Finite Group Databasebeta

← #14283423

Subgroups — #14283423

Order 1920.

572 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65858C211yesmaximal, characteristic
2#11752424C211yesmaximal, characteristic, complemented
3#11752415C211yesmaximal, characteristic, complemented
4#11746740C212yesmaximal, complemented
5#11746739C212yesmaximal, complemented
6#10604348—31nomaximal, Hall
7C2×C3×#709C411yescharacteristic, complemented
8C22×C3×C4×D5C2211yescharacteristic
9C2×#6763—21no—
10C2×C4⋊ΓL(1,𝔽16)—21no—
11C2×#6758—21no—
12C3×C5×#117C411yescharacteristic, complemented
13C3×#4043C411yescharacteristic, complemented
14C3×#4043C2211yescharacteristic
15C3×#4041C411yescharacteristic, complemented
16C3×#4041C2211yescharacteristic
17C3×#4032—21no—
18C3×#4028—21no—
19C3×C4×D20—21no—
20C3×C4×D20—21no—
21#91241—21no—
22#91235C2212yes—
23#91232C2212yes—
24#71709—51nomaximal, Hall
25#65858D311yescharacteristic
26#65765—31no—
27#65756—31no—
28#64869—32no—
29#64868—32no—
30C23×C3×D5D411yescharacteristic
31C22×ΓL(1,𝔽16)—22no—
32C22×C3×C4×C5C2×C411yescharacteristic
33C2×C3×C22⋊D5—21no—
34C22×C3×Dic5D411yescharacteristic
35C3×#709—42no—
36C2×C3×D20—21no—
37C2×C3×D20—21no—
38C2×C3×D20C2×C411yescharacteristic
39C2×C3×C4×D5—21no—
40C2×C3×C4×D5—42no—
41C2×C3×C4×D5C2×C411yescharacteristic
42C2×C3×C4×D5D411yescharacteristic
43C2×C3×C4×D5D411yescharacteristic
44C2×C3×C4×D5C2311yescharacteristic
45C2×C3×Dic10C2×C411yescharacteristic
46#6763—42no—
47C2×C3⋊#170—22no—
48C4⋊ΓL(1,𝔽16)—42no—
49#6758—42no—
50C3×C5×C4⋊C4C2×C411yescharacteristic
51C3×C5×C4⋊C4C2×C411yescharacteristic
52C3×C5×C22⋊C4—21no—
53C3×C42×C5—21no—
54C3×#685—21no—
55C3×#685—21no—
56C3×#685—21no—
57C3×#685—21no—
58C3×C4⋊Dic5—21no—
59C3×C4⋊Dic5—21no—
60C3×#525—51no—
61#5212—51no—
62#5207—51no—
63#4538—52no—
64#4537—52no—
65C2×#709Dic311yescharacteristic, complemented
66C22×C4×D5C2×D311yescharacteristic
67C2×C22⋊AGL(1,𝔽5)—61no—
68C2×C4⋊AGL(1,𝔽5)—61no—
69C2×#700—61no—
70C5×#117Dic311yescharacteristic, complemented
71#4043Dic311yescharacteristic, complemented
72#4043C2×D311yescharacteristic
73#4041Dic311yescharacteristic, complemented
74#4041C2×D311yescharacteristic
75#4032—21no—
76#4028—21no—
77C4×D20—21no—
78C4×D20—21no—
79C22⋊#170—61no—
80(C2×AGL(1,𝔽5))⋊C4—32no—
81#4003—32no—
82#1513—151noSylow
83C23×C3×C5C22⋊C411yescharacteristic
84C22×C3×D5—21no—
85C22×C3×D5—21no—
86C22×C3×D5—21no—
87C22×C3×D5—41no—
88C22×C3×D5—41no—
89C22×C3×D5C22⋊C411yescharacteristic
90C22×C3×D5C2×D411yescharacteristic
91C2×ΓL(1,𝔽16)—42no—
92C2×ΓL(1,𝔽16)—42no—
93C2×ΓL(1,𝔽16)—42no—
94C2×C3×C4×C5—22no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5C22⋊C411yescharacteristic
98C2×C3×C4×C5C22×C411yescharacteristic
99C3×C22⋊D5—42no—
100C2×C3×Dic5—21no—
101C2×C3×Dic5—41no—
102C2×C3×Dic5C22⋊C411yescharacteristic
103C2×C3×Dic5C2×D411yescharacteristic
104C3×D20—41no—
105C3×D20—42no—
106C3×D20—41no—
107C3×D20—41no—
108C3×C4×D5—22no—
109C3×C4×D5—22no—
110C3×C4×D5—41no—
111C3×C4×D5—41no—
112C3×C4×D5—42no—
113C3×Dic10—41no—
114C3⋊#170—42no—
115C2×C3×#41—51no—
116C23×C3×C4—51no—
117C3×#123—101no—
118C3×C22⋊Q8—51no—
119C3×C22⋊Q8—51no—
120C3×#121—51no—
121C3×#121—51no—
122C3×#120—101no—
123C3×C4×D4—101no—
124C3×C4×D4—101no—
125C3×#117AGL(1,𝔽5)11yescharacteristic, complemented
126C2×#217—101no—
127C2×C4⋊Dic3—101no—
128C2×#208—101no—
129C3⋊#95—52no—
130#830—52no—
131#830—101no—
132C23×D5#6711yescharacteristic
133C22×AGL(1,𝔽5)—62no—
134C22×C4×C5C2×Dic311yescharacteristic
135C2×C22⋊D5—21no—
136C22×Dic5#6711yescharacteristic
137#709—42no—
138C2×D20—21no—
139C2×D20—21no—
140C2×D20C2×Dic311yescharacteristic
141C2×C4×D5—21no—
142C2×C4×D5—42no—
143C2×C4×D5C2×Dic311yescharacteristic
144C2×C4×D5#6711yescharacteristic
145C2×C4×D5#6711yescharacteristic
146C2×C4×D5C22×D311yescharacteristic
147C2×Dic10C2×Dic311yescharacteristic
148C22⋊AGL(1,𝔽5)—122no—
149C2×#170—62no—
150C4⋊AGL(1,𝔽5)—122no—
151#700—122no—
152C5×C4⋊C4C2×Dic311yescharacteristic
153C5×C4⋊C4C2×Dic311yescharacteristic
154C5×C22⋊C4—21no—
155C42×C5—21no—
156#685—21no—
157#685—21no—
158#685—21no—
159#685—21no—
160C4⋊Dic5—21no—
161C4⋊Dic5—21no—
162#525—51no—
163#407—151no—
164#380—151no—
165#328—152no—
166#327—152no—
167C22×C3×C5—21no—
168C22×C3×C5—21no—
169C22×C3×C5—21no—
170C22×C3×C5C2×C22⋊C411yescharacteristic
171C2×C3×D5—21no—
172C2×C3×D5—21no—
173C2×C3×D5—21no—
174C2×C3×D5—21no—
175C2×C3×D5—21no—
176C2×C3×D5—42no—
177C2×C3×D5—42no—
178C2×C3×D5—41no—
179C2×C3×D5—41no—
180C2×C3×D5—41no—
181C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
182C2×C3×D5#13711yescharacteristic
183ΓL(1,𝔽16)—82no—
184C3×C4×C5—21no—
185C3×C4×C5—41no—
186C3×C4×C5—41no—
187C3×C4×C5#9912yes—
188C3×Dic5—21no—
189C3×Dic5—21no—
190C3×Dic5—41no—
191C24×C3—51no—
192C3×#41—202no—
193C2×C3×Q8—51no—
194C2×C3×D4—51no—
195C2×C3×D4—101no—
196C2×C3×D4—101no—
197C2×C3×D4—101no—
198C22×C3×C4—51no—
199C22×C3×C4—51no—
200C22×C3×C4—51no—
201C22×C3×C4—51no—
202C22×C3×C4—51no—
203C22×C3×C4—101no—
204C22×C3×C4—202no—
205C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
206C22×Dic3—102no—
207C3×C4⋊C4—101no—
208C3×C4⋊C4—101no—
209C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
210C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
211C3×C22⋊C4—21no—
212C3×C22⋊C4—101no—
213C3×C22⋊C4—101no—
214C3×C22⋊C4—101no—
215C3×C22⋊C4—101no—
216C3×C42—21no—
217#217—202no—
218C4⋊Dic3—202no—
219#208—202no—
220C2×C3⋊C8—102no—
221C23×C5#21711yescharacteristic
222C22×D5—21no—
223C22×D5—21no—
224C22×D5—21no—
225C22×D5—41no—
226C22×D5—41no—
227C22×D5#21711yescharacteristic
228C22×D5C2×#6711yescharacteristic
229C2×AGL(1,𝔽5)—122no—
230C2×AGL(1,𝔽5)—122no—
231C2×AGL(1,𝔽5)—122no—
232C2×C4×C5—22no—
233C2×C4×C5—21no—
234C2×C4×C5—21no—
235C2×C4×C5#21711yescharacteristic
236C2×C4×C5C22×Dic311yescharacteristic
237C22⋊D5—42no—
238C2×Dic5—21no—
239C2×Dic5—41no—
240C2×Dic5#21711yescharacteristic
241C2×Dic5C2×#6711yescharacteristic
242D20—41no—
243D20—42no—
244D20—41no—
245D20—41no—
246C4×D5—22no—
247C4×D5—22no—
248C4×D5—41no—
249C4×D5—41no—
250C4×D5—42no—
251Dic10—41no—
252#170—122no—
253C2×#41—51no—
254C23×C4—51no—
255C2×M16—301no—
256#123—101no—
257C22⋊Q8—51no—
258C22⋊Q8—51no—
259#121—51no—
260#121—51no—
261#120—101no—
262C4×D4—101no—
263C4×D4—101no—
264#117ΓL(1,𝔽16)11yescharacteristic, complemented
265C2×C4⋊C4—301no—
266C2×C22⋊C4—301no—
267C22⋊C8—152no—
268C22⋊C8—301no—
269#95—152no—
270C2×C3×C5—41no—
271C2×C3×C5C2×#9911yescharacteristic
272C2×C3×C5#41711yescharacteristic
273C2×C3×C5#42111yescharacteristic
274C3×D5—21no—
275C3×D5—21no—
276C3×D5—41no—
277C3×D5—81no—
278C23×C3—51no—
279C23×C3—51no—
280C23×C3—101no—
281C23×C3—101no—
282C23×C3—101no—
283C23×C3—201no—
284C23×C3—201no—
285C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
286C3×Q8—201no—
287C3×D4—201no—
288C3×D4—201no—
289C3×D4—201no—
290C3×D4—202no—
291C3×D4—202no—
292C2×C3×C4—22no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—51no—
296C2×C3×C4—51no—
297C2×C3×C4—102no—
298C2×C3×C4—102no—
299C2×C3×C4—101no—
300C2×C3×C4—201no—
301C2×C3×C4—201no—
302C2×C3×C4—201no—
303C2×C3×C4—202no—
304C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
305C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
306C2×Dic3—202no—
307C2×Dic3—202no—
308C2×Dic3—202no—
309C3⋊C8—202no—
310C22×C5—21no—
311C22×C5—21no—
312C22×C5—21no—
313C22×C5C2×#21711yescharacteristic
314C2×D5—21no—
315C2×D5—21no—
316C2×D5—21no—
317C2×D5—21no—
318C2×D5—21no—
319C2×D5—42no—
320C2×D5—42no—
321C2×D5—41no—
322C2×D5—41no—
323C2×D5—41no—
324C2×D5#94911yescharacteristic
325C2×D5#95111yescharacteristic
326AGL(1,𝔽5)—242no—
327C4×C5—21no—
328C4×C5—41no—
329C4×C5—41no—
330C4×C5#83412yes—
331Dic5—21no—
332Dic5—21no—
333Dic5—41no—
334C24—51no—
335#41—202no—
336C2×Q8—51no—
337C2×D4—51no—
338C2×D4—101no—
339C2×D4—101no—
340C2×D4—101no—
341C22×C4—51no—
342C22×C4—51no—
343C22×C4—51no—
344C22×C4—51no—
345C22×C4—51no—
346C22×C4—101no—
347C22×C4—202no—
348C22×C4—302no—
349C22×C4C2×ΓL(1,𝔽16)11yescharacteristic
350M16—602no—
351C2×C8—302no—
352C4⋊C4—101no—
353C4⋊C4—101no—
354C4⋊C4—602no—
355C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
356C4⋊C4C2×ΓL(1,𝔽16)11yescharacteristic
357C22⋊C4—21no—
358C22⋊C4—101no—
359C22⋊C4—101no—
360C22⋊C4—101no—
361C22⋊C4—101no—
362C22⋊C4—602no—
363C42—21no—
364C3×C5#151311yescharacteristic, complemented, Hall
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—51no—
369C22×C3—51no—
370C22×C3—101no—
371C22×C3—101no—
372C22×C3—101no—
373C22×C3—101no—
374C22×C3—101no—
375C22×C3—202no—
376C22×C3—202no—
377C22×C3—201no—
378C22×C3—201no—
379C22×C3—201no—
380C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
381C3×C4—21no—
382C3×C4—41no—
383C3×C4—41no—
384C3×C4—101no—
385C3×C4—101no—
386C3×C4—201no—
387C3×C4#399912yes—
388Dic3—402no—
389C2×C5—41no—
390C2×C5C2×#83411yescharacteristic
391C2×C5#523211yescharacteristic
392C2×C5#523611yescharacteristic
393D5—21no—
394D5—21no—
395D5—41no—
396D5—81no—
397C23—51no—
398C23—51no—
399C23—101no—
400C23—101no—
401C23—101no—
402C23—201no—
403C23—201no—
404C23#676311yescharacteristic
405Q8—201no—
406D4—201no—
407D4—201no—
408D4—201no—
409D4—202no—
410D4—202no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—51no—
415C2×C4—51no—
416C2×C4—102no—
417C2×C4—102no—
418C2×C4—101no—
419C2×C4—201no—
420C2×C4—201no—
421C2×C4—201no—
422C2×C4—202no—
423C2×C4—602no—
424C2×C4—602no—
425C2×C4—602no—
426C2×C4#676311yescharacteristic
427C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
428C8—602no—
429C2×C3—41no—
430C2×C3—101no—
431C2×C3—101no—
432C2×C3—201no—
433C2×C3—401no—
434C2×C3C2×#399911yescharacteristic
435C2×C3#6579311yescharacteristic
436C2×C3#6579511yescharacteristic
437C5#7170911yescharacteristic, complemented, Sylow
438C22—21no—
439C22—21no—
440C22—21no—
441C22—51no—
442C22—51no—
443C22—101no—
444C22—101no—
445C22—101no—
446C22—101no—
447C22—101no—
448C22—202no—
449C22—202no—
450C22—201no—
451C22—201no—
452C22—201no—
453C22C2×#676311yescharacteristic
454C4—21no—
455C4—41no—
456C4—41no—
457C4—101no—
458C4—101no—
459C4—201no—
460C4—1202no—
461C4#9122812yes—
462C3#1060434811yescharacteristic, complemented, Sylow
463C2—41no—
464C2—101no—
465C2—101no—
466C2—201no—
467C2—401no—
468C2C2×#9122811yescharacteristic
469C2#1175245211yescharacteristic
470C2#1175245411yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#117C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#65858C2
cosocleC2×C3×C4×D5C23
hypercenter#117ΓL(1,𝔽16)
hypocenterC3×C5#1513
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #11752424C2semidirect
3. #11752415C2semidirect
4. #11746740C2semidirect
5. #11746739C2semidirect
7. C2×C3×#709C4semidirect
12. C3×C5×#117C4semidirect
13. C3×#4043C4semidirect
15. C3×#4041C4semidirect
65. C2×#709Dic3semidirect
70. C5×#117Dic3semidirect
71. #4043Dic3semidirect
73. #4041Dic3semidirect
125. C3×#117AGL(1,𝔽5)semidirect
264. #117ΓL(1,𝔽16)semidirect
364. C3×C5#1513semidirect
437. C5#71709semidirect
462. C3#10604348semidirect

Maximal subgroups

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

subgroupindex
C3×#658582
#117524242
#117524152
#117467402
#117467392
#106043483
#717095