Finite Group Databasebeta

← #14289006

Subgroups — #14289006

Order 1920.

472 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 90 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#11752657C211yesmaximal, characteristic, complemented
2C5×#4848C211yesmaximal, characteristic, complemented
3#11747066C211yesmaximal, characteristic, complemented
4#11747026C211yesmaximal, characteristic
5#11746302C211yesmaximal, characteristic, complemented
6C3×#64703C211yesmaximal, characteristic, complemented
7#11745843C211yesmaximal, characteristic, complemented
8#10601821—31nomaximal, Hall
9#91758C2211yescharacteristic
10C2×C5×#204—21no—
11C5×#874C2211yescharacteristic
12C4×C5×Dic6C2211yescharacteristic
13C3×C4×D20C2211yescharacteristic
14#91431—22no—
15Dic3×D20—21no—
16D5×C4⋊Dic3—21no—
17#91381—21no—
18#91359—21no—
19C2×#6659—22no—
20C2×#6652—21no—
21C3×C5×C4⋊C8C2211yescharacteristic
22C4⋊(C3⋊#168)C2211yescharacteristic
23C5×#821—21no—
24C3×C4×#168C2211yescharacteristic
25#90966—21no—
26#68198—51nomaximal, Hall
27#65913—31no—
28C5×#460—31no—
29#65365—31no—
30#65361—31no—
31Q8⋊#168—31no—
32#64703D311yescharacteristic, complemented
33#64701—31no—
34C2×Dic30D412yes—
35C2×C5×D12—21no—
36C2×C5×D12—21no—
37C2×C5×D12—21no—
38C2×C5×Dic6D411yescharacteristic
39C2×C3×D20D411yescharacteristic
40C2×C3×C4×D5—21no—
41C2×D5×Dic3—22no—
42C2×C3×C5×C8—21no—
43C3×C42×C5C2311yescharacteristic
44C4⋊Dic15—21no—
45C4⋊Dic15—21no—
46C2×C3⋊#168—21no—
47C5×C4⋊Dic3D411yescharacteristic
48C5×Dic3⋊C4—21no—
49C4×C5×Dic3—21no—
50C5×#204—42no—
51C3×#685—21no—
52C3×C4⋊Dic5D411yescharacteristic
53C2×C3×#168—21no—
54#6669—22no—
55#6668—22no—
56#6665—22no—
57#6659—44no—
58#6652—42no—
59#5543—51no—
60#5015—51no—
61#4848D511yescharacteristic, complemented
62#4693—51no—
63C3×D4⋊C8—51no—
64#4481—51no—
65#4456—51no—
66C2×C5×SD16—61no—
67C5×#127—31no—
68C4×C5×Q8—31no—
69#4092—61no—
70C2×#687—62no—
71C2×#686—61no—
72#4042—62no—
73D5×C4⋊C4—61no—
74C4×D20—61no—
75C4×D20C2×D311yescharacteristic
76#4016—61no—
77#4015—31no—
78C5×C4⋊C8C2×D311yescharacteristic
79C5×#102—61no—
80#3964—61no—
81C4⋊#168—31no—
82C4×#168C2×D311yescharacteristic
83#1622—151noSylow
84C22×C5×D3—41no—
85C22×C3×D5—21no—
86C2×C3×C4×C5C2×D411yescharacteristic
87C2×C3×C4×C5C2×D411yescharacteristic
88C2×C3×C4×C5C2×D411yescharacteristic
89C2×Dic15—22no—
90Dic30—42no—
91C2×C5×Dic3—21no—
92C2×C5×Dic3—21no—
93C5×D12—42no—
94C5×D12—41no—
95C5×D12—41no—
96C5×D12—41no—
97C5×D12—41no—
98C5×Dic6—21no—
99C5×Dic6—21no—
100C2×C3×Dic5—21no—
101C3×D20—21no—
102C3×D20SD1612yescomplemented
103C3×C4×D5—41no—
104C3×C4×D5—41no—
105D5×Dic3—82no—
106C3×C5×C8—41no—
107C3⋊#168—41no—
108C3×#168—22no—
109C3×C4×D4—51no—
110D4×Dic3—101no—
111C2×#213—101no—
112C2×C4⋊Dic3—101no—
113#924—101no—
114C2×#204—21no—
115C2×#204—102no—
116#890—101no—
117#878—102no—
118#874C2×D511yescharacteristic
119#869—51no—
120C4×Dic6C2×D511yescharacteristic
121C3×C4⋊C8C2×D511yescharacteristic
122C3×C4×C8—51no—
123#821—21no—
124#809—101no—
125C4⋊(C3⋊C8)—51no—
126C2×C5×Q8—31no—
127C2×C5×D4—61no—
128C2×C5×D4—61no—
129C2×C5×D4—61no—
130C2×D20D1211yescharacteristic
131C2×C4×D5—21no—
132C2×C4×D5—62no—
133C2×Dic10—32no—
134C5×SD16—122no—
135C2×C5×C8—21no—
136C5×C4⋊C4—31no—
137C5×C4⋊C4—61no—
138C42×C5—61no—
139C42×C5C22×D311yescharacteristic
140#687—124no—
141#686—122no—
142#685—21no—
143#685—62no—
144C4⋊Dic5—61no—
145C4⋊Dic5—61no—
146C4⋊Dic5D1211yescharacteristic
147#683—62no—
148#683—62no—
149C2×#168—21no—
150C2×#168—61no—
151#554—151no—
152#492—151no—
153#460—31no—
154#459—151no—
155#333—151no—
156Q8⋊C8—151no—
157D4⋊C8—51no—
158C22×C3×C5#12011yescharacteristic
159C2×C5×D3—81no—
160C2×C5×D3—81no—
161C2×C5×D3—81no—
162C2×C3×D5—22no—
163C2×C3×D5—41no—
164C2×C3×D5—41no—
165C3×C4×C5—21no—
166C3×C4×C5C2×SD1611yescharacteristic
167C3×C4×C5#13511yescharacteristic
168C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
169C3×C4×C5#13711yescharacteristic
170Dic15—42no—
171C3×Dic5—41no—
172C5×Dic3—41no—
173C5×Dic3—41no—
174C2×C3×D4—51no—
175C22×C3×C4—101no—
176C22×Dic3—102no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×Dic6—52no—
181C2×Dic6C22⋊D511yescharacteristic
182C2×C3×C8—21no—
183C2×C3×C8—101no—
184C3×C4⋊C4—51no—
185C3×C22⋊C4—101no—
186C3×C42C22×D511yescharacteristic
187#217—102no—
188#213—202no—
189C4⋊Dic3—102no—
190C4⋊Dic3—101no—
191C4⋊Dic3—101no—
192C4⋊Dic3C22⋊D511yescharacteristic
193Dic3⋊C4—21no—
194Dic3⋊C4—102no—
195C4×Dic3—21no—
196C2×C3⋊C8—101no—
197#204—42no—
198#204—204no—
199C23×C5—121no—
200C22×D5—21no—
201C5×Q8—61no—
202C5×Q8—61no—
203C5×D4—122no—
204C5×D4—121no—
205C5×D4—121no—
206C5×D4—121no—
207C5×D4—121no—
208C2×C4×C5—61no—
209C2×C4×C5—61no—
210C2×C4×C5C2×D1211yescharacteristic
211C2×C4×C5D3×D411yescharacteristic
212C2×C4×C5D3×D411yescharacteristic
213C2×Dic5—21no—
214C2×Dic5—62no—
215D20—21no—
216D20#20412yescomplemented
217C4×D5—41no—
218C4×D5—41no—
219C4×D5—242no—
220Dic10—122no—
221C5×C8—41no—
222#168—22no—
223#168—121no—
224C2×SD16—61no—
225C2×SD16—302no—
226C2×D8—301no—
227C4⋊Q8—151no—
228#127—31no—
229#125—301no—
230C22⋊Q8—302no—
231C22⋊Q8—301no—
232C4×Q8—31no—
233C4×D4—51no—
234C4×D4—301no—
235C2×C4⋊C4—301no—
236C4⋊C8—151no—
237C4⋊C8D3×D511yescharacteristic
238#102—61no—
239#102—301no—
240C4×C8—51no—
241C2×C3×C5#44811yescharacteristic
242C2×C3×C5#44911yescharacteristic
243C2×C3×C5#45311yescharacteristic
244C3×D5—42no—
245C5×D3—161no—
246C23×C3—101no—
247C22×D3—41no—
248C3×D4—52no—
249C3×D4—101no—
250C2×C3×C4—101no—
251C2×C3×C4—201no—
252C2×C3×C4—201no—
253C2×C3×C4D4×D511yescharacteristic
254C2×C3×C4D4×D511yescharacteristic
255C2×C3×C4C2×C22⋊D511yescharacteristic
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—102no—
259C2×Dic3—402no—
260D12—42no—
261D12—41no—
262D12—41no—
263D12—41no—
264D12—41no—
265Dic6—21no—
266Dic6—21no—
267Dic6—202no—
268C3×C8—41no—
269C3×C8—102no—
270C3⋊C8—201no—
271C22×C5—241no—
272C22×C5—241no—
273C22×C5—241no—
274C22×C5#88211yescharacteristic
275C2×D5—22no—
276C2×D5—41no—
277C2×D5—41no—
278C4×C5—21no—
279C4×C5—121no—
280C4×C5—121no—
281C4×C5C2×#20411yescharacteristic
282C4×C5#90911yescharacteristic
283C4×C5#91111yescharacteristic
284C4×C5#91611yescharacteristic
285Dic5—41no—
286Dic5—122no—
287C2×Q8—31no—
288C2×Q8—152no—
289C2×D4—51no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—61no—
293C22×C4—101no—
294C22×C4—302no—
295SD16—122no—
296SD16—604no—
297D8—602no—
298C2×C8—21no—
299C2×C8—101no—
300C2×C8—301no—
301C4⋊C4—31no—
302C4⋊C4—51no—
303C4⋊C4—61no—
304C4⋊C4—302no—
305C4⋊C4—302no—
306C4⋊C4—301no—
307C4⋊C4—301no—
308C22⋊C4—101no—
309C22⋊C4—302no—
310C42—61no—
311C42C2×D3×D511yescharacteristic
312C3×C5#162211yescharacteristic, complemented, Hall
313C22×C3—102no—
314C22×C3—201no—
315C22×C3—201no—
316C22×C3#408311yescharacteristic
317C2×D3—81no—
318C2×D3—81no—
319C2×D3—81no—
320C3×C4—21no—
321C3×C4—201no—
322C3×C4D5×SD1611yescharacteristic
323C3×C4#406111yescharacteristic
324C3×C4#407811yescharacteristic
325C3×C4#409611yescharacteristic
326Dic3—41no—
327Dic3—41no—
328Dic3—202no—
329C2×C5—481no—
330C2×C5#473811yescharacteristic
331C2×C5#478111yescharacteristic
332C2×C5#482211yescharacteristic
333D5—42no—
334C23—101no—
335C23—121no—
336Q8—61no—
337Q8—61no—
338Q8—602no—
339D4—52no—
340D4—101no—
341D4—122no—
342D4—121no—
343D4—121no—
344D4—121no—
345D4—121no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—101no—
349C2×C4—201no—
350C2×C4—201no—
351C2×C4—302no—
352C2×C4—1202no—
353C2×C4D5×D1211yescharacteristic
354C2×C4#677711yescharacteristic
355C2×C4D3×C22⋊D511yescharacteristic
356C8—41no—
357C8—102no—
358C8—601no—
359C2×C3—202no—
360C2×C3#6532611yescharacteristic
361C2×C3#6545911yescharacteristic
362C2×C3#6548711yescharacteristic
363D3—161no—
364C5#6819811yescharacteristic, complemented, Sylow
365C22—102no—
366C22—201no—
367C22—201no—
368C22—241no—
369C22—241no—
370C22—241no—
371C22#9147511yescharacteristic
372C4—21no—
373C4—121no—
374C4—121no—
375C4—201no—
376C4—602no—
377C4D5×#20411yescharacteristic
378C4#9126011yescharacteristic
379C4#9149411yescharacteristic
380C4#9151711yescharacteristic
381C3#1060182111yescharacteristic, complemented, Sylow
382C2—202no—
383C2—481no—
384C2#1174751511yescharacteristic
385C2#1174812711yescharacteristic
386C2#1174824711yescharacteristic
subgroupG/H
centerC22#91475
commutator, cosocleC3×C42×C5C23
FrattiniC42C2×D3×D5
FittingC3×C5×C4⋊C8C22
socleC22×C3×C5#120
prime radical#11747026C2
hypercenterC4⋊C8D3×D5
hypocenterC3×C5#1622
LCSG ▷ C3×C42×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
1. #11752657C2semidirect
2. C5×#4848C2semidirect
3. #11747066C2semidirect
5. #11746302C2semidirect
6. C3×#64703C2semidirect
7. #11745843C2semidirect
32. #64703D3semidirect
61. #4848D5semidirect
102. C3×D20SD16semidirect
216. D20#204semidirect
312. C3×C5#1622semidirect
364. C5#68198semidirect
381. C3#10601821semidirect

Maximal subgroups

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

subgroupindex
#117526572
C5×#48482
#117470662
#117470262
#117463022
C3×#647032
#117458432
#106018213
#681985