Finite Group Databasebeta

← #28481672

Subgroups — #28481672

Order 15104.

456 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432790C211yesmaximal, characteristic
2#21424125C211yesmaximal, characteristic, complemented
3#21422640C212yesmaximal
4#21422003C212yesmaximal, complemented
5C59×#1810C211yesmaximal, characteristic, complemented
6C2×#14273733C411yescharacteristic, complemented
7#16837913C2211yescharacteristic
8#16837473—21no—
9#16837473—21no—
10C8×C22⋊D59—21no—
11C8×C22⋊D59—21no—
12C2×#14273661—21no—
13#16837428C411yescharacteristic, complemented
14#16837368C411yescharacteristic, complemented
15#16837366C2211yescharacteristic
16#16837357C411yescharacteristic, complemented
17#16837262C2212yes—
18#16837165—21no—
19#16837140—22no—
20#16837137—22no—
21C23×C8×C59C2211yescharacteristic
22C2×#108×C59—21no—
23C59×#407C2212yes—
24C59×#341—21no—
25C2×#11745685C2×C411yescharacteristic
26#14273733—42no—
27C2×#11745664C2×C411yescharacteristic
28C2×#11745664C2×C411yescharacteristic
29C2×#11745659—21no—
30C2×C8×D59—21no—
31#14273710—21no—
32#14273706D411yescharacteristic
33#14273706Q811yescharacteristic
34#14273701D411yescharacteristic
35#14273701Q811yescharacteristic
36#14273696—21no—
37C4×C22⋊D59—21no—
38C4×C22⋊D59—21no—
39#14273686C2×C411yescharacteristic
40C2×C4⋊Dic59—22no—
41C2×C22⋊Dic59—22no—
42#14273661—42no—
43Dic59⋊C8—21no—
44Dic59⋊C8—21no—
45C22⋊(C59⋊C8)—21no—
46C22⋊(C59⋊C8)—21no—
47(C2×D59)⋊C8—21no—
48(C2×D59)⋊C8—21no—
49#14273637—21no—
50C8×Dic59—21no—
51C59⋊#95—22no—
52C23×C4×C59C2311yescharacteristic
53C2×M16×C59—22no—
54C22×C8×C59—42no—
55C22×C8×C59C2×C411yescharacteristic
56C22×C8×C59C2×C411yescharacteristic
57C22×C8×C59D411yescharacteristic
58C22×C8×C59D411yescharacteristic
59C22×C8×C59D411yescharacteristic
60C22×C8×C59D411yescharacteristic
61#108×C59—42no—
62C22⋊C8×C59—22no—
63C22⋊C8×C59—22no—
64C24×C59#4111yescharacteristic
65C22×C4×C59—41no—
66C22×C4×C59—41no—
67C22×C4×C59C22×C411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×Q811yescharacteristic
72C22×C4×C59#4111yescharacteristic
73C2×C22⋊D59—21no—
74C2×C22⋊D59—21no—
75C2×C22⋊D59—21no—
76C22×Dic59—42no—
77#11745685—41no—
78#11745685—41no—
79C2×D236C4⋊C411yescharacteristic
80C2×C4×D59—21no—
81C2×Dic118C4⋊C411yescharacteristic
82M16×C59—42no—
83M16×C59—42no—
84C2×C8×C59—22no—
85C2×C8×C59—22no—
86C2×C8×C59—22no—
87C2×C8×C59—41no—
88C2×C8×C59—41no—
89C2×C8×C59—41no—
90C2×C8×C59—42no—
91C2×C8×C59C22⋊C411yescharacteristic
92C2×C8×C59C22⋊C411yescharacteristic
93C2×C8×C59C22⋊C411yescharacteristic
94C2×C8×C59C22⋊C411yescharacteristic
95C22⋊Dic59—21no—
96C22⋊Dic59—21no—
97C22⋊Dic59—21no—
98C22⋊Dic59—82no—
99(C2×D59)⋊C4—21no—
100(C2×D59)⋊C4—21no—
101C4⋊Dic59—42no—
102C4⋊Dic59—42no—
103C4⋊Dic59C4⋊C411yescharacteristic
104C4⋊Dic59C4⋊C411yescharacteristic
105Dic59⋊C4—21no—
106Dic59⋊C4—21no—
107C4×Dic59—21no—
108#11745664—41no—
109#11745664—41no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—21no—
112#11745659—42no—
113C8×D59—42no—
114C23×C59—21no—
115C23×C59—21no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59#11711yescharacteristic
119C23×C59#12111yescharacteristic
120C23×C59C22⋊Q811yescharacteristic
121C22×D59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59C2×C22⋊C411yescharacteristic
132C2×C4×C59C2×C4⋊C411yescharacteristic
133C2×C4×C59#12111yescharacteristic
134C2×C4×C59C22⋊Q811yescharacteristic
135C22⋊D59—42no—
136C22⋊D59—42no—
137C22⋊D59—41no—
138C22⋊D59—41no—
139C2×Dic59—21no—
140C2×Dic59—21no—
141C2×Dic59—21no—
142C2×Dic59—82no—
143C2×Dic59—82no—
144C2×Dic59—82no—
145D236—41no—
146C4×D59—41no—
147C4×D59—41no—
148Dic118—41no—
149C8×C59—21no—
150C8×C59—21no—
151C8×C59—21no—
152C8×C59—82no—
153C8×C59#9912yes—
154C59⋊C8—41no—
155C59⋊C8—41no—
156#8628—591nomaximal, Sylow
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—21no—
165C22×C59—42no—
166C22×C59—42no—
167C22×C59#38011yescharacteristic
168C22×C59#47211yescharacteristic
169C22×C59#47311yescharacteristic
170C2×D59—41no—
171C2×D59—41no—
172C2×D59—41no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—21no—
176C4×C59C2×#9911yescharacteristic
177C4×C59#41011yescharacteristic
178Dic59—41no—
179Dic59—41no—
180Dic59—41no—
181Dic59—162no—
182#2929—591no—
183#1810D5911yescharacteristic, complemented
184#1808—591no—
185#1513—592no—
186#1305—592no—
187C2×C59—21no—
188C2×C59—21no—
189C2×C59—41no—
190C2×C59—41no—
191C2×C59#179911yescharacteristic
192C2×C59#181411yescharacteristic
193C2×C59#181611yescharacteristic
194D59—81no—
195C2×#131—591no—
196C23×C8C2×D5911yescharacteristic
197#525—591no—
198#435—1181no—
199#435—1181no—
200C8×D4—1181no—
201C8×D4—1181no—
202C2×#108—21no—
203C2×#107—1181no—
204#424—591no—
205#408—591no—
206#407—591no—
207#407C2×D5912yes—
208#405—591no—
209#380—592no—
210#341—21no—
211#340—1181no—
212#328—1182no—
213#327—1182no—
214C59#862811yescharacteristic, complemented, Sylow
215C2×#41—591no—
216C23×C4C22×D5911yescharacteristic
217#131—2362no—
218C2×M16—22no—
219C2×M16—591no—
220C2×M16—591no—
221C2×M16—1181no—
222C22×C8—42no—
223C22×C8—1181no—
224C22×C8C4×D5911yescharacteristic
225C22×C8C4×D5911yescharacteristic
226C22×C8D23611yescharacteristic
227C22×C8D23611yescharacteristic
228C22×C8C22⋊D5911yescharacteristic
229C22×C8C22⋊D5911yescharacteristic
230#123—1181no—
231C22⋊Q8—591no—
232C22⋊Q8—591no—
233#121—591no—
234#121—591no—
235#120—1181no—
236C4×D4—1181no—
237C4×D4—1181no—
238#117—591no—
239C2×C4⋊C4—1182no—
240C2×C22⋊C4—1182no—
241#108—42no—
242#107—2362no—
243C4⋊C8—1181no—
244C4⋊C8—1181no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—1181no—
248C22⋊C8—1181no—
249C22⋊C8—1181no—
250C22⋊C8—1181no—
251#97—1181no—
252C4×C8—1181no—
253#95—1182no—
254C24#1174568911yescharacteristic
255#41—2361no—
256#41—2361no—
257C2×Q8—591no—
258C2×D4—591no—
259C2×D4—1181no—
260C2×D4—1181no—
261C2×D4—1181no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—1181no—
265C22×C4—2362no—
266C22×C4C2×C4×D5911yescharacteristic
267C22×C4C2×D23611yescharacteristic
268C22×C4D4×D5911yescharacteristic
269C22×C4#1174568711yescharacteristic
270C22×C4Q8×D5911yescharacteristic
271C22×C4C2×C22⋊D5911yescharacteristic
272M16—42no—
273M16—42no—
274M16—2362no—
275M16—2361no—
276M16—2361no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—1181no—
285C2×C8—1181no—
286C2×C8—2362no—
287C2×C8(C2×D59)⋊C411yescharacteristic
288C2×C8(C2×D59)⋊C411yescharacteristic
289C2×C8(C2×D59)⋊C411yescharacteristic
290C2×C8(C2×D59)⋊C411yescharacteristic
291C4⋊C4—591no—
292C4⋊C4—591no—
293C4⋊C4—1181no—
294C4⋊C4—1181no—
295C4⋊C4—2362no—
296C4⋊C4—2362no—
297C22⋊C4—1181no—
298C22⋊C4—1181no—
299C22⋊C4—1181no—
300C22⋊C4—1181no—
301C22⋊C4—1181no—
302C22⋊C4—4722no—
303C42—1181no—
304C23—21no—
305C23—21no—
306C23—41no—
307C23—41no—
308C23—1181no—
309C23#1427368511yescharacteristic
310C23#1427370211yescharacteristic
311C23#1427370311yescharacteristic
312Q8—2361no—
313D4—2361no—
314D4—2362no—
315D4—2362no—
316D4—2361no—
317D4—2361no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—1181no—
328C2×C4—1181no—
329C2×C4—1181no—
330C2×C4—4722no—
331C2×C4—4722no—
332C2×C4—4722no—
333C2×C4—2361no—
334C2×C4—2361no—
335C2×C4C2×(C2×D59)⋊C411yescharacteristic
336C2×C4C4⋊C4×D5911yescharacteristic
337C2×C4#1427369811yescharacteristic
338C2×C4#1427370711yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—82no—
343C8—2361no—
344C8—2361no—
345C8#1427364112yes—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22—2361no—
357C22—2361no—
358C22—2361no—
359C22#1683725911yescharacteristic
360C22#1683767311yescharacteristic
361C22#1683767811yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—9442no—
366C4—2361no—
367C4—2361no—
368C4—2361no—
369C4C2×#1427364111yescharacteristic
370C4#1683737511yescharacteristic
371C2—21no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2—4721no—
376C2#2142408111yescharacteristic
377C2#2142415111yescharacteristic
378C2#2142415711yescharacteristic
subgroupG/H
centerC22#16837259
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1810C2
socleC22×C59#380
prime radical#16837913C22
cosocleC23×C4×C59C23
hypercenter#1810D59
hypocenterC59#8628
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1810

Complements

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

NH ≅ G/Ntype
2. #21424125C2semidirect
4. #21422003C2semidirect
5. C59×#1810C2semidirect
6. C2×#14273733C4semidirect
13. #16837428C4semidirect
14. #16837368C4semidirect
16. #16837357C4semidirect
183. #1810D59semidirect
214. C59#8628semidirect

Maximal subgroups

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

subgroupindex
#214327902
#214241252
#214226402
#214220032
C59×#18102
#862859