Finite Group Databasebeta

← #28570015

Subgroups — #28570015

Order 15104.

938 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 171 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#21435099C212yesmaximal, complemented
2#21434645C211yesmaximal, characteristic, complemented
3#21423413C218yesmaximal, complemented
4#21422723C211yesmaximal, characteristic, complemented
5#21422694C212yesmaximal, complemented
6C2×C59×#562C211yesmaximal, characteristic
7#16837959C2211yescharacteristic
8#16837812C2214yescomplemented
9#16837801—22no—
10#16837797C2214yescomplemented
11#16837761C2214yescomplemented
12#16837621—24no—
13#16837558—22no—
14#16837553—22no—
15#16837459C2212yes—
16#16837442—22no—
17C4×#11745664C2211yescharacteristic
18#16837144C2214yescomplemented
19#16837142C2218yescomplemented
20C59×#562—22no—
21C59×#562—24no—
22C59×#562C2214yes—
23C2×#125×C59C2211yescharacteristic
24C2×#124×C59C2212yes—
25C2×#123×C59—22no—
26C2×C22⋊Q8×C59—22no—
27C2×#121×C59—22no—
28C2×#11745672—24no—
29C2×#11745670—24no—
30C2×#11745664—21no—
31#14273727C2312yes—
32#14273719—21no—
33#14273706—22no—
34C4×Dic118—22no—
35#14273686—21no—
36#14273660—24no—
37#14273658—24no—
38C4⋊(C59⋊C8)—22no—
39#14273651—24no—
40#14273648—24no—
41C22⋊(C59⋊C8)—22no—
42#14273638C2312yes—
43#14273638C2314yes—
44C4×C59⋊C8—21no—
45C22×Q8×C59D412yes—
46C22×D4×C59—22no—
47C22×D4×C59D412yes—
48C23×C4×C59—22no—
49#125×C59—22no—
50#125×C59D412yes—
51#125×C59C2312yes—
52#124×C59—22no—
53#124×C59—22no—
54#124×C59D414yes—
55#124×C59C2314yes—
56#123×C59—28no—
57#123×C59—28no—
58C22⋊Q8×C59—28no—
59C22⋊Q8×C59—28no—
60#121×C59—28no—
61#121×C59—28no—
62C2×C4⋊C4×C59—22no—
63C2×C4⋊C4×C59—22no—
64C2×C4⋊C4×C59D412yes—
65C2×C22⋊C4×C59—22no—
66C2×C22⋊C4×C59—22no—
67C2×C22⋊C4×C59—22no—
68C2×C42×C59C2311yescharacteristic
69C24×C59—22no—
70C2×Q8×C59—22no—
71C2×Q8×C59—82no—
72C2×Q8×C59C2×D414yes—
73C2×D4×C59—28no—
74C2×D4×C59—22no—
75C2×D4×C59—82no—
76C2×D4×C59—84no—
77C2×D4×C59C2×D414yes—
78C22×C4×C59—22no—
79C22×C4×C59—22no—
80C22×C4×C59—28no—
81C22×C4×C59—84no—
82C22×C4×C59C2×D411yescharacteristic
83C22×C4×C59C2×D412yes—
84C2×Dic118—22no—
85C4⋊C4×C59—28no—
86C4⋊C4×C59—24no—
87C4⋊C4×C59—22no—
88C4⋊C4×C59—24no—
89C4⋊C4×C59C2×D414yes—
90C22⋊C4×C59—28no—
91C22⋊C4×C59—24no—
92C22⋊C4×C59—24no—
93C22⋊C4×C59—24no—
94C22⋊C4×C59—24no—
95C42×C59C2×D411yescharacteristic
96C42×C59C2×D412yes—
97C42×C59C2411yescharacteristic
98C22⋊Dic59—41no—
99#11745672—84no—
100#11745670—84no—
101C4⋊Dic59—22no—
102C4⋊Dic59—22no—
103Dic59⋊C4—42no—
104C4×Dic59—41no—
105#11745664—24no—
106C2×C59⋊C8—22no—
107C2×C59⋊C8—24no—
108C23×C59—24no—
109C23×C59—44no—
110C23×C59—44no—
111C23×C59—44no—
112C23×C59#12011yescharacteristic
113Q8×C59—44no—
114Q8×C59—82no—
115D4×C59—44no—
116D4×C59—82no—
117D4×C59—88no—
118C2×C4×C59—22no—
119C2×C4×C59—24no—
120C2×C4×C59—24no—
121C2×C4×C59—42no—
122C2×C4×C59—44no—
123C2×C4×C59—44no—
124C2×C4×C59—88no—
125C2×C4×C59#12011yescharacteristic
126C2×C4×C59#12012yes—
127C2×C4×C59C22×D411yescharacteristic
128C2×C4×C59C22×D412yes—
129C2×Dic59—42no—
130Dic118—44no—
131C59⋊C8—24no—
132C59⋊C8—44no—
133#19006—591nomaximal, Sylow
134C22×C59—22no—
135C22×C59—44no—
136C22×C59—44no—
137C22×C59—44no—
138C22×C59—88no—
139C22×C59#45612yes—
140C22×C59C2×#12011yescharacteristic
141C4×C59—21no—
142C4×C59—21no—
143C4×C59—22no—
144C4×C59—22no—
145C4×C59—44no—
146C4×C59—44no—
147Dic59—82no—
148C2×#562D5911yescharacteristic
149#3215—592no—
150#3166—591no—
151#1661—598no—
152#1534—591no—
153#1527—592no—
154C2×C59—22no—
155C2×C59—84no—
156C2×C59C2×#45611yescharacteristic
157C2×C59#301912yes—
158#562—22no—
159#562—24no—
160#562C2×D5914yes—
161#533—591no—
162C2×#125C2×D5911yescharacteristic
163C2×#124C2×D5912yes—
164C2×#123—22no—
165C2×C22⋊Q8—22no—
166C2×#121—22no—
167#501—594no—
168#499—1182no—
169#497—594no—
170#489—594no—
171#464—1184no—
172#451—1182no—
173#450—1182no—
174#432—592no—
175#428—1182no—
176C4×M16—591no—
177#330—594no—
178#329—598no—
179C59#1900611yescharacteristic, complemented, Sylow
180C22×Q8C22⋊D5912yes—
181C22×D4—22no—
182C22×D4C22⋊D5912yes—
183C23×C4—22no—
184C2×Q16—1184no—
185C2×SD16—1184no—
186C2×M16—1181no—
187C4⋊Q8—592no—
188#125—22no—
189#125—1181no—
190#125C22⋊D5912yes—
191#125C22×D5912yes—
192#124—22no—
193#124—22no—
194#124C22⋊D5914yes—
195#124C22×D5914yes—
196#123—28no—
197#123—28no—
198C22⋊Q8—28no—
199C22⋊Q8—28no—
200C22⋊Q8—1182no—
201#121—28no—
202#121—28no—
203C4×Q8—1182no—
204#117—1181no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4C22⋊D5912yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C42C22×D5911yescharacteristic
212#107—1184no—
213#106—1184no—
214C4⋊C8—1182no—
215#103—1184no—
216#102—1184no—
217C22⋊C8—1182no—
218#97—592no—
219#97—594no—
220C4×C8—1181no—
221C24—22no—
222C2×Q8—22no—
223C2×Q8—82no—
224C2×Q8—1182no—
225C2×Q8C2×C22⋊D5914yes—
226C2×D4—28no—
227C2×D4—22no—
228C2×D4—82no—
229C2×D4—84no—
230C2×D4C2×C22⋊D5914yes—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—28no—
234C22×C4—84no—
235C22×C4C2×C22⋊D5911yescharacteristic
236C22×C4C2×C22⋊D5912yes—
237Q16—4724no—
238SD16—4724no—
239M16—1184no—
240C2×C8—1182no—
241C2×C8—1184no—
242C4⋊C4—28no—
243C4⋊C4—24no—
244C4⋊C4—22no—
245C4⋊C4—24no—
246C4⋊C4—1182no—
247C4⋊C4—1182no—
248C4⋊C4—2362no—
249C4⋊C4C2×C22⋊D5914yes—
250C22⋊C4—28no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—2361no—
256C42—2361no—
257C42C2×C22⋊D5911yescharacteristic
258C42C2×C22⋊D5912yes—
259C42C23×D5911yescharacteristic
260C23—24no—
261C23—44no—
262C23—44no—
263C23—44no—
264C23#1427369611yescharacteristic
265Q8—44no—
266Q8—82no—
267Q8—2364no—
268D4—44no—
269D4—82no—
270D4—88no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—88no—
278C2×C4—2362no—
279C2×C4#1427369611yescharacteristic
280C2×C4#1427369612yes—
281C2×C4C22×C22⋊D5911yescharacteristic
282C2×C4C22×C22⋊D5912yes—
283C8—1184no—
284C8—2364no—
285C22—22no—
286C22—44no—
287C22—44no—
288C22—44no—
289C22—88no—
290C22#1683758512yes—
291C22C2×#1427369611yescharacteristic
292C4—21no—
293C4—21no—
294C4—22no—
295C4—22no—
296C4—44no—
297C4—44no—
298C4—4722no—
299C2—22no—
300C2—84no—
301C2C2×#1683758511yescharacteristic
302C2#2143345312yes—
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
Fitting, prime radicalC2×C59×#562C2
socleC22×C59C2×#120
hypercenterC2×#562D59
hypocenterC59#19006
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #21435099C2semidirect
2. #21434645C2semidirect
3. #21423413C2semidirect
4. #21422723C2semidirect
5. #21422694C2semidirect
8. #16837812C22semidirect
10. #16837797C22semidirect
11. #16837761C22semidirect
18. #16837144C22semidirect
19. #16837142C22semidirect
179. C59#19006semidirect

Maximal subgroups

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

subgroupindex
#214350992
#214346452
#214234132
#214227232
#214226942
C2×C59×#5622
#1900659