Finite Group Databasebeta

← #28499301

Subgroups — #28499301

Order 15104.

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

normal subgroups only

This group has 462 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#21428413C211yesmaximal, characteristic
2#21422437C211yesmaximal, characteristic
3#21422119C214yesmaximal, complemented
4C4×#99×C59C211yesmaximal, characteristic
5C22×C4⋊Dic59C411yescharacteristic, complemented
6#16837459—21no—
7#16837341C412yescomplemented
8#16837338C2212yes—
9#16837300—21no—
10#16837282—21no—
11#16837274—21no—
12#16837262—21no—
13#16837262C411yescharacteristic, complemented
14#16837126—21no—
15#16837126C2212yes—
16C2×C4×D4×C59C2211yescharacteristic
17C2×#99×C59—21no—
18C4×C22⋊C4×C59—21no—
19C59×#342C2212yes—
20C23×Dic59—21no—
21C2×C4⋊Dic59—21no—
22C2×C4⋊Dic59—21no—
23C2×C4⋊Dic59—41no—
24C2×C4⋊Dic59—41no—
25C2×C4⋊Dic59C2×C412yes—
26C2×Dic59⋊C4—22no—
27C2×C22⋊Dic59—21no—
28C2×C22⋊Dic59—22no—
29C4⋊(C59⋊C8)—41no—
30C22⋊(C59⋊C8)—22no—
31#14273638—41no—
32C59⋊#95—22no—
33C59⋊#95—22no—
34C59⋊#95C2×C412yes—
35C22×D4×C59D411yescharacteristic
36C23×C4×C59C2×C411yescharacteristic
37C23×C4×C59D411yescharacteristic
38C4×D4×C59—41no—
39C4×D4×C59—41no—
40C2×C4⋊C4×C59C2×C411yescharacteristic
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59D411yescharacteristic
43C2×C22⋊C4×C59C2311yescharacteristic
44C2×C42×C59—41no—
45C2×C42×C59D411yescharacteristic
46#99×C59—24no—
47#95×C59—41no—
48C24×C59C22⋊C411yescharacteristic
49C24×C59C2×D411yescharacteristic
50C2×D4×C59—22no—
51C2×D4×C59—22no—
52C2×D4×C59—41no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—42no—
58C22×C4×C59—42no—
59C22×C4×C59—42no—
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22×C411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C22×Dic59—22no—
66C22×Dic59—22no—
67C22×Dic59—42no—
68C22×Dic59—82no—
69C4⋊C4×C59—41no—
70C22⋊C4×C59—24no—
71C22⋊C4×C59—41no—
72C22⋊C4×C59—41no—
73C22⋊C4×C59—42no—
74C42×C59—21no—
75C42×C59—21no—
76C42×C59—82no—
77C22⋊Dic59—42no—
78C22⋊Dic59—82no—
79C4⋊Dic59—42no—
80C4⋊Dic59—41no—
81C4⋊Dic59—41no—
82C4⋊Dic59—42no—
83Dic59⋊C4—82no—
84C2×C59⋊C8—42no—
85C23×C59—21no—
86C23×C59—21no—
87C23×C59—21no—
88C23×C59—21no—
89C23×C59—41no—
90C23×C59—41no—
91C23×C59—41no—
92C23×C59—41no—
93C23×C59#10412yes—
94C23×C59C2×C22⋊C411yescharacteristic
95C23×C59#13712yes—
96D4×C59—44no—
97C2×C4×C59—22no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—44no—
111C2×C4×C59—81no—
112C2×C4×C59—82no—
113C2×C4×C59—82no—
114C2×C4×C59#9911yescharacteristic
115C2×C4×C59#9911yescharacteristic
116C2×Dic59—42no—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—42no—
120C2×Dic59—42no—
121C2×Dic59—42no—
122C2×Dic59—84no—
123C59⋊C8—82no—
124#11786—591nomaximal, Sylow
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—22no—
128C22×C59—22no—
129C22×C59—42no—
130C22×C59—42no—
131C22×C59—42no—
132C22×C59—41no—
133C22×C59—41no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—81no—
137C22×C59C2×#9911yescharacteristic
138C22×C59#41911yescharacteristic
139C22×C59C2×#10411yescharacteristic
140C4×C59—21no—
141C4×C59—41no—
142C4×C59—41no—
143C4×C59—41no—
144C4×C59—41no—
145C4×C59—84no—
146C4×C59#35212yes—
147Dic59—82no—
148Dic59—82no—
149#2487—591no—
150C4×#99D5911yescharacteristic
151#1466—591no—
152#1352—594no—
153C2×C59—21no—
154C2×C59—21no—
155C2×C59—42no—
156C2×C59—42no—
157C2×C59#150811yescharacteristic
158C2×C59C2×#35211yescharacteristic
159C2×C59#212811yescharacteristic
160C2×C4×D4C2×D5911yescharacteristic
161C22×C4⋊C4—591no—
162#432—1181no—
163C2×#99—21no—
164#400—592no—
165#399—592no—
166#389—1181no—
167#386—1181no—
168#385—1181no—
169#380—591no—
170#380—1181no—
171C4×C22⋊C4—21no—
172#342C2×D5912yes—
173#323—592no—
174#323—1181no—
175C59#1178611yescharacteristic, complemented, Sylow
176C22×D4C22⋊D5911yescharacteristic
177C23×C4—1181no—
178C23×C4C4×D5911yescharacteristic
179C23×C4C22⋊D5911yescharacteristic
180C4×D4—41no—
181C4×D4—41no—
182C2×C4⋊C4—592no—
183C2×C4⋊C4—1182no—
184C2×C4⋊C4—1181no—
185C2×C4⋊C4—1181no—
186C2×C4⋊C4—2361no—
187C2×C4⋊C4—2361no—
188C2×C4⋊C4C4×D5911yescharacteristic
189C2×C22⋊C4—22no—
190C2×C22⋊C4—1182no—
191C2×C22⋊C4—1181no—
192C2×C22⋊C4D23611yescharacteristic
193C2×C22⋊C4C22×D5911yescharacteristic
194C2×C42—41no—
195C2×C42D23611yescharacteristic
196C4⋊C8—2361no—
197#99—24no—
198C22⋊C8—1182no—
199#97—2361no—
200#95—41no—
201#95—592no—
202#95—1182no—
203#95—1182no—
204C24(C2×D59)⋊C411yescharacteristic
205C24C2×C22⋊D5911yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—1182no—
217C22×C4—1182no—
218C22×C4—4722no—
219C22×C4—2362no—
220C22×C4(C2×D59)⋊C411yescharacteristic
221C22×C4(C2×D59)⋊C411yescharacteristic
222C22×C4(C2×D59)⋊C411yescharacteristic
223C22×C4C2×C4×D5911yescharacteristic
224C22×C4C2×D23611yescharacteristic
225C2×C8—2362no—
226C4⋊C4—41no—
227C4⋊C4—4722no—
228C4⋊C4—2362no—
229C4⋊C4—2362no—
230C4⋊C4—2361no—
231C4⋊C4—2361no—
232C22⋊C4—24no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—42no—
236C22⋊C4—4722no—
237C22⋊C4—2362no—
238C42—21no—
239C42—21no—
240C42—82no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23#1427365512yes—
250C23C2×(C2×D59)⋊C411yescharacteristic
251C23#1427375712yes—
252D4—44no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—44no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—4724no—
271C2×C4—2362no—
272C2×C4—2362no—
273C2×C4—2362no—
274C2×C4—2362no—
275C2×C4—2362no—
276C2×C4—2362no—
277C2×C4#1427364111yescharacteristic
278C2×C4#1427364111yescharacteristic
279C8—4722no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22C2×#1427364111yescharacteristic
293C22#1683741011yescharacteristic
294C22C2×#1427365511yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—84no—
301C4—4722no—
302C4—4722no—
303C4#1683719112yes—
304C2—21no—
305C2—21no—
306C2—42no—
307C2—42no—
308C2#2142261511yescharacteristic
309C2C2×#1683719111yescharacteristic
310C2#2142576711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC4×#99×C59C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenterC4×#99D59
hypocenterC59#11786
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #21422119C2semidirect
5. C22×C4⋊Dic59C4semidirect
7. #16837341C4semidirect
13. #16837262C4semidirect
175. C59#11786semidirect

Maximal subgroups

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

subgroupindex
#214284132
#214224372
#214221192
C4×#99×C592
#1178659