Finite Group Databasebeta

← #28483534

Subgroups — #28483534

Order 15104.

540 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434104C211yesmaximal, characteristic
2#21425040C211yesmaximal, characteristic, complemented
3C4×#14273648C211yesmaximal, characteristic
4#21422550C211yesmaximal, characteristic, complemented
5#21421997C211yesmaximal, characteristic, complemented
6#21421934C211yesmaximal, characteristic, complemented
7D4×M16×C59C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837732—21no—
10#16837692C411yescharacteristic, complemented
11#16837692C411yescharacteristic, complemented
12#16837607C411yescharacteristic, complemented
13#16837607C411yescharacteristic, complemented
14#16837553—21no—
15C4×#11745670—21no—
16C4×#11745670—21no—
17#16837455C2211yescharacteristic
18C2×#14273648—21no—
19C2×C4×C59⋊C8C2211yescharacteristic
20C4×C4⋊Dic59C2211yescharacteristic
21#16837164—21no—
22#16837163—21no—
23#16837163—21no—
24#16837132—22no—
25C22×M16×C59—21no—
26C2×C4×D4×C59C2211yescharacteristic
27C59×#436—21no—
28C59×#435—22no—
29C8×D4×C59—21no—
30C59×#423C2211yescharacteristic
31C59×#407—21no—
32C4×M16×C59C2211yescharacteristic
33C2×#11745670—21no—
34C2×#11745670—21no—
35C22×C59⋊C8—21no—
36#14273727C2×C411yescharacteristic
37#14273727C2×C411yescharacteristic
38#14273719—21no—
39#14273706—21no—
40#14273706—21no—
41C4×Dic118—21no—
42C4×Dic118—21no—
43#14273686—21no—
44C2×C4⋊Dic59—21no—
45C2×C4×Dic59—21no—
46#14273658—21no—
47#14273658—21no—
48C4⋊(C59⋊C8)C2×C411yescharacteristic
49C4⋊(C59⋊C8)C2×C411yescharacteristic
50#14273650—21no—
51#14273650—21no—
52#14273648—24no—
53#14273648—21no—
54#14273648—21no—
55C22⋊(C59⋊C8)—21no—
56C4×C59⋊C8—21no—
57C4×C59⋊C8—21no—
58C4×C59⋊C8—21no—
59C59⋊#95—21no—
60C22×D4×C59D411yescharacteristic
61C23×C4×C59—21no—
62C2×M16×C59—22no—
63C2×M16×C59—21no—
64C2×M16×C59—42no—
65C2×M16×C59—42no—
66C2×M16×C59D411yescharacteristic
67C2×M16×C59D411yescharacteristic
68C22×C8×C59—22no—
69C4×D4×C59—21no—
70C4×D4×C59—21no—
71C4×D4×C59C2×C411yescharacteristic
72C4×D4×C59C2×C411yescharacteristic
73C4×D4×C59D411yescharacteristic
74C4×D4×C59D411yescharacteristic
75C2×C4⋊C4×C59D411yescharacteristic
76C2×C22⋊C4×C59—21no—
77C2×C42×C59C2311yescharacteristic
78C4⋊C8×C59—21no—
79C4⋊C8×C59D412yes—
80C22⋊C8×C59—22no—
81C22⋊C8×C59—22no—
82#97×C59—21no—
83C4×C8×C59—21no—
84C24×C59—21no—
85C2×D4×C59—22no—
86C2×D4×C59—21no—
87C2×D4×C59—41no—
88C2×D4×C59C22⋊C411yescharacteristic
89C2×D4×C59C22⋊C411yescharacteristic
90C22×C4×C59—21no—
91C22×C4×C59—21no—
92C22×C4×C59—21no—
93C22×C4×C59—21no—
94C22×C4×C59—21no—
95C22×C4×C59—41no—
96C22×C4×C59—41no—
97C22×C4×C59—41no—
98C22×C4×C59—41no—
99C22×C4×C59C2×D411yescharacteristic
100C22×C4×C59C2×D411yescharacteristic
101C22×C4×C59#4111yescharacteristic
102C22×Dic59—41no—
103C2×Dic118—21no—
104C2×Dic118—21no—
105M16×C59—21no—
106M16×C59—21no—
107M16×C59—21no—
108M16×C59—44no—
109M16×C59—41no—
110M16×C59—41no—
111M16×C59SD1612yescomplemented
112C2×C8×C59—21no—
113C2×C8×C59—21no—
114C2×C8×C59—22no—
115C2×C8×C59—44no—
116C4⋊C4×C59—21no—
117C4⋊C4×C59C22⋊C411yescharacteristic
118C4⋊C4×C59C22⋊C411yescharacteristic
119C22⋊C4×C59—21no—
120C22⋊C4×C59—21no—
121C22⋊C4×C59—21no—
122C22⋊C4×C59—21no—
123C42×C59C22×C411yescharacteristic
124C42×C59C2×D411yescharacteristic
125C42×C59C2×D411yescharacteristic
126C42×C59#4111yescharacteristic
127C22⋊Dic59—41no—
128#11745670—42no—
129#11745670—42no—
130C4⋊Dic59—22no—
131C4⋊Dic59—21no—
132C4⋊Dic59—21no—
133C4⋊Dic59—21no—
134C4⋊Dic59—21no—
135C4⋊Dic59—41no—
136Dic59⋊C4—41no—
137Dic59⋊C4—41no—
138C4×Dic59—41no—
139C4×Dic59—41no—
140C4×Dic59—41no—
141C2×C59⋊C8—22no—
142C2×C59⋊C8—21no—
143C2×C59⋊C8—21no—
144C2×C59⋊C8—41no—
145C2×C59⋊C8—41no—
146C23×C59—21no—
147C23×C59—21no—
148C23×C59—42no—
149C23×C59—41no—
150C23×C59—41no—
151C23×C59—81no—
152C23×C59#12111yescharacteristic
153D4×C59—21no—
154D4×C59—21no—
155D4×C59—21no—
156D4×C59—42no—
157D4×C59#10412yes—
158C2×C4×C59—21no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—41no—
168C2×C4×C59—41no—
169C2×C4×C59—41no—
170C2×C4×C59—41no—
171C2×C4×C59—41no—
172C2×C4×C59—41no—
173C2×C4×C59C2×C22⋊C411yescharacteristic
174C2×C4×C59C4×D411yescharacteristic
175C2×C4×C59C4×D411yescharacteristic
176C2×C4×C59#12011yescharacteristic
177C2×C4×C59#12111yescharacteristic
178C2×C4×C59#12311yescharacteristic
179C2×C4×C59C2×SD1611yescharacteristic
180C2×C4×C59#13511yescharacteristic
181C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C59#13711yescharacteristic
183C2×Dic59—42no—
184C2×Dic59—41no—
185C2×Dic59—41no—
186C2×Dic59—81no—
187C2×Dic59—81no—
188Dic118—42no—
189Dic118—41no—
190Dic118—41no—
191C8×C59—22no—
192C8×C59—42no—
193C8×C59—41no—
194C59⋊C8—42no—
195C59⋊C8—41no—
196C59⋊C8—41no—
197#8989—591nomaximal, Sylow
198C22×C59—21no—
199C22×C59—21no—
200C22×C59—22no—
201C22×C59—41no—
202C22×C59—41no—
203C22×C59—41no—
204C22×C59—41no—
205C22×C59—41no—
206C22×C59—82no—
207C22×C59—82no—
208C22×C59#38611yescharacteristic
209C22×C59#46111yescharacteristic
210C22×C59#46311yescharacteristic
211C4×C59—21no—
212C4×C59—21no—
213C4×C59—21no—
214C4×C59—41no—
215C4×C59—41no—
216C4×C59C2×#10411yescharacteristic
217C4×C59#42111yescharacteristic
218C4×C59C4×SD1611yescharacteristic
219C4×C59#44011yescharacteristic
220C4×C59#45011yescharacteristic
221C4×C59#45211yescharacteristic
222Dic59—82no—
223Dic59—81no—
224Dic59—81no—
225#3101—591no—
226D4×M16D5911yescharacteristic
227#1986—591no—
228C4×#102—591no—
229#1493—591no—
230#1303—591no—
231#1278—591no—
232C2×C59—21no—
233C2×C59—21no—
234C2×C59—42no—
235C2×C59—81no—
236C2×C59#187111yescharacteristic
237C2×C59#187711yescharacteristic
238C2×C59#188111yescharacteristic
239C22×M16—21no—
240#533—591no—
241C2×C4×D4C2×D5911yescharacteristic
242#483—1181no—
243#476—591no—
244#476—591no—
245#461—591no—
246#461—591no—
247#450—1181no—
248C4×SD16—1181no—
249C4×SD16—1181no—
250#436—21no—
251#435—22no—
252C8×D4—21no—
253#431—591no—
254#423C2×D5911yescharacteristic
255C2×#102—1181no—
256#407—21no—
257C4×M16C2×D5911yescharacteristic
258C2×C4×C8—591no—
259C4×C4⋊C4—591no—
260#340—1181no—
261#339—1181no—
262#339—1181no—
263Q8⋊C8—1182no—
264C59#898911yescharacteristic, complemented, Sylow
265C22×D4D23611yescharacteristic
266C23×C4—21no—
267C2×SD16—1181no—
268C2×SD16—1181no—
269C2×M16—22no—
270C2×M16—21no—
271C2×M16—42no—
272C2×M16—42no—
273C2×M16C22⋊D5911yescharacteristic
274C2×M16C22⋊D5911yescharacteristic
275C22×C8—22no—
276C22×C8—1181no—
277C4⋊Q8—591no—
278C4⋊Q8—591no—
279#125—1181no—
280C22⋊Q8—1181no—
281C22⋊Q8—1181no—
282C4×Q8—1181no—
283C4×Q8—1181no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4C4×D5911yescharacteristic
287C4×D4C4×D5911yescharacteristic
288C4×D4C22⋊D5911yescharacteristic
289C4×D4C22⋊D5911yescharacteristic
290#117—1181no—
291C2×C4⋊C4—1181no—
292C2×C4⋊C4D23611yescharacteristic
293C2×C22⋊C4—21no—
294C2×C42—1181no—
295C2×C42C22×D5911yescharacteristic
296#106—1181no—
297#106—1181no—
298C4⋊C8—21no—
299C4⋊C8—591no—
300C4⋊C8—591no—
301C4⋊C8C22⋊D5912yes—
302#103—1181no—
303#103—1181no—
304#102—1184no—
305#102—1181no—
306#102—1181no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—1181no—
310#97—21no—
311C4×C8—21no—
312C4×C8—1181no—
313C4×C8—1181no—
314C4×C8—1181no—
315#95—1181no—
316C24—21no—
317C2×Q8—1181no—
318C2×Q8—1181no—
319C2×D4—22no—
320C2×D4—21no—
321C2×D4—41no—
322C2×D4(C2×D59)⋊C411yescharacteristic
323C2×D4(C2×D59)⋊C411yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—2361no—
334C22×C4C2×D23611yescharacteristic
335C22×C4#1174568511yescharacteristic
336C22×C4C2×C22⋊D5911yescharacteristic
337SD16—2362no—
338SD16—2362no—
339M16—21no—
340M16—21no—
341M16—21no—
342M16—44no—
343M16—41no—
344M16—41no—
345M16#1174567012yescomplemented
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—44no—
350C2×C8—1182no—
351C2×C8—1181no—
352C2×C8—1181no—
353C2×C8—2361no—
354C2×C8—2361no—
355C4⋊C4—21no—
356C4⋊C4—1182no—
357C4⋊C4—1181no—
358C4⋊C4—1181no—
359C4⋊C4—1181no—
360C4⋊C4—1181no—
361C4⋊C4—2361no—
362C4⋊C4—2361no—
363C4⋊C4—2361no—
364C4⋊C4(C2×D59)⋊C411yescharacteristic
365C4⋊C4(C2×D59)⋊C411yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—21no—
369C22⋊C4—21no—
370C22⋊C4—2361no—
371C42—2361no—
372C42—2361no—
373C42—2361no—
374C42C2×C4×D5911yescharacteristic
375C42#1174568511yescharacteristic
376C42C2×C22⋊D5911yescharacteristic
377C42C2×C22⋊D5911yescharacteristic
378C23—21no—
379C23—21no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—81no—
384C23#1427370111yescharacteristic
385Q8—2362no—
386Q8—2361no—
387Q8—2361no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—42no—
392D4#1427365312yes—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—4721no—
409C2×C4—4721no—
410C2×C4—2362no—
411C2×C4—2361no—
412C2×C4—2361no—
413C2×C4C2×(C2×D59)⋊C411yescharacteristic
414C2×C4C4×C22⋊D5911yescharacteristic
415C2×C4C4×C22⋊D5911yescharacteristic
416C2×C4#1427369611yescharacteristic
417C2×C4#1427370111yescharacteristic
418C2×C4#1427371011yescharacteristic
419C2×C4C2×#1174567011yescharacteristic
420C2×C4#1427374711yescharacteristic
421C2×C4#1427375311yescharacteristic
422C2×C4#1427375711yescharacteristic
423C8—22no—
424C8—42no—
425C8—41no—
426C8—2362no—
427C8—2361no—
428C8—2361no—
429C22—21no—
430C22—21no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—82no—
438C22—82no—
439C22#1683728111yescharacteristic
440C22#1683760711yescharacteristic
441C22#1683761711yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—41no—
446C4—41no—
447C4—4722no—
448C4—4721no—
449C4—4721no—
450C4C2×#1427365311yescharacteristic
451C4#1683741811yescharacteristic
452C4C4×#1174567011yescharacteristic
453C4#1683750211yescharacteristic
454C4#1683755311yescharacteristic
455C4#1683756411yescharacteristic
456C2—21no—
457C2—21no—
458C2—42no—
459C2—81no—
460C2#2142439811yescharacteristic
461C2#2142443011yescharacteristic
462C2#2142445811yescharacteristic
subgroupG/H
centerC2×C4#14273696
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingD4×M16×C59C2
socleC22×C59#386
prime radicalC22×D4×C59D4
cosocleC2×C42×C59C23
hypercenterD4×M16D59
hypocenterC59#8989
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
2. #21425040C2semidirect
4. #21422550C2semidirect
5. #21421997C2semidirect
6. #21421934C2semidirect
10. #16837692C4semidirect
11. #16837692C4semidirect
12. #16837607C4semidirect
13. #16837607C4semidirect
111. M16×C59SD16semidirect
264. C59#8989semidirect
345. M16#11745670semidirect

Maximal subgroups

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

subgroupindex
#214341042
#214250402
C4×#142736482
#214225502
#214219972
#214219342
D4×M16×C592
#898959