Finite Group Databasebeta

← #28481663

Subgroups — #28481663

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#21424134C211yesmaximal, characteristic, complemented
3#21422637C212yesmaximal
4#21422002C212yesmaximal, complemented
5C59×#1808C211yesmaximal, characteristic, complemented
6C2×#14273733C411yescharacteristic, complemented
7#16837913C2211yescharacteristic
8#16837473—21no—
9#16837473—21no—
10C8×C22⋊D59—21no—
11C8×C22⋊D59—21no—
12C2×#14273663—21no—
13#16837428C411yescharacteristic, complemented
14#16837368C411yescharacteristic, complemented
15#16837366C2212yes—
16#16837366C2211yescharacteristic
17#16837357C411yescharacteristic, complemented
18#16837167—21no—
19#16837138—22no—
20#16837135—22no—
21C23×C8×C59C2211yescharacteristic
22C2×#107×C59—21no—
23C59×#380C2212yes—
24C59×#340—21no—
25C2×#11745685C2×C411yescharacteristic
26#14273733—42no—
27C2×#11745664—22no—
28C2×#11745664C2×C411yescharacteristic
29C2×#11745664C2×C411yescharacteristic
30C2×#11745659—21no—
31C2×C8×D59—21no—
32#14273710—21no—
33#14273706D411yescharacteristic
34#14273706Q811yescharacteristic
35#14273701D411yescharacteristic
36#14273701Q811yescharacteristic
37#14273696—21no—
38C4×C22⋊D59—21no—
39C4×C22⋊D59—21no—
40#14273686C2×C411yescharacteristic
41#14273663—42no—
42Dic59⋊C8—21no—
43Dic59⋊C8—21no—
44C22⋊(C59⋊C8)—21no—
45C22⋊(C59⋊C8)—21no—
46C22⋊(C59⋊C8)—22no—
47C22⋊(C59⋊C8)—22no—
48(C2×D59)⋊C8—21no—
49(C2×D59)⋊C8—21no—
50#14273637—21no—
51C8×Dic59—21no—
52C23×C4×C59C2311yescharacteristic
53C22×C8×C59—42no—
54C22×C8×C59C2×C411yescharacteristic
55C22×C8×C59C2×C411yescharacteristic
56C22×C8×C59D411yescharacteristic
57C22×C8×C59D411yescharacteristic
58C22×C8×C59D411yescharacteristic
59C22×C8×C59D411yescharacteristic
60C2×C4⋊C4×C59—22no—
61C2×C22⋊C4×C59—22no—
62#107×C59—42no—
63#95×C59—22no—
64C24×C59#4111yescharacteristic
65C22×C4×C59—41no—
66C22×C4×C59—41no—
67C22×C4×C59—42no—
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×Q811yescharacteristic
73C22×C4×C59#4111yescharacteristic
74C2×C22⋊D59—21no—
75C2×C22⋊D59—21no—
76C2×C22⋊D59—21no—
77#11745685—41no—
78#11745685—41no—
79C2×D236C4⋊C411yescharacteristic
80C2×C4×D59—21no—
81C2×Dic118C4⋊C411yescharacteristic
82C2×C8×C59—22no—
83C2×C8×C59—22no—
84C2×C8×C59—22no—
85C2×C8×C59—41no—
86C2×C8×C59—41no—
87C2×C8×C59—41no—
88C2×C8×C59C22⋊C411yescharacteristic
89C2×C8×C59C22⋊C411yescharacteristic
90C2×C8×C59C22⋊C411yescharacteristic
91C2×C8×C59C22⋊C411yescharacteristic
92C4⋊C4×C59—42no—
93C4⋊C4×C59—42no—
94C22⋊C4×C59—82no—
95C22⋊Dic59—21no—
96C22⋊Dic59—21no—
97C22⋊Dic59—21no—
98(C2×D59)⋊C4—21no—
99(C2×D59)⋊C4—21no—
100C4⋊Dic59C4⋊C411yescharacteristic
101C4⋊Dic59C4⋊C411yescharacteristic
102Dic59⋊C4—21no—
103Dic59⋊C4—21no—
104C4×Dic59—21no—
105#11745664—41no—
106#11745664—41no—
107#11745664—42no—
108#11745664—42no—
109C2×C59⋊C8—21no—
110C2×C59⋊C8—21no—
111C2×C59⋊C8—42no—
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×C59—82no—
132C2×C4×C59—82no—
133C2×C4×C59—82no—
134C2×C4×C59C2×C22⋊C411yescharacteristic
135C2×C4×C59C2×C4⋊C411yescharacteristic
136C2×C4×C59#12111yescharacteristic
137C2×C4×C59C22⋊Q811yescharacteristic
138C22⋊D59—42no—
139C22⋊D59—42no—
140C22⋊D59—41no—
141C22⋊D59—41no—
142C2×Dic59—21no—
143C2×Dic59—21no—
144C2×Dic59—21no—
145D236—41no—
146C4×D59—41no—
147C4×D59—41no—
148Dic118—41no—
149C8×C59—21no—
150C8×C59—21no—
151C8×C59—21no—
152C8×C59#9912yes—
153C59⋊C8—41no—
154C59⋊C8—41no—
155C59⋊C8—82no—
156#8627—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×C59—162no—
177C4×C59C2×#9911yescharacteristic
178C4×C59#41011yescharacteristic
179Dic59—41no—
180Dic59—41no—
181Dic59—41no—
182#2929—591no—
183#1810—591no—
184#1808D5911yescharacteristic, complemented
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—1181no—
203C2×#107—21no—
204#424—591no—
205#408—591no—
206#407—591no—
207#407—592no—
208#405—591no—
209#380C2×D5912yes—
210#341—1181no—
211#340—21no—
212#328—1182no—
213#327—1182no—
214C59#862711yescharacteristic, complemented, Sylow
215C2×#41—591no—
216C23×C4C22×D5911yescharacteristic
217#131—2362no—
218C2×M16—591no—
219C2×M16—591no—
220C2×M16—1181no—
221C2×M16—1182no—
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—22no—
240C2×C22⋊C4—22no—
241#108—2362no—
242#107—42no—
243C4⋊C8—1181no—
244C4⋊C8—1181no—
245C22⋊C8—1181no—
246C22⋊C8—1181no—
247C22⋊C8—1181no—
248C22⋊C8—1181no—
249C22⋊C8—1182no—
250C22⋊C8—1182no—
251#97—1181no—
252C4×C8—1181no—
253#95—22no—
254C24#1174568711yescharacteristic
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—42no—
265C22×C4—1181no—
266C22×C4C2×C4×D5911yescharacteristic
267C22×C4C2×D23611yescharacteristic
268C22×C4D4×D5911yescharacteristic
269C22×C4Q8×D5911yescharacteristic
270C22×C4#1174568911yescharacteristic
271C22×C4C2×C22⋊D5911yescharacteristic
272M16—2362no—
273M16—2362no—
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—1181no—
284C2×C8—1181no—
285C2×C8—2362no—
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—42no—
292C4⋊C4—42no—
293C4⋊C4—591no—
294C4⋊C4—591no—
295C4⋊C4—1181no—
296C4⋊C4—1181no—
297C22⋊C4—82no—
298C22⋊C4—1181no—
299C22⋊C4—1181no—
300C22⋊C4—1181no—
301C22⋊C4—1181no—
302C22⋊C4—1181no—
303C42—1181no—
304C23—21no—
305C23—21no—
306C23—41no—
307C23—41no—
308C23—1181no—
309C23#1427368511yescharacteristic
310C23#1427369811yescharacteristic
311C23#1427370711yescharacteristic
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—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—1181no—
331C2×C4—1181no—
332C2×C4—1181no—
333C2×C4—2361no—
334C2×C4—2361no—
335C2×C4C2×(C2×D59)⋊C411yescharacteristic
336C2×C4C4⋊C4×D5911yescharacteristic
337C2×C4#1427370211yescharacteristic
338C2×C4#1427370311yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—4722no—
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#1683766911yescharacteristic
361C22#1683767411yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—162no—
366C4—2361no—
367C4—2361no—
368C4—2361no—
369C4C2×#1427364111yescharacteristic
370C4#1683737511yescharacteristic
371C2—21no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2—4721no—
376C2#2142408111yescharacteristic
377C2#2142414711yescharacteristic
378C2#2142416111yescharacteristic
subgroupG/H
centerC22#16837259
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1808C2
socleC22×C59#380
prime radical#16837913C22
cosocleC23×C4×C59C23
hypercenter#1808D59
hypocenterC59#8627
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1808

Complements

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

NH ≅ G/Ntype
2. #21424134C2semidirect
4. #21422002C2semidirect
5. C59×#1808C2semidirect
6. C2×#14273733C4semidirect
13. #16837428C4semidirect
14. #16837368C4semidirect
17. #16837357C4semidirect
184. #1808D59semidirect
214. C59#8627semidirect

Maximal subgroups

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

subgroupindex
#214327902
#214241342
#214226372
#214220022
C59×#18082
#862759