Finite Group Databasebeta

← #28499620

Subgroups — #28499620

Order 15104.

412 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 83 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#21429476C211yesmaximal, characteristic
2C2×#16837128C211yesmaximal, characteristic
3#21422125C212yesmaximal, complemented
4#21422101C212yesmaximal, complemented
5C59×#1821C211yesmaximal, characteristic
6C2×#14273706C411yescharacteristic
7C2×C22⋊(C59⋊C8)—21no—
8#16837332C411yescharacteristic
9#16837332C2211yescharacteristic
10#16837322C411yescharacteristic
11#16837322C2211yescharacteristic
12#16837297—21no—
13#16837297—21no—
14#16837275—21no—
15#16837275—21no—
16C4×C22⋊Dic59C411yescharacteristic
17#16837128—21no—
18#16837128C2212yes—
19C22×C4⋊C4×C59C2211yescharacteristic
20C59×#407—21no—
21C59×#343—21no—
22C59×#343C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C22×C59⋊C8—41no—
25#14273706—41no—
26#14273706—41no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×C4×Dic59—21no—
36C22⋊(C59⋊C8)—22no—
37C22⋊(C59⋊C8)—41no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C23×C4×C59C2×C411yescharacteristic
45C23×C4×C59D411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C2×M16×C59—22no—
48C2×C4⋊C4×C59—41no—
49C2×C4⋊C4×C59—41no—
50C2×C4⋊C4×C59C2×C411yescharacteristic
51C2×C4⋊C4×C59D411yescharacteristic
52C2×C4⋊C4×C59D411yescharacteristic
53C2×C4⋊C4×C59C2311yescharacteristic
54C22⋊C8×C59—41no—
55C22⋊C8×C59—41no—
56C24×C59C22⋊C411yescharacteristic
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—22no—
64C22×C4×C59—41no—
65C22×C4×C59—81no—
66C22×C4×C59—81no—
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—
77C2×Dic118—41no—
78C2×Dic118—81no—
79M16×C59—42no—
80M16×C59—42no—
81C2×C8×C59—42no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—41no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—41no—
86C22⋊Dic59—24no—
87C22⋊Dic59—41no—
88C22⋊Dic59—41no—
89C4⋊Dic59—41no—
90Dic59⋊C4—42no—
91Dic59⋊C4—42no—
92Dic59⋊C4—42no—
93Dic59⋊C4—81no—
94Dic59⋊C4—81no—
95C4×Dic59—81no—
96C2×C59⋊C8—42no—
97C2×C59⋊C8—82no—
98C23×C59—22no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—41no—
102C23×C59#9911yescharacteristic
103C23×C59#9911yescharacteristic
104C23×C59C2×C22⋊C411yescharacteristic
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—42no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—81no—
122C2×C4×C59—81no—
123C2×C4×C59—81no—
124C2×C4×C59#13512yes—
125C2×Dic59—22no—
126C2×Dic59—22no—
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—
137Dic118—82no—
138C8×C59—82no—
139C59⋊C8—82no—
140#11854—591nomaximal, Sylow
141C22×C59—21no—
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—42no—
149C22×C59—42no—
150C22×C59#35612yes—
151C22×C59C2×#9911yescharacteristic
152C22×C59#41611yescharacteristic
153C22×C59#42111yescharacteristic
154C4×C59—22no—
155C4×C59—41no—
156C4×C59—41no—
157C4×C59—41no—
158C4×C59—41no—
159C4×C59—41no—
160Dic59—42no—
161Dic59—42no—
162Dic59—81no—
163Dic59—81no—
164#2583—591no—
165#1821D5911yescharacteristic
166C2×#324—591no—
167#1354—592no—
168#1346—592no—
169C2×C59—21no—
170C2×C59—21no—
171C2×C59—22no—
172C2×C59—41no—
173C2×C59#151611yescharacteristic
174C2×C59#212111yescharacteristic
175C2×C59C2×#35611yescharacteristic
176C2×C22⋊Q8—591no—
177C22×C4⋊C4C2×D5911yescharacteristic
178#407—21no—
179C2×C22⋊C8—1181no—
180#398—591no—
181#398—591no—
182#395—591no—
183#395—591no—
184#389—1181no—
185#389—1181no—
186#385—1181no—
187#385—1181no—
188C4×C22⋊C4—591no—
189#343—21no—
190#343C2×D5912yes—
191#324—592no—
192#324—1181no—
193C59#1185411yescharacteristic, complemented, Sylow
194C22×Q8—591no—
195C23×C4C4×D5911yescharacteristic
196C23×C4D23611yescharacteristic
197C23×C4C22⋊D5911yescharacteristic
198C2×M16—22no—
199C22×C8—2361no—
200C22⋊Q8—2361no—
201C22⋊Q8—2361no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—591no—
205C2×C4⋊C4—1181no—
206C2×C4⋊C4—1181no—
207C2×C4⋊C4—1181no—
208C2×C4⋊C4—1181no—
209C2×C4⋊C4—1181no—
210C2×C4⋊C4C4×D5911yescharacteristic
211C2×C4⋊C4D23611yescharacteristic
212C2×C4⋊C4C22⋊D5911yescharacteristic
213C2×C4⋊C4C22×D5911yescharacteristic
214C2×C22⋊C4—1181no—
215C2×C22⋊C4—1181no—
216C2×C42—1181no—
217C22⋊C8—41no—
218C22⋊C8—41no—
219C22⋊C8—1182no—
220C22⋊C8—2361no—
221#95—591no—
222#95—591no—
223#95—1181no—
224#95—1181no—
225#95—1181no—
226#95—1181no—
227C24(C2×D59)⋊C411yescharacteristic
228C2×Q8—4721no—
229C2×Q8—2361no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4—1181no—
241C22×C4—1181no—
242C22×C4—1181no—
243C22×C4—1181no—
244C22×C4(C2×D59)⋊C411yescharacteristic
245C22×C4(C2×D59)⋊C411yescharacteristic
246C22×C4(C2×D59)⋊C411yescharacteristic
247C22×C4C2×C4×D5911yescharacteristic
248C22×C4C2×D23611yescharacteristic
249C22×C4C2×C22⋊D5911yescharacteristic
250M16—42no—
251M16—42no—
252C2×C8—42no—
253C2×C8—4722no—
254C2×C8—2362no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—4721no—
260C4⋊C4—4721no—
261C4⋊C4—2361no—
262C4⋊C4—2362no—
263C4⋊C4—2362no—
264C4⋊C4—2362no—
265C22⋊C4—1184no—
266C22⋊C4—2361no—
267C22⋊C4—2361no—
268C42—4721no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—41no—
273C23#1427364111yescharacteristic
274C23#1427364111yescharacteristic
275C23C2×(C2×D59)⋊C411yescharacteristic
276Q8—4722no—
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—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—1182no—
297C2×C4—1182no—
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—2361no—
308C2×C4#1427374712yes—
309C8—82no—
310C8—4722no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22#1683720012yes—
321C22C2×#1427364111yescharacteristic
322C22#1683739611yescharacteristic
323C22#1683742011yescharacteristic
324C4—22no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—4721no—
331C4—4721no—
332C4—2362no—
333C4—2362no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—41no—
338C2#2142265011yescharacteristic
339C2#2142574311yescharacteristic
340C2C2×#1683720011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1821C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1821D59
hypocenterC59#11854
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #21422125C2semidirect
4. #21422101C2semidirect
193. C59#11854semidirect

Maximal subgroups

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

subgroupindex
#214294762
C2×#168371282
#214221252
#214221012
C59×#18212
#1185459