Finite Group Databasebeta

← #28499708

Subgroups — #28499708

Order 15104.

472 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 107 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#21424084C211yesmaximal, characteristic
2#21422417C211yesmaximal, characteristic
3#21422096C214yesmaximal, complemented
4C59×#2486C211yesmaximal, characteristic
5#16837455—21no—
6C2×#14273642—21no—
7#16837262—21no—
8#16837169C2212yes—
9#16837126—21no—
10#16837126C2212yes—
11C2×#123×C59—21no—
12C2×C4×D4×C59C2211yescharacteristic
13C22×C4⋊C4×C59C411yescharacteristic, complemented
14C59×#398C412yescomplemented
15C59×#397C2212yes—
16C59×#389—21no—
17C59×#381—21no—
18C59×#380—21no—
19C59×#380C411yescharacteristic, complemented
20C2×C4⋊Dic59—41no—
21C2×C22⋊Dic59—22no—
22C4⋊(C59⋊C8)—41no—
23#14273642—42no—
24C22⋊(C59⋊C8)—22no—
25C4×C59⋊C8—41no—
26C59⋊#95—41no—
27C22×D4×C59D411yescharacteristic
28C23×C4×C59—21no—
29C23×C4×C59C2×C411yescharacteristic
30C23×C4×C59D411yescharacteristic
31#123×C59—42no—
32#123×C59—42no—
33C4×D4×C59—41no—
34C4×D4×C59—41no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59—22no—
38C2×C4⋊C4×C59—41no—
39C2×C4⋊C4×C59—41no—
40C2×C4⋊C4×C59C2×C412yes—
41C2×C4⋊C4×C59C2×C411yescharacteristic
42C2×C22⋊C4×C59—21no—
43C2×C22⋊C4×C59—22no—
44C2×C22⋊C4×C59D411yescharacteristic
45C2×C22⋊C4×C59C2311yescharacteristic
46C2×C42×C59D411yescharacteristic
47#95×C59—22no—
48#95×C59—22no—
49#95×C59C2×C412yes—
50C24×C59C22⋊C411yescharacteristic
51C24×C59C2×D411yescharacteristic
52C2×D4×C59—41no—
53C2×D4×C59—41no—
54C2×D4×C59D812yes—
55C2×D4×C59SD1612yes—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—22no—
61C22×C4×C59—22no—
62C22×C4×C59—42no—
63C22×C4×C59—42no—
64C22×C4×C59—42no—
65C22×C4×C59—42no—
66C22×C4×C59—42no—
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×Dic59—42no—
73C4⋊C4×C59—41no—
74C4⋊C4×C59—42no—
75C4⋊C4×C59—42no—
76C4⋊C4×C59—41no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—44no—
79C22⋊C4×C59—41no—
80C22⋊C4×C59—41no—
81C22⋊C4×C59—42no—
82C22⋊C4×C59—42no—
83C22⋊C4×C59—42no—
84C42×C59—21no—
85C42×C59—21no—
86C22⋊Dic59—42no—
87C22⋊Dic59—42no—
88C4⋊Dic59—82no—
89C2×C59⋊C8—42no—
90C23×C59—21no—
91C23×C59—21no—
92C23×C59—21no—
93C23×C59—21no—
94C23×C59—41no—
95C23×C59—41no—
96C23×C59—41no—
97C23×C59—41no—
98C23×C59#10212yes—
99C23×C59C2×C22⋊C411yescharacteristic
100C23×C59C2×D811yescharacteristic
101C23×C59C2×SD1611yescharacteristic
102D4×C59—42no—
103D4×C59—42no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—22no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—42no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59—81no—
123C2×C4×C59—84no—
124C2×C4×C59#9911yescharacteristic
125C2×C4×C59#9911yescharacteristic
126C2×C4×C59#10212yes—
127C2×Dic59—82no—
128C2×Dic59—82no—
129C2×Dic59—82no—
130C59⋊C8—82no—
131#11874—591nomaximal, Sylow
132C22×C59—21no—
133C22×C59—21no—
134C22×C59—21no—
135C22×C59—22no—
136C22×C59—21no—
137C22×C59—42no—
138C22×C59—41no—
139C22×C59—41no—
140C22×C59—42no—
141C22×C59—41no—
142C22×C59—41no—
143C22×C59—41no—
144C22×C59—41no—
145C22×C59—81no—
146C22×C59C2×#9911yescharacteristic
147C22×C59C2×#10211yescharacteristic
148C22×C59#42111yescharacteristic
149C4×C59—21no—
150C4×C59—22no—
151C4×C59—41no—
152C4×C59—41no—
153C4×C59—41no—
154C4×C59—82no—
155C4×C59—82no—
156C4×C59#35212yes—
157Dic59—162no—
158#2486D5911yescharacteristic
159#1799—591no—
160#1460—591no—
161#1344—594no—
162C2×C59—21no—
163C2×C59—21no—
164C2×C59—41no—
165C2×C59—42no—
166C2×C59—41no—
167C2×C59#151911yescharacteristic
168C2×C59C2×#35211yescharacteristic
169C2×C59#212911yescharacteristic
170C2×#123—21no—
171C2×C4×D4C2×D5911yescharacteristic
172C22×C4⋊C4Dic5911yescharacteristic, complemented
173#431—1181no—
174C2×#99—1181no—
175#398Dic5912yescomplemented
176#397C2×D5912yes—
177#389—21no—
178#381—21no—
179#380—21no—
180#380—1181no—
181#380Dic5911yescharacteristic, complemented
182#342—592no—
183#323—592no—
184#323—1181no—
185C59#1187411yescharacteristic, complemented, Sylow
186C22×D4C22⋊D5911yescharacteristic
187C23×C4—21no—
188C23×C4C2×Dic5911yescharacteristic
189C23×C4C22⋊D5911yescharacteristic
190#123—42no—
191#123—42no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—2361no—
200C2×C4⋊C4C2×Dic5912yes—
201C2×C4⋊C4C2×Dic5911yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—1182no—
205C2×C22⋊C4C22⋊D5911yescharacteristic
206C2×C22⋊C4C22×D5911yescharacteristic
207C2×C42C22⋊D5911yescharacteristic
208C4⋊C8—2361no—
209#99—2362no—
210C22⋊C8—1182no—
211C4×C8—2361no—
212#95—22no—
213#95—22no—
214#95—2361no—
215#95C2×Dic5912yes—
216C24C22⋊Dic5911yescharacteristic
217C24C2×C22⋊D5911yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4#1174566912yes—
221C2×D4#1174567012yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—2362no—
234C22×C4C22⋊Dic5911yescharacteristic
235C22×C4C22⋊Dic5911yescharacteristic
236C22×C4C22⋊Dic5911yescharacteristic
237C22×C4C22×Dic5911yescharacteristic
238C22×C4C2×C22⋊D5911yescharacteristic
239C2×C8—2362no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—44no—
246C4⋊C4—4722no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—2362no—
253C22⋊C4—2362no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23#1427364812yes—
265C23C2×C22⋊Dic5911yescharacteristic
266C23C2×#1174566911yescharacteristic
267C23C2×#1174567011yescharacteristic
268D4—42no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—84no—
290C2×C4—4722no—
291C2×C4—4722no—
292C2×C4—4722no—
293C2×C4#1427364211yescharacteristic
294C2×C4#1427364211yescharacteristic
295C2×C4#1427364812yes—
296C8—4722no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#1427364211yescharacteristic
312C22C2×#1427364811yescharacteristic
313C22#1683741911yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—82no—
320C4—82no—
321C4—9442no—
322C4#1683719212yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2#2142266411yescharacteristic
329C2C2×#1683719211yescharacteristic
330C2#2142577111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2486C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenter#2486D59
hypocenterC59#11874
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
3. #21422096C2semidirect
13. C22×C4⋊C4×C59C4semidirect
14. C59×#398C4semidirect
19. C59×#380C4semidirect
172. C22×C4⋊C4Dic59semidirect
175. #398Dic59semidirect
181. #380Dic59semidirect
185. C59#11874semidirect

Maximal subgroups

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

subgroupindex
#214240842
#214224172
#214220962
C59×#24862
#1187459