Finite Group Databasebeta

← #28484640

Subgroups — #28484640

Order 15104.

548 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427069C211yesmaximal, characteristic, complemented
2#21422656C211yesmaximal, characteristic, complemented
3#21422591C211yesmaximal, characteristic, complemented
4#21422487C211yesmaximal, characteristic, complemented
5#21421997C211yesmaximal, characteristic, complemented
6#21421956C211yesmaximal, characteristic, complemented
7C59×#3036C211yesmaximal, characteristic
8#16837455C2211yescharacteristic
9#16837455C2211yescharacteristic
10#16837305—21no—
11#16837295—22no—
12#16837268C2211yescharacteristic
13#16837265—21no—
14C4⋊C4×Dic59—21no—
15C4×C4⋊Dic59C2211yescharacteristic
16C4×C22⋊Dic59—21no—
17#16837136—22no—
18#16837136—21no—
19#16837136—21no—
20D4⋊(C59⋊C8)—22no—
21C2×AGL(1,ℤ/8)×C59—21no—
22C59×#532C2211yescharacteristic
23C2×C4×D4×C59C2211yescharacteristic
24C59×#463—22no—
25C59×#461C412yescomplemented
26C59×#459C412yescomplemented
27C59×#423C2211yescharacteristic
28C59×#417—21no—
29C2×C4⋊Dic59—21no—
30C2×C4⋊Dic59—21no—
31C2×C22⋊Dic59—22no—
32C2×C22⋊Dic59—22no—
33C2×C4×Dic59—21no—
34C2×C4×Dic59—22no—
35C4⋊(C59⋊C8)—21no—
36C4⋊(C59⋊C8)—21no—
37C22⋊(C59⋊C8)—21no—
38C22⋊(C59⋊C8)—21no—
39C4×C59⋊C8—21no—
40C4×C59⋊C8—21no—
41C59⋊#95—22no—
42C59⋊#95—21no—
43C59⋊#95D411yescharacteristic
44C59⋊#95D411yescharacteristic
45C59⋊#95D412yes—
46C2×#41×C59—21no—
47C22×D4×C59D411yescharacteristic
48C23×C4×C59—21no—
49AGL(1,ℤ/8)×C59—42no—
50AGL(1,ℤ/8)×C59—42no—
51C2×SD16×C59—22no—
52C2×D8×C59—22no—
53C2×M16×C59—21no—
54C4⋊Q8×C59C2×C411yescharacteristic
55#127×C59C2×C411yescharacteristic
56#124×C59—21no—
57#121×C59—21no—
58#121×C59—21no—
59C4×D4×C59—21no—
60C4×D4×C59—21no—
61C4×D4×C59—21no—
62C4×D4×C59—21no—
63C4×D4×C59C2×C412yes—
64C4×D4×C59D412yes—
65C2×C4⋊C4×C59D411yescharacteristic
66C2×C22⋊C4×C59—21no—
67C2×C42×C59C2311yescharacteristic
68C4⋊C8×C59—21no—
69C4⋊C8×C59C2×C412yes—
70#103×C59—22no—
71#102×C59—22no—
72C24×C59—21no—
73#41×C59—41no—
74#41×C59—41no—
75#41×C59—41no—
76#41×C59—41no—
77C2×Q8×C59—21no—
78C2×D4×C59—21no—
79C2×D4×C59—21no—
80C2×D4×C59—21no—
81C2×D4×C59—21no—
82C2×D4×C59—21no—
83C2×D4×C59—21no—
84C2×D4×C59—21no—
85C2×D4×C59—21no—
86C2×D4×C59—41no—
87C2×D4×C59C22⋊C412yes—
88C22×C4×C59—21no—
89C22×C4×C59—22no—
90C22×C4×C59—22no—
91C22×C4×C59—41no—
92C22×C4×C59—82no—
93C22×C4×C59C2×D411yescharacteristic
94C22×C4×C59C2×D411yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C22×Dic59—21no—
97C22×Dic59—22no—
98C22×Dic59—21no—
99SD16×C59—44no—
100D8×C59—44no—
101M16×C59—41no—
102M16×C59—41no—
103C2×C8×C59—22no—
104C4⋊C4×C59—21no—
105C4⋊C4×C59—21no—
106C4⋊C4×C59—21no—
107C4⋊C4×C59C22⋊C412yes—
108C22⋊C4×C59—22no—
109C22⋊C4×C59—22no—
110C22⋊C4×C59—41no—
111C22⋊C4×C59—41no—
112C42×C59C22×C411yescharacteristic
113C42×C59C2×D411yescharacteristic
114C42×C59#4111yescharacteristic
115C42×C59#4111yescharacteristic
116C22⋊Dic59—44no—
117C22⋊Dic59—82no—
118C4⋊Dic59—42no—
119C4⋊Dic59—41no—
120C4⋊Dic59—41no—
121C4×Dic59—41no—
122C4×Dic59—41no—
123C4×Dic59—82no—
124C2×C59⋊C8—41no—
125C2×C59⋊C8—41no—
126C23×C59—22no—
127C23×C59—42no—
128C23×C59—42no—
129C23×C59—41no—
130C23×C59—41no—
131C23×C59—41no—
132C23×C59#12011yescharacteristic
133Q8×C59—41no—
134Q8×C59—41no—
135D4×C59—22no—
136D4×C59—22no—
137D4×C59—41no—
138D4×C59—41no—
139D4×C59—41no—
140D4×C59—41no—
141D4×C59—81no—
142D4×C59—81no—
143D4×C59—81no—
144D4×C59—81no—
145C2×C4×C59—22no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—42no—
149C2×C4×C59—41no—
150C2×C4×C59—82no—
151C2×C4×C59—82no—
152C2×C4×C59—81no—
153C2×C4×C59—81no—
154C2×C4×C59C2×C22⋊C411yescharacteristic
155C2×C4×C59C4×D411yescharacteristic
156C2×C4×C59C4×D411yescharacteristic
157C2×C4×C59#12111yescharacteristic
158C2×C4×C59#12111yescharacteristic
159C2×C4×C59#12311yescharacteristic
160C2×C4×C59#13511yescharacteristic
161C2×C4×C59#13511yescharacteristic
162C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
164C2×Dic59—22no—
165C2×Dic59—41no—
166C2×Dic59—41no—
167C2×Dic59—42no—
168C2×Dic59—41no—
169C2×Dic59—42no—
170C2×Dic59—41no—
171C2×Dic59—41no—
172C2×Dic59—42no—
173C8×C59—42no—
174C59⋊C8—81no—
175C59⋊C8—81no—
176#9181—591nomaximal, Sylow
177C22×C59—21no—
178C22×C59—21no—
179C22×C59—42no—
180C22×C59—42no—
181C22×C59—42no—
182C22×C59—82no—
183C22×C59—82no—
184C22×C59—81no—
185C22×C59—81no—
186C22×C59—81no—
187C22×C59#38611yescharacteristic
188C22×C59#44911yescharacteristic
189C22×C59#44911yescharacteristic
190C4×C59—21no—
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—21no—
194C4×C59—42no—
195C4×C59—81no—
196C4×C59#41611yescharacteristic
197C4×C59#41811yescharacteristic
198C4×C59#48211yescharacteristic
199C4×C59#48211yescharacteristic
200Dic59—42no—
201Dic59—81no—
202Dic59—82no—
203#3036D5911yescharacteristic
204#2357—591no—
205#1517—591no—
206#1502—591no—
207#1478—591no—
208#1303—591no—
209#1286—591no—
210C2×C59—21no—
211C2×C59—21no—
212C2×C59—82no—
213C2×C59—161no—
214C2×C59#187511yescharacteristic
215C2×C59#190411yescharacteristic
216C2×C59#190611yescharacteristic
217C2×AGL(1,ℤ/8)—21no—
218#532C2×D5911yescharacteristic
219C2×C4×D4C2×D5911yescharacteristic
220#463—22no—
221#461Dic5912yescomplemented
222#459Dic5912yescomplemented
223#431—591no—
224#431—591no—
225#423C2×D5911yescharacteristic
226#417—21no—
227#390—1181no—
228#388—1182no—
229#382—591no—
230#381—1181no—
231C4×C4⋊C4—591no—
232C4×C4⋊C4—1181no—
233C4×C22⋊C4—1181no—
234#327—1181no—
235#327—1181no—
236#327—1182no—
237D4⋊C8—1182no—
238C59#918111yescharacteristic, complemented, Sylow
239C2×#41—21no—
240C22×D4C22⋊D5911yescharacteristic
241C23×C4—21no—
242AGL(1,ℤ/8)—42no—
243AGL(1,ℤ/8)—42no—
244C2×SD16—22no—
245C2×D8—22no—
246C2×M16—21no—
247C4⋊Q8C2×Dic5911yescharacteristic
248#127C2×Dic5911yescharacteristic
249#124—21no—
250#121—21no—
251#121—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4C2×Dic5912yes—
257C4×D4C22⋊D5912yes—
258C2×C4⋊C4—1181no—
259C2×C4⋊C4—1181no—
260C2×C4⋊C4C22⋊D5911yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—1182no—
263C2×C22⋊C4—1182no—
264C2×C42—1182no—
265C2×C42—1181no—
266C2×C42C22×D5911yescharacteristic
267C4⋊C8—21no—
268C4⋊C8—1181no—
269C4⋊C8—1181no—
270C4⋊C8C2×Dic5912yes—
271#103—22no—
272#102—22no—
273C22⋊C8—1181no—
274C22⋊C8—1181no—
275C4×C8—1181no—
276C4×C8—1181no—
277#95—591no—
278#95—591no—
279#95—592no—
280#95—1182no—
281#95—1181no—
282C24—21no—
283#41—41no—
284#41—41no—
285#41—41no—
286#41—41no—
287C2×Q8—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4C22⋊Dic5912yes—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4—82no—
303C22×C4—1181no—
304C22×C4—1182no—
305C22×C4—1181no—
306C22×C4D4×D5911yescharacteristic
307C22×C4D4×D5911yescharacteristic
308C22×C4C2×C22⋊D5911yescharacteristic
309SD16—44no—
310D8—44no—
311M16—41no—
312M16—41no—
313C2×C8—22no—
314C2×C8—2361no—
315C2×C8—2361no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—2362no—
320C4⋊C4—2361no—
321C4⋊C4—2361no—
322C4⋊C4C22⋊Dic5912yes—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—4722no—
328C22⋊C4—2364no—
329C42—4722no—
330C42—2361no—
331C42—2361no—
332C42#1174568711yescharacteristic
333C42#1174568711yescharacteristic
334C42C22×Dic5911yescharacteristic
335C42C2×C22⋊D5911yescharacteristic
336C23—22no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23#1427369511yescharacteristic
343Q8—41no—
344Q8—41no—
345D4—22no—
346D4—22no—
347D4—41no—
348D4—41no—
349D4—41no—
350D4—41no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—81no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—1182no—
365C2×C4—2362no—
366C2×C4—2361no—
367C2×C4—2362no—
368C2×C4—2361no—
369C2×C4—2361no—
370C2×C4—2362no—
371C2×C4—2361no—
372C2×C4—2361no—
373C2×C4C2×C22⋊Dic5911yescharacteristic
374C2×C4D4×Dic5911yescharacteristic
375C2×C4D4×Dic5911yescharacteristic
376C2×C4#1427369911yescharacteristic
377C2×C4#1427369911yescharacteristic
378C2×C4#1427370811yescharacteristic
379C2×C4#1427374711yescharacteristic
380C2×C4#1427374711yescharacteristic
381C2×C4#1427375311yescharacteristic
382C2×C4#1427375311yescharacteristic
383C8—42no—
384C8—4721no—
385C8—4721no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—82no—
392C22—82no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22#1683728211yescharacteristic
397C22#1683754511yescharacteristic
398C22#1683754511yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—81no—
405C4—4722no—
406C4—4721no—
407C4—2362no—
408C4#1683739811yescharacteristic
409C4#1683740711yescharacteristic
410C4#1683772311yescharacteristic
411C4#1683772311yescharacteristic
412C2—21no—
413C2—21no—
414C2—82no—
415C2—161no—
416C2#2142441911yescharacteristic
417C2#2142458511yescharacteristic
418C2#2142459911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3036C2
socleC22×C59#386
cosocleC2×C42×C59C23
hypercenter#3036D59
hypocenterC59#9181
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #21427069C2semidirect
2. #21422656C2semidirect
3. #21422591C2semidirect
4. #21422487C2semidirect
5. #21421997C2semidirect
6. #21421956C2semidirect
25. C59×#461C4semidirect
26. C59×#459C4semidirect
221. #461Dic59semidirect
222. #459Dic59semidirect
238. C59#9181semidirect

Maximal subgroups

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

subgroupindex
#214270692
#214226562
#214225912
#214224872
#214219972
#214219562
C59×#30362
#918159