Finite Group Databasebeta

← #28498339

Subgroups — #28498339

Order 15104.

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

normal subgroups only

This group has 480 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#21429049C211yesmaximal, characteristic
2#21422434C211yesmaximal, characteristic
3#21422041C214yesmaximal, complemented
4C59×#1459C211yesmaximal, characteristic
5C2×#14273727C411yescharacteristic
6C2×#14273706—21no—
7C2×#14273706—21no—
8#16837366—21no—
9#16837338—21no—
10#16837338—21no—
11#16837305C412yes—
12#16837271C2212yes—
13C4×C4⋊Dic59C411yescharacteristic
14#16837128C2212yes—
15#16837126—21no—
16C2×C4×D4×C59C2211yescharacteristic
17C2×C22⋊C8×C59—21no—
18C59×#324C2212yes—
19C59×#323—21no—
20C22×Dic118—21no—
21C2×#11745664—41no—
22#14273727—41no—
23#14273727—41no—
24#14273706—44no—
25#14273706—44no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59C2×C412yes—
29C2×Dic59⋊C4—22no—
30C2×Dic59⋊C4—22no—
31C2×C22⋊Dic59—22no—
32C2×C22⋊Dic59—22no—
33C2×C4×Dic59C2×C412yes—
34C22⋊(C59⋊C8)—22no—
35C22⋊(C59⋊C8)—41no—
36C59⋊#95—21no—
37C22×D4×C59C2×C411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C23×C4×C59D411yescharacteristic
40C22×C8×C59—41no—
41C4×D4×C59—41no—
42C4×D4×C59—41no—
43C2×C4⋊C4×C59C2311yescharacteristic
44C2×C22⋊C4×C59D411yescharacteristic
45C2×C22⋊C4×C59D411yescharacteristic
46C2×C42×C59C2×C411yescharacteristic
47C22⋊C8×C59—22no—
48C22⋊C8×C59—41no—
49C24×C59C22⋊C411yescharacteristic
50C24×C59C22⋊C411yescharacteristic
51C2×D4×C59—22no—
52C2×D4×C59—22no—
53C2×D4×C59—41no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—42no—
59C22×C4×C59—42no—
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22×C411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C22×Dic59—22no—
66C22×Dic59—22no—
67C2×Dic118—42no—
68C2×Dic118—42no—
69C2×Dic118—41no—
70C2×Dic118—41no—
71C2×C8×C59—42no—
72C2×C8×C59—82no—
73C4⋊C4×C59—41no—
74C22⋊C4×C59—41no—
75C22⋊C4×C59—41no—
76C42×C59—21no—
77C42×C59—21no—
78C22⋊Dic59—44no—
79C22⋊Dic59—44no—
80C4⋊Dic59—24no—
81C4⋊Dic59—42no—
82C4⋊Dic59—42no—
83Dic59⋊C4—44no—
84Dic59⋊C4—44no—
85C4×Dic59—24no—
86#11745664—82no—
87C2×C59⋊C8—42no—
88C23×C59—21no—
89C23×C59—21no—
90C23×C59—21no—
91C23×C59—21no—
92C23×C59—21no—
93C23×C59—41no—
94C23×C59—41no—
95C23×C59—41no—
96C23×C59—41no—
97C23×C59#10112yes—
98C23×C59C2×C22⋊C411yescharacteristic
99D4×C59—44no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—22no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—81no—
112C2×C4×C59—81no—
113C2×C4×C59#9911yescharacteristic
114C2×C4×C59#9911yescharacteristic
115C2×C4×C59#13112yes—
116C2×Dic59—24no—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—42no—
120C2×Dic59—42no—
121C2×Dic59—42no—
122Dic118—82no—
123Dic118—82no—
124C8×C59—82no—
125C59⋊C8—82no—
126#11570—591nomaximal, Sylow
127C22×C59—21no—
128C22×C59—21no—
129C22×C59—22no—
130C22×C59—42no—
131C22×C59—42no—
132C22×C59—41no—
133C22×C59—41no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—41no—
137C22×C59—81no—
138C22×C59—81no—
139C22×C59#40811yescharacteristic
140C22×C59C2×#9911yescharacteristic
141C22×C59C2×#10111yescharacteristic
142C4×C59—21no—
143C4×C59—22no—
144C4×C59—41no—
145C4×C59—41no—
146C4×C59—41no—
147C4×C59#35312yes—
148Dic59—44no—
149Dic59—82no—
150#2542—591no—
151#1465—591no—
152#1459D5911yescharacteristic
153#1325—594no—
154C2×C59—21no—
155C2×C59—21no—
156C2×C59—42no—
157C2×C59—81no—
158C2×C59#147211yescharacteristic
159C2×C59C2×#35311yescharacteristic
160C2×C59#213511yescharacteristic
161C2×C4⋊Q8—591no—
162C2×C22⋊Q8—1181no—
163C2×C22⋊Q8—1181no—
164C2×C4×D4C2×D5911yescharacteristic
165#407—1181no—
166C2×C22⋊C8—21no—
167#399—1181no—
168#399—1181no—
169#390—592no—
170#384—592no—
171C4×C4⋊C4—591no—
172#324—592no—
173#324C2×D5912yes—
174#323—21no—
175#323—1181no—
176C59#1157011yescharacteristic, complemented, Sylow
177C22×Q8—1181no—
178C22×D4C4×D5911yescharacteristic
179C23×C4D23611yescharacteristic
180C23×C4C22⋊D5911yescharacteristic
181C2×M16—2361no—
182C22×C8—41no—
183C4⋊Q8—2361no—
184C4⋊Q8—2361no—
185C22⋊Q8—2364no—
186C22⋊Q8—2364no—
187C4×D4—41no—
188C4×D4—41no—
189C2×C4⋊C4—592no—
190C2×C4⋊C4—1181no—
191C2×C4⋊C4—1181no—
192C2×C4⋊C4—1182no—
193C2×C4⋊C4—1182no—
194C2×C4⋊C4C22×D5911yescharacteristic
195C2×C22⋊C4—1182no—
196C2×C22⋊C4—1182no—
197C2×C22⋊C4D23611yescharacteristic
198C2×C22⋊C4C22⋊D5911yescharacteristic
199C2×C42—592no—
200C2×C42C4×D5911yescharacteristic
201C22⋊C8—22no—
202C22⋊C8—41no—
203C22⋊C8—1182no—
204C22⋊C8—2361no—
205#95—1181no—
206C24(C2×D59)⋊C411yescharacteristic
207C24(C2×D59)⋊C411yescharacteristic
208C2×Q8—2362no—
209C2×Q8—2362no—
210C2×Q8—2361no—
211C2×Q8—2361no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—1182no—
222C22×C4—1182no—
223C22×C4(C2×D59)⋊C411yescharacteristic
224C22×C4(C2×D59)⋊C411yescharacteristic
225C22×C4C2×C4×D5911yescharacteristic
226C22×C4C2×D23611yescharacteristic
227C22×C4C2×C22⋊D5911yescharacteristic
228M16—4722no—
229C2×C8—42no—
230C2×C8—82no—
231C2×C8—2362no—
232C4⋊C4—41no—
233C4⋊C4—1184no—
234C4⋊C4—2362no—
235C4⋊C4—2362no—
236C4⋊C4—2364no—
237C4⋊C4—2364no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—2364no—
241C22⋊C4—2364no—
242C42—21no—
243C42—21no—
244C42—1184no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23#1427364512yes—
255C23C2×(C2×D59)⋊C411yescharacteristic
256Q8—4722no—
257Q8—4722no—
258D4—44no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—1184no—
273C2×C4—2362no—
274C2×C4—2362no—
275C2×C4—2362no—
276C2×C4—2362no—
277C2×C4—2362no—
278C2×C4#1427364111yescharacteristic
279C2×C4#1427364111yescharacteristic
280C2×C4#1427373312yes—
281C8—82no—
282C8—4722no—
283C22—21no—
284C22—21no—
285C22—22no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—81no—
295C22#1683736811yescharacteristic
296C22C2×#1427364111yescharacteristic
297C22C2×#1427364511yescharacteristic
298C4—21no—
299C4—22no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—4722no—
304C4—2364no—
305C4#1683719412yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#2142245911yescharacteristic
311C2C2×#1683719411yescharacteristic
312C2#2142579011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1459C2
socleC22×C59C2×#99
prime radicalC22×D4×C59C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1459D59
hypocenterC59#11570
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #21422041C2semidirect
176. C59#11570semidirect

Maximal subgroups

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

subgroupindex
#214290492
#214224342
#214220412
C59×#14592
#1157059