Finite Group Databasebeta

← #28505500

Subgroups — #28505500

Order 15104.

516 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 103 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#21427270C211yesmaximal, characteristic
2#21424485C211yesmaximal, characteristic
3#21424379C212yesmaximal
4#21422213C212yesmaximal
5C2×C59×#354C211yesmaximal, characteristic
6#16837877C2211yescharacteristic
7#16837411—21no—
8C2×#14273644C2211yescharacteristic
9#16837309—21no—
10#16837295—22no—
11#16837282C2212yes—
12#16837277—22no—
13#16837277—22no—
14#16837268—21no—
15#16837168—22no—
16#16837162—21no—
17C2×#124×C59C2211yescharacteristic
18C2×#99×C59C2212yes—
19C59×#354C414yescomplemented
20C23×Dic59D412yes—
21C2×#11745664—21no—
22D4×Dic59—41no—
23D4×Dic59—41no—
24#14273686—21no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59—21no—
28C2×C4⋊Dic59—42no—
29C2×C22⋊Dic59—22no—
30C2×C22⋊Dic59—22no—
31C2×C22⋊Dic59D412yes—
32C2×C4×Dic59—21no—
33#14273651—41no—
34#14273651—41no—
35#14273644—22no—
36C59⋊#95—22no—
37C59⋊#95—22no—
38C59⋊#95—22no—
39C59⋊#95—42no—
40C22×Q8×C59D411yescharacteristic
41C22×D4×C59C2311yescharacteristic
42#124×C59—22no—
43#124×C59C2×C412yes—
44#124×C59Q812yes—
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59—22no—
47C2×C42×C59D411yescharacteristic
48#99×C59—22no—
49#99×C59C2×C414yes—
50C24×C59C2×D412yes—
51C2×Q8×C59—21no—
52C2×Q8×C59—41no—
53C2×Q8×C59—41no—
54C2×Q8×C59C4⋊C412yescomplemented
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C2×D4×C59C22×C411yescharacteristic
58C2×D4×C59C2×Q811yescharacteristic
59C2×D4×C59#4112yes—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—42no—
63C22×C4×C59C2×D411yescharacteristic
64C22×Dic59—22no—
65C22×Dic59—22no—
66C22×Dic59—22no—
67C22×Dic59—22no—
68C22×Dic59—42no—
69C22×Dic59—82no—
70C22⋊C4×C59—24no—
71C22⋊C4×C59—24no—
72C22⋊C4×C59—24no—
73C22⋊C4×C59—42no—
74C42×C59—21no—
75C42×C59C4⋊C412yescomplemented
76C22⋊Dic59—42no—
77C22⋊Dic59—41no—
78C22⋊Dic59—42no—
79C22⋊Dic59—82no—
80C4⋊Dic59—21no—
81C4⋊Dic59—21no—
82C4⋊Dic59—81no—
83C4⋊Dic59—81no—
84C4⋊Dic59—81no—
85C4⋊Dic59—84no—
86C4×Dic59—41no—
87C4×Dic59—81no—
88#11745664—41no—
89#11745664—41no—
90C2×C59⋊C8—41no—
91C23×C59—22no—
92C23×C59—22no—
93C23×C59—42no—
94C23×C59—42no—
95C23×C59C4×D412yes—
96C23×C59#12011yescharacteristic
97C23×C59C22⋊Q812yes—
98Q8×C59—42no—
99Q8×C59—81no—
100D4×C59—41no—
101D4×C59—41no—
102D4×C59—42no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—22no—
106C2×C4×C59—22no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—44no—
112C2×C4×C59—82no—
113C2×C4×C59—82no—
114C2×C4×C59C2×C4⋊C411yescharacteristic
115C2×C4×C59#12311yescharacteristic
116C2×Dic59—42no—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—42no—
120C2×Dic59—44no—
121C2×Dic59—41no—
122C2×Dic59—41no—
123C2×Dic59—42no—
124C2×Dic59—42no—
125C2×Dic59—82no—
126C2×Dic59—82no—
127C2×Dic59—82no—
128C59⋊C8—81no—
129#12967—591nomaximal, Sylow
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—22no—
133C22×C59—22no—
134C22×C59—42no—
135C22×C59—42no—
136C22×C59—42no—
137C22×C59—42no—
138C22×C59—82no—
139C22×C59#38511yescharacteristic
140C22×C59B(4,𝔽2)11yescharacteristic
141C22×C59#45811yescharacteristic
142C4×C59—21no—
143C4×C59—21no—
144C4×C59—42no—
145C4×C59—42no—
146C4×C59—84no—
147Dic59—82no—
148Dic59—82no—
149Dic59—81no—
150Dic59—81no—
151#2379—591no—
152C2×#354D5911yescharacteristic
153#1885—591no—
154#1867—592no—
155#1390—592no—
156C2×C59—21no—
157C2×C59—21no—
158C2×C59—42no—
159C2×C59—42no—
160C2×C59#186711yescharacteristic
161C2×C59#219911yescharacteristic
162C2×C59#220111yescharacteristic
163C2×#124C2×D5911yescharacteristic
164#518—591no—
165#419—1181no—
166C2×#100—591no—
167C2×#99C2×D5912yes—
168#391—1181no—
169#388—1182no—
170#386—592no—
171#385—1182no—
172#385—1182no—
173#382—1181no—
174#354Dic5914yescomplemented
175#342—1182no—
176#339—1181no—
177C59#1296711yescharacteristic, complemented, Sylow
178C22×Q8D23611yescharacteristic
179C22×D4C22×D5911yescharacteristic
180C23×C4—592no—
181C2×M16—1181no—
182#124—22no—
183#124Dic11812yes—
184#124C2×Dic5912yes—
185C4×D4—2361no—
186C4×D4—2361no—
187#117—1181no—
188C2×C4⋊C4—1181no—
189C2×C4⋊C4—1181no—
190C2×C4⋊C4—1181no—
191C2×C4⋊C4—2362no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—592no—
195C2×C22⋊C4—1182no—
196C2×C22⋊C4—1182no—
197C2×C42—1181no—
198C2×C42D23611yescharacteristic
199#103—2361no—
200#103—2361no—
201#100—1182no—
202#99—22no—
203#99C2×Dic5914yes—
204#95—1182no—
205#95—1182no—
206#95—1182no—
207#95—2362no—
208C24D4×D5912yes—
209C2×Q8—21no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8C4⋊Dic5912yescomplemented
213C2×D4—41no—
214C2×D4—41no—
215C2×D4C2×Dic11811yescharacteristic
216C2×D4#1174568712yes—
217C2×D4C22×Dic5911yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—1182no—
222C22×C4—1182no—
223C22×C4—1182no—
224C22×C4—1182no—
225C22×C4—4722no—
226C22×C4—2362no—
227C22×C4C2×D23611yescharacteristic
228M16—2361no—
229M16—2361no—
230C2×C8—2361no—
231C4⋊C4—1181no—
232C4⋊C4—1181no—
233C4⋊C4—4721no—
234C4⋊C4—4721no—
235C4⋊C4—4721no—
236C4⋊C4—4724no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—42no—
241C22⋊C4—4722no—
242C22⋊C4—2362no—
243C22⋊C4—2361no—
244C22⋊C4—2362no—
245C42—21no—
246C42—4721no—
247C42—2361no—
248C42C4⋊Dic5912yescomplemented
249C23—22no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23D4×Dic5912yes—
254C23#1427369711yescharacteristic
255C23#1427370412yes—
256Q8—42no—
257Q8—81no—
258D4—41no—
259D4—42no—
260D4—41no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—44no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—4722no—
273C2×C4—4722no—
274C2×C4—4722no—
275C2×C4—2362no—
276C2×C4—2362no—
277C2×C4—2362no—
278C2×C4—2362no—
279C2×C4—2364no—
280C2×C4—2361no—
281C2×C4—2361no—
282C2×C4—2362no—
283C2×C4—2362no—
284C2×C4C2×C4⋊Dic5911yescharacteristic
285C2×C4#1427371111yescharacteristic
286C8—4721no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—82no—
296C22#1683727711yescharacteristic
297C22#1683759011yescharacteristic
298C22#1683759311yescharacteristic
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—84no—
304C4—4722no—
305C4—4721no—
306C4—4722no—
307C4—4721no—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#2142437911yescharacteristic
313C2#2142609911yescharacteristic
314C2#2142611111yescharacteristic
subgroupG/H
centerC22#16837590
commutatorC2×D4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC2×C59×#354C2
socleC22×C59B(4,𝔽2)
prime radical, cosocleC22×D4×C59C23
hypercenterC2×#354D59
hypocenterC59#12967
LCSG ▷ C2×D4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
19. C59×#354C4semidirect
54. C2×Q8×C59C4⋊C4semidirect
75. C42×C59C4⋊C4semidirect
174. #354Dic59semidirect
177. C59#12967semidirect
212. C2×Q8C4⋊Dic59semidirect
248. C42C4⋊Dic59semidirect

Maximal subgroups

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

subgroupindex
#214272702
#214244852
#214243792
#214222132
C2×C59×#3542
#1296759