Finite Group Databasebeta

← #28477372

Subgroups — #28477372

Order 15104.

450 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 107 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×#1817C211yesmaximal, 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×C42×C59C2211yescharacteristic
20C2×C22⋊C8×C59—21no—
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
46C22×C8×C59—22no—
47C2×C42×C59—42no—
48C2×C42×C59C2×C411yescharacteristic
49C2×C42×C59D411yescharacteristic
50C2×C42×C59D411yescharacteristic
51C2×C42×C59C2311yescharacteristic
52C22⋊C8×C59—42no—
53C24×C59C22⋊C411yescharacteristic
54C22×C4×C59—22no—
55C22×C4×C59—22no—
56C22×C4×C59—22no—
57C22×C4×C59—22no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—41no—
61C22×C4×C59—41no—
62C22×C4×C59—41no—
63C22×C4×C59—41no—
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22×C411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×Dic59—21no—
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74C2×C8×C59—42no—
75C2×C8×C59—42no—
76C2×C8×C59—42no—
77C42×C59—22no—
78C42×C59—22no—
79C42×C59—41no—
80C42×C59—41no—
81C22⋊Dic59—42no—
82C22⋊Dic59—42no—
83C4⋊Dic59—22no—
84C4⋊Dic59—22no—
85C4⋊Dic59—41no—
86C4⋊Dic59—41no—
87Dic59⋊C4—42no—
88Dic59⋊C4—42no—
89Dic59⋊C4—42no—
90Dic59⋊C4—41no—
91Dic59⋊C4—41no—
92C2×C59⋊C8—42no—
93C2×C59⋊C8—42no—
94C2×C59⋊C8—42no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59#10112yes—
102C23×C59C2×C22⋊C411yescharacteristic
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
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—44no—
118C2×C4×C59—44no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59#9912yes—
122C2×C4×C59#13112yes—
123C2×C4×C59#13512yes—
124C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C59#13711yescharacteristic
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C8×C59—82no—
139C59⋊C8—82no—
140#7731—591nomaximal, Sylow
141C22×C59—22no—
142C22×C59—22no—
143C22×C59—22no—
144C22×C59—41no—
145C22×C59—41no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59#40811yescharacteristic
149C22×C59C2×#9911yescharacteristic
150C22×C59C2×#10111yescharacteristic
151C22×C59#41611yescharacteristic
152C22×C59#41711yescharacteristic
153C22×C59#42111yescharacteristic
154C22×C59#42111yescharacteristic
155C4×C59—22no—
156C4×C59—21no—
157C4×C59—21no—
158C4×C59—22no—
159C4×C59—22no—
160C4×C59—41no—
161C4×C59—41no—
162Dic59—81no—
163Dic59—81no—
164Dic59—81no—
165Dic59—81no—
166#2851—591no—
167#1817D5911yescharacteristic
168#1797—591no—
169#1271—592no—
170#1271—592no—
171C2×C59—42no—
172C2×C59#147211yescharacteristic
173C2×C59#148211yescharacteristic
174C2×C59#149511yescharacteristic
175C2×C59#151311yescharacteristic
176C2×C59#151611yescharacteristic
177C2×C59#153411yescharacteristic
178C2×C59#155211yescharacteristic
179C2×#125—591no—
180C22×C42C2×D5911yescharacteristic
181C2×C4⋊C8—1181no—
182C2×C22⋊C8—21no—
183C2×C22⋊C8—1181no—
184#389—591no—
185#389—591no—
186#387—1181no—
187#387—1181no—
188#385—1181no—
189#385—1181no—
190#382—591no—
191#380—591no—
192#380—591no—
193#336—21no—
194#336—592no—
195#336C2×D5912yes—
196C59#773111yescharacteristic, complemented, Sylow
197C23×C4C4×D5911yescharacteristic
198C23×C4D23611yescharacteristic
199C23×C4C22⋊D5911yescharacteristic
200C22×C8—22no—
201C22×C8—1182no—
202#125—2362no—
203C2×C4⋊C4—591no—
204C2×C4⋊C4—591no—
205C2×C4⋊C4—591no—
206C2×C4⋊C4—591no—
207C2×C4⋊C4—1181no—
208C2×C4⋊C4—1181no—
209C2×C4⋊C4—1181no—
210C2×C4⋊C4—1181no—
211C2×C22⋊C4—1181no—
212C2×C22⋊C4—1181no—
213C2×C42—42no—
214C2×C42C4×D5911yescharacteristic
215C2×C42D23611yescharacteristic
216C2×C42C22⋊D5911yescharacteristic
217C2×C42C22×D5911yescharacteristic
218C4⋊C8—2362no—
219C22⋊C8—42no—
220C22⋊C8—2362no—
221#95—1181no—
222#95—1181no—
223#95—1181no—
224#95—1181no—
225#95—1181no—
226#95—1181no—
227C24(C2×D59)⋊C411yescharacteristic
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—1181no—
239C22×C4—1181no—
240C22×C4—1181no—
241C22×C4—1181no—
242C22×C4(C2×D59)⋊C411yescharacteristic
243C22×C4(C2×D59)⋊C411yescharacteristic
244C22×C4(C2×D59)⋊C411yescharacteristic
245C22×C4C2×C4×D5911yescharacteristic
246C22×C4C2×D23611yescharacteristic
247C22×C4C2×C22⋊D5911yescharacteristic
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—2362no—
252C2×C8—2362no—
253C2×C8—2362no—
254C4⋊C4—1182no—
255C4⋊C4—1182no—
256C4⋊C4—2361no—
257C4⋊C4—2361no—
258C4⋊C4—2362no—
259C4⋊C4—2362no—
260C4⋊C4—2362no—
261C4⋊C4—2361no—
262C4⋊C4—2361no—
263C22⋊C4—2362no—
264C22⋊C4—2362no—
265C42—22no—
266C42—22no—
267C42—41no—
268C42—41no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23#1427364512yes—
276C23C2×(C2×D59)⋊C411yescharacteristic
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—2361no—
296C2×C4—2361no—
297C2×C4—2361no—
298C2×C4—2361no—
299C2×C4—2361no—
300C2×C4—2361no—
301C2×C4—2361no—
302C2×C4—2361no—
303C2×C4—2361no—
304C2×C4—2361no—
305C2×C4—2361no—
306C2×C4—2361no—
307C2×C4#1427364112yes—
308C2×C4#1427373312yes—
309C2×C4#1427374712yes—
310C2×C4#1427375511yescharacteristic
311C2×C4#1427376011yescharacteristic
312C8—82no—
313C8—4722no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22#1683736811yescharacteristic
322C22C2×#1427364111yescharacteristic
323C22C2×#1427364511yescharacteristic
324C22#1683739611yescharacteristic
325C22#1683740411yescharacteristic
326C22#1683741811yescharacteristic
327C22#1683742011yescharacteristic
328C4—21no—
329C4—21no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—4721no—
336C4—4721no—
337C4—4721no—
338C4—4721no—
339C2—42no—
340C2#2142245911yescharacteristic
341C2#2142250211yescharacteristic
342C2#2142256211yescharacteristic
343C2#2142264011yescharacteristic
344C2#2142265011yescharacteristic
345C2#2142272211yescharacteristic
346C2#2142279711yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1817C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1817D59
hypocenterC59#7731
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
3. #21421920C2semidirect
4. #21421918C2semidirect
196. C59#7731semidirect

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×#18172
#773159