Finite Group Databasebeta

← #28498178

Subgroups — #28498178

Order 15104.

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

normal subgroups only

This group has 440 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#21426810C211yesmaximal, characteristic
2#21422433C212yesmaximal
3#21422041C212yesmaximal, complemented
4#21422030C211yesmaximal, characteristic, complemented
5C59×#1319C211yesmaximal, characteristic, complemented
6C2×#14273716C411yescharacteristic
7C2×Q8×Dic59C411yescharacteristic
8C2×#14273689C2211yescharacteristic
9#16837364—22no—
10#16837304C411yescharacteristic
11#16837291—22no—
12#16837276—22no—
13#16837271C2211yescharacteristic
14#16837248C411yescharacteristic
15#16837128C2212yes—
16#16837125—22no—
17C59×#391C2211yescharacteristic
18C59×#324C2212yes—
19C22×D236C2×C411yescharacteristic
20C22×C4×D59D412yes—
21C2×#11745659—42no—
22#14273716—22no—
23#14273716—22no—
24Q8×Dic59—42no—
25#14273689—42no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—22no—
29C2×(C2×D59)⋊C4—22no—
30C2×(C2×D59)⋊C4—22no—
31C2×(C2×D59)⋊C4D412yes—
32C2×C4×Dic59—21no—
33C2×C4×Dic59C2×C411yescharacteristic
34C2×C4×Dic59C2×C411yescharacteristic
35C22⋊(C59⋊C8)—22no—
36(C2×D59)⋊C8—42no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C22×Q8×C59C2×C411yescharacteristic
40C2×C4⋊C4×C59C2311yescharacteristic
41C2×C42×C59C2×C411yescharacteristic
42C22⋊C8×C59—22no—
43#95×C59—22no—
44C23×D59C22⋊C412yes—
45C2×Q8×C59—21no—
46C2×Q8×C59—41no—
47C2×Q8×C59—41no—
48C2×Q8×C59M1612yescomplemented
49C22×C4×C59—21no—
50C22×C4×C59—21no—
51C22×C4×C59C22×C411yescharacteristic
52C22×C4×C59C2×D412yes—
53C22×Dic59—21no—
54C22×Dic59—21no—
55C22×Dic59C22⋊C411yescharacteristic
56C22×Dic59C22⋊C411yescharacteristic
57C2×D236—21no—
58C2×D236—21no—
59C2×D236—41no—
60C2×D236—41no—
61C2×C4×D59—22no—
62C2×C4×D59—22no—
63C2×C4×D59—82no—
64C2×C8×C59—42no—
65C4⋊C4×C59—41no—
66C42×C59—41no—
67(C2×D59)⋊C4—28no—
68(C2×D59)⋊C4—42no—
69(C2×D59)⋊C4—82no—
70C4⋊Dic59—42no—
71C4⋊Dic59—41no—
72Dic59⋊C4—82no—
73C4×Dic59—21no—
74C4×Dic59—22no—
75C4×Dic59—42no—
76C4×Dic59M1612yescomplemented
77C2×C59⋊C8—42no—
78#11745659—84no—
79C23×C59C2×C22⋊C411yescharacteristic
80C22×D59—22no—
81C22×D59—22no—
82C22×D59—22no—
83C22×D59—42no—
84C22×D59—42no—
85Q8×C59—42no—
86Q8×C59—42no—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—22no—
90C2×C4×C59—22no—
91C2×C4×C59—41no—
92C2×C4×C59—41no—
93C2×C4×C59—42no—
94C2×C4×C59—41no—
95C2×C4×C59—41no—
96C2×C4×C59—41no—
97C2×C4×C59C2×M1611yescharacteristic
98C2×C4×C59C2×M1611yescharacteristic
99C2×Dic59—22no—
100C2×Dic59—21no—
101C2×Dic59—41no—
102C2×Dic59—41no—
103C2×Dic59—41no—
104C2×Dic59—41no—
105C2×Dic59—42no—
106C2×Dic59—41no—
107C2×Dic59#9912yes—
108D236—42no—
109D236—42no—
110C4×D59—44no—
111C4×D59—82no—
112C8×C59—82no—
113C59⋊C8—82no—
114#11535—591nomaximal, Sylow
115C22×C59—21no—
116C22×C59—21no—
117C22×C59#40711yescharacteristic
118C22×C59C2×#9911yescharacteristic
119C22×C59#41311yescharacteristic
120C2×D59—44no—
121C2×D59—42no—
122C2×D59—42no—
123C2×D59—42no—
124C2×D59—82no—
125C4×C59—21no—
126C4×C59—21no—
127C4×C59—42no—
128C4×C59—42no—
129C4×C59—81no—
130Dic59—22no—
131Dic59—42no—
132Dic59—41no—
133Dic59—81no—
134#2326—591no—
135#1465—592no—
136#1325—592no—
137#1319—591no—
138#1319D5911yescharacteristic, complemented
139C2×C59—21no—
140C2×C59—21no—
141C2×C59#146511yescharacteristic
142C2×C59#212711yescharacteristic
143C2×C59#213511yescharacteristic
144D59—82no—
145C2×#124—591no—
146C2×C4×Q8—591no—
147C2×C4×D4—591no—
148#407—1182no—
149#391C2×D5911yescharacteristic
150#390—591no—
151#388—1182no—
152#385—1182no—
153#384—591no—
154#376—591no—
155#324—592no—
156#324C2×D5912yes—
157#323—1182no—
158C59#1153511yescharacteristic, complemented, Sylow
159C22×Q8C4×D5911yescharacteristic
160C22×D4—591no—
161C23×C4—592no—
162C2×M16—2362no—
163#124—1182no—
164#124—1182no—
165C4×Q8—2362no—
166C4×D4—2362no—
167C2×C4⋊C4—591no—
168C2×C4⋊C4—1182no—
169C2×C4⋊C4—1181no—
170C2×C4⋊C4C22×D5911yescharacteristic
171C2×C22⋊C4—592no—
172C2×C22⋊C4—1182no—
173C2×C22⋊C4—1182no—
174C2×C42—591no—
175C2×C42—591no—
176C2×C42—1181no—
177C2×C42C4×D5911yescharacteristic
178C22⋊C8—22no—
179C22⋊C8—1182no—
180C22⋊C8—2362no—
181#95—22no—
182#95—1181no—
183#95—1181no—
184C24—592no—
185C2×Q8—21no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8#1174565912yescomplemented
189C2×D4—1181no—
190C2×D4—1181no—
191C2×D4—2361no—
192C2×D4—2361no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—591no—
196C22×C4—591no—
197C22×C4—1182no—
198C22×C4—1182no—
199C22×C4—1181no—
200C22×C4—1181no—
201C22×C4—4722no—
202C22×C4C2×C4×D5911yescharacteristic
203C22×C4D4×D5912yes—
204M16—4724no—
205C2×C8—42no—
206C2×C8—2362no—
207C4⋊C4—41no—
208C4⋊C4—4722no—
209C4⋊C4—2361no—
210C4⋊C4—2362no—
211C22⋊C4—1188no—
212C22⋊C4—4722no—
213C22⋊C4—2362no—
214C42—41no—
215C42—592no—
216C42—1182no—
217C42—1181no—
218C42—2362no—
219C23—1182no—
220C23—1182no—
221C23—1182no—
222C23—2362no—
223C23—2362no—
224C23C22⋊C4×D5911yescharacteristic
225Q8—42no—
226Q8—42no—
227D4—2362no—
228D4—2362no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—592no—
240C2×C4—1182no—
241C2×C4—1181no—
242C2×C4—4722no—
243C2×C4—2364no—
244C2×C4—2361no—
245C2×C4—2362no—
246C2×C4—2361no—
247C2×C4—2361no—
248C2×C4—2361no—
249C2×C4—2361no—
250C2×C4M16×D5911yescharacteristic
251C2×C4C2×#1174565911yescharacteristic
252C8—82no—
253C8—4722no—
254C22—21no—
255C22—21no—
256C22—4722no—
257C22—2364no—
258C22—2362no—
259C22—2362no—
260C22—2362no—
261C22#1683736411yescharacteristic
262C22#99×D5911yescharacteristic
263C22#1683738311yescharacteristic
264C4—21no—
265C4—21no—
266C4—42no—
267C4—42no—
268C4—81no—
269C4—1182no—
270C4—4721no—
271C4—2362no—
272C4—2361no—
273C2—21no—
274C2—21no—
275C2—4722no—
276C2#2142243111yescharacteristic
277C2#2142576411yescharacteristic
278C2#2142578911yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1319C2
socleC22×C59C2×#99
prime radicalC22×D236C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1319D59
hypocenterC59#11535
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
3. #21422041C2semidirect
4. #21422030C2semidirect
5. C59×#1319C2semidirect
48. C2×Q8×C59M16semidirect
76. C4×Dic59M16semidirect
138. #1319D59semidirect
158. C59#11535semidirect
188. C2×Q8#11745659semidirect

Maximal subgroups

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

subgroupindex
#214268102
#214224332
#214220412
#214220302
C59×#13192
#1153559