Finite Group Databasebeta

← #14289159

Subgroups — #14289159

Order 1920.

420 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#557C211yesmaximal, characteristic, complemented
2#11751189C211yesmaximal, characteristic
3#11751128C211yesmaximal, characteristic
4#11751128C211yesmaximal, characteristic
5#11746368C211yesmaximal, characteristic, complemented
6#11746356C211yesmaximal, characteristic, complemented
7#11746356C211yesmaximal, characteristic, complemented
8#10602728—31nomaximal, Hall
9C3×C5×C4⋊Q8—21no—
10C3×C5×C4⋊Q8C2211yescharacteristic
11C3×C5×#125—22no—
12C3×C5×#125—21no—
13C3×C4×C5×Q8—21no—
14C3×C4×C5×Q8—21no—
15C3×C4×C5×Q8C2211yescharacteristic
16C3×C4×C5×Q8C2211yescharacteristic
17C2×#6717—21no—
18C2×#6717—21no—
19#91763C2211yescharacteristic
20#91093—22no—
21#91093—21no—
22#91093—21no—
23C4⋊(C3⋊#168)C2211yescharacteristic
24C4⋊(C3⋊#168)C2211yescharacteristic
25C4×C3⋊#168C2211yescharacteristic
26#71935—51nomaximal, Hall
27C5×#557D311yescharacteristic, complemented
28#65377—31no—
29#65316—31no—
30#65316—31no—
31#64767—31no—
32Q8⋊#168—31no—
33Q8⋊#168—31no—
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8D411yescharacteristic
36C2×C3×C5×Q8D411yescharacteristic
37C2×D60—21no—
38C2×D60—21no—
39C2×D60—21no—
40C3×C5×C4⋊C4—22no—
41C3×C5×C4⋊C4—22no—
42C3×C5×C4⋊C4—22no—
43C3×C5×C4⋊C4—21no—
44C3×C5×C4⋊C4—21no—
45C3×C5×C4⋊C4—21no—
46C3×C5×C4⋊C4D411yescharacteristic
47C3×C5×C4⋊C4D411yescharacteristic
48C3×C5×C4⋊C4D412yes—
49C3×C42×C5—22no—
50C3×C42×C5—21no—
51C3×C42×C5—21no—
52C3×C42×C5C2311yescharacteristic
53#6717—42no—
54#6717—42no—
55C2×C3⋊#168—21no—
56C2×C3⋊#168—21no—
57C2×C3⋊#168—21no—
58C3×#557D511yescharacteristic, complemented
59#5088—51no—
60#5027—51no—
61#5027—51no—
62#4547—51no—
63#4535—51no—
64#4535—51no—
65C5×C4⋊Q8—21no—
66C5×C4⋊Q8C2×D311yescharacteristic
67C5×#125—22no—
68C5×#125—21no—
69C4×C5×Q8—21no—
70C4×C5×Q8—21no—
71C4×C5×Q8C2×D311yescharacteristic
72C4×C5×Q8C2×D311yescharacteristic
73C2×#688—61no—
74C2×#688—61no—
75#4020—31no—
76#3941—61no—
77#3941—61no—
78#3941—62no—
79C4⋊#168—31no—
80C4⋊#168—31no—
81C4×#168—31no—
82#1626—151noSylow
83C22×D15—41no—
84C3×C5×Q8—21no—
85C3×C5×Q8—21no—
86C3×C5×Q8—21no—
87C3×C5×Q8—41no—
88C3×C5×Q8—41no—
89C3×C5×Q8SD1612yescomplemented
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5—21no—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—22no—
95C2×C3×C4×C5C2×D411yescharacteristic
96C2×C3×C4×C5C2×D411yescharacteristic
97C2×C3×C4×C5C2×D411yescharacteristic
98D60—42no—
99D60—41no—
100D60—41no—
101D60—41no—
102D60—41no—
103C3⋊#168—41no—
104C3⋊#168—41no—
105C3⋊#168—41no—
106C3×C4⋊Q8—21no—
107C3×C4⋊Q8C2×D511yescharacteristic
108C3×#125—22no—
109C3×#125—21no—
110C3×C4×Q8—21no—
111C3×C4×Q8—21no—
112C3×C4×Q8C2×D511yescharacteristic
113C3×C4×Q8C2×D511yescharacteristic
114C2×#215—101no—
115C2×#215—101no—
116#874—51no—
117#809—101no—
118#809—101no—
119#809—102no—
120C4⋊(C3⋊C8)—51no—
121C4⋊(C3⋊C8)—51no—
122C4×C3⋊C8—51no—
123C2×C5×Q8—21no—
124C2×C5×Q8#6711yescharacteristic
125C2×C5×Q8#6711yescharacteristic
126C2×D20—61no—
127C2×D20—61no—
128C2×D20—61no—
129C5×C4⋊C4—22no—
130C5×C4⋊C4—22no—
131C5×C4⋊C4—22no—
132C5×C4⋊C4—21no—
133C5×C4⋊C4—21no—
134C5×C4⋊C4—21no—
135C5×C4⋊C4#6711yescharacteristic
136C5×C4⋊C4#6711yescharacteristic
137C5×C4⋊C4#6712yes—
138C42×C5—22no—
139C42×C5—21no—
140C42×C5—21no—
141C42×C5C22×D311yescharacteristic
142#688—122no—
143#688—122no—
144C2×#168—61no—
145C2×#168—61no—
146C2×#168—61no—
147#557D1511yescharacteristic, complemented
148#486—151no—
149#460—151no—
150#460—151no—
151#332—151no—
152Q8⋊C8—151no—
153Q8⋊C8—151no—
154C22×C3×C5#12011yescharacteristic
155C2×D15—81no—
156C2×D15—81no—
157C2×D15—81no—
158C3×C4×C5—22no—
159C3×C4×C5—21no—
160C3×C4×C5—41no—
161C3×C4×C5—42no—
162C3×C4×C5—41no—
163C3×C4×C5—41no—
164C3×C4×C5C2×SD1611yescharacteristic
165C3×C4×C5#13511yescharacteristic
166C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
167C3×C4×C5#13711yescharacteristic
168C2×C3×Q8—21no—
169C2×C3×Q8C22⋊D511yescharacteristic
170C2×C3×Q8C22⋊D511yescharacteristic
171C2×D12—101no—
172C2×D12—101no—
173C2×D12—101no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4C22⋊D511yescharacteristic
181C3×C4⋊C4C22⋊D511yescharacteristic
182C3×C4⋊C4C22⋊D512yes—
183C3×C42—22no—
184C3×C42—21no—
185C3×C42—21no—
186C3×C42C22×D511yescharacteristic
187#215—202no—
188#215—202no—
189C2×C3⋊C8—101no—
190C2×C3⋊C8—101no—
191C2×C3⋊C8—101no—
192C22×D5—121no—
193C5×Q8—21no—
194C5×Q8—21no—
195C5×Q8—21no—
196C5×Q8—41no—
197C5×Q8—41no—
198C5×Q8#21512yescomplemented
199C2×C4×C5—21no—
200C2×C4×C5—21no—
201C2×C4×C5—21no—
202C2×C4×C5—21no—
203C2×C4×C5—22no—
204C2×C4×C5C2×#6711yescharacteristic
205C2×C4×C5C2×#6711yescharacteristic
206C2×C4×C5C2×#6711yescharacteristic
207D20—122no—
208D20—121no—
209D20—121no—
210D20—121no—
211D20—121no—
212#168—121no—
213#168—121no—
214#168—121no—
215C2×SD16—301no—
216C2×SD16—301no—
217C4⋊Q8—21no—
218C4⋊Q8C2×D1511yescharacteristic
219#127—151no—
220#125—22no—
221#125—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8C2×D1511yescharacteristic
225C4×Q8C2×D1511yescharacteristic
226C4⋊C8—151no—
227C4⋊C8—151no—
228#102—301no—
229#102—301no—
230#102—302no—
231C4×C8—151no—
232C2×C3×C5#44811yescharacteristic
233C2×C3×C5#44911yescharacteristic
234C2×C3×C5#45411yescharacteristic
235D15—161no—
236C22×D3—201no—
237C3×Q8—21no—
238C3×Q8—21no—
239C3×Q8—21no—
240C3×Q8—41no—
241C3×Q8—41no—
242C3×Q8#68812yescomplemented
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—22no—
248C2×C3×C4C2×C22⋊D511yescharacteristic
249C2×C3×C4C2×C22⋊D511yescharacteristic
250C2×C3×C4C2×C22⋊D511yescharacteristic
251D12—202no—
252D12—201no—
253D12—201no—
254D12—201no—
255D12—201no—
256C3⋊C8—201no—
257C3⋊C8—201no—
258C3⋊C8—201no—
259C22×C5#95311yescharacteristic
260C2×D5—241no—
261C2×D5—241no—
262C2×D5—241no—
263C4×C5—22no—
264C4×C5—21no—
265C4×C5—41no—
266C4×C5—42no—
267C4×C5—41no—
268C4×C5—41no—
269C4×C5C2×#21511yescharacteristic
270C4×C5#94211yescharacteristic
271C4×C5#94911yescharacteristic
272C4×C5#95011yescharacteristic
273C2×Q8—21no—
274C2×Q8#122011yescharacteristic
275C2×Q8#122011yescharacteristic
276C2×D4—301no—
277C2×D4—301no—
278C2×D4—301no—
279SD16—602no—
280SD16—602no—
281C2×C8—301no—
282C2×C8—301no—
283C2×C8—301no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4#122011yescharacteristic
291C4⋊C4#122011yescharacteristic
292C4⋊C4#122012yes—
293C42—22no—
294C42—21no—
295C42—21no—
296C42C22×D1511yescharacteristic
297C3×C5#162611yescharacteristic, complemented, Hall
298C22×C3#409911yescharacteristic
299C2×D3—401no—
300C2×D3—401no—
301C2×D3—401no—
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—41no—
305C3×C4—42no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4C2×#68811yescharacteristic
309C3×C4#408811yescharacteristic
310C3×C4#409511yescharacteristic
311C3×C4#409611yescharacteristic
312C2×C5#522711yescharacteristic
313C2×C5#523811yescharacteristic
314C2×C5#524511yescharacteristic
315D5—481no—
316C23—601no—
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—41no—
321Q8—41no—
322Q8#671712yescomplemented
323D4—602no—
324D4—601no—
325D4—601no—
326D4—601no—
327D4—601no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4C2×#122011yescharacteristic
334C2×C4C2×#122011yescharacteristic
335C2×C4C2×#122011yescharacteristic
336C8—601no—
337C8—601no—
338C8—601no—
339C2×C3#6551611yescharacteristic
340C2×C3#6552711yescharacteristic
341C2×C3#6553411yescharacteristic
342D3—801no—
343C5#7193511yescharacteristic, complemented, Sylow
344C22—1201no—
345C22—1201no—
346C22—1201no—
347C22#9184211yescharacteristic
348C4—22no—
349C4—21no—
350C4—41no—
351C4—42no—
352C4—41no—
353C4—41no—
354C4C2×#671711yescharacteristic
355C4#9183111yescharacteristic
356C4#9183811yescharacteristic
357C4#9183911yescharacteristic
358C3#1060272811yescharacteristic, complemented, Sylow
359C2—2401no—
360C2#1175132811yescharacteristic
361C2#1175133911yescharacteristic
362C2#1175134611yescharacteristic
subgroupG/H
centerC22#91842
commutator, cosocleC3×C42×C5C23
FrattiniC42C22×D15
FittingC3×C5×#557C2
socleC22×C3×C5#120
prime radical#91763C22
hypercenter#557D15
hypocenterC3×C5#1626
LCSG ▷ C3×C42×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #557

Complements

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

NH ≅ G/Ntype
1. C3×C5×#557C2semidirect
5. #11746368C2semidirect
6. #11746356C2semidirect
7. #11746356C2semidirect
27. C5×#557D3semidirect
58. C3×#557D5semidirect
89. C3×C5×Q8SD16semidirect
147. #557D15semidirect
198. C5×Q8#215semidirect
242. C3×Q8#688semidirect
297. C3×C5#1626semidirect
322. Q8#6717semidirect
343. C5#71935semidirect
358. C3#10602728semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5572
#117511892
#117511282
#117463682
#117463562
#106027283
#719355