Finite Group Databasebeta

← #28499470

Subgroups — #28499470

Order 15104.

464 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#21424084C211yesmaximal, characteristic, complemented
2#21422437C211yesmaximal, characteristic, complemented
3#21422120C212yesmaximal, complemented
4#21422093C212yesmaximal, complemented
5C59×#2635C211yesmaximal, characteristic
6#16837459—21no—
7C2×#14273642—21no—
8#16837262—21no—
9#16837169C2212yes—
10#16837126—21no—
11#16837126C2212yes—
12C2×#121×C59C411yescharacteristic, complemented
13C2×C4×D4×C59C2211yescharacteristic
14C59×#400C411yescharacteristic, complemented
15C59×#399C2211yescharacteristic
16C59×#394—21no—
17C59×#394C411yescharacteristic, complemented
18C59×#392—21no—
19C59×#392C2211yescharacteristic
20C59×#388—21no—
21C59×#386—21no—
22C59×#381C411yescharacteristic, complemented
23C2×C4⋊Dic59—41no—
24C2×C22⋊Dic59—22no—
25C4⋊(C59⋊C8)—41no—
26#14273642—42no—
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×C59C2×C411yescharacteristic
35C23×C4×C59D411yescharacteristic
36#121×C59—41no—
37#121×C59—41no—
38C4×D4×C59—41no—
39C4×D4×C59—41no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59C2×C411yescharacteristic
42C2×C4⋊C4×C59C2×C411yescharacteristic
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59—21no—
45C2×C22⋊C4×C59—21no—
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59D411yescharacteristic
50C2×C22⋊C4×C59C2311yescharacteristic
51C2×C42×C59D411yescharacteristic
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—21no—
55#95×C59C2×C411yescharacteristic
56#95×C59C2×C411yescharacteristic
57C24×C59—21no—
58C24×C59C22⋊C411yescharacteristic
59C24×C59C2×D411yescharacteristic
60C2×D4×C59—21no—
61C2×D4×C59—21no—
62C2×D4×C59—41no—
63C2×D4×C59—41no—
64C2×D4×C59—41no—
65C2×D4×C59—42no—
66C2×D4×C59—41no—
67C2×D4×C59—41no—
68C2×D4×C59—81no—
69C2×D4×C59—81no—
70C2×D4×C59—81no—
71C2×D4×C59—81no—
72C2×D4×C59—81no—
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—21no—
80C22×C4×C59—81no—
81C22×C4×C59—81no—
82C22×C4×C59C22⋊C411yescharacteristic
83C22×C4×C59C22⋊C411yescharacteristic
84C22×C4×C59C22⋊C411yescharacteristic
85C22×C4×C59C22×C411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×Dic59—42no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—41no—
90C4⋊C4×C59—81no—
91C22⋊C4×C59—41no—
92C22⋊C4×C59—41no—
93C22⋊C4×C59—42no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—81no—
97C22⋊C4×C59—81no—
98C22⋊C4×C59—81no—
99C22⋊C4×C59—81no—
100C42×C59—21no—
101C42×C59—21no—
102C22⋊Dic59—42no—
103C22⋊Dic59—42no—
104C4⋊Dic59—82no—
105C2×C59⋊C8—42no—
106C23×C59—21no—
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×C59C2×C22⋊C411yescharacteristic
123C23×C59AGL(1,ℤ/8)11yescharacteristic
124C23×C59#13711yescharacteristic
125D4×C59—42no—
126D4×C59—42no—
127D4×C59—82no—
128D4×C59—82no—
129D4×C59—82no—
130D4×C59—84no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—41no—
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—81no—
152C2×C4×C59—81no—
153C2×C4×C59—81no—
154C2×C4×C59#9911yescharacteristic
155C2×C4×C59#9911yescharacteristic
156C2×Dic59—82no—
157C2×Dic59—82no—
158C2×Dic59—82no—
159C59⋊C8—82no—
160#11824—591nomaximal, Sylow
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—21no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—41no—
168C22×C59—41no—
169C22×C59—41no—
170C22×C59—41no—
171C22×C59—41no—
172C22×C59—81no—
173C22×C59—81no—
174C22×C59—81no—
175C22×C59—81no—
176C22×C59—81no—
177C22×C59—81no—
178C22×C59—81no—
179C22×C59—81no—
180C22×C59—81no—
181C22×C59—81no—
182C22×C59C2×#9911yescharacteristic
183C22×C59#41711yescharacteristic
184C22×C59#42111yescharacteristic
185C4×C59—21no—
186C4×C59—21no—
187C4×C59—41no—
188C4×C59—41no—
189C4×C59—41no—
190C4×C59—41no—
191C4×C59—81no—
192C4×C59—81no—
193C4×C59—81no—
194Dic59—162no—
195#2635D5911yescharacteristic
196#1799—591no—
197#1466—591no—
198#1352—592no—
199#1343—592no—
200C2×C59—21no—
201C2×C59—21no—
202C2×C59—41no—
203C2×C59—41no—
204C2×C59—81no—
205C2×C59—161no—
206C2×C59#151311yescharacteristic
207C2×C59#212111yescharacteristic
208C2×C59#212811yescharacteristic
209C2×#121Dic5911yescharacteristic, complemented
210C2×C4×D4C2×D5911yescharacteristic
211#432—1181no—
212C2×#99—1181no—
213#400Dic5911yescharacteristic, complemented
214#399C2×D5911yescharacteristic
215#394—21no—
216#394Dic5911yescharacteristic, complemented
217#392—21no—
218#392C2×D5911yescharacteristic
219#388—21no—
220#386—21no—
221#381Dic5911yescharacteristic, complemented
222#380—1181no—
223#342—592no—
224#323—592no—
225#323—1181no—
226C59#1182411yescharacteristic, complemented, Sylow
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C2×Dic5911yescharacteristic
230C22×D4C22⋊D5911yescharacteristic
231C23×C4C2×Dic5911yescharacteristic
232C23×C4C22⋊D5911yescharacteristic
233#121—41no—
234#121—41no—
235C4×D4—41no—
236C4×D4—41no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—2361no—
239C2×C4⋊C4C2×Dic5911yescharacteristic
240C2×C4⋊C4C2×Dic5911yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—1182no—
248C2×C22⋊C4C22⋊D5911yescharacteristic
249C2×C22⋊C4C22×D5911yescharacteristic
250C2×C42C22⋊D5911yescharacteristic
251C4⋊C8—2361no—
252#99—2362no—
253C22⋊C8—1182no—
254#97—2361no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—2361no—
259#95C2×Dic5911yescharacteristic
260#95C2×Dic5911yescharacteristic
261C24—21no—
262C24C22⋊Dic5911yescharacteristic
263C24C2×C22⋊D5911yescharacteristic
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—81no—
273C2×D4—81no—
274C2×D4—81no—
275C2×D4—81no—
276C2×D4—81no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—2362no—
287C22×C4C22⋊Dic5911yescharacteristic
288C22×C4C22⋊Dic5911yescharacteristic
289C22×C4C22⋊Dic5911yescharacteristic
290C22×C4C22×Dic5911yescharacteristic
291C22×C4C2×C22⋊D5911yescharacteristic
292C2×C8—2362no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—81no—
296C4⋊C4—4722no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—81no—
303C22⋊C4—81no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C22⋊C4—2362no—
307C22⋊C4—2362no—
308C42—21no—
309C42—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23C2×C22⋊Dic5911yescharacteristic
327C23#1427375211yescharacteristic
328C23#1427375711yescharacteristic
329D4—42no—
330D4—42no—
331D4—82no—
332D4—82no—
333D4—82no—
334D4—84no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—4722no—
359C2×C4—4722no—
360C2×C4—4722no—
361C2×C4#1427364211yescharacteristic
362C2×C4#1427364211yescharacteristic
363C8—4722no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C2×#1427364211yescharacteristic
386C22#1683740311yescharacteristic
387C22#1683741911yescharacteristic
388C4—21no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—9442no—
398C2—21no—
399C2—21no—
400C2—41no—
401C2—41no—
402C2—81no—
403C2—161no—
404C2#2142263911yescharacteristic
405C2#2142574211yescharacteristic
406C2#2142576811yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
Fitting, prime radicalC59×#2635C2
socleC22×C59C2×#99
cosocleC2×C22⋊C4×C59C23
hypercenter#2635D59
hypocenterC59#11824
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2635

Complements

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

NH ≅ G/Ntype
1. #21424084C2semidirect
2. #21422437C2semidirect
3. #21422120C2semidirect
4. #21422093C2semidirect
12. C2×#121×C59C4semidirect
14. C59×#400C4semidirect
17. C59×#394C4semidirect
22. C59×#381C4semidirect
209. C2×#121Dic59semidirect
213. #400Dic59semidirect
216. #394Dic59semidirect
221. #381Dic59semidirect
226. C59#11824semidirect

Maximal subgroups

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

subgroupindex
#214240842
#214224372
#214221202
#214220932
C59×#26352
#1182459