Finite Group Databasebeta

← #14301580

Subgroups — #14301580

Order 1920.

688 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#11756991C211yesmaximal, characteristic, complemented
2C3×D5×#124C211yesmaximal, characteristic
3D5⋊#836C211yesmaximal, characteristic, complemented
4#11752426C212yesmaximal
5#11746733C212yesmaximal, complemented
6#10604563—31nomaximal, Hall
7C2×C3×Q8×D5C2211yescharacteristic
8C2×C3×D4×D5C2211yescharacteristic
9Q8×ΓL(1,𝔽16)—24no—
10#91989—22no—
11#91989C2212yes—
12D5⋊#208C2212yes—
13C3×C5×#124C411yescharacteristic, complemented
14C3×#4093C411yescharacteristic, complemented
15C3×#4082C411yescharacteristic, complemented
16C3×#4031—22no—
17C3×D5×C22⋊C4—22no—
18C3×#4021C411yescharacteristic, complemented
19C3×C42×D5C2211yescharacteristic
20#91236—24no—
21#91234—22no—
22#71724—51nomaximal, Hall
23#66262—31no—
24D5×#124D311yescharacteristic
25D5⋊#101—31no—
26#65767—32no—
27#64862—32no—
28C23×C3×D5—21no—
29C2×C3×C5×Q8C2×C411yescharacteristic
30C2×C3×C5×D4C2×C411yescharacteristic
31C2×C3×C22⋊D5—21no—
32C3×Q8×D5—22no—
33C3×Q8×D5—22no—
34C3×D4×D5—22no—
35C3×D4×D5—22no—
36C2×C3×D20C2×C411yescharacteristic
37C2×C3×C4×D5—21no—
38C2×C3×C4×D5D412yes—
39C2×C3×C4×D5C2311yescharacteristic
40C2×C3×Dic10C2×C411yescharacteristic
41C2×C3×Dic10D412yes—
42#6763—22no—
43#6762—22no—
44C4⋊ΓL(1,𝔽16)—24no—
45C4⋊ΓL(1,𝔽16)D414yes—
46C4×ΓL(1,𝔽16)—24no—
47C4×ΓL(1,𝔽16)—22no—
48#6758—22no—
49D5⋊(C3⋊C8)—22no—
50C3×C5×C22⋊C4—22no—
51C3×C42×C5C2×C411yescharacteristic
52C3×C22⋊Dic5—22no—
53C3×#685—22no—
54C3×#685—22no—
55C3×C4×Dic5—21no—
56C3×C4×Dic5C2×C411yescharacteristic
57C2×C3×#124—51no—
58#5812—51no—
59C2×#836—51no—
60#5215—52no—
61#4523—52no—
62C2×Q8×D5C2×D311yescharacteristic
63C2×D4×D5C2×D311yescharacteristic
64Q8×AGL(1,𝔽5)—64no—
65#4130—32no—
66#4130—62no—
67D5⋊M16—32no—
68C5×#124Dic311yescharacteristic, complemented
69#4093Dic311yescharacteristic, complemented
70#4082Dic311yescharacteristic, complemented
71#4031—22no—
72D5×C22⋊C4—22no—
73#4021Dic311yescharacteristic, complemented
74C42×D5C2×D311yescharacteristic
75#4007—64no—
76#4005—62no—
77#1888—151noSylow
78C23×C3×C5—21no—
79C22×C3×D5—21no—
80C22×C3×D5—44no—
81C22×C3×D5—41no—
82C22×C3×D5—41no—
83C22×C3×D5C2×D411yescharacteristic
84C2×ΓL(1,𝔽16)—24no—
85C3×C5×Q8—22no—
86C3×C5×D4—22no—
87C2×C3×C4×C5—21no—
88C2×C3×C4×C5C22⋊C412yes—
89C2×C3×C4×C5C22×C411yescharacteristic
90C3×C22⋊D5—42no—
91C2×C3×Dic5—21no—
92C2×C3×Dic5C22⋊C412yes—
93C2×C3×Dic5C2×D411yescharacteristic
94C3×D20—22no—
95C3×C4×D5—22no—
96C3×C4×D5—22no—
97C3×C4×D5—42no—
98C3×C4×D5C2×D412yes—
99C3×C4×D5#4112yes—
100C3×Dic10—22no—
101C3×Dic10—24no—
102C3⋊#170—42no—
103C22×C3×Q8—51no—
104C22×C3×D4—51no—
105C3×#124—51no—
106C3×#124—51no—
107C3×#124—51no—
108C3×#124—102no—
109C3×#124AGL(1,𝔽5)11yescharacteristic, complemented
110C2×C3×C22⋊C4—102no—
111C2×C3×C42—51no—
112Q8×Dic3—104no—
113#926—52no—
114#926—102no—
115C2×#208—52no—
116#836—102no—
117#832—104no—
118C23×D5—21no—
119C2×C5×Q8C2×Dic311yescharacteristic
120C2×C5×D4C2×Dic311yescharacteristic
121C2×C22⋊D5—21no—
122Q8×D5—22no—
123Q8×D5—22no—
124D4×D5—22no—
125D4×D5—22no—
126C2×D20C2×Dic311yescharacteristic
127C2×C4×D5—21no—
128C2×C4×D5#6712yes—
129C2×C4×D5C22×D311yescharacteristic
130C2×Dic10C2×Dic311yescharacteristic
131C2×Dic10#6712yes—
132C22⋊AGL(1,𝔽5)—62no—
133#704—62no—
134C4⋊AGL(1,𝔽5)—34no—
135C4⋊AGL(1,𝔽5)—64no—
136C4×AGL(1,𝔽5)—64no—
137C4×AGL(1,𝔽5)—62no—
138#700—62no—
139D5⋊C8—62no—
140C5×C22⋊C4—22no—
141C42×C5C2×Dic311yescharacteristic
142C22⋊Dic5—22no—
143#685—22no—
144#685—22no—
145C4×Dic5—21no—
146C4×Dic5C2×Dic311yescharacteristic
147C2×#124—51no—
148#519—151no—
149#418—152no—
150C2×#101—151no—
151#339—152no—
152C22×C3×C5—42no—
153C22×C3×C5—41no—
154C22×C3×C5C2×C22⋊C411yescharacteristic
155C2×C3×D5—22no—
156C2×C3×D5—42no—
157C2×C3×D5—41no—
158C2×C3×D5—82no—
159C2×C3×D5—82no—
160C2×C3×D5#12011yescharacteristic
161C2×C3×D5#12311yescharacteristic
162ΓL(1,𝔽16)—44no—
163C3×C4×C5—22no—
164C3×C4×C5—41no—
165C3×C4×C5C4×D412yes—
166C3×Dic5—22no—
167C3×Dic5—41no—
168C3×Dic5#12112yes—
169C24×C3—101no—
170C2×C3×Q8—52no—
171C2×C3×Q8—51no—
172C2×C3×Q8—102no—
173C2×C3×Q8—102no—
174C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
175C2×C3×D4—51no—
176C2×C3×D4—102no—
177C2×C3×D4—102no—
178C2×C3×D4—101no—
179C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
180C22×C3×C4—51no—
181C22×C3×C4—52no—
182C22×C3×C4—101no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—102no—
185C3×C22⋊C4—102no—
186C3×C22⋊C4—102no—
187C3×C42—51no—
188C3×C42—101no—
189C3×C42C2×AGL(1,𝔽5)11yescharacteristic
190#217—102no—
191C4⋊Dic3—54no—
192C4⋊Dic3—104no—
193C4×Dic3—102no—
194C4×Dic3—104no—
195#208—102no—
196#208—102no—
197C2×C3⋊C8—102no—
198C23×C5—21no—
199C22×D5—21no—
200C22×D5—44no—
201C22×D5—41no—
202C22×D5—41no—
203C22×D5C2×#6711yescharacteristic
204C2×AGL(1,𝔽5)—64no—
205C5×Q8—22no—
206C5×D4—22no—
207C2×C4×C5—21no—
208C2×C4×C5#21712yes—
209C2×C4×C5C22×Dic311yescharacteristic
210C22⋊D5—42no—
211C2×Dic5—21no—
212C2×Dic5#21712yes—
213C2×Dic5C2×#6711yescharacteristic
214D20—22no—
215C4×D5—22no—
216C4×D5—22no—
217C4×D5—42no—
218C4×D5D3×D412yes—
219C4×D5#23712yes—
220Dic10—22no—
221Dic10—24no—
222#170—122no—
223C22×Q8—51no—
224C22×D4—51no—
225C2×M16—152no—
226#124—51no—
227#124—51no—
228#124—51no—
229#124—102no—
230#124ΓL(1,𝔽16)11yescharacteristic, complemented
231C4×Q8—304no—
232#117—152no—
233#117—302no—
234C2×C22⋊C4—102no—
235C2×C42—51no—
236#102—304no—
237#101—302no—
238C2×C3×C5—21no—
239C2×C3×C5—81no—
240C2×C3×C5#38611yescharacteristic
241C3×D5—21no—
242C3×D5—81no—
243C3×D5#45412yes—
244C23×C3—21no—
245C23×C3—51no—
246C23×C3—101no—
247C23×C3—201no—
248C23×C3—201no—
249C23×C3—204no—
250C3×Q8—22no—
251C3×Q8—104no—
252C3×Q8—102no—
253C3×D4—22no—
254C3×D4—102no—
255C3×D4—202no—
256C2×C3×C4—21no—
257C2×C3×C4—51no—
258C2×C3×C4—52no—
259C2×C3×C4—52no—
260C2×C3×C4—52no—
261C2×C3×C4—102no—
262C2×C3×C4—102no—
263C2×C3×C4—101no—
264C2×C3×C4—202no—
265C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
266C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
267C2×Dic3—104no—
268C3⋊C8—202no—
269C22×C5—42no—
270C22×C5—41no—
271C22×C5C2×#21711yescharacteristic
272C2×D5—22no—
273C2×D5—42no—
274C2×D5—41no—
275C2×D5—82no—
276C2×D5—82no—
277C2×D5#93511yescharacteristic
278C2×D5#93711yescharacteristic
279AGL(1,𝔽5)—124no—
280C4×C5—22no—
281C4×C5—41no—
282C4×C5D4×Dic312yes—
283Dic5—22no—
284Dic5—41no—
285Dic5#93912yes—
286C24—101no—
287C2×Q8—52no—
288C2×Q8—51no—
289C2×Q8—102no—
290C2×Q8—102no—
291C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
292C2×D4—51no—
293C2×D4—102no—
294C2×D4—102no—
295C2×D4—101no—
296C2×D4C2×ΓL(1,𝔽16)11yescharacteristic
297C22×C4—51no—
298C22×C4—52no—
299C22×C4—101no—
300M16—302no—
301M16—302no—
302C2×C8—302no—
303C4⋊C4—154no—
304C4⋊C4—304no—
305C22⋊C4—22no—
306C22⋊C4—102no—
307C22⋊C4—102no—
308C22⋊C4—102no—
309C22⋊C4—302no—
310C42—51no—
311C42—101no—
312C42—302no—
313C42—304no—
314C42C2×ΓL(1,𝔽16)11yescharacteristic
315C3×C5#188811yescharacteristic, complemented, Hall
316C22×C3—42no—
317C22×C3—41no—
318C22×C3—51no—
319C22×C3—51no—
320C22×C3—102no—
321C22×C3—202no—
322C22×C3—201no—
323C22×C3—402no—
324C22×C3—402no—
325C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
326C3×C4—22no—
327C3×C4—41no—
328C3×C4—52no—
329C3×C4—102no—
330C3×C4—201no—
331C3×C4D4×AGL(1,𝔽5)12yes—
332Dic3—204no—
333C2×C5—21no—
334C2×C5—81no—
335C2×C5#522011yescharacteristic
336D5—21no—
337D5—81no—
338D5#506212yes—
339C23—21no—
340C23—51no—
341C23—101no—
342C23—201no—
343C23—201no—
344C23—204no—
345Q8—22no—
346Q8—104no—
347Q8—102no—
348D4—22no—
349D4—102no—
350D4—202no—
351C2×C4—21no—
352C2×C4—51no—
353C2×C4—52no—
354C2×C4—52no—
355C2×C4—52no—
356C2×C4—102no—
357C2×C4—102no—
358C2×C4—101no—
359C2×C4—202no—
360C2×C4—304no—
361C2×C4#676312yes—
362C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
363C8—602no—
364C2×C3—21no—
365C2×C3—52no—
366C2×C3—81no—
367C2×C3—101no—
368C2×C3—401no—
369C2×C3#6577911yescharacteristic
370C5#7172411yescharacteristic, complemented, Sylow
371C22—42no—
372C22—41no—
373C22—51no—
374C22—51no—
375C22—102no—
376C22—202no—
377C22—201no—
378C22—402no—
379C22—402no—
380C22C2×#676311yescharacteristic
381C4—22no—
382C4—41no—
383C4—52no—
384C4—102no—
385C4—201no—
386C4—604no—
387C4D4×ΓL(1,𝔽16)12yes—
388C3#1060456311yescharacteristic, complemented, Sylow
389C2—21no—
390C2—52no—
391C2—81no—
392C2—101no—
393C2—401no—
394C2#1175243811yescharacteristic
subgroupG/H
centerC2#11752438
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#124C4
socleC2×C3×C5#386
prime radicalC2×C3×D4×D5C22
cosocleC2×C3×C4×D5C23
hypercenter#124ΓL(1,𝔽16)
hypocenterC3×C5#1888
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #11756991C2semidirect
3. D5⋊#836C2semidirect
5. #11746733C2semidirect
13. C3×C5×#124C4semidirect
14. C3×#4093C4semidirect
15. C3×#4082C4semidirect
18. C3×#4021C4semidirect
68. C5×#124Dic3semidirect
69. #4093Dic3semidirect
70. #4082Dic3semidirect
73. #4021Dic3semidirect
109. C3×#124AGL(1,𝔽5)semidirect
230. #124ΓL(1,𝔽16)semidirect
315. C3×C5#1888semidirect
370. C5#71724semidirect
388. C3#10604563semidirect

Maximal subgroups

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

subgroupindex
#117569912
C3×D5×#1242
D5⋊#8362
#117524262
#117467332
#106045633
#717245