Finite Group Databasebeta

← #28498365

Subgroups — #28498365

Order 15104.

426 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 87 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#21428413C211yesmaximal, characteristic
2#21422437C211yesmaximal, characteristic
3#21422045C214yesmaximal, complemented
4C59×#1459C211yesmaximal, characteristic
5C22×C4⋊Dic59C411yescharacteristic
6#16837459—21no—
7#16837341—21no—
8#16837338—21no—
9#16837300—21no—
10#16837282C412yes—
11#16837274C2212yes—
12#16837262—21no—
13#16837262C411yescharacteristic
14#16837126—21no—
15#16837126C2212yes—
16C2×C4×D4×C59C2211yescharacteristic
17C2×C22⋊C8×C59—21no—
18C59×#324—21no—
19C59×#323C2212yes—
20C23×Dic59C2×C412yes—
21C2×C4⋊Dic59—21no—
22C2×C4⋊Dic59—21no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59—41no—
25C2×C4⋊Dic59—41no—
26C2×Dic59⋊C4—22no—
27C2×C22⋊Dic59—22no—
28C2×C22⋊Dic59C2×C412yes—
29C4⋊(C59⋊C8)—41no—
30C22⋊(C59⋊C8)—22no—
31#14273638—41no—
32C59⋊#95—21no—
33C59⋊#95—22no—
34C59⋊#95—22no—
35C22×D4×C59C2×C411yescharacteristic
36C23×C4×C59C2×C411yescharacteristic
37C23×C4×C59D411yescharacteristic
38C22×C8×C59—41no—
39C4×D4×C59—41no—
40C4×D4×C59—41no—
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C22⋊C4×C59D411yescharacteristic
43C2×C22⋊C4×C59C2311yescharacteristic
44C2×C42×C59D411yescharacteristic
45C22⋊C8×C59—22no—
46C22⋊C8×C59—41no—
47C24×C59C22⋊C411yescharacteristic
48C24×C59C22×C411yescharacteristic
49C2×D4×C59—22no—
50C2×D4×C59—22no—
51C2×D4×C59—41no—
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—42no—
57C22×C4×C59—42no—
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59C22⋊C411yescharacteristic
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59C2×D411yescharacteristic
62C22×C4×C59C2×D411yescharacteristic
63C22×Dic59—22no—
64C22×Dic59—22no—
65C22×Dic59—42no—
66C22×Dic59—82no—
67C2×C8×C59—42no—
68C2×C8×C59—82no—
69C4⋊C4×C59—41no—
70C22⋊C4×C59—41no—
71C22⋊C4×C59—41no—
72C42×C59—21no—
73C42×C59—21no—
74C22⋊Dic59—24no—
75C22⋊Dic59—82no—
76C4⋊Dic59—42no—
77C4⋊Dic59—42no—
78C4⋊Dic59—41no—
79C4⋊Dic59—41no—
80Dic59⋊C4—82no—
81C2×C59⋊C8—42no—
82C23×C59—21no—
83C23×C59—21no—
84C23×C59—21no—
85C23×C59—21no—
86C23×C59—41no—
87C23×C59—41no—
88C23×C59—41no—
89C23×C59—41no—
90C23×C59#10112yes—
91C23×C59C2×C22⋊C411yescharacteristic
92C23×C59#13112yes—
93D4×C59—44no—
94C2×C4×C59—22no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—42no—
102C2×C4×C59—42no—
103C2×C4×C59—42no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—81no—
108C2×C4×C59#9911yescharacteristic
109C2×C4×C59#9911yescharacteristic
110C2×Dic59—42no—
111C2×Dic59—42no—
112C2×Dic59—42no—
113C2×Dic59—42no—
114C2×Dic59—42no—
115C2×Dic59—42no—
116C2×Dic59—84no—
117C8×C59—82no—
118C59⋊C8—82no—
119#11576—591nomaximal, Sylow
120C22×C59—21no—
121C22×C59—21no—
122C22×C59—22no—
123C22×C59—22no—
124C22×C59—42no—
125C22×C59—42no—
126C22×C59—42no—
127C22×C59—41no—
128C22×C59—41no—
129C22×C59—41no—
130C22×C59—41no—
131C22×C59—81no—
132C22×C59#40811yescharacteristic
133C22×C59C2×#9911yescharacteristic
134C22×C59C2×#10111yescharacteristic
135C4×C59—21no—
136C4×C59—41no—
137C4×C59—41no—
138C4×C59—41no—
139C4×C59—41no—
140C4×C59#35212yes—
141Dic59—82no—
142Dic59—82no—
143#2487—591no—
144#1466—591no—
145#1459D5911yescharacteristic
146#1326—594no—
147C2×C59—21no—
148C2×C59—21no—
149C2×C59—42no—
150C2×C59—42no—
151C2×C59#147211yescharacteristic
152C2×C59C2×#35211yescharacteristic
153C2×C59#212211yescharacteristic
154C2×C4×D4C2×D5911yescharacteristic
155C22×C4⋊C4—591no—
156#432—1181no—
157C2×C22⋊C8—21no—
158#400—1181no—
159#399—1181no—
160#389—1181no—
161#386—592no—
162#385—592no—
163#380—591no—
164#380—1181no—
165#324—21no—
166#323—592no—
167#323—1181no—
168#323C2×D5912yes—
169C59#1157611yescharacteristic, complemented, Sylow
170C22×D4C4×D5911yescharacteristic
171C23×C4—592no—
172C23×C4C4×D5911yescharacteristic
173C23×C4C22⋊D5911yescharacteristic
174C22×C8—41no—
175C4×D4—41no—
176C4×D4—41no—
177C2×C4⋊C4—1182no—
178C2×C4⋊C4—1181no—
179C2×C4⋊C4—1181no—
180C2×C4⋊C4—1181no—
181C2×C4⋊C4—2361no—
182C2×C4⋊C4—2361no—
183C2×C4⋊C4C22⋊D5911yescharacteristic
184C2×C22⋊C4—592no—
185C2×C22⋊C4—1182no—
186C2×C22⋊C4D23611yescharacteristic
187C2×C22⋊C4C22×D5911yescharacteristic
188C2×C42D23611yescharacteristic
189C4⋊C8—2361no—
190C22⋊C8—22no—
191C22⋊C8—41no—
192C22⋊C8—1182no—
193#97—2361no—
194#95—1181no—
195#95—1182no—
196#95—1182no—
197C24(C2×D59)⋊C411yescharacteristic
198C24C2×C4×D5911yescharacteristic
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—41no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—1182no—
209C22×C4—1182no—
210C22×C4—4722no—
211C22×C4—2362no—
212C22×C4(C2×D59)⋊C411yescharacteristic
213C22×C4(C2×D59)⋊C411yescharacteristic
214C22×C4(C2×D59)⋊C411yescharacteristic
215C22×C4C2×D23611yescharacteristic
216C22×C4C2×C22⋊D5911yescharacteristic
217C2×C8—42no—
218C2×C8—82no—
219C2×C8—2362no—
220C4⋊C4—41no—
221C4⋊C4—4722no—
222C4⋊C4—2362no—
223C4⋊C4—2362no—
224C4⋊C4—2361no—
225C4⋊C4—2361no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—1184no—
229C22⋊C4—4722no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23#1427364512yes—
241C23C2×(C2×D59)⋊C411yescharacteristic
242C23#1427373312yes—
243D4—44no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—81no—
258C2×C4—4724no—
259C2×C4—2362no—
260C2×C4—2362no—
261C2×C4—2362no—
262C2×C4—2362no—
263C2×C4—2362no—
264C2×C4—2362no—
265C2×C4#1427364111yescharacteristic
266C2×C4#1427364111yescharacteristic
267C8—82no—
268C8—4722no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—81no—
281C22#1683736811yescharacteristic
282C22C2×#1427364111yescharacteristic
283C22C2×#1427364511yescharacteristic
284C4—21no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—4722no—
290C4—4722no—
291C4#1683719112yes—
292C2—21no—
293C2—21no—
294C2—42no—
295C2—42no—
296C2#2142245911yescharacteristic
297C2C2×#1683719111yescharacteristic
298C2#2142574711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC24×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1459C2
socleC22×C59C2×#99
prime radicalC22×D4×C59C2×C4
cosocleC2×C22⋊C4×C59C23
hypercenter#1459D59
hypocenterC59#11576
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #21422045C2semidirect
169. C59#11576semidirect

Maximal subgroups

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

subgroupindex
#214284132
#214224372
#214220452
C59×#14592
#1157659