Finite Group Databasebeta

← #28498255

Subgroups — #28498255

Order 15104.

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

normal subgroups only

This group has 398 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#21422040C212yesmaximal, complemented
4#21422030C212yesmaximal, complemented
5C59×#2503C211yesmaximal, 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×#400—21no—
15C59×#400—21no—
16C59×#391C411yescharacteristic
17C59×#391C2211yescharacteristic
18C59×#382C411yescharacteristic
19C59×#382C2211yescharacteristic
20C59×#381—21no—
21C59×#381—21no—
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—41no—
37C2×C4⋊C4×C59—41no—
38C2×C4⋊C4×C59C2×C411yescharacteristic
39C2×C4⋊C4×C59C2×C411yescharacteristic
40C2×C4⋊C4×C59D411yescharacteristic
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C4⋊C4×C59C2311yescharacteristic
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59—21no—
45C2×C42×C59C2×C411yescharacteristic
46C2×C42×C59C2×C411yescharacteristic
47#95×C59—21no—
48#95×C59—21no—
49#95×C59—21no—
50#95×C59—21no—
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—22no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—41no—
72C22×C4×C59—42no—
73C22×C4×C59—42no—
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—
86C22⋊C4×C59—42no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—41no—
89C42×C59—22no—
90C42×C59—41no—
91#11745664—82no—
92C2×C59⋊C8—42no—
93C2×C59⋊C8—42no—
94C2×C59⋊C8—82no—
95C23×C59—22no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—41no—
99C23×C59#9911yescharacteristic
100C23×C59#9911yescharacteristic
101C23×C59C2×C22⋊C411yescharacteristic
102Q8×C59—82no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—42no—
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—81no—
132C2×C4×C59—81no—
133C2×C4×C59#13112yes—
134C59⋊C8—82no—
135C59⋊C8—82no—
136#11551—591nomaximal, Sylow
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—21no—
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—42no—
145C22×C59—42no—
146C22×C59#35412yes—
147C22×C59#40811yescharacteristic
148C22×C59C2×#9911yescharacteristic
149C22×C59#41311yescharacteristic
150C4×C59—22no—
151C4×C59—41no—
152C4×C59—41no—
153C4×C59—41no—
154C4×C59—41no—
155C4×C59—41no—
156C4×C59—81no—
157C4×C59—81no—
158C4×C59—81no—
159C4×C59—81no—
160#2503D5911yescharacteristic
161#1463—591no—
162C2×#324—591no—
163#1324—592no—
164#1319—592no—
165C2×C59—21no—
166C2×C59—21no—
167C2×C59—22no—
168C2×C59—41no—
169C2×C59#146811yescharacteristic
170C2×C59C2×#35411yescharacteristic
171C2×C59#213311yescharacteristic
172C2×C22⋊Q8Dic5911yescharacteristic
173C22×C4⋊C4C2×D5911yescharacteristic
174#407—1181no—
175C2×C22⋊C8—1181no—
176#400—21no—
177#400—21no—
178#391Dic5911yescharacteristic
179#391C2×D5911yescharacteristic
180#382Dic5911yescharacteristic
181#382C2×D5911yescharacteristic
182#381—21no—
183#381—21no—
184C4×C22⋊C4Dic5911yescharacteristic
185#324—592no—
186#324—592no—
187#324—1181no—
188#324—1181no—
189C59#1155111yescharacteristic, complemented, Sylow
190C22×Q8C2×Dic5911yescharacteristic
191C23×C4C2×Dic5911yescharacteristic
192C23×C4C22⋊D5911yescharacteristic
193C23×C4C22⋊D5911yescharacteristic
194C2×M16—2361no—
195C22×C8—2361no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4C2×Dic5911yescharacteristic
202C2×C4⋊C4C2×Dic5911yescharacteristic
203C2×C4⋊C4C22⋊D5911yescharacteristic
204C2×C4⋊C4C22⋊D5911yescharacteristic
205C2×C4⋊C4C22×D5911yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C42C2×Dic5911yescharacteristic
209C2×C42C2×Dic5911yescharacteristic
210C22⋊C8—1182no—
211C22⋊C8—1182no—
212C22⋊C8—2361no—
213C22⋊C8—2361no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223C24C22⋊Dic5911yescharacteristic
224C2×Q8—22no—
225C2×Q8—41no—
226C2×Q8—41no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4C22⋊Dic5911yescharacteristic
242C22×C4C22⋊Dic5911yescharacteristic
243C22×C4C22⋊Dic5911yescharacteristic
244C22×C4C22×Dic5911yescharacteristic
245C22×C4C2×C22⋊D5911yescharacteristic
246C22×C4C2×C22⋊D5911yescharacteristic
247M16—4722no—
248C2×C8—4722no—
249C2×C8—2362no—
250C2×C8—2362no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C42—22no—
261C42—41no—
262C23—21no—
263C23—22no—
264C23—21no—
265C23—41no—
266C23#1427364211yescharacteristic
267C23#1427364211yescharacteristic
268C23C2×C22⋊Dic5911yescharacteristic
269Q8—82no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4#1427373412yes—
301C8—4722no—
302C8—4722no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—42no—
312C22#1683719712yes—
313C22#1683736911yescharacteristic
314C22C2×#1427364211yescharacteristic
315C22#1683738611yescharacteristic
316C4—22no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4—81no—
326C2—21no—
327C2—21no—
328C2—22no—
329C2—41no—
330C2#2142244411yescharacteristic
331C2C2×#1683719711yescharacteristic
332C2#2142578511yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2503C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#2503D59
hypocenterC59#11551
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
3. #21422040C2semidirect
4. #21422030C2semidirect
189. C59#11551semidirect

Maximal subgroups

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

subgroupindex
#214224262
C2×#168371282
#214220402
#214220302
C59×#25032
#1155159