Finite Group Databasebeta

← #28489794

Subgroups — #28489794

Order 15104.

634 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
1C2×#16837750C211yesmaximal, characteristic
2C2×#16837602C211yesmaximal, characteristic
3#21425023C211yesmaximal, characteristic
4C2×#16837148C211yesmaximal, characteristic, complemented
5C2×Q8⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
6#21421937C211yesmaximal, characteristic, complemented
7C59×#2568C211yesmaximal, characteristic, complemented
8C22×#11745671—21no—
9C2×#14273724C2211yescharacteristic
10#16837750—22no—
11#16837750—22no—
12#16837602—22no—
13#16837602—22no—
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15C2×C4⋊(C59⋊C8)C2211yescharacteristic
16C2×#14273647—22no—
17C2×#14273647—21no—
18C2×#14273647—22no—
19C2×C4×C59⋊C8C2211yescharacteristic
20#16837148—22no—
21Q8⋊(C59⋊C8)—22no—
22C2×C4⋊Q8×C59C2211yescharacteristic
23C2×C4×Q8×C59C2211yescharacteristic
24C59×#400—22no—
25C59×#391—21no—
26C59×#391—22no—
27C59×#382—21no—
28C4×C4⋊C4×C59C2211yescharacteristic
29C22×D236—21no—
30C22×D236—21no—
31C22×D236—21no—
32C2×#11745671—24no—
33C2×#11745671—42no—
34C2×#11745671—42no—
35C22×C59⋊C8—21no—
36C22×C59⋊C8—21no—
37C22×C59⋊C8—21no—
38#14273724—21no—
39#14273724—21no—
40#14273724—21no—
41#14273724D412yes—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)D412yes—
46#14273647—28no—
47#14273647—24no—
48#14273647—44no—
49C4×C59⋊C8—21no—
50C4×C59⋊C8D412yes—
51C22×Q8×C59—21no—
52C22×Q8×C59D411yescharacteristic
53C4⋊Q8×C59—21no—
54C4⋊Q8×C59—21no—
55C4⋊Q8×C59—21no—
56C4⋊Q8×C59—21no—
57C4×Q8×C59—21no—
58C4×Q8×C59—21no—
59C4×Q8×C59—21no—
60C4×Q8×C59—21no—
61C2×C4⋊C4×C59—21no—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59D412yes—
64C2×C4⋊C4×C59D411yescharacteristic
65C2×C4⋊C4×C59D412yes—
66C2×C42×C59—21no—
67C2×C42×C59—21no—
68C2×C42×C59C2311yescharacteristic
69#95×C59—24no—
70#95×C59—24no—
71#95×C59—21no—
72C23×D59—41no—
73C2×Q8×C59—21no—
74C2×Q8×C59—21no—
75C2×Q8×C59—21no—
76C2×Q8×C59—21no—
77C2×Q8×C59—21no—
78C2×Q8×C59—41no—
79C2×Q8×C59—41no—
80C2×Q8×C59—42no—
81C2×Q8×C59—42no—
82C2×Q8×C59SD1612yes—
83C22×C4×C59—21no—
84C22×C4×C59—22no—
85C22×C4×C59—21no—
86C22×C4×C59—22no—
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C2×D236—22no—
91C2×D236—22no—
92C2×D236—22no—
93C2×D236—44no—
94C2×D236—41no—
95C2×D236—41no—
96C2×D236—42no—
97C2×D236—42no—
98C2×D236—41no—
99C2×D236—41no—
100C2×D236—41no—
101C2×D236—41no—
102C2×D236—41no—
103C4⋊C4×C59—21no—
104C4⋊C4×C59—22no—
105C4⋊C4×C59—21no—
106C4⋊C4×C59—22no—
107C4⋊C4×C59—22no—
108C4⋊C4×C59—22no—
109C4⋊C4×C59—41no—
110C4⋊C4×C59—41no—
111C4⋊C4×C59—41no—
112C4⋊C4×C59—41no—
113C42×C59—41no—
114C42×C59—41no—
115C42×C59—41no—
116C42×C59—41no—
117C42×C59C2×D411yescharacteristic
118C42×C59C2×D411yescharacteristic
119C42×C59C2×D411yescharacteristic
120C42×C59#4111yescharacteristic
121#11745671—48no—
122C2×C59⋊C8—22no—
123C2×C59⋊C8—22no—
124C2×C59⋊C8—42no—
125C2×C59⋊C8—41no—
126C2×C59⋊C8—41no—
127C2×C59⋊C8—41no—
128C2×C59⋊C8—42no—
129C23×C59#12011yescharacteristic
130C22×D59—42no—
131C22×D59—81no—
132C22×D59—81no—
133C22×D59—82no—
134C22×D59—82no—
135Q8×C59—22no—
136Q8×C59—22no—
137Q8×C59—41no—
138Q8×C59—41no—
139Q8×C59—82no—
140Q8×C59—81no—
141Q8×C59—81no—
142C2×C4×C59—22no—
143C2×C4×C59—22no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—42no—
148C2×C4×C59—42no—
149C2×C4×C59—42no—
150C2×C4×C59—44no—
151C2×C4×C59—42no—
152C2×C4×C59—42no—
153C2×C4×C59#12011yescharacteristic
154C2×C4×C59#12011yescharacteristic
155C2×C4×C59#12111yescharacteristic
156C2×C4×C59#12111yescharacteristic
157C2×C4×C59#12111yescharacteristic
158C2×C4×C59#12711yescharacteristic
159C2×C4×C59C2×SD1611yescharacteristic
160C2×C4×C59#13511yescharacteristic
161C2×C4×C59#13512yes—
162C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C59AGL(1,ℤ/8)12yes—
164C2×C4×C59#13711yescharacteristic
165D236—44no—
166D236—42no—
167D236—42no—
168D236—44no—
169D236—82no—
170D236—81no—
171D236—81no—
172D236—81no—
173D236—81no—
174C59⋊C8—42no—
175C59⋊C8—42no—
176C59⋊C8—81no—
177#10053—591nomaximal, Sylow
178C22×C59#39211yescharacteristic
179C22×C59#44811yescharacteristic
180C22×C59#44912yes—
181C22×C59#44911yescharacteristic
182C22×C59#45412yes—
183C2×D59—82no—
184C2×D59—82no—
185C2×D59—82no—
186C2×D59—162no—
187C2×D59—162no—
188C4×C59—21no—
189C4×C59—21no—
190C4×C59—42no—
191C4×C59—82no—
192C4×C59—81no—
193C4×C59—82no—
194C4×C59#45011yescharacteristic
195C4×C59#45211yescharacteristic
196C4×C59#46312yes—
197C4×C59#46511yescharacteristic
198C4×C59#46811yescharacteristic
199C4×C59#49511yescharacteristic
200C4×C59#49611yescharacteristic
201C2×#486—591no—
202C2×#460—591no—
203#2568D5911yescharacteristic, complemented
204#1982—591no—
205C2×#332—591no—
206C2×Q8⋊C8—591no—
207#1279—591no—
208C2×C59#162612yes—
209C2×C59#162812yes—
210C2×C59#200011yescharacteristic
211C2×C59#200311yescharacteristic
212C2×C59#201211yescharacteristic
213D59—162no—
214C22×SD16—1181no—
215C2×C4⋊Q8C2×D5911yescharacteristic
216C2×#127—591no—
217C2×C4×Q8C2×D5911yescharacteristic
218#486—1182no—
219#486—1182no—
220#460—1182no—
221#460—1182no—
222C2×C4⋊C8—591no—
223C2×C4⋊C8—591no—
224C2×#102—1182no—
225C2×#102—1181no—
226C2×#102—1182no—
227C2×C4×C8—591no—
228#400—22no—
229#391—21no—
230#391—22no—
231#382—21no—
232C4×C4⋊C4C2×D5911yescharacteristic
233#332—1182no—
234Q8⋊C8—1182no—
235C59#1005311yescharacteristic, complemented, Sylow
236C22×Q8—21no—
237C22×Q8C22⋊D5911yescharacteristic
238C22×D4—1181no—
239C22×D4—1181no—
240C22×D4—1181no—
241C2×SD16—1184no—
242C2×SD16—2362no—
243C2×SD16—2362no—
244C22×C8—1181no—
245C22×C8—1181no—
246C22×C8—1181no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251#127—592no—
252#127—1181no—
253#127—1181no—
254#127—1181no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C22⋊D5911yescharacteristic
262C2×C4⋊C4C22⋊D5912yes—
263C2×C4⋊C4C22⋊D5912yes—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42C22×D5911yescharacteristic
267C4⋊C8—592no—
268C4⋊C8—1181no—
269C4⋊C8—1181no—
270C4⋊C8—1181no—
271#102—1184no—
272#102—1188no—
273#102—2364no—
274C4×C8—592no—
275C4×C8—1181no—
276#95—24no—
277#95—24no—
278#95—21no—
279C24—2361no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8#1174567112yes—
290C2×D4—1182no—
291C2×D4—1182no—
292C2×D4—1182no—
293C2×D4—2364no—
294C2×D4—2361no—
295C2×D4—2361no—
296C2×D4—2361no—
297C2×D4—2361no—
298C2×D4—2361no—
299C2×D4—2362no—
300C2×D4—2362no—
301C2×D4—2361no—
302C2×D4—2361no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4C2×C22⋊D5911yescharacteristic
308C22×C4C2×C22⋊D5911yescharacteristic
309C22×C4C2×C22⋊D5911yescharacteristic
310SD16—2368no—
311C2×C8—1182no—
312C2×C8—1182no—
313C2×C8—2361no—
314C2×C8—2362no—
315C2×C8—2362no—
316C2×C8—2361no—
317C2×C8—2361no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42D4×D5911yescharacteristic
333C42D4×D5911yescharacteristic
334C42D4×D5911yescharacteristic
335C42#1174568711yescharacteristic
336C23—4721no—
337C23—4722no—
338C23—4721no—
339C23—4722no—
340C23—2362no—
341C23#1427369611yescharacteristic
342Q8—22no—
343Q8—22no—
344Q8—41no—
345Q8—41no—
346Q8—81no—
347Q8—81no—
348Q8—82no—
349D4—4722no—
350D4—4721no—
351D4—4721no—
352D4—4721no—
353D4—4721no—
354D4—2364no—
355D4—2362no—
356D4—2362no—
357D4—2364no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4#1427369511yescharacteristic
370C2×C4#1427369511yescharacteristic
371C2×C4#1427369811yescharacteristic
372C2×C4#1427369911yescharacteristic
373C2×C4#1427369911yescharacteristic
374C2×C4#1427372311yescharacteristic
375C2×C4C2×#1174567111yescharacteristic
376C2×C4#1427374711yescharacteristic
377C2×C4#1427374712yes—
378C2×C4#1427375211yescharacteristic
379C2×C4#1427375212yes—
380C2×C4#1427375811yescharacteristic
381C8—4721no—
382C8—2362no—
383C8—2362no—
384C22—9442no—
385C22—9442no—
386C22—4722no—
387C22—4722no—
388C22—4722no—
389C22#1683731011yescharacteristic
390C22#1683754111yescharacteristic
391C22#1683754712yes—
392C22#1683754711yescharacteristic
393C22#1683757512yes—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—82no—
398C4—82no—
399C4—81no—
400C4#1683755211yescharacteristic
401C4#1683756211yescharacteristic
402C4#1683761412yes—
403C4#1683763311yescharacteristic
404C4#1683765011yescharacteristic
405C4#1683778411yescharacteristic
406C4#1683779211yescharacteristic
407C2—9442no—
408C2#2142319212yes—
409C2#2142320612yes—
410C2#2142509711yescharacteristic
411C2#2142511611yescharacteristic
412C2#2142517311yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2568C2
socleC23×C59#120
prime radicalC2×#14273724C22
hypercenter#2568D59
hypocenterC59#10053
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2568

Complements

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

NH ≅ G/Ntype
4. C2×#16837148C2semidirect
5. C2×Q8⋊(C59⋊C8)C2semidirect
6. #21421937C2semidirect
7. C59×#2568C2semidirect
203. #2568D59semidirect
235. C59#10053semidirect

Maximal subgroups

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

subgroupindex
C2×#168377502
C2×#168376022
#214250232
C2×#168371482
C2×Q8⋊(C59⋊C8)2
#214219372
C59×#25682
#1005359