Finite Group Databasebeta

← #28498585

Subgroups — #28498585

Order 15104.

328 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 93 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#21433061C211yesmaximal, characteristic
2#21425897C211yesmaximal, characteristic, complemented
3C4⋊C16×D59C211yesmaximal, characteristic, complemented
4#21422078C211yesmaximal, characteristic, complemented
5#21422055C211yesmaximal, characteristic, complemented
6#21422053C211yesmaximal, characteristic, complemented
7C59×#1338C211yesmaximal, characteristic, complemented
8C4×#11745689C411yescharacteristic
9C2×C16×D59—21no—
10#16837533C411yescharacteristic
11#16837481C2211yescharacteristic
12#16837479C411yescharacteristic
13#16837475—21no—
14#16837472—21no—
15#16837454—21no—
16#16837454C411yescharacteristic
17#16837354—21no—
18C4×C8×D59C2211yescharacteristic
19C4⋊(C59⋊C16)C2211yescharacteristic
20Dic59⋊C16—21no—
21Dic59⋊C16—21no—
22(C2×D59)⋊C16—21no—
23#16837178C2211yescharacteristic
24C59×#446C2211yescharacteristic
25C59×C4⋊C16C2211yescharacteristic
26C59×#346C2211yescharacteristic
27C2×#11745689C811yescharacteristic
28C2×#11745659—21no—
29C2×#11745659—21no—
30C2×C8×D59D411yescharacteristic
31C2×C8×D59D411yescharacteristic
32Q8×Dic59C811yescharacteristic
33C4×D236—21no—
34C4×D236C2×C411yescharacteristic
35#14273689—21no—
36#14273689C811yescharacteristic
37#14273685—21no—
38#14273685C811yescharacteristic
39C42×D59—21no—
40C42×D59C2×C411yescharacteristic
41C2×C59⋊C16—21no—
42C2×C59⋊C16—21no—
43C16×D59—42no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)C2×C411yescharacteristic
46(C2×D59)⋊C8—21no—
47(C2×D59)⋊C8—21no—
48(C2×D59)⋊C8—21no—
49#14273638—21no—
50C4×C59⋊C8C2×C411yescharacteristic
51C8×Dic59—21no—
52C8×Dic59D411yescharacteristic
53C8×Dic59D411yescharacteristic
54C4×Q8×C59C2×C411yescharacteristic
55C2×C16×C59—21no—
56C2×C16×C59—21no—
57C4⋊C8×C59—21no—
58C4⋊C8×C59C2×C411yescharacteristic
59#97×C59—21no—
60C4×C8×C59C2311yescharacteristic
61C2×Q8×C59C2×C811yescharacteristic
62#11745689—42no—
63C2×D236—21no—
64C2×D236C2×C811yescharacteristic
65C2×C4×D59—21no—
66C2×C4×D59—21no—
67C2×C4×D59—21no—
68C2×C4×D59C22⋊C411yescharacteristic
69C2×C4×D59C2×C811yescharacteristic
70C2×C4×D59M1611yescharacteristic
71C2×C8×C59—21no—
72C2×C8×C59C2×D411yescharacteristic
73C2×C8×C59C2×D411yescharacteristic
74C4⋊C4×C59—21no—
75C4⋊C4×C59C2×C811yescharacteristic
76C42×C59—21no—
77C42×C59C22×C411yescharacteristic
78(C2×D59)⋊C4—21no—
79(C2×D59)⋊C4—21no—
80(C2×D59)⋊C4—21no—
81C4⋊Dic59—21no—
82C4⋊Dic59C2×C811yescharacteristic
83C4×Dic59—21no—
84C4×Dic59—21no—
85C4×Dic59C22⋊C411yescharacteristic
86C4×Dic59C2×C811yescharacteristic
87C4×Dic59M1611yescharacteristic
88C2×C59⋊C8—21no—
89C2×C59⋊C8C22⋊C411yescharacteristic
90C2×C59⋊C8C22⋊C411yescharacteristic
91#11745659—42no—
92#11745659—41no—
93#11745659—41no—
94C8×D59—21no—
95C8×D59—21no—
96C8×D59—41no—
97C16×C59—41no—
98C16×C59—41no—
99C59⋊C16—41no—
100C59⋊C16—41no—
101C22×D59—21no—
102C22×D59C22⋊C811yescharacteristic
103Q8×C59—41no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59C2×C22⋊C411yescharacteristic
107C2×C4×C59C22×C811yescharacteristic
108C2×C4×C59C2×M1611yescharacteristic
109C2×Dic59—21no—
110C2×Dic59C22⋊C811yescharacteristic
111C2×Dic59C22⋊C811yescharacteristic
112C2×Dic59C22⋊C811yescharacteristic
113D236—41no—
114D236—42no—
115C4×D59—22no—
116C4×D59—21no—
117C4×D59—21no—
118C4×D59—22no—
119C4×D59—42no—
120C4×D59—41no—
121C4×D59—41no—
122C4×D59—42no—
123C8×C59—21no—
124C8×C59—41no—
125C8×C59AGL(1,ℤ/8)11yescharacteristic
126C8×C59#13711yescharacteristic
127C59⋊C8—21no—
128C59⋊C8—21no—
129C59⋊C8—41no—
130#11626—591nomaximal, Sylow
131C22×C59C2×C22⋊C811yescharacteristic
132C2×D59—21no—
133C2×D59—21no—
134C2×D59—41no—
135C2×D59—41no—
136C2×D59—41no—
137C2×D59#34912yes—
138C4×C59—21no—
139C4×C59—21no—
140C4×C59—41no—
141C4×C59—41no—
142C4×C59#41711yescharacteristic
143C4×C59#42111yescharacteristic
144Dic59—21no—
145Dic59—21no—
146Dic59—41no—
147Dic59#35012yes—
148#2965—591no—
149#2163—591no—
150C2×C4⋊C16—591no—
151#1338—591no—
152#1338D5911yescharacteristic, complemented
153#1330—591no—
154#1330—591no—
155C2×C59#147711yescharacteristic
156C2×C59C2×#34911yescharacteristic
157C2×C59C2×#35011yescharacteristic
158D59—22no—
159D59—81no—
160C4×#41—591no—
161C22×C16—1181no—
162#446—591no—
163#446C2×D5911yescharacteristic
164#436—591no—
165#436—591no—
166#435—1181no—
167#435—1181no—
168#431—591no—
169#431—1181no—
170C4×M16—1181no—
171C2×C4×C8—591no—
172C4⋊C16—591no—
173C4⋊C16—1181no—
174C4⋊C16—1181no—
175C4⋊C16C2×D5911yescharacteristic
176C22⋊C16—1181no—
177#346—591no—
178#346C2×D5911yescharacteristic
179C59#1162611yescharacteristic, complemented, Sylow
180C2×#41—591no—
181C2×M16—1181no—
182C2×M16—1181no—
183C22×C8—591no—
184C22×C8—591no—
185C4×Q8—591no—
186C4×Q8C4×D5911yescharacteristic
187C4×D4—591no—
188C4×D4—591no—
189C4×D4—1181no—
190C4×D4—1181no—
191#117—591no—
192#117—1181no—
193C2×C42—591no—
194C2×C42—1181no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—1181no—
198C2×C16—1181no—
199C2×C16—2362no—
200C4⋊C8—21no—
201C4⋊C8—591no—
202C4⋊C8—1181no—
203C4⋊C8C4×D5911yescharacteristic
204C22⋊C8—1181no—
205C22⋊C8—1181no—
206C22⋊C8—1181no—
207#97—21no—
208#97—1181no—
209C4×C8—591no—
210C4×C8—591no—
211C4×C8—591no—
212C4×C8—1181no—
213C4×C8C22×D5911yescharacteristic
214#41—2362no—
215C2×Q8C8×D5911yescharacteristic
216C2×D4—591no—
217C2×D4—1181no—
218C22×C4—591no—
219C22×C4—591no—
220C22×C4—591no—
221C22×C4—1181no—
222C22×C4—1181no—
223C22×C4—1181no—
224M16—2362no—
225M16—2361no—
226M16—2361no—
227C2×C8—21no—
228C2×C8—591no—
229C2×C8—591no—
230C2×C8—1181no—
231C2×C8—1181no—
232C2×C8—1181no—
233C2×C8—2361no—
234C2×C8D4×D5911yescharacteristic
235C2×C8D4×D5911yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—591no—
238C4⋊C4—1181no—
239C4⋊C4C8×D5911yescharacteristic
240C22⋊C4—1181no—
241C22⋊C4—1181no—
242C22⋊C4—1181no—
243C42—21no—
244C42—591no—
245C42—591no—
246C42—591no—
247C42—1181no—
248C42—1181no—
249C42C2×C4×D5911yescharacteristic
250C16—41no—
251C16—41no—
252C16—2361no—
253C16—2361no—
254C23—591no—
255C23—1181no—
256Q8—41no—
257D4—2361no—
258D4—2362no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—591no—
262C2×C4—591no—
263C2×C4—591no—
264C2×C4—1182no—
265C2×C4—1182no—
266C2×C4—1181no—
267C2×C4—1181no—
268C2×C4—1181no—
269C2×C4—2362no—
270C2×C4—2362no—
271C2×C4—2361no—
272C2×C4—2361no—
273C2×C4C22⋊C4×D5911yescharacteristic
274C2×C4C2×C8×D5911yescharacteristic
275C2×C4M16×D5911yescharacteristic
276C8—21no—
277C8—41no—
278C8—1181no—
279C8—1181no—
280C8—2361no—
281C8#1427375411yescharacteristic
282C8#1427375911yescharacteristic
283C22—592no—
284C22—1181no—
285C22—1181no—
286C22—2361no—
287C22—2361no—
288C22—2361no—
289C22C22⋊C8×D5911yescharacteristic
290C4—21no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—592no—
295C4—1181no—
296C4—1181no—
297C4—2361no—
298C4#1683740211yescharacteristic
299C4#1683741711yescharacteristic
300C2—1182no—
301C2—4721no—
302C2#2142248211yescharacteristic
303C2#349×D5911yescharacteristic
304C2#350×D5911yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D59
commutatorC2×C4×C59C22×C8
FrattiniC4×C8C22×D59
FittingC59×#1338C2
socleC22×C59C2×C22⋊C8
prime radicalC2×#11745689C8
cosocleC4×C8×C59C23
hypercenter#1338D59
hypocenterC59#11626
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1338

Complements

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

NH ≅ G/Ntype
2. #21425897C2semidirect
3. C4⋊C16×D59C2semidirect
4. #21422078C2semidirect
5. #21422055C2semidirect
6. #21422053C2semidirect
7. C59×#1338C2semidirect
152. #1338D59semidirect
179. C59#11626semidirect

Maximal subgroups

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

subgroupindex
#214330612
#214258972
C4⋊C16×D592
#214220782
#214220552
#214220532
C59×#13382
#1162659