Finite Group Databasebeta

← #14298654

Subgroups — #14298654

Order 1920.

592 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 102 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#11754488C211yesmaximal, characteristic
2C2×#91228C211yesmaximal, characteristic
3C2×C3×#3999C211yesmaximal, characteristic
4#11746572C212yesmaximal, complemented
5#11746571C212yesmaximal, complemented
6#10604385—31nomaximal, Hall
7C22×C3×D20C2211yescharacteristic
8C2×#6763C2212yes—
9C2×C3×C22⋊AGL(1,𝔽5)C2212yes—
10#91815C411yescharacteristic, complemented
11C5×#926C411yescharacteristic, complemented
12D5×#217C411yescharacteristic, complemented
13D5×#217C411yescharacteristic, complemented
14D5×#217C2211yescharacteristic
15D5×#217C2211yescharacteristic
16#91434—21no—
17#91434—21no—
18Dic3×D20—21no—
19Dic3×D20—21no—
20#91228—21no—
21#91228—21no—
22C3×#3999—21no—
23C3×#3999—21no—
24#70020—51nomaximal, Hall
25#65855—31no—
26C2×#3999—31no—
27C2×#3999D311yescharacteristic
28#64866—32no—
29#64866—32no—
30C23×C3×D5C2×C411yescharacteristic
31C23×C3×D5C2311yescharacteristic
32C22×ΓL(1,𝔽16)—22no—
33C22×C3×AGL(1,𝔽5)—22no—
34C22×C3×C4×C5C2×C411yescharacteristic
35C2×C3×D20—22no—
36C2×C3×D20—22no—
37C2×C3×D20D411yescharacteristic
38C2×C3×D20D411yescharacteristic
39C2×C3×D20D411yescharacteristic
40C2×C3×D20D411yescharacteristic
41C2×D5×Dic3—21no—
42C2×D5×Dic3—21no—
43C2×D5×Dic3—21no—
44C2×D5×Dic3—21no—
45#6763—22no—
46#6763—22no—
47C3×C22⋊AGL(1,𝔽5)—22no—
48C3×C22⋊AGL(1,𝔽5)—22no—
49#6719C2×C411yescharacteristic
50#6719C2×C411yescharacteristic
51C4⋊Dic15—21no—
52C4×Dic15—21no—
53C5×#217C2×C411yescharacteristic
54C5×#217C2×C411yescharacteristic
55C5×C4⋊Dic3—21no—
56C4×C5×Dic3—21no—
57#6665—21no—
58#6665—21no—
59#6665—21no—
60#6665—21no—
61#5798—51no—
62C2×C3×#99—51no—
63C2×#834—51no—
64#4527—52no—
65#4526—52no—
66C22×D20C2×D311yescharacteristic
67C2×C22⋊AGL(1,𝔽5)—32no—
68C2×C22⋊AGL(1,𝔽5)C2×D312yes—
69C5×#117—31no—
70#4072—31no—
71#4039—61no—
72#4039—61no—
73D5×C22⋊C4—31no—
74D5×C22⋊C4—31no—
75D5×C22⋊C4—31no—
76D5×C22⋊C4—31no—
77C4×D20—61no—
78C4×D20—61no—
79#3999—21no—
80#3999—21no—
81#3999—61no—
82#3999—61no—
83#1828—151noSylow
84C23×C3×C5C22×C411yescharacteristic
85C22×C3×D5—21no—
86C22×C3×D5—21no—
87C22×C3×D5—81no—
88C22×C3×D5—81no—
89C22×C3×D5C22⋊C411yescharacteristic
90C22×C3×D5C22⋊C411yescharacteristic
91C22×C3×D5C2×D411yescharacteristic
92C22×C3×D5C2×D411yescharacteristic
93C22×C3×D5#4112yes—
94C22×C3×D5#4112yes—
95C2×ΓL(1,𝔽16)—42no—
96C2×ΓL(1,𝔽16)—82no—
97C2×C3×AGL(1,𝔽5)—42no—
98C2×C3×AGL(1,𝔽5)—82no—
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—21no—
101C2×C3×C4×C5C22⋊C411yescharacteristic
102C2×C3×C4×C5C22⋊C411yescharacteristic
103C2×Dic15—21no—
104C2×Dic15—21no—
105C2×C5×Dic3—21no—
106C2×C5×Dic3—21no—
107C3×D20—22no—
108C3×D20—22no—
109C3×D20—22no—
110C3×D20—22no—
111D5×Dic3—81no—
112D5×Dic3—81no—
113D5×Dic3—81no—
114D5×Dic3—81no—
115C22×C3×D4—51no—
116C2×C3×C22⋊C4—52no—
117C2×#217—51no—
118C2×#217—51no—
119C2×#217—51no—
120C2×#217—51no—
121C2×#217—52no—
122D4×Dic3—101no—
123D4×Dic3—101no—
124D4×Dic3—101no—
125D4×Dic3—101no—
126#926—51no—
127#926AGL(1,𝔽5)11yescharacteristic, complemented
128C3×#99—101no—
129C3×#99—101no—
130#834—101no—
131#834—101no—
132C23×D5C4×D311yescharacteristic
133C23×D5C22×D311yescharacteristic
134C22×AGL(1,𝔽5)—22no—
135C22×AGL(1,𝔽5)—62no—
136C22×C4×C5C4×D311yescharacteristic
137C2×D20—22no—
138C2×D20—22no—
139C2×D20D1211yescharacteristic
140C2×D20D1211yescharacteristic
141C2×D20#6711yescharacteristic
142C2×D20#6711yescharacteristic
143C2×C4×D5—61no—
144C2×C4×D5—61no—
145C2×C4×D5—61no—
146C2×C4×D5—61no—
147C22⋊AGL(1,𝔽5)—22no—
148C22⋊AGL(1,𝔽5)—22no—
149C22⋊AGL(1,𝔽5)—62no—
150C22⋊AGL(1,𝔽5)—62no—
151C5×C4⋊C4—61no—
152C5×C22⋊C4—31no—
153C5×C22⋊C4—31no—
154C42×C5—61no—
155C22⋊Dic5—31no—
156C22⋊Dic5—31no—
157#685—61no—
158#685—61no—
159#685—61no—
160#685—61no—
161C4⋊Dic5—61no—
162C4×Dic5—61no—
163#518—151no—
164C2×#99—51no—
165C2×#99—151no—
166#342—152no—
167#342—152no—
168C22×C3×C5—41no—
169C22×C3×C5C2×C22⋊C411yescharacteristic
170C22×C3×C5#11711yescharacteristic
171C22×C3×C5#11711yescharacteristic
172C2×C3×D5—21no—
173C2×C3×D5—21no—
174C2×C3×D5—41no—
175C2×C3×D5—41no—
176C2×C3×D5—42no—
177C2×C3×D5—42no—
178C2×C3×D5—41no—
179C2×C3×D5—41no—
180C2×C3×D5—81no—
181C2×C3×D5—81no—
182C2×C3×D5#12312yes—
183C2×C3×D5#12312yes—
184ΓL(1,𝔽16)—82no—
185C3×AGL(1,𝔽5)—82no—
186C3×C4×C5—21no—
187C3×C4×C5—21no—
188Dic15—41no—
189Dic15—41no—
190C5×Dic3—41no—
191C5×Dic3—41no—
192C24×C3—51no—
193C24×C3—51no—
194C2×C3×D4—51no—
195C2×C3×D4—51no—
196C2×C3×D4—51no—
197C2×C3×D4—51no—
198C2×C3×D4—102no—
199C2×C3×D4—102no—
200C22×C3×C4—102no—
201C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
202C22×Dic3—101no—
203C22×Dic3—101no—
204C22×Dic3—101no—
205C22×Dic3—101no—
206C22×Dic3—102no—
207C3×C22⋊C4—102no—
208C3×C22⋊C4—102no—
209#217—51no—
210#217—51no—
211#217—101no—
212#217—101no—
213#217—101no—
214#217—101no—
215#217—102no—
216#217—102no—
217#217C2×AGL(1,𝔽5)11yescharacteristic
218#217C2×AGL(1,𝔽5)11yescharacteristic
219C4⋊Dic3—21no—
220C4⋊Dic3—101no—
221C4×Dic3—21no—
222C4×Dic3—101no—
223C23×C5C2×C4×D311yescharacteristic
224C22×D5—21no—
225C22×D5—21no—
226C22×D5—81no—
227C22×D5—81no—
228C22×D5(C2×D3)⋊C411yescharacteristic
229C22×D5(C2×D3)⋊C411yescharacteristic
230C22×D5C2×D1211yescharacteristic
231C22×D5#23712yes—
232C22×D5#23712yes—
233C22×D5C2×#6711yescharacteristic
234C2×AGL(1,𝔽5)—42no—
235C2×AGL(1,𝔽5)—82no—
236C2×AGL(1,𝔽5)—122no—
237C2×AGL(1,𝔽5)—242no—
238C2×C4×C5—21no—
239C2×C4×C5—21no—
240C2×C4×C5—61no—
241C2×C4×C5—61no—
242C2×C4×C5(C2×D3)⋊C411yescharacteristic
243C2×C4×C5(C2×D3)⋊C411yescharacteristic
244C2×Dic5—61no—
245C2×Dic5—61no—
246D20—22no—
247D20—22no—
248D20—22no—
249D20—22no—
250C4×D5—241no—
251C4×D5—241no—
252C4×D5—241no—
253C4×D5—241no—
254C22×D4—51no—
255C4×D4—301no—
256C4×D4—301no—
257C4×D4—301no—
258C4×D4—301no—
259#117—31no—
260#117—151no—
261C2×C22⋊C4—52no—
262C2×C22⋊C4—151no—
263C2×C22⋊C4—151no—
264C2×C22⋊C4—151no—
265C2×C22⋊C4—151no—
266C2×C22⋊C4—152no—
267#99—101no—
268#99—101no—
269#99—301no—
270#99—301no—
271C2×C3×C5—21no—
272C2×C3×C5—21no—
273C2×C3×C5—21no—
274C2×C3×C5#38111yescharacteristic
275C3×D5—22no—
276C3×D5—22no—
277C3×D5—41no—
278C3×D5—41no—
279C23×C3—52no—
280C23×C3—52no—
281C23×C3—51no—
282C23×C3—51no—
283C23×C3—51no—
284C23×C3—51no—
285C23×C3—101no—
286C23×C3—101no—
287C23×C3—401no—
288C23×C3—401no—
289C23×C3C22×AGL(1,𝔽5)11yescharacteristic
290C3×D4—102no—
291C3×D4—102no—
292C3×D4—102no—
293C3×D4—102no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—202no—
297C2×C3×C4—402no—
298C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
299C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—101no—
303C2×Dic3—101no—
304C2×Dic3—202no—
305C2×Dic3—401no—
306C2×Dic3—401no—
307C2×Dic3—401no—
308C2×Dic3—401no—
309C2×Dic3—402no—
310C22×C5—41no—
311C22×C5#87711yescharacteristic
312C22×C5#87711yescharacteristic
313C22×C5C2×(C2×D3)⋊C411yescharacteristic
314C2×D5—21no—
315C2×D5—21no—
316C2×D5—41no—
317C2×D5—41no—
318C2×D5—42no—
319C2×D5—42no—
320C2×D5—41no—
321C2×D5—41no—
322C2×D5—81no—
323C2×D5—81no—
324C2×D5#88612yes—
325C2×D5#93512yes—
326AGL(1,𝔽5)—82no—
327AGL(1,𝔽5)—242no—
328C4×C5—21no—
329C4×C5—21no—
330C4×C5—121no—
331C4×C5—121no—
332Dic5—121no—
333Dic5—121no—
334C24—51no—
335C24—51no—
336C2×D4—51no—
337C2×D4—51no—
338C2×D4—51no—
339C2×D4—51no—
340C2×D4—102no—
341C2×D4—102no—
342C22×C4—102no—
343C22×C4—301no—
344C22×C4—301no—
345C22×C4—301no—
346C22×C4—301no—
347C22×C4—302no—
348C22×C4D3×AGL(1,𝔽5)11yescharacteristic
349C4⋊C4—61no—
350C4⋊C4—301no—
351C22⋊C4—31no—
352C22⋊C4—31no—
353C22⋊C4—102no—
354C22⋊C4—102no—
355C22⋊C4—151no—
356C22⋊C4—151no—
357C22⋊C4—301no—
358C22⋊C4—301no—
359C22⋊C4—301no—
360C22⋊C4—301no—
361C22⋊C4—302no—
362C22⋊C4—302no—
363C42—61no—
364C42—301no—
365C3×C5#182811yescharacteristic, complemented, Hall
366C22×C3—41no—
367C22×C3—52no—
368C22×C3—52no—
369C22×C3—101no—
370C22×C3—101no—
371C22×C3—201no—
372C22×C3—201no—
373C22×C3—202no—
374C22×C3—202no—
375C22×C3—201no—
376C22×C3—201no—
377C22×C3—401no—
378C22×C3—401no—
379C22×C3#413011yescharacteristic
380C22×C3#413011yescharacteristic
381C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—402no—
385Dic3—41no—
386Dic3—41no—
387Dic3—201no—
388Dic3—201no—
389Dic3—402no—
390C2×C5—21no—
391C2×C5—21no—
392C2×C5—21no—
393C2×C5#494311yescharacteristic
394D5—22no—
395D5—22no—
396D5—41no—
397D5—41no—
398C23—52no—
399C23—52no—
400C23—51no—
401C23—51no—
402C23—51no—
403C23—51no—
404C23—101no—
405C23—101no—
406C23—401no—
407C23—401no—
408C23C2×D3×AGL(1,𝔽5)11yescharacteristic
409D4—102no—
410D4—102no—
411D4—102no—
412D4—102no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—202no—
418C2×C4—301no—
419C2×C4—301no—
420C2×C4—402no—
421C2×C4—602no—
422C2×C4—1201no—
423C2×C4—1201no—
424C2×C4—1201no—
425C2×C4—1201no—
426C2×C4—1202no—
427C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
428C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—102no—
433C2×C3—102no—
434C2×C3—201no—
435C2×C3—201no—
436C2×C3#6576411yescharacteristic
437C5#7002011yescharacteristic, complemented, Sylow
438C22—41no—
439C22—52no—
440C22—52no—
441C22—101no—
442C22—101no—
443C22—202no—
444C22—201no—
445C22—201no—
446C22—202no—
447C22—201no—
448C22—201no—
449C22—401no—
450C22—401no—
451C22#9192311yescharacteristic
452C22#9192311yescharacteristic
453C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
454C4—21no—
455C4—21no—
456C4—121no—
457C4—121no—
458C4—402no—
459C4—601no—
460C4—601no—
461C4—1202no—
462C3#1060438511yescharacteristic, complemented, Sylow
463C2—21no—
464C2—21no—
465C2—21no—
466C2—102no—
467C2—102no—
468C2—201no—
469C2—201no—
470C2#1175198411yescharacteristic
subgroupG/H
centerC2#11751984
commutatorC23×C3×C5C22×C4
FrattiniC23C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC2×C3×C5#381
prime radicalC22×C3×D20C22
cosocleC23×C3×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1828
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11746572C2semidirect
5. #11746571C2semidirect
10. #91815C4semidirect
11. C5×#926C4semidirect
12. D5×#217C4semidirect
13. D5×#217C4semidirect
127. #926AGL(1,𝔽5)semidirect
365. C3×C5#1828semidirect
437. C5#70020semidirect
462. C3#10604385semidirect

Maximal subgroups

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

subgroupindex
#117544882
C2×#912282
C2×C3×#39992
#117465722
#117465712
#106043853
#700205