Finite Group Databasebeta

← #14283425

Subgroups — #14283425

Order 1920.

572 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 105 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×#65940C211yesmaximal, characteristic
2#11752424C211yesmaximal, characteristic, complemented
3#11752414C211yesmaximal, characteristic, complemented
4#11746791C212yesmaximal, complemented
5#11746740C212yesmaximal, complemented
6#10604470—31nomaximal, Hall
7C2×C3×#711C411yescharacteristic, complemented
8C22×C3×C4×D5C2211yescharacteristic
9C2×#6763—21no—
10C2×#6762—21no—
11C2×C4⋊ΓL(1,𝔽16)—21no—
12C3×C5×#121C411yescharacteristic, complemented
13C3×#4084C2211yescharacteristic
14C3×#4083—21no—
15C3×D4×Dic5—21no—
16C3×#4070C411yescharacteristic, complemented
17C3×#4043C2211yescharacteristic
18C3×#4038C411yescharacteristic, complemented
19C3×#4029—21no—
20C3×#4027—21no—
21#91235C2212yes—
22#91232—21no—
23#91232C2212yes—
24#71709—51nomaximal, Hall
25#65940D311yescharacteristic
26#65765—31no—
27#65755—31no—
28#64920—32no—
29#64869—32no—
30C23×C3×D5D411yescharacteristic
31C22×ΓL(1,𝔽16)—22no—
32C2×C3×C5×D4—21no—
33C2×C3×C5×D4C2×C411yescharacteristic
34C22×C3×C4×C5C2×C411yescharacteristic
35C2×C3×C22⋊D5—21no—
36C2×C3×C22⋊D5—21no—
37C22×C3×Dic5—21no—
38C22×C3×Dic5D411yescharacteristic
39C3×#711—42no—
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—
47#6762—42no—
48C2×C3⋊#170—22no—
49C4⋊ΓL(1,𝔽16)—42no—
50C3×C5×C4⋊C4C2×C411yescharacteristic
51C3×C5×C22⋊C4—21no—
52C3×C22⋊Dic5—21no—
53C3×C22⋊Dic5—21no—
54C3×#685—21no—
55C3×#685—21no—
56C3×C4⋊Dic5—21no—
57C3×C4⋊Dic5C2×C411yescharacteristic
58C3×#683—21no—
59C3×C4×Dic5—21no—
60C3×#525—51no—
61#5212—51no—
62#5207—51no—
63#4538—52no—
64#4537—52no—
65C2×#711Dic311yescharacteristic, complemented
66C22×C4×D5C2×D311yescharacteristic
67C2×C22⋊AGL(1,𝔽5)—61no—
68C2×#704—61no—
69C2×C4⋊AGL(1,𝔽5)—61no—
70C5×#121Dic311yescharacteristic, complemented
71#4084C2×D311yescharacteristic
72#4083—21no—
73D4×Dic5—21no—
74#4070Dic311yescharacteristic, complemented
75#4043C2×D311yescharacteristic
76#4038Dic311yescharacteristic, complemented
77#4029—21no—
78#4027—21no—
79(C2×AGL(1,𝔽5))⋊C4—32no—
80#4003—32no—
81#4003—61no—
82#1513—151noSylow
83C23×C3×C5—21no—
84C23×C3×C5C22⋊C411yescharacteristic
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—
94C3×C5×D4—41no—
95C3×C5×D4—42no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—41no—
98C2×C3×C4×C5C22⋊C411yescharacteristic
99C2×C3×C4×C5C22×C411yescharacteristic
100C3×C22⋊D5—42no—
101C3×C22⋊D5—41no—
102C3×C22⋊D5—41no—
103C2×C3×Dic5—22no—
104C2×C3×Dic5—21no—
105C2×C3×Dic5—21no—
106C2×C3×Dic5—42no—
107C2×C3×Dic5C22⋊C411yescharacteristic
108C2×C3×Dic5C2×D411yescharacteristic
109C3×C4×D5—22no—
110C3×C4×D5—22no—
111C3×C4×D5—41no—
112C3×C4×D5—41no—
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×#121AGL(1,𝔽5)11yescharacteristic, complemented
122C3×#120—101no—
123C3×C4×D4—101no—
124C3×C4×D4—101no—
125C3×#117—51no—
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—
134C2×C5×D4—21no—
135C2×C5×D4C2×Dic311yescharacteristic
136C22×C4×C5C2×Dic311yescharacteristic
137C2×C22⋊D5—21no—
138C2×C22⋊D5—21no—
139C22×Dic5—21no—
140C22×Dic5#6711yescharacteristic
141#711—42no—
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—
149#704—122no—
150C2×#170—62no—
151C4⋊AGL(1,𝔽5)—122no—
152C5×C4⋊C4C2×Dic311yescharacteristic
153C5×C22⋊C4—21no—
154C22⋊Dic5—21no—
155C22⋊Dic5—21no—
156#685—21no—
157#685—21no—
158C4⋊Dic5—21no—
159C4⋊Dic5C2×Dic311yescharacteristic
160#683—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×C5—41no—
171C22×C3×C5—41no—
172C22×C3×C5—41no—
173C22×C3×C5C2×C22⋊C411yescharacteristic
174C2×C3×D5—21no—
175C2×C3×D5—21no—
176C2×C3×D5—21no—
177C2×C3×D5—21no—
178C2×C3×D5—21no—
179C2×C3×D5—42no—
180C2×C3×D5—42no—
181C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
182C2×C3×D5#13711yescharacteristic
183ΓL(1,𝔽16)—82no—
184C3×C4×C5—21no—
185C3×C4×C5—21no—
186C3×C4×C5—41no—
187C3×Dic5—21no—
188C3×Dic5—41no—
189C3×Dic5—41no—
190C3×Dic5#9912yes—
191C24×C3—51no—
192C3×#41—202no—
193C2×C3×Q8—51no—
194C2×C3×D4—21no—
195C2×C3×D4—101no—
196C2×C3×D4—101no—
197C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
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—51no—
208C3×C4⋊C4—101no—
209C3×C4⋊C4—101no—
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—101no—
217#217—202no—
218C4⋊Dic3—202no—
219#208—202no—
220C2×C3⋊C8—102no—
221C23×C5—21no—
222C23×C5#21711yescharacteristic
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—
232C5×D4—41no—
233C5×D4—42no—
234C2×C4×C5—21no—
235C2×C4×C5—41no—
236C2×C4×C5#21711yescharacteristic
237C2×C4×C5C22×Dic311yescharacteristic
238C22⋊D5—42no—
239C22⋊D5—41no—
240C22⋊D5—41no—
241C2×Dic5—22no—
242C2×Dic5—21no—
243C2×Dic5—21no—
244C2×Dic5—42no—
245C2×Dic5#21711yescharacteristic
246C2×Dic5C2×#6711yescharacteristic
247C4×D5—22no—
248C4×D5—22no—
249C4×D5—41no—
250C4×D5—41no—
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#121ΓL(1,𝔽16)11yescharacteristic, complemented
261#120—101no—
262C4×D4—101no—
263C4×D4—101no—
264#117—51no—
265C2×C4⋊C4—301no—
266C2×C22⋊C4—301no—
267C22⋊C8—152no—
268C22⋊C8—301no—
269#95—152no—
270C2×C3×C5—41no—
271C2×C3×C5—81no—
272C2×C3×C5C2×#9911yescharacteristic
273C2×C3×C5#41711yescharacteristic
274C2×C3×C5#42111yescharacteristic
275C3×D5—21no—
276C3×D5—21no—
277C3×D5—41no—
278C23×C3—21no—
279C23×C3—51no—
280C23×C3—51no—
281C23×C3—101no—
282C23×C3—101no—
283C23×C3—201no—
284C23×C3—201no—
285C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
286C3×Q8—201no—
287C3×D4—41no—
288C3×D4—42no—
289C3×D4—201no—
290C3×D4—201no—
291C3×D4—202no—
292C2×C3×C4—21no—
293C2×C3×C4—41no—
294C2×C3×C4—51no—
295C2×C3×C4—51no—
296C2×C3×C4—102no—
297C2×C3×C4—102no—
298C2×C3×C4—102no—
299C2×C3×C4—101no—
300C2×C3×C4—101no—
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×C5—41no—
314C22×C5—41no—
315C22×C5—41no—
316C22×C5C2×#21711yescharacteristic
317C2×D5—21no—
318C2×D5—21no—
319C2×D5—21no—
320C2×D5—21no—
321C2×D5—21no—
322C2×D5—42no—
323C2×D5—42no—
324C2×D5#94911yescharacteristic
325C2×D5#95111yescharacteristic
326AGL(1,𝔽5)—242no—
327C4×C5—21no—
328C4×C5—21no—
329C4×C5—41no—
330Dic5—21no—
331Dic5—41no—
332Dic5—41no—
333Dic5#83412yes—
334C24—51no—
335#41—202no—
336C2×Q8—51no—
337C2×D4—21no—
338C2×D4—101no—
339C2×D4—101no—
340C2×D4C2×ΓL(1,𝔽16)11yescharacteristic
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—51no—
353C4⋊C4—101no—
354C4⋊C4—101no—
355C4⋊C4—602no—
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—101no—
364C3×C5#151311yescharacteristic, complemented, Hall
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—51no—
372C22×C3—51no—
373C22×C3—101no—
374C22×C3—101no—
375C22×C3—101no—
376C22×C3—101no—
377C22×C3—101no—
378C22×C3—202no—
379C22×C3—202no—
380C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—41no—
384C3×C4—52no—
385C3×C4—101no—
386C3×C4—201no—
387C3×C4—201no—
388Dic3—402no—
389C2×C5—41no—
390C2×C5—81no—
391C2×C5C2×#83411yescharacteristic
392C2×C5#523211yescharacteristic
393C2×C5#523611yescharacteristic
394D5—21no—
395D5—21no—
396D5—41no—
397C23—21no—
398C23—51no—
399C23—51no—
400C23—101no—
401C23—101no—
402C23—201no—
403C23—201no—
404C23#676311yescharacteristic
405Q8—201no—
406D4—41no—
407D4—42no—
408D4—201no—
409D4—201no—
410D4—202no—
411C2×C4—21no—
412C2×C4—41no—
413C2×C4—51no—
414C2×C4—51no—
415C2×C4—102no—
416C2×C4—102no—
417C2×C4—102no—
418C2×C4—101no—
419C2×C4—101no—
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—81no—
431C2×C3—101no—
432C2×C3—101no—
433C2×C3—201no—
434C2×C3#6576911yescharacteristic
435C2×C3D5⋊#9911yescharacteristic
436C2×C3#6579311yescharacteristic
437C5#7170911yescharacteristic, complemented, Sylow
438C22—21no—
439C22—21no—
440C22—21no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—51no—
445C22—51no—
446C22—101no—
447C22—101no—
448C22—101no—
449C22—101no—
450C22—101no—
451C22—202no—
452C22—202no—
453C22C2×#676311yescharacteristic
454C4—21no—
455C4—21no—
456C4—41no—
457C4—52no—
458C4—101no—
459C4—201no—
460C4—201no—
461C4—1202no—
462C3#1060447011yescharacteristic, complemented, Sylow
463C2—41no—
464C2—81no—
465C2—101no—
466C2—101no—
467C2—201no—
468C2#1175242811yescharacteristic
469C2D5⋊#83411yescharacteristic
470C2#1175245211yescharacteristic
subgroupG/H
centerC22C2×#6763
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×#121C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#65940C2
cosocleC2×C3×C4×D5C23
hypercenter#121ΓL(1,𝔽16)
hypocenterC3×C5#1513
LCSG ▷ C2×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. #11752424C2semidirect
3. #11752414C2semidirect
4. #11746791C2semidirect
5. #11746740C2semidirect
7. C2×C3×#711C4semidirect
12. C3×C5×#121C4semidirect
16. C3×#4070C4semidirect
18. C3×#4038C4semidirect
65. C2×#711Dic3semidirect
70. C5×#121Dic3semidirect
74. #4070Dic3semidirect
76. #4038Dic3semidirect
121. C3×#121AGL(1,𝔽5)semidirect
260. #121ΓL(1,𝔽16)semidirect
364. C3×C5#1513semidirect
437. C5#71709semidirect
462. C3#10604470semidirect

Maximal subgroups

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

subgroupindex
C3×#659402
#117524242
#117524142
#117467912
#117467402
#106044703
#717095