Finite Group Databasebeta

← #28498181

Subgroups — #28498181

Order 15104.

440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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#21422434C212yesmaximal
2#21422042C212yesmaximal, complemented
3#21422030C212yesmaximal, complemented
4C59×#2326C211yesmaximal, characteristic
5#16837366—22no—
6#16837128C2214yes—
7#16837126—22no—
8C2×#124×C59C411yescharacteristic
9C2×C4×Q8×C59C411yescharacteristic
10C2×C4×D4×C59C2211yescharacteristic
11C59×#391C2211yescharacteristic
12C59×#390C411yescharacteristic
13C59×#388—22no—
14C59×#385—22no—
15C59×#384C2211yescharacteristic
16C59×#376C411yescharacteristic
17C2×#11745664—42no—
18C22⋊(C59⋊C8)—24no—
19C22⋊(C59⋊C8)—42no—
20C22×Q8×C59C2×C411yescharacteristic
21C22×D4×C59C2×C411yescharacteristic
22C23×C4×C59D412yes—
23#124×C59—22no—
24#124×C59—22no—
25C4×Q8×C59—42no—
26C4×D4×C59—42no—
27C2×C4⋊C4×C59—21no—
28C2×C4⋊C4×C59—22no—
29C2×C4⋊C4×C59C2×C411yescharacteristic
30C2×C4⋊C4×C59C2311yescharacteristic
31C2×C22⋊C4×C59—22no—
32C2×C22⋊C4×C59—22no—
33C2×C22⋊C4×C59D412yes—
34C2×C42×C59—21no—
35C2×C42×C59C2×C411yescharacteristic
36C2×C42×C59C2×C411yescharacteristic
37C2×C42×C59C2×C411yescharacteristic
38#95×C59—21no—
39#95×C59—21no—
40#95×C59—22no—
41C24×C59C22⋊C412yes—
42C2×Q8×C59—21no—
43C2×Q8×C59—41no—
44C2×Q8×C59—41no—
45C2×Q8×C59M1612yescomplemented
46C2×D4×C59—21no—
47C2×D4×C59—21no—
48C2×D4×C59—41no—
49C2×D4×C59—41no—
50C22×C4×C59—21no—
51C22×C4×C59—21no—
52C22×C4×C59—22no—
53C22×C4×C59—22no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—82no—
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59C22×C411yescharacteristic
60C22×C4×C59C2×D412yes—
61C4⋊C4×C59—41no—
62C4⋊C4×C59—41no—
63C4⋊C4×C59—42no—
64C4⋊C4×C59—82no—
65C22⋊C4×C59—28no—
66C22⋊C4×C59—42no—
67C22⋊C4×C59—82no—
68C42×C59—22no—
69C42×C59—21no—
70C42×C59—42no—
71C42×C59—41no—
72C42×C59M1612yescomplemented
73#11745664—84no—
74C2×C59⋊C8—44no—
75C23×C59—22no—
76C23×C59—22no—
77C23×C59—22no—
78C23×C59—42no—
79C23×C59—42no—
80C23×C59C2×C22⋊C411yescharacteristic
81Q8×C59—42no—
82Q8×C59—42no—
83D4×C59—42no—
84D4×C59—42no—
85C2×C4×C59—21no—
86C2×C4×C59—21no—
87C2×C4×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—21no—
91C2×C4×C59—41no—
92C2×C4×C59—44no—
93C2×C4×C59—41no—
94C2×C4×C59—41no—
95C2×C4×C59—41no—
96C2×C4×C59—41no—
97C2×C4×C59—42no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—42no—
101C2×C4×C59—41no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59—82no—
105C2×C4×C59#9912yes—
106C2×C4×C59C2×M1611yescharacteristic
107C2×C4×C59C2×M1611yescharacteristic
108C59⋊C8—84no—
109#11535—591nomaximal, Sylow
110C22×C59—21no—
111C22×C59—21no—
112C22×C59—44no—
113C22×C59—42no—
114C22×C59—42no—
115C22×C59—42no—
116C22×C59—82no—
117C22×C59#40711yescharacteristic
118C22×C59C2×#9911yescharacteristic
119C22×C59#41311yescharacteristic
120C4×C59—22no—
121C4×C59—21no—
122C4×C59—21no—
123C4×C59—42no—
124C4×C59—42no—
125C4×C59—41no—
126C4×C59—42no—
127C4×C59—81no—
128C4×C59—81no—
129#2326D5911yescharacteristic
130#1465—592no—
131#1325—592no—
132#1319—592no—
133C2×C59—21no—
134C2×C59—21no—
135C2×C59—82no—
136C2×C59#146511yescharacteristic
137C2×C59#212711yescharacteristic
138C2×C59#213511yescharacteristic
139C2×#124Dic5911yescharacteristic
140C2×C4×Q8Dic5911yescharacteristic
141C2×C4×D4C2×D5911yescharacteristic
142#407—1182no—
143#391C2×D5911yescharacteristic
144#390Dic5911yescharacteristic
145#388—22no—
146#385—22no—
147#384C2×D5911yescharacteristic
148#376Dic5911yescharacteristic
149#324—594no—
150#323—1182no—
151C59#1153511yescharacteristic, complemented, Sylow
152C22×Q8C2×Dic5911yescharacteristic
153C22×D4C2×Dic5911yescharacteristic
154C23×C4C22⋊D5912yes—
155C2×M16—2362no—
156#124—22no—
157#124—22no—
158C4×Q8—42no—
159C4×D4—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C2×Dic5911yescharacteristic
163C2×C4⋊C4C22×D5911yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4C22⋊D5912yes—
167C2×C42—21no—
168C2×C42C2×Dic5911yescharacteristic
169C2×C42C2×Dic5911yescharacteristic
170C2×C42C2×Dic5911yescharacteristic
171C22⋊C8—1184no—
172C22⋊C8—2362no—
173#95—21no—
174#95—21no—
175#95—22no—
176C24C22⋊Dic5912yes—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8#1174566412yescomplemented
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—41no—
184C2×D4—41no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—82no—
192C22×C4C22⋊Dic5911yescharacteristic
193C22×C4C22⋊Dic5911yescharacteristic
194C22×C4C22×Dic5911yescharacteristic
195C22×C4C2×C22⋊D5912yes—
196M16—4724no—
197C2×C8—2364no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—82no—
202C22⋊C4—28no—
203C22⋊C4—42no—
204C22⋊C4—82no—
205C42—22no—
206C42—21no—
207C42—41no—
208C42—42no—
209C42#1174566412yescomplemented
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×C22⋊Dic5911yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—82no—
240C2×C4#1427364212yes—
241C2×C4C2×#1174566411yescharacteristic
242C2×C4C2×#1174566411yescharacteristic
243C8—4724no—
244C22—21no—
245C22—21no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22#1683736611yescharacteristic
252C22C2×#1427364211yescharacteristic
253C22#1683738611yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—42no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—81no—
262C4—81no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2142243411yescharacteristic
267C2#2142576511yescharacteristic
268C2#2142579111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2326C2
socleC22×C59C2×#99
prime radicalC22×D4×C59C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenter#2326D59
hypocenterC59#11535
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #21422042C2semidirect
3. #21422030C2semidirect
45. C2×Q8×C59M16semidirect
72. C42×C59M16semidirect
151. C59#11535semidirect
180. C2×Q8#11745664semidirect
209. C42#11745664semidirect

Maximal subgroups

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

subgroupindex
#214224342
#214220422
#214220302
C59×#23262
#1153559