Finite Group Databasebeta

← #28499825

Subgroups — #28499825

Order 15104.

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

normal subgroups only

This group has 508 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#21422414C211yesmaximal, characteristic
3#21422120C214yesmaximal, complemented
4C59×#2542C211yesmaximal, characteristic
5C2×#14273642—21no—
6C2×C22⋊(C59⋊C8)—21no—
7#16837262—21no—
8#16837169C2212yes—
9#16837128—21no—
10#16837126C2212yes—
11C2×C4⋊Q8×C59C411yescharacteristic, complemented
12C2×C22⋊Q8×C59—21no—
13C2×C22⋊Q8×C59—21no—
14C2×C4×D4×C59C2211yescharacteristic
15C59×#399C412yescomplemented
16C59×#399C2212yes—
17C59×#390—21no—
18C59×#384—21no—
19C4×C4⋊C4×C59C411yescharacteristic, complemented
20C22×C59⋊C8—41no—
21C2×C4⋊Dic59—41no—
22C2×C22⋊Dic59—22no—
23#14273642—42no—
24C22⋊(C59⋊C8)—22no—
25C22⋊(C59⋊C8)—41no—
26C59⋊#95—41no—
27C22×Q8×C59—21no—
28C22×D4×C59D411yescharacteristic
29C23×C4×C59D411yescharacteristic
30C23×C4×C59D411yescharacteristic
31C4⋊Q8×C59—41no—
32C4⋊Q8×C59—41no—
33C22⋊Q8×C59—44no—
34C22⋊Q8×C59—44no—
35C4×D4×C59—41no—
36C4×D4×C59—41no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59—22no—
40C2×C4⋊C4×C59—22no—
41C2×C4⋊C4×C59C2×C412yes—
42C2×C4⋊C4×C59D411yescharacteristic
43C2×C22⋊C4×C59—22no—
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59C2×C411yescharacteristic
46C2×C22⋊C4×C59C2311yescharacteristic
47C2×C42×C59—21no—
48C2×C42×C59C2×C411yescharacteristic
49#95×C59C2×C412yes—
50C24×C59C22⋊C411yescharacteristic
51C24×C59C2×D411yescharacteristic
52C2×Q8×C59—42no—
53C2×Q8×C59—42no—
54C2×Q8×C59—41no—
55C2×Q8×C59—41no—
56C2×D4×C59—41no—
57C2×D4×C59—41no—
58C2×D4×C59SD1612yes—
59C2×D4×C59Q1612yes—
60C22×C4×C59—22no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—22no—
66C22×C4×C59—42no—
67C22×C4×C59—42no—
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×Dic59—42no—
74C4⋊C4×C59—24no—
75C4⋊C4×C59—41no—
76C4⋊C4×C59—44no—
77C4⋊C4×C59—44no—
78C4⋊C4×C59—42no—
79C4⋊C4×C59—42no—
80C22⋊C4×C59—41no—
81C22⋊C4×C59—41no—
82C22⋊C4×C59—44no—
83C22⋊C4×C59—44no—
84C42×C59—21no—
85C42×C59—21no—
86C42×C59—42no—
87C22⋊Dic59—42no—
88C22⋊Dic59—42no—
89C4⋊Dic59—82no—
90C2×C59⋊C8—42no—
91C2×C59⋊C8—82no—
92C23×C59—21no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—21no—
96C23×C59—41no—
97C23×C59—41no—
98C23×C59—41no—
99C23×C59—41no—
100C23×C59#10312yes—
101C23×C59C2×C22⋊C411yescharacteristic
102C23×C59C2×SD1611yescharacteristic
103C23×C59C2×Q1611yescharacteristic
104Q8×C59—82no—
105Q8×C59—82no—
106D4×C59—42no—
107D4×C59—42no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—22no—
111C2×C4×C59—24no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—42no—
123C2×C4×C59—42no—
124C2×C4×C59—42no—
125C2×C4×C59—81no—
126C2×C4×C59—81no—
127C2×C4×C59#9911yescharacteristic
128C2×C4×C59#9911yescharacteristic
129C2×C4×C59#10312yes—
130C2×Dic59—82no—
131C2×Dic59—82no—
132C2×Dic59—82no—
133C59⋊C8—82no—
134#11901—591nomaximal, Sylow
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—22no—
139C22×C59—21no—
140C22×C59—42no—
141C22×C59—41no—
142C22×C59—41no—
143C22×C59—41no—
144C22×C59—41no—
145C22×C59—41no—
146C22×C59—41no—
147C22×C59—81no—
148C22×C59—81no—
149C22×C59C2×#9911yescharacteristic
150C22×C59C2×#10311yescharacteristic
151C22×C59#42111yescharacteristic
152C4×C59—21no—
153C4×C59—22no—
154C4×C59—44no—
155C4×C59—41no—
156C4×C59—41no—
157C4×C59—41no—
158C4×C59—82no—
159C4×C59#35312yes—
160Dic59—162no—
161#2542D5911yescharacteristic
162#1799—591no—
163#1459—591no—
164#1352—594no—
165C2×C59—21no—
166C2×C59—21no—
167C2×C59—41no—
168C2×C59—42no—
169C2×C59—41no—
170C2×C59#152011yescharacteristic
171C2×C59#212311yescharacteristic
172C2×C59C2×#35311yescharacteristic
173C2×C4⋊Q8Dic5911yescharacteristic, complemented
174C2×C22⋊Q8—21no—
175C2×C22⋊Q8—21no—
176C2×C4×D4C2×D5911yescharacteristic
177C2×#99—1181no—
178C2×C22⋊C8—1181no—
179#399Dic5912yescomplemented
180#399C2×D5912yes—
181#390—21no—
182#384—21no—
183#380—1181no—
184C4×C4⋊C4Dic5911yescharacteristic, complemented
185#342—592no—
186#324—1181no—
187#323—592no—
188C59#1190111yescharacteristic, complemented, Sylow
189C22×Q8—21no—
190C22×D4C22⋊D5911yescharacteristic
191C23×C4C22⋊D5911yescharacteristic
192C23×C4C22⋊D5911yescharacteristic
193C22×C8—2361no—
194C4⋊Q8—41no—
195C4⋊Q8—41no—
196C22⋊Q8—44no—
197C22⋊Q8—44no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—2361no—
205C2×C4⋊C4C2×Dic5912yes—
206C2×C4⋊C4C22⋊D5911yescharacteristic
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—1182no—
210C2×C22⋊C4C2×Dic5911yescharacteristic
211C2×C22⋊C4C22×D5911yescharacteristic
212C2×C42—21no—
213C2×C42C2×Dic5911yescharacteristic
214#99—2362no—
215C22⋊C8—1182no—
216C22⋊C8—2361no—
217#95—2361no—
218#95C2×Dic5912yes—
219C24C22⋊Dic5911yescharacteristic
220C24C2×C22⋊D5911yescharacteristic
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4#1174567112yes—
228C2×D4#1174567212yes—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—2362no—
238C22×C4C22⋊Dic5911yescharacteristic
239C22×C4C22⋊Dic5911yescharacteristic
240C22×C4C22⋊Dic5911yescharacteristic
241C22×C4C22×Dic5911yescharacteristic
242C22×C4C2×C22⋊D5911yescharacteristic
243C2×C8—4722no—
244C2×C8—2362no—
245C4⋊C4—24no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—4722no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—2362no—
257C22⋊C4—2362no—
258C42—21no—
259C42—21no—
260C42—42no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1427365112yes—
270C23C2×C22⋊Dic5911yescharacteristic
271C23C2×#1174567111yescharacteristic
272C23C2×#1174567211yescharacteristic
273Q8—82no—
274Q8—82no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—4722no—
297C2×C4—4722no—
298C2×C4—4722no—
299C2×C4#1427364211yescharacteristic
300C2×C4#1427364211yescharacteristic
301C2×C4#1427365112yes—
302C8—4722no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22C2×#1427364211yescharacteristic
318C22C2×#1427365111yescharacteristic
319C22#1683741911yescharacteristic
320C4—22no—
321C4—21no—
322C4—44no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—9442no—
328C4#1683719512yes—
329C2—21no—
330C2—21no—
331C2—41no—
332C2—42no—
333C2—41no—
334C2#2142266811yescharacteristic
335C2#2142575011yescharacteristic
336C2C2×#1683719511yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2542C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenter#2542D59
hypocenterC59#11901
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
3. #21422120C2semidirect
11. C2×C4⋊Q8×C59C4semidirect
15. C59×#399C4semidirect
19. C4×C4⋊C4×C59C4semidirect
173. C2×C4⋊Q8Dic59semidirect
179. #399Dic59semidirect
184. C4×C4⋊C4Dic59semidirect
188. C59#11901semidirect

Maximal subgroups

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

subgroupindex
#214240842
#214224142
#214221202
C59×#25422
#1190159