Finite Group Databasebeta

← #28498176

Subgroups — #28498176

Order 15104.

424 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 75 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#21426804C211yesmaximal, characteristic
2#21422438C212yesmaximal, complemented
3#21422042C211yesmaximal, characteristic, complemented
4#21422027C212yesmaximal
5C59×#1325C211yesmaximal, characteristic, complemented
6C2×#14273715C411yescharacteristic
7C2×#14273692C2211yescharacteristic
8C2×#14273689C411yescharacteristic
9#16837456—22no—
10#16837307C411yescharacteristic
11#16837303C2211yescharacteristic
12#16837292—22no—
13#16837273—22no—
14#16837248C411yescharacteristic
15#16837128C2212yes—
16#16837127—22no—
17C59×#384C2211yescharacteristic
18C59×#324C2212yes—
19C22×Dic118C2×C411yescharacteristic
20C22×D236C2×C411yescharacteristic
21C22×C4×D59—21no—
22#14273715—22no—
23#14273715—22no—
24#14273692—42no—
25#14273689—42no—
26C2×Dic59⋊C4D412yes—
27C2×(C2×D59)⋊C4—22no—
28C2×(C2×D59)⋊C4—21no—
29C2×(C2×D59)⋊C4—22no—
30C2×C4×Dic59C2×C411yescharacteristic
31C2×C4×Dic59C2×C411yescharacteristic
32C2×C4×Dic59D412yes—
33Dic59⋊C8—42no—
34C22⋊(C59⋊C8)—22no—
35#14273637—42no—
36C59⋊#95—21no—
37C59⋊#95—21no—
38C59⋊#95—22no—
39C2×C4⋊C4×C59—22no—
40C2×C4⋊C4×C59C2×C411yescharacteristic
41C2×C4⋊C4×C59C2311yescharacteristic
42C2×C42×C59C2×C411yescharacteristic
43C22⋊C8×C59—22no—
44C23×D59—21no—
45C22×C4×C59—21no—
46C22×C4×C59—21no—
47C22×C4×C59C22×C411yescharacteristic
48C22×C4×C59C2×D412yes—
49C22×Dic59—21no—
50C22×Dic59C22⋊C412yes—
51C22×Dic59C22⋊C411yescharacteristic
52C22×Dic59C22⋊C411yescharacteristic
53C2×D236—21no—
54C2×D236—41no—
55C2×D236—41no—
56C2×D236M1612yes—
57C2×C4×D59—42no—
58C2×C4×D59—82no—
59C2×Dic118—21no—
60C2×Dic118—21no—
61C2×Dic118—41no—
62C2×Dic118—41no—
63C2×C8×C59—42no—
64C4⋊C4×C59—41no—
65C4⋊C4×C59—41no—
66C4⋊C4×C59—82no—
67C42×C59—21no—
68C42×C59M1612yes—
69(C2×D59)⋊C4—28no—
70(C2×D59)⋊C4—42no—
71(C2×D59)⋊C4—82no—
72Dic59⋊C4—42no—
73C4×Dic59—22no—
74C4×Dic59—22no—
75C4×Dic59—22no—
76C4×Dic59—41no—
77C2×C59⋊C8—42no—
78C23×C59C2×C22⋊C411yescharacteristic
79C22×D59—22no—
80C22×D59—42no—
81C22×D59—42no—
82C22×D59—41no—
83C22×D59—41no—
84C2×C4×C59—21no—
85C2×C4×C59—21no—
86C2×C4×C59—22no—
87C2×C4×C59—22no—
88C2×C4×C59—41no—
89C2×C4×C59—41no—
90C2×C4×C59—42no—
91C2×C4×C59—42no—
92C2×C4×C59—41no—
93C2×C4×C59C2×M1611yescharacteristic
94C2×C4×C59C2×M1611yescharacteristic
95C2×Dic59—22no—
96C2×Dic59—21no—
97C2×Dic59—22no—
98C2×Dic59—22no—
99C2×Dic59—41no—
100C2×Dic59—41no—
101C2×Dic59—41no—
102C2×Dic59—41no—
103C2×Dic59—41no—
104C2×Dic59#9912yes—
105D236—42no—
106D236—42no—
107C4×D59—84no—
108Dic118—42no—
109Dic118—42no—
110C8×C59—82no—
111C59⋊C8—82no—
112#11534—591nomaximal, Sylow
113C22×C59—21no—
114C22×C59—21no—
115C22×C59#40711yescharacteristic
116C22×C59C2×#9911yescharacteristic
117C22×C59#41311yescharacteristic
118C2×D59—44no—
119C2×D59—42no—
120C2×D59—82no—
121C2×D59—81no—
122C2×D59—81no—
123C4×C59—21no—
124C4×C59—21no—
125C4×C59—42no—
126C4×C59—42no—
127C4×C59—81no—
128Dic59—22no—
129Dic59—41no—
130Dic59—42no—
131Dic59—81no—
132#2326—591no—
133#1467—592no—
134#1325—591no—
135#1325D5911yescharacteristic, complemented
136#1318—592no—
137C2×C59—21no—
138C2×C59—21no—
139C2×C59#146511yescharacteristic
140C2×C59#212811yescharacteristic
141C2×C59#213311yescharacteristic
142D59—82no—
143C2×#124—591no—
144C2×C4×Q8—591no—
145C2×C4×D4—591no—
146#432—1182no—
147#391—591no—
148#390—591no—
149#388—1182no—
150#385—1182no—
151#384C2×D5911yescharacteristic
152#376—591no—
153#324—592no—
154#324—1182no—
155#324C2×D5912yes—
156C59#1153411yescharacteristic, complemented, Sylow
157C22×Q8—591no—
158C22×D4—591no—
159C23×C4—1181no—
160#124—1182no—
161#124—1182no—
162C4×Q8—2362no—
163C4×D4—2362no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—592no—
166C2×C4⋊C4C4×D5911yescharacteristic
167C2×C4⋊C4C22×D5911yescharacteristic
168C2×C22⋊C4—1182no—
169C2×C22⋊C4—1182no—
170C2×C22⋊C4—1181no—
171C2×C42—591no—
172C2×C42—592no—
173C2×C42—591no—
174C2×C42C4×D5911yescharacteristic
175C4⋊C8—2362no—
176C22⋊C8—22no—
177C22⋊C8—1182no—
178#97—2362no—
179#95—1181no—
180#95—1181no—
181#95—1182no—
182C24—1181no—
183C2×Q8—1181no—
184C2×Q8—1181no—
185C2×Q8—2361no—
186C2×Q8—2361no—
187C2×D4—592no—
188C2×D4—1181no—
189C2×D4—2361no—
190C2×D4—2361no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—591no—
194C22×C4—592no—
195C22×C4—591no—
196C22×C4—1181no—
197C22×C4—4722no—
198C22×C4—2362no—
199C22×C4C2×C4×D5911yescharacteristic
200C22×C4D4×D5912yes—
201C2×C8—42no—
202C2×C8—2362no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—82no—
206C4⋊C4—2362no—
207C22⋊C4—1188no—
208C22⋊C4—4722no—
209C22⋊C4—2362no—
210C42—21no—
211C42—1182no—
212C42—1182no—
213C42—1182no—
214C42—2361no—
215C42#1174565912yes—
216C23—1182no—
217C23—2362no—
218C23—2362no—
219C23—2361no—
220C23—2361no—
221C23C22⋊C4×D5911yescharacteristic
222Q8—2362no—
223Q8—2362no—
224D4—2362no—
225D4—2362no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—592no—
236C2×C4—1182no—
237C2×C4—1182no—
238C2×C4—1181no—
239C2×C4—1182no—
240C2×C4—4724no—
241C2×C4—2361no—
242C2×C4—2361no—
243C2×C4—2361no—
244C2×C4—2361no—
245C2×C4—2361no—
246C2×C4M16×D5911yescharacteristic
247C2×C4C2×#1174565911yescharacteristic
248C8—82no—
249C8—4722no—
250C22—21no—
251C22—21no—
252C22—4722no—
253C22—4721no—
254C22—4721no—
255C22—2364no—
256C22—2362no—
257C22#1683736411yescharacteristic
258C22#99×D5911yescharacteristic
259C22#1683738511yescharacteristic
260C4—21no—
261C4—21no—
262C4—42no—
263C4—42no—
264C4—81no—
265C4—1182no—
266C4—4721no—
267C4—2362no—
268C4—2361no—
269C2—21no—
270C2—21no—
271C2—4722no—
272C2#2142243311yescharacteristic
273C2#2142576611yescharacteristic
274C2#2142578411yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1325C2
socleC22×C59C2×#99
prime radicalC22×D236C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1325D59
hypocenterC59#11534
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
2. #21422438C2semidirect
3. #21422042C2semidirect
5. C59×#1325C2semidirect
135. #1325D59semidirect
156. C59#11534semidirect

Maximal subgroups

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

subgroupindex
#214268042
#214224382
#214220422
#214220272
C59×#13252
#1153459