Finite Group Databasebeta

← #28498174

Subgroups — #28498174

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#21426806C211yesmaximal, characteristic
2#21422438C212yesmaximal, complemented
3#21422041C212yesmaximal, complemented
4#21422028C211yesmaximal, characteristic
5C59×#1318C211yesmaximal, characteristic
6C2×#14273714C411yescharacteristic
7C2×#14273692C2211yescharacteristic
8C2×D4×Dic59C411yescharacteristic
9#16837456—22no—
10#16837308C411yescharacteristic
11#16837296—22no—
12#16837274—22no—
13#16837271C2211yescharacteristic
14#16837248C411yescharacteristic
15#16837128C2212yes—
16#16837127—22no—
17C59×#390C2211yescharacteristic
18C59×#324C2212yes—
19C22×Dic118C2×C411yescharacteristic
20C23×Dic59—21no—
21#14273714—22no—
22#14273714—22no—
23#14273692—42no—
24D4×Dic59—42no—
25C2×C4⋊Dic59C2×C411yescharacteristic
26C2×Dic59⋊C4—22no—
27C2×Dic59⋊C4D412yes—
28C2×C22⋊Dic59—22no—
29C2×C22⋊Dic59—21no—
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—
39C22×D4×C59C2×C411yescharacteristic
40C2×C4⋊C4×C59C2311yescharacteristic
41C2×C22⋊C4×C59—22no—
42C2×C42×C59C2×C411yescharacteristic
43C22⋊C8×C59—22no—
44C24×C59—21no—
45C2×D4×C59—21no—
46C2×D4×C59—41no—
47C2×D4×C59—41no—
48C2×D4×C59M1612yes—
49C22×C4×C59—21no—
50C22×C4×C59C22×C411yescharacteristic
51C22×C4×C59C2×D412yes—
52C22×Dic59—21no—
53C22×Dic59—21no—
54C22×Dic59—42no—
55C22×Dic59—82no—
56C22×Dic59C22⋊C412yes—
57C22×Dic59C22⋊C411yescharacteristic
58C22×Dic59C22⋊C411yescharacteristic
59C2×Dic118—21no—
60C2×Dic118—21no—
61C2×Dic118—41no—
62C2×Dic118—41no—
63C2×C8×C59—42no—
64C4⋊C4×C59—41no—
65C22⋊C4×C59—82no—
66C42×C59—41no—
67C22⋊Dic59—28no—
68C22⋊Dic59—42no—
69C4⋊Dic59—41no—
70Dic59⋊C4—42no—
71Dic59⋊C4—82no—
72C4×Dic59—21no—
73C4×Dic59—22no—
74C4×Dic59—22no—
75C4×Dic59—22no—
76C4×Dic59M1612yes—
77C2×C59⋊C8—42no—
78C23×C59—22no—
79C23×C59—42no—
80C23×C59—42no—
81C23×C59—41no—
82C23×C59—41no—
83C23×C59C2×C22⋊C411yescharacteristic
84D4×C59—42no—
85D4×C59—42no—
86C2×C4×C59—21no—
87C2×C4×C59—21no—
88C2×C4×C59—22no—
89C2×C4×C59—42no—
90C2×C4×C59—41no—
91C2×C4×C59—41no—
92C2×C4×C59—41no—
93C2×C4×C59C2×M1611yescharacteristic
94C2×C4×C59C2×M1611yescharacteristic
95C2×Dic59—22no—
96C2×Dic59—22no—
97C2×Dic59—21no—
98C2×Dic59—22no—
99C2×Dic59—22no—
100C2×Dic59—41no—
101C2×Dic59—41no—
102C2×Dic59—41no—
103C2×Dic59—41no—
104C2×Dic59—42no—
105C2×Dic59—41no—
106C2×Dic59—84no—
107C2×Dic59#9912yes—
108Dic118—42no—
109Dic118—42no—
110C8×C59—82no—
111C59⋊C8—82no—
112#11534—591nomaximal, Sylow
113C22×C59—21no—
114C22×C59—21no—
115C22×C59—44no—
116C22×C59—42no—
117C22×C59—82no—
118C22×C59—81no—
119C22×C59—81no—
120C22×C59#40711yescharacteristic
121C22×C59C2×#9911yescharacteristic
122C22×C59#41311yescharacteristic
123C4×C59—21no—
124C4×C59—21no—
125C4×C59—42no—
126C4×C59—81no—
127Dic59—22no—
128Dic59—42no—
129Dic59—41no—
130Dic59—42no—
131Dic59—81no—
132#2326—591no—
133#1467—592no—
134#1325—592no—
135#1318—591no—
136#1318D5911yescharacteristic
137C2×C59—21no—
138C2×C59—21no—
139C2×C59—82no—
140C2×C59#146511yescharacteristic
141C2×C59#212811yescharacteristic
142C2×C59#213311yescharacteristic
143C2×#124—591no—
144C2×C4×Q8—591no—
145C2×C4×D4—591no—
146#432—1182no—
147#391—591no—
148#390C2×D5911yescharacteristic
149#388—1182no—
150#385—1182no—
151#384—591no—
152#376—591no—
153#324—592no—
154#324—1182no—
155#324C2×D5912yes—
156C59#1153411yescharacteristic, complemented, Sylow
157C22×Q8—591no—
158C22×D4C4×D5911yescharacteristic
159C23×C4—1181no—
160#124—1182no—
161#124—1182no—
162C4×Q8—2362no—
163C4×D4—2362no—
164C2×C4⋊C4—592no—
165C2×C4⋊C4—591no—
166C2×C4⋊C4—1182no—
167C2×C4⋊C4C22×D5911yescharacteristic
168C2×C22⋊C4—22no—
169C2×C22⋊C4—1182no—
170C2×C22⋊C4—1181no—
171C2×C42—591no—
172C2×C42—591no—
173C2×C42—592no—
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—21no—
183C2×Q8—1181no—
184C2×Q8—1181no—
185C2×Q8—2361no—
186C2×Q8—2361no—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4#1174565912yes—
191C22×C4—21no—
192C22×C4—591no—
193C22×C4—592no—
194C22×C4—591no—
195C22×C4—1181no—
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—4722no—
205C4⋊C4—2362no—
206C4⋊C4—2361no—
207C22⋊C4—82no—
208C22⋊C4—1188no—
209C22⋊C4—2362no—
210C42—41no—
211C42—592no—
212C42—1182no—
213C42—1182no—
214C42—1182no—
215C42—1181no—
216C23—22no—
217C23—42no—
218C23—42no—
219C23—41no—
220C23—41no—
221C23C22⋊C4×D5911yescharacteristic
222Q8—2362no—
223Q8—2362no—
224D4—42no—
225D4—42no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—592no—
234C2×C4—1182no—
235C2×C4—1182no—
236C2×C4—1182no—
237C2×C4—1181no—
238C2×C4—1182no—
239C2×C4—4724no—
240C2×C4—2361no—
241C2×C4—2362no—
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—44no—
253C22—42no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22#1683736411yescharacteristic
258C22#99×D5911yescharacteristic
259C22#1683738311yescharacteristic
260C4—21no—
261C4—21no—
262C4—42no—
263C4—81no—
264C4—1182no—
265C4—4721no—
266C4—2362no—
267C4—2362no—
268C4—2361no—
269C2—21no—
270C2—21no—
271C2—82no—
272C2#2142243111yescharacteristic
273C2#2142576611yescharacteristic
274C2#2142578211yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1318C2
socleC22×C59C2×#99
prime radicalC22×D4×C59C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1318D59
hypocenterC59#11534
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1318

Complements

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

NH ≅ G/Ntype
2. #21422438C2semidirect
3. #21422041C2semidirect
156. C59#11534semidirect

Maximal subgroups

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

subgroupindex
#214268062
#214224382
#214220412
#214220282
C59×#13182
#1153459