Finite Group Databasebeta

← #28507044

Subgroups — #28507044

Order 15104.

448 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425791C211yesmaximal, characteristic, complemented
2#21425788C211yesmaximal, characteristic
3#21425750C211yesmaximal, characteristic, complemented
4#21425746C211yesmaximal, characteristic
5#21424146C211yesmaximal, characteristic, complemented
6#21423896C211yesmaximal, characteristic, complemented
7C59×#3057C211yesmaximal, characteristic
8C2×#14273663—21no—
9#16837386C2211yescharacteristic
10#16837386C2211yescharacteristic
11#16837386C2211yescharacteristic
12#16837376C2211yescharacteristic
13#16837357—21no—
14#16837201—22no—
15#16837199—22no—
16#16837192—22no—
17#16837189—22no—
18#16837173—21no—
19C59⋊#338—21no—
20C22×SD16×C59C411yescharacteristic, complemented
21C2×#131×C59C2211yescharacteristic
22C59×#536C2211yescharacteristic
23C59×#535C2211yescharacteristic
24C59×#468—21no—
25C59×#468—21no—
26C59×#465—21no—
27C59×#465—21no—
28C59×#427C411yescharacteristic, complemented
29C59×#419C411yescharacteristic, complemented
30C59×#418C411yescharacteristic, complemented
31C2×#11745664—21no—
32C2×#11745664—21no—
33C2×#11745664—21no—
34#14273686—21no—
35#14273663—41no—
36#14273663—41no—
37#14273646—21no—
38#14273646—21no—
39#14273646D412yes—
40#14273644—21no—
41#14273644—21no—
42#14273644—21no—
43#14273642—21no—
44#14273642D412yes—
45#14273637—41no—
46C8×Dic59—41no—
47C2×#41×C59C2311yescharacteristic
48C22×Q8×C59C2×C411yescharacteristic
49C22×D4×C59C2×C411yescharacteristic
50C2×SD16×C59—21no—
51C2×SD16×C59—21no—
52C2×SD16×C59—41no—
53C2×SD16×C59—41no—
54#131×C59—41no—
55#131×C59—41no—
56C2×M16×C59C2×C411yescharacteristic
57C2×M16×C59D411yescharacteristic
58C2×M16×C59D411yescharacteristic
59C22×C8×C59C2×C411yescharacteristic
60C22×C8×C59D411yescharacteristic
61C22×C8×C59D411yescharacteristic
62C4⋊Q8×C59—21no—
63#127×C59—21no—
64#124×C59—21no—
65#124×C59—21no—
66#123×C59—21no—
67#123×C59—21no—
68C22⋊Q8×C59—21no—
69C22⋊Q8×C59—21no—
70#121×C59—21no—
71#121×C59—21no—
72#120×C59—21no—
73#120×C59—21no—
74#117×C59C2×C411yescharacteristic
75#117×C59C2×C411yescharacteristic
76#107×C59—21no—
77#106×C59—21no—
78#103×C59—21no—
79#103×C59—21no—
80#102×C59—21no—
81#102×C59—21no—
82C24×C59—21no—
83#41×C59—41no—
84#41×C59—41no—
85C2×Q8×C59—21no—
86C2×Q8×C59—21no—
87C2×Q8×C59—41no—
88C2×Q8×C59—41no—
89C2×Q8×C59C2×D411yescharacteristic
90C2×D4×C59—21no—
91C2×D4×C59—21no—
92C2×D4×C59—41no—
93C2×D4×C59—41no—
94C2×D4×C59—41no—
95C2×D4×C59—41no—
96C2×D4×C59C2×D411yescharacteristic
97C2×D4×C59#4111yescharacteristic
98C2×D4×C59#4111yescharacteristic
99C22×C4×C59—21no—
100C22×C4×C59C22×C411yescharacteristic
101C22×C4×C59C2×D411yescharacteristic
102C22×C4×C59C2×D411yescharacteristic
103SD16×C59—42no—
104SD16×C59—41no—
105SD16×C59—41no—
106M16×C59—41no—
107M16×C59—41no—
108M16×C59—41no—
109C2×C8×C59—21no—
110C2×C8×C59—21no—
111C2×C8×C59—21no—
112C2×C8×C59—21no—
113C2×C8×C59—21no—
114C2×C8×C59—21no—
115C2×C8×C59C22⋊C411yescharacteristic
116C2×C8×C59C22⋊C411yescharacteristic
117C2×C8×C59C22⋊C411yescharacteristic
118C2×C8×C59C22⋊C411yescharacteristic
119C4⋊C4×C59—21no—
120C4⋊C4×C59—21no—
121C4⋊C4×C59—41no—
122C4⋊C4×C59—41no—
123C22⋊C4×C59—21no—
124C22⋊C4×C59—21no—
125C22⋊C4×C59—41no—
126C22⋊C4×C59—41no—
127C22⋊C4×C59—41no—
128C22⋊C4×C59—41no—
129C42×C59—21no—
130C42×C59—21no—
131C22⋊Dic59—22no—
132C4⋊Dic59—41no—
133C4×Dic59—41no—
134#11745664—22no—
135#11745664—41no—
136#11745664—41no—
137#11745664—41no—
138#11745664—41no—
139#11745664—41no—
140C2×C59⋊C8—41no—
141C2×C59⋊C8—41no—
142C2×C59⋊C8—41no—
143C23×C59—41no—
144C23×C59—81no—
145C23×C59—161no—
146C23×C59C4×D411yescharacteristic
147C23×C59#12111yescharacteristic
148C23×C59#12411yescharacteristic
149Q8×C59—41no—
150Q8×C59—41no—
151Q8×C59—41no—
152Q8×C59—81no—
153D4×C59—41no—
154D4×C59—41no—
155D4×C59—41no—
156D4×C59—41no—
157D4×C59—41no—
158D4×C59—81no—
159D4×C59—161no—
160D4×C59—161no—
161C2×C4×C59—21no—
162C2×C4×C59—21no—
163C2×C4×C59—21no—
164C2×C4×C59—21no—
165C2×C4×C59—21no—
166C2×C4×C59—21no—
167C2×C4×C59—41no—
168C2×C4×C59—41no—
169C2×C4×C59—41no—
170C2×C4×C59—81no—
171C2×C4×C59C2×C22⋊C411yescharacteristic
172C2×C4×C59C4×D411yescharacteristic
173C2×C4×C59#12111yescharacteristic
174C2×C4×C59#12711yescharacteristic
175C2×Dic59—42no—
176C8×C59—21no—
177C8×C59—21no—
178C8×C59—21no—
179C8×C59#9911yescharacteristic
180C8×C59#9911yescharacteristic
181C59⋊C8—42no—
182C59⋊C8—81no—
183C59⋊C8—81no—
184#13232—591nomaximal, Sylow
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—41no—
189C22×C59—41no—
190C22×C59—41no—
191C22×C59—81no—
192C22×C59—81no—
193C22×C59—81no—
194C22×C59—162no—
195C22×C59#39011yescharacteristic
196C4×C59—21no—
197C4×C59—21no—
198C4×C59—21no—
199C4×C59—81no—
200C4×C59—81no—
201C4×C59—81no—
202C4×C59C2×#9911yescharacteristic
203C4×C59#41011yescharacteristic
204Dic59—82no—
205#3057D5911yescharacteristic
206#2135—591no—
207#2134—591no—
208#2123—591no—
209#2122—591no—
210#1813—591no—
211#1757—591no—
212C2×C59—21no—
213C2×C59—41no—
214C2×C59—41no—
215C2×C59—41no—
216C2×C59—161no—
217C2×C59#196411yescharacteristic
218C22×SD16Dic5911yescharacteristic, complemented
219C2×#131C2×D5911yescharacteristic
220#536C2×D5911yescharacteristic
221#535C2×D5911yescharacteristic
222#468—21no—
223#468—21no—
224#465—21no—
225#465—21no—
226C2×#108—1181no—
227#427Dic5911yescharacteristic, complemented
228#419Dic5911yescharacteristic, complemented
229#418Dic5911yescharacteristic, complemented
230#413—591no—
231#413—591no—
232#413—591no—
233#410—591no—
234#405—1181no—
235#356—1182no—
236#355—1182no—
237#352—1182no—
238AΣL(1,𝔽16)—1182no—
239#344—1181no—
240#338—1181no—
241C59#1323211yescharacteristic, complemented, Sylow
242C2×#41C22×D5911yescharacteristic
243C22×Q8C2×Dic5911yescharacteristic
244C22×D4C2×Dic5911yescharacteristic
245C2×SD16—21no—
246C2×SD16—21no—
247C2×SD16—41no—
248C2×SD16—41no—
249#131—41no—
250#131—41no—
251C2×M16—1181no—
252C2×M16—1181no—
253C2×M16—1181no—
254C2×M16C2×Dic5911yescharacteristic
255C2×M16C22⋊D5911yescharacteristic
256C2×M16C22⋊D5911yescharacteristic
257C22×C8C2×Dic5911yescharacteristic
258C22×C8C22⋊D5911yescharacteristic
259C22×C8C22⋊D5911yescharacteristic
260C4⋊Q8—21no—
261#127—21no—
262#124—21no—
263#124—21no—
264#123—21no—
265#123—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268#121—21no—
269#121—21no—
270#120—21no—
271#120—21no—
272#117—1181no—
273#117C2×Dic5911yescharacteristic
274#117C2×Dic5911yescharacteristic
275#108—2361no—
276#108—2361no—
277#107—21no—
278#106—21no—
279#103—21no—
280#103—21no—
281#102—21no—
282#102—21no—
283#101—592no—
284#101—1181no—
285#101—1181no—
286#100—1181no—
287#100—1181no—
288#100—1181no—
289#99—592no—
290#99—1181no—
291#97—2361no—
292C4×C8—2361no—
293C24—21no—
294#41—41no—
295#41—41no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8D4×D5911yescharacteristic
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4D4×D5911yescharacteristic
308C2×D4#1174568711yescharacteristic
309C2×D4#1174568711yescharacteristic
310C22×C4—21no—
311C22×C4C22×Dic5911yescharacteristic
312C22×C4C2×C22⋊D5911yescharacteristic
313C22×C4C2×C22⋊D5911yescharacteristic
314SD16—42no—
315SD16—41no—
316SD16—41no—
317M16—41no—
318M16—41no—
319M16—41no—
320M16—1182no—
321M16—2361no—
322M16—2361no—
323M16—2361no—
324M16—2361no—
325M16—2361no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—2361no—
333C2×C8—2361no—
334C2×C8—2361no—
335C2×C8C22⋊Dic5911yescharacteristic
336C2×C8C22⋊Dic5911yescharacteristic
337C2×C8C22⋊Dic5911yescharacteristic
338C2×C8C22⋊Dic5911yescharacteristic
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—2361no—
344C22⋊C4—21no—
345C22⋊C4—21no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—1182no—
351C42—21no—
352C42—21no—
353C42—2361no—
354C23—41no—
355C23—81no—
356C23—161no—
357C23D4×Dic5911yescharacteristic
358C23#1427369811yescharacteristic
359C23#1427371411yescharacteristic
360Q8—41no—
361Q8—41no—
362Q8—41no—
363Q8—81no—
364D4—41no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—81no—
370D4—161no—
371D4—161no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—2362no—
383C2×C4C2×C22⋊Dic5911yescharacteristic
384C2×C4D4×Dic5911yescharacteristic
385C2×C4#1427369811yescharacteristic
386C2×C4#1427372311yescharacteristic
387C8—21no—
388C8—21no—
389C8—21no—
390C8—4721no—
391C8—4721no—
392C8—2362no—
393C8#1427364211yescharacteristic
394C8#1427364211yescharacteristic
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—162no—
405C22#1683730511yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—81no—
410C4—81no—
411C4—81no—
412C4—4722no—
413C4C2×#1427364211yescharacteristic
414C4#1683737611yescharacteristic
415C2—21no—
416C2—41no—
417C2—41no—
418C2—41no—
419C2—161no—
420C2#2142493311yescharacteristic
subgroupG/H
centerC2#21424933
commutatorC22×C4×C59C22×C4
FrattiniC2×#41C22×D59
FittingC59×#3057C2
socleC2×C59#1964
prime radicalC59×#535C22
cosocleC2×#41×C59C23
hypercenter#3057D59
hypocenterC59#13232
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #3057

Complements

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

NH ≅ G/Ntype
1. #21425791C2semidirect
3. #21425750C2semidirect
5. #21424146C2semidirect
6. #21423896C2semidirect
20. C22×SD16×C59C4semidirect
28. C59×#427C4semidirect
29. C59×#419C4semidirect
30. C59×#418C4semidirect
218. C22×SD16Dic59semidirect
227. #427Dic59semidirect
228. #419Dic59semidirect
229. #418Dic59semidirect
241. C59#13232semidirect

Maximal subgroups

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

subgroupindex
#214257912
#214257882
#214257502
#214257462
#214241462
#214238962
C59×#30572
#1323259