Finite Group Databasebeta

← #28481248

Subgroups — #28481248

Order 15104.

728 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 139 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#21425074C212yesmaximal, complemented
2#21424925C211yesmaximal, characteristic
3#21422834C212yesmaximal, complemented
4#21422831C211yesmaximal, characteristic
5C59×#3550C211yesmaximal, characteristic
6#16837426C2211yescharacteristic
7#16837426C2212yes—
8C2×#14273654—22no—
9C42×Dic59C2211yescharacteristic
10#16837161—24no—
11#16837151—21no—
12#16837148—22no—
13#16837148—22no—
14#16837146—21no—
15C59×#563C414yescomplemented
16C59×#559—21no—
17Q82×C59—21no—
18C59×#556—22no—
19D4×Q8×C59—22no—
20C59×#542—24no—
21C59×#536—22no—
22C59×#533C2212yes—
23C59×#532C2211yescharacteristic
24C2×#11745664—21no—
25C2×#11745664—22no—
26C2×C4×Dic59—21no—
27C2×C4×Dic59—22no—
28C4⋊(C59⋊C8)—22no—
29C4⋊(C59⋊C8)—22no—
30C4⋊(C59⋊C8)—22no—
31#14273654—44no—
32C2×#41×C59D412yes—
33C22×Q8×C59—22no—
34C4⋊Q8×C59—24no—
35C4⋊Q8×C59—24no—
36C4⋊Q8×C59D411yescharacteristic
37C4⋊Q8×C59D412yes—
38C4⋊Q8×C59D412yes—
39#127×C59D411yescharacteristic
40#126×C59—28no—
41#125×C59C2×C414yes—
42#124×C59—24no—
43#124×C59—24no—
44#124×C59C2×C412yes—
45#123×C59—24no—
46#123×C59—24no—
47C22⋊Q8×C59—24no—
48C22⋊Q8×C59—22no—
49C22⋊Q8×C59—22no—
50C22⋊Q8×C59—28no—
51#121×C59—22no—
52C4×Q8×C59—24no—
53C4×Q8×C59—22no—
54C4×Q8×C59—22no—
55C4×D4×C59—24no—
56C4×D4×C59—22no—
57#117×C59—22no—
58#117×C59D414yes—
59C2×C42×C59C2311yescharacteristic
60#41×C59—42no—
61#41×C59—42no—
62C2×Q8×C59—24no—
63C2×Q8×C59—22no—
64C2×Q8×C59—22no—
65C2×Q8×C59—82no—
66C2×Q8×C59—82no—
67C2×Q8×C59C22⋊C412yes—
68C2×D4×C59—22no—
69C2×D4×C59—22no—
70C2×D4×C59C22⋊C412yes—
71C22×C4×C59—24no—
72C22×C4×C59—22no—
73C22×C4×C59C2×D412yes—
74C22×C4×C59C2×D411yescharacteristic
75C22×Dic59—41no—
76C4⋊C4×C59—28no—
77C4⋊C4×C59—24no—
78C4⋊C4×C59—24no—
79C4⋊C4×C59—22no—
80C4⋊C4×C59—22no—
81C4⋊C4×C59—22no—
82C4⋊C4×C59C22⋊C414yes—
83C4⋊C4×C59C22⋊C414yes—
84C22⋊C4×C59—24no—
85C22⋊C4×C59—28no—
86C22⋊C4×C59—24no—
87C22⋊C4×C59—24no—
88C42×C59—24no—
89C42×C59—24no—
90C42×C59C22×C411yescharacteristic
91C42×C59C2×D412yes—
92C42×C59C2×D411yescharacteristic
93C4×Dic59—22no—
94C4×Dic59—24no—
95C4×Dic59—42no—
96C4×Dic59—41no—
97#11745664—42no—
98#11745664—44no—
99C2×C59⋊C8—41no—
100C2×C59⋊C8—42no—
101C23×C59—22no—
102C23×C59#12011yescharacteristic
103Q8×C59—42no—
104Q8×C59—42no—
105Q8×C59—42no—
106Q8×C59—42no—
107Q8×C59—42no—
108Q8×C59—84no—
109D4×C59—42no—
110D4×C59—42no—
111D4×C59—42no—
112D4×C59—82no—
113C2×C4×C59—21no—
114C2×C4×C59—22no—
115C2×C4×C59—21no—
116C2×C4×C59—22no—
117C2×C4×C59—24no—
118C2×C4×C59—22no—
119C2×C4×C59—24no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59—84no—
123C2×C4×C59C2×C22⋊C412yes—
124C2×C4×C59C2×C22⋊C411yescharacteristic
125C2×C4×C59#12011yescharacteristic
126C2×C4×C59#12012yes—
127C2×Dic59—42no—
128C2×Dic59—44no—
129C59⋊C8—81no—
130C59⋊C8—82no—
131#8531—591nomaximal, Sylow
132C22×C59—21no—
133C22×C59—22no—
134C22×C59—42no—
135C22×C59—44no—
136C22×C59#37911yescharacteristic
137C4×C59—21no—
138C4×C59—22no—
139C4×C59—42no—
140C4×C59—44no—
141C4×C59—44no—
142C4×C59#45311yescharacteristic
143C4×C59#45412yes—
144C4×C59#45511yescharacteristic
145C4×C59#45612yes—
146Dic59—44no—
147#3550D5911yescharacteristic
148#1994—592no—
149#1962—591no—
150#1561—592no—
151#1560—591no—
152C2×C59—41no—
153C2×C59—82no—
154C2×C59#179211yescharacteristic
155C2×C59#179312yes—
156#563Dic5914yescomplemented
157#559—21no—
158Q82—21no—
159#556—22no—
160D4×Q8—22no—
161#542—24no—
162#536—22no—
163#533C2×D5912yes—
164#532C2×D5911yescharacteristic
165#423—592no—
166#423—591no—
167C2×#104—1182no—
168C43—591no—
169#339—1184no—
170#333—1181no—
171#332—1182no—
172#332—1182no—
173#331—1181no—
174C59#853111yescharacteristic, complemented, Sylow
175C2×#41C22⋊D5912yes—
176C22×Q8—22no—
177C2×M16—1182no—
178C2×M16—1181no—
179C4⋊Q8—24no—
180C4⋊Q8—24no—
181C4⋊Q8C22⋊D5911yescharacteristic
182C4⋊Q8C22⋊D5912yes—
183C4⋊Q8C22⋊D5912yes—
184#127C22⋊D5911yescharacteristic
185#126—28no—
186#125C2×Dic5914yes—
187#124—24no—
188#124—24no—
189#124C2×Dic5912yes—
190#123—24no—
191#123—24no—
192C22⋊Q8—24no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—28no—
196#121—22no—
197C4×Q8—24no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×D4—24no—
201C4×D4—22no—
202#117—22no—
203#117C22⋊D5914yes—
204C2×C42—1182no—
205C2×C42—1181no—
206C2×C42C22×D5911yescharacteristic
207C4⋊C8—1182no—
208C4⋊C8—1182no—
209C4⋊C8—1182no—
210#104—2364no—
211#41—42no—
212#41—42no—
213C2×Q8—24no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—82no—
217C2×Q8—82no—
218C2×Q8C22⋊Dic5912yes—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4C22⋊Dic5912yes—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—2361no—
225C22×C4C2×C22⋊D5912yes—
226C22×C4C2×C22⋊D5911yescharacteristic
227M16—2362no—
228M16—2364no—
229C2×C8—2361no—
230C2×C8—2362no—
231C4⋊C4—28no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4C22⋊Dic5914yes—
238C4⋊C4C22⋊Dic5914yes—
239C22⋊C4—24no—
240C22⋊C4—28no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C42—24no—
244C42—24no—
245C42—1182no—
246C42—1184no—
247C42—2362no—
248C42—2361no—
249C42C22×Dic5911yescharacteristic
250C42C2×C22⋊D5912yes—
251C42C2×C22⋊D5911yescharacteristic
252C23—22no—
253C23#1427369611yescharacteristic
254Q8—42no—
255Q8—42no—
256Q8—42no—
257Q8—42no—
258Q8—42no—
259Q8—84no—
260D4—42no—
261D4—42no—
262D4—42no—
263D4—82no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—84no—
274C2×C4—2362no—
275C2×C4—2364no—
276C2×C4C2×C22⋊Dic5912yes—
277C2×C4C2×C22⋊Dic5911yescharacteristic
278C2×C4#1427369611yescharacteristic
279C2×C4#1427369612yes—
280C8—4721no—
281C8—4722no—
282C22—21no—
283C22—22no—
284C22—42no—
285C22—44no—
286C22#1683725811yescharacteristic
287C4—21no—
288C4—22no—
289C4—42no—
290C4—44no—
291C4—44no—
292C4—2364no—
293C4#1683757011yescharacteristic
294C4#1683757512yes—
295C4#1683758011yescharacteristic
296C4#1683758512yes—
297C2—41no—
298C2—82no—
299C2#2142405011yescharacteristic
300C2#2142405512yes—
subgroupG/H
centerC22#16837258
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3550C2
socleC22×C59#379
prime radicalC59×#532C22
cosocleC2×C42×C59C23
hypercenter#3550D59
hypocenterC59#8531
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3550

Complements

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

NH ≅ G/Ntype
1. #21425074C2semidirect
3. #21422834C2semidirect
15. C59×#563C4semidirect
156. #563Dic59semidirect
174. C59#8531semidirect

Maximal subgroups

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

subgroupindex
#214250742
#214249252
#214228342
#214228312
C59×#35502
#853159