Finite Group Databasebeta

← #28486010

Subgroups — #28486010

Order 15104.

640 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433120C211yesmaximal, characteristic, complemented
2#21424084C211yesmaximal, characteristic, complemented
3#21422664C212yesmaximal, complemented
4#21421975C212yesmaximal, complemented
5C59×#2543C211yesmaximal, characteristic
6D4×C59⋊C8—24no—
7#16837455—21no—
8C2×#14273642—21no—
9C2×C22⋊(C59⋊C8)—21no—
10C2×C22⋊(C59⋊C8)C2212yes—
11#16837262C2212yes—
12#16837169—21no—
13#16837136—24no—
14C2×#121×C59—21no—
15C2×#121×C59C2211yescharacteristic
16C2×C4×D4×C59C2211yescharacteristic
17C59×#398C411yescharacteristic, complemented
18C59×#397C411yescharacteristic, complemented
19C59×#394C411yescharacteristic, complemented
20C59×#392C411yescharacteristic, complemented
21C59×#388—21no—
22C59×#386—21no—
23C59×#386—21no—
24C2×#95×C59C2211yescharacteristic
25C22×C59⋊C8—22no—
26C22×C59⋊C8—22no—
27C2×C4⋊Dic59—22no—
28C2×C22⋊Dic59—22no—
29C4⋊(C59⋊C8)—22no—
30#14273642—42no—
31C22⋊(C59⋊C8)—22no—
32C22⋊(C59⋊C8)—22no—
33C22⋊(C59⋊C8)D414yescomplemented
34C4×C59⋊C8—22no—
35C59⋊#95—22no—
36C22×D4×C59—21no—
37C22×D4×C59—21no—
38C22×D4×C59C2×C411yescharacteristic
39C22×D4×C59D411yescharacteristic
40C23×C4×C59—21no—
41C23×C4×C59D411yescharacteristic
42C23×C4×C59C2311yescharacteristic
43#121×C59—22no—
44#121×C59—22no—
45#121×C59—42no—
46#121×C59—42no—
47C4×D4×C59—22no—
48C4×D4×C59—22no—
49C2×C4⋊C4×C59—21no—
50C2×C4⋊C4×C59C2×C411yescharacteristic
51C2×C4⋊C4×C59C2×C411yescharacteristic
52C2×C22⋊C4×C59—21no—
53C2×C22⋊C4×C59—21no—
54C2×C22⋊C4×C59—21no—
55C2×C22⋊C4×C59—21no—
56C2×C22⋊C4×C59—21no—
57C2×C22⋊C4×C59C2×C411yescharacteristic
58C2×C22⋊C4×C59D411yescharacteristic
59C2×C42×C59D411yescharacteristic
60#95×C59—21no—
61#95×C59—21no—
62#95×C59—21no—
63#95×C59C2×C411yescharacteristic
64#95×C59C2×C411yescharacteristic
65C24×C59—21no—
66C24×C59C22⋊C411yescharacteristic
67C24×C59C2×D411yescharacteristic
68C2×D4×C59—22no—
69C2×D4×C59—22no—
70C2×D4×C59—22no—
71C2×D4×C59—42no—
72C2×D4×C59—42no—
73C2×D4×C59—42no—
74C2×D4×C59—41no—
75C2×D4×C59—41no—
76C2×D4×C59—42no—
77C2×D4×C59—81no—
78C2×D4×C59—81no—
79C2×D4×C59D812yes—
80C2×D4×C59SD1612yes—
81C22×C4×C59—21no—
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—21no—
85C22×C4×C59—21no—
86C22×C4×C59—42no—
87C22×C4×C59—41no—
88C22×C4×C59—41no—
89C22×C4×C59—81no—
90C22×C4×C59—81no—
91C22×C4×C59—81no—
92C22×C4×C59C22⋊C411yescharacteristic
93C22×C4×C59C22⋊C411yescharacteristic
94C22×C4×C59C22⋊C411yescharacteristic
95C22×C4×C59C22×C411yescharacteristic
96C22×C4×C59C2×D412yes—
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59#4112yes—
99C22×Dic59—42no—
100C4⋊C4×C59—22no—
101C4⋊C4×C59—22no—
102C4⋊C4×C59—42no—
103C22⋊C4×C59—22no—
104C22⋊C4×C59—22no—
105C22⋊C4×C59—42no—
106C22⋊C4×C59—42no—
107C22⋊C4×C59—42no—
108C22⋊C4×C59—41no—
109C22⋊C4×C59—41no—
110C22⋊C4×C59—81no—
111C42×C59—21no—
112C42×C59—21no—
113C22⋊Dic59—42no—
114C22⋊Dic59—42no—
115C4⋊Dic59—82no—
116C2×C59⋊C8—24no—
117C2×C59⋊C8—82no—
118C2×C59⋊C8—82no—
119C23×C59—21no—
120C23×C59—21no—
121C23×C59—41no—
122C23×C59—41no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59—41no—
126C23×C59—41no—
127C23×C59—41no—
128C23×C59—41no—
129C23×C59—41no—
130C23×C59—41no—
131C23×C59—41no—
132C23×C59#10212yes—
133C23×C59C2×C22⋊C411yescharacteristic
134C23×C59#12112yes—
135C23×C59C2×D811yescharacteristic
136C23×C59C2×SD1611yescharacteristic
137C23×C59#13512yes—
138D4×C59—24no—
139D4×C59—24no—
140D4×C59—82no—
141D4×C59—84no—
142D4×C59—82no—
143D4×C59—82no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—22no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—42no—
150C2×C4×C59—42no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—42no—
154C2×C4×C59—41no—
155C2×C4×C59—41no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—81no—
165C2×C4×C59—82no—
166C2×C4×C59—82no—
167C2×C4×C59#10212yes—
168C2×C4×C59C4×D412yes—
169C2×C4×C59#12011yescharacteristic
170C2×C4×C59#12311yescharacteristic
171C2×Dic59—82no—
172C2×Dic59—82no—
173C2×Dic59—82no—
174C59⋊C8—44no—
175#9384—591nomaximal, Sylow
176C22×C59—22no—
177C22×C59—21no—
178C22×C59—22no—
179C22×C59—21no—
180C22×C59—22no—
181C22×C59—41no—
182C22×C59—41no—
183C22×C59—42no—
184C22×C59—42no—
185C22×C59—41no—
186C22×C59—41no—
187C22×C59—81no—
188C22×C59—81no—
189C22×C59—81no—
190C22×C59—81no—
191C22×C59—81no—
192C22×C59—81no—
193C22×C59—81no—
194C22×C59—81no—
195C22×C59#38611yescharacteristic
196C22×C59C2×#10211yescharacteristic
197C22×C59#41611yescharacteristic
198C22×C59#46512yes—
199C22×C59#46612yes—
200C4×C59—22no—
201C4×C59—21no—
202C4×C59—21no—
203C4×C59—22no—
204C4×C59—42no—
205C4×C59—41no—
206C4×C59—41no—
207C4×C59—81no—
208C4×C59—81no—
209Dic59—162no—
210#2972—591no—
211#2543D5911yescharacteristic
212#1799—591no—
213#1519—592no—
214#1294—592no—
215C2×C59—22no—
216C2×C59—21no—
217C2×C59—21no—
218C2×C59—22no—
219C2×C59—42no—
220C2×C59—161no—
221C2×C59#188411yescharacteristic
222C2×C59#189311yescharacteristic
223C2×C59#191311yescharacteristic
224C2×#121—21no—
225C2×#121C2×D5911yescharacteristic
226C2×C4×D4C2×D5911yescharacteristic
227C8×D4—1184no—
228#431—1181no—
229C2×#99—1181no—
230C2×C22⋊C8—592no—
231C2×C22⋊C8—1181no—
232#398Dic5911yescharacteristic, complemented
233#397Dic5911yescharacteristic, complemented
234#394Dic5911yescharacteristic, complemented
235#392Dic5911yescharacteristic, complemented
236#388—21no—
237#386—21no—
238#386—21no—
239#380—592no—
240C2×#95C2×D5911yescharacteristic
241#342—1181no—
242#327—1184no—
243C59#938411yescharacteristic, complemented, Sylow
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2×Dic5911yescharacteristic
247C22×D4C22⋊D5911yescharacteristic
248C23×C4—21no—
249C23×C4C22⋊D5911yescharacteristic
250C23×C4C22×D5911yescharacteristic
251C22×C8—1182no—
252C22×C8—1182no—
253#121—22no—
254#121—22no—
255#121—42no—
256#121—42no—
257C4×D4—22no—
258C4×D4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—1182no—
261C2×C4⋊C4C2×Dic5911yescharacteristic
262C2×C4⋊C4C2×Dic5911yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—1182no—
269C2×C22⋊C4C2×Dic5911yescharacteristic
270C2×C22⋊C4C22⋊D5911yescharacteristic
271C2×C42C22⋊D5911yescharacteristic
272C4⋊C8—1182no—
273#99—2362no—
274C22⋊C8—594no—
275C22⋊C8—1182no—
276C22⋊C8—1182no—
277C4×C8—1182no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—1182no—
282#95C2×Dic5911yescharacteristic
283#95C2×Dic5911yescharacteristic
284C24—21no—
285C24C22⋊Dic5911yescharacteristic
286C24C2×C22⋊D5911yescharacteristic
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—81no—
297C2×D4—81no—
298C2×D4#1174566912yes—
299C2×D4#1174567012yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4—2362no—
312C22×C4C22⋊Dic5911yescharacteristic
313C22×C4C22⋊Dic5911yescharacteristic
314C22×C4C22⋊Dic5911yescharacteristic
315C22×C4D4×D5912yes—
316C22×C4#1174568712yes—
317C22×C4C22×Dic5911yescharacteristic
318C22×C4C2×C22⋊D5911yescharacteristic
319C2×C8—1184no—
320C2×C8—4722no—
321C2×C8—4722no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—42no—
325C4⋊C4—4722no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—81no—
334C22⋊C4—2362no—
335C22⋊C4—2362no—
336C42—21no—
337C42—21no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23#1427364812yes—
352C23C2×C22⋊Dic5911yescharacteristic
353C23#1427369912yes—
354C23C2×#1174566911yescharacteristic
355C23C2×#1174567011yescharacteristic
356C23#1427374712yes—
357D4—24no—
358D4—24no—
359D4—82no—
360D4—84no—
361D4—82no—
362D4—82no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—4722no—
387C2×C4—4722no—
388C2×C4—4722no—
389C2×C4#1427364812yes—
390C2×C4D4×Dic5912yes—
391C2×C4#1427369511yescharacteristic
392C2×C4#1427370811yescharacteristic
393C8—2364no—
394C22—21no—
395C22—22no—
396C22—21no—
397C22—22no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22#1683728211yescharacteristic
414C22C2×#1427364811yescharacteristic
415C22#1683739811yescharacteristic
416C22#1683762912yes—
417C22#1683763612yes—
418C4—22no—
419C4—21no—
420C4—21no—
421C4—22no—
422C4—42no—
423C4—41no—
424C4—41no—
425C4—81no—
426C4—81no—
427C4—9442no—
428C2—21no—
429C2—22no—
430C2—21no—
431C2—22no—
432C2—42no—
433C2—161no—
434C2#2142448011yescharacteristic
435C2#2142452511yescharacteristic
436C2#2142463911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
Fitting, prime radicalC59×#2543C2
socleC22×C59#386
cosocleC23×C4×C59C23
hypercenter#2543D59
hypocenterC59#9384
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2543

Complements

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

NH ≅ G/Ntype
1. #21433120C2semidirect
2. #21424084C2semidirect
3. #21422664C2semidirect
4. #21421975C2semidirect
17. C59×#398C4semidirect
18. C59×#397C4semidirect
19. C59×#394C4semidirect
20. C59×#392C4semidirect
33. C22⋊(C59⋊C8)D4semidirect
232. #398Dic59semidirect
233. #397Dic59semidirect
234. #394Dic59semidirect
235. #392Dic59semidirect
243. C59#9384semidirect

Maximal subgroups

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

subgroupindex
#214331202
#214240842
#214226642
#214219752
C59×#25432
#938459