Finite Group Databasebeta

← #28498275

Subgroups — #28498275

Order 15104.

414 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 91 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#21422426C211yesmaximal, characteristic
2C2×#16837128C211yesmaximal, characteristic
3#21422042C212yesmaximal, complemented
4#21422030C212yesmaximal, complemented
5C59×#2502C211yesmaximal, characteristic
6#16837366—21no—
7C2×C22⋊(C59⋊C8)—21no—
8#16837128—21no—
9#16837128—21no—
10#16837128C2212yes—
11#16837128C2212yes—
12C2×C22⋊Q8×C59C411yescharacteristic
13C22×C4⋊C4×C59C2211yescharacteristic
14C59×#398—21no—
15C59×#398—21no—
16C59×#391C411yescharacteristic
17C59×#391C2211yescharacteristic
18C59×#385—21no—
19C59×#385—21no—
20C59×#384C411yescharacteristic
21C59×#384C2211yescharacteristic
22C4×C22⋊C4×C59C411yescharacteristic
23C2×#11745664—41no—
24C22×C59⋊C8—41no—
25C22⋊(C59⋊C8)—22no—
26C22⋊(C59⋊C8)—22no—
27C22⋊(C59⋊C8)—41no—
28C22⋊(C59⋊C8)—41no—
29C22×Q8×C59C2×C411yescharacteristic
30C23×C4×C59C2×C411yescharacteristic
31C23×C4×C59D411yescharacteristic
32C23×C4×C59D411yescharacteristic
33C22⋊Q8×C59—41no—
34C22⋊Q8×C59—41no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59—41no—
41C2×C4⋊C4×C59—41no—
42C2×C4⋊C4×C59C2×C411yescharacteristic
43C2×C4⋊C4×C59C2×C411yescharacteristic
44C2×C4⋊C4×C59D411yescharacteristic
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C4⋊C4×C59C2311yescharacteristic
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49C2×C42×C59C2×C411yescharacteristic
50C2×C42×C59C2×C411yescharacteristic
51#95×C59—21no—
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—21no—
55#95×C59—21no—
56C24×C59C22⋊C411yescharacteristic
57C2×Q8×C59—22no—
58C2×Q8×C59—41no—
59C2×Q8×C59—41no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—22no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—41no—
72C22×C4×C59—81no—
73C22×C4×C59—81no—
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22×C411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—41no—
84C4⋊C4×C59—42no—
85C4⋊C4×C59—41no—
86C4⋊C4×C59—81no—
87C4⋊C4×C59—81no—
88C4⋊C4×C59—81no—
89C4⋊C4×C59—81no—
90C22⋊C4×C59—24no—
91C22⋊C4×C59—41no—
92C22⋊C4×C59—41no—
93C42×C59—22no—
94C42×C59—41no—
95#11745664—82no—
96C2×C59⋊C8—42no—
97C2×C59⋊C8—42no—
98C2×C59⋊C8—82no—
99C23×C59—22no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—41no—
103C23×C59#9911yescharacteristic
104C23×C59#9911yescharacteristic
105C23×C59C2×C22⋊C411yescharacteristic
106Q8×C59—82no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—42no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—81no—
137C2×C4×C59—81no—
138C2×C4×C59#13112yes—
139C59⋊C8—82no—
140C59⋊C8—82no—
141#11555—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—21no—
149C22×C59—42no—
150C22×C59—42no—
151C22×C59#35612yes—
152C22×C59#40811yescharacteristic
153C22×C59C2×#9911yescharacteristic
154C22×C59#41311yescharacteristic
155C4×C59—22no—
156C4×C59—42no—
157C4×C59—42no—
158C4×C59—41no—
159C4×C59—41no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—41no—
163C4×C59—81no—
164C4×C59—81no—
165#2502D5911yescharacteristic
166#1463—591no—
167C2×#324—591no—
168#1325—592no—
169#1319—592no—
170C2×C59—21no—
171C2×C59—21no—
172C2×C59—22no—
173C2×C59—41no—
174C2×C59#146811yescharacteristic
175C2×C59#212711yescharacteristic
176C2×C59C2×#35611yescharacteristic
177C2×C22⋊Q8Dic5911yescharacteristic
178C22×C4⋊C4C2×D5911yescharacteristic
179#407—1181no—
180C2×C22⋊C8—1181no—
181#398—21no—
182#398—21no—
183#391Dic5911yescharacteristic
184#391C2×D5911yescharacteristic
185#385—21no—
186#385—21no—
187#384Dic5911yescharacteristic
188#384C2×D5911yescharacteristic
189C4×C22⋊C4Dic5911yescharacteristic
190#324—592no—
191#324—592no—
192#324—1181no—
193#324—1181no—
194C59#1155511yescharacteristic, complemented, Sylow
195C22×Q8C2×Dic5911yescharacteristic
196C23×C4C2×Dic5911yescharacteristic
197C23×C4C22⋊D5911yescharacteristic
198C23×C4C22⋊D5911yescharacteristic
199C2×M16—2361no—
200C22×C8—2361no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4C2×Dic5911yescharacteristic
211C2×C4⋊C4C2×Dic5911yescharacteristic
212C2×C4⋊C4C22⋊D5911yescharacteristic
213C2×C4⋊C4C22⋊D5911yescharacteristic
214C2×C4⋊C4C22×D5911yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C42C2×Dic5911yescharacteristic
218C2×C42C2×Dic5911yescharacteristic
219C22⋊C8—1182no—
220C22⋊C8—1182no—
221C22⋊C8—2361no—
222C22⋊C8—2361no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228C24C22⋊Dic5911yescharacteristic
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4C22⋊Dic5911yescharacteristic
247C22×C4C22⋊Dic5911yescharacteristic
248C22×C4C22⋊Dic5911yescharacteristic
249C22×C4C22×Dic5911yescharacteristic
250C22×C4C2×C22⋊D5911yescharacteristic
251C22×C4C2×C22⋊D5911yescharacteristic
252M16—4722no—
253C2×C8—4722no—
254C2×C8—2362no—
255C2×C8—2362no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C4⋊C4—81no—
266C22⋊C4—24no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C42—22no—
270C42—41no—
271C23—21no—
272C23—22no—
273C23—21no—
274C23—41no—
275C23#1427364211yescharacteristic
276C23#1427364211yescharacteristic
277C23C2×C22⋊Dic5911yescharacteristic
278Q8—82no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4#1427373412yes—
311C8—4722no—
312C8—4722no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—42no—
321C22—42no—
322C22#1683720112yes—
323C22#1683736911yescharacteristic
324C22C2×#1427364211yescharacteristic
325C22#1683738611yescharacteristic
326C4—22no—
327C4—42no—
328C4—42no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C2—21no—
337C2—21no—
338C2—22no—
339C2—41no—
340C2#2142244411yescharacteristic
341C2#2142576511yescharacteristic
342C2C2×#1683720111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2502C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#2502D59
hypocenterC59#11555
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
3. #21422042C2semidirect
4. #21422030C2semidirect
194. C59#11555semidirect

Maximal subgroups

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

subgroupindex
#214224262
C2×#168371282
#214220422
#214220302
C59×#25022
#1155559