Finite Group Databasebeta

← #28505498

Subgroups — #28505498

Order 15104.

516 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 99 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×#16837197C211yesmaximal, characteristic
2#21424485C211yesmaximal, characteristic
3#21424376C212yesmaximal
4#21422214C212yesmaximal
5C59×#2379C211yesmaximal, characteristic
6#16837877C2211yescharacteristic
7#16837411—21no—
8C2×#14273644C2211yescharacteristic
9C2×#14273642C2212yes—
10#16837275—22no—
11#16837197C414yescomplemented
12#16837168—22no—
13#16837162—21no—
14C2×#124×C59C2211yescharacteristic
15C59×#391—21no—
16C59×#388—22no—
17C59×#386C2212yes—
18C59×#385—22no—
19C59×#382—21no—
20C2×#11745664—21no—
21D4×Dic59—41no—
22D4×Dic59—41no—
23#14273686—21no—
24C2×Dic59⋊C4—42no—
25C2×C22⋊Dic59—22no—
26C2×C22⋊Dic59—22no—
27#14273651—41no—
28#14273651—41no—
29#14273644—22no—
30#14273642—22no—
31#14273642C2×C414yes—
32C59⋊#95—42no—
33C22×Q8×C59D411yescharacteristic
34C22×D4×C59C2311yescharacteristic
35C23×C4×C59D412yes—
36#124×C59—22no—
37#124×C59C2×C412yes—
38#124×C59Q812yes—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59—21no—
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59—22no—
44C2×C22⋊C4×C59D412yes—
45C2×C42×C59—21no—
46C2×C42×C59D411yescharacteristic
47#95×C59—22no—
48#95×C59—22no—
49#95×C59—22no—
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—22no—
63C22×C4×C59—22no—
64C22×C4×C59—42no—
65C22×C4×C59—82no—
66C22×C4×C59C2×D411yescharacteristic
67C22×Dic59—22no—
68C22×Dic59—22no—
69C22×Dic59—42no—
70C4⋊C4×C59—81no—
71C4⋊C4×C59—81no—
72C4⋊C4×C59—81no—
73C22⋊C4×C59—24no—
74C22⋊C4×C59—24no—
75C22⋊C4×C59—42no—
76C22⋊C4×C59—82no—
77C42×C59—21no—
78C42×C59—81no—
79C42×C59C4⋊C412yescomplemented
80C22⋊Dic59—24no—
81C22⋊Dic59—42no—
82C22⋊Dic59—42no—
83C22⋊Dic59—41no—
84C4⋊Dic59—21no—
85C4⋊Dic59—21no—
86Dic59⋊C4—84no—
87C4×Dic59—41no—
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—42no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—44no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—82no—
119C2×C4×C59C2×C4⋊C411yescharacteristic
120C2×C4×C59#12311yescharacteristic
121C2×Dic59—44no—
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—82no—
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—82no—
147C4×C59—82no—
148Dic59—84no—
149Dic59—81no—
150Dic59—81no—
151#2379D5911yescharacteristic
152C2×#354—591no—
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×#99—592no—
168#391—21no—
169#388—22no—
170#386C2×D5912yes—
171#385—22no—
172#385—1182no—
173#382—21no—
174#354—594no—
175#342—1182no—
176#339—1181no—
177C59#1296711yescharacteristic, complemented, Sylow
178C22×Q8C22⋊D5911yescharacteristic
179C22×D4C22×D5911yescharacteristic
180C23×C4C22⋊D5912yes—
181C2×M16—1181no—
182#124—22no—
183#124Dic11812yes—
184#124C4×D5912yes—
185C4×D4—2361no—
186C4×D4—2361no—
187#117—1181no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—2362no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—1182no—
195C2×C22⋊C4—1182no—
196C2×C22⋊C4C22⋊D5912yes—
197C2×C42—21no—
198C2×C42C22⋊D5911yescharacteristic
199#103—2361no—
200#103—2361no—
201#100—1182no—
202#99—594no—
203#99—1182no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—2362no—
208C24C2×C22⋊D5912yes—
209C2×Q8—21no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8Dic59⋊C412yescomplemented
213C2×D4—41no—
214C2×D4—41no—
215C2×D4C2×Dic11811yescharacteristic
216C2×D4C2×C4×D5911yescharacteristic
217C2×D4#1174568512yes—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—82no—
224C22×C4—1182no—
225C22×C4—1182no—
226C22×C4—2362no—
227C22×C4C2×C22⋊D5911yescharacteristic
228M16—2361no—
229M16—2361no—
230C2×C8—2361no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C4⋊C4—1181no—
235C4⋊C4—1181no—
236C4⋊C4—4724no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—42no—
240C22⋊C4—82no—
241C22⋊C4—1184no—
242C22⋊C4—2362no—
243C22⋊C4—2362no—
244C22⋊C4—2361no—
245C42—21no—
246C42—81no—
247C42—2361no—
248C42Dic59⋊C412yescomplemented
249C23—22no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23C4×C22⋊D5912yes—
254C23#1427369611yescharacteristic
255C23#1427370612yes—
256Q8—42no—
257Q8—81no—
258D4—41no—
259D4—41no—
260D4—42no—
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—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—82no—
277C2×C4—4722no—
278C2×C4—4722no—
279C2×C4—4722no—
280C2×C4—4722no—
281C2×C4—2364no—
282C2×C4—2361no—
283C2×C4—2361no—
284C2×C4C2×Dic59⋊C411yescharacteristic
285C2×C4#1427371011yescharacteristic
286C8—4721no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—82no—
296C22#1683727511yescharacteristic
297C22#1683758911yescharacteristic
298C22#1683759211yescharacteristic
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—82no—
304C4—82no—
305C4—4724no—
306C4—4721no—
307C4—4721no—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#2142437711yescharacteristic
313C2#2142609711yescharacteristic
314C2#2142610911yescharacteristic
subgroupG/H
centerC22#16837589
commutatorC2×D4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#2379C2
socleC22×C59B(4,𝔽2)
prime radical, cosocleC22×D4×C59C23
hypercenter#2379D59
hypocenterC59#12967
LCSG ▷ C2×D4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2379

Complements

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

NH ≅ G/Ntype
11. #16837197C4semidirect
54. C2×Q8×C59C4⋊C4semidirect
79. C42×C59C4⋊C4semidirect
177. C59#12967semidirect
212. C2×Q8Dic59⋊C4semidirect
248. C42Dic59⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#168371972
#214244852
#214243762
#214222142
C59×#23792
#1296759