Finite Group Databasebeta

← #14289249

Subgroups — #14289249

Order 1920.

500 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×D20C211yesmaximal, characteristic, complemented
2#11751128C211yesmaximal, characteristic, complemented
3#11747028C211yesmaximal, characteristic
4#11746910C211yesmaximal, characteristic, complemented
5C5×#4535C211yesmaximal, characteristic, complemented
6#11745855C211yesmaximal, characteristic, complemented
7#11745835C211yesmaximal, characteristic, complemented
8#10599059—31nomaximal, Hall
9C2×C3×Q8×D5—21no—
10C3×C4×C5×Q8C2211yescharacteristic
11C2×#6717—21no—
12#91763C2211yescharacteristic
13C3×#4043—21no—
14C3×#4043—22no—
15C3×C4×D20—21no—
16C3×C4×D20C2211yescharacteristic
17C3×#4015—21no—
18C3×#4015C2211yescharacteristic
19C2×#6653—21no—
20#91093—21no—
21C4⋊(C3⋊#168)C2211yescharacteristic
22C5×C4⋊(C3⋊C8)C2211yescharacteristic
23C4×C5×C3⋊C8C2211yescharacteristic
24#90968—22no—
25#90968—21no—
26#71914—51nomaximal, Hall
27Q8×D20D311yescharacteristic, complemented
28#65316—31no—
29#65132—31no—
30#64983—31no—
31C5×Q8⋊C8—31no—
32#64706—31no—
33#64679—31no—
34C2×C3×C5×Q8D411yescharacteristic
35C2×D60—21no—
36C2×D60—21no—
37C2×D60—21no—
38C3×Q8×D5—42no—
39C3×Q8×D5—42no—
40C2×C3×D20D411yescharacteristic
41C2×C3×C4×D5—22no—
42C2×C3×C4×D5—21no—
43C2×C3×Dic10—22no—
44C2×C3×Dic10—21no—
45C3×C5×C4⋊C4—21no—
46C3×C5×C4⋊C4D411yescharacteristic
47C3×C42×C5—21no—
48C3×C42×C5C2311yescharacteristic
49#6717—42no—
50C2×C3⋊#168—21no—
51C2×C5×C3⋊C8—21no—
52C2×C5×C3⋊C8—21no—
53C3×#685—22no—
54C3×#685—21no—
55C3×C4⋊Dic5—22no—
56C3×C4⋊Dic5—21no—
57C3×C4⋊Dic5D411yescharacteristic
58C3×C4⋊Dic5D412yes—
59#6653—42no—
60C3×D4×Q8—51no—
61#5088—51no—
62#5027—51no—
63#5015—51no—
64#4547—51no—
65#4535D511yescharacteristic, complemented
66D4⋊(C3⋊C8)—51no—
67C2×Q8×D5—21no—
68C4×C5×Q8C2×D311yescharacteristic
69C2×#688—61no—
70C2×D40—61no—
71#4043—21no—
72#4043—22no—
73#4020—31no—
74C4×D20—21no—
75C4×D20C2×D311yescharacteristic
76#4015—21no—
77#4015C2×D311yescharacteristic
78C5×C4⋊C8—31no—
79C4×C5×C8—31no—
80#3953—61no—
81#3953—62no—
82#3941—61no—
83C4⋊#168—31no—
84#1627—151noSylow
85C22×D15—41no—
86C22×C3×D5—21no—
87C3×C5×Q8—21no—
88C3×C5×Q8D812yescomplemented
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×C3×C4×C5C2×D411yescharacteristic
93C2×C3×C4×C5C2×D411yescharacteristic
94D60—42no—
95D60—41no—
96D60—41no—
97D60—41no—
98D60—41no—
99C2×C3×Dic5—21no—
100C2×C3×Dic5—22no—
101C3×D20—21no—
102C3×D20SD1612yescomplemented
103C3×C4×D5—44no—
104C3×C4×D5—41no—
105C3×C4×D5—41no—
106C3×Dic10—44no—
107C3×Dic10—41no—
108C3×Dic10—41no—
109C3⋊#168—41no—
110C5×C3⋊C8—41no—
111C5×C3⋊C8—41no—
112C22×C3×Q8—101no—
113C3×C4⋊Q8—51no—
114C3×C4⋊Q8—101no—
115C3×C22⋊Q8—102no—
116C3×C22⋊Q8—101no—
117C3×C4×Q8C2×D511yescharacteristic
118C3×C4×D4—51no—
119C3×C4×D4—101no—
120C2×#215—101no—
121C2×#213—101no—
122#874—51no—
123#809—101no—
124#809—102no—
125#809—101no—
126C4⋊(C3⋊C8)—51no—
127C4⋊(C3⋊C8)C2×D511yescharacteristic
128C4×C3⋊C8C2×D511yescharacteristic
129C2×C5×Q8#6711yescharacteristic
130Q8×D5—42no—
131Q8×D5—42no—
132C2×D20—61no—
133C2×D20—61no—
134C2×D20—61no—
135C2×D20#6711yescharacteristic
136C2×C4×D5—22no—
137C2×C4×D5—21no—
138C2×Dic10—22no—
139C2×Dic10—21no—
140C2×C5×C8—61no—
141C2×C5×C8—61no—
142C5×C4⋊C4—21no—
143C5×C4⋊C4#6711yescharacteristic
144C42×C5—21no—
145C42×C5C22×D311yescharacteristic
146#688—122no—
147#685—22no—
148#685—21no—
149C4⋊Dic5—22no—
150C4⋊Dic5—21no—
151C4⋊Dic5#6711yescharacteristic
152C4⋊Dic5#6712yes—
153C2×#168—61no—
154D40—122no—
155D4×Q8—51no—
156#486—151no—
157#460—151no—
158#459—151no—
159#332—151no—
160Q8⋊C8—31no—
161D4⋊C8—151no—
162C22×C3×C5#12011yescharacteristic
163C2×D15—81no—
164C2×D15—81no—
165C2×D15—81no—
166C2×C3×D5—22no—
167C2×C3×D5—41no—
168C2×C3×D5—41no—
169C3×C4×C5—22no—
170C3×C4×C5—21no—
171C3×C4×C5—41no—
172C3×C4×C5C2×D811yescharacteristic
173C3×C4×C5C2×SD1611yescharacteristic
174C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
175C3×C4×C5#13711yescharacteristic
176C3×Dic5—41no—
177C3×Dic5—42no—
178C2×C3×Q8—102no—
179C2×C3×Q8—101no—
180C2×C3×Q8—202no—
181C2×C3×Q8—202no—
182C2×C3×Q8D2011yescharacteristic
183C2×C3×D4—51no—
184C22×C3×C4—102no—
185C22×C3×C4—101no—
186C2×D12—101no—
187C2×D12—101no—
188C2×D12—101no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—51no—
191C3×C4⋊C4—52no—
192C3×C4⋊C4—102no—
193C3×C4⋊C4—101no—
194C3×C4⋊C4D2011yescharacteristic
195C3×C22⋊C4—102no—
196C3×C22⋊C4—101no—
197C3×C42—21no—
198C3×C42C22×D511yescharacteristic
199#215—202no—
200#213—202no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—101no—
204C22×D5—21no—
205C22×D5—121no—
206C5×Q8—21no—
207C5×Q8#21312yescomplemented
208C2×C4×C5—21no—
209C2×C4×C5—21no—
210C2×C4×C5C2×#6711yescharacteristic
211C2×C4×C5C2×#6711yescharacteristic
212C2×C4×C5C2×#6711yescharacteristic
213C2×Dic5—21no—
214C2×Dic5—22no—
215D20—21no—
216D20—122no—
217D20—121no—
218D20—121no—
219D20—121no—
220D20—121no—
221D20#21512yescomplemented
222C4×D5—44no—
223C4×D5—41no—
224C4×D5—41no—
225Dic10—44no—
226Dic10—41no—
227Dic10—41no—
228C5×C8—121no—
229C5×C8—121no—
230#168—121no—
231C22×Q8—101no—
232C2×SD16—301no—
233C2×D8—301no—
234C4⋊Q8—51no—
235C4⋊Q8—101no—
236#127—151no—
237C22⋊Q8—102no—
238C22⋊Q8—101no—
239C4×Q8D3×D511yescharacteristic
240C4×D4—51no—
241C4×D4—101no—
242C4⋊C8—31no—
243C4⋊C8—151no—
244#102—301no—
245#102—302no—
246#102—301no—
247C4×C8—31no—
248C2×C3×C5#44711yescharacteristic
249C2×C3×C5#44811yescharacteristic
250C2×C3×C5#45411yescharacteristic
251D15—161no—
252C3×D5—42no—
253C23×C3—101no—
254C22×D3—201no—
255C3×Q8—21no—
256C3×Q8—204no—
257C3×Q8—201no—
258C3×Q8—201no—
259C3×Q8D4012yescomplemented
260C3×D4—52no—
261C3×D4—101no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—102no—
265C2×C3×C4—101no—
266C2×C3×C4—204no—
267C2×C3×C4—201no—
268C2×C3×C4—201no—
269C2×C3×C4C2×D2011yescharacteristic
270C2×C3×C4D4×D511yescharacteristic
271C2×C3×C4D4×D511yescharacteristic
272D12—202no—
273D12—201no—
274D12—201no—
275D12—201no—
276D12—201no—
277C3⋊C8—41no—
278C3⋊C8—41no—
279C3⋊C8—201no—
280C22×C5#95311yescharacteristic
281C2×D5—22no—
282C2×D5—41no—
283C2×D5—41no—
284C2×D5—241no—
285C2×D5—241no—
286C2×D5—241no—
287C4×C5—22no—
288C4×C5—21no—
289C4×C5—41no—
290C4×C5C2×#21311yescharacteristic
291C4×C5#93211yescharacteristic
292C4×C5C2×#21511yescharacteristic
293C4×C5#94211yescharacteristic
294Dic5—41no—
295Dic5—42no—
296C2×Q8—102no—
297C2×Q8—101no—
298C2×Q8—202no—
299C2×Q8—202no—
300C2×Q8#120211yescharacteristic
301C2×D4—51no—
302C2×D4—301no—
303C2×D4—301no—
304C2×D4—301no—
305C22×C4—102no—
306C22×C4—101no—
307SD16—602no—
308D8—602no—
309C2×C8—61no—
310C2×C8—61no—
311C2×C8—301no—
312C4⋊C4—21no—
313C4⋊C4—51no—
314C4⋊C4—52no—
315C4⋊C4—102no—
316C4⋊C4—101no—
317C4⋊C4#120211yescharacteristic
318C22⋊C4—102no—
319C22⋊C4—101no—
320C42—21no—
321C42C2×D3×D511yescharacteristic
322C3×C5#162711yescharacteristic, complemented, Hall
323C22×C3—102no—
324C22×C3—201no—
325C22×C3—201no—
326C22×C3#402811yescharacteristic
327C2×D3—401no—
328C2×D3—401no—
329C2×D3—401no—
330C3×C4—21no—
331C3×C4—22no—
332C3×C4—41no—
333C3×C4—202no—
334C3×C4—201no—
335C3×C4C2×D4011yescharacteristic
336C3×C4#405411yescharacteristic
337C3×C4D5×SD1611yescharacteristic
338C3×C4#406411yescharacteristic
339C2×C5#521711yescharacteristic
340C2×C5#522711yescharacteristic
341C2×C5#524511yescharacteristic
342D5—42no—
343D5—481no—
344C23—101no—
345C23—601no—
346Q8—21no—
347Q8—204no—
348Q8—201no—
349Q8—201no—
350Q8#665312yescomplemented
351D4—52no—
352D4—101no—
353D4—602no—
354D4—601no—
355D4—601no—
356D4—601no—
357D4—601no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—102no—
361C2×C4—101no—
362C2×C4—204no—
363C2×C4—201no—
364C2×C4—201no—
365C2×C4C2×#120211yescharacteristic
366C2×C4D5×#6711yescharacteristic
367C2×C4D5×#6711yescharacteristic
368C8—121no—
369C8—121no—
370C8—601no—
371C2×C3—202no—
372C2×C3#6502111yescharacteristic
373C2×C3#6509511yescharacteristic
374C2×C3#6511611yescharacteristic
375D3—801no—
376C5#7191411yescharacteristic, complemented, Sylow
377C22—102no—
378C22—201no—
379C22—201no—
380C22—1201no—
381C22—1201no—
382C22—1201no—
383C22#9156411yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—202no—
388C4—201no—
389C4C2×#665311yescharacteristic
390C4#9131211yescharacteristic
391C4D5×#21511yescharacteristic
392C4#9151511yescharacteristic
393C3#1059905911yescharacteristic, complemented, Sylow
394C2—202no—
395C2—2401no—
396C2#1174851211yescharacteristic
397C2#1174926811yescharacteristic
398C2#1174945711yescharacteristic
subgroupG/H
centerC22#91564
commutator, cosocleC3×C42×C5C23
FrattiniC42C2×D3×D5
FittingC3×C4×C5×Q8C22
socleC22×C3×C5#120
prime radical#11747028C2
hypercenterC4×Q8D3×D5
hypocenterC3×C5#1627
LCSG ▷ C3×C42×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×D20C2semidirect
2. #11751128C2semidirect
4. #11746910C2semidirect
5. C5×#4535C2semidirect
6. #11745855C2semidirect
7. #11745835C2semidirect
27. Q8×D20D3semidirect
65. #4535D5semidirect
88. C3×C5×Q8D8semidirect
102. C3×D20SD16semidirect
207. C5×Q8#213semidirect
221. D20#215semidirect
259. C3×Q8D40semidirect
322. C3×C5#1627semidirect
350. Q8#6653semidirect
376. C5#71914semidirect
393. C3#10599059semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×D202
#117511282
#117470282
#117469102
C5×#45352
#117458552
#117458352
#105990593
#719145