Finite Group Databasebeta

← #28564990

Subgroups — #28564990

Order 15104.

918 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 195 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#21434476C211yesmaximal, characteristic
2#21434226C212yesmaximal
3#21423317C218yesmaximal, complemented
4#21422822C211yesmaximal, characteristic
5#21422543C212yesmaximal
6C2×Q82×C59C211yesmaximal, characteristic
7#16837959C2211yescharacteristic
8#16837770—22no—
9#16837752C2214yes—
10#16837732—22no—
11#16837687C2214yes—
12#16837612C2214yes—
13#16837558—22no—
14#16837507—24no—
15#16837459C2212yes—
16#16837411—22no—
17C4×#11745664C2211yescharacteristic
18#16837151C2214yes—
19Q8⋊(C59⋊C8)C2218yes—
20Q82×C59—22no—
21Q82×C59—24no—
22Q82×C59C2214yes—
23C2×C4⋊Q8×C59—22no—
24C2×C4⋊Q8×C59—22no—
25C2×C4⋊Q8×C59C2211yescharacteristic
26C2×C4×Q8×C59—22no—
27C2×C4×Q8×C59C2212yes—
28C2×#11745672—24no—
29C2×#11745664—21no—
30#14273727C2312yes—
31#14273719—21no—
32#14273706—22no—
33C4×Dic118—22no—
34#14273686—21no—
35#14273658—24no—
36C4⋊(C59⋊C8)C2314yes—
37#14273651—28no—
38#14273651—24no—
39#14273650—24no—
40C22⋊(C59⋊C8)—22no—
41#14273638—22no—
42#14273638—21no—
43C4×C59⋊C8C2312yes—
44C22×Q8×C59—22no—
45C22×Q8×C59D412yes—
46C22×Q8×C59D412yes—
47C4⋊Q8×C59—28no—
48C4⋊Q8×C59—28no—
49C4⋊Q8×C59—28no—
50C4⋊Q8×C59—28no—
51C4⋊Q8×C59—22no—
52C4⋊Q8×C59D412yes—
53C4⋊Q8×C59C2312yes—
54C4×Q8×C59—28no—
55C4×Q8×C59—28no—
56C4×Q8×C59—22no—
57C4×Q8×C59—22no—
58C4×Q8×C59D414yes—
59C4×Q8×C59C2314yes—
60C2×C4⋊C4×C59—24no—
61C2×C4⋊C4×C59—22no—
62C2×C4⋊C4×C59—22no—
63C2×C4⋊C4×C59D412yes—
64C2×C42×C59—22no—
65C2×C42×C59—22no—
66C2×C42×C59C2311yescharacteristic
67C2×Q8×C59—28no—
68C2×Q8×C59—28no—
69C2×Q8×C59—22no—
70C2×Q8×C59—22no—
71C2×Q8×C59—44no—
72C2×Q8×C59—84no—
73C2×Q8×C59C2×D414yes—
74C2×Q8×C59C2×D414yes—
75C22×C4×C59—22no—
76C22×C4×C59—22no—
77C22×C4×C59—22no—
78C22×C4×C59C2×D412yes—
79C22×C4×C59C2×D411yescharacteristic
80C2×Dic118—22no—
81C4⋊C4×C59—216no—
82C4⋊C4×C59—24no—
83C4⋊C4×C59—24no—
84C4⋊C4×C59—22no—
85C4⋊C4×C59—24no—
86C4⋊C4×C59—24no—
87C4⋊C4×C59C2×D414yes—
88C42×C59—28no—
89C42×C59—24no—
90C42×C59—24no—
91C42×C59C2×D411yescharacteristic
92C42×C59C2×D412yes—
93C42×C59C2411yescharacteristic
94C22⋊Dic59—41no—
95#11745672—48no—
96C4⋊Dic59—22no—
97C4⋊Dic59—22no—
98Dic59⋊C4—42no—
99C4×Dic59—41no—
100#11745664—42no—
101C2×C59⋊C8—22no—
102C2×C59⋊C8—24no—
103C23×C59#12011yescharacteristic
104Q8×C59—28no—
105Q8×C59—24no—
106Q8×C59—44no—
107Q8×C59—82no—
108Q8×C59—88no—
109Q8×C59#13718yes—
110C2×C4×C59—28no—
111C2×C4×C59—24no—
112C2×C4×C59—22no—
113C2×C4×C59—24no—
114C2×C4×C59—24no—
115C2×C4×C59—24no—
116C2×C4×C59—44no—
117C2×C4×C59—44no—
118C2×C4×C59#12011yescharacteristic
119C2×C4×C59#12012yes—
120C2×C4×C59C22×D411yescharacteristic
121C2×C4×C59C22×D412yes—
122C2×Dic59—42no—
123Dic118—44no—
124C59⋊C8—42no—
125C59⋊C8—44no—
126#18551—591nomaximal, Sylow
127C22×C59—22no—
128C22×C59#45512yes—
129C22×C59C2×#12011yescharacteristic
130C4×C59—28no—
131C4×C59—22no—
132C4×C59—21no—
133C4×C59—21no—
134C4×C59—44no—
135C4×C59—44no—
136C4×C59C2×#13714yes—
137Dic59—82no—
138C2×Q82D5911yescharacteristic
139#3139—591no—
140#3113—592no—
141#1645—598no—
142#1558—591no—
143#1491—592no—
144C2×C59—22no—
145C2×C59#300412yes—
146C2×C59C2×#45511yescharacteristic
147Q82—22no—
148Q82—24no—
149Q82C2×D5914yes—
150#533—591no—
151C2×C4⋊Q8—22no—
152C2×C4⋊Q8—22no—
153C2×C4⋊Q8C2×D5911yescharacteristic
154C2×C4×Q8—22no—
155C2×C4×Q8C2×D5912yes—
156#491—1182no—
157#487—594no—
158#483—1182no—
159#475—594no—
160#462—594no—
161#451—1182no—
162#441—1184no—
163#432—592no—
164#419—1182no—
165C4×M16—591no—
166#333—594no—
167Q8⋊C8—598no—
168C59#1855111yescharacteristic, complemented, Sylow
169C22×Q8—22no—
170C22×Q8C22⋊D5912yes—
171C22×Q8C22⋊D5912yes—
172C2×Q16—1184no—
173C2×M16—1181no—
174C4⋊Q8—28no—
175C4⋊Q8—28no—
176C4⋊Q8—28no—
177C4⋊Q8—28no—
178C4⋊Q8—22no—
179C4⋊Q8—592no—
180C4⋊Q8C22⋊D5912yes—
181C4⋊Q8C22×D5912yes—
182#125—1181no—
183C22⋊Q8—1182no—
184C4×Q8—28no—
185C4×Q8—28no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—1182no—
189C4×Q8C22⋊D5914yes—
190C4×Q8C22×D5914yes—
191#117—1181no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4C22⋊D5912yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42C22×D5911yescharacteristic
199#106—1184no—
200C4⋊C8—594no—
201#103—1188no—
202#103—1184no—
203#103—1184no—
204C22⋊C8—1182no—
205#97—1181no—
206#97—1182no—
207C4×C8—592no—
208C2×Q8—28no—
209C2×Q8—28no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—44no—
213C2×Q8—84no—
214C2×Q8—1182no—
215C2×Q8C2×C22⋊D5914yes—
216C2×Q8C2×C22⋊D5914yes—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4C2×C22⋊D5912yes—
221C22×C4C2×C22⋊D5911yescharacteristic
222Q16—2368no—
223M16—2362no—
224C2×C8—1182no—
225C2×C8—1184no—
226C4⋊C4—216no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—1182no—
233C4⋊C4—1182no—
234C4⋊C4—2362no—
235C4⋊C4C2×C22⋊D5914yes—
236C22⋊C4—2361no—
237C42—28no—
238C42—24no—
239C42—24no—
240C42—2361no—
241C42C2×C22⋊D5911yescharacteristic
242C42C2×C22⋊D5912yes—
243C42C23×D5911yescharacteristic
244C23#1427369611yescharacteristic
245Q8—28no—
246Q8—24no—
247Q8—44no—
248Q8—82no—
249Q8—88no—
250Q8—2364no—
251Q8#1427375818yes—
252C2×C4—28no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—2362no—
261C2×C4#1427369612yes—
262C2×C4#1427369611yescharacteristic
263C2×C4C22×C22⋊D5911yescharacteristic
264C2×C4C22×C22⋊D5912yes—
265C8—2362no—
266C8—2364no—
267C22—22no—
268C22#1683758012yes—
269C22C2×#1427369611yescharacteristic
270C4—21no—
271C4—21no—
272C4—28no—
273C4—22no—
274C4—44no—
275C4—44no—
276C4—4722no—
277C4C2×#1427375814yes—
278C2—22no—
279C2#2143333012yes—
280C2C2×#1683758011yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC2×Q82×C59C2
socleC22×C59C2×#120
prime radicalC23×C59#120
hypercenterC2×Q82D59
hypocenterC59#18551
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
3. #21423317C2semidirect
168. C59#18551semidirect

Maximal subgroups

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

subgroupindex
#214344762
#214342262
#214233172
#214228222
#214225432
C2×Q82×C592
#1855159