Finite Group Databasebeta

← #14288971

Subgroups — #14288971

Order 1920.

472 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 96 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×#66064C211yesmaximal, characteristic, complemented
2C5×#5085C211yesmaximal, characteristic, complemented
3#11747060C211yesmaximal, characteristic, complemented
4#11747026C211yesmaximal, characteristic
5#11746367C211yesmaximal, characteristic, complemented
6#11745841C211yesmaximal, characteristic, complemented
7#11745835C211yesmaximal, characteristic, complemented
8#10602057—31nomaximal, Hall
9C3×C5×C4⋊Q8C2211yescharacteristic
10C2×C5×#215—22no—
11C5×#874C2211yescharacteristic
12C3×#4092—22no—
13C3×#4043—21no—
14C3×#4039—21no—
15C3×D5×C4⋊C4—21no—
16C3×#4036—21no—
17C3×C4×D20C2211yescharacteristic
18C3×C4×Dic10C2211yescharacteristic
19C2×#6656—21no—
20C2×#6652—21no—
21C4⋊(C3⋊#168)C2211yescharacteristic
22C4⋊(C3⋊#168)C2211yescharacteristic
23C4×C5×C3⋊C8C2211yescharacteristic
24#90966—21no—
25#90966—21no—
26#71923—51nomaximal, Hall
27#66064D311yescharacteristic, complemented
28C5×#492—31no—
29#65364—31no—
30#65361—31no—
31#64766—31no—
32#64679—31no—
33#64676—31no—
34C2×C3×C5×Q8D412yes—
35C2×C5×D12—21no—
36C2×C5×D12—21no—
37C2×C5×D12—21no—
38C2×C3×D20D411yescharacteristic
39C2×C3×C4×D5—21no—
40C2×C3×C4×D5—22no—
41C2×C3×Dic10D411yescharacteristic
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4—21no—
44C3×C42×C5C2311yescharacteristic
45C2×C3⋊#168—21no—
46C2×C3⋊#168—21no—
47C5×#215—44no—
48C2×C5×C3⋊C8—21no—
49C3×#685—22no—
50C3×#685—21no—
51C3×C4⋊Dic5—22no—
52C3×C4⋊Dic5D411yescharacteristic
53C3×C4⋊Dic5D411yescharacteristic
54C3×#683—22no—
55C3×#683—21no—
56C3×C4×Dic5—21no—
57#6656—42no—
58#6652—42no—
59C3×#554—51no—
60#5085D511yescharacteristic, complemented
61#5027—51no—
62#5015—51no—
63#4546—51no—
64#4535—51no—
65D4⋊(C3⋊C8)—51no—
66C2×C5×SD16—62no—
67C5×C4⋊Q8C2×D311yescharacteristic
68C5×#127—31no—
69#4092—22no—
70C2×#687—61no—
71C2×#686—61no—
72#4043—21no—
73#4039—21no—
74D5×C4⋊C4—21no—
75#4036—21no—
76C4×D20C2×D311yescharacteristic
77C4×Dic10C2×D311yescharacteristic
78C4×C5×C8—31no—
79#3964—61no—
80#3964—61no—
81C4⋊#168—31no—
82C4⋊#168—31no—
83#1622—151noSylow
84C22×C5×D3—41no—
85C22×C3×D5—21no—
86C3×C5×Q8—42no—
87C2×C3×C4×C5—22no—
88C2×C3×C4×C5C2×D411yescharacteristic
89C2×C3×C4×C5C2×D411yescharacteristic
90C2×C3×C4×C5C2×D411yescharacteristic
91C5×D12—42no—
92C5×D12—41no—
93C5×D12—41no—
94C5×D12—41no—
95C5×D12—41no—
96C2×C3×Dic5—21no—
97C2×C3×Dic5—21no—
98C2×C3×Dic5—21no—
99C3×D20—21no—
100C3×D20SD1612yescomplemented
101C3×C4×D5—41no—
102C3×C4×D5—41no—
103C3×C4×D5—82no—
104C3×Dic10—21no—
105C3×Dic10—21no—
106C3⋊#168—41no—
107C3⋊#168—41no—
108C5×C3⋊C8—22no—
109C3×C4⋊Q8C2×D511yescharacteristic
110C3×#125—101no—
111C3×C22⋊Q8—101no—
112C3×C22⋊Q8—102no—
113C3×C4×Q8—51no—
114C3×C4×D4—51no—
115C3×C4×D4—101no—
116C2×C3×C4⋊C4—101no—
117C2×#215—22no—
118C2×#215—101no—
119C2×#213—101no—
120#874C2×D511yescharacteristic
121#809—101no—
122#809—101no—
123C4⋊(C3⋊C8)—51no—
124C4⋊(C3⋊C8)—51no—
125C4×C3⋊C8C2×D511yescharacteristic
126C2×C5×Q8#6712yes—
127C2×C5×D4—61no—
128C2×C5×D4—61no—
129C2×C5×D4—61no—
130C2×D20#6711yescharacteristic
131C2×C4×D5—21no—
132C2×C4×D5—22no—
133C2×Dic10#6711yescharacteristic
134C5×SD16—124no—
135C2×C5×C8—61no—
136C5×C4⋊C4—21no—
137C5×C4⋊C4—21no—
138C42×C5C22×D311yescharacteristic
139#687—122no—
140#686—122no—
141#685—22no—
142#685—21no—
143C4⋊Dic5—22no—
144C4⋊Dic5#6711yescharacteristic
145C4⋊Dic5#6711yescharacteristic
146#683—22no—
147#683—21no—
148C4×Dic5—21no—
149C2×#168—61no—
150C2×#168—61no—
151#554—51no—
152#492—31no—
153#460—151no—
154#459—151no—
155#333—151no—
156Q8⋊C8—151no—
157D4⋊C8—151no—
158C22×C3×C5#12011yescharacteristic
159C2×C5×D3—81no—
160C2×C5×D3—81no—
161C2×C5×D3—81no—
162C2×C3×D5—22no—
163C2×C3×D5—41no—
164C2×C3×D5—41no—
165C3×C4×C5—21no—
166C3×C4×C5—42no—
167C3×C4×C5C2×SD1611yescharacteristic
168C3×C4×C5#13511yescharacteristic
169C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
170C3×C4×C5#13711yescharacteristic
171C3×Dic5—41no—
172C3×Dic5—41no—
173C3×Dic5—41no—
174C2×C3×Q8—51no—
175C2×C3×Q8C22⋊D512yes—
176C2×C3×D4—51no—
177C22×C3×C4—102no—
178C22×C3×C4—101no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—51no—
185C3×C4⋊C4—51no—
186C3×C4⋊C4—102no—
187C3×C4⋊C4—102no—
188C3×C4⋊C4—101no—
189C3×C22⋊C4—102no—
190C3×C22⋊C4—101no—
191C3×C42—101no—
192C3×C42C22×D511yescharacteristic
193#215—44no—
194#215—202no—
195#213—202no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—101no—
198C2×C3⋊C8—101no—
199C23×C5—121no—
200C22×D5—21no—
201C5×Q8—42no—
202C5×D4—122no—
203C5×D4—121no—
204C5×D4—121no—
205C5×D4—121no—
206C5×D4—121no—
207C2×C4×C5—22no—
208C2×C4×C5C2×#6711yescharacteristic
209C2×C4×C5C2×#6711yescharacteristic
210C2×C4×C5C2×#6711yescharacteristic
211C2×Dic5—21no—
212C2×Dic5—21no—
213C2×Dic5—21no—
214D20—21no—
215D20#21512yescomplemented
216C4×D5—41no—
217C4×D5—41no—
218C4×D5—82no—
219Dic10—21no—
220Dic10—21no—
221C5×C8—62no—
222#168—121no—
223#168—121no—
224C2×SD16—62no—
225C2×SD16—301no—
226C2×D8—301no—
227C4⋊Q8D3×D511yescharacteristic
228#127—31no—
229#125—101no—
230C22⋊Q8—101no—
231C22⋊Q8—102no—
232C4×Q8—51no—
233C4×D4—51no—
234C4×D4—101no—
235C2×C4⋊C4—101no—
236C4⋊C8—151no—
237C4⋊C8—151no—
238#102—301no—
239#102—301no—
240C4×C8—31no—
241C2×C3×C5#44811yescharacteristic
242C2×C3×C5#44911yescharacteristic
243C2×C3×C5#45311yescharacteristic
244C3×D5—42no—
245C5×D3—161no—
246C23×C3—101no—
247C22×D3—41no—
248C3×Q8—42no—
249C3×Q8—101no—
250C3×Q8—101no—
251C3×D4—52no—
252C3×D4—101no—
253C2×C3×C4—22no—
254C2×C3×C4—101no—
255C2×C3×C4—101no—
256C2×C3×C4—101no—
257C2×C3×C4—201no—
258C2×C3×C4—201no—
259C2×C3×C4—402no—
260C2×C3×C4D4×D511yescharacteristic
261C2×C3×C4D4×D511yescharacteristic
262C2×C3×C4C2×C22⋊D511yescharacteristic
263D12—42no—
264D12—41no—
265D12—41no—
266D12—41no—
267D12—41no—
268C3⋊C8—22no—
269C3⋊C8—201no—
270C3⋊C8—201no—
271C22×C5—241no—
272C22×C5—241no—
273C22×C5—241no—
274C22×C5#95311yescharacteristic
275C2×D5—22no—
276C2×D5—41no—
277C2×D5—41no—
278C4×C5—21no—
279C4×C5—42no—
280C4×C5C2×#21511yescharacteristic
281C4×C5#94211yescharacteristic
282C4×C5#94911yescharacteristic
283C4×C5#95011yescharacteristic
284Dic5—41no—
285Dic5—41no—
286Dic5—41no—
287C2×Q8—51no—
288C2×Q8#120112yes—
289C2×D4—51no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—61no—
293C22×C4—102no—
294C22×C4—101no—
295SD16—124no—
296SD16—602no—
297D8—602no—
298C2×C8—61no—
299C2×C8—301no—
300C2×C8—301no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—51no—
304C4⋊C4—51no—
305C4⋊C4—102no—
306C4⋊C4—102no—
307C4⋊C4—101no—
308C22⋊C4—102no—
309C22⋊C4—101no—
310C42—101no—
311C42C2×D3×D511yescharacteristic
312C3×C5#162211yescharacteristic, complemented, Hall
313C22×C3—102no—
314C22×C3—201no—
315C22×C3—201no—
316C22×C3#408311yescharacteristic
317C2×D3—81no—
318C2×D3—81no—
319C2×D3—81no—
320C3×C4—21no—
321C3×C4—42no—
322C3×C4—201no—
323C3×C4—201no—
324C3×C4—201no—
325C3×C4D5×SD1611yescharacteristic
326C3×C4#406011yescharacteristic
327C3×C4#406111yescharacteristic
328C3×C4#406211yescharacteristic
329C2×C5—481no—
330C2×C5#522711yescharacteristic
331C2×C5#523811yescharacteristic
332C2×C5#524111yescharacteristic
333D5—42no—
334C23—101no—
335C23—121no—
336Q8—42no—
337Q8—101no—
338Q8—101no—
339D4—52no—
340D4—101no—
341D4—122no—
342D4—121no—
343D4—121no—
344D4—121no—
345D4—121no—
346C2×C4—22no—
347C2×C4—101no—
348C2×C4—101no—
349C2×C4—101no—
350C2×C4—201no—
351C2×C4—201no—
352C2×C4—402no—
353C2×C4C2×#120111yescharacteristic
354C2×C4D5×#6711yescharacteristic
355C2×C4D5×#6711yescharacteristic
356C8—62no—
357C8—601no—
358C8—601no—
359C2×C3—202no—
360C2×C3#6536611yescharacteristic
361C2×C3#6545911yescharacteristic
362C2×C3#6546111yescharacteristic
363D3—161no—
364C5#7192311yescharacteristic, complemented, Sylow
365C22—102no—
366C22—201no—
367C22—201no—
368C22—241no—
369C22—241no—
370C22—241no—
371C22#9156311yescharacteristic
372C4—21no—
373C4—42no—
374C4—201no—
375C4—201no—
376C4—201no—
377C4D5×#21511yescharacteristic
378C4#9150211yescharacteristic
379C4#9151311yescharacteristic
380C4#9151411yescharacteristic
381C3#1060205711yescharacteristic, complemented, Sylow
382C2—202no—
383C2—481no—
384C2#1174913711yescharacteristic
385C2#1174926711yescharacteristic
386C2#1174927011yescharacteristic
subgroupG/H
centerC22#91563
commutator, cosocleC3×C42×C5C23
FrattiniC42C2×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5#120
prime radical#11747026C2
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#1622
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×#66064C2semidirect
2. C5×#5085C2semidirect
3. #11747060C2semidirect
5. #11746367C2semidirect
6. #11745841C2semidirect
7. #11745835C2semidirect
27. #66064D3semidirect
60. #5085D5semidirect
100. C3×D20SD16semidirect
215. D20#215semidirect
312. C3×C5#1622semidirect
364. C5#71923semidirect
381. C3#10602057semidirect

Maximal subgroups

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

subgroupindex
C3×#660642
C5×#50852
#117470602
#117470262
#117463672
#117458412
#117458352
#106020573
#719235