42×D5—31no—
74C5×C4⋊C8—31no—
75C5×C4⋊C8—31no—
76#3952—61no—
77#3947—61no—
78#3946—61no—
79C4⋊#168C2×D311yescharacteristic
80C4⋊#168C2×D311yescharacteristic
81#1551—151noSylow
82C22×D15—22no—
83C22×D15C22⋊C411yescharacteristic
84C3×C5×Q8—42no—
85C2×C3×C4×C5—22no—
86C2×C3×C4×C5C22×C411yescharacteristic
87C2×C3×C4×C5C2×D411yescharacteristic
88C2×C3×C4×C5C2×D411yescharacteristic
89C2×Dic15C22⋊C411yescharacteristic
90C2×Dic15C22⋊C411yescharacteristic
91C2×Dic15C22⋊C411yescharacteristic
92D60—42no—
93D60—44no—
94D60—41no—
95D60—41no—
96D60—41no—
97D60—41no—
98C4×D15—22no—
99C4×D15—21no—
100C4×D15—21no—
101C4×D15—21no—
102C4×D15—21no—
103C4×D15—44no—
104C3×#168—41no—
105C3×#168—41no—
106C5×C3⋊C8—41no—
107C5×C3⋊C8—41no—
108C2×#239—52no—
109C3×C4⋊Q8C2×D511yescharacteristic
110#947—52no—
111C2×D3×C8—101no—
112#895—101no—
113#895—101no—
114#893—101no—
115#893—101no—
116#874—51no—
117C42×D3—51no—
118C3×C4⋊C8—51no—
119C3×C4⋊C8—51no—
120(C2×D3)⋊C8—101no—
121#815—101no—
122Dic3⋊C8—101no—
123C4⋊(C3⋊C8)C2×D511yescharacteristic
124C4⋊(C3⋊C8)C2×D511yescharacteristic
125C2×C5×Q8C4×D312yes—
126#713—124no—
127C2×D20—32no—
128C2×D20—62no—
129C2×D20—61no—
130C2×D20—61no—
131C2×C4×D5—31no—
132C2×C4×D5—31no—
133C2×C4×D5—31no—
134C2×C4×D5—62no—
135C2×C5×C8—61no—
136C2×C5×C8—61no—
137C5×C4⋊C4—21no—
138C5×C4⋊C4—21no—
139C42×C5C22×D311yescharacteristic
140#685—62no—
141#685—62no—
142C4×Dic5—31no—
143C4×Dic5—31no—
144C4×Dic5—31no—
145C2×#168—21no—
146C2×#168—21no—
147C8×D5—122no—
148#532—151no—
149#432—151no—
150C2×C4⋊C8—151no—
151#333—31no—
152#333—51no—
153#331—151no—
154#331—151no—
155C22×C3×C5C2×C22⋊C411yescharacteristic
156C2×D15—21no—
157C2×D15—21no—
158C2×D15—42no—
159C2×D15—42no—
160C2×D15—42no—
161C2×D15#10012yes—
162C3×C4×C5—21no—
163C3×C4×C5—42no—
164C3×C4×C5#13511yescharacteristic
165C3×C4×C5#13511yescharacteristic
166C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
167C3×C4×C5#13711yescharacteristic
168Dic15—21no—
169Dic15—21no—
170Dic15—21no—
171C2×C3×Q8C4×D512yes—
172#239—204no—
173C2×D12—52no—
174C2×D12—102no—
175C2×D12—101no—
176C2×D12—101no—
177C2×C4×D3—51no—
178C2×C4×D3—51no—
179C2×C4×D3—51no—
180C2×C4×D3—102no—
181C2×C3×C8—101no—
182C2×C3×C8—101no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C42C22×D511yescharacteristic
186(C2×D3)⋊C4—102no—
187(C2×D3)⋊C4—102no—
188C4×Dic3—51no—
189C4×Dic3—51no—
190C4×Dic3—51no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8—21no—
193D3×C8—202no—
194C22×D5—31no—
195C22×D5—62no—
196C5×Q8—42no—
197C2×C4×C5—22no—
198C2×C4×C5C2×C4×D311yescharacteristic
199C2×C4×C5D3×D411yescharacteristic
200C2×C4×C5D3×D411yescharacteristic
201C2×Dic5—31no—
202C2×Dic5—31no—
203C2×Dic5—31no—
204D20—122no—
205D20—124no—
206D20—121no—
207D20—121no—
208D20—121no—
209D20—121no—
210C4×D5—62no—
211C4×D5—61no—
212C4×D5—61no—
213C4×D5—61no—
214C4×D5—61no—
215C4×D5—124no—
216C5×C8—121no—
217C5×C8—121no—
218#168—41no—
219#168—41no—
220C2×#41—152no—
221C22×C8—301no—
222C4⋊Q8D3×D511yescharacteristic
223#127—151no—
224#124—152no—
225#121—301no—
226#121—301no—
227C4×D4—301no—
228C4×D4—301no—
229C2×C42—151no—
230C4⋊C8—31no—
231C4⋊C8—31no—
232C4⋊C8—51no—
233C4⋊C8—51no—
234C4⋊C8—301no—
235C22⋊C8—301no—
236#97—301no—
237C2×C3×C5C2×#10011yescharacteristic
238C2×C3×C5#41611yescharacteristic
239C2×C3×C5#41711yescharacteristic
240D15—22no—
241D15—82no—
242C22×D3—51no—
243C22×D3—102no—
244C3×Q8—42no—
245C2×C3×C4—22no—
246C2×C3×C4C2×C4×D511yescharacteristic
247C2×C3×C4D4×D511yescharacteristic
248C2×C3×C4D4×D511yescharacteristic
249C2×Dic3—51no—
250C2×Dic3—51no—
251C2×Dic3—51no—
252D12—202no—
253D12—204no—
254D12—201no—
255D12—201no—
256D12—201no—
257D12—201no—
258C4×D3—102no—
259C4×D3—101no—
260C4×D3—101no—
261C4×D3—101no—
262C4×D3—101no—
263C4×D3—204no—
264C3×C8—201no—
265C3×C8—201no—
266C3⋊C8—41no—
267C3⋊C8—41no—
268C22×C5D3×C22⋊C411yescharacteristic
269C2×D5—32no—
270C2×D5—61no—
271C2×D5—61no—
272C2×D5—122no—
273C2×D5—122no—
274C2×D5—122no—
275C4×C5—21no—
276C4×C5—42no—
277C4×C5#91411yescharacteristic
278C4×C5#91611yescharacteristic
279C4×C5#91811yescharacteristic
280C4×C5#91911yescharacteristic
281Dic5—61no—
282Dic5—61no—
283Dic5—61no—
284#41—604no—
285C2×Q8#120012yes—
286C2×D4—152no—
287C2×D4—302no—
288C2×D4—301no—
289C2×D4—301no—
290C22×C4—151no—
291C22×C4—151no—
292C22×C4—151no—
293C22×C4—302no—
294C2×C8—61no—
295C2×C8—61no—
296C2×C8—101no—
297C2×C8—101no—
298C2×C8—602no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C22⋊C4—302no—
302C22⋊C4—302no—
303C42—151no—
304C42—151no—
305C42—151no—
306C42C2×D3×D511yescharacteristic
307C3×C5#155111yescharacteristic, complemented, Hall
308C22×C3D5×C22⋊C411yescharacteristic
309C2×D3—52no—
310C2×D3—101no—
311C2×D3—101no—
312C2×D3—202no—
313C2×D3—202no—
314C2×D3—202no—
315C3×C4—21no—
316C3×C4—42no—
317C3×C4#406011yescharacteristic
318C3×C4#406211yescharacteristic
319C3×C4#406411yescharacteristic
320C3×C4#406511yescharacteristic
321Dic3—101no—
322Dic3—101no—
323Dic3—101no—
324C2×C5D3×#10011yescharacteristic
325C2×C5#480311yescharacteristic
326C2×C5#480411yescharacteristic
327D5—62no—
328D5—242no—
329C23—151no—
330C23—302no—
331Q8—42no—
332D4—602no—
333D4—604no—
334D4—601no—
335D4—601no—
336D4—601no—
337D4—601no—
338C2×C4—22no—
339C2×C4—151no—
340C2×C4—151no—
341C2×C4—151no—
342C2×C4—302no—
343C2×C4—301no—
344C2×C4—301no—
345C2×C4—301no—
346C2×C4—301no—
347C2×C4—604no—
348C2×C4C2×#120011yescharacteristic
349C2×C4#679011yescharacteristic
350C2×C4#679011yescharacteristic
351C8—121no—
352C8—121no—
353C8—201no—
354C8—201no—
355C2×C3D5×#10011yescharacteristic
356C2×C3#6509211yescharacteristic
357C2×C3#6509311yescharacteristic
358D3—102no—
359D3—402no—
360C5#6848511yescharacteristic, complemented, Sylow
361C22—152no—
362C22—301no—
363C22—301no—
364C22—602no—
365C22—602no—
366C22—602no—
367C22#9156111yescharacteristic
368C4—21no—
369C4—42no—
370C4—301no—
371C4—301no—
372C4—301no—
373C4#9150611yescharacteristic
374C4#9151211yescharacteristic
375C4#9152111yescharacteristic
376C4#9152311yescharacteristic
377C3#1059927811yescharacteristic, complemented, Sylow
378C2—302no—
379C2—1202no—
380C2#1174912711yescharacteristic
381C2#1174920511yescharacteristic
382C2#1174921511yescharacteristic
subgroupG/H
centerC22#91561
commutatorC2×C3×C4×C5C22×C4
FrattiniC42C2×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C2×C22⋊C4
prime radical#11756484C2
cosocleC3×C42×C5C23
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#1551
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #11747000C2semidirect
3. #11746867C2semidirect
4. C5×#4546C2semidirect
5. C3×#64766C2semidirect
6. #11745854C2semidirect
7. #11745854C2semidirect
31. #64766D3semidirect
61. #4546D5semidirect
307. C3×C5#1551semidirect
360. C5#68485semidirect
377. C3#10599278semidirect

Maximal subgroups

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

subgroupindex
#117564842
#117470002
#117468672
C5×#45462
C3×#647662
#117458542
#105992783
#684855