Finite Group Databasebeta

← #28477283

Subgroups — #28477283

Order 15104.

500 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 111 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#21432406C211yesmaximal, characteristic
2#21424076C211yesmaximal, characteristic
3#21421920C212yesmaximal, complemented
4#21421918C212yesmaximal, complemented
5C59×#1728C211yesmaximal, characteristic
6C2×#14273719C411yescharacteristic
7C2×C4⋊(C59⋊C8)—21no—
8C2×C22⋊(C59⋊C8)—21no—
9#16837300C2211yescharacteristic
10#16837300C2211yescharacteristic
11#16837286—21no—
12#16837286—21no—
13#16837275—21no—
14#16837275—21no—
15#16837268C411yescharacteristic
16#16837262C411yescharacteristic
17#16837262C411yescharacteristic
18#16837157C2212yes—
19C22×M16×C59—21no—
20C22×C42×C59C2211yescharacteristic
21C59×#336—21no—
22C59×#336C2212yes—
23C22×C59⋊C8—22no—
24#14273719—42no—
25C2×C4⋊Dic59C2×C411yescharacteristic
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C4⋊(C59⋊C8)—42no—
36C22⋊(C59⋊C8)—42no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C23×C4×C59C2×C411yescharacteristic
44C23×C4×C59D411yescharacteristic
45C23×C4×C59D411yescharacteristic
46C2×M16×C59—24no—
47C2×M16×C59—42no—
48C2×M16×C59—42no—
49C22×C8×C59—22no—
50C2×C42×C59—42no—
51C2×C42×C59C2×C411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C2×C42×C59C2311yescharacteristic
55C24×C59C22⋊C411yescharacteristic
56C22×C4×C59—22no—
57C22×C4×C59—22no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—22no—
61C22×C4×C59—22no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—41no—
65C22×C4×C59—41no—
66C22×C4×C59—41no—
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77M16×C59—48no—
78C2×C8×C59—24no—
79C2×C8×C59—42no—
80C2×C8×C59—42no—
81C42×C59—22no—
82C42×C59—22no—
83C42×C59—41no—
84C42×C59—41no—
85C22⋊Dic59—42no—
86C22⋊Dic59—42no—
87C4⋊Dic59—22no—
88C4⋊Dic59—22no—
89C4⋊Dic59—41no—
90C4⋊Dic59—41no—
91Dic59⋊C4—42no—
92Dic59⋊C4—42no—
93Dic59⋊C4—41no—
94Dic59⋊C4—41no—
95Dic59⋊C4—41no—
96Dic59⋊C4—41no—
97C2×C59⋊C8—42no—
98C2×C59⋊C8—42no—
99C2×C59⋊C8—42no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59#10412yes—
107C23×C59C2×C22⋊C411yescharacteristic
108C2×C4×C59—24no—
109C2×C4×C59—24no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—44no—
124C2×C4×C59—42no—
125C2×C4×C59—42no—
126C2×C4×C59#9912yes—
127C2×C4×C59#10112yes—
128C2×C4×C59#13112yes—
129C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C59#13711yescharacteristic
132C2×C4×C59#13711yescharacteristic
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C8×C59—44no—
146C59⋊C8—82no—
147#7711—591nomaximal, Sylow
148C22×C59—22no—
149C22×C59—22no—
150C22×C59—22no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59#40811yescharacteristic
156C22×C59C2×#9911yescharacteristic
157C22×C59C2×#10111yescharacteristic
158C22×C59#41711yescharacteristic
159C22×C59#41711yescharacteristic
160C22×C59C2×#10411yescharacteristic
161C22×C59#42111yescharacteristic
162C4×C59—22no—
163C4×C59—22no—
164C4×C59—21no—
165C4×C59—22no—
166C4×C59—21no—
167C4×C59—21no—
168C4×C59—21no—
169C4×C59—41no—
170Dic59—81no—
171Dic59—81no—
172Dic59—81no—
173Dic59—81no—
174#2851—591no—
175#1797—591no—
176#1728D5911yescharacteristic
177#1271—592no—
178#1271—592no—
179C2×C59—42no—
180C2×C59#147211yescharacteristic
181C2×C59#148311yescharacteristic
182C2×C59#149511yescharacteristic
183C2×C59#150411yescharacteristic
184C2×C59#151311yescharacteristic
185C2×C59#152811yescharacteristic
186C2×C59#156111yescharacteristic
187C22×M16—21no—
188C2×#125—591no—
189C22×C42C2×D5911yescharacteristic
190C2×C4⋊C8—1181no—
191C2×C22⋊C8—1181no—
192#389—591no—
193#389—591no—
194#387—1181no—
195#387—1181no—
196#385—1181no—
197#385—1181no—
198#382—591no—
199#380—591no—
200#380—591no—
201#336—21no—
202#336—592no—
203#336C2×D5912yes—
204C59#771111yescharacteristic, complemented, Sylow
205C23×C4C4×D5911yescharacteristic
206C23×C4D23611yescharacteristic
207C23×C4C22⋊D5911yescharacteristic
208C2×M16—24no—
209C2×M16—42no—
210C2×M16—42no—
211C22×C8—22no—
212C22×C8—1182no—
213#125—2362no—
214C2×C4⋊C4—591no—
215C2×C4⋊C4—591no—
216C2×C4⋊C4—591no—
217C2×C4⋊C4—591no—
218C2×C4⋊C4—1181no—
219C2×C4⋊C4—1181no—
220C2×C4⋊C4—1181no—
221C2×C4⋊C4—1181no—
222C2×C22⋊C4—1181no—
223C2×C22⋊C4—1181no—
224C2×C42—42no—
225C2×C42C4×D5911yescharacteristic
226C2×C42D23611yescharacteristic
227C2×C42C22⋊D5911yescharacteristic
228C2×C42C22×D5911yescharacteristic
229C4⋊C8—2362no—
230C22⋊C8—2362no—
231#95—1181no—
232#95—1181no—
233#95—1181no—
234#95—1181no—
235#95—1181no—
236#95—1181no—
237C24(C2×D59)⋊C411yescharacteristic
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—1181no—
250C22×C4—1181no—
251C22×C4—1181no—
252C22×C4—1181no—
253C22×C4(C2×D59)⋊C411yescharacteristic
254C22×C4(C2×D59)⋊C411yescharacteristic
255C22×C4(C2×D59)⋊C411yescharacteristic
256C22×C4C2×C4×D5911yescharacteristic
257C22×C4C2×D23611yescharacteristic
258C22×C4C2×C22⋊D5911yescharacteristic
259M16—48no—
260C2×C8—24no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—2362no—
264C2×C8—2362no—
265C2×C8—2362no—
266C4⋊C4—1182no—
267C4⋊C4—1182no—
268C4⋊C4—2361no—
269C4⋊C4—2361no—
270C4⋊C4—2362no—
271C4⋊C4—2362no—
272C4⋊C4—2361no—
273C4⋊C4—2361no—
274C4⋊C4—2361no—
275C4⋊C4—2361no—
276C22⋊C4—2362no—
277C22⋊C4—2362no—
278C42—22no—
279C42—22no—
280C42—41no—
281C42—41no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23#1427365512yes—
289C23C2×(C2×D59)⋊C411yescharacteristic
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—2361no—
309C2×C4—2361no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4—2361no—
314C2×C4—2361no—
315C2×C4—2361no—
316C2×C4—2361no—
317C2×C4—2361no—
318C2×C4—2361no—
319C2×C4—2361no—
320C2×C4#1427364112yes—
321C2×C4#1427364512yes—
322C2×C4#1427373512yes—
323C2×C4#1427375311yescharacteristic
324C2×C4#1427375511yescharacteristic
325C2×C4#1427375811yescharacteristic
326C2×C4#1427376011yescharacteristic
327C8—44no—
328C8—4722no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22#1683737011yescharacteristic
337C22C2×#1427364111yescharacteristic
338C22C2×#1427364511yescharacteristic
339C22#1683740111yescharacteristic
340C22#1683740411yescharacteristic
341C22C2×#1427365511yescharacteristic
342C22#1683741811yescharacteristic
343C4—21no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—22no—
349C4—22no—
350C4—41no—
351C4—4721no—
352C4—4721no—
353C4—4721no—
354C4—4721no—
355C2—42no—
356C2#2142246111yescharacteristic
357C2#2142251011yescharacteristic
358C2#2142255911yescharacteristic
359C2#2142259811yescharacteristic
360C2#2142264011yescharacteristic
361C2#2142269711yescharacteristic
362C2#2142283311yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1728C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1728D59
hypocenterC59#7711
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1728

Complements

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

NH ≅ G/Ntype
3. #21421920C2semidirect
4. #21421918C2semidirect
204. C59#7711semidirect

Maximal subgroups

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

subgroupindex
#214324062
#214240762
#214219202
#214219182
C59×#17282
#771159