Finite Group Databasebeta

← #28498261

Subgroups — #28498261

Order 15104.

366 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 79 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#21428545C211yesmaximal, characteristic
2#21422410C211yesmaximal, characteristic
3#21422041C212yesmaximal, complemented
4#21422029C212yesmaximal, complemented
5C59×#1464C211yesmaximal, characteristic
6C2×#14273706C411yescharacteristic
7#16837455—21no—
8#16837330—21no—
9#16837330—21no—
10#16837307C411yescharacteristic
11#16837307C2211yescharacteristic
12#16837286—21no—
13#16837286—21no—
14#16837271C2211yescharacteristic
15#16837267C411yescharacteristic
16C4×C22⋊Dic59C411yescharacteristic
17#16837128C2212yes—
18#16837126—21no—
19C2×#117×C59C2211yescharacteristic
20C59×#432—21no—
21C59×#324C2212yes—
22C59×#323—21no—
23C22×Dic118C2×C411yescharacteristic
24#14273706—41no—
25#14273706—41no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×Dic59⋊C4—21no—
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59—21no—
32C2×C4×Dic59C2×C411yescharacteristic
33C2×C4×Dic59C2×C411yescharacteristic
34C4⋊(C59⋊C8)—41no—
35C22⋊(C59⋊C8)—22no—
36C4×C59⋊C8—41no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C23×C4×C59C2×C411yescharacteristic
45#117×C59—41no—
46#117×C59—41no—
47C2×C4⋊C4×C59C2×C411yescharacteristic
48C2×C4⋊C4×C59C2311yescharacteristic
49C2×C22⋊C4×C59D411yescharacteristic
50C2×C22⋊C4×C59D411yescharacteristic
51C2×C42×C59D411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C4⋊C8×C59—41no—
54C22⋊C8×C59—22no—
55#97×C59—41no—
56C24×C59C22⋊C411yescharacteristic
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—22no—
60C22×C4×C59—41no—
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22×C411yescharacteristic
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×Dic59—21no—
68C22×Dic59—21no—
69C22×Dic59—21no—
70C22×Dic59—21no—
71C2×Dic118—22no—
72C2×Dic118—41no—
73C2×Dic118—41no—
74C2×C8×C59—42no—
75C4⋊C4×C59—41no—
76C4⋊C4×C59—41no—
77C22⋊C4×C59—41no—
78C22⋊C4×C59—41no—
79C42×C59—21no—
80C42×C59—21no—
81C42×C59—21no—
82C42×C59—21no—
83C22⋊Dic59—42no—
84C22⋊Dic59—42no—
85C4⋊Dic59—41no—
86Dic59⋊C4—42no—
87Dic59⋊C4—42no—
88Dic59⋊C4—41no—
89Dic59⋊C4—41no—
90C4×Dic59—22no—
91C4×Dic59—41no—
92C2×C59⋊C8—42no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—21no—
96C23×C59—41no—
97C23×C59—41no—
98C23×C59C2×C22⋊C411yescharacteristic
99C2×C4×C59—21no—
100C2×C4×C59—21no—
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—42no—
108C2×C4×C59—42no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59#9911yescharacteristic
114C2×C4×C59#9911yescharacteristic
115C2×C4×C59#13112yes—
116C2×Dic59—22no—
117C2×Dic59—41no—
118C2×Dic59—41no—
119C2×Dic59—41no—
120C2×Dic59—41no—
121C2×Dic59—41no—
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128Dic118—82no—
129C8×C59—82no—
130C59⋊C8—82no—
131#11552—591nomaximal, Sylow
132C22×C59—21no—
133C22×C59—21no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—41no—
137C22×C59—81no—
138C22×C59—81no—
139C22×C59#40811yescharacteristic
140C22×C59C2×#9911yescharacteristic
141C22×C59#41311yescharacteristic
142C4×C59—22no—
143C4×C59—21no—
144C4×C59—21no—
145C4×C59—41no—
146C4×C59—41no—
147C4×C59—41no—
148C4×C59—41no—
149Dic59—42no—
150Dic59—81no—
151Dic59—81no—
152Dic59—81no—
153#2499—591no—
154#1464D5911yescharacteristic
155#1458—591no—
156#1325—592no—
157#1319—592no—
158C2×C59—21no—
159C2×C59—21no—
160C2×C59—81no—
161C2×C59#146811yescharacteristic
162C2×C59#212611yescharacteristic
163C2×C59#213511yescharacteristic
164C2×C22⋊Q8—591no—
165C2×#117C2×D5911yescharacteristic
166#432—21no—
167#431—1181no—
168#397—1181no—
169#397—1181no—
170#391—591no—
171#391—591no—
172#387—1181no—
173#387—1181no—
174#384—591no—
175#382—591no—
176C4×C22⋊C4—591no—
177#324—592no—
178#324C2×D5912yes—
179#323—21no—
180#323—1181no—
181C59#1155211yescharacteristic, complemented, Sylow
182C22×Q8—591no—
183C23×C4C4×D5911yescharacteristic
184C22⋊Q8—2361no—
185C22⋊Q8—2361no—
186#117—41no—
187#117—41no—
188C2×C4⋊C4—591no—
189C2×C4⋊C4—1181no—
190C2×C4⋊C4—1181no—
191C2×C4⋊C4—1181no—
192C2×C4⋊C4C4×D5911yescharacteristic
193C2×C4⋊C4C22×D5911yescharacteristic
194C2×C22⋊C4—1181no—
195C2×C22⋊C4—1181no—
196C2×C22⋊C4D23611yescharacteristic
197C2×C22⋊C4C22⋊D5911yescharacteristic
198C2×C42—591no—
199C2×C42—591no—
200C2×C42D23611yescharacteristic
201C2×C42C22⋊D5911yescharacteristic
202C4⋊C8—41no—
203C4⋊C8—2361no—
204C22⋊C8—22no—
205C22⋊C8—1182no—
206#97—41no—
207C4×C8—2361no—
208#95—1181no—
209#95—1181no—
210#95—1181no—
211#95—1181no—
212#95—1181no—
213#95—1181no—
214#95—1181no—
215C24(C2×D59)⋊C411yescharacteristic
216C2×Q8—1182no—
217C2×Q8—2361no—
218C2×Q8—2361no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—1181no—
224C22×C4—1181no—
225C22×C4—1181no—
226C22×C4—1181no—
227C22×C4(C2×D59)⋊C411yescharacteristic
228C22×C4(C2×D59)⋊C411yescharacteristic
229C22×C4(C2×D59)⋊C411yescharacteristic
230C22×C4C2×C4×D5911yescharacteristic
231C22×C4C2×D23611yescharacteristic
232C22×C4C2×C22⋊D5911yescharacteristic
233C2×C8—42no—
234C2×C8—2362no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—2361no—
238C4⋊C4—2362no—
239C4⋊C4—2362no—
240C4⋊C4—2361no—
241C4⋊C4—2361no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—2362no—
245C22⋊C4—2362no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—1182no—
251C42—2361no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23C2×(C2×D59)⋊C411yescharacteristic
258Q8—4722no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—1182no—
274C2×C4—2361no—
275C2×C4—2361no—
276C2×C4—2361no—
277C2×C4—2361no—
278C2×C4—2361no—
279C2×C4—2361no—
280C2×C4—2361no—
281C2×C4—2361no—
282C2×C4—2361no—
283C2×C4—2361no—
284C2×C4—2361no—
285C2×C4#1427364111yescharacteristic
286C2×C4#1427364111yescharacteristic
287C2×C4#1427373512yes—
288C8—82no—
289C8—4722no—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1683737011yescharacteristic
298C22C2×#1427364111yescharacteristic
299C22#1683738411yescharacteristic
300C4—21no—
301C4—21no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—4721no—
308C4—4721no—
309C4—4721no—
310C4—2362no—
311C2—21no—
312C2—21no—
313C2—81no—
314C2#2142244511yescharacteristic
315C2#2142575911yescharacteristic
316C2#2142579011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1464C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1464D59
hypocenterC59#11552
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #21422041C2semidirect
4. #21422029C2semidirect
181. C59#11552semidirect

Maximal subgroups

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

subgroupindex
#214285452
#214224102
#214220412
#214220292
C59×#14642
#1155259