Finite Group Databasebeta

← #28503848

Subgroups — #28503848

Order 15104.

456 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432797C211yesmaximal, characteristic
2#21425738C211yesmaximal, characteristic
3#21425735C211yesmaximal, characteristic
4#21423896C211yesmaximal, characteristic, complemented
5#21422218C211yesmaximal, characteristic, complemented
6#21422216C211yesmaximal, characteristic, complemented
7C59×#1757C211yesmaximal, characteristic, complemented
8C22×#11745659C411yescharacteristic, complemented
9#16837909C2211yescharacteristic
10#16837478—21no—
11#16837478—21no—
12#16837473—21no—
13#16837473—21no—
14#16837430C411yescharacteristic, complemented
15#16837386C2211yescharacteristic
16#16837376C2211yescharacteristic
17#16837369C2211yescharacteristic
18#16837365C411yescharacteristic, complemented
19#16837357—21no—
20#16837357C411yescharacteristic, complemented
21#16837193C412yescomplemented
22#16837191C412yescomplemented
23#16837190C412yescomplemented
24#16837188C412yescomplemented
25#16837173—21no—
26C2×#131×C59C2211yescharacteristic
27C59×#413C2211yescharacteristic
28C59×#410C2211yescharacteristic
29C59×#405—21no—
30C59×#344—21no—
31C22×C4×D59C2×C411yescharacteristic
32C2×#11745664—21no—
33C2×#11745664C2×C411yescharacteristic
34C2×#11745659—21no—
35C2×#11745659—21no—
36C2×#11745659—41no—
37C2×#11745659—41no—
38C22×C59⋊C8C2×C411yescharacteristic
39#14273708C2×C411yescharacteristic
40#14273708C2×C411yescharacteristic
41#14273703—21no—
42#14273698—21no—
43#14273695C2×C411yescharacteristic
44#14273695C2×C411yescharacteristic
45C4×C22⋊D59—21no—
46C4×C22⋊D59—21no—
47#14273686—21no—
48#14273686C2×C411yescharacteristic
49Dic59⋊C8—21no—
50Dic59⋊C8—21no—
51#14273646—21no—
52#14273644C2×C412yes—
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—41no—
61C8×Dic59—21no—
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×C59—21no—
75#100×C59C2×C412yes—
76#99×C59—21no—
77#99×C59C2×C412yes—
78#97×C59—41no—
79C4×C8×C59—41no—
80C23×D59C4211yescharacteristic
81#41×C59—41no—
82#41×C59—41no—
83C2×Q8×C59#4111yescharacteristic
84C2×D4×C59C22×C411yescharacteristic
85C2×D4×C59C22×C411yescharacteristic
86C2×D4×C59#4111yescharacteristic
87C22×C4×C59C22×C411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C2×C22⋊D59—21no—
91C2×C22⋊D59—21no—
92C22×Dic59C4211yescharacteristic
93C2×C4×D59—21no—
94C2×C4×D59—21no—
95C2×C4×D59—41no—
96C2×C4×D59—41no—
97M16×C59—22no—
98M16×C59—41no—
99M16×C59—41no—
100M16×C59—41no—
101M16×C59—41no—
102C2×C8×C59—21no—
103C2×C8×C59—21no—
104C2×C8×C59—21no—
105C2×C8×C59—21no—
106C2×C8×C59—21no—
107C2×C8×C59—21no—
108C2×C8×C59—41no—
109C2×C8×C59C22⋊C411yescharacteristic
110C2×C8×C59C22⋊C411yescharacteristic
111C2×C8×C59C22⋊C411yescharacteristic
112C2×C8×C59C22⋊C411yescharacteristic
113C4⋊C4×C59—41no—
114C22⋊C4×C59—22no—
115C42×C59—41no—
116C22⋊Dic59—21no—
117C22⋊Dic59—21no—
118C22⋊Dic59—22no—
119C22⋊Dic59C4211yescharacteristic
120C22⋊Dic59C4211yescharacteristic
121(C2×D59)⋊C4—21no—
122(C2×D59)⋊C4—21no—
123C4⋊Dic59—21no—
124C4⋊Dic59—41no—
125Dic59⋊C4—21no—
126Dic59⋊C4—21no—
127C4×Dic59—21no—
128C4×Dic59—41no—
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—42no—
137#11745659—41no—
138#11745659—41no—
139C23×C59C2×C4211yescharacteristic
140C23×C59C4×D411yescharacteristic
141C23×C59C4×D411yescharacteristic
142C22×D59—21no—
143C22×D59—41no—
144C22×D59—81no—
145Q8×C59—41no—
146D4×C59—41no—
147D4×C59—41no—
148D4×C59—41no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—42no—
156C2×C4×C59C2×C22⋊C411yescharacteristic
157C2×C4×C59#11711yescharacteristic
158C2×C4×C59C4×D411yescharacteristic
159C2×C4×C59C4×D411yescharacteristic
160C22⋊D59—81no—
161C22⋊D59—81no—
162C2×Dic59—21no—
163C2×Dic59—21no—
164C2×Dic59—21no—
165C2×Dic59—41no—
166C2×Dic59—42no—
167C4×D59—42no—
168C4×D59—41no—
169C4×D59—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#12624—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—
190C2×D59—82no—
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—21no—
194C4×C59—82no—
195C4×C59C2×#9911yescharacteristic
196C4×C59#41011yescharacteristic
197Dic59—41no—
198Dic59—41no—
199Dic59—41no—
200Dic59—82no—
201#2930—591no—
202#2120—591no—
203#2120—591no—
204#1757—591no—
205#1757D5911yescharacteristic, complemented
206#1392—591no—
207#1391—591no—
208C2×C59—21no—
209C2×C59—41no—
210C2×C59—41no—
211C2×C59—41no—
212C2×C59C4×#9911yescharacteristic
213D59—81no—
214C2×#131C2×D5911yescharacteristic
215C22×M16—591no—
216#525—591no—
217#436—1181no—
218#436—1181no—
219#435—1181no—
220#435—1181no—
221#424—591no—
222#413—591no—
223#413C2×D5911yescharacteristic
224#410—591no—
225#410C2×D5911yescharacteristic
226#408—591no—
227#407—591no—
228#405—21no—
229#405—591no—
230#405—1181no—
231#352—592no—
232#352—592no—
233AΣL(1,𝔽16)—592no—
234AΣL(1,𝔽16)—592no—
235#344—21no—
236#344—1181no—
237C59#1262411yescharacteristic, complemented, Sylow
238C2×#41C22×D5911yescharacteristic
239C23×C4—591no—
240#131—41no—
241#131—41no—
242C2×M16—21no—
243C2×M16—591no—
244C2×M16—1181no—
245C2×M16—1181no—
246C2×M16—1181no—
247C2×M16—2361no—
248C2×M16—2361no—
249C2×M16C4×D5911yescharacteristic
250C2×M16D23611yescharacteristic
251C2×M16C22⋊D5911yescharacteristic
252C22×C8—591no—
253C22×C8C4×D5911yescharacteristic
254C22×C8D23611yescharacteristic
255C22×C8C22⋊D5911yescharacteristic
256#123—591no—
257#123—591no—
258C22⋊Q8—1181no—
259#121—1181no—
260#120—591no—
261#120—591no—
262C4×D4—1181no—
263C4×D4—1181no—
264#117—21no—
265#117—591no—
266#117—1181no—
267C4⋊C8—1181no—
268C4⋊C8—1181no—
269#101—21no—
270#101—1181no—
271#100—592no—
272#100C4×D5912yes—
273#99—21no—
274#99—592no—
275#99—1181no—
276#99C4×D5912yes—
277C22⋊C8—1181no—
278C22⋊C8—1181no—
279C22⋊C8—1181no—
280C22⋊C8—1181no—
281#97—41no—
282#97—1181no—
283#97—2361no—
284C4×C8—41no—
285C4×C8—1181no—
286C4×C8—2361no—
287C24—591no—
288#41—41no—
289#41—41no—
290C2×Q8#1174568511yescharacteristic
291C2×D4—1181no—
292C2×D4—1181no—
293C2×D4C2×C4×D5911yescharacteristic
294C2×D4C2×C4×D5911yescharacteristic
295C2×D4#1174568511yescharacteristic
296C22×C4—591no—
297C22×C4—1181no—
298C22×C4—1181no—
299C22×C4—2361no—
300C22×C4—2361no—
301C22×C4C2×C4×D5911yescharacteristic
302C22×C4C2×D23611yescharacteristic
303C22×C4C2×C22⋊D5911yescharacteristic
304M16—22no—
305M16—41no—
306M16—41no—
307M16—41no—
308M16—41no—
309M16—1182no—
310M16—2362no—
311M16—2361no—
312M16—2361no—
313M16—2361no—
314M16—2361no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—41no—
322C2×C8—1181no—
323C2×C8—1181no—
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—1181no—
334C4⋊C4—2361no—
335C22⋊C4—22no—
336C22⋊C4—591no—
337C22⋊C4—591no—
338C22⋊C4—1181no—
339C22⋊C4—1181no—
340C22⋊C4—1181no—
341C22⋊C4—1181no—
342C22⋊C4—1182no—
343C42—41no—
344C42—1181no—
345C42—2361no—
346C23—1181no—
347C23—4721no—
348C23—2361no—
349C23C42×D5911yescharacteristic
350C23C4×C22⋊D5911yescharacteristic
351C23C4×D23611yescharacteristic
352Q8—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—4721no—
357D4—4721no—
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—2362no—
369C2×C4—2362no—
370C2×C4—2361no—
371C2×C4—2361no—
372C2×C4—2361no—
373C2×C4C2×(C2×D59)⋊C411yescharacteristic
374C2×C4#1427368411yescharacteristic
375C2×C4C4×C22⋊D5911yescharacteristic
376C2×C4C4×D23611yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8—42no—
381C8—2362no—
382C8—2361no—
383C8—2361no—
384C8#1427364111yescharacteristic
385C8#1427364111yescharacteristic
386C22—21no—
387C22—21no—
388C22—21no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—4722no—
393C22—2361no—
394C22—2361no—
395C22—2361no—
396C22C4×(C2×D59)⋊C411yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—82no—
401C4—4722no—
402C4—2361no—
403C4—2361no—
404C4—2361no—
405C4C2×#1427364111yescharacteristic
406C4#1683737511yescharacteristic
407C2—21no—
408C2—41no—
409C2—41no—
410C2—41no—
411C2—4721no—
412C2C4×#1427364111yescharacteristic
subgroupG/H
centerC4C2×#14273641
commutatorC23×C59C2×C42
FrattiniC2×#41C22×D59
FittingC59×#1757C2
socleC2×C59C4×#99
prime radical#16837909C22
cosocleC2×#41×C59C23
hypercenter#1757D59
hypocenterC59#12624
LCSG ▷ C23×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. #21422218C2semidirect
6. #21422216C2semidirect
7. C59×#1757C2semidirect
8. C22×#11745659C4semidirect
14. #16837430C4semidirect
18. #16837365C4semidirect
20. #16837357C4semidirect
21. #16837193C4semidirect
22. #16837191C4semidirect
23. #16837190C4semidirect
24. #16837188C4semidirect
205. #1757D59semidirect
237. C59#12624semidirect

Maximal subgroups

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

subgroupindex
#214327972
#214257382
#214257352
#214238962
#214222182
#214222162
C59×#17572
#1262459