Finite Group Databasebeta

← #28499600

Subgroups — #28499600

Order 15104.

396 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 83 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#21428383C211yesmaximal, characteristic
2C2×#16837128C211yesmaximal, characteristic
3#21422122C212yesmaximal, complemented
4#21422101C212yesmaximal, complemented
5C59×#1821C211yesmaximal, characteristic
6C2×#14273686C411yescharacteristic
7C2×C22⋊(C59⋊C8)—21no—
8#16837341C411yescharacteristic
9#16837341C2211yescharacteristic
10#16837332C411yescharacteristic
11#16837332C2211yescharacteristic
12#16837283—21no—
13#16837283—21no—
14#16837275—21no—
15#16837265—21no—
16#16837265C411yescharacteristic
17#16837128—21no—
18#16837128C2212yes—
19C22×C4⋊C4×C59C2211yescharacteristic
20C59×#407—21no—
21C59×#343—21no—
22C59×#343C2212yes—
23C22×C59⋊C8—41no—
24#14273686—41no—
25#14273686—41no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59—21no—
32C2×C4×Dic59—21no—
33C22⋊(C59⋊C8)—22no—
34C22⋊(C59⋊C8)—41no—
35C59⋊#95—21no—
36C59⋊#95—21no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C23×C4×C59C2×C411yescharacteristic
45C23×C4×C59D411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C2×M16×C59—22no—
48C2×C4⋊C4×C59—41no—
49C2×C4⋊C4×C59—41no—
50C2×C4⋊C4×C59C2×C411yescharacteristic
51C2×C4⋊C4×C59D411yescharacteristic
52C2×C4⋊C4×C59D411yescharacteristic
53C2×C4⋊C4×C59C2311yescharacteristic
54C22⋊C8×C59—41no—
55C22⋊C8×C59—41no—
56C24×C59C22⋊C411yescharacteristic
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—22no—
64C22×C4×C59—41no—
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×C4×C59C2×D411yescharacteristic
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77M16×C59—42no—
78M16×C59—42no—
79C2×C8×C59—42no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—41no—
84C22⋊Dic59—42no—
85C22⋊Dic59—41no—
86C22⋊Dic59—41no—
87C4⋊Dic59—41no—
88C4⋊Dic59—41no—
89Dic59⋊C4—42no—
90Dic59⋊C4—42no—
91C4×Dic59—42no—
92C2×C59⋊C8—42no—
93C2×C59⋊C8—82no—
94C23×C59—22no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—41no—
98C23×C59#9911yescharacteristic
99C23×C59#9911yescharacteristic
100C23×C59C2×C22⋊C411yescharacteristic
101C2×C4×C59—21no—
102C2×C4×C59—21no—
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—42no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—81no—
120C2×C4×C59#13512yes—
121C2×Dic59—41no—
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—42no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C8×C59—82no—
133C59⋊C8—82no—
134#11850—591nomaximal, Sylow
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—21no—
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—42no—
143C22×C59—42no—
144C22×C59#35212yes—
145C22×C59C2×#9911yescharacteristic
146C22×C59#41611yescharacteristic
147C22×C59#42111yescharacteristic
148C4×C59—22no—
149C4×C59—41no—
150C4×C59—41no—
151C4×C59—41no—
152C4×C59—41no—
153C4×C59—41no—
154Dic59—81no—
155Dic59—81no—
156Dic59—81no—
157Dic59—81no—
158#2484—591no—
159#1821D5911yescharacteristic
160C2×#324—591no—
161#1353—592no—
162#1346—592no—
163C2×C59—21no—
164C2×C59—21no—
165C2×C59—22no—
166C2×C59—41no—
167C2×C59#151611yescharacteristic
168C2×C59C2×#35211yescharacteristic
169C2×C59#213311yescharacteristic
170C2×#117—591no—
171C22×C4⋊C4C2×D5911yescharacteristic
172#407—21no—
173C2×C22⋊C8—1181no—
174#400—591no—
175#400—591no—
176#398—591no—
177#398—591no—
178#387—1181no—
179#387—1181no—
180#385—1181no—
181#381—591no—
182#381—1181no—
183#343—21no—
184#343C2×D5912yes—
185#324—592no—
186#324—1181no—
187C59#1185011yescharacteristic, complemented, Sylow
188C23×C4C4×D5911yescharacteristic
189C23×C4D23611yescharacteristic
190C23×C4C22⋊D5911yescharacteristic
191C2×M16—22no—
192C22×C8—2361no—
193#117—2361no—
194#117—2361no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—591no—
198C2×C4⋊C4—591no—
199C2×C4⋊C4—1181no—
200C2×C4⋊C4—1181no—
201C2×C4⋊C4C4×D5911yescharacteristic
202C2×C4⋊C4D23611yescharacteristic
203C2×C4⋊C4C22⋊D5911yescharacteristic
204C2×C4⋊C4C22×D5911yescharacteristic
205C2×C22⋊C4—1181no—
206C2×C22⋊C4—1181no—
207C2×C42—1181no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—1182no—
211C22⋊C8—2361no—
212#95—591no—
213#95—591no—
214#95—1181no—
215#95—1181no—
216#95—1181no—
217#95—1181no—
218#95—1181no—
219#95—1181no—
220#95—1181no—
221C24(C2×D59)⋊C411yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—1181no—
233C22×C4—1181no—
234C22×C4—1181no—
235C22×C4—1181no—
236C22×C4(C2×D59)⋊C411yescharacteristic
237C22×C4(C2×D59)⋊C411yescharacteristic
238C22×C4(C2×D59)⋊C411yescharacteristic
239C22×C4C2×C4×D5911yescharacteristic
240C22×C4C2×D23611yescharacteristic
241C22×C4C2×C22⋊D5911yescharacteristic
242M16—42no—
243M16—42no—
244C2×C8—42no—
245C2×C8—4722no—
246C2×C8—2362no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—2361no—
252C4⋊C4—2361no—
253C4⋊C4—2362no—
254C4⋊C4—2362no—
255C22⋊C4—2362no—
256C22⋊C4—2361no—
257C22⋊C4—2361no—
258C42—2362no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—41no—
263C23#1427364111yescharacteristic
264C23#1427364111yescharacteristic
265C23C2×(C2×D59)⋊C411yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—2361no—
286C2×C4—2361no—
287C2×C4—2361no—
288C2×C4—2361no—
289C2×C4—2362no—
290C2×C4—2361no—
291C2×C4—2361no—
292C2×C4—2361no—
293C2×C4—2361no—
294C2×C4—2361no—
295C2×C4—2361no—
296C2×C4#1427374712yes—
297C8—82no—
298C8—4722no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—42no—
307C22—42no—
308C22#1683719312yes—
309C22C2×#1427364111yescharacteristic
310C22#1683739611yescharacteristic
311C22#1683742011yescharacteristic
312C4—22no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—4721no—
319C4—4721no—
320C4—4721no—
321C4—4721no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—41no—
326C2#2142265011yescharacteristic
327C2C2×#1683719311yescharacteristic
328C2#2142578311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1821C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1821D59
hypocenterC59#11850
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #21422122C2semidirect
4. #21422101C2semidirect
187. C59#11850semidirect

Maximal subgroups

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

subgroupindex
#214283832
C2×#168371282
#214221222
#214221012
C59×#18212
#1185059