Finite Group Databasebeta

← #28488769

Subgroups — #28488769

Order 15104.

598 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21423858C211yesmaximal, characteristic
2#21422326C211yesmaximal, characteristic
3#21421974C212yesmaximal
4#21421974C212yesmaximal
5C59×#2851C211yesmaximal, characteristic
6C22×C4⋊Dic59—21no—
7C2×C22⋊(C59⋊C8)C2212yes—
8C2×#14273638—21no—
9C2×C59⋊#95—21no—
10C2×C59⋊#95C2212yes—
11#16837157—21no—
12C2×#125×C59C411yescharacteristic, complemented
13C22×C42×C59C2211yescharacteristic
14C59×#389C411yescharacteristic, complemented
15C59×#389C411yescharacteristic, complemented
16C59×#387—21no—
17C59×#387—21no—
18C59×#385—21no—
19C59×#385—21no—
20C59×#382C411yescharacteristic, complemented
21C59×#380C2211yescharacteristic
22C59×#380C2211yescharacteristic
23C23×Dic59—22no—
24C22×C59⋊C8—22no—
25C2×C4⋊Dic59—24no—
26C2×C4⋊Dic59—42no—
27C2×C4⋊Dic59—42no—
28C22⋊(C59⋊C8)—22no—
29C22⋊(C59⋊C8)Q814yes—
30#14273638—42no—
31C59⋊#95—24no—
32C59⋊#95—42no—
33C23×C4×C59D411yescharacteristic
34C23×C4×C59D411yescharacteristic
35C23×C4×C59C2311yescharacteristic
36#125×C59—42no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59C2×C411yescharacteristic
42C2×C4⋊C4×C59C2×C411yescharacteristic
43C2×C4⋊C4×C59C2×C411yescharacteristic
44C2×C4⋊C4×C59C2×C411yescharacteristic
45C2×C22⋊C4×C59—21no—
46C2×C22⋊C4×C59—21no—
47C2×C42×C59—42no—
48C2×C42×C59C2×C411yescharacteristic
49C2×C42×C59C2×C411yescharacteristic
50C2×C42×C59D411yescharacteristic
51C2×C42×C59D411yescharacteristic
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—21no—
55#95×C59—21no—
56#95×C59—21no—
57#95×C59—21no—
58C24×C59C2×D411yescharacteristic
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—22no—
62C22×C4×C59—22no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—41no—
70C22×C4×C59—41no—
71C22×C4×C59—41no—
72C22×C4×C59—41no—
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22×C411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×Q812yes—
80C22×C4×C59#4112yes—
81C22×Dic59—24no—
82C22×Dic59—42no—
83C22×Dic59—42no—
84C22×Dic59—42no—
85C22×Dic59—42no—
86C22×Dic59—42no—
87C22×Dic59—42no—
88C4⋊C4×C59—42no—
89C4⋊C4×C59—42no—
90C4⋊C4×C59—41no—
91C4⋊C4×C59—41no—
92C4⋊C4×C59—41no—
93C4⋊C4×C59—41no—
94C4⋊C4×C59—42no—
95C4⋊C4×C59—42no—
96C22⋊C4×C59—41no—
97C22⋊C4×C59—41no—
98C22⋊C4×C59—41no—
99C22⋊C4×C59—41no—
100C42×C59—22no—
101C42×C59—22no—
102C42×C59—41no—
103C42×C59—41no—
104C4⋊Dic59—48no—
105C2×C59⋊C8—24no—
106C2×C59⋊C8—42no—
107C2×C59⋊C8—42no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59—21no—
112C23×C59C2×C22⋊C411yescharacteristic
113C23×C59#12411yescharacteristic
114C23×C59C4⋊Q811yescharacteristic
115C23×C59AGL(1,ℤ/8)11yescharacteristic
116C23×C59AGL(1,ℤ/8)11yescharacteristic
117C23×C59#13711yescharacteristic
118C23×C59#13711yescharacteristic
119C2×C4×C59—24no—
120C2×C4×C59—24no—
121C2×C4×C59—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—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—42no—
136C2×C4×C59—42no—
137C2×C4×C59—42no—
138C2×C4×C59—42no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59#9912yes—
152C2×C4×C59#10412yes—
153C2×C4×C59C4×Q812yes—
154C2×C4×C59C22⋊Q812yes—
155C2×Dic59—44no—
156C2×Dic59—44no—
157C2×Dic59—44no—
158C2×Dic59—82no—
159C2×Dic59—82no—
160C2×Dic59—82no—
161C2×Dic59—82no—
162C59⋊C8—44no—
163#9887—591nomaximal, Sylow
164C22×C59—21no—
165C22×C59—21no—
166C22×C59—21no—
167C22×C59—21no—
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—21no—
171C22×C59—21no—
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—21no—
176C22×C59#39111yescharacteristic
177C22×C59C2×#9911yescharacteristic
178C22×C59#41011yescharacteristic
179C22×C59#41711yescharacteristic
180C22×C59#41711yescharacteristic
181C22×C59C2×#10411yescharacteristic
182C22×C59#42111yescharacteristic
183C22×C59#49011yescharacteristic
184C22×C59#49111yescharacteristic
185C22×C59#50111yescharacteristic
186C22×C59#50111yescharacteristic
187C4×C59—22no—
188C4×C59—22no—
189C4×C59—22no—
190C4×C59—22no—
191C4×C59—41no—
192C4×C59—41no—
193C4×C59—81no—
194C4×C59—81no—
195C4×C59—81no—
196C4×C59—81no—
197Dic59—84no—
198#2851D5911yescharacteristic
199#1749—591no—
200C4⋊#95—591no—
201#1294—592no—
202#1294—592no—
203C2×C59—21no—
204C2×C59—21no—
205C2×C59—21no—
206C2×C59—21no—
207C2×C59#150411yescharacteristic
208C2×C59#150911yescharacteristic
209C2×C59#151211yescharacteristic
210C2×C59#151311yescharacteristic
211C2×C59#198611yescharacteristic
212C2×C59#199211yescharacteristic
213C2×C59#199611yescharacteristic
214C2×#125Dic5911yescharacteristic, complemented
215C22×C4⋊C4—1181no—
216C22×C42C2×D5911yescharacteristic
217C2×C22⋊C8—592no—
218C2×#97—1181no—
219#389Dic5911yescharacteristic, complemented
220#389Dic5911yescharacteristic, complemented
221#387—21no—
222#387—21no—
223#385—21no—
224#385—21no—
225#382Dic5911yescharacteristic, complemented
226#380C2×D5911yescharacteristic
227#380C2×D5911yescharacteristic
228C2×#95—592no—
229C2×#95—1181no—
230#336—1181no—
231C59#988711yescharacteristic, complemented, Sylow
232C23×C4—1182no—
233C23×C4C22⋊D5911yescharacteristic
234C23×C4C22⋊D5911yescharacteristic
235C23×C4C22×D5911yescharacteristic
236C22×C8—1182no—
237#125—42no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—1184no—
243C2×C4⋊C4—2362no—
244C2×C4⋊C4—2362no—
245C2×C4⋊C4C2×Dic5911yescharacteristic
246C2×C4⋊C4C2×Dic5911yescharacteristic
247C2×C4⋊C4C2×Dic5911yescharacteristic
248C2×C4⋊C4C2×Dic5911yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42—42no—
252C2×C42C2×Dic5911yescharacteristic
253C2×C42C2×Dic5911yescharacteristic
254C2×C42C22⋊D5911yescharacteristic
255C2×C42C22⋊D5911yescharacteristic
256C22⋊C8—594no—
257C22⋊C8—1182no—
258#97—2362no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—1184no—
266#95—2362no—
267C24C2×C22⋊D5911yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—1184no—
283C22×C4—2362no—
284C22×C4—2362no—
285C22×C4—2362no—
286C22×C4—2362no—
287C22×C4—2362no—
288C22×C4—2362no—
289C22×C4C22⋊Dic5911yescharacteristic
290C22×C4C22⋊Dic5911yescharacteristic
291C22×C4C22⋊Dic5911yescharacteristic
292C22×C4C22⋊Dic5911yescharacteristic
293C22×C4Q8×D5912yes—
294C22×C4#1174568912yes—
295C22×C4C22×Dic5911yescharacteristic
296C22×C4C2×C22⋊D5911yescharacteristic
297C2×C8—1184no—
298C2×C8—2362no—
299C2×C8—2362no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—2368no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—22no—
314C42—22no—
315C42—41no—
316C42—41no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×C22⋊Dic5911yescharacteristic
322C23#1427371611yescharacteristic
323C23#1427372511yescharacteristic
324C23#1427375211yescharacteristic
325C23#1427375211yescharacteristic
326C23#1427375711yescharacteristic
327C23#1427375711yescharacteristic
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—4722no—
361C2×C4—4722no—
362C2×C4—4722no—
363C2×C4—4722no—
364C2×C4—2364no—
365C2×C4—2364no—
366C2×C4—2364no—
367C2×C4#1427364212yes—
368C2×C4#1427365412yes—
369C2×C4Q8×Dic5912yes—
370C2×C4#1427370312yes—
371C8—2364no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22#1683730911yescharacteristic
385C22C2×#1427364211yescharacteristic
386C22#1683737611yescharacteristic
387C22#1683740311yescharacteristic
388C22#1683740311yescharacteristic
389C22C2×#1427365411yescharacteristic
390C22#1683741911yescharacteristic
391C22#1683776811yescharacteristic
392C22#1683777211yescharacteristic
393C22#1683781211yescharacteristic
394C22#1683781211yescharacteristic
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—41no—
400C4—41no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—4724no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2#2142260011yescharacteristic
411C2#2142262111yescharacteristic
412C2#2142263411yescharacteristic
413C2#2142263911yescharacteristic
414C2#2142504111yescharacteristic
415C2#2142506611yescharacteristic
416C2#2142508211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2851C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenter#2851D59
hypocenterC59#9887
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
12. C2×#125×C59C4semidirect
14. C59×#389C4semidirect
15. C59×#389C4semidirect
20. C59×#382C4semidirect
214. C2×#125Dic59semidirect
219. #389Dic59semidirect
220. #389Dic59semidirect
225. #382Dic59semidirect
231. C59#9887semidirect

Maximal subgroups

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

subgroupindex
#214238582
#214223262
#214219742
C59×#28512
#988759