Finite Group Databasebeta

← #28486369

Subgroups — #28486369

Order 15104.

518 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 111 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#21433125C211yesmaximal, characteristic
2#21422822C212yesmaximal, complemented
3#21422437C211yesmaximal, characteristic
4#21421952C212yesmaximal, complemented
5C59×#2547C211yesmaximal, characteristic
6#16837474—24no—
7#16837459C2212yes—
8#16837459C2212yes—
9#16837366—21no—
10C2×C22⋊(C59⋊C8)—21no—
11#16837151—24no—
12#16837126—21no—
13#16837126—21no—
14C2×C4⋊Q8×C59C2211yescharacteristic
15C2×#124×C59C412yes—
16C2×C4×D4×C59C2211yescharacteristic
17C59×#397—21no—
18C59×#397—21no—
19C59×#391C412yes—
20C59×#380—21no—
21C59×#380—21no—
22C4×C4⋊C4×C59C2211yescharacteristic
23C2×#11745664—22no—
24C22×C59⋊C8—22no—
25C4⋊(C59⋊C8)—22no—
26C4⋊(C59⋊C8)D414yes—
27C22⋊(C59⋊C8)—22no—
28C22⋊(C59⋊C8)—22no—
29C22⋊(C59⋊C8)—22no—
30C22⋊(C59⋊C8)—22no—
31#14273638—22no—
32#14273638—22no—
33C22×Q8×C59C2×C412yes—
34C22×D4×C59C2×C411yescharacteristic
35C23×C4×C59D411yescharacteristic
36C23×C4×C59D411yescharacteristic
37C4⋊Q8×C59—22no—
38C4⋊Q8×C59—22no—
39#124×C59—44no—
40C4×D4×C59—22no—
41C4×D4×C59—22no—
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59C2×C411yescharacteristic
46C2×C22⋊C4×C59—22no—
47C2×C22⋊C4×C59—22no—
48C2×C22⋊C4×C59D411yescharacteristic
49C2×C22⋊C4×C59D411yescharacteristic
50C2×C42×C59C2×C412yes—
51C2×C42×C59C2311yescharacteristic
52#95×C59—22no—
53#95×C59—22no—
54#95×C59—21no—
55C24×C59C22⋊C411yescharacteristic
56C24×C59C22⋊C411yescharacteristic
57C2×Q8×C59—24no—
58C2×Q8×C59—42no—
59C2×Q8×C59—42no—
60C2×D4×C59—22no—
61C2×D4×C59—22no—
62C2×D4×C59—41no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—22no—
68C22×C4×C59—22no—
69C22×C4×C59—42no—
70C22×C4×C59—42no—
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22×C411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C4⋊C4×C59—22no—
77C4⋊C4×C59—42no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—42no—
80C4⋊C4×C59—41no—
81C22⋊C4×C59—22no—
82C22⋊C4×C59—22no—
83C22⋊C4×C59—44no—
84C22⋊C4×C59—44no—
85C42×C59—42no—
86C42×C59C2×D412yes—
87C42×C59#4112yes—
88#11745664—82no—
89C2×C59⋊C8—24no—
90C2×C59⋊C8—42no—
91C2×C59⋊C8—82no—
92C23×C59—21no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—41no—
98C23×C59—41no—
99C23×C59—41no—
100C23×C59—41no—
101C23×C59#10012yes—
102C23×C59C2×C22⋊C411yescharacteristic
103Q8×C59—84no—
104D4×C59—44no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—22no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—41no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59—81no—
122C2×C4×C59—81no—
123C2×C4×C59C4×D412yes—
124C2×C4×C59#12011yescharacteristic
125C2×C4×C59#12112yes—
126C2×C4×C59#12311yescharacteristic
127C59⋊C8—44no—
128C59⋊C8—82no—
129#9451—591nomaximal, Sylow
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—22no—
133C22×C59—42no—
134C22×C59—42no—
135C22×C59—41no—
136C22×C59—41no—
137C22×C59—41no—
138C22×C59—41no—
139C22×C59—41no—
140C22×C59—81no—
141C22×C59—81no—
142C22×C59#38611yescharacteristic
143C22×C59C2×#10011yescharacteristic
144C22×C59#41311yescharacteristic
145C4×C59—21no—
146C4×C59—22no—
147C4×C59—22no—
148C4×C59—41no—
149C4×C59—41no—
150C4×C59—82no—
151C4×C59—82no—
152C4×C59#45512yes—
153#2973—591no—
154#2547D5911yescharacteristic
155#1558—592no—
156#1466—591no—
157#1285—592no—
158C2×C59—21no—
159C2×C59—21no—
160C2×C59—42no—
161C2×C59—81no—
162C2×C59#188511yescharacteristic
163C2×C59#190011yescharacteristic
164C2×C59#190911yescharacteristic
165C2×C4⋊Q8C2×D5911yescharacteristic
166C2×#124Dic5912yes—
167C2×C4×D4C2×D5911yescharacteristic
168#435—1184no—
169#432—592no—
170#432—592no—
171#407—1181no—
172C2×C22⋊C8—1181no—
173#397—21no—
174#397—21no—
175#391Dic5912yes—
176#380—21no—
177#380—21no—
178C4×C4⋊C4C2×D5911yescharacteristic
179#333—1184no—
180#323—1181no—
181#323—1181no—
182C59#945111yescharacteristic, complemented, Sylow
183C22×Q8C2×Dic5912yes—
184C22×D4C2×Dic5911yescharacteristic
185C23×C4C22⋊D5911yescharacteristic
186C23×C4C22⋊D5911yescharacteristic
187C2×M16—1182no—
188C22×C8—1182no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191#124—44no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×Dic5911yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C22⋊D5911yescharacteristic
201C2×C22⋊C4C22⋊D5911yescharacteristic
202C2×C42C2×Dic5912yes—
203C2×C42C22×D5911yescharacteristic
204C4⋊C8—594no—
205C4⋊C8—1182no—
206C22⋊C8—1182no—
207C22⋊C8—1182no—
208C22⋊C8—1182no—
209C22⋊C8—1182no—
210#97—1182no—
211#97—1182no—
212#95—22no—
213#95—22no—
214#95—21no—
215C24C22⋊Dic5911yescharacteristic
216C24C22⋊Dic5911yescharacteristic
217C2×Q8—24no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C22⋊Dic5911yescharacteristic
232C22×C4C22⋊Dic5911yescharacteristic
233C22×C4C22×Dic5911yescharacteristic
234C22×C4C2×C22⋊D5911yescharacteristic
235C22×C4C2×C22⋊D5911yescharacteristic
236M16—4722no—
237C2×C8—1184no—
238C2×C8—4722no—
239C2×C8—2362no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C42—42no—
250C42D4×D5912yes—
251C42#1174568712yes—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#1427364412yes—
262C23C2×C22⋊Dic5911yescharacteristic
263Q8—84no—
264D4—44no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4D4×Dic5912yes—
284C2×C4#1427369511yescharacteristic
285C2×C4#1427369912yes—
286C2×C4#1427370811yescharacteristic
287C8—4722no—
288C8—2364no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22#1683728211yescharacteristic
302C22C2×#1427364411yescharacteristic
303C22#1683738611yescharacteristic
304C4—21no—
305C4—22no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—82no—
310C4—82no—
311C4#1683757912yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2#2142448511yescharacteristic
317C2#2142456011yescharacteristic
318C2#2142461911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2547C2
socleC22×C59#386
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenter#2547D59
hypocenterC59#9451
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
2. #21422822C2semidirect
4. #21421952C2semidirect
182. C59#9451semidirect

Maximal subgroups

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

subgroupindex
#214331252
#214228222
#214224372
#214219522
C59×#25472
#945159