Finite Group Databasebeta

← #28484455

Subgroups — #28484455

Order 15104.

586 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#16837270C211yesmaximal, characteristic
2#21424164C211yesmaximal, characteristic
3C2×#16837139C212yesmaximal
4#21421974C212yesmaximal
5C59×#2855C211yesmaximal, characteristic
6C2×C22⋊(C59⋊C8)C2212yes—
7#16837270—24no—
8C2×C59⋊#95—21no—
9C2×C59⋊#95—21no—
10C2×C59⋊#95C2212yes—
11#16837157—22no—
12#16837139—24no—
13C2×C22⋊Q8×C59C2211yescharacteristic
14C22×C42×C59C2211yescharacteristic
15C59×#391C412yescomplemented
16C59×#389C412yescomplemented
17C59×#387—22no—
18C59×#385—21no—
19C59×#381—21no—
20C59×#380C2211yescharacteristic
21C23×Dic59—22no—
22C22×C59⋊C8—22no—
23C22⋊(C59⋊C8)—22no—
24C22⋊(C59⋊C8)—22no—
25C59⋊#95—24no—
26C59⋊#95—24no—
27C59⋊#95—22no—
28C59⋊#95D414yes—
29C22×Q8×C59C2×C411yescharacteristic
30C23×C4×C59D411yescharacteristic
31C23×C4×C59D411yescharacteristic
32C23×C4×C59C2311yescharacteristic
33C22⋊Q8×C59—22no—
34C22⋊Q8×C59—22no—
35C2×C4⋊C4×C59—22no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59C2×C411yescharacteristic
38C2×C4⋊C4×C59C2×C412yes—
39C2×C22⋊C4×C59—21no—
40C2×C22⋊C4×C59—21no—
41C2×C42×C59—22no—
42C2×C42×C59—22no—
43C2×C42×C59C2×C412yes—
44C2×C42×C59D412yes—
45#95×C59—22no—
46#95×C59—22no—
47#95×C59—22no—
48#95×C59—21no—
49C24×C59C2×D411yescharacteristic
50C2×Q8×C59—22no—
51C2×Q8×C59—42no—
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—22no—
57C22×C4×C59—22no—
58C22×C4×C59—22no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—42no—
62C22×C4×C59—42no—
63C22×C4×C59C22⋊C412yes—
64C22×C4×C59C22⋊C412yes—
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59C2×D412yes—
68C22×C4×C59#4112yes—
69C22×Dic59—24no—
70C22×Dic59—44no—
71C22×Dic59—44no—
72C22×Dic59—42no—
73C22×Dic59—42no—
74C4⋊C4×C59—22no—
75C4⋊C4×C59—44no—
76C4⋊C4×C59—42no—
77C4⋊C4×C59—42no—
78C22⋊C4×C59—41no—
79C22⋊C4×C59—41no—
80C22⋊C4×C59—42no—
81C42×C59—24no—
82C42×C59—24no—
83C2×C59⋊C8—44no—
84C2×C59⋊C8—42no—
85C23×C59—22no—
86C23×C59—22no—
87C23×C59C2×C22⋊C411yescharacteristic
88C23×C59#12011yescharacteristic
89C23×C59#12311yescharacteristic
90C23×C59#13512yes—
91C23×C59#13712yes—
92Q8×C59—82no—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—42no—
102C2×C4×C59—42no—
103C2×C4×C59—42no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59C4×D412yes—
115C2×C4×C59#12112yes—
116C2×Dic59—44no—
117C2×Dic59—44no—
118C2×Dic59—44no—
119C2×Dic59—84no—
120C2×Dic59—84no—
121C59⋊C8—82no—
122#9153—591nomaximal, Sylow
123C22×C59—21no—
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—24no—
128C22×C59—24no—
129C22×C59#38611yescharacteristic
130C22×C59#41012yes—
131C22×C59#41611yescharacteristic
132C22×C59#41911yescharacteristic
133C22×C59#42112yes—
134C22×C59#45212yes—
135C22×C59#48512yes—
136C4×C59—22no—
137C4×C59—22no—
138C4×C59—42no—
139C4×C59—42no—
140C4×C59—81no—
141C4×C59—81no—
142C4×C59—82no—
143Dic59—84no—
144#2855D5911yescharacteristic
145C2×#383—591no—
146#1817—591no—
147C2×#328—592no—
148#1294—592no—
149C2×C59—22no—
150C2×C59—22no—
151C2×C59#151512yes—
152C2×C59#152112yes—
153C2×C59#187311yescharacteristic
154C2×C59#189711yescharacteristic
155C2×C59#189911yescharacteristic
156C2×C22⋊Q8C2×D5911yescharacteristic
157C22×C42C2×D5911yescharacteristic
158C2×C22⋊C8—592no—
159#391Dic5912yescomplemented
160#389Dic5912yescomplemented
161#387—22no—
162#385—21no—
163#383—1184no—
164#381—21no—
165#380C2×D5911yescharacteristic
166C2×#95—592no—
167C2×#95—1181no—
168C2×#95—1181no—
169#336—1182no—
170#328—1184no—
171C59#915311yescharacteristic, complemented, Sylow
172C22×Q8C2×Dic5911yescharacteristic
173C23×C4—1182no—
174C23×C4C22⋊D5911yescharacteristic
175C23×C4C22⋊D5911yescharacteristic
176C23×C4C22×D5911yescharacteristic
177C22×C8—1182no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C2×Dic5911yescharacteristic
183C2×C4⋊C4C2×Dic5912yes—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42C2×Dic5912yes—
189C2×C42C22⋊D5912yes—
190C22⋊C8—1182no—
191C22⋊C8—1182no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—21no—
196#95—594no—
197#95—1184no—
198#95—1184no—
199#95—1182no—
200C24C2×C22⋊D5911yescharacteristic
201C2×Q8—22no—
202C2×Q8—42no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—1184no—
215C22×C4—2364no—
216C22×C4—2364no—
217C22×C4—2362no—
218C22×C4—2362no—
219C22×C4C22⋊Dic5912yes—
220C22×C4C22⋊Dic5912yes—
221C22×C4D4×D5912yes—
222C22×C4#1174568712yes—
223C22×C4C22×Dic5911yescharacteristic
224C22×C4C2×C22⋊D5911yescharacteristic
225C2×C8—2364no—
226C2×C8—2362no—
227C4⋊C4—22no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—42no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—42no—
234C42—24no—
235C42—24no—
236C23—22no—
237C23—22no—
238C23C2×C22⋊Dic5911yescharacteristic
239C23#1427369511yescharacteristic
240C23#1427370811yescharacteristic
241C23#1427374712yes—
242C23#1427375812yes—
243Q8—82no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—4724no—
266C2×C4—4724no—
267C2×C4—2364no—
268C2×C4—2364no—
269C2×C4—2364no—
270C2×C4D4×Dic5912yes—
271C2×C4#1427369912yes—
272C8—4722no—
273C22—24no—
274C22—24no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22#1683728211yescharacteristic
280C22#1683737612yes—
281C22#1683739811yescharacteristic
282C22#1683741111yescharacteristic
283C22#1683741912yes—
284C22#1683756212yes—
285C22#1683774012yes—
286C4—22no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—82no—
291C4—81no—
292C4—81no—
293C4—4724no—
294C2—22no—
295C2—22no—
296C2#2142264712yes—
297C2#2142267312yes—
298C2#2142440911yescharacteristic
299C2#2142454511yescharacteristic
300C2#2142455511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2855C2
socleC23×C59C2×C22⋊C4
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenter#2855D59
hypocenterC59#9153
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
15. C59×#391C4semidirect
16. C59×#389C4semidirect
159. #391Dic59semidirect
160. #389Dic59semidirect
171. C59#9153semidirect

Maximal subgroups

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

subgroupindex
C2×#168372702
#214241642
C2×#168371392
#214219742
C59×#28552
#915359