Finite Group Databasebeta

← #28498903

Subgroups — #28498903

Order 15104.

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

normal subgroups only

This group has 452 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#21429862C211yesmaximal, characteristic
2#21422410C211yesmaximal, characteristic
3#21422101C214yesmaximal, complemented
4C59×#1742C211yesmaximal, characteristic
5C2×#14273727C411yescharacteristic
6C2×#14273706—21no—
7#16837455—21no—
8#16837332—21no—
9#16837332C2212yes—
10#16837327—21no—
11#16837327C412yes—
12#16837308—21no—
13#16837268C411yescharacteristic
14#16837128C2212yes—
15#16837126—21no—
16C22×M16×C59—21no—
17C2×#117×C59C2211yescharacteristic
18C59×#343—21no—
19C59×#343C2212yes—
20C22×Dic118—21no—
21#14273727—41no—
22#14273727—41no—
23#14273706—42no—
24#14273706—42no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59C2×C412yes—
28C2×Dic59⋊C4—22no—
29C2×C22⋊Dic59—22no—
30C4⋊(C59⋊C8)—41no—
31C22⋊(C59⋊C8)—22no—
32C4×C59⋊C8—41no—
33C59⋊#95—21no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95C2×C412yes—
37C23×C4×C59D411yescharacteristic
38C2×M16×C59—24no—
39C2×M16×C59—22no—
40C2×M16×C59—41no—
41C2×M16×C59—42no—
42C22×C8×C59—41no—
43#117×C59—41no—
44#117×C59—41no—
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C4⋊C4×C59C2311yescharacteristic
47C2×C22⋊C4×C59C2×C411yescharacteristic
48C2×C22⋊C4×C59D411yescharacteristic
49C2×C42×C59C2×C411yescharacteristic
50C2×C42×C59D411yescharacteristic
51C24×C59C22⋊C411yescharacteristic
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—22no—
55C22×C4×C59C22⋊C411yescharacteristic
56C22×C4×C59C22⋊C411yescharacteristic
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59D812yes—
59C22×C4×C59SD1612yes—
60C22×C4×C59C22×C411yescharacteristic
61C22×C4×C59C2×D411yescharacteristic
62C22×C4×C59C2×D411yescharacteristic
63C22×Dic59—22no—
64C22×Dic59—22no—
65C2×Dic118—41no—
66C2×Dic118—41no—
67C2×Dic118—42no—
68C2×Dic118—42no—
69M16×C59—24no—
70M16×C59—44no—
71M16×C59—42no—
72C2×C8×C59—42no—
73C2×C8×C59—44no—
74C4⋊C4×C59—41no—
75C4⋊C4×C59—41no—
76C22⋊C4×C59—41no—
77C22⋊C4×C59—41no—
78C42×C59—21no—
79C42×C59—21no—
80C42×C59—21no—
81C42×C59—21no—
82C22⋊Dic59—44no—
83C4⋊Dic59—42no—
84C4⋊Dic59—42no—
85C4⋊Dic59—42no—
86Dic59⋊C4—42no—
87Dic59⋊C4—42no—
88C2×C59⋊C8—42no—
89C23×C59—21no—
90C23×C59—21no—
91C23×C59—41no—
92C23×C59—41no—
93C23×C59#10212yes—
94C23×C59C2×C22⋊C411yescharacteristic
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59#9911yescharacteristic
110C2×C4×C59#9911yescharacteristic
111C2×C4×C59#10212yes—
112C2×C4×C59#10412yes—
113C2×C4×C59C2×D811yescharacteristic
114C2×C4×C59C2×SD1611yescharacteristic
115C2×Dic59—42no—
116C2×Dic59—42no—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—42no—
120C2×Dic59—42no—
121Dic118—82no—
122Dic118—82no—
123C8×C59—44no—
124C59⋊C8—82no—
125#11700—591nomaximal, Sylow
126C22×C59—21no—
127C22×C59—21no—
128C22×C59—22no—
129C22×C59—42no—
130C22×C59—41no—
131C22×C59—41no—
132C22×C59—81no—
133C22×C59C2×#9911yescharacteristic
134C22×C59C2×#10211yescharacteristic
135C22×C59C2×#10411yescharacteristic
136C4×C59—21no—
137C4×C59—21no—
138C4×C59—41no—
139C4×C59—41no—
140C4×C59—41no—
141C4×C59—41no—
142C4×C59#32712yes—
143C4×C59#32712yes—
144Dic59—82no—
145Dic59—82no—
146#2621—591no—
147#1742D5911yescharacteristic
148#1458—591no—
149#1346—594no—
150C2×C59—21no—
151C2×C59—21no—
152C2×C59—42no—
153C2×C59C2×#32711yescharacteristic
154C2×C59#212311yescharacteristic
155C2×C59#213511yescharacteristic
156C22×M16—21no—
157C2×C4⋊Q8—591no—
158C2×C22⋊Q8—1181no—
159C2×#117C2×D5911yescharacteristic
160#431—1181no—
161#398—592no—
162#398—1181no—
163#397—592no—
164#397—1181no—
165#391—1181no—
166#382—591no—
167#343—21no—
168#343C2×D5912yes—
169#324—592no—
170#323—1181no—
171C59#1170011yescharacteristic, complemented, Sylow
172C22×Q8—1181no—
173C23×C4C22⋊D5911yescharacteristic
174C2×M16—24no—
175C2×M16—22no—
176C2×M16—41no—
177C2×M16—42no—
178C22×C8—41no—
179C4⋊Q8—2361no—
180C4⋊Q8—2361no—
181C22⋊Q8—2362no—
182C22⋊Q8—2362no—
183#117—41no—
184#117—41no—
185C2×C4⋊C4—592no—
186C2×C4⋊C4—1181no—
187C2×C4⋊C4—1181no—
188C2×C4⋊C4—1182no—
189C2×C4⋊C4C22⋊D5911yescharacteristic
190C2×C4⋊C4C22×D5911yescharacteristic
191C2×C22⋊C4—1182no—
192C2×C22⋊C4C4×D5911yescharacteristic
193C2×C22⋊C4D23611yescharacteristic
194C2×C42C4×D5911yescharacteristic
195C2×C42D23611yescharacteristic
196C4⋊C8—2361no—
197C22⋊C8—1182no—
198C4×C8—2361no—
199#95—592no—
200#95—1181no—
201#95—1182no—
202#95—1182no—
203C24(C2×D59)⋊C411yescharacteristic
204C2×Q8—2361no—
205C2×Q8—2361no—
206C2×Q8—2362no—
207C2×Q8—2362no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—1182no—
212C22×C4—1182no—
213C22×C4(C2×D59)⋊C411yescharacteristic
214C22×C4(C2×D59)⋊C411yescharacteristic
215C22×C4(C2×D59)⋊C411yescharacteristic
216C22×C4#1174566912yes—
217C22×C4#1174567112yes—
218C22×C4C2×C4×D5911yescharacteristic
219C22×C4C2×D23611yescharacteristic
220C22×C4C2×C22⋊D5911yescharacteristic
221M16—24no—
222M16—44no—
223M16—42no—
224C2×C8—42no—
225C2×C8—44no—
226C2×C8—2362no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—2362no—
230C4⋊C4—2362no—
231C4⋊C4—2362no—
232C4⋊C4—2362no—
233C4⋊C4—2362no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—2364no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23#1427364712yes—
246C23C2×(C2×D59)⋊C411yescharacteristic
247Q8—4722no—
248Q8—4722no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—2362no—
264C2×C4—2362no—
265C2×C4—2362no—
266C2×C4—2362no—
267C2×C4—2362no—
268C2×C4—2362no—
269C2×C4#1427364111yescharacteristic
270C2×C4#1427364111yescharacteristic
271C2×C4#1427364712yes—
272C2×C4#1427365512yes—
273C2×C4C2×#1174566911yescharacteristic
274C2×C4C2×#1174567111yescharacteristic
275C8—44no—
276C8—4722no—
277C22—21no—
278C22—21no—
279C22—22no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—81no—
284C22C2×#1427364111yescharacteristic
285C22C2×#1427364711yescharacteristic
286C22C2×#1427365511yescharacteristic
287C4—21no—
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—4722no—
294C4—4722no—
295C4#1683713512yes—
296C4#1683713512yes—
297C2—21no—
298C2—21no—
299C2—42no—
300C2C2×#1683713511yescharacteristic
301C2#2142575111yescharacteristic
302C2#2142579011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1742C2
socleC22×C59C2×#99
prime radicalC24×C59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1742D59
hypocenterC59#11700
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
3. #21422101C2semidirect
171. C59#11700semidirect

Maximal subgroups

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

subgroupindex
#214298622
#214224102
#214221012
C59×#17422
#1170059