Finite Group Databasebeta

← #28498575

Subgroups — #28498575

Order 15104.

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

normal subgroups only

This group has 314 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#21433073C211yesmaximal, characteristic
2#21425893C211yesmaximal, characteristic, complemented
3#21425850C211yesmaximal, characteristic, complemented
4Q8⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
5#21422052C211yesmaximal, characteristic, complemented
6#21422050C211yesmaximal, characteristic, complemented
7C59×Q8⋊C16C211yesmaximal, characteristic, complemented
8#16837939C411yescharacteristic
9C2×#14273666—21no—
10#16837533C411yescharacteristic
11#16837479C411yescharacteristic
12#16837472—21no—
13C8×D236—21no—
14C8×D236C2211yescharacteristic
15#16837462—21no—
16#16837458C411yescharacteristic
17#16837356—21no—
18C4×C8×D59C2211yescharacteristic
19C4⋊(C59⋊C16)C2211yescharacteristic
20Dic59⋊C16—21no—
21Dic59⋊C16—21no—
22(C2×D59)⋊C16—21no—
23C4×C59⋊C16C2211yescharacteristic
24C8×Q8×C59C2211yescharacteristic
25C59×C4⋊C16C2211yescharacteristic
26C4×C16×C59C2211yescharacteristic
27C2×#11745659—21no—
28C2×C8×D59—21no—
29C2×C8×D59D411yescharacteristic
30C2×C8×D59D411yescharacteristic
31#14273722—21no—
32#14273718C811yescharacteristic
33#14273716C811yescharacteristic
34#14273712—21no—
35#14273703C811yescharacteristic
36#14273702C811yescharacteristic
37C4×D236—21no—
38C4×D236C2×C411yescharacteristic
39#14273684—21no—
40C42×D59C2×C411yescharacteristic
41#14273666—42no—
42C2×C59⋊C16—21no—
43C2×C59⋊C16—21no—
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—
50#14273637—21no—
51C4×C59⋊C8C2×C411yescharacteristic
52C8×Dic59D411yescharacteristic
53C8×Dic59D411yescharacteristic
54C4×Q8×C59C2×C411yescharacteristic
55C2×C16×C59—21no—
56C2×C16×C59—21no—
57C4⋊C8×C59—21no—
58C4⋊C8×C59C2×C411yescharacteristic
59C4×C8×C59—21no—
60C4×C8×C59C2311yescharacteristic
61C2×Q8×C59C2×C811yescharacteristic
62C2×D236—21no—
63C2×D236C2×C811yescharacteristic
64C2×C4×D59—21no—
65C2×C4×D59C22⋊C411yescharacteristic
66C2×C4×D59C2×C811yescharacteristic
67C2×C4×D59M1611yescharacteristic
68C2×C8×C59—21no—
69C2×C8×C59C2×D411yescharacteristic
70C2×C8×C59C2×D411yescharacteristic
71C4⋊C4×C59—21no—
72C4⋊C4×C59C2×C811yescharacteristic
73C42×C59—21no—
74C42×C59C22×C411yescharacteristic
75(C2×D59)⋊C4—21no—
76(C2×D59)⋊C4—21no—
77(C2×D59)⋊C4—21no—
78(C2×D59)⋊C4—21no—
79(C2×D59)⋊C4—21no—
80C4⋊Dic59—21no—
81C4⋊Dic59C2×C811yescharacteristic
82Dic59⋊C4—21no—
83Dic59⋊C4—21no—
84C4×Dic59C22⋊C411yescharacteristic
85C4×Dic59C2×C811yescharacteristic
86C4×Dic59M1611yescharacteristic
87C2×C59⋊C8—21no—
88C2×C59⋊C8C22⋊C411yescharacteristic
89C2×C59⋊C8C22⋊C411yescharacteristic
90#11745659—42no—
91C8×D59—21no—
92C8×D59—21no—
93C8×D59—41no—
94C8×D59—41no—
95C8×D59—41no—
96C16×C59—41no—
97C16×C59—41no—
98C59⋊C16—41no—
99C59⋊C16—41no—
100C22×D59—21no—
101C22×D59C22⋊C811yescharacteristic
102Q8×C59—21no—
103Q8×C59—21no—
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—21no—
114D236—21no—
115D236—42no—
116C4×D59—21no—
117C4×D59—21no—
118C4×D59—21no—
119C4×D59—21no—
120C4×D59—41no—
121C4×D59—41no—
122C4×D59—41no—
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#11624—591nomaximal, Sylow
131C22×C59C2×C22⋊C811yescharacteristic
132C2×D59—21no—
133C2×D59—21no—
134C2×D59—21no—
135C2×D59—41no—
136C2×D59—41no—
137C2×D59—41no—
138C4×C59—21no—
139C4×C59—41no—
140C4×C59—41no—
141C4×C59#41711yescharacteristic
142C4×C59#42111yescharacteristic
143C4×C59#50411yescharacteristic
144C4×C59#50411yescharacteristic
145Dic59—21no—
146Dic59—21no—
147Dic59—21no—
148Dic59—41no—
149#2966—591no—
150#2162—591no—
151#2150—591no—
152Q8⋊C16—591no—
153Q8⋊C16D5911yescharacteristic, complemented
154D4⋊C16—591no—
155D4⋊C16—591no—
156C2×C59#147711yescharacteristic
157C2×C59#211311yescharacteristic
158C2×C59#211711yescharacteristic
159D59—41no—
160D59—81no—
161#529—591no—
162C2×M32—1181no—
163#446—591no—
164C8×Q8C2×D5911yescharacteristic
165#436—591no—
166#435—1181no—
167C8×D4—591no—
168C8×D4—1181no—
169#433—1181no—
170#432—591no—
171#405—1181no—
172C2×C4×C8—591no—
173C4⋊C16—591no—
174C4⋊C16—1181no—
175C4⋊C16—1181no—
176C4⋊C16C2×D5911yescharacteristic
177C22⋊C16—1181no—
178C4×C16—591no—
179C4×C16C2×D5911yescharacteristic
180C59#1162411yescharacteristic, complemented, Sylow
181C2×M16—1181no—
182C22×C8—591no—
183C22×C8—591no—
184C22×C8—1181no—
185#126—1181no—
186#125—591no—
187#124—591no—
188#123—1181no—
189C22⋊Q8—591no—
190#121—591no—
191C4×Q8C4×D5911yescharacteristic
192C4×D4—591no—
193C4×D4—1181no—
194#117—1181no—
195C2×C42—591no—
196M32—2362no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—1181no—
200C2×C16—1181no—
201C4⋊C8—21no—
202C4⋊C8—591no—
203C4⋊C8—1181no—
204C4⋊C8C4×D5911yescharacteristic
205C22⋊C8—1181no—
206C22⋊C8—1181no—
207C22⋊C8—1181no—
208#97—1181no—
209#97—1181no—
210C4×C8—21no—
211C4×C8—591no—
212C4×C8—591no—
213C4×C8—591no—
214C4×C8C22×D5911yescharacteristic
215C2×Q8C8×D5911yescharacteristic
216C2×D4—591no—
217C2×D4—1181no—
218C22×C4—591no—
219C22×C4—591no—
220C22×C4—591no—
221C22×C4—1181no—
222M16—2362no—
223C2×C8—21no—
224C2×C8—591no—
225C2×C8—591no—
226C2×C8—1181no—
227C2×C8—1181no—
228C2×C8—1181no—
229C2×C8—2361no—
230C2×C8—2361no—
231C2×C8—2361no—
232C2×C8D4×D5911yescharacteristic
233C2×C8D4×D5911yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—591no—
236C4⋊C4—1181no—
237C4⋊C4—1181no—
238C4⋊C4—1181no—
239C4⋊C4C8×D5911yescharacteristic
240C22⋊C4—1181no—
241C22⋊C4—1181no—
242C22⋊C4—1181no—
243C22⋊C4—1181no—
244C22⋊C4—1181no—
245C42—21no—
246C42—591no—
247C42—591no—
248C42—591no—
249C42C2×C4×D5911yescharacteristic
250C16—41no—
251C16—41no—
252C16—2361no—
253C16—2361no—
254C23—591no—
255C23—1181no—
256Q8—21no—
257Q8—21no—
258D4—1181no—
259D4—1181no—
260D4—2362no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—591no—
264C2×C4—591no—
265C2×C4—591no—
266C2×C4—1181no—
267C2×C4—1181no—
268C2×C4—1181no—
269C2×C4—1181no—
270C2×C4—1181no—
271C2×C4—2361no—
272C2×C4—2361no—
273C2×C4—2361no—
274C2×C4C22⋊C4×D5911yescharacteristic
275C2×C4C2×C8×D5911yescharacteristic
276C2×C4M16×D5911yescharacteristic
277C8—21no—
278C8—41no—
279C8—1181no—
280C8—1181no—
281C8—2361no—
282C8#1427375411yescharacteristic
283C8#1427375911yescharacteristic
284C22—1181no—
285C22—1181no—
286C22—1181no—
287C22—2361no—
288C22—2361no—
289C22—2361no—
290C22C22⋊C8×D5911yescharacteristic
291C4—21no—
292C4—41no—
293C4—41no—
294C4—1181no—
295C4—1181no—
296C4—1181no—
297C4—2361no—
298C4#1683740211yescharacteristic
299C4#1683741711yescharacteristic
300C4#1683781811yescharacteristic
301C4#1683782011yescharacteristic
302C2—4721no—
303C2—2361no—
304C2#2142248211yescharacteristic
305C2#2142570811yescharacteristic
306C2#2142572211yescharacteristic
subgroupG/H
centerC2×C8D4×D59
commutatorC2×C4×C59C22×C8
FrattiniC4×C8C22×D59
FittingC59×Q8⋊C16C2
socleC22×C59C2×C22⋊C8
prime radical#14273702C8
cosocleC4×C8×C59C23
hypercenterQ8⋊C16D59
hypocenterC59#11624
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ Q8⋊C16

Complements

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

NH ≅ G/Ntype
2. #21425893C2semidirect
3. #21425850C2semidirect
4. Q8⋊(C59⋊C16)C2semidirect
5. #21422052C2semidirect
6. #21422050C2semidirect
7. C59×Q8⋊C16C2semidirect
153. Q8⋊C16D59semidirect
180. C59#11624semidirect

Maximal subgroups

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

subgroupindex
#214330732
#214258932
#214258502
Q8⋊(C59⋊C16)2
#214220522
#214220502
C59×Q8⋊C162
#1162459