Finite Group Databasebeta

← #28503849

Subgroups — #28503849

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#21432800C211yesmaximal, characteristic
2#21425737C211yesmaximal, characteristic
3#21423899C211yesmaximal, characteristic, complemented
4#21423898C211yesmaximal, characteristic, complemented
5#21422218C211yesmaximal, characteristic, complemented
6#21422216C211yesmaximal, characteristic, complemented
7C59×#2120C211yesmaximal, characteristic
8C2×#14273734C2211yescharacteristic
9C22×#11745664C411yescharacteristic, complemented
10#16837480—21no—
11#16837480—21no—
12#16837474—21no—
13#16837474—21no—
14#16837429C411yescharacteristic, complemented
15#16837386C2211yescharacteristic
16#16837376C2211yescharacteristic
17#16837369C2211yescharacteristic
18#16837366C411yescharacteristic, complemented
19#16837360—21no—
20#16837358—21no—
21#16837358C411yescharacteristic, complemented
22#16837192C412yescomplemented
23#16837189C412yescomplemented
24#16837174—21no—
25#16837172—21no—
26C59×#525C2211yescharacteristic
27C59×#413C2211yescharacteristic
28C59×#410C2211yescharacteristic
29C59×#352C412yescomplemented
30C59×AΣL(1,𝔽16)C412yescomplemented
31#14273734—41no—
32#14273734—41no—
33C2×#11745664—21no—
34C2×#11745664—21no—
35C2×#11745664—21no—
36C2×#11745664—41no—
37C2×#11745664—41no—
38C2×#11745664C2×C411yescharacteristic
39C2×#11745664C2×C411yescharacteristic
40C2×#11745664D411yescharacteristic
41C2×#11745664D411yescharacteristic
42C22×C59⋊C8C2×C411yescharacteristic
43C22×C59⋊C8C2×C411yescharacteristic
44C22×C59⋊C8D411yescharacteristic
45C22×C59⋊C8D411yescharacteristic
46#14273686—21no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)—21no—
49#14273646—21no—
50#14273644C2×C412yes—
51#14273642—21no—
52#14273642C2×C412yes—
53C22⋊(C59⋊C8)—21no—
54C22⋊(C59⋊C8)—21no—
55C22⋊(C59⋊C8)—21no—
56C22⋊(C59⋊C8)—21no—
57#14273638—21no—
58#14273638—41no—
59#14273637—41no—
60C4×C59⋊C8—21no—
61C4×C59⋊C8—41no—
62C8×Dic59—41no—
63C2×#41×C59C2311yescharacteristic
64C23×C4×C59C2×C411yescharacteristic
65C2×M16×C59—21no—
66#123×C59C2×C411yescharacteristic
67#123×C59C2×C411yescharacteristic
68C22⋊Q8×C59—21no—
69#121×C59—21no—
70#120×C59C2×C411yescharacteristic
71#120×C59C2×C411yescharacteristic
72C4×D4×C59—21no—
73C4×D4×C59—21no—
74#117×C59—21no—
75#117×C59C2×C411yescharacteristic
76#101×C59—21no—
77#100×C59C2×C412yes—
78#99×C59—21no—
79#99×C59C2×C412yes—
80C24×C59C4211yescharacteristic
81#41×C59—41no—
82#41×C59—41no—
83C2×Q8×C59#4111yescharacteristic
84C2×D4×C59—21no—
85C2×D4×C59—21no—
86C2×D4×C59C22×C411yescharacteristic
87C2×D4×C59C22×C411yescharacteristic
88C2×D4×C59#4111yescharacteristic
89C22×C4×C59—21no—
90C22×C4×C59—21no—
91C22×C4×C59—41no—
92C22×C4×C59—41no—
93C22×C4×C59C4211yescharacteristic
94C22×C4×C59C22×C411yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59C2×D411yescharacteristic
97M16×C59—22no—
98M16×C59—41no—
99C2×C8×C59—41no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—21no—
102C4⋊C4×C59—21no—
103C4⋊C4×C59—41no—
104C22⋊C4×C59—21no—
105C22⋊C4×C59—21no—
106C22⋊C4×C59—21no—
107C22⋊C4×C59—21no—
108C22⋊C4×C59—22no—
109C22⋊C4×C59C4211yescharacteristic
110C22⋊C4×C59C4211yescharacteristic
111C42×C59—21no—
112C42×C59—41no—
113C22⋊Dic59—22no—
114C4⋊Dic59—41no—
115C4×Dic59—41no—
116#11745664—22no—
117#11745664—41no—
118#11745664—41no—
119#11745664—41no—
120#11745664—42no—
121#11745664—41no—
122#11745664—41no—
123#11745664—41no—
124#11745664—41no—
125C2×C59⋊C8—21no—
126C2×C59⋊C8—21no—
127C2×C59⋊C8—21no—
128C2×C59⋊C8—21no—
129C2×C59⋊C8—21no—
130C2×C59⋊C8—21no—
131C2×C59⋊C8—21no—
132C2×C59⋊C8—21no—
133C2×C59⋊C8—41no—
134C2×C59⋊C8—41no—
135C2×C59⋊C8C22⋊C411yescharacteristic
136C2×C59⋊C8C22⋊C411yescharacteristic
137C2×C59⋊C8C22⋊C411yescharacteristic
138C2×C59⋊C8C22⋊C411yescharacteristic
139C23×C59—21no—
140C23×C59—41no—
141C23×C59—81no—
142C23×C59C2×C4211yescharacteristic
143C23×C59C4×D411yescharacteristic
144C23×C59C4×D411yescharacteristic
145Q8×C59—41no—
146D4×C59—41no—
147D4×C59—41no—
148D4×C59—41no—
149D4×C59—81no—
150D4×C59—81no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—21no—
160C2×C4×C59—42no—
161C2×C4×C59—41no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—42no—
165C2×C4×C59C2×C22⋊C411yescharacteristic
166C2×C4×C59#11711yescharacteristic
167C2×C4×C59C4×D411yescharacteristic
168C2×C4×C59C4×D411yescharacteristic
169C2×Dic59—42no—
170C8×C59—42no—
171C59⋊C8—21no—
172C59⋊C8—21no—
173C59⋊C8—21no—
174C59⋊C8—42no—
175C59⋊C8—41no—
176C59⋊C8—41no—
177C59⋊C8#9911yescharacteristic
178C59⋊C8#9911yescharacteristic
179#12624—591nomaximal, Sylow
180C22×C59—21no—
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—41no—
184C22×C59—41no—
185C22×C59—41no—
186C22×C59—41no—
187C22×C59—41no—
188C22×C59—41no—
189C22×C59—82no—
190C22×C59C4×C22⋊C411yescharacteristic
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—21no—
194C4×C59—41no—
195C4×C59—41no—
196C4×C59—41no—
197C4×C59—82no—
198C4×C59C2×#9911yescharacteristic
199C4×C59#41011yescharacteristic
200Dic59—82no—
201#2930—591no—
202#2120—591no—
203#2120D5911yescharacteristic
204#1757—591no—
205#1757—591no—
206#1392—591no—
207#1391—591no—
208C2×C59—21no—
209C2×C59—41no—
210C2×C59—41no—
211C2×C59—41no—
212C2×C59—81no—
213C2×C59C4×#9911yescharacteristic
214C2×#131—591no—
215C22×M16—591no—
216#525C2×D5911yescharacteristic
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—591no—
229#405—1181no—
230#405—1181no—
231#352—592no—
232#352Dic5912yescomplemented
233AΣL(1,𝔽16)—592no—
234AΣL(1,𝔽16)Dic5912yescomplemented
235#344—1181no—
236#344—1181no—
237C59#1262411yescharacteristic, complemented, Sylow
238C2×#41C22×D5911yescharacteristic
239C23×C4C4×D5911yescharacteristic
240#131—2361no—
241#131—2361no—
242C2×M16—21no—
243C2×M16—591no—
244C2×M16—591no—
245C2×M16—591no—
246C2×M16—591no—
247C2×M16—1181no—
248C2×M16—1181no—
249C2×M16—1181no—
250C2×M16—2361no—
251C2×M16—2361no—
252C22×C8—591no—
253C22×C8—591no—
254C22×C8—591no—
255C22×C8—591no—
256#123C4×D5911yescharacteristic
257#123C2×Dic5911yescharacteristic
258C22⋊Q8—21no—
259#121—21no—
260#120C4×D5911yescharacteristic
261#120C2×Dic5911yescharacteristic
262C4×D4—21no—
263C4×D4—21no—
264#117—21no—
265#117—1181no—
266#117C4×D5911yescharacteristic
267C4⋊C8—1181no—
268C4⋊C8—1181no—
269#101—21no—
270#101—1181no—
271#100—592no—
272#100C2×Dic5912yes—
273#99—21no—
274#99—592no—
275#99—1181no—
276#99C2×Dic5912yes—
277C22⋊C8—1181no—
278C22⋊C8—1181no—
279C22⋊C8—1181no—
280C22⋊C8—1181no—
281#97—1181no—
282#97—2361no—
283#97—2361no—
284C4×C8—1181no—
285C4×C8—2361no—
286C4×C8—2361no—
287C24C4×Dic5911yescharacteristic
288#41—41no—
289#41—41no—
290C2×Q8#1174568711yescharacteristic
291C2×D4—21no—
292C2×D4—21no—
293C2×D4C2×C4×D5911yescharacteristic
294C2×D4#1174568711yescharacteristic
295C2×D4C22×Dic5911yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4C4×Dic5911yescharacteristic
301C22×C4C2×C4×D5911yescharacteristic
302C22×C4D4×D5911yescharacteristic
303C22×C4D4×D5911yescharacteristic
304M16—22no—
305M16—41no—
306M16—1182no—
307M16—2361no—
308M16—2361no—
309M16—2361no—
310M16—2361no—
311M16—2361no—
312M16—2362no—
313M16—2361no—
314M16—2361no—
315C2×C8—41no—
316C2×C8—591no—
317C2×C8—591no—
318C2×C8—591no—
319C2×C8—591no—
320C2×C8—1181no—
321C2×C8—1181no—
322C2×C8—1181no—
323C2×C8—1181no—
324C2×C8—1181no—
325C2×C8—1181no—
326C2×C8—1181no—
327C2×C8—1181no—
328C2×C8—2361no—
329C2×C8—2361no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—2361no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C22⋊C4—21no—
338C22⋊C4—21no—
339C22⋊C4—22no—
340C22⋊C4—1182no—
341C22⋊C4C4×Dic5911yescharacteristic
342C22⋊C4C4×Dic5911yescharacteristic
343C42—21no—
344C42—41no—
345C42—2361no—
346C23—21no—
347C23—41no—
348C23—81no—
349C23C2×C4×Dic5911yescharacteristic
350C23(C22⋊D59)⋊C411yescharacteristic
351C23D4×Dic5911yescharacteristic
352Q8—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—81no—
357D4—81no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—2362no—
373C2×C4C22⋊C4×D5911yescharacteristic
374C2×C4#1427368311yescharacteristic
375C2×C4(C22⋊D59)⋊C411yescharacteristic
376C2×C4D4×Dic5911yescharacteristic
377C8—42no—
378C8—591no—
379C8—591no—
380C8—1181no—
381C8—1181no—
382C8—1181no—
383C8—2362no—
384C8—2361no—
385C8—2361no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—82no—
396C22C22⋊C4×Dic5911yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—82no—
404C4—4722no—
405C4#99×D5911yescharacteristic
406C4#1683737411yescharacteristic
407C2—21no—
408C2—41no—
409C2—41no—
410C2—41no—
411C2—81no—
412C2#99×Dic5911yescharacteristic
subgroupG/H
centerC4#99×D59
commutatorC23×C59C2×C42
FrattiniC2×#41C22×D59
FittingC59×#2120C2
socleC2×C59C4×#99
prime radicalC59×#525C22
cosocleC2×#41×C59C23
hypercenter#2120D59
hypocenterC59#12624
LCSG ▷ C23×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2120

Complements

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

NH ≅ G/Ntype
3. #21423899C2semidirect
4. #21423898C2semidirect
5. #21422218C2semidirect
6. #21422216C2semidirect
9. C22×#11745664C4semidirect
14. #16837429C4semidirect
18. #16837366C4semidirect
21. #16837358C4semidirect
22. #16837192C4semidirect
23. #16837189C4semidirect
29. C59×#352C4semidirect
30. C59×AΣL(1,𝔽16)C4semidirect
232. #352Dic59semidirect
234. AΣL(1,𝔽16)Dic59semidirect
237. C59#12624semidirect

Maximal subgroups

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

subgroupindex
#214328002
#214257372
#214238992
#214238982
#214222182
#214222162
C59×#21202
#1262459