Finite Group Databasebeta

← #28498101

Subgroups — #28498101

Order 15104.

418 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 77 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#21426717C211yesmaximal, characteristic
2#21422429C212yesmaximal
3#21422042C211yesmaximal, characteristic, complemented
4#21422041C212yesmaximal, complemented
5C59×#1325C211yesmaximal, characteristic, complemented
6C2×#14273685C2211yescharacteristic
7C2×#14273684C411yescharacteristic
8C2×C4⋊C4×D59C411yescharacteristic
9#16837456—22no—
10#16837297—22no—
11#16837273—22no—
12#16837271C2211yescharacteristic
13#16837268C411yescharacteristic
14#16837267C411yescharacteristic
15#16837128C2212yes—
16#16837125—22no—
17C59×#384C2211yescharacteristic
18C59×#324C2212yes—
19C22×C4×D59—21no—
20C22×C4×D59C2×C411yescharacteristic
21#14273685—42no—
22#14273684—22no—
23#14273684—22no—
24C2×C4⋊Dic59C2×C411yescharacteristic
25C2×C4⋊Dic59C2×C411yescharacteristic
26C2×Dic59⋊C4—21no—
27C2×Dic59⋊C4—21no—
28C2×Dic59⋊C4—22no—
29C4⋊C4×D59—42no—
30C2×(C2×D59)⋊C4—21no—
31C2×(C2×D59)⋊C4D412yes—
32C2×C4×Dic59C2×C411yescharacteristic
33C2×C4×Dic59D412yes—
34Dic59⋊C8—42no—
35C22⋊(C59⋊C8)—22no—
36#14273637—42no—
37C59⋊#95—22no—
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×D59C22⋊C411yescharacteristic
45C22×C4×C59—21no—
46C22×C4×C59—21no—
47C22×C4×C59C22×C411yescharacteristic
48C22×C4×C59C2×D412yes—
49C22×Dic59—21no—
50C22×Dic59—21no—
51C22×Dic59C22⋊C411yescharacteristic
52C22×Dic59C22⋊C412yes—
53C2×C4×D59—22no—
54C2×C4×D59—21no—
55C2×C4×D59—21no—
56C2×C4×D59—22no—
57C2×C4×D59—42no—
58C2×C4×D59—82no—
59C2×C8×C59—42no—
60C4⋊C4×C59—41no—
61C4⋊C4×C59—41no—
62C4⋊C4×C59—82no—
63C42×C59—21no—
64C42×C59M1612yes—
65(C2×D59)⋊C4—24no—
66(C2×D59)⋊C4—42no—
67C4⋊Dic59—41no—
68C4⋊Dic59—41no—
69Dic59⋊C4—24no—
70Dic59⋊C4—42no—
71Dic59⋊C4—82no—
72C4×Dic59—21no—
73C4×Dic59—22no—
74C4×Dic59—22no—
75C4×Dic59M1612yes—
76C2×C59⋊C8—42no—
77C23×C59C2×C22⋊C411yescharacteristic
78C22×D59—21no—
79C22×D59—21no—
80C22×D59—41no—
81C22×D59—41no—
82C22×D59#10112yes—
83C2×C4×C59—21no—
84C2×C4×C59—21no—
85C2×C4×C59—22no—
86C2×C4×C59—22no—
87C2×C4×C59—41no—
88C2×C4×C59—41no—
89C2×C4×C59—42no—
90C2×C4×C59—42no—
91C2×C4×C59—41no—
92C2×C4×C59C2×M1611yescharacteristic
93C2×C4×C59C2×M1611yescharacteristic
94C2×Dic59—21no—
95C2×Dic59—21no—
96C2×Dic59—22no—
97C2×Dic59—22no—
98C2×Dic59—41no—
99C2×Dic59—41no—
100C2×Dic59—42no—
101C2×Dic59—42no—
102C2×Dic59—41no—
103C2×Dic59#9912yes—
104C4×D59—24no—
105C4×D59—24no—
106C4×D59—84no—
107C8×C59—82no—
108C59⋊C8—82no—
109#11516—591nomaximal, Sylow
110C22×C59—21no—
111C22×C59—21no—
112C22×C59#40711yescharacteristic
113C22×C59C2×#9911yescharacteristic
114C22×C59C2×#10111yescharacteristic
115C2×D59—22no—
116C2×D59—42no—
117C2×D59—41no—
118C2×D59—41no—
119C2×D59—42no—
120C4×C59—21no—
121C4×C59—21no—
122C4×C59—42no—
123C4×C59—42no—
124C4×C59—81no—
125Dic59—22no—
126Dic59—42no—
127Dic59—42no—
128Dic59—81no—
129#2317—591no—
130#1464—592no—
131#1325—591no—
132#1325—592no—
133#1325D5911yescharacteristic, complemented
134C2×C59—21no—
135C2×C59—21no—
136C2×C59#146311yescharacteristic
137C2×C59#212811yescharacteristic
138C2×C59#213511yescharacteristic
139D59—42no—
140C2×#117—591no—
141C2×#117—591no—
142C22×C4⋊C4—591no—
143#432—1182no—
144#389—1182no—
145#385—1182no—
146#384—591no—
147#384C2×D5911yescharacteristic
148#382—591no—
149#382—591no—
150#324—592no—
151#324C2×D5912yes—
152#323—1182no—
153C59#1151611yescharacteristic, complemented, Sylow
154C23×C4—591no—
155C23×C4—1181no—
156#117—1182no—
157#117—1182no—
158#117—2362no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—591no—
161C2×C4⋊C4—591no—
162C2×C4⋊C4—1181no—
163C2×C4⋊C4—1182no—
164C2×C4⋊C4—1181no—
165C2×C4⋊C4—2362no—
166C2×C4⋊C4C4×D5911yescharacteristic
167C2×C4⋊C4C22×D5911yescharacteristic
168C2×C22⋊C4—592no—
169C2×C22⋊C4—1181no—
170C2×C42—591no—
171C2×C42—592no—
172C2×C42C4×D5911yescharacteristic
173C4⋊C8—2362no—
174C22⋊C8—22no—
175C22⋊C8—1182no—
176#97—2362no—
177#95—1182no—
178#95—1182no—
179C24—591no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—591no—
183C22×C4—592no—
184C22×C4—1182no—
185C22×C4—1182no—
186C22×C4—1181no—
187C22×C4—1181no—
188C22×C4—1181no—
189C22×C4—1181no—
190C22×C4—4722no—
191C22×C4—2362no—
192C22×C4C2×C4×D5911yescharacteristic
193C22×C4D4×D5912yes—
194C2×C8—42no—
195C2×C8—2362no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—82no—
199C4⋊C4—1184no—
200C4⋊C4—4722no—
201C4⋊C4—2361no—
202C4⋊C4—2361no—
203C4⋊C4—2362no—
204C22⋊C4—1184no—
205C22⋊C4—2362no—
206C42—21no—
207C42—592no—
208C42—1182no—
209C42—1182no—
210C42—1181no—
211C42#1174565912yes—
212C23—592no—
213C23—1181no—
214C23—1181no—
215C23—2361no—
216C23—2361no—
217C23C22⋊C4×D5911yescharacteristic
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—592no—
228C2×C4—1184no—
229C2×C4—1184no—
230C2×C4—1181no—
231C2×C4—1181no—
232C2×C4—1182no—
233C2×C4—1182no—
234C2×C4—4724no—
235C2×C4—2362no—
236C2×C4—2361no—
237C2×C4—2361no—
238C2×C4—2361no—
239C2×C4—2362no—
240C2×C4M16×D5911yescharacteristic
241C2×C4C2×#1174565911yescharacteristic
242C8—82no—
243C8—4722no—
244C22—21no—
245C22—21no—
246C22—1182no—
247C22—2362no—
248C22—2362no—
249C22—2361no—
250C22—2361no—
251C22#1683736411yescharacteristic
252C22#99×D5911yescharacteristic
253C22#101×D5911yescharacteristic
254C4—21no—
255C4—21no—
256C4—42no—
257C4—42no—
258C4—81no—
259C4—1182no—
260C4—4721no—
261C4—2362no—
262C4—2362no—
263C2—21no—
264C2—21no—
265C2—2362no—
266C2#2142242411yescharacteristic
267C2#2142576611yescharacteristic
268C2#2142578911yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1325C2
socleC22×C59C2×#99
prime radicalC23×D59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1325D59
hypocenterC59#11516
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
3. #21422042C2semidirect
4. #21422041C2semidirect
5. C59×#1325C2semidirect
133. #1325D59semidirect
153. C59#11516semidirect

Maximal subgroups

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

subgroupindex
#214267172
#214224292
#214220422
#214220412
C59×#13252
#1151659