Finite Group Databasebeta

← #28503889

Subgroups — #28503889

Order 15104.

452 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432812C211yesmaximal, characteristic
2#21425779C211yesmaximal, characteristic
3#21425759C211yesmaximal, characteristic
4#21423896C211yesmaximal, characteristic, complemented
5#21422226C211yesmaximal, characteristic, complemented
6#21422224C211yesmaximal, characteristic, complemented
7C59×#1757C211yesmaximal, characteristic, complemented
8C2×#14273733C411yescharacteristic, complemented
9#16837948C2211yescharacteristic
10#16837473—21no—
11#16837473—21no—
12#16837471—21no—
13#16837471—21no—
14#16837425C411yescharacteristic, complemented
15#16837386C2211yescharacteristic
16#16837376C2211yescharacteristic
17#16837370C411yescharacteristic, complemented
18#16837369C2211yescharacteristic
19#16837357—21no—
20C2×#14273637C411yescharacteristic, complemented
21#16837200C412yescomplemented
22#16837198C412yescomplemented
23#16837196C412yes—
24#16837194C412yes—
25#16837173—21no—
26C2×#131×C59C2211yescharacteristic
27C59×#413C2211yescharacteristic
28C59×#410C2211yescharacteristic
29C59×#405—21no—
30C59×#344—21no—
31C2×#11745685C2×C411yescharacteristic
32#14273733—41no—
33#14273733—41no—
34C2×#11745664—21no—
35C2×#11745664C2×C411yescharacteristic
36C2×#11745659—21no—
37C22×C59⋊C8C2×C411yescharacteristic
38C2×C8×D59—21no—
39#14273725C2×C411yescharacteristic
40#14273723C2×C411yescharacteristic
41#14273716C2×C411yescharacteristic
42#14273714C2×C411yescharacteristic
43#14273699—21no—
44#14273699—21no—
45C4×C22⋊D59—21no—
46C4×C22⋊D59—21no—
47#14273686—21no—
48C2×C4×Dic59C2×C411yescharacteristic
49Dic59⋊C8—21no—
50Dic59⋊C8—21no—
51#14273646C2×C412yes—
52#14273644—21no—
53#14273642—21no—
54#14273642C2×C412yes—
55C22⋊(C59⋊C8)—21no—
56C22⋊(C59⋊C8)—21no—
57(C2×D59)⋊C8—21no—
58(C2×D59)⋊C8—21no—
59#14273637—21no—
60#14273637—21no—
61#14273637—41no—
62C8×Dic59—41no—
63C2×#41×C59C2311yescharacteristic
64#131×C59—41no—
65#131×C59—41no—
66C2×M16×C59—21no—
67C2×M16×C59C2×C411yescharacteristic
68C2×M16×C59D411yescharacteristic
69C2×M16×C59D411yescharacteristic
70C22×C8×C59C2×C411yescharacteristic
71C22×C8×C59D411yescharacteristic
72C22×C8×C59D411yescharacteristic
73#117×C59—21no—
74#101×C59C2×C412yes—
75#100×C59—21no—
76#99×C59—21no—
77#99×C59C2×C412yes—
78#97×C59—41no—
79C4×C8×C59—41no—
80#41×C59—41no—
81#41×C59—41no—
82C2×Q8×C59C22×C411yescharacteristic
83C2×D4×C59C22×C411yescharacteristic
84C2×D4×C59#4111yescharacteristic
85C2×D4×C59#4111yescharacteristic
86C22×C4×C59C22×C411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C2×C22⋊D59—21no—
90C2×C22⋊D59—21no—
91C2×C22⋊D59—21no—
92C22×Dic59—21no—
93#11745685—41no—
94#11745685—41no—
95C2×D236C4211yescharacteristic
96C2×C4×D59—21no—
97C2×Dic118C4211yescharacteristic
98M16×C59—22no—
99M16×C59—41no—
100M16×C59—41no—
101M16×C59—41no—
102M16×C59—41no—
103C2×C8×C59—21no—
104C2×C8×C59—21no—
105C2×C8×C59—21no—
106C2×C8×C59—21no—
107C2×C8×C59—21no—
108C2×C8×C59—21no—
109C2×C8×C59—41no—
110C2×C8×C59C22⋊C411yescharacteristic
111C2×C8×C59C22⋊C411yescharacteristic
112C2×C8×C59C22⋊C411yescharacteristic
113C2×C8×C59C22⋊C411yescharacteristic
114C4⋊C4×C59—41no—
115C22⋊C4×C59—22no—
116C42×C59—41no—
117C22⋊Dic59—21no—
118C22⋊Dic59—21no—
119C22⋊Dic59—22no—
120(C2×D59)⋊C4—21no—
121(C2×D59)⋊C4—21no—
122C4⋊Dic59—41no—
123Dic59⋊C4—21no—
124Dic59⋊C4—21no—
125C4×Dic59—21no—
126C4×Dic59—41no—
127C4×Dic59C4211yescharacteristic
128C4×Dic59C4211yescharacteristic
129#11745664—22no—
130#11745664—41no—
131#11745664—41no—
132C2×C59⋊C8—21no—
133C2×C59⋊C8—21no—
134C2×C59⋊C8—41no—
135C2×C59⋊C8—41no—
136#11745659—41no—
137#11745659—41no—
138C8×D59—42no—
139C23×C59#11711yescharacteristic
140C23×C59C4×D411yescharacteristic
141C23×C59C4×D411yescharacteristic
142C22×D59—21no—
143Q8×C59—41no—
144D4×C59—41no—
145D4×C59—41no—
146D4×C59—41no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—42no—
154C2×C4×C59C2×C4211yescharacteristic
155C2×C4×C59C2×C22⋊C411yescharacteristic
156C2×C4×C59C4×D411yescharacteristic
157C2×C4×C59C4×D411yescharacteristic
158C22⋊D59—42no—
159C22⋊D59—81no—
160C22⋊D59—81no—
161C2×Dic59—21no—
162C2×Dic59—21no—
163C2×Dic59—21no—
164C2×Dic59—42no—
165C2×Dic59—81no—
166D236—41no—
167C4×D59—41no—
168C4×D59—41no—
169Dic118—41no—
170C8×C59—21no—
171C8×C59—21no—
172C8×C59—21no—
173C8×C59—42no—
174C8×C59#9911yescharacteristic
175C8×C59#9911yescharacteristic
176C59⋊C8—42no—
177C59⋊C8—41no—
178C59⋊C8—41no—
179#12634—591nomaximal, Sylow
180C22×C59—21no—
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—41no—
184C22×C59—41no—
185C22×C59—41no—
186C22×C59C4×C22⋊C411yescharacteristic
187C2×D59—41no—
188C2×D59—41no—
189C2×D59—41no—
190C4×C59—21no—
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—82no—
194C4×C59C2×#9911yescharacteristic
195C4×C59#41011yescharacteristic
196Dic59—41no—
197Dic59—41no—
198Dic59—41no—
199Dic59—82no—
200#2932—591no—
201#2132—591no—
202#2126—591no—
203#1757—591no—
204#1757D5911yescharacteristic, complemented
205#1395—591no—
206#1395—591no—
207C2×C59—21no—
208C2×C59—41no—
209C2×C59—41no—
210C2×C59—41no—
211C2×C59C4×#9911yescharacteristic
212D59—81no—
213C2×#131—591no—
214C2×#131C2×D5911yescharacteristic
215#532—591no—
216#435—1181no—
217#435—1181no—
218#435—1181no—
219#435—1181no—
220#423—591no—
221#413—591no—
222#413C2×D5911yescharacteristic
223#410—591no—
224#410C2×D5911yescharacteristic
225#408—591no—
226#408—591no—
227#405—21no—
228#405—1181no—
229C2×#97—591no—
230#356—592no—
231#355—592no—
232#354—592no—
233#353—592no—
234#344—21no—
235#344—1181no—
236C59#1263411yescharacteristic, complemented, Sylow
237C2×#41—591no—
238C2×#41C22×D5911yescharacteristic
239#131—41no—
240#131—41no—
241#131—2361no—
242#131—2361no—
243C2×M16—21no—
244C2×M16—591no—
245C2×M16—1181no—
246C2×M16—1181no—
247C2×M16C4×D5911yescharacteristic
248C2×M16D23611yescharacteristic
249C2×M16C22⋊D5911yescharacteristic
250C22×C8—591no—
251C22×C8—1181no—
252C22×C8C4×D5911yescharacteristic
253C22×C8D23611yescharacteristic
254C22×C8C22⋊D5911yescharacteristic
255C4⋊Q8—591no—
256#127—591no—
257#124—591no—
258#124—591no—
259#121—1181no—
260#121—1181no—
261C4×D4—1181no—
262C4×D4—1181no—
263#117—21no—
264#117—1181no—
265C2×C42—591no—
266C4⋊C8—1181no—
267C4⋊C8—1181no—
268#101—592no—
269#101C4×D5912yes—
270#100—21no—
271#100—1181no—
272#99—21no—
273#99—592no—
274#99—1181no—
275#99C4×D5912yes—
276C22⋊C8—1181no—
277C22⋊C8—1181no—
278C22⋊C8—1181no—
279C22⋊C8—1181no—
280#97—41no—
281#97—1181no—
282#97—1181no—
283#97—2361no—
284C4×C8—41no—
285C4×C8—2361no—
286#41—41no—
287#41—41no—
288#41—2361no—
289#41—2361no—
290C2×Q8—591no—
291C2×Q8C2×C4×D5911yescharacteristic
292C2×D4—591no—
293C2×D4—1181no—
294C2×D4—1181no—
295C2×D4—1181no—
296C2×D4C2×C4×D5911yescharacteristic
297C2×D4#1174568511yescharacteristic
298C2×D4#1174568511yescharacteristic
299C22×C4—1181no—
300C22×C4—1181no—
301C22×C4C2×C4×D5911yescharacteristic
302C22×C4C2×D23611yescharacteristic
303C22×C4C2×C22⋊D5911yescharacteristic
304M16—22no—
305M16—41no—
306M16—41no—
307M16—41no—
308M16—41no—
309M16—1182no—
310M16—2361no—
311M16—2361no—
312M16—2361no—
313M16—2361no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—41no—
321C2×C8—1181no—
322C2×C8—1181no—
323C2×C8—2362no—
324C2×C8—2361no—
325C2×C8—2361no—
326C2×C8(C2×D59)⋊C411yescharacteristic
327C2×C8(C2×D59)⋊C411yescharacteristic
328C2×C8(C2×D59)⋊C411yescharacteristic
329C2×C8(C2×D59)⋊C411yescharacteristic
330C4⋊C4—41no—
331C4⋊C4—1181no—
332C4⋊C4—1181no—
333C4⋊C4—2361no—
334C22⋊C4—22no—
335C22⋊C4—1181no—
336C22⋊C4—1181no—
337C22⋊C4—1181no—
338C22⋊C4—1181no—
339C22⋊C4—1182no—
340C42—41no—
341C42—591no—
342C42—591no—
343C42—1181no—
344C42—2361no—
345C23—1181no—
346C23#1427368411yescharacteristic
347C23C4×C22⋊D5911yescharacteristic
348C23C4×D23611yescharacteristic
349Q8—41no—
350Q8—2361no—
351D4—41no—
352D4—41no—
353D4—41no—
354D4—4721no—
355D4—4721no—
356D4—2361no—
357D4—2362no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—1181no—
366C2×C4—1181no—
367C2×C4—1181no—
368C2×C4—4721no—
369C2×C4—2362no—
370C2×C4—2361no—
371C2×C4—2361no—
372C2×C4C42×D5911yescharacteristic
373C2×C4C2×(C2×D59)⋊C411yescharacteristic
374C2×C4C4×C22⋊D5911yescharacteristic
375C2×C4C4×D23611yescharacteristic
376C8—21no—
377C8—21no—
378C8—21no—
379C8—42no—
380C8—2362no—
381C8—2361no—
382C8—2361no—
383C8#1427364111yescharacteristic
384C8#1427364111yescharacteristic
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—2361no—
392C22—2361no—
393C22—2361no—
394C22C4×(C2×D59)⋊C411yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—82no—
399C4—4722no—
400C4—2361no—
401C4—2361no—
402C4—2361no—
403C4C2×#1427364111yescharacteristic
404C4#1683737511yescharacteristic
405C2—21no—
406C2—41no—
407C2—41no—
408C2—41no—
409C2—4721no—
410C2C4×#1427364111yescharacteristic
subgroupG/H
centerC4C2×#14273641
commutatorC2×C4×C59C2×C42
FrattiniC2×#41C22×D59
FittingC59×#1757C2
socleC2×C59C4×#99
prime radical#16837948C22
cosocleC2×#41×C59C23
hypercenter#1757D59
hypocenterC59#12634
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ #1757

Complements

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

NH ≅ G/Ntype
4. #21423896C2semidirect
5. #21422226C2semidirect
6. #21422224C2semidirect
7. C59×#1757C2semidirect
8. C2×#14273733C4semidirect
14. #16837425C4semidirect
17. #16837370C4semidirect
20. C2×#14273637C4semidirect
21. #16837200C4semidirect
22. #16837198C4semidirect
204. #1757D59semidirect
236. C59#12634semidirect

Maximal subgroups

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

subgroupindex
#214328122
#214257792
#214257592
#214238962
#214222262
#214222242
C59×#17572
#1263459