Finite Group Databasebeta

← #14289489

Subgroups — #14289489

Order 1920.

428 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#554C211yesmaximal, characteristic
2#11751193C211yesmaximal, characteristic, complemented
3#11751122C211yesmaximal, characteristic, complemented
4#11751109C211yesmaximal, characteristic
5#11746367C211yesmaximal, characteristic, complemented
6#11746356C211yesmaximal, characteristic, complemented
7D4⋊(C3⋊#168)C211yesmaximal, characteristic
8#10602717—31nomaximal, Hall
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×C5×#125—21no—
11C3×C5×C22⋊Q8—21no—
12C3×C5×C22⋊Q8—22no—
13C3×C4×C5×Q8C2211yescharacteristic
14C3×C4×C5×D4—21no—
15C3×C4×C5×D4C2211yescharacteristic
16C2×C3×C5×C4⋊C4—21no—
17C2×#6718—22no—
18#91758C2211yescharacteristic
19#91119—21no—
20#91116—21no—
21#91092—21no—
22#91091—21no—
23C4⋊(C3⋊#168)C2211yescharacteristic
24C4⋊(C3⋊#168)C2211yescharacteristic
25C4×C3⋊#168C2211yescharacteristic
26#71924—51nomaximal, Hall
27C5×#554D311yescharacteristic
28#65381—31no—
29#65310—31no—
30#65297—31no—
31#64766—31no—
32Q8⋊#168—31no—
33D4⋊#168—31no—
34C2×C3×C5×Q8D411yescharacteristic
35C2×C3×C5×Q8D412yes—
36C2×C3×C5×D4D411yescharacteristic
37C22×C3×C4×C5—22no—
38C22×C3×C4×C5—21no—
39C2×Dic30—21no—
40C3×C5×C4⋊C4—22no—
41C3×C5×C4⋊C4—22no—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4—21no—
44C3×C5×C4⋊C4—21no—
45C3×C5×C4⋊C4D411yescharacteristic
46C3×C5×C4⋊C4D411yescharacteristic
47C3×C5×C22⋊C4—22no—
48C3×C5×C22⋊C4—21no—
49C3×C42×C5—21no—
50C3×C42×C5C2311yescharacteristic
51#6718—44no—
52C4⋊Dic15—21no—
53C4⋊Dic15—21no—
54C2×C3⋊#168—21no—
55C2×C3⋊#168—21no—
56C2×C3⋊#168—21no—
57C3×#554D511yescharacteristic
58#5092—51no—
59#5021—51no—
60#5008—51no—
61#4546—51no—
62#4535—51no—
63D4⋊(C3⋊C8)—51no—
64C5×C4⋊Q8C2×D311yescharacteristic
65C5×#125—21no—
66C5×C22⋊Q8—21no—
67C5×C22⋊Q8—22no—
68C4×C5×Q8C2×D311yescharacteristic
69C4×C5×D4—21no—
70C4×C5×D4C2×D311yescharacteristic
71C2×C5×C4⋊C4—21no—
72C2×#689—62no—
73#4015—31no—
74#3967—61no—
75#3964—61no—
76#3940—61no—
77#3939—61no—
78C4⋊#168—31no—
79C4⋊#168—31no—
80C4×#168—31no—
81#1632—151noSylow
82C23×C3×C5—21no—
83C3×C5×Q8—21no—
84C3×C5×Q8—21no—
85C3×C5×Q8—42no—
86C3×C5×D4—21no—
87C3×C5×D4Q1612yescomplemented
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5—22no—
92C2×C3×C4×C5—41no—
93C2×C3×C4×C5—41no—
94C2×C3×C4×C5—82no—
95C2×C3×C4×C5C2×D411yescharacteristic
96C2×C3×C4×C5C2×D411yescharacteristic
97C2×C3×C4×C5C2×D411yescharacteristic
98C2×Dic15—41no—
99Dic30—42no—
100C3⋊#168—22no—
101C3⋊#168—41no—
102C3⋊#168—41no—
103C3×C4⋊Q8C2×D511yescharacteristic
104C3×#125—21no—
105C3×C22⋊Q8—21no—
106C3×C22⋊Q8—22no—
107C3×C4×Q8C2×D511yescharacteristic
108C3×C4×D4—21no—
109C3×C4×D4C2×D511yescharacteristic
110C2×C3×C4⋊C4—21no—
111C2×#216—102no—
112#869—51no—
113#835—101no—
114#832—101no—
115#808—101no—
116#807—101no—
117C4⋊(C3⋊C8)—51no—
118C4⋊(C3⋊C8)—51no—
119C4×C3⋊C8—51no—
120C2×C5×Q8#6711yescharacteristic
121C2×C5×Q8#6712yes—
122C2×C5×D4#6711yescharacteristic
123C22×C4×C5—22no—
124C22×C4×C5—21no—
125C2×Dic10—61no—
126C5×C4⋊C4—22no—
127C5×C4⋊C4—22no—
128C5×C4⋊C4—21no—
129C5×C4⋊C4—21no—
130C5×C4⋊C4—21no—
131C5×C4⋊C4#6711yescharacteristic
132C5×C4⋊C4#6711yescharacteristic
133C5×C22⋊C4—22no—
134C5×C22⋊C4—21no—
135C42×C5—21no—
136C42×C5C22×D311yescharacteristic
137#689—124no—
138C4⋊Dic5—61no—
139C4⋊Dic5—61no—
140C2×#168—61no—
141C2×#168—61no—
142C2×#168—61no—
143#554D1511yescharacteristic
144#494—151no—
145#477—151no—
146#476—151no—
147#333—151no—
148Q8⋊C8—151no—
149D4⋊C8—151no—
150C22×C3×C5—22no—
151C22×C3×C5—41no—
152C22×C3×C5—41no—
153C22×C3×C5#12011yescharacteristic
154C3×C4×C5—21no—
155C3×C4×C5—41no—
156C3×C4×C5—41no—
157C3×C4×C5—41no—
158C3×C4×C5—42no—
159C3×C4×C5C2×Q1611yescharacteristic
160C3×C4×C5#13511yescharacteristic
161C3×C4×C5#13711yescharacteristic
162C3×C4×C5#13711yescharacteristic
163Dic15—81no—
164C2×C3×Q8C22⋊D511yescharacteristic
165C2×C3×Q8C22⋊D512yes—
166C2×C3×D4C22⋊D511yescharacteristic
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C2×Dic6—101no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4C22⋊D511yescharacteristic
176C3×C4⋊C4C22⋊D511yescharacteristic
177C3×C22⋊C4—22no—
178C3×C22⋊C4—21no—
179C3×C42—21no—
180C3×C42C22×D511yescharacteristic
181#216—204no—
182C4⋊Dic3—101no—
183C4⋊Dic3—101no—
184C2×C3⋊C8—101no—
185C2×C3⋊C8—101no—
186C2×C3⋊C8—101no—
187C23×C5—21no—
188C5×Q8—21no—
189C5×Q8—21no—
190C5×Q8—42no—
191C5×D4—21no—
192C5×D4#21612yescomplemented
193C2×C4×C5—21no—
194C2×C4×C5—21no—
195C2×C4×C5—21no—
196C2×C4×C5—22no—
197C2×C4×C5—41no—
198C2×C4×C5—41no—
199C2×C4×C5—82no—
200C2×C4×C5C2×#6711yescharacteristic
201C2×C4×C5C2×#6711yescharacteristic
202C2×C4×C5C2×#6711yescharacteristic
203C2×Dic5—121no—
204Dic10—122no—
205#168—62no—
206#168—121no—
207#168—121no—
208C2×Q16—302no—
209C4⋊Q8—151no—
210C4⋊Q8C2×D1511yescharacteristic
211#125—21no—
212C22⋊Q8—21no—
213C22⋊Q8—22no—
214C4×Q8C2×D1511yescharacteristic
215C4×D4—21no—
216C4×D4C2×D1511yescharacteristic
217C2×C4⋊C4—21no—
218#107—301no—
219#106—301no—
220C4⋊C8—151no—
221C4⋊C8—151no—
222#103—301no—
223#102—301no—
224C4×C8—151no—
225C2×C3×C5—42no—
226C2×C3×C5#45111yescharacteristic
227C2×C3×C5#45211yescharacteristic
228C2×C3×C5#45311yescharacteristic
229C23×C3—21no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—42no—
233C3×D4—21no—
234C3×D4#68912yescomplemented
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—22no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—82no—
242C2×C3×C4C2×C22⋊D511yescharacteristic
243C2×C3×C4C2×C22⋊D511yescharacteristic
244C2×C3×C4C2×C22⋊D511yescharacteristic
245C2×Dic3—201no—
246Dic6—202no—
247C3⋊C8—102no—
248C3⋊C8—201no—
249C3⋊C8—201no—
250C22×C5—22no—
251C22×C5—41no—
252C22×C5—41no—
253C22×C5#95311yescharacteristic
254C4×C5—21no—
255C4×C5—41no—
256C4×C5—41no—
257C4×C5—41no—
258C4×C5—42no—
259C4×C5#94211yescharacteristic
260C4×C5C2×#21611yescharacteristic
261C4×C5#95011yescharacteristic
262C4×C5#95111yescharacteristic
263Dic5—241no—
264C2×Q8—301no—
265C2×Q8#122011yescharacteristic
266C2×Q8#122012yes—
267C2×D4#122011yescharacteristic
268C22×C4—22no—
269C22×C4—21no—
270Q16—604no—
271C2×C8—301no—
272C2×C8—301no—
273C2×C8—301no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—301no—
280C4⋊C4—301no—
281C4⋊C4#122011yescharacteristic
282C4⋊C4#122011yescharacteristic
283C22⋊C4—22no—
284C22⋊C4—21no—
285C42—21no—
286C42C22×D1511yescharacteristic
287C3×C5#163211yescharacteristic, complemented, Hall
288C22×C3—22no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3#409911yescharacteristic
292C3×C4—21no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—42no—
297C3×C4#408811yescharacteristic
298C3×C4C2×#68911yescharacteristic
299C3×C4#409611yescharacteristic
300C3×C4#409711yescharacteristic
301Dic3—401no—
302C2×C5—42no—
303C2×C5#522811yescharacteristic
304C2×C5#523911yescharacteristic
305C2×C5#524111yescharacteristic
306C23—21no—
307Q8—21no—
308Q8—21no—
309Q8—42no—
310Q8—602no—
311D4—21no—
312D4#671812yescomplemented
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—82no—
320C2×C4—601no—
321C2×C4C2×#122011yescharacteristic
322C2×C4C2×#122011yescharacteristic
323C2×C4C2×#122011yescharacteristic
324C8—302no—
325C8—601no—
326C8—601no—
327C2×C3—42no—
328C2×C3#6551711yescharacteristic
329C2×C3#6552811yescharacteristic
330C2×C3#6553011yescharacteristic
331C5#7192411yescharacteristic, complemented, Sylow
332C22—22no—
333C22—41no—
334C22—41no—
335C22#9184211yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—42no—
341C4—1201no—
342C4#9183111yescharacteristic
343C4C2×#671811yescharacteristic
344C4#9183911yescharacteristic
345C4#9184011yescharacteristic
346C3#1060271711yescharacteristic, complemented, Sylow
347C2—42no—
348C2#1175132911yescharacteristic
349C2#1175134011yescharacteristic
350C2#1175134211yescharacteristic
subgroupG/H
centerC22#91842
commutator, cosocleC3×C42×C5C23
FrattiniC42C22×D15
FittingC3×C5×#554C2
socleC22×C3×C5#120
prime radicalC2×C3×C5×D4D4
hypercenter#554D15
hypocenterC3×C5#1632
LCSG ▷ C3×C42×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #11751193C2semidirect
3. #11751122C2semidirect
5. #11746367C2semidirect
6. #11746356C2semidirect
87. C3×C5×D4Q16semidirect
192. C5×D4#216semidirect
234. C3×D4#689semidirect
287. C3×C5#1632semidirect
312. D4#6718semidirect
331. C5#71924semidirect
346. C3#10602717semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5542
#117511932
#117511222
#117511092
#117463672
#117463562
D4⋊(C3⋊#168)2
#106027173
#719245