Finite Group Databasebeta

← #14313698

Subgroups — #14313698

Order 1920.

460 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5816C211yesmaximal, characteristic
2C2×#91120C211yesmaximal, characteristic, complemented
3C2×C3×#3968C211yesmaximal, characteristic, complemented
4#11746016C212yesmaximal, complemented
5#11746013C212yesmaximal, complemented
6#10602665—31nomaximal, Hall
7C22×C3×C5×Q8C2211yescharacteristic
8C2×C5×#235C411yescharacteristic
9C2×#6710—21no—
10C5×#947C411yescharacteristic
11C5×#947C2211yescharacteristic
12C5×#946—21no—
13C5×#946—21no—
14C5×#944C411yescharacteristic
15C5×#944C2211yescharacteristic
16C5×#929—21no—
17C5×#929—21no—
18C5×#926C411yescharacteristic
19C2×C3×#681—21no—
20#91120—21no—
21#91120C2212yes—
22C3×#3968—21no—
23C3×#3968C2212yes—
24#70128—51nomaximal, Hall
25C5×#536—31no—
26C2×#3968—31no—
27C2×#3968D311yescharacteristic, complemented
28#64768—32no—
29#64762—32no—
30C2×C3×C5×Q8—41no—
31C2×C3×C5×Q8—41no—
32C2×C3×C5×Q8C2×C411yescharacteristic
33C2×C3×C5×Q8D411yescharacteristic
34C2×C3×C5×Q8D411yescharacteristic
35C2×C3×C5×Q8C2311yescharacteristic
36C22×C3×C4×C5C2×C411yescharacteristic
37C22×C3×C4×C5D411yescharacteristic
38C22×C3×C4×C5D411yescharacteristic
39C2×C5×#67—21no—
40C5×#235—41no—
41C5×#235—41no—
42C2×C5×D12C2×C411yescharacteristic
43C2×C4×C5×D3—21no—
44C2×C5×Dic6C2×C411yescharacteristic
45#6710—22no—
46#6710—41no—
47C2×C3⋊#168—41no—
48C5×#217—21no—
49C5×(C2×D3)⋊C4—21no—
50C5×(C2×D3)⋊C4—21no—
51C5×(C2×D3)⋊C4—21no—
52C5×(C2×D3)⋊C4—21no—
53C5×C4⋊Dic3—21no—
54C5×Dic3⋊C4—21no—
55C5×Dic3⋊C4—21no—
56C5×Dic3⋊C4—21no—
57C5×Dic3⋊C4—21no—
58C4×C5×Dic3C2×C411yescharacteristic
59C4×C5×Dic3C2×C411yescharacteristic
60C3×#681—22no—
61C3×#681—41no—
62C2×C3×#168—41no—
63#5816D511yescharacteristic
64C2×C3×#101—51no—
65C2×#836—51no—
66#4478—52no—
67#4477—52no—
68C2×C5×#41—31no—
69C22×C5×Q8C2×D311yescharacteristic
70C5×C4⋊Q8—31no—
71C5×C4⋊Q8—31no—
72C5×#124—31no—
73C5×#124—31no—
74C5×#123—61no—
75C5×#123—61no—
76C5×C22⋊Q8—61no—
77C5×C22⋊Q8—61no—
78C5×#117—31no—
79C2×#681—21no—
80C2×#681—61no—
81#3968—21no—
82#3968—32no—
83#3968—61no—
84#3968C2×D312yes—
85#2133—151noSylow
86C23×C3×C5C22⋊C411yescharacteristic
87C22×C5×D3—21no—
88C3×C5×Q8—41no—
89C3×C5×Q8—41no—
90C3×C5×Q8—41no—
91C3×C5×Q8—41no—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—21no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5C22⋊C411yescharacteristic
99C2×C3×C4×C5C22⋊C411yescharacteristic
100C2×C3×C4×C5C22⋊C411yescharacteristic
101C2×C3×C4×C5C22×C411yescharacteristic
102C2×C3×C4×C5C2×D411yescharacteristic
103C2×C3×C4×C5C2×D411yescharacteristic
104C5×#67—41no—
105C5×#67—41no—
106C2×C5×Dic3—21no—
107C2×C5×Dic3—21no—
108C2×C5×Dic3—21no—
109C5×D12—41no—
110C4×C5×D3—42no—
111C5×Dic6—41no—
112C3⋊#168—42no—
113C3×#168—42no—
114C22×C3×Q8C2×D511yescharacteristic
115C2×#235Dic511yescharacteristic
116C2×C3×M16—101no—
117#947Dic511yescharacteristic
118#947C2×D511yescharacteristic
119#946—21no—
120#946—21no—
121#944Dic511yescharacteristic
122#944C2×D511yescharacteristic
123#929—21no—
124#929—21no—
125#926Dic511yescharacteristic
126C2×#208—101no—
127C3×#101—52no—
128C3×#101—101no—
129#836—52no—
130#836—101no—
131C5×#41—121no—
132C5×#41—121no—
133C2×C5×Q8—31no—
134C2×C5×Q8—41no—
135C2×C5×Q8—41no—
136C2×C5×Q8C4×D311yescharacteristic
137C2×C5×Q8D1211yescharacteristic
138C2×C5×Q8#6711yescharacteristic
139C2×C5×Q8C22×D311yescharacteristic
140C2×C5×D4—31no—
141C2×C5×D4—61no—
142C22×C4×C5—61no—
143C22×C4×C5C4×D311yescharacteristic
144C22×C4×C5D1211yescharacteristic
145C22×C4×C5#6711yescharacteristic
146C5×C4⋊C4—61no—
147C5×C4⋊C4—61no—
148C5×C4⋊C4—61no—
149C5×C4⋊C4—61no—
150C5×C4⋊C4—61no—
151C5×C22⋊C4—61no—
152C5×C22⋊C4—61no—
153C5×C22⋊C4—61no—
154C5×C22⋊C4—61no—
155C5×C22⋊C4—61no—
156C42×C5—31no—
157C42×C5—31no—
158#681—22no—
159#681—41no—
160#681—62no—
161#681—121no—
162C2×#168—41no—
163C2×#168—121no—
164#536—31no—
165C2×#101—51no—
166C2×#101—151no—
167#356—152no—
168#355—152no—
169C22×C3×C5—41no—
170C22×C3×C5#9911yescharacteristic
171C22×C3×C5#9911yescharacteristic
172C22×C3×C5C2×C22⋊C411yescharacteristic
173C2×C5×D3—42no—
174C2×C5×D3—41no—
175C3×C4×C5—21no—
176C3×C4×C5—21no—
177C3×C4×C5—21no—
178C3×C4×C5—21no—
179C3×C4×C5—21no—
180C3×C4×C5—21no—
181C5×Dic3—41no—
182C5×Dic3—41no—
183C5×Dic3—41no—
184C2×C3×Q8—41no—
185C2×C3×Q8—41no—
186C2×C3×Q8C2×Dic511yescharacteristic
187C2×C3×Q8C22⋊D511yescharacteristic
188C2×C3×Q8C22⋊D511yescharacteristic
189C2×C3×Q8C22×D511yescharacteristic
190C22×C3×C4C2×Dic511yescharacteristic
191C22×C3×C4C22⋊D511yescharacteristic
192C22×C3×C4C22⋊D511yescharacteristic
193C2×#67—21no—
194#235—41no—
195#235—41no—
196C2×D12C2×Dic511yescharacteristic
197C2×C4×D3—21no—
198C2×Dic6C2×Dic511yescharacteristic
199C3×M16—102no—
200C3×M16—201no—
201C2×C3×C8—201no—
202#217—21no—
203(C2×D3)⋊C4—21no—
204(C2×D3)⋊C4—21no—
205(C2×D3)⋊C4—21no—
206(C2×D3)⋊C4—21no—
207C4⋊Dic3—21no—
208Dic3⋊C4—21no—
209Dic3⋊C4—21no—
210Dic3⋊C4—21no—
211Dic3⋊C4—21no—
212C4×Dic3C2×Dic511yescharacteristic
213C4×Dic3C2×Dic511yescharacteristic
214#208—102no—
215#208—201no—
216C2×C3⋊C8—201no—
217C23×C5—61no—
218C23×C5(C2×D3)⋊C411yescharacteristic
219C5×Q8—41no—
220C5×Q8—41no—
221C5×Q8—41no—
222C5×Q8—41no—
223C5×Q8—121no—
224C5×D4—121no—
225C5×D4—121no—
226C5×D4—121no—
227C2×C4×C5—21no—
228C2×C4×C5—21no—
229C2×C4×C5—21no—
230C2×C4×C5—21no—
231C2×C4×C5—21no—
232C2×C4×C5—21no—
233C2×C4×C5—61no—
234C2×C4×C5—61no—
235C2×C4×C5—61no—
236C2×C4×C5—122no—
237C2×C4×C5(C2×D3)⋊C411yescharacteristic
238C2×C4×C5(C2×D3)⋊C411yescharacteristic
239C2×C4×C5(C2×D3)⋊C411yescharacteristic
240C2×C4×C5C2×C4×D311yescharacteristic
241C2×C4×C5C2×D1211yescharacteristic
242C2×C4×C5C2×#6711yescharacteristic
243#168—42no—
244#168—122no—
245C2×#41—31no—
246C22×Q8D3×D511yescharacteristic
247C2×M16—101no—
248C2×M16—301no—
249C4⋊Q8—31no—
250C4⋊Q8—31no—
251#124—31no—
252#124—31no—
253#123—61no—
254#123—61no—
255C22⋊Q8—61no—
256C22⋊Q8—61no—
257#117—31no—
258#101—52no—
259#101—101no—
260#101—152no—
261#101—301no—
262C2×C3×C5—21no—
263C2×C3×C5—21no—
264C2×C3×C5—21no—
265C2×C3×C5C2×#9911yescharacteristic
266C5×D3—81no—
267C23×C3C22⋊Dic511yescharacteristic
268C22×D3—21no—
269C3×Q8—41no—
270C3×Q8—41no—
271C3×Q8—41no—
272C3×Q8—41no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4C22⋊Dic511yescharacteristic
280C2×C3×C4C22⋊Dic511yescharacteristic
281C2×C3×C4C22⋊Dic511yescharacteristic
282C2×C3×C4C22×Dic511yescharacteristic
283C2×C3×C4C2×C22⋊D511yescharacteristic
284C2×C3×C4C2×C22⋊D511yescharacteristic
285#67—41no—
286#67—41no—
287C2×Dic3—21no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290D12—41no—
291C4×D3—42no—
292Dic6—41no—
293C3×C8—202no—
294C3⋊C8—202no—
295C22×C5—41no—
296C22×C5—122no—
297C22×C5—121no—
298C22×C5#80611yescharacteristic
299C22×C5#80611yescharacteristic
300C22×C5C2×(C2×D3)⋊C411yescharacteristic
301C4×C5—21no—
302C4×C5—21no—
303C4×C5—21no—
304C4×C5—21no—
305C4×C5—21no—
306C4×C5—21no—
307C4×C5—121no—
308C4×C5—121no—
309C4×C5—121no—
310#41—121no—
311#41—121no—
312C2×Q8—31no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8D3×Dic511yescharacteristic
316C2×Q8#120111yescharacteristic
317C2×Q8#120311yescharacteristic
318C2×Q8C2×D3×D511yescharacteristic
319C2×D4—31no—
320C2×D4—61no—
321C22×C4—61no—
322C22×C4D3×Dic511yescharacteristic
323C22×C4#120111yescharacteristic
324C22×C4#120311yescharacteristic
325M16—102no—
326M16—201no—
327M16—302no—
328M16—601no—
329C2×C8—201no—
330C2×C8—601no—
331C4⋊C4—61no—
332C4⋊C4—61no—
333C4⋊C4—61no—
334C4⋊C4—61no—
335C4⋊C4—61no—
336C22⋊C4—61no—
337C22⋊C4—61no—
338C22⋊C4—61no—
339C22⋊C4—61no—
340C22⋊C4—61no—
341C42—31no—
342C42—31no—
343C3×C5#213311yescharacteristic, complemented, Hall
344C22×C3—41no—
345C22×C3#396611yescharacteristic
346C22×C3#396611yescharacteristic
347C22×C3C2×C22⋊Dic511yescharacteristic
348C2×D3—42no—
349C2×D3—41no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356Dic3—41no—
357Dic3—41no—
358Dic3—41no—
359C2×C5—21no—
360C2×C5—21no—
361C2×C5—21no—
362C2×C5—241no—
363C2×C5C2×#80611yescharacteristic
364C23—61no—
365C23(C2×D3)⋊Dic511yescharacteristic
366Q8—41no—
367Q8—41no—
368Q8—41no—
369Q8—41no—
370Q8—121no—
371D4—121no—
372D4—121no—
373D4—121no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—122no—
384C2×C4(C2×D3)⋊Dic511yescharacteristic
385C2×C4(C2×D3)⋊Dic511yescharacteristic
386C2×C4(C2×D3)⋊Dic511yescharacteristic
387C2×C4C2×D3×Dic511yescharacteristic
388C2×C4C2×#120111yescharacteristic
389C2×C4C2×#120311yescharacteristic
390C8—202no—
391C8—602no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3C2×#396611yescharacteristic
396D3—81no—
397C5#7012811yescharacteristic, complemented, Sylow
398C22—41no—
399C22—122no—
400C22—121no—
401C22#9099611yescharacteristic
402C22#9099611yescharacteristic
403C22C2×(C2×D3)⋊Dic511yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—121no—
411C4—121no—
412C4—121no—
413C3#1060266511yescharacteristic, complemented, Sylow
414C2—21no—
415C2—21no—
416C2—21no—
417C2—241no—
418C2C2×#9099611yescharacteristic
subgroupG/H
centerC2C2×#90996
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×Q8C2×D3×D5
FittingC22×C3×C5×Q8C22
socleC2×C3×C5C2×#99
prime radicalC2×C5×#235C4
cosocleC2×C3×C5×Q8C23
hypercenterC22×Q8D3×D5
hypocenterC3×C5#2133
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#91120C2semidirect
3. C2×C3×#3968C2semidirect
4. #11746016C2semidirect
5. #11746013C2semidirect
27. C2×#3968D3semidirect
343. C3×C5#2133semidirect
397. C5#70128semidirect
413. C3#10602665semidirect

Maximal subgroups

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

subgroupindex
C5×#58162
C2×#911202
C2×C3×#39682
#117460162
#117460132
#106026653
#701285