Finite Group Databasebeta

← #28476922

Subgroups — #28476922

Order 15104.

658 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 109 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#391×D59C211yesmaximal, characteristic
2#21423857C212yesmaximal
3#21421922C211yesmaximal, characteristic, complemented
4#21421921C212yesmaximal, complemented
5C59×#1272C211yesmaximal, characteristic, complemented
6C22×Q8×D59C411yescharacteristic
7C2×C4⋊C4×D59C411yescharacteristic
8C2×C42×D59C2211yescharacteristic
9C2×(C2×D59)⋊C8—22no—
10C2×#14273637—22no—
11#16837309C411yescharacteristic
12#16837308C2211yescharacteristic
13#16837307C411yescharacteristic
14#16837306—22no—
15#95×D59—22no—
16#16837157C2212yes—
17C59×#391C2211yescharacteristic
18C59×#336C2212yes—
19C22×Dic118—21no—
20C22×Dic118C2×C411yescharacteristic
21C2×Q8×D59—22no—
22C2×Q8×D59—22no—
23C2×Q8×D59—42no—
24C2×Q8×D59—42no—
25C22×C4×D59—21no—
26C22×C4×D59—21no—
27C22×C4×D59C2×C411yescharacteristic
28C22×C4×D59D412yes—
29C22×C59⋊C8—22no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×Dic59⋊C4—21no—
32C4⋊C4×D59—42no—
33C2×C4×Dic59C2×C411yescharacteristic
34C2×C4×Dic59D412yes—
35C42×D59—42no—
36(C2×D59)⋊C8—28no—
37#14273637—44no—
38C59⋊#95—22no—
39C59⋊#95—22no—
40C59⋊#95—22no—
41C22×Q8×C59C2×C411yescharacteristic
42C22×C8×C59—22no—
43C2×C4⋊C4×C59C2×C411yescharacteristic
44C2×C42×C59C2311yescharacteristic
45#95×C59—22no—
46C23×D59C22⋊C411yescharacteristic
47C2×Q8×C59—21no—
48C2×Q8×C59—41no—
49C2×Q8×C59—41no—
50C2×Q8×C59M1612yes—
51C22×C4×C59—21no—
52C22×C4×C59—21no—
53C22×C4×C59C22×C411yescharacteristic
54C22×C4×C59C2×D412yes—
55C22×Dic59—21no—
56C22×Dic59—21no—
57C22×Dic59C22⋊C412yes—
58C22×Dic59C22⋊C411yescharacteristic
59Q8×D59—48no—
60Q8×D59—48no—
61C2×C4×D59—22no—
62C2×C4×D59—21no—
63C2×C4×D59—21no—
64C2×C4×D59—22no—
65C2×C4×D59—24no—
66C2×C4×D59—24no—
67C2×C4×D59—22no—
68C2×C4×D59—22no—
69C2×C4×D59—42no—
70C2×C4×D59—42no—
71C2×C4×D59—42no—
72C2×C4×D59—42no—
73C2×C4×D59—44no—
74C2×Dic118—24no—
75C2×Dic118—21no—
76C2×Dic118—42no—
77C2×Dic118—42no—
78C2×Dic118—41no—
79C2×Dic118—41no—
80C2×Dic118M1612yes—
81C2×C8×C59—24no—
82C2×C8×C59—42no—
83C2×C8×C59—42no—
84C4⋊C4×C59—41no—
85C42×C59—41no—
86C4⋊Dic59—41no—
87Dic59⋊C4—42no—
88C4×Dic59—24no—
89C4×Dic59—41no—
90C2×C59⋊C8—24no—
91C2×C59⋊C8—42no—
92C2×C59⋊C8—42no—
93C23×C59C2×C22⋊C411yescharacteristic
94C22×D59—21no—
95C22×D59—21no—
96C22×D59—22no—
97C22×D59#10012yes—
98C22×D59#10414yes—
99Q8×C59—42no—
100Q8×C59—42no—
101C2×C4×C59—22no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—22no—
105C2×C4×C59—22no—
106C2×C4×C59—41no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—42no—
110C2×C4×C59C2×M1611yescharacteristic
111C2×C4×C59C2×M1611yescharacteristic
112C2×C4×C59#13712yes—
113C2×C4×C59#13712yes—
114C2×Dic59—21no—
115C2×Dic59—21no—
116C2×Dic59—22no—
117C2×Dic59—22no—
118C2×Dic59—22no—
119C2×Dic59—22no—
120C2×Dic59—41no—
121C2×Dic59—41no—
122C2×Dic59—41no—
123C2×Dic59—42no—
124C2×Dic59#9912yes—
125C4×D59—24no—
126C4×D59—24no—
127C4×D59—44no—
128C4×D59—44no—
129C4×D59—48no—
130C4×D59—84no—
131Dic118—48no—
132Dic118—42no—
133Dic118—42no—
134C8×C59—44no—
135C59⋊C8—44no—
136#7629—591nomaximal, Sylow
137C22×C59#40711yescharacteristic
138C22×C59C2×#9911yescharacteristic
139C22×C59C2×#10011yescharacteristic
140C22×C59#41912yes—
141C22×C59C2×#10412yes—
142C2×D59—22no—
143C2×D59—22no—
144C2×D59—24no—
145C2×D59—24no—
146C2×D59#32914yes—
147C4×C59—21no—
148C4×C59—21no—
149C4×C59—22no—
150C4×C59—22no—
151C4×C59—42no—
152C4×C59—81no—
153Dic59—22no—
154Dic59—42no—
155Dic59—42no—
156Dic59—81no—
157C2×#391—591no—
158#1749—592no—
159#1272—591no—
160#1272—592no—
161#1272D5911yescharacteristic, complemented
162C2×C59#146211yescharacteristic
163C2×C59#149112yes—
164C2×C59#150812yes—
165C2×C59C2×#32911yescharacteristic
166C2×C59#155811yescharacteristic
167D59—24no—
168C23×Q8—591no—
169C22×C4⋊C4—591no—
170C22×C42—591no—
171C2×C22⋊C8—1182no—
172C2×#97—1182no—
173#391—591no—
174#391—591no—
175#391—591no—
176#391—1182no—
177#391C2×D5911yescharacteristic
178C2×#95—1182no—
179#336—592no—
180#336C2×D5912yes—
181C59#762911yescharacteristic, complemented, Sylow
182C22×Q8—591no—
183C22×Q8—1181no—
184C22×Q8—1182no—
185C22×Q8—1182no—
186C22×Q8—2362no—
187C22×Q8—2362no—
188C22×Q8C4×D5911yescharacteristic
189C23×C4—591no—
190C23×C4—592no—
191C23×C4—1181no—
192C23×C4—1181no—
193C22×C8—22no—
194C22×C8—1182no—
195C2×C4⋊C4—591no—
196C2×C4⋊C4—1181no—
197C2×C4⋊C4—2362no—
198C2×C4⋊C4C4×D5911yescharacteristic
199C2×C42—592no—
200C2×C42—591no—
201C2×C42—2362no—
202C2×C42C22×D5911yescharacteristic
203C22⋊C8—1188no—
204#97—2364no—
205#95—22no—
206#95—1182no—
207#95—1182no—
208#95—1182no—
209C24—591no—
210C2×Q8—21no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—592no—
214C2×Q8—1184no—
215C2×Q8—1181no—
216C2×Q8—2361no—
217C2×Q8—2361no—
218C2×Q8—2362no—
219C2×Q8—2362no—
220C2×Q8—2368no—
221C2×Q8—2368no—
222C2×Q8#1174565912yes—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—591no—
226C22×C4—592no—
227C22×C4—1182no—
228C22×C4—1182no—
229C22×C4—1184no—
230C22×C4—1184no—
231C22×C4—1181no—
232C22×C4—1181no—
233C22×C4—1181no—
234C22×C4—1182no—
235C22×C4—1182no—
236C22×C4—1181no—
237C22×C4—2362no—
238C22×C4—2362no—
239C22×C4—2364no—
240C22×C4—2362no—
241C22×C4—2362no—
242C22×C4C2×C4×D5911yescharacteristic
243C22×C4D4×D5912yes—
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—1184no—
248C2×C8—2362no—
249C2×C8—2362no—
250C4⋊C4—41no—
251C4⋊C4—2361no—
252C4⋊C4—2362no—
253C42—41no—
254C42—1184no—
255C42—2361no—
256C23—592no—
257C23—594no—
258C23—1182no—
259C23—1181no—
260C23—1181no—
261C23C22⋊C4×D5911yescharacteristic
262Q8—42no—
263Q8—42no—
264Q8—2362no—
265Q8—2362no—
266Q8—2368no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—592no—
277C2×C4—1184no—
278C2×C4—1184no—
279C2×C4—1181no—
280C2×C4—1181no—
281C2×C4—1182no—
282C2×C4—1182no—
283C2×C4—1182no—
284C2×C4—1182no—
285C2×C4—4724no—
286C2×C4—2364no—
287C2×C4—2364no—
288C2×C4—2368no—
289C2×C4—2361no—
290C2×C4—2362no—
291C2×C4—2361no—
292C2×C4—2361no—
293C2×C4M16×D5911yescharacteristic
294C2×C4C2×#1174565911yescharacteristic
295C2×C4#1427375612yes—
296C2×C4#1427375912yes—
297C8—44no—
298C8—2364no—
299C22—594no—
300C22—1184no—
301C22—1182no—
302C22—1184no—
303C22—1182no—
304C22#1683736411yescharacteristic
305C22#99×D5911yescharacteristic
306C22#100×D5911yescharacteristic
307C22#1683740912yes—
308C22#104×D5912yes—
309C4—22no—
310C4—22no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—81no—
315C4—1182no—
316C4—4721no—
317C4—2362no—
318C4—2362no—
319C2—1184no—
320C2#2142242111yescharacteristic
321C2#2142254112yes—
322C2#2142261412yes—
323C2#329×D5911yescharacteristic
324C2#2142282011yescharacteristic
subgroupG/H
centerC23C22⋊C4×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1272C2
socleC23×C59C2×C22⋊C4
prime radicalC23×D59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1272D59
hypocenterC59#7629
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
3. #21421922C2semidirect
4. #21421921C2semidirect
5. C59×#1272C2semidirect
161. #1272D59semidirect
181. C59#7629semidirect

Maximal subgroups

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

subgroupindex
#391×D592
#214238572
#214219222
#214219212
C59×#12722
#762959