Finite Group Databasebeta

← #28484638

Subgroups — #28484638

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#21433560C211yesmaximal, characteristic
2#21422655C211yesmaximal, characteristic, complemented
3#21422590C211yesmaximal, characteristic, complemented
4#21422487C211yesmaximal, characteristic, complemented
5#21421999C211yesmaximal, characteristic, complemented
6#21421954C211yesmaximal, characteristic, complemented
7C59×#2357C211yesmaximal, characteristic, complemented
8C2×#14273753—21no—
9#16837952C2211yescharacteristic
10#16837617—22no—
11#16837607C412yescomplemented
12#16837597C412yescomplemented
13#16837455C2211yescharacteristic
14#16837455C2211yescharacteristic
15#16837426C2211yescharacteristic
16#16837401—21no—
17#16837135—22no—
18#16837135—21no—
19#16837135—21no—
20D4⋊(C59⋊C8)—22no—
21C2×C4×D4×C59C2211yescharacteristic
22C59×#390—21no—
23C59×#388—22no—
24C59×#382C2211yescharacteristic
25C59×#381—21no—
26C4×C4⋊C4×C59—21no—
27C4×C4⋊C4×C59C2211yescharacteristic
28C4×C22⋊C4×C59—21no—
29C2×#11745685—21no—
30#14273753—42no—
31#14273753—42no—
32C2×#11745670—22no—
33C2×#11745669—22no—
34C2×#11745664—21no—
35#14273727C2×C411yescharacteristic
36#14273724C2×C411yescharacteristic
37#14273715—21no—
38#14273701—21no—
39#14273701—21no—
40C4×D236—21no—
41C4×D236—21no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)C2×C412yes—
46#14273650—22no—
47#14273647—22no—
48C22⋊(C59⋊C8)—21no—
49C22⋊(C59⋊C8)—21no—
50C4×C59⋊C8—21no—
51C4×C59⋊C8—21no—
52C22×D4×C59D411yescharacteristic
53C23×C4×C59—21no—
54C4×D4×C59—21no—
55C4×D4×C59—21no—
56C4×D4×C59C2×C412yes—
57C4×D4×C59D412yes—
58C2×C4⋊C4×C59—21no—
59C2×C4⋊C4×C59—21no—
60C2×C4⋊C4×C59D411yescharacteristic
61C2×C22⋊C4×C59—21no—
62C2×C22⋊C4×C59—22no—
63C2×C22⋊C4×C59—22no—
64C2×C42×C59—21no—
65C2×C42×C59—22no—
66C2×C42×C59C2311yescharacteristic
67#95×C59—22no—
68#95×C59—21no—
69#95×C59D411yescharacteristic
70#95×C59D411yescharacteristic
71#95×C59D412yes—
72C24×C59—21no—
73C2×D4×C59—21no—
74C2×D4×C59—21no—
75C2×D4×C59—21no—
76C2×D4×C59—21no—
77C2×D4×C59—21no—
78C2×D4×C59C22⋊C412yes—
79C22×C4×C59—21no—
80C22×C4×C59—21no—
81C22×C4×C59—22no—
82C22×C4×C59—22no—
83C22×C4×C59—22no—
84C22×C4×C59—21no—
85C22×C4×C59—82no—
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C2×C22⋊D59—41no—
90#11745685—41no—
91#11745685—41no—
92#11745685—41no—
93#11745685—41no—
94C2×D236—21no—
95C2×D236—21no—
96C2×D236—21no—
97C2×C4×D59—41no—
98C2×Dic118—21no—
99C4⋊C4×C59—21no—
100C4⋊C4×C59—42no—
101C4⋊C4×C59—41no—
102C4⋊C4×C59—41no—
103C4⋊C4×C59C22⋊C412yes—
104C22⋊C4×C59—22no—
105C22⋊C4×C59—22no—
106C22⋊C4×C59—44no—
107C22⋊C4×C59—82no—
108C42×C59—41no—
109C42×C59—41no—
110C42×C59—82no—
111C42×C59C22×C411yescharacteristic
112C42×C59C2×D411yescharacteristic
113C42×C59#4111yescharacteristic
114C42×C59#4111yescharacteristic
115#11745670—44no—
116#11745669—44no—
117(C2×D59)⋊C4—41no—
118(C2×D59)⋊C4—41no—
119C4⋊Dic59—21no—
120C4⋊Dic59—21no—
121#11745664—41no—
122#11745664—41no—
123C2×C59⋊C8—22no—
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#12011yescharacteristic
132C22×D59—41no—
133D4×C59—22no—
134D4×C59—22no—
135D4×C59—41no—
136D4×C59—41no—
137C2×C4×C59—22no—
138C2×C4×C59—22no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—42no—
143C2×C4×C59—41no—
144C2×C4×C59—42no—
145C2×C4×C59—42no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—42no—
150C2×C4×C59—82no—
151C2×C4×C59—82no—
152C2×C4×C59C2×C22⋊C411yescharacteristic
153C2×C4×C59C4×D411yescharacteristic
154C2×C4×C59C4×D411yescharacteristic
155C2×C4×C59#12111yescharacteristic
156C2×C4×C59#12111yescharacteristic
157C2×C4×C59#12311yescharacteristic
158C2×C4×C59#13511yescharacteristic
159C2×C4×C59#13511yescharacteristic
160C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
162C22⋊D59—81no—
163C22⋊D59—81no—
164C2×Dic59—41no—
165D236—41no—
166D236—41no—
167D236—81no—
168D236—81no—
169C4×D59—81no—
170C4×D59—81no—
171Dic118—41no—
172Dic118—41no—
173C59⋊C8—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#38611yescharacteristic
185C22×C59#44911yescharacteristic
186C22×C59#44911yescharacteristic
187C2×D59—81no—
188C2×D59—81no—
189C2×D59—81no—
190C4×C59—21no—
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—21no—
194C4×C59—42no—
195C4×C59—42no—
196C4×C59—81no—
197C4×C59—82no—
198C4×C59#41611yescharacteristic
199C4×C59#41811yescharacteristic
200C4×C59#48211yescharacteristic
201C4×C59#48211yescharacteristic
202Dic59—81no—
203#3036—591no—
204#2357D5911yescharacteristic, complemented
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#187511yescharacteristic
214C2×C59#190411yescharacteristic
215C2×C59#190611yescharacteristic
216D59—161no—
217C2×AGL(1,ℤ/8)—1181no—
218#532—591no—
219C2×C4×D4C2×D5911yescharacteristic
220#463—1182no—
221#461—592no—
222#459—592no—
223#431—591no—
224#431—591no—
225#423—591no—
226#417—1181no—
227#390—21no—
228#388—22no—
229#382C2×D5911yescharacteristic
230#381—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4C2×D5911yescharacteristic
233C4×C22⋊C4—21no—
234#327—1181no—
235#327—1181no—
236#327—1182no—
237D4⋊C8—1182no—
238C59#918111yescharacteristic, complemented, Sylow
239C2×#41—1181no—
240C22×D4C22⋊D5911yescharacteristic
241C23×C4—21no—
242AGL(1,ℤ/8)—2362no—
243AGL(1,ℤ/8)—2362no—
244C2×SD16—1182no—
245C2×D8—1182no—
246C2×M16—1181no—
247C4⋊Q8—591no—
248#127—591no—
249#124—1181no—
250#121—1181no—
251#121—1181no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—1181no—
255C4×D4—1181no—
256C4×D4C4×D5912yes—
257C4×D4D23612yes—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C22⋊D5911yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42C22×D5911yescharacteristic
267C4⋊C8—592no—
268C4⋊C8—1181no—
269C4⋊C8—1181no—
270C4⋊C8—1181no—
271#103—1182no—
272#102—1182no—
273C22⋊C8—1181no—
274C22⋊C8—1181no—
275C4×C8—1181no—
276C4×C8—1181no—
277#95—22no—
278#95—21no—
279#95C22⋊D5911yescharacteristic
280#95C22⋊D5911yescharacteristic
281#95C22⋊D5912yes—
282C24—21no—
283#41—2361no—
284#41—2361no—
285#41—2361no—
286#41—2361no—
287C2×Q8—1181no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—1181no—
294C2×D4—1181no—
295C2×D4—1181no—
296C2×D4—2361no—
297C2×D4(C2×D59)⋊C412yes—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—82no—
305C22×C4—2361no—
306C22×C4C2×C22⋊D5911yescharacteristic
307C22×C4C2×C22⋊D5911yescharacteristic
308C22×C4C2×C22⋊D5911yescharacteristic
309SD16—2364no—
310D8—2364no—
311M16—2361no—
312M16—2361no—
313C2×C8—1182no—
314C2×C8—2361no—
315C2×C8—2361no—
316C4⋊C4—21no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—1181no—
321C4⋊C4—1181no—
322C4⋊C4(C2×D59)⋊C412yes—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—44no—
326C22⋊C4—82no—
327C22⋊C4—2361no—
328C22⋊C4—2361no—
329C42—41no—
330C42—41no—
331C42—82no—
332C42C2×C4×D5911yescharacteristic
333C42C2×D23611yescharacteristic
334C42#1174568511yescharacteristic
335C42#1174568511yescharacteristic
336C23—22no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—2361no—
342C23#1427369611yescharacteristic
343Q8—2361no—
344Q8—2361no—
345D4—22no—
346D4—22no—
347D4—41no—
348D4—41no—
349D4—4721no—
350D4—4721no—
351D4—4721no—
352D4—4721no—
353D4—2361no—
354D4—2361no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—4721no—
371C2×C4—4721no—
372C2×C4—2361no—
373C2×C4C2×(C2×D59)⋊C411yescharacteristic
374C2×C4C4×C22⋊D5911yescharacteristic
375C2×C4C4×C22⋊D5911yescharacteristic
376C2×C4#1427370111yescharacteristic
377C2×C4#1427370111yescharacteristic
378C2×C4#1427371011yescharacteristic
379C2×C4#1427374711yescharacteristic
380C2×C4#1427374711yescharacteristic
381C2×C4#1427375211yescharacteristic
382C2×C4#1427375211yescharacteristic
383C8—4721no—
384C8—4721no—
385C8—2362no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—82no—
392C22—82no—
393C22—4721no—
394C22—4721no—
395C22—4721no—
396C22#1683728111yescharacteristic
397C22#1683754711yescharacteristic
398C22#1683754711yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—42no—
405C4—81no—
406C4—82no—
407C4—4721no—
408C4#1683739611yescharacteristic
409C4#1683740611yescharacteristic
410C4#1683772611yescharacteristic
411C4#1683772611yescharacteristic
412C2—21no—
413C2—21no—
414C2—82no—
415C2—9441no—
416C2#2142441811yescharacteristic
417C2#2142458311yescharacteristic
418C2#2142459711yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2357C2
socleC22×C59#386
prime radical#21433560C2
cosocleC2×C42×C59C23
hypercenter#2357D59
hypocenterC59#9181
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. #21422655C2semidirect
3. #21422590C2semidirect
4. #21422487C2semidirect
5. #21421999C2semidirect
6. #21421954C2semidirect
7. C59×#2357C2semidirect
11. #16837607C4semidirect
12. #16837597C4semidirect
204. #2357D59semidirect
238. C59#9181semidirect

Maximal subgroups

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

subgroupindex
#214335602
#214226552
#214225902
#214224872
#214219992
#214219542
C59×#23572
#918159