Finite Group Databasebeta

← #14298617

Subgroups — #14298617

Order 1920.

524 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C5×#235C211yesmaximal, characteristic
2#11748625C211yesmaximal, characteristic
3#11748355C211yesmaximal, characteristic
4#11746344C211yesmaximal, characteristic, complemented
5C3×#64743C211yesmaximal, characteristic, complemented
6#11745971C211yesmaximal, characteristic, complemented
7#11745969C211yesmaximal, characteristic, complemented
8#10602437—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C2×C3×C42×C5C2211yescharacteristic
11#91815C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C4×C5×#67—22no—
14C5×#926C2211yescharacteristic
15C4×C5×D12—21no—
16C5×#872C412yescomplemented
17C42×C5×D3C412yescomplemented
18C4×C5×Dic6—21no—
19C3×#4072C2211yescharacteristic
20C2×C3×#681C2211yescharacteristic
21#90996—22no—
22#90962—21no—
23#90959—21no—
24#68737—51nomaximal, Hall
25C4×C5×#41—31no—
26#65529—31no—
27#65526—31no—
28#64751—31no—
29#64751—31no—
30#64743—31no—
31#64743D311yescharacteristic, complemented
32C22×C3×C4×C5—21no—
33C22×C3×C4×C5C2311yescharacteristic
34C2×C5×#67D412yes—
35C5×#235—21no—
36C5×#235—21no—
37C5×#235—21no—
38C5×#235—21no—
39C2×C5×D12D411yescharacteristic
40C2×C4×C5×D3—22no—
41C2×C4×C5×D3C2×C412yes—
42C2×C5×Dic6D411yescharacteristic
43C3×C42×C5—21no—
44C3×C42×C5C2×C411yescharacteristic
45C3×C42×C5C2×C411yescharacteristic
46#6719—21no—
47C4⋊Dic15—21no—
48C4×Dic15—21no—
49#6710—21no—
50#6710—21no—
51C2×C3⋊#168—21no—
52C5×#217—21no—
53C5×(C2×D3)⋊C4—22no—
54C5×C4⋊Dic3—21no—
55C5×Dic3⋊C4—22no—
56C4×C5×Dic3—22no—
57C4×C5×Dic3C2×C412yes—
58C3×C22⋊Dic5—21no—
59C3×C4⋊Dic5—21no—
60C3×C4×Dic5—21no—
61C3×#681—21no—
62C3×#681—21no—
63C2×C3×#168—21no—
64C4×#235D511yescharacteristic
65#5132—51no—
66#5006—51no—
67C3×#339—51no—
68#4557—51no—
69#4531—51no—
70#4523—51no—
71C2×C5×#41—31no—
72C4×C5×Q8—61no—
73C4×C5×D4—61no—
74C4×C5×D4—62no—
75C5×#117—31no—
76C5×#117—32no—
77C2×C42×C5—32no—
78C2×C42×C5C2×D311yescharacteristic
79#4072—31no—
80#4072C2×D311yescharacteristic
81C2×#681—31no—
82C2×#681C2×D311yescharacteristic
83#3968—61no—
84#3966—62no—
85#3965—61no—
86#1827—151noSylow
87C23×C3×C5C2×D411yescharacteristic
88C22×C5×D3C22⋊C412yes—
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5—42no—
92C2×C3×C4×C5C22×C411yescharacteristic
93C2×C3×C4×C5C2×D411yescharacteristic
94C2×C3×C4×C5#4111yescharacteristic
95C2×C3×C4×C5#4111yescharacteristic
96C2×C3×C4×C5#4111yescharacteristic
97C2×C3×C4×C5#4111yescharacteristic
98C2×Dic15—41no—
99C5×#67—22no—
100C5×#67—22no—
101C2×C5×Dic3—22no—
102C2×C5×Dic3C22⋊C412yes—
103C5×D12—21no—
104C5×D12—21no—
105C4×C5×D3—22no—
106C4×C5×D3—22no—
107C4×C5×D3—44no—
108C5×Dic6—21no—
109C5×Dic6—21no—
110C2×C3×Dic5—41no—
111C3⋊#168—41no—
112C3×#168—41no—
113C2×#235C2×D511yescharacteristic
114C2×C3×M16—51no—
115C3×#117—51no—
116C2×C3×C42C2×D511yescharacteristic
117C4×#67—22no—
118#926—51no—
119#926C2×D511yescharacteristic
120C2×#208—51no—
121C4×D12—21no—
122#872Dic512yescomplemented
123C42×D3Dic512yescomplemented
124C4×Dic6—21no—
125#824—101no—
126#823—101no—
127#806—102no—
128C5×#41—61no—
129C5×#41—61no—
130C5×#41—61no—
131C5×#41—61no—
132C2×C5×Q8—31no—
133C2×C5×D4—32no—
134C2×C5×D4—31no—
135C22×C4×C5—21no—
136C22×C4×C5—32no—
137C22×C4×C5—62no—
138C22×C4×C5C22×D311yescharacteristic
139C5×C4⋊C4—62no—
140C5×C4⋊C4—61no—
141C5×C22⋊C4—62no—
142C5×C22⋊C4—61no—
143C42×C5—21no—
144C42×C5—32no—
145C42×C5—62no—
146C42×C5C4×D311yescharacteristic
147C42×C5C4×D311yescharacteristic
148C22⋊Dic5—21no—
149C22⋊Dic5—61no—
150C4⋊Dic5—21no—
151C4⋊Dic5—61no—
152C4×Dic5—21no—
153C4×Dic5—61no—
154#681—21no—
155#681—21no—
156#681—61no—
157#681—61no—
158C2×#168—21no—
159C2×#168—61no—
160C4×#41—31no—
161#413—151no—
162#410—151no—
163#339—51no—
164#339—151no—
165H(3,ℤ/4)—151no—
166H(3,ℤ/4)—151no—
167C22×C3×C5—41no—
168C22×C3×C5C2×C22⋊C411yescharacteristic
169C22×C3×C5#12311yescharacteristic
170C22×C3×C5#12311yescharacteristic
171C2×C5×D3—22no—
172C2×C5×D3—42no—
173C3×C4×C5—22no—
174C3×C4×C5—41no—
175C3×C4×C5#11711yescharacteristic
176C3×C4×C5#11711yescharacteristic
177C3×C4×C5#12311yescharacteristic
178C3×C4×C5#12311yescharacteristic
179Dic15—81no—
180C3×Dic5—81no—
181C5×Dic3—22no—
182C5×Dic3—42no—
183C22×C3×C4—21no—
184C22×C3×C4C22×D511yescharacteristic
185C2×#67C22⋊D512yes—
186#235—21no—
187#235—21no—
188#235—21no—
189#235—21no—
190C2×D12C22⋊D511yescharacteristic
191C2×C4×D3—22no—
192C2×C4×D3C2×Dic512yes—
193C2×Dic6C22⋊D511yescharacteristic
194C3×M16—101no—
195C3×M16—101no—
196C2×C3×C8—101no—
197C3×C4⋊C4—101no—
198C3×C22⋊C4—101no—
199C3×C42—21no—
200C3×C42—101no—
201C3×C42C2×Dic511yescharacteristic
202C3×C42C2×Dic511yescharacteristic
203#217—21no—
204#217—101no—
205(C2×D3)⋊C4—22no—
206C4⋊Dic3—21no—
207C4⋊Dic3—101no—
208Dic3⋊C4—22no—
209C4×Dic3—22no—
210C4×Dic3—101no—
211C4×Dic3C2×Dic512yes—
212#208—101no—
213#208—101no—
214C2×C3⋊C8—101no—
215C23×C5—32no—
216C23×C5D3×D411yescharacteristic
217C5×Q8—61no—
218C5×Q8—61no—
219C5×D4—62no—
220C5×D4—62no—
221C5×D4—61no—
222C5×D4—61no—
223C2×C4×C5—21no—
224C2×C4×C5—21no—
225C2×C4×C5—32no—
226C2×C4×C5—42no—
227C2×C4×C5—62no—
228C2×C4×C5—62no—
229C2×C4×C5—62no—
230C2×C4×C5—124no—
231C2×C4×C5C2×C4×D311yescharacteristic
232C2×C4×C5#23511yescharacteristic
233C2×C4×C5#23511yescharacteristic
234C2×C4×C5D3×D411yescharacteristic
235C2×C4×C5#23711yescharacteristic
236C2×C4×C5#23911yescharacteristic
237C2×Dic5—41no—
238C2×Dic5—121no—
239#168—41no—
240#168—121no—
241C2×#41—31no—
242C2×M16—51no—
243C2×M16—151no—
244C4×Q8—61no—
245C4×D4—61no—
246C4×D4—62no—
247#117—31no—
248#117—32no—
249#117—51no—
250#117—151no—
251C2×C42—32no—
252C2×C42D3×D511yescharacteristic
253#101—301no—
254#100—301no—
255#99—302no—
256C2×C3×C5—21no—
257C2×C3×C5—21no—
258C2×C3×C5—21no—
259C2×C3×C5#38111yescharacteristic
260C5×D3—42no—
261C23×C3C2×C22⋊D511yescharacteristic
262C22×D3C22⋊Dic512yes—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—42no—
266C2×C3×C4—201no—
267C2×C3×C4#70911yescharacteristic
268C2×C3×C4#70911yescharacteristic
269C2×C3×C4#70911yescharacteristic
270C2×C3×C4#70911yescharacteristic
271C2×C3×C4C22×Dic511yescharacteristic
272C2×C3×C4C2×C22⋊D511yescharacteristic
273#67—22no—
274#67—22no—
275C2×Dic3—22no—
276C2×Dic3—201no—
277C2×Dic3C22⋊Dic512yes—
278D12—21no—
279D12—21no—
280C4×D3—22no—
281C4×D3—22no—
282C4×D3—44no—
283Dic6—21no—
284Dic6—21no—
285C3×C8—201no—
286C3⋊C8—201no—
287C22×C5—41no—
288C22×C5—62no—
289C22×C5—122no—
290C22×C5D3×C22⋊C411yescharacteristic
291C22×C5#88311yescharacteristic
292C22×C5#89411yescharacteristic
293C4×C5—22no—
294C4×C5—41no—
295C4×C5—62no—
296C4×C5—122no—
297C4×C5#87211yescharacteristic
298C4×C5#88311yescharacteristic
299C4×C5#89211yescharacteristic
300C4×C5#89411yescharacteristic
301Dic5—81no—
302Dic5—241no—
303#41—61no—
304#41—61no—
305#41—61no—
306#41—61no—
307C2×Q8—31no—
308C2×D4—32no—
309C2×D4—31no—
310C22×C4—21no—
311C22×C4—32no—
312C22×C4—62no—
313C22×C4C2×D3×D511yescharacteristic
314M16—101no—
315M16—101no—
316M16—301no—
317M16—301no—
318C2×C8—101no—
319C2×C8—301no—
320C4⋊C4—62no—
321C4⋊C4—61no—
322C4⋊C4—101no—
323C4⋊C4—301no—
324C22⋊C4—62no—
325C22⋊C4—61no—
326C22⋊C4—101no—
327C22⋊C4—301no—
328C42—21no—
329C42—32no—
330C42—62no—
331C42—101no—
332C42—301no—
333C42D3×Dic511yescharacteristic
334C42D3×Dic511yescharacteristic
335C3×C5#182711yescharacteristic, complemented, Hall
336C22×C3—41no—
337C22×C3#407511yescharacteristic
338C22×C3#407511yescharacteristic
339C22×C3C2×C22⋊Dic511yescharacteristic
340C2×D3—22no—
341C2×D3—42no—
342C3×C4—22no—
343C3×C4—41no—
344C3×C4—401no—
345C3×C4#407211yescharacteristic
346C3×C4#407211yescharacteristic
347C3×C4#407511yescharacteristic
348C3×C4#407511yescharacteristic
349Dic3—22no—
350Dic3—42no—
351Dic3—401no—
352C2×C5—21no—
353C2×C5—21no—
354C2×C5—21no—
355C2×C5—122no—
356C2×C5#466411yescharacteristic
357C23—32no—
358C23D3×C22⋊D511yescharacteristic
359Q8—61no—
360Q8—61no—
361D4—62no—
362D4—62no—
363D4—61no—
364D4—61no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—32no—
368C2×C4—42no—
369C2×C4—62no—
370C2×C4—62no—
371C2×C4—62no—
372C2×C4—124no—
373C2×C4—201no—
374C2×C4—601no—
375C2×C4#676511yescharacteristic
376C2×C4#676911yescharacteristic
377C2×C4#677111yescharacteristic
378C2×C4#677111yescharacteristic
379C2×C4C2×D3×Dic511yescharacteristic
380C2×C4D3×C22⋊D511yescharacteristic
381C8—201no—
382C8—601no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3#6550011yescharacteristic
387D3—42no—
388C5#6873711yescharacteristic, complemented, Sylow
389C22—41no—
390C22—62no—
391C22—122no—
392C22#9135111yescharacteristic
393C22#9145211yescharacteristic
394C22D3×C22⋊Dic511yescharacteristic
395C4—22no—
396C4—41no—
397C4—62no—
398C4—122no—
399C4—401no—
400C4—1201no—
401C4#9133811yescharacteristic
402C4#9135111yescharacteristic
403C4#9137111yescharacteristic
404C4#9145211yescharacteristic
405C3#1060243711yescharacteristic, complemented, Sylow
406C2—21no—
407C2—21no—
408C2—21no—
409C2—122no—
410C2#1174877711yescharacteristic
subgroupG/H
centerC4#91338
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C42×C5C22
socleC2×C3×C5#381
prime radicalC2×C5×#235C22
cosocleC22×C3×C4×C5C23
hypercenterC2×C42D3×D5
hypocenterC3×C5#1827
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
4. #11746344C2semidirect
5. C3×#64743C2semidirect
6. #11745971C2semidirect
7. #11745969C2semidirect
16. C5×#872C4semidirect
17. C42×C5×D3C4semidirect
31. #64743D3semidirect
122. #872Dic5semidirect
123. C42×D3Dic5semidirect
335. C3×C5#1827semidirect
388. C5#68737semidirect
405. C3#10602437semidirect

Maximal subgroups

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

subgroupindex
C4×C5×#2352
#117486252
#117483552
#117463442
C3×#647432
#117459712
#117459692
#106024373
#687375