Finite Group Databasebeta

← #28745864

Subgroups — #28745864

Order 15104.

548 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×Dic236C211yesmaximal, characteristic, complemented
2#21426343C211yesmaximal, characteristic, complemented
3#21426304C211yesmaximal, characteristic, complemented
4#21426298C211yesmaximal, characteristic, complemented
5#21426257C211yesmaximal, characteristic, complemented
6#21426207C211yesmaximal, characteristic, complemented
7#21426190C211yesmaximal, characteristic, complemented
8#21426141C211yesmaximal, characteristic, complemented
9#21426012C211yesmaximal, characteristic, complemented
10#21426012C211yesmaximal, characteristic, complemented
11#21425960C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425815C211yesmaximal, characteristic, complemented
14C2×#16837205C211yesmaximal, characteristic, complemented
15C59×#3339C211yesmaximal, characteristic
16C22×Dic236C2211yescharacteristic
17C22×Dic236C2211yescharacteristic
18D4×Dic118—21no—
19C2×#14273671C2211yescharacteristic
20C2×#14273671C2211yescharacteristic
21C2×#14273671C2211yescharacteristic
22C2×#14273667C2211yescharacteristic
23C22×C59⋊C16C2211yescharacteristic
24#16837782C2211yescharacteristic
25#16837643C2211yescharacteristic
26#16837643C2211yescharacteristic
27#16837613—21no—
28#16837560—21no—
29#16837560—21no—
30C4×Dic236C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837227C2211yescharacteristic
33#16837227C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35#16837205C2211yescharacteristic
36#16837205C2211yescharacteristic
37#16837205C2211yescharacteristic
38#16837205C2211yescharacteristic
39#16837205C2211yescharacteristic
40#16837205C2211yescharacteristic
41#16837205C2211yescharacteristic
42#16837205C2211yescharacteristic
43#16837205C2211yescharacteristic
44#16837203C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47#16837178C2211yescharacteristic
48C59×#553—21no—
49C59×#499C2211yescharacteristic
50C59×#482—21no—
51C59×#481—21no—
52C59×#478—21no—
53C59×#466C2211yescharacteristic
54C59×#466C2211yescharacteristic
55C4×D8×C59C2211yescharacteristic
56C8×D4×C59C2211yescharacteristic
57C59×#427C2211yescharacteristic
58C2×#107×C59C2211yescharacteristic
59C22×Dic118—21no—
60C22×Dic118—21no—
61C2×Dic236—22no—
62C2×Dic236—22no—
63C2×Dic236—21no—
64C2×Dic236—21no—
65C2×Dic236—21no—
66C2×Dic236—41no—
67C2×Dic236—41no—
68C2×Dic236C2311yescharacteristic
69C2×Dic236C2311yescharacteristic
70C2×Dic236C2311yescharacteristic
71#14273727—21no—
72#14273726—41no—
73#14273706—21no—
74#14273706—21no—
75#14273704—41no—
76#14273704—41no—
77C4×Dic118—21no—
78D4×Dic59—41no—
79#14273671—22no—
80#14273671—41no—
81#14273671—41no—
82#14273667—41no—
83C2×C59⋊C16—41no—
84C2×C59⋊C16C2311yescharacteristic
85C2×C59⋊C16C2311yescharacteristic
86C2×C59⋊C16C2311yescharacteristic
87C2×C59⋊C16C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273652—21no—
90#14273652—21no—
91#14273652—21no—
92C2×D8×C59C2311yescharacteristic
93C22×C8×C59C2311yescharacteristic
94C22×C8×C59C2311yescharacteristic
95#126×C59—41no—
96#125×C59—21no—
97#123×C59—41no—
98#121×C59—21no—
99#121×C59—21no—
100#121×C59—41no—
101C4×D4×C59—21no—
102C4×D4×C59—41no—
103C4×D4×C59D411yescharacteristic
104#117×C59—21no—
105C2×C4⋊C4×C59—21no—
106#107×C59—21no—
107#107×C59C2311yescharacteristic
108#107×C59C2311yescharacteristic
109#107×C59C2311yescharacteristic
110#106×C59—21no—
111#106×C59—21no—
112C4⋊C8×C59D411yescharacteristic
113#102×C59—21no—
114#102×C59—21no—
115#102×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C2×D4×C59—21no—
120C2×D4×C59—41no—
121C2×D4×C59—41no—
122C2×D4×C59D811yescharacteristic
123C22×C4×C59—41no—
124C22×C4×C59—41no—
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×Dic59—41no—
128C22×Dic59—41no—
129C2×Dic118—21no—
130C2×Dic118—21no—
131C2×Dic118—21no—
132C2×Dic118—42no—
133C2×Dic118—42no—
134C2×Dic118—41no—
135C2×Dic118—41no—
136C2×Dic118—81no—
137C2×Dic118—81no—
138D8×C59—21no—
139D8×C59—21no—
140C2×C8×C59—21no—
141C2×C8×C59—21no—
142C2×C8×C59—21no—
143C2×C8×C59—21no—
144C2×C8×C59C2×D411yescharacteristic
145C2×C8×C59C2×D411yescharacteristic
146C2×C8×C59C2×D411yescharacteristic
147C2×C8×C59C2411yescharacteristic
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—41no—
152C4⋊C4×C59—41no—
153C4⋊C4×C59—41no—
154C4⋊C4×C59D811yescharacteristic
155C22⋊C4×C59—41no—
156C22⋊C4×C59—41no—
157C22⋊C4×C59—41no—
158C22⋊C4×C59—41no—
159C22⋊C4×C59D811yescharacteristic
160C22⋊C4×C59D811yescharacteristic
161C42×C59—41no—
162C42×C59C2×D411yescharacteristic
163C22⋊Dic59—41no—
164C22⋊Dic59—41no—
165C4⋊Dic59—21no—
166C4⋊Dic59—41no—
167C4⋊Dic59—41no—
168Dic59⋊C4—41no—
169Dic59⋊C4—41no—
170Dic59⋊C4—41no—
171C4×Dic59—41no—
172Dic236—21no—
173Dic236—21no—
174Dic236—21no—
175Dic236—21no—
176Dic236—21no—
177Dic236—42no—
178Dic236—42no—
179Dic236—81no—
180Dic236#4112yes—
181C59⋊C16—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59—41no—
186C23×C59C2×D811yescharacteristic
187C23×C59C2×D811yescharacteristic
188D4×C59—22no—
189D4×C59—41no—
190D4×C59—41no—
191D4×C59—82no—
192D4×C59—161no—
193C2×C4×C59—21no—
194C2×C4×C59—21no—
195C2×C4×C59—21no—
196C2×C4×C59—21no—
197C2×C4×C59—41no—
198C2×C4×C59—41no—
199C2×C4×C59—41no—
200C2×C4×C59—161no—
201C2×C4×C59—161no—
202C2×C4×C59C2×D811yescharacteristic
203C2×C4×C59C2×D811yescharacteristic
204C2×C4×C59C2×D811yescharacteristic
205C2×C4×C59C2×D811yescharacteristic
206C2×C4×C59C22×D411yescharacteristic
207C2×Dic59—41no—
208C2×Dic59—41no—
209C2×Dic59—41no—
210C2×Dic59—41no—
211C2×Dic59—82no—
212C2×Dic59—82no—
213Dic118—22no—
214Dic118—41no—
215Dic118—41no—
216Dic118—41no—
217Dic118—41no—
218Dic118—41no—
219Dic118—82no—
220Dic118—82no—
221C8×C59—21no—
222C8×C59—21no—
223C8×C59—21no—
224C8×C59C2×#4111yescharacteristic
225C8×C59H(5,𝔽2)11yescharacteristic
226#33887—591nomaximal, Sylow
227C22×C59—21no—
228C22×C59—21no—
229C22×C59—21no—
230C22×C59—21no—
231C22×C59—21no—
232C22×C59—81no—
233C22×C59—81no—
234C22×C59—81no—
235C22×C59#51012yes—
236C22×C59C22×D811yescharacteristic
237C4×C59—21no—
238C4×C59—21no—
239C4×C59—21no—
240C4×C59—21no—
241C4×C59—81no—
242C4×C59—81no—
243C4×C59—81no—
244C4×C59#54611yescharacteristic
245C4×C59#57711yescharacteristic
246Dic59—42no—
247Dic59—81no—
248Dic59—81no—
249Dic59—81no—
250#3339D5911yescharacteristic
251D4×Q16—591no—
252#2245—591no—
253#2237—591no—
254#2236—591no—
255#2229—591no—
256#2220—591no—
257#2217—591no—
258#2208—591no—
259#2185—591no—
260#2185—591no—
261#2175—591no—
262#2168—591no—
263#2141—591no—
264C2×#358—591no—
265C2×C59—22no—
266C2×C59—41no—
267C2×C59—161no—
268C2×C59#329411yescharacteristic
269C2×C59C2×#51011yescharacteristic
270C2×C59#341611yescharacteristic
271C22×Q16—591no—
272C22×Q16—591no—
273#553—21no—
274D4×Q8—1181no—
275C2×SD32—591no—
276C2×SD32—591no—
277C2×SD32—591no—
278C2×M32—591no—
279C22×C16—591no—
280#499C2×D5911yescharacteristic
281#494—591no—
282#482—21no—
283#481—21no—
284#478—21no—
285#467—591no—
286#467—591no—
287#466C2×D5911yescharacteristic
288#466C2×D5911yescharacteristic
289#462—1181no—
290#451—1181no—
291#451—1181no—
292C4×Q16—591no—
293C4×D8C2×D5911yescharacteristic
294C8×D4C2×D5911yescharacteristic
295#427C2×D5911yescharacteristic
296C2×#107C2×D5911yescharacteristic
297#367—591no—
298#366—591no—
299#366—591no—
300C4⋊C16—591no—
301#358—591no—
302#358—591no—
303#358—591no—
304#358—591no—
305#358—591no—
306#358—591no—
307#358—591no—
308#358—591no—
309#358—591no—
310#357—591no—
311C22⋊C16—591no—
312C22⋊C16—591no—
313#346—591no—
314C59#3388711yescharacteristic, complemented, Sylow
315C22×Q8—1181no—
316C22×Q8—1181no—
317C2×Q16—591no—
318C2×Q16—591no—
319C2×Q16—591no—
320C2×Q16—1182no—
321C2×Q16—1182no—
322C2×Q16—1181no—
323C2×Q16—1181no—
324C2×Q16—1181no—
325C2×Q16—2361no—
326C2×Q16—2361no—
327C2×D8C22×D5911yescharacteristic
328C22×C8C22×D5911yescharacteristic
329C22×C8C22×D5911yescharacteristic
330C4⋊Q8—1181no—
331C4⋊Q8—2361no—
332#126—41no—
333#125—21no—
334#123—41no—
335C22⋊Q8—1181no—
336C22⋊Q8—1181no—
337C22⋊Q8—2361no—
338C22⋊Q8—2361no—
339#121—21no—
340#121—21no—
341#121—41no—
342C4×Q8—1181no—
343C4×D4—21no—
344C4×D4—41no—
345C4×D4—2361no—
346C4×D4C22⋊D5911yescharacteristic
347#117—21no—
348C2×C4⋊C4—21no—
349SD32—1182no—
350SD32—2361no—
351SD32—2361no—
352M32—2361no—
353C2×C16—591no—
354C2×C16—591no—
355C2×C16—591no—
356C2×C16—591no—
357C2×C16—2361no—
358#107—21no—
359#107—591no—
360#107C22×D5911yescharacteristic
361#107C22×D5911yescharacteristic
362#107C22×D5911yescharacteristic
363#106—21no—
364#106—21no—
365C4⋊C8C22⋊D5911yescharacteristic
366#103—1181no—
367#103—1181no—
368#103—1181no—
369#102—21no—
370#102—21no—
371#102—21no—
372C22⋊C8C22⋊D5911yescharacteristic
373C22⋊C8C22⋊D5911yescharacteristic
374C4×C8C22×D5911yescharacteristic
375C2×Q8—1181no—
376C2×Q8—1181no—
377C2×Q8—1181no—
378C2×Q8—4721no—
379C2×Q8—4721no—
380C2×Q8—2362no—
381C2×Q8—2362no—
382C2×Q8—2361no—
383C2×Q8—2361no—
384C2×D4—21no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4#1174566911yescharacteristic
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—2361no—
391C22×C4—2361no—
392C22×C4C2×C22⋊D5911yescharacteristic
393C22×C4C2×C22⋊D5911yescharacteristic
394Q16—592no—
395Q16—1181no—
396Q16—1181no—
397Q16—1181no—
398Q16—1181no—
399Q16—1181no—
400Q16—4721no—
401Q16—2362no—
402Q16—2362no—
403D8—21no—
404D8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8C2×C22⋊D5911yescharacteristic
410C2×C8C2×C22⋊D5911yescharacteristic
411C2×C8C2×C22⋊D5911yescharacteristic
412C2×C8C23×D5911yescharacteristic
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—1181no—
420C4⋊C4—2361no—
421C4⋊C4—2361no—
422C4⋊C4—2361no—
423C4⋊C4—2361no—
424C4⋊C4—2361no—
425C4⋊C4#1174566911yescharacteristic
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—2361no—
431C22⋊C4—2361no—
432C22⋊C4#1174566911yescharacteristic
433C22⋊C4#1174566911yescharacteristic
434C42—41no—
435C42—2361no—
436C42C2×C22⋊D5911yescharacteristic
437C16—1181no—
438C16—1181no—
439C16—1181no—
440C16—1181no—
441C23—41no—
442C23C2×#1174566911yescharacteristic
443C23C2×#1174566911yescharacteristic
444Q8—1182no—
445Q8—4722no—
446Q8—4722no—
447Q8—2361no—
448Q8—2361no—
449Q8—2361no—
450Q8—2361no—
451Q8—2361no—
452D4—22no—
453D4—41no—
454D4—41no—
455D4—82no—
456D4—161no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—161no—
465C2×C4—161no—
466C2×C4—4722no—
467C2×C4—4722no—
468C2×C4—2361no—
469C2×C4—2361no—
470C2×C4—2361no—
471C2×C4—2361no—
472C2×C4C2×#1174566911yescharacteristic
473C2×C4C2×#1174566911yescharacteristic
474C2×C4C2×#1174566911yescharacteristic
475C2×C4C2×#1174566911yescharacteristic
476C2×C4C22×C22⋊D5911yescharacteristic
477C8—21no—
478C8—21no—
479C8—21no—
480C8#41×D5911yescharacteristic
481C8#1427377311yescharacteristic
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22#1683784212yes—
491C22C22×#1174566911yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—4721no—
500C4—4721no—
501C4—4721no—
502C4—2362no—
503C4#1683804011yescharacteristic
504C4#1683821011yescharacteristic
505C2—22no—
506C2—41no—
507C2—161no—
508C2#2143590311yescharacteristic
509C2C2×#1683784211yescharacteristic
510C2#2143724311yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
Fitting, prime radicalC59×#3339C2
socleC22×C59C22×D8
hypercenter#3339D59
hypocenterC59#33887
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3339

Complements

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

NH ≅ G/Ntype
1. D4×Dic236C2semidirect
2. #21426343C2semidirect
3. #21426304C2semidirect
4. #21426298C2semidirect
5. #21426257C2semidirect
6. #21426207C2semidirect
7. #21426190C2semidirect
8. #21426141C2semidirect
9. #21426012C2semidirect
10. #21426012C2semidirect
11. #21425960C2semidirect
12. #21425922C2semidirect
13. #21425815C2semidirect
14. C2×#16837205C2semidirect
314. C59#33887semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2362
#214263432
#214263042
#214262982
#214262572
#214262072
#214261902
#214261412
#214260122
#214259602
#214259222
#214258152
C2×#168372052
C59×#33392
#3388759