Finite Group Databasebeta

← #28746169

Subgroups — #28746169

Order 15104.

516 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436499C211yesmaximal, characteristic, complemented
2#21426379C211yesmaximal, characteristic, complemented
3#21426284C211yesmaximal, characteristic, complemented
4#21426281C211yesmaximal, characteristic, complemented
5#21426279C211yesmaximal, characteristic, complemented
6#21426276C211yesmaximal, characteristic, complemented
7#21426253C211yesmaximal, characteristic, complemented
8#21426240C211yesmaximal, characteristic, complemented
9#21426197C211yesmaximal, characteristic, complemented
10#21426180C211yesmaximal, characteristic, complemented
11#21425970C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425882C211yesmaximal, characteristic, complemented
14C2×#16837229C211yesmaximal, characteristic, complemented
15C59×#3334C211yesmaximal, characteristic, complemented
16#16838098—21no—
17C2×#14273668C2211yescharacteristic
18C2×#14273667C2211yescharacteristic
19C22×C59⋊C16C2211yescharacteristic
20#16837804C2211yescharacteristic
21#16837727—21no—
22#16837721—21no—
23#16837703—21no—
24#16837638C2211yescharacteristic
25#16837638C2211yescharacteristic
26C4×D472C2211yescharacteristic
27#16837440C2211yescharacteristic
28C2×#14273661C2211yescharacteristic
29#16837229C2211yescharacteristic
30#16837229C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837229C2211yescharacteristic
33#16837229C2211yescharacteristic
34#16837229C2211yescharacteristic
35#16837227C2211yescharacteristic
36#16837227C2211yescharacteristic
37#16837227C2211yescharacteristic
38C4⋊(C59⋊C16)C2211yescharacteristic
39#16837203C2211yescharacteristic
40#16837203C2211yescharacteristic
41#16837203C2211yescharacteristic
42#16837202C2211yescharacteristic
43#16837202C2211yescharacteristic
44#16837202C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47#16837178C2211yescharacteristic
48C59×#547—21no—
49C59×#500C2211yescharacteristic
50C59×#480—21no—
51C59×#480—21no—
52C59×#474—21no—
53C59×#466C2211yescharacteristic
54C59×#466C2211yescharacteristic
55C4×D8×C59C2211yescharacteristic
56C8×D4×C59C2211yescharacteristic
57C2×#107×C59C2211yescharacteristic
58C2×#107×C59C2211yescharacteristic
59C2×D472C2311yescharacteristic
60#14273722—41no—
61#14273719—21no—
62#14273709—41no—
63#14273701—21no—
64#14273701—21no—
65#14273699—41no—
66C4×D236—21no—
67(C22⋊D59)⋊C4—41no—
68#14273686—21no—
69C2×C4⋊Dic59—21no—
70#14273668—22no—
71#14273667—22no—
72C2×C59⋊C16—22no—
73C2×C59⋊C16C2311yescharacteristic
74C2×C59⋊C16C2311yescharacteristic
75C2×C59⋊C16C2311yescharacteristic
76C2×C59⋊C16C2311yescharacteristic
77#14273661—21no—
78#14273661C2311yescharacteristic
79#14273661C2311yescharacteristic
80#14273661C2311yescharacteristic
81#14273659—21no—
82#14273659—21no—
83#14273649—21no—
84#14273649—21no—
85#14273649—21no—
86C2×#41×C59—41no—
87C2×D8×C59C2311yescharacteristic
88C22×C8×C59C2311yescharacteristic
89C22×C8×C59C2311yescharacteristic
90C4⋊Q8×C59—21no—
91#123×C59—41no—
92#123×C59—41no—
93C22⋊Q8×C59—41no—
94#121×C59—21no—
95#121×C59—21no—
96C4×D4×C59—21no—
97C4×D4×C59D411yescharacteristic
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—21no—
100#107×C59—21no—
101#107×C59—21no—
102#107×C59—21no—
103#107×C59C2311yescharacteristic
104#107×C59C2311yescharacteristic
105#107×C59C2311yescharacteristic
106C4⋊C8×C59D411yescharacteristic
107#102×C59—21no—
108#102×C59—21no—
109#102×C59—21no—
110C22⋊C8×C59D411yescharacteristic
111C22⋊C8×C59D411yescharacteristic
112C4×C8×C59C2311yescharacteristic
113#41×C59—84no—
114C2×Q8×C59—41no—
115C2×D4×C59—21no—
116C2×D4×C59—41no—
117C2×D4×C59—41no—
118C2×D4×C59D811yescharacteristic
119C22×C4×C59—41no—
120C22×C4×C59—41no—
121C22×C4×C59—41no—
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C2×C22⋊D59—41no—
125C2×C22⋊D59—41no—
126C22×Dic59—41no—
127C2×D236—21no—
128C2×C4×D59—41no—
129D8×C59—21no—
130D8×C59—21no—
131C2×C8×C59—21no—
132C2×C8×C59—21no—
133C2×C8×C59—21no—
134C2×C8×C59—21no—
135C2×C8×C59C2×D411yescharacteristic
136C2×C8×C59C2×D411yescharacteristic
137C2×C8×C59C2×D411yescharacteristic
138C2×C8×C59C2411yescharacteristic
139C4⋊C4×C59—21no—
140C4⋊C4×C59—21no—
141C4⋊C4×C59—21no—
142C4⋊C4×C59—41no—
143C4⋊C4×C59—41no—
144C4⋊C4×C59—41no—
145C4⋊C4×C59D811yescharacteristic
146C22⋊C4×C59—41no—
147C22⋊C4×C59—41no—
148C22⋊C4×C59—41no—
149C22⋊C4×C59D811yescharacteristic
150C22⋊C4×C59D811yescharacteristic
151C42×C59C2×D411yescharacteristic
152C22⋊Dic59—41no—
153(C2×D59)⋊C4—41no—
154(C2×D59)⋊C4—41no—
155(C2×D59)⋊C4—41no—
156C4⋊Dic59—21no—
157C4⋊Dic59—21no—
158C4⋊Dic59—21no—
159C4⋊Dic59—41no—
160Dic59⋊C4—41no—
161Dic59⋊C4—41no—
162C4×Dic59—41no—
163D472—21no—
164D472—21no—
165C59⋊C16—21no—
166C59⋊C16—21no—
167C59⋊C16—21no—
168C59⋊C16#4112yescomplemented
169C23×C59—41no—
170C23×C59C2×D811yescharacteristic
171C23×C59C2×D811yescharacteristic
172C22×D59—41no—
173Q8×C59—82no—
174D4×C59—22no—
175D4×C59—41no—
176D4×C59—41no—
177D4×C59—82no—
178D4×C59—82no—
179C2×C4×C59—21no—
180C2×C4×C59—21no—
181C2×C4×C59—21no—
182C2×C4×C59—21no—
183C2×C4×C59—41no—
184C2×C4×C59—41no—
185C2×C4×C59—41no—
186C2×C4×C59—82no—
187C2×C4×C59—82no—
188C2×C4×C59—82no—
189C2×C4×C59C2×D811yescharacteristic
190C2×C4×C59C2×D811yescharacteristic
191C2×C4×C59C2×D811yescharacteristic
192C2×C4×C59C2×D811yescharacteristic
193C2×C4×C59C22×D411yescharacteristic
194C22⋊D59—82no—
195C22⋊D59—161no—
196C2×Dic59—41no—
197C2×Dic59—41no—
198C2×Dic59—41no—
199C2×Dic59—161no—
200D236—41no—
201D236—41no—
202C4×D59—161no—
203C8×C59—21no—
204C8×C59—21no—
205C8×C59—21no—
206C8×C59C2×#4111yescharacteristic
207C8×C592−1+411yescharacteristic
208#33912—591nomaximal, Sylow
209C22×C59—21no—
210C22×C59—21no—
211C22×C59—21no—
212C22×C59—21no—
213C22×C59—21no—
214C22×C59—81no—
215C22×C59—81no—
216C22×C59—81no—
217C22×C59#50912yes—
218C22×C59C22×D811yescharacteristic
219C2×D59—81no—
220C2×D59—81no—
221C2×D59—81no—
222C4×C59—21no—
223C4×C59—21no—
224C4×C59—21no—
225C4×C59—21no—
226C4×C59—81no—
227C4×C59—81no—
228C4×C59—81no—
229C4×C59#54711yescharacteristic
230C4×C59#57811yescharacteristic
231Dic59—81no—
232Dic59—81no—
233Dic59—81no—
234#3339—591no—
235#3334D5911yescharacteristic, complemented
236#2255—591no—
237#2234—591no—
238#2233—591no—
239#2233—591no—
240#2232—591no—
241#2228—591no—
242#2226—591no—
243#2218—591no—
244#2215—591no—
245#2177—591no—
246#2168—591no—
247#2159—591no—
248C2×#367—591no—
249C2×C59—22no—
250C2×C59—41no—
251C2×C59—161no—
252C2×C59#333411yescharacteristic
253C2×C59C2×#50911yescharacteristic
254C2×C59#341611yescharacteristic
255D59—161no—
256#553—1181no—
257#547—21no—
258C2×D16—591no—
259C2×M32—591no—
260C22×C16—591no—
261#500C2×D5911yescharacteristic
262#499—591no—
263#482—1181no—
264#481—1181no—
265#480—21no—
266#480—21no—
267#478—1181no—
268#474—21no—
269#466—591no—
270#466—591no—
271#466C2×D5911yescharacteristic
272#466C2×D5911yescharacteristic
273C4×D8—591no—
274C4×D8C2×D5911yescharacteristic
275C8×D4C2×D5911yescharacteristic
276#427—591no—
277C2×#107—591no—
278C2×#107C2×D5911yescharacteristic
279C2×#107C2×D5911yescharacteristic
280#367—591no—
281#367—591no—
282#367—591no—
283#367—591no—
284#367—591no—
285#367—591no—
286#366—591no—
287#366—591no—
288#366—591no—
289C4⋊C16—591no—
290#357—591no—
291#357—591no—
292#357—591no—
293#357—591no—
294#357—591no—
295#357—591no—
296C22⋊C16—591no—
297C22⋊C16—591no—
298#346—591no—
299C59#3391211yescharacteristic, complemented, Sylow
300C2×#41—41no—
301C2×D8—591no—
302C2×D8C22×D5911yescharacteristic
303C22×C8C22×D5911yescharacteristic
304C22×C8C22×D5911yescharacteristic
305C4⋊Q8—21no—
306#126—2361no—
307#125—1181no—
308#123—41no—
309#123—41no—
310#123—2361no—
311C22⋊Q8—41no—
312#121—21no—
313#121—21no—
314#121—1181no—
315#121—1181no—
316#121—2361no—
317C4×D4—21no—
318C4×D4—1181no—
319C4×D4—2361no—
320C4×D4C22⋊D5911yescharacteristic
321#117—1181no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—1181no—
325D16—1182no—
326M32—1182no—
327C2×C16—591no—
328C2×C16—591no—
329C2×C16—591no—
330C2×C16—591no—
331C2×C16—1182no—
332#107—21no—
333#107—21no—
334#107—21no—
335#107—591no—
336#107—591no—
337#107—591no—
338#107—1181no—
339#107C22×D5911yescharacteristic
340#107C22×D5911yescharacteristic
341#107C22×D5911yescharacteristic
342#106—1181no—
343#106—1181no—
344C4⋊C8C22⋊D5911yescharacteristic
345#102—21no—
346#102—21no—
347#102—21no—
348#102—1181no—
349#102—1181no—
350#102—1181no—
351C22⋊C8C22⋊D5911yescharacteristic
352C22⋊C8C22⋊D5911yescharacteristic
353C4×C8C22×D5911yescharacteristic
354#41—84no—
355C2×Q8—41no—
356C2×D4—21no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—1181no—
360C2×D4—2361no—
361C2×D4—2361no—
362C2×D4#1174566911yescharacteristic
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—2361no—
367C22×C4—2361no—
368C22×C4C2×C22⋊D5911yescharacteristic
369C22×C4C2×C22⋊D5911yescharacteristic
370D8—21no—
371D8—21no—
372D8—1181no—
373D8—1181no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8C2×C22⋊D5911yescharacteristic
379C2×C8C2×C22⋊D5911yescharacteristic
380C2×C8C2×C22⋊D5911yescharacteristic
381C2×C8C23×D5911yescharacteristic
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—1181no—
389C4⋊C4—1181no—
390C4⋊C4—1181no—
391C4⋊C4—2361no—
392C4⋊C4—2361no—
393C4⋊C4—2361no—
394C4⋊C4#1174566911yescharacteristic
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—2361no—
399C22⋊C4—2361no—
400C22⋊C4—2361no—
401C22⋊C4—2361no—
402C22⋊C4#1174566911yescharacteristic
403C22⋊C4#1174566911yescharacteristic
404C42—2361no—
405C42C2×C22⋊D5911yescharacteristic
406C16—592no—
407C16—1181no—
408C16—1181no—
409C16—1181no—
410C23—41no—
411C23—2361no—
412C23C2×#1174566911yescharacteristic
413C23C2×#1174566911yescharacteristic
414Q8—82no—
415D4—22no—
416D4—41no—
417D4—41no—
418D4—82no—
419D4—82no—
420D4—9441no—
421D4—4722no—
422D4—2361no—
423D4—2361no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—9441no—
435C2×C4—9441no—
436C2×C4—2361no—
437C2×C4—2361no—
438C2×C4—2361no—
439C2×C4C2×#1174566911yescharacteristic
440C2×C4C2×#1174566911yescharacteristic
441C2×C4C2×#1174566911yescharacteristic
442C2×C4C2×#1174566911yescharacteristic
443C2×C4C22×C22⋊D5911yescharacteristic
444C8—21no—
445C8—21no—
446C8—21no—
447C8#41×D5911yescharacteristic
448C8#1427377411yescharacteristic
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—4721no—
458C22—4721no—
459C22—4721no—
460C22#1683783812yes—
461C22C22×#1174566911yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—4721no—
470C4—4721no—
471C4—4721no—
472C4#1683805111yescharacteristic
473C4#1683821711yescharacteristic
474C2—22no—
475C2—41no—
476C2—161no—
477C2—9441no—
478C2#2143643911yescharacteristic
479C2C2×#1683783811yescharacteristic
480C2#2143724011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3334C2
socleC22×C59C22×D8
hypercenter#3334D59
hypocenterC59#33912
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. #21436499C2semidirect
2. #21426379C2semidirect
3. #21426284C2semidirect
4. #21426281C2semidirect
5. #21426279C2semidirect
6. #21426276C2semidirect
7. #21426253C2semidirect
8. #21426240C2semidirect
9. #21426197C2semidirect
10. #21426180C2semidirect
11. #21425970C2semidirect
12. #21425922C2semidirect
13. #21425882C2semidirect
14. C2×#16837229C2semidirect
15. C59×#3334C2semidirect
168. C59⋊C16#41semidirect
235. #3334D59semidirect
299. C59#33912semidirect

Maximal subgroups

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

subgroupindex
#214364992
#214263792
#214262842
#214262812
#214262792
#214262762
#214262532
#214262402
#214261972
#214261802
#214259702
#214259222
#214258822
C2×#168372292
C59×#33342
#3391259