Finite Group Databasebeta

← #28499255

Subgroups — #28499255

Order 15104.

532 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14273642C211yesmaximal, characteristic, complemented
2#21422437C211yesmaximal, characteristic, complemented
3#21422096C212yesmaximal, complemented
4#21422093C212yesmaximal, complemented
5C59×#2543C211yesmaximal, characteristic
6#16837459—21no—
7C2×#14273642—21no—
8C4×C22⋊Dic59—21no—
9#16837169C2212yes—
10#16837126—21no—
11#16837126C2212yes—
12C2×#121×C59—21no—
13C2×#121×C59C411yescharacteristic, complemented
14C2×C4×D4×C59C2211yescharacteristic
15C59×#398C411yescharacteristic, complemented
16C59×#397C2211yescharacteristic
17C59×#394C411yescharacteristic, complemented
18C59×#392C2211yescharacteristic
19C59×#388—21no—
20C59×#386—21no—
21C59×#386—21no—
22C2×#95×C59C411yescharacteristic, complemented
23C2×C22⋊Dic59—22no—
24C2×C4×Dic59—41no—
25C4⋊(C59⋊C8)—41no—
26#14273642—24no—
27C22⋊(C59⋊C8)—22no—
28#14273638—41no—
29C59⋊#95—41no—
30C22×D4×C59—21no—
31C22×D4×C59—21no—
32C22×D4×C59C2×C411yescharacteristic
33C22×D4×C59D411yescharacteristic
34C23×C4×C59—21no—
35C23×C4×C59C2×C411yescharacteristic
36C23×C4×C59D411yescharacteristic
37#121×C59—44no—
38#121×C59—41no—
39#121×C59—41no—
40C4×D4×C59—41no—
41C4×D4×C59—41no—
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59C2×C411yescharacteristic
44C2×C4⋊C4×C59C2×C411yescharacteristic
45C2×C22⋊C4×C59—21no—
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59D411yescharacteristic
51C2×C22⋊C4×C59C2311yescharacteristic
52C2×C42×C59D411yescharacteristic
53#95×C59—21no—
54#95×C59—21no—
55#95×C59—21no—
56#95×C59C2×C411yescharacteristic
57#95×C59C2×C411yescharacteristic
58C24×C59—21no—
59C24×C59C22⋊C411yescharacteristic
60C24×C59C2×D411yescharacteristic
61C2×D4×C59—22no—
62C2×D4×C59—22no—
63C2×D4×C59—41no—
64C2×D4×C59—42no—
65C2×D4×C59—41no—
66C2×D4×C59—42no—
67C2×D4×C59—42no—
68C2×D4×C59—42no—
69C2×D4×C59—42no—
70C2×D4×C59—81no—
71C2×D4×C59—81no—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—21no—
78C22×C4×C59—21no—
79C22×C4×C59—42no—
80C22×C4×C59—42no—
81C22×C4×C59—81no—
82C22×C4×C59—81no—
83C22×C4×C59—81no—
84C22×C4×C59C22⋊C411yescharacteristic
85C22×C4×C59C22⋊C411yescharacteristic
86C22×C4×C59C22⋊C411yescharacteristic
87C22×C4×C59C22×C411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×Dic59—42no—
90C4⋊C4×C59—41no—
91C4⋊C4×C59—42no—
92C4⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—42no—
96C22⋊C4×C59—42no—
97C22⋊C4×C59—42no—
98C22⋊C4×C59—41no—
99C22⋊C4×C59—41no—
100C22⋊C4×C59—81no—
101C42×C59—21no—
102C42×C59—21no—
103C22⋊Dic59—24no—
104C22⋊Dic59—42no—
105C4×Dic59—82no—
106C2×C59⋊C8—42no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59—41no—
121C23×C59—41no—
122C23×C59#10412yes—
123C23×C59C2×C22⋊C411yescharacteristic
124C23×C59AGL(1,ℤ/8)12yes—
125D4×C59—44no—
126D4×C59—84no—
127D4×C59—82no—
128D4×C59—84no—
129C2×C4×C59—22no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—41no—
137C2×C4×C59—42no—
138C2×C4×C59—41no—
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—81no—
150C2×C4×C59—81no—
151C2×C4×C59—81no—
152C2×C4×C59—84no—
153C2×C4×C59#9911yescharacteristic
154C2×C4×C59#9911yescharacteristic
155C2×Dic59—44no—
156C2×Dic59—82no—
157C2×Dic59—82no—
158C59⋊C8—82no—
159#11775—591nomaximal, Sylow
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—22no—
163C22×C59—22no—
164C22×C59—42no—
165C22×C59—42no—
166C22×C59—42no—
167C22×C59—41no—
168C22×C59—41no—
169C22×C59—41no—
170C22×C59—41no—
171C22×C59—81no—
172C22×C59—81no—
173C22×C59—81no—
174C22×C59—81no—
175C22×C59—81no—
176C22×C59—81no—
177C22×C59—81no—
178C22×C59—81no—
179C22×C59C2×#9911yescharacteristic
180C22×C59#41811yescharacteristic
181C22×C59C2×#10411yescharacteristic
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—42no—
185C4×C59—41no—
186C4×C59—41no—
187C4×C59—41no—
188C4×C59—41no—
189C4×C59—81no—
190C4×C59—81no—
191Dic59—84no—
192#2543D5911yescharacteristic
193C4×#99—591no—
194#1466—591no—
195#1344—592no—
196#1343—592no—
197C2×C59—21no—
198C2×C59—21no—
199C2×C59—42no—
200C2×C59—42no—
201C2×C59—161no—
202C2×C59#150711yescharacteristic
203C2×C59#212111yescharacteristic
204C2×C59#212611yescharacteristic
205C2×#121—21no—
206C2×#121Dic5911yescharacteristic, complemented
207C2×C4×D4C2×D5911yescharacteristic
208#432—1181no—
209C2×#99—1181no—
210#398Dic5911yescharacteristic, complemented
211#397C2×D5911yescharacteristic
212#394Dic5911yescharacteristic, complemented
213#392C2×D5911yescharacteristic
214#388—21no—
215#386—21no—
216#386—21no—
217C4×C22⋊C4—1181no—
218C2×#95Dic5911yescharacteristic, complemented
219#342—592no—
220#323—592no—
221#323—1181no—
222C59#1177511yescharacteristic, complemented, Sylow
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C2×Dic5911yescharacteristic
226C22×D4C22⋊D5911yescharacteristic
227C23×C4—21no—
228C23×C4C2×Dic5911yescharacteristic
229C23×C4C22⋊D5911yescharacteristic
230#121—44no—
231#121—41no—
232#121—41no—
233C4×D4—41no—
234C4×D4—41no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C2×Dic5911yescharacteristic
237C2×C4⋊C4C2×Dic5911yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—1182no—
244C2×C22⋊C4C22⋊D5911yescharacteristic
245C2×C22⋊C4C22×D5911yescharacteristic
246C2×C42—2361no—
247C2×C42C22⋊D5911yescharacteristic
248C4⋊C8—2361no—
249#99—1184no—
250C22⋊C8—1182no—
251#97—2361no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—2361no—
256#95C2×Dic5911yescharacteristic
257#95C2×Dic5911yescharacteristic
258C24—21no—
259C24C22⋊Dic5911yescharacteristic
260C24C2×C22⋊D5911yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—81no—
271C2×D4—81no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—2362no—
285C22×C4C22⋊Dic5911yescharacteristic
286C22×C4C22⋊Dic5911yescharacteristic
287C22×C4C22⋊Dic5911yescharacteristic
288C22×C4C22×Dic5911yescharacteristic
289C22×C4C2×C22⋊D5911yescharacteristic
290C2×C8—2362no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—81no—
302C22⋊C4—1184no—
303C22⋊C4—2362no—
304C42—21no—
305C42—21no—
306C42—4722no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23#1427365412yes—
323C23C2×C22⋊Dic5911yescharacteristic
324C23#1427375312yes—
325D4—44no—
326D4—84no—
327D4—82no—
328D4—84no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—84no—
353C2×C4—4722no—
354C2×C4—4722no—
355C2×C4—2364no—
356C2×C4#1427364211yescharacteristic
357C2×C4#1427364211yescharacteristic
358C8—4722no—
359C22—21no—
360C22—21no—
361C22—22no—
362C22—22no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22C2×#1427364211yescharacteristic
379C22#1683740711yescharacteristic
380C22C2×#1427365411yescharacteristic
381C4—21no—
382C4—21no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—81no—
390C4—4724no—
391C2—21no—
392C2—21no—
393C2—42no—
394C2—42no—
395C2—161no—
396C2#2142261211yescharacteristic
397C2#2142574211yescharacteristic
398C2#2142576111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
Fitting, prime radicalC59×#2543C2
socleC22×C59C2×#99
cosocleC2×C22⋊C4×C59C23
hypercenter#2543D59
hypocenterC59#11775
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2543

Complements

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

NH ≅ G/Ntype
1. C4×#14273642C2semidirect
2. #21422437C2semidirect
3. #21422096C2semidirect
4. #21422093C2semidirect
13. C2×#121×C59C4semidirect
15. C59×#398C4semidirect
17. C59×#394C4semidirect
22. C2×#95×C59C4semidirect
206. C2×#121Dic59semidirect
210. #398Dic59semidirect
212. #394Dic59semidirect
218. C2×#95Dic59semidirect
222. C59#11775semidirect

Maximal subgroups

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

subgroupindex
C4×#142736422
#214224372
#214220962
#214220932
C59×#25432
#1177559