Finite Group Databasebeta

← #14309036

Subgroups — #14309036

Order 1920.

488 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
1C5×#5517C211yesmaximal, characteristic, complemented
2C3×#65294C211yesmaximal, characteristic
3#11748624C211yesmaximal, characteristic
4C2×#90977C211yesmaximal, characteristic
5#11746352C211yesmaximal, characteristic, complemented
6#11745988C211yesmaximal, characteristic, complemented
7#11745976C211yesmaximal, characteristic, complemented
8#10602635—31nomaximal, Hall
9C2×C3×#709C2211yescharacteristic
10C3×C5×#117C2211yescharacteristic
11#91815C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C5×#926C2211yescharacteristic
14C2×C4×C5×Dic3C2211yescharacteristic
15C5×#890—21no—
16C5×#888—21no—
17C5×#878—22no—
18C4×C5×Dic6—22no—
19C2×C3×#681C2211yescharacteristic
20C3×#3937—22no—
21#90995—22no—
22#90977—21no—
23#90977—21no—
24#70394—51nomaximal, Hall
25C5×#533—31no—
26#65295—31no—
27#65294D311yescharacteristic
28C2×#3937—31no—
29#64753—31no—
30#64751—31no—
31#64743—31no—
32C22×C3×C4×C5C2311yescharacteristic
33C22×C5×Dic3—21no—
34C2×C5×Dic6—22no—
35C2×C3×C22⋊D5—21no—
36C3×#709—21no—
37C3×#709—21no—
38C3×#709—21no—
39C3×#709—21no—
40C2×C3×D20Q811yescharacteristic, complemented
41C2×C3×C4×D5—21no—
42C2×C3×Dic10Q811yescharacteristic, complemented
43C3×C5×C4⋊C4—21no—
44C3×C5×C22⋊C4—21no—
45C3×C42×C5D412yes—
46#6719—21no—
47C4⋊Dic15—21no—
48C4×Dic15—21no—
49#6710—21no—
50#6710—21no—
51C2×C3⋊#168—21no—
52C5×#217D412yes—
53C5×C4⋊Dic3—22no—
54C5×C4⋊Dic3—21no—
55C5×Dic3⋊C4—22no—
56C5×Dic3⋊C4—22no—
57C4×C5×Dic3—21no—
58C4×C5×Dic3—21no—
59C4×C5×Dic3D411yescharacteristic
60C4×C5×Dic3D411yescharacteristic
61C3×#681—21no—
62C3×#681—21no—
63C2×C3×#168—21no—
64#5517D511yescharacteristic, complemented
65C3×#421—51no—
66#5240—51no—
67C2×#837—51no—
68#4556—51no—
69#4555—51no—
70#4531—51no—
71C2×#709C2×D311yescharacteristic
72C5×C4⋊Q8—61no—
73C5×#125—61no—
74C5×C22⋊Q8—62no—
75C4×C5×Q8—62no—
76C5×#117—31no—
77C5×#117C2×D311yescharacteristic
78C2×C42×C5—31no—
79#4072—31no—
80C2×#681—31no—
81C2×#681C2×D311yescharacteristic
82#3938—62no—
83#3937—22no—
84#3937—61no—
85#3937—61no—
86#2031—151noSylow
87C23×C3×C5C2×D411yescharacteristic
88C22×C3×D5—21no—
89C2×C3×C4×C5—22no—
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×C3×C4×C5C2×Q811yescharacteristic
93C2×C3×C4×C5#4111yescharacteristic
94C2×C3×C4×C5#4111yescharacteristic
95C2×C3×C4×C5#4111yescharacteristic
96C2×Dic15—41no—
97C2×C5×Dic3—21no—
98C2×C5×Dic3—21no—
99C2×C5×Dic3—22no—
100C2×C5×Dic3—42no—
101C5×Dic6—44no—
102C3×C22⋊D5—41no—
103C3×C22⋊D5—41no—
104C2×C3×Dic5—21no—
105C3×D20—21no—
106C3×D20—21no—
107C3×C4×D5—41no—
108C3×C4×D5—41no—
109C3×Dic10—21no—
110C3×Dic10—21no—
111C3⋊#168—41no—
112C3×#168—41no—
113C2×C3×#41—51no—
114C2×C3×M16—51no—
115C3×#117C2×D511yescharacteristic
116#926—51no—
117#926C2×D511yescharacteristic
118C2×C4×Dic3C2×D511yescharacteristic
119C2×#208—51no—
120#890—21no—
121#888—21no—
122#878—22no—
123C4×Dic6—22no—
124C3×#104—102no—
125#837—101no—
126#837—101no—
127#834—102no—
128C2×C5×Q8—62no—
129C22×C4×C5—61no—
130C22×C4×C5C22×D311yescharacteristic
131C2×C22⋊D5—21no—
132#709—21no—
133#709—21no—
134#709—21no—
135#709—21no—
136C2×D20Dic611yescharacteristic, complemented
137C2×C4×D5—21no—
138C2×Dic10Dic611yescharacteristic, complemented
139C5×C4⋊C4—21no—
140C5×C4⋊C4—62no—
141C5×C4⋊C4—62no—
142C5×C4⋊C4—62no—
143C5×C4⋊C4—61no—
144C5×C22⋊C4—21no—
145C5×C22⋊C4—32no—
146C42×C5—31no—
147C42×C5—31no—
148C42×C5—61no—
149C42×C5—61no—
150C42×C5#6712yes—
151C22⋊Dic5—61no—
152C4⋊Dic5—61no—
153C4×Dic5—61no—
154#681—21no—
155#681—21no—
156#681—61no—
157#681—61no—
158C2×#168—21no—
159C2×#168—61no—
160#533—31no—
161#421—51no—
162C2×#104—151no—
163#410—151no—
164#344—151no—
165#339—151no—
166H(3,ℤ/4)—151no—
167C22×C3×C5—41no—
168C22×C3×C5#12011yescharacteristic
169C22×C3×C5C22⋊Q811yescharacteristic
170C22×C3×C5#12411yescharacteristic
171C2×C3×D5—41no—
172C2×C3×D5—81no—
173C3×C4×C5—42no—
174C3×C4×C5C22⋊Q811yescharacteristic
175C3×C4×C5C22⋊Q811yescharacteristic
176C3×C4×C5#12411yescharacteristic
177C3×C4×C5#12411yescharacteristic
178Dic15—81no—
179C3×Dic5—41no—
180C5×Dic3—22no—
181C5×Dic3—42no—
182C5×Dic3—41no—
183C3×#41—101no—
184C3×#41—101no—
185C3×#41—101no—
186C3×#41—101no—
187C2×C3×Q8—51no—
188C2×C3×D4—51no—
189C2×C3×D4—101no—
190C22×C3×C4—101no—
191C22×C3×C4C22×D511yescharacteristic
192C22×Dic3—21no—
193C2×Dic6—22no—
194C3×M16—101no—
195C3×M16—101no—
196C2×C3×C8—101no—
197C3×C4⋊C4—21no—
198C3×C22⋊C4—21no—
199C3×C42C22⋊D512yes—
200#217—101no—
201#217C22⋊D512yes—
202C4⋊Dic3—22no—
203C4⋊Dic3—21no—
204C4⋊Dic3—101no—
205Dic3⋊C4—22no—
206Dic3⋊C4—22no—
207C4×Dic3—21no—
208C4×Dic3—21no—
209C4×Dic3—101no—
210C4×Dic3C22⋊D511yescharacteristic
211C4×Dic3C22⋊D511yescharacteristic
212#208—101no—
213#208—101no—
214C2×C3⋊C8—101no—
215C23×C5D3×D411yescharacteristic
216C22×D5—21no—
217C5×Q8—124no—
218C2×C4×C5—22no—
219C2×C4×C5—61no—
220C2×C4×C5—61no—
221C2×C4×C5—62no—
222C2×C4×C5—122no—
223C2×C4×C5C2×Dic611yescharacteristic
224C2×C4×C5#23511yescharacteristic
225C2×C4×C5D3×D411yescharacteristic
226C2×C4×C5#23711yescharacteristic
227C2×C4×C5#23711yescharacteristic
228C2×C4×C5C2×#6711yescharacteristic
229C22⋊D5—41no—
230C22⋊D5—41no—
231C2×Dic5—21no—
232C2×Dic5—121no—
233D20—21no—
234D20—21no—
235C4×D5—41no—
236C4×D5—41no—
237Dic10—21no—
238Dic10—21no—
239#168—41no—
240#168—121no—
241C2×#41—51no—
242C2×M16—51no—
243C2×M16—151no—
244C4⋊Q8—61no—
245#125—61no—
246C22⋊Q8—62no—
247C4×Q8—62no—
248#117—31no—
249#117—151no—
250#117D3×D511yescharacteristic
251C2×C42—31no—
252#104—102no—
253#104—301no—
254#104—301no—
255#99—302no—
256C2×C3×C5—21no—
257C2×C3×C5—21no—
258C2×C3×C5—21no—
259C2×C3×C5#39311yescharacteristic
260C3×D5—81no—
261C23×C3—101no—
262C23×C3C2×C22⋊D511yescharacteristic
263C3×Q8—101no—
264C3×Q8—101no—
265C3×D4—101no—
266C3×D4—101no—
267C3×D4—201no—
268C3×D4—201no—
269C2×C3×C4—22no—
270C2×C3×C4—101no—
271C2×C3×C4—201no—
272C2×C3×C4—201no—
273C2×C3×C4Q8×D511yescharacteristic
274C2×C3×C4#71311yescharacteristic
275C2×C3×C4#71311yescharacteristic
276C2×C3×C4#71311yescharacteristic
277C2×C3×C4C2×C22⋊D511yescharacteristic
278C2×C3×C4C2×C22⋊D511yescharacteristic
279C2×Dic3—21no—
280C2×Dic3—21no—
281C2×Dic3—22no—
282C2×Dic3—42no—
283C2×Dic3—201no—
284Dic6—44no—
285C3×C8—201no—
286C3⋊C8—201no—
287C22×C5—41no—
288C22×C5#87811yescharacteristic
289C22×C5#88511yescharacteristic
290C22×C5#93711yescharacteristic
291C2×D5—41no—
292C2×D5—81no—
293C4×C5—42no—
294C4×C5—62no—
295C4×C5—121no—
296C4×C5—122no—
297C4×C5#87811yescharacteristic
298C4×C5#88511yescharacteristic
299C4×C5#92411yescharacteristic
300C4×C5#93611yescharacteristic
301Dic5—41no—
302Dic5—241no—
303#41—101no—
304#41—101no—
305#41—101no—
306#41—101no—
307C2×Q8—51no—
308C2×Q8—62no—
309C2×D4—51no—
310C2×D4—101no—
311C22×C4—61no—
312C22×C4—101no—
313C22×C4C2×D3×D511yescharacteristic
314M16—101no—
315M16—101no—
316M16—301no—
317M16—301no—
318C2×C8—101no—
319C2×C8—301no—
320C4⋊C4—21no—
321C4⋊C4—62no—
322C4⋊C4—62no—
323C4⋊C4—62no—
324C4⋊C4—61no—
325C4⋊C4—301no—
326C22⋊C4—21no—
327C22⋊C4—32no—
328C22⋊C4—301no—
329C42—31no—
330C42—31no—
331C42—61no—
332C42—61no—
333C42—301no—
334C42#120112yes—
335C3×C5#203111yescharacteristic, complemented, Hall
336C22×C3—41no—
337C22×C3—201no—
338C22×C3—401no—
339C22×C3#409211yescharacteristic
340C22×C3#409311yescharacteristic
341C22×C3#409911yescharacteristic
342C3×C4—42no—
343C3×C4—201no—
344C3×C4#409211yescharacteristic
345C3×C4#409211yescharacteristic
346C3×C4#409311yescharacteristic
347C3×C4#409311yescharacteristic
348Dic3—22no—
349Dic3—42no—
350Dic3—41no—
351Dic3—401no—
352C2×C5—21no—
353C2×C5—21no—
354C2×C5—21no—
355C2×C5#494711yescharacteristic
356D5—81no—
357C23—101no—
358C23D3×C22⋊D511yescharacteristic
359Q8—101no—
360Q8—101no—
361Q8—124no—
362D4—101no—
363D4—101no—
364D4—201no—
365D4—201no—
366C2×C4—22no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—62no—
370C2×C4—101no—
371C2×C4—122no—
372C2×C4—201no—
373C2×C4—201no—
374C2×C4—601no—
375C2×C4D5×Dic611yescharacteristic
376C2×C4#676611yescharacteristic
377C2×C4#676611yescharacteristic
378C2×C4#677311yescharacteristic
379C2×C4C2×#120111yescharacteristic
380C2×C4D3×C22⋊D511yescharacteristic
381C8—201no—
382C8—601no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—401no—
387C2×C3#6552011yescharacteristic
388C5#7039411yescharacteristic, complemented, Sylow
389C22—41no—
390C22—201no—
391C22—401no—
392C22#9142111yescharacteristic
393C22#9145511yescharacteristic
394C22#9156811yescharacteristic
395C4—42no—
396C4—62no—
397C4—121no—
398C4—122no—
399C4—201no—
400C4—1201no—
401C4#9135911yescharacteristic
402C4#9136811yescharacteristic
403C4#9142111yescharacteristic
404C4#9145511yescharacteristic
405C3#1060263511yescharacteristic, complemented, Sylow
406C2—21no—
407C2—21no—
408C2—21no—
409C2—401no—
410C2#1174877311yescharacteristic
subgroupG/H
centerC4#91368
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC2×C3×C5#393
prime radicalC2×C3×#709C22
hypercenter#117D3×D5
hypocenterC3×C5#2031
LCSG ▷ C22×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C5×#5517C2semidirect
5. #11746352C2semidirect
6. #11745988C2semidirect
7. #11745976C2semidirect
40. C2×C3×D20Q8semidirect
42. C2×C3×Dic10Q8semidirect
64. #5517D5semidirect
136. C2×D20Dic6semidirect
138. C2×Dic10Dic6semidirect
335. C3×C5#2031semidirect
388. C5#70394semidirect
405. C3#10602635semidirect

Maximal subgroups

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

subgroupindex
C5×#55172
C3×#652942
#117486242
C2×#909772
#117463522
#117459882
#117459762
#106026353
#703945