Finite Group Databasebeta

← #28486468

Subgroups — #28486468

Order 15104.

716 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 119 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#21427461C211yesmaximal, characteristic
2#21424084C211yesmaximal, characteristic
3#21424032C212yesmaximal, complemented
4#21421964C212yesmaximal, complemented
5C59×#2973C211yesmaximal, characteristic
6C23×Dic118C2211yescharacteristic
7C2×#14273706—21no—
8C2×#14273652—24no—
9C2×#14273642—21no—
10#16837322C412yescomplemented
11#16837314C412yescomplemented
12#16837308—21no—
13#16837282—21no—
14#16837274—21no—
15#16837262C2212yes—
16#16837262C2211yescharacteristic
17#16837169—21no—
18C2×C4×D4×C59C2211yescharacteristic
19C59×#435—24no—
20C59×#432—21no—
21C59×#407—21no—
22C2×C22⋊C8×C59C2212yes—
23C22×Dic118—22no—
24C22×Dic118—21no—
25C22×Dic118—22no—
26C22×Dic118—44no—
27C22×Dic118C2×C412yes—
28C23×Dic59—21no—
29#14273706—44no—
30C2×C4⋊Dic59—21no—
31C2×C4⋊Dic59—22no—
32C2×Dic59⋊C4—22no—
33C2×C22⋊Dic59—22no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—22no—
36#14273652—48no—
37#14273642—42no—
38C59⋊#95—22no—
39C59⋊#95—22no—
40C59⋊#95—22no—
41C59⋊#95C2×C412yes—
42C22×D4×C59D411yescharacteristic
43C23×C4×C59D411yescharacteristic
44C23×C4×C59C2311yescharacteristic
45C2×M16×C59—22no—
46C22×C8×C59D414yes—
47C4×D4×C59—22no—
48C4×D4×C59—22no—
49C2×C4⋊C4×C59C2×C411yescharacteristic
50C2×C22⋊C4×C59C2×C411yescharacteristic
51C2×C22⋊C4×C59D411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C4⋊C8×C59—22no—
54C22⋊C8×C59—22no—
55C22⋊C8×C59—22no—
56C22⋊C8×C59—22no—
57#97×C59—22no—
58C24×C59C22⋊C411yescharacteristic
59C24×C59C2×D411yescharacteristic
60C2×D4×C59—22no—
61C2×D4×C59—21no—
62C2×D4×C59—21no—
63C2×D4×C59—22no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—42no—
67C22×C4×C59—42no—
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C2×D412yes—
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59#4112yes—
75C22×Dic59—22no—
76C22×Dic59—22no—
77C22×Dic59—44no—
78C22×Dic59—42no—
79C22×Dic59—42no—
80C2×Dic118—28no—
81C2×Dic118—24no—
82C2×Dic118—24no—
83C2×Dic118—44no—
84C2×Dic118—48no—
85C2×Dic118—44no—
86C2×Dic118—42no—
87C2×Dic118—41no—
88C2×Dic118—41no—
89M16×C59—82no—
90C2×C8×C59—24no—
91C2×C8×C59—44no—
92C4⋊C4×C59—22no—
93C22⋊C4×C59—22no—
94C22⋊C4×C59—22no—
95C42×C59—21no—
96C42×C59—21no—
97C22⋊Dic59—44no—
98C22⋊Dic59—41no—
99C22⋊Dic59—41no—
100C22⋊Dic59—42no—
101C22⋊Dic59—42no—
102C4⋊Dic59—42no—
103C4⋊Dic59—44no—
104Dic59⋊C4—44no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59C2×C22⋊C411yescharacteristic
113C23×C59#12112yes—
114C23×C59AGL(1,ℤ/8)11yescharacteristic
115C23×C59#13711yescharacteristic
116C23×C59#13712yes—
117D4×C59—42no—
118D4×C59—42no—
119C2×C4×C59—24no—
120C2×C4×C59—22no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—41no—
127C2×C4×C59—42no—
128C2×C4×C59—42no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—81no—
132C2×C4×C59C4×D412yes—
133C2×C4×C59#12011yescharacteristic
134C2×C4×C59#12311yescharacteristic
135C2×Dic59—24no—
136C2×Dic59—48no—
137C2×Dic59—42no—
138C2×Dic59—42no—
139C2×Dic59—42no—
140C2×Dic59—42no—
141C2×Dic59—42no—
142C2×Dic59—82no—
143C2×Dic59—82no—
144C2×Dic59—82no—
145Dic118—28no—
146Dic118—44no—
147Dic118—44no—
148Dic118—44no—
149C8×C59—44no—
150#9471—591nomaximal, Sylow
151C22×C59—21no—
152C22×C59—22no—
153C22×C59—21no—
154C22×C59—22no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—42no—
158C22×C59—42no—
159C22×C59—41no—
160C22×C59—41no—
161C22×C59—41no—
162C22×C59—81no—
163C22×C59#38611yescharacteristic
164C22×C59#41711yescharacteristic
165C22×C59#41911yescharacteristic
166C22×C59#46812yes—
167C22×C59#47012yes—
168C4×C59—22no—
169C4×C59—21no—
170C4×C59—41no—
171C4×C59—41no—
172C4×C59—41no—
173C4×C59B(4,𝔽2)12yes—
174Dic59—44no—
175Dic59—82no—
176Dic59—162no—
177#2973D5911yescharacteristic
178#2409—591no—
179#1799—591no—
180#1787—592no—
181#1289—592no—
182C2×C59—21no—
183C2×C59—22no—
184C2×C59—21no—
185C2×C59—22no—
186C2×C59—81no—
187C2×C59#188911yescharacteristic
188C2×C59#189411yescharacteristic
189C2×C59#190911yescharacteristic
190C23×Q8—591no—
191C2×C22⋊Q8—1181no—
192C2×C4×D4C2×D5911yescharacteristic
193#435—24no—
194#432—21no—
195C2×#103—1184no—
196C2×#99—1181no—
197#407—21no—
198C2×C22⋊C8C2×D5912yes—
199#395—592no—
200#393—592no—
201#391—1181no—
202#386—1181no—
203#385—1181no—
204#380—591no—
205#380—592no—
206#342—1181no—
207C59#947111yescharacteristic, complemented, Sylow
208C22×Q8—592no—
209C22×Q8—1181no—
210C22×Q8—1182no—
211C22×Q8—1182no—
212C22×Q8—2364no—
213C22×D4D23611yescharacteristic
214C23×C4—1181no—
215C23×C4C22⋊D5911yescharacteristic
216C23×C4C22×D5911yescharacteristic
217C2×M16—22no—
218C22×C8C22⋊D5914yes—
219C22⋊Q8—2364no—
220C4×D4—22no—
221C4×D4—22no—
222C2×C4⋊C4—1182no—
223C2×C4⋊C4—1181no—
224C2×C4⋊C4—1182no—
225C2×C4⋊C4C4×D5911yescharacteristic
226C2×C22⋊C4—1182no—
227C2×C22⋊C4—1181no—
228C2×C22⋊C4—1182no—
229C2×C22⋊C4C4×D5911yescharacteristic
230C2×C22⋊C4D23611yescharacteristic
231C2×C42C22⋊D5911yescharacteristic
232C4⋊C8—22no—
233#103—2368no—
234#99—2362no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238#97—22no—
239#95—592no—
240#95—1182no—
241#95—1182no—
242#95—1182no—
243C24(C2×D59)⋊C411yescharacteristic
244C24C2×D23611yescharacteristic
245C2×Q8—1184no—
246C2×Q8—1188no—
247C2×Q8—1184no—
248C2×Q8—2364no—
249C2×Q8—2362no—
250C2×Q8—2364no—
251C2×Q8—2368no—
252C2×Q8—2361no—
253C2×Q8—2361no—
254C2×D4—22no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—1182no—
263C22×C4—1182no—
264C22×C4—2364no—
265C22×C4—2362no—
266C22×C4—2362no—
267C22×C4(C2×D59)⋊C411yescharacteristic
268C22×C4(C2×D59)⋊C411yescharacteristic
269C22×C4(C2×D59)⋊C411yescharacteristic
270C22×C4C2×C4×D5911yescharacteristic
271C22×C4#1174568512yes—
272C22×C4C2×C22⋊D5912yes—
273C22×C4C2×C22⋊D5911yescharacteristic
274M16—82no—
275C2×C8—24no—
276C2×C8—44no—
277C4⋊C4—22no—
278C4⋊C4—2364no—
279C4⋊C4—2362no—
280C4⋊C4—2364no—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C22⋊C4—2364no—
284C22⋊C4—2362no—
285C22⋊C4—2362no—
286C22⋊C4—2361no—
287C22⋊C4—2361no—
288C42—21no—
289C42—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×(C2×D59)⋊C411yescharacteristic
298C23#1427370112yes—
299C23#1427375511yescharacteristic
300C23#1427376012yes—
301C23#1427376011yescharacteristic
302Q8—1188no—
303Q8—2364no—
304Q8—2364no—
305Q8—2364no—
306D4—42no—
307D4—42no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—1184no—
322C2×C4—4722no—
323C2×C4—4722no—
324C2×C4—4722no—
325C2×C4—2362no—
326C2×C4—2362no—
327C2×C4—2362no—
328C2×C4—2362no—
329C2×C4—2368no—
330C2×C4—2362no—
331C2×C4C4×C22⋊D5912yes—
332C2×C4#1427369611yescharacteristic
333C2×C4#1427371011yescharacteristic
334C8—44no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—41no—
340C22—42no—
341C22—41no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22#1683728111yescharacteristic
348C22#1683740411yescharacteristic
349C22#1683741211yescharacteristic
350C22#1683764812yes—
351C22#1683766012yes—
352C4—21no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—9442no—
358C4—4722no—
359C4—2364no—
360C4#1683758912yes—
361C2—22no—
362C2—21no—
363C2—21no—
364C2—22no—
365C2—81no—
366C2#2142450411yescharacteristic
367C2#2142452911yescharacteristic
368C2#2142461811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2973C2
socleC22×C59#386
prime radicalC22×D4×C59D4
cosocleC23×C4×C59C23
hypercenter#2973D59
hypocenterC59#9471
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
3. #21424032C2semidirect
4. #21421964C2semidirect
10. #16837322C4semidirect
11. #16837314C4semidirect
207. C59#9471semidirect

Maximal subgroups

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

subgroupindex
#214274612
#214240842
#214240322
#214219642
C59×#29732
#947159