Finite Group Databasebeta

← #28507027

Subgroups — #28507027

Order 15104.

452 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425771C211yesmaximal, characteristic, complemented
2#21425768C211yesmaximal, characteristic
3#21425750C211yesmaximal, characteristic, complemented
4#21425746C211yesmaximal, characteristic
5#21424142C211yesmaximal, characteristic, complemented
6#21423896C211yesmaximal, characteristic, complemented
7C59×#3057C211yesmaximal, characteristic
8#16837440—21no—
9#16837386C2211yescharacteristic
10#16837376C2211yescharacteristic
11#16837376C2211yescharacteristic
12#16837376C2211yescharacteristic
13#16837357—21no—
14#16837197—22no—
15#16837195—22no—
16#16837192—22no—
17#16837189—22no—
18#16837173—21no—
19#16837159—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—
32#14273686—21no—
33#14273686—21no—
34#14273686—21no—
35#14273661—41no—
36#14273659—41no—
37#14273646—21no—
38#14273644D412yes—
39#14273642—21no—
40#14273642—21no—
41#14273642—21no—
42#14273642—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×C59#4111yescharacteristic
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×C59C2×D411yescharacteristic
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—
132C22⋊Dic59—41no—
133C22⋊Dic59—41no—
134C4⋊Dic59—41no—
135C4⋊Dic59—41no—
136C4⋊Dic59—41no—
137C4×Dic59—41no—
138C4×Dic59—41no—
139C4×Dic59—41no—
140#11745664—22no—
141#11745664—41no—
142C2×C59⋊C8—41no—
143C23×C59—41no—
144C23×C59—81no—
145C23×C59—161no—
146C23×C59C4×D411yescharacteristic
147C23×C59#12111yescharacteristic
148C23×C59#12711yescharacteristic
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#12411yescharacteristic
175C2×Dic59—42no—
176C2×Dic59—81no—
177C2×Dic59—81no—
178C8×C59—21no—
179C8×C59—21no—
180C8×C59—21no—
181C8×C59#9911yescharacteristic
182C8×C59#9911yescharacteristic
183C59⋊C8—42no—
184#13229—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—
205Dic59—161no—
206Dic59—161no—
207#3057D5911yescharacteristic
208#2129—591no—
209#2128—591no—
210#2123—591no—
211#2122—591no—
212#1812—591no—
213#1757—591no—
214C2×C59—21no—
215C2×C59—41no—
216C2×C59—41no—
217C2×C59—41no—
218C2×C59—161no—
219C2×C59#196411yescharacteristic
220C22×SD16Dic5911yescharacteristic, complemented
221C2×#131C2×D5911yescharacteristic
222#536C2×D5911yescharacteristic
223#535C2×D5911yescharacteristic
224#468—21no—
225#468—21no—
226#465—21no—
227#465—21no—
228#427—1181no—
229#427Dic5911yescharacteristic, complemented
230#419Dic5911yescharacteristic, complemented
231#418Dic5911yescharacteristic, complemented
232#413—591no—
233#410—591no—
234#410—591no—
235#410—591no—
236#405—1181no—
237#354—1182no—
238#353—1182no—
239#352—1182no—
240AΣL(1,𝔽16)—1182no—
241#344—1181no—
242H(3,ℤ/4)—1181no—
243C59#1322911yescharacteristic, complemented, Sylow
244C2×#41C22×D5911yescharacteristic
245C22×Q8C2×Dic5911yescharacteristic
246C22×D4C2×Dic5911yescharacteristic
247C2×SD16—21no—
248C2×SD16—21no—
249C2×SD16—41no—
250C2×SD16—41no—
251#131—41no—
252#131—41no—
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#117—1181no—
274#117—1181no—
275#117C2×Dic5911yescharacteristic
276#117C2×Dic5911yescharacteristic
277#107—21no—
278#107—2361no—
279#106—21no—
280#106—2361no—
281#103—21no—
282#103—21no—
283#102—21no—
284#102—21no—
285#101—1181no—
286#100—592no—
287#99—592no—
288#99—1181no—
289#99—1181no—
290#99—1181no—
291#99—1181no—
292#99—1181no—
293#97—2361no—
294C4×C8—2361no—
295C24—21no—
296#41—41no—
297#41—41no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8#1174568711yescharacteristic
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4D4×D5911yescharacteristic
310C2×D4D4×D5911yescharacteristic
311C2×D4#1174568711yescharacteristic
312C22×C4—21no—
313C22×C4C22×Dic5911yescharacteristic
314C22×C4C2×C22⋊D5911yescharacteristic
315C22×C4C2×C22⋊D5911yescharacteristic
316SD16—42no—
317SD16—41no—
318SD16—41no—
319M16—41no—
320M16—41no—
321M16—41no—
322M16—1182no—
323M16—2361no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—2361no—
331C2×C8C22⋊Dic5911yescharacteristic
332C2×C8C22⋊Dic5911yescharacteristic
333C2×C8C22⋊Dic5911yescharacteristic
334C2×C8C22⋊Dic5911yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—2361no—
340C4⋊C4—2361no—
341C4⋊C4—2361no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—1182no—
349C22⋊C4—2361no—
350C22⋊C4—2361no—
351C42—21no—
352C42—21no—
353C42—2361no—
354C42—2361no—
355C42—2361no—
356C23—41no—
357C23—81no—
358C23—161no—
359C23D4×Dic5911yescharacteristic
360C23#1427369811yescharacteristic
361C23#1427372311yescharacteristic
362Q8—41no—
363Q8—41no—
364Q8—41no—
365Q8—81no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—41no—
370D4—41no—
371D4—81no—
372D4—161no—
373D4—161no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—4721no—
385C2×C4—4721no—
386C2×C4—2362no—
387C2×C4C2×C22⋊Dic5911yescharacteristic
388C2×C4D4×Dic5911yescharacteristic
389C2×C4#1427369811yescharacteristic
390C2×C4#1427371411yescharacteristic
391C8—21no—
392C8—21no—
393C8—21no—
394C8—2362no—
395C8#1427364211yescharacteristic
396C8#1427364211yescharacteristic
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—162no—
407C22#1683730511yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—9441no—
415C4—9441no—
416C4—4722no—
417C4C2×#1427364211yescharacteristic
418C4#1683737611yescharacteristic
419C2—21no—
420C2—41no—
421C2—41no—
422C2—41no—
423C2—161no—
424C2#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#13229
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. #21425771C2semidirect
3. #21425750C2semidirect
5. #21424142C2semidirect
6. #21423896C2semidirect
20. C22×SD16×C59C4semidirect
28. C59×#427C4semidirect
29. C59×#419C4semidirect
30. C59×#418C4semidirect
220. C22×SD16Dic59semidirect
229. #427Dic59semidirect
230. #419Dic59semidirect
231. #418Dic59semidirect
243. C59#13229semidirect

Maximal subgroups

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

subgroupindex
#214257712
#214257682
#214257502
#214257462
#214241422
#214238962
C59×#30572
#1322959