Finite Group Databasebeta

← #28481437

Subgroups — #28481437

Order 15104.

614 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21424076C211yesmaximal, characteristic
2C2×#16837139C212yesmaximal
3#21422323C211yesmaximal, characteristic
4#21421974C212yesmaximal
5C4×C22⋊Q8×C59C211yesmaximal, characteristic
6C2×C4⋊(C59⋊C8)—21no—
7C2×C22⋊(C59⋊C8)C2212yes—
8C2×C59⋊#95—21no—
9C2×C59⋊#95—21no—
10C2×C59⋊#95C2212yes—
11#16837157—21no—
12#16837139—24no—
13C2×C22⋊Q8×C59C2211yescharacteristic
14C2×C4×Q8×C59C411yescharacteristic, complemented
15C22×C42×C59C2211yescharacteristic
16C59×#391C411yescharacteristic, complemented
17C59×#389C411yescharacteristic, complemented
18C59×#387—21no—
19C59×#385—21no—
20C59×#380C2211yescharacteristic
21C4×C4⋊C4×C59—21no—
22C4×C4⋊C4×C59C411yescharacteristic, complemented
23C4×C22⋊C4×C59—21no—
24C23×Dic59—22no—
25C22×C59⋊C8—22no—
26C4⋊(C59⋊C8)—42no—
27C22⋊(C59⋊C8)—22no—
28C22⋊(C59⋊C8)—22no—
29C59⋊#95—24no—
30C59⋊#95—22no—
31C59⋊#95—22no—
32C59⋊#95—42no—
33C22×Q8×C59C2×C411yescharacteristic
34C23×C4×C59D411yescharacteristic
35C23×C4×C59D411yescharacteristic
36C23×C4×C59C2311yescharacteristic
37C22⋊Q8×C59—21no—
38C22⋊Q8×C59—21no—
39C22⋊Q8×C59D412yes—
40C22⋊Q8×C59Q812yes—
41C4×Q8×C59—42no—
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59C2×C411yescharacteristic
46C2×C4⋊C4×C59C2×C411yescharacteristic
47C2×C4⋊C4×C59C2×C411yescharacteristic
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C42×C59—21no—
51C2×C42×C59—21no—
52C2×C42×C59—42no—
53C2×C42×C59C2×C411yescharacteristic
54C2×C42×C59C2×C411yescharacteristic
55C2×C42×C59D411yescharacteristic
56C2×C42×C59D411yescharacteristic
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60#95×C59—21no—
61#95×C59—21no—
62C24×C59C2×D411yescharacteristic
63C2×Q8×C59—21no—
64C2×Q8×C59—41no—
65C2×Q8×C59—41no—
66C2×Q8×C59C4⋊C412yes—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—22no—
70C22×C4×C59—22no—
71C22×C4×C59—22no—
72C22×C4×C59—22no—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—41no—
76C22×C4×C59—41no—
77C22×C4×C59—41no—
78C22×C4×C59—41no—
79C22×C4×C59C22⋊C411yescharacteristic
80C22×C4×C59C22⋊C411yescharacteristic
81C22×C4×C59C22⋊C411yescharacteristic
82C22×C4×C59C22⋊C411yescharacteristic
83C22×C4×C59C22×C411yescharacteristic
84C22×C4×C59C2×D411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×Q811yescharacteristic
87C22×C4×C59#4111yescharacteristic
88C22×C4×C59#4111yescharacteristic
89C22×Dic59—24no—
90C22×Dic59—42no—
91C22×Dic59—42no—
92C22×Dic59—42no—
93C22×Dic59—42no—
94C22×Dic59—42no—
95C22×Dic59—42no—
96C4⋊C4×C59—21no—
97C4⋊C4×C59—22no—
98C4⋊C4×C59—22no—
99C4⋊C4×C59—42no—
100C4⋊C4×C59—41no—
101C4⋊C4×C59—41no—
102C4⋊C4×C59—42no—
103C4⋊C4×C59C4⋊C412yes—
104C22⋊C4×C59—22no—
105C22⋊C4×C59—22no—
106C22⋊C4×C59—42no—
107C42×C59—22no—
108C42×C59—22no—
109C42×C59—41no—
110C42×C59—41no—
111C42×C59—42no—
112C42×C59—42no—
113C2×C59⋊C8—42no—
114C2×C59⋊C8—42no—
115C2×C59⋊C8—42no—
116C23×C59—21no—
117C23×C59—21no—
118C23×C59—21no—
119C23×C59—21no—
120C23×C59C2×C22⋊C411yescharacteristic
121C23×C59#12111yescharacteristic
122C23×C59C22⋊Q811yescharacteristic
123C23×C59#13512yes—
124C23×C59#13712yes—
125Q8×C59—42no—
126Q8×C59—42no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—22no—
136C2×C4×C59—22no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—44no—
148C2×C4×C59—44no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—41no—
154C2×C4×C59—41no—
155C2×C4×C59—41no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59#9912yes—
160C2×C4×C59#10412yes—
161C2×C4×C59C2×C4⋊C411yescharacteristic
162C2×C4×C59#11711yescharacteristic
163C2×C4×C59#12111yescharacteristic
164C2×C4×C59C22⋊Q811yescharacteristic
165C2×Dic59—44no—
166C2×Dic59—44no—
167C2×Dic59—44no—
168C2×Dic59—82no—
169C2×Dic59—82no—
170C2×Dic59—82no—
171C2×Dic59—82no—
172C59⋊C8—82no—
173#8576—591nomaximal, Sylow
174C22×C59—21no—
175C22×C59—21no—
176C22×C59—21no—
177C22×C59—21no—
178C22×C59—22no—
179C22×C59—22no—
180C22×C59—22no—
181C22×C59—22no—
182C22×C59#38011yescharacteristic
183C22×C59C2×#9911yescharacteristic
184C22×C59#41011yescharacteristic
185C22×C59#41611yescharacteristic
186C22×C59#41911yescharacteristic
187C22×C59C2×#10411yescharacteristic
188C22×C59#42111yescharacteristic
189C22×C59#46412yes—
190C22×C59#47912yes—
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—21no—
194C4×C59—21no—
195C4×C59—22no—
196C4×C59—22no—
197C4×C59—42no—
198C4×C59—42no—
199C4×C59—41no—
200C4×C59—41no—
201C4×C59—81no—
202C4×C59—81no—
203Dic59—84no—
204C4×C22⋊Q8D5911yescharacteristic
205#1797—591no—
206C2×#328—592no—
207#1434—591no—
208#1294—592no—
209C2×C59—22no—
210C2×C59—22no—
211C2×C59#150111yescharacteristic
212C2×C59#150611yescharacteristic
213C2×C59#151111yescharacteristic
214C2×C59#152111yescharacteristic
215C2×C59#179911yescharacteristic
216C2×C59#180511yescharacteristic
217C2×C59#180611yescharacteristic
218C2×C22⋊Q8C2×D5911yescharacteristic
219C2×C4×Q8Dic5911yescharacteristic, complemented
220C22×C42C2×D5911yescharacteristic
221C2×C4⋊C8—1181no—
222C2×C22⋊C8—592no—
223#391Dic5911yescharacteristic, complemented
224#389Dic5911yescharacteristic, complemented
225#387—21no—
226#385—21no—
227#380C2×D5911yescharacteristic
228C4×C4⋊C4—21no—
229C4×C4⋊C4Dic5911yescharacteristic, complemented
230C4×C22⋊C4—21no—
231C2×#95—592no—
232C2×#95—1181no—
233C2×#95—1181no—
234#336—1181no—
235#328—1184no—
236C59#857611yescharacteristic, complemented, Sylow
237C22×Q8C2×Dic5911yescharacteristic
238C23×C4—1182no—
239C23×C4C22⋊D5911yescharacteristic
240C23×C4C22⋊D5911yescharacteristic
241C23×C4C22×D5911yescharacteristic
242C22×C8—1182no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C22⋊Q8Dic11812yes—
246C22⋊Q8D23612yes—
247C4×Q8—42no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C2×Dic5911yescharacteristic
252C2×C4⋊C4C2×Dic5911yescharacteristic
253C2×C4⋊C4C2×Dic5911yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—42no—
259C2×C42C2×Dic5911yescharacteristic
260C2×C42C2×Dic5911yescharacteristic
261C2×C42C22⋊D5911yescharacteristic
262C2×C42C22⋊D5911yescharacteristic
263C4⋊C8—2362no—
264C22⋊C8—1182no—
265C22⋊C8—1182no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—1184no—
272#95—1182no—
273#95—1182no—
274#95—2362no—
275C24C2×C22⋊D5911yescharacteristic
276C2×Q8—21no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8C4⋊Dic5912yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—1184no—
293C22×C4—2362no—
294C22×C4—2362no—
295C22×C4—2362no—
296C22×C4—2362no—
297C22×C4—2362no—
298C22×C4—2362no—
299C22×C4C22⋊Dic5911yescharacteristic
300C22×C4C22⋊Dic5911yescharacteristic
301C22×C4C22⋊Dic5911yescharacteristic
302C22×C4C22⋊Dic5911yescharacteristic
303C22×C4C2×Dic11811yescharacteristic
304C22×C4C2×D23611yescharacteristic
305C22×C4#1174568511yescharacteristic
306C22×C4#1174568511yescharacteristic
307C22×C4C22×Dic5911yescharacteristic
308C22×C4C2×C22⋊D5911yescharacteristic
309C2×C8—2362no—
310C2×C8—2362no—
311C2×C8—2362no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4C4⋊Dic5912yes—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—42no—
323C42—22no—
324C42—22no—
325C42—41no—
326C42—41no—
327C42—42no—
328C42—42no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23C2×C22⋊Dic5911yescharacteristic
334C23#1427370111yescharacteristic
335C23#1427370611yescharacteristic
336C23#1427374712yes—
337C23#1427375812yes—
338Q8—42no—
339Q8—42no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—4722no—
373C2×C4—4722no—
374C2×C4—4722no—
375C2×C4—4722no—
376C2×C4—2364no—
377C2×C4—2364no—
378C2×C4—2364no—
379C2×C4#1427364212yes—
380C2×C4#1427365412yes—
381C2×C4C2×C4⋊Dic5911yescharacteristic
382C2×C4#1427368611yescharacteristic
383C2×C4#1427370111yescharacteristic
384C2×C4#1427370611yescharacteristic
385C8—4722no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22#1683726211yescharacteristic
395C22C2×#1427364211yescharacteristic
396C22#1683737611yescharacteristic
397C22#1683739811yescharacteristic
398C22#1683741111yescharacteristic
399C22C2×#1427365411yescharacteristic
400C22#1683741911yescharacteristic
401C22#1683762412yes—
402C22#1683770912yes—
403C4—22no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—22no—
409C4—41no—
410C4—42no—
411C4—42no—
412C4—41no—
413C4—81no—
414C4—81no—
415C4—4724no—
416C2—22no—
417C2—22no—
418C2#2142258711yescharacteristic
419C2#2142260811yescharacteristic
420C2#2142262911yescharacteristic
421C2#2142267311yescharacteristic
422C2#2142408411yescharacteristic
423C2#2142411111yescharacteristic
424C2#2142411611yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC4×C22⋊Q8×C59C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenterC4×C22⋊Q8D59
hypocenterC59#8576
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
14. C2×C4×Q8×C59C4semidirect
16. C59×#391C4semidirect
17. C59×#389C4semidirect
22. C4×C4⋊C4×C59C4semidirect
219. C2×C4×Q8Dic59semidirect
223. #391Dic59semidirect
224. #389Dic59semidirect
229. C4×C4⋊C4Dic59semidirect
236. C59#8576semidirect

Maximal subgroups

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

subgroupindex
#214240762
C2×#168371392
#214223232
#214219742
C4×C22⋊Q8×C592
#857659