Finite Group Databasebeta

← #28497798

Subgroups — #28497798

Order 15104.

594 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 93 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#21426687C211yesmaximal, characteristic
2#324×D59C212yesmaximal, complemented
3#21422030C211yesmaximal, characteristic, complemented
4#21422027C212yesmaximal, complemented
5C59×#1319C211yesmaximal, characteristic, complemented
6C22×#11745689C411yescharacteristic
7C2×#14273684C411yescharacteristic
8C2×C4⋊C4×D59C2211yescharacteristic
9C22⋊C8×D59—22no—
10#16837309C411yescharacteristic
11#16837304C411yescharacteristic
12#16837303C2211yescharacteristic
13#16837280—22no—
14#16837261—22no—
15#16837128C2212yes—
16#16837127—22no—
17C59×#391C2211yescharacteristic
18C59×#324C2212yes—
19C2×#11745689—22no—
20C2×#11745689—42no—
21C2×#11745689—42no—
22C2×#11745689C814yescomplemented
23C22×D236—21no—
24C22×D236C2×C411yescharacteristic
25C22×C4×D59—21no—
26C22×C4×D59C2×C411yescharacteristic
27C22×C4×D59D412yes—
28C2×C8×D59—42no—
29#14273684—42no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4D412yes—
33C4⋊C4×D59—42no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—22no—
36C2×(C2×D59)⋊C4—22no—
37C2×C4×Dic59C2×C411yescharacteristic
38C22⋊(C59⋊C8)—22no—
39(C2×D59)⋊C8—42no—
40C59⋊#95—22no—
41C22×Q8×C59C2×C411yescharacteristic
42C2×C4⋊C4×C59C2311yescharacteristic
43C2×C42×C59C2×C411yescharacteristic
44C22⋊C8×C59—22no—
45#95×C59—22no—
46C23×D59—21no—
47C23×D59C22⋊C411yescharacteristic
48C2×Q8×C59—21no—
49C2×Q8×C59—41no—
50C2×Q8×C59—41no—
51C2×Q8×C59C2×C812yescomplemented
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59C22×C411yescharacteristic
55C22×C4×C59C2×D412yes—
56C22×Dic59—21no—
57C22×Dic59C22⋊C411yescharacteristic
58C22×Dic59C22⋊C412yes—
59#11745689—48no—
60#11745689—48no—
61C2×D236—24no—
62C2×D236—21no—
63C2×D236—41no—
64C2×D236—41no—
65C2×D236—42no—
66C2×D236—42no—
67C2×D236C2×C812yescomplemented
68C2×C4×D59—22no—
69C2×C4×D59—22no—
70C2×C4×D59—22no—
71C2×C4×D59—24no—
72C2×C4×D59—22no—
73C2×C4×D59—42no—
74C2×C4×D59—42no—
75C2×C4×D59—82no—
76C2×C4×D59C2×C812yes—
77C2×C4×D59M1612yes—
78C2×C8×C59—42no—
79C4⋊C4×C59—41no—
80C42×C59—41no—
81(C2×D59)⋊C4—42no—
82(C2×D59)⋊C4—82no—
83(C2×D59)⋊C4—82no—
84C4⋊Dic59—41no—
85Dic59⋊C4—42no—
86Dic59⋊C4—42no—
87C4×Dic59—41no—
88C2×C59⋊C8—42no—
89C8×D59—84no—
90C23×C59C2×C22⋊C411yescharacteristic
91C22×D59—22no—
92C22×D59—41no—
93C22×D59—41no—
94C22×D59—42no—
95C22×D59—42no—
96C22×D59—41no—
97C22×D59—41no—
98C22×D59C22⋊C812yes—
99C22×D59#9912yes—
100C22×D59#10012yes—
101Q8×C59—42no—
102Q8×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—42no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59C22×C811yescharacteristic
114C2×C4×C59C2×M1611yescharacteristic
115C2×Dic59—21no—
116C2×Dic59—21no—
117C2×Dic59—22no—
118C2×Dic59—42no—
119C2×Dic59—41no—
120C2×Dic59—41no—
121C2×Dic59—41no—
122C2×Dic59C22⋊C812yes—
123D236—42no—
124D236—42no—
125D236—48no—
126C4×D59—24no—
127C4×D59—24no—
128C4×D59—48no—
129C4×D59—44no—
130C4×D59—82no—
131C8×C59—82no—
132C59⋊C8—82no—
133#11432—591nomaximal, Sylow
134C22×C59—21no—
135C22×C59—21no—
136C22×C59C2×C22⋊C811yescharacteristic
137C22×C59C2×#9911yescharacteristic
138C22×C59C2×#10011yescharacteristic
139C2×D59—22no—
140C2×D59—22no—
141C2×D59—42no—
142C2×D59—42no—
143C2×D59—44no—
144C2×D59—42no—
145C2×D59—82no—
146C2×D59—81no—
147C2×D59—81no—
148C2×D59#32314yes—
149C4×C59—21no—
150C4×C59—21no—
151C4×C59—42no—
152C4×C59—42no—
153C4×C59—81no—
154Dic59—22no—
155Dic59—42no—
156Dic59—81no—
157#2311—591no—
158C2×#324—592no—
159#1319—591no—
160#1319D5911yescharacteristic, complemented
161#1318—592no—
162C2×C59—21no—
163C2×C59—21no—
164C2×C59C2×#32311yescharacteristic
165C2×C59#212711yescharacteristic
166C2×C59#213311yescharacteristic
167D59—24no—
168D59—82no—
169C22×#41—591no—
170C2×#117—591no—
171C22×C4⋊C4—591no—
172C2×C22⋊C8—1182no—
173#391—591no—
174#391C2×D5911yescharacteristic
175#390—591no—
176#390—591no—
177#386—1182no—
178#380—1182no—
179#324—592no—
180#324—1182no—
181#324C2×D5912yes—
182C59#1143211yescharacteristic, complemented, Sylow
183C2×#41—594no—
184C2×#41—1182no—
185C2×#41—2362no—
186C2×#41—2362no—
187C22×Q8C4×D5911yescharacteristic
188C22×D4—591no—
189C22×D4—1181no—
190C23×C4—591no—
191C23×C4—592no—
192C23×C4—1181no—
193C22×C8—2362no—
194#117—2362no—
195C2×C4⋊C4—591no—
196C2×C4⋊C4—592no—
197C2×C4⋊C4—1181no—
198C2×C4⋊C4—2362no—
199C2×C4⋊C4C22×D5911yescharacteristic
200C2×C22⋊C4—1182no—
201C2×C22⋊C4—1182no—
202C2×C22⋊C4—1181no—
203C2×C42—591no—
204C2×C42C4×D5911yescharacteristic
205C22⋊C8—22no—
206C22⋊C8—1182no—
207C22⋊C8—2362no—
208#95—22no—
209#95—1182no—
210C24—591no—
211C24—1181no—
212#41—2368no—
213#41—2368no—
214C2×Q8—21no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8C8×D5912yescomplemented
218C2×D4—592no—
219C2×D4—1184no—
220C2×D4—1181no—
221C2×D4—2361no—
222C2×D4—2361no—
223C2×D4—2362no—
224C2×D4—2362no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—591no—
228C22×C4—592no—
229C22×C4—592no—
230C22×C4—592no—
231C22×C4—1184no—
232C22×C4—1182no—
233C22×C4—1182no—
234C22×C4—1182no—
235C22×C4—1182no—
236C22×C4—1181no—
237C22×C4—4722no—
238C22×C4—2362no—
239C22×C4—2362no—
240C22×C4C2×C4×D5911yescharacteristic
241C22×C4D4×D5912yes—
242C2×C8—42no—
243C2×C8—4724no—
244C2×C8—2362no—
245C4⋊C4—41no—
246C4⋊C4—2362no—
247C4⋊C4—2361no—
248C4⋊C4—2362no—
249C22⋊C4—4722no—
250C22⋊C4—4722no—
251C22⋊C4—2362no—
252C42—41no—
253C42—2361no—
254C23—592no—
255C23—592no—
256C23—592no—
257C23—1182no—
258C23—2361no—
259C23—2361no—
260C23—2362no—
261C23—2362no—
262C23—2361no—
263C23—2361no—
264C23C22⋊C4×D5911yescharacteristic
265Q8—42no—
266Q8—42no—
267D4—2362no—
268D4—2362no—
269D4—2368no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—592no—
281C2×C4—1184no—
282C2×C4—1184no—
283C2×C4—1181no—
284C2×C4—1181no—
285C2×C4—1182no—
286C2×C4—4722no—
287C2×C4—2368no—
288C2×C4—2364no—
289C2×C4—2361no—
290C2×C4—2361no—
291C2×C4—2361no—
292C2×C4—2362no—
293C2×C4C2×C8×D5911yescharacteristic
294C2×C4M16×D5911yescharacteristic
295C8—82no—
296C8—4722no—
297C22—21no—
298C22—21no—
299C22—594no—
300C22—1182no—
301C22—1182no—
302C22—4722no—
303C22—4721no—
304C22—4721no—
305C22—2362no—
306C22—2362no—
307C22—2364no—
308C22—2362no—
309C22C22⋊C8×D5911yescharacteristic
310C22#99×D5911yescharacteristic
311C22#100×D5911yescharacteristic
312C4—21no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—81no—
317C4—1182no—
318C4—4721no—
319C4—2362no—
320C2—21no—
321C2—21no—
322C2—1184no—
323C2—4722no—
324C2#323×D5911yescharacteristic
325C2#2142576411yescharacteristic
326C2#2142578411yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC2×C4×C59C22×C8
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1319C2
socleC22×C59C2×#99
prime radicalC22×#11745689C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1319D59
hypocenterC59#11432
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
2. #324×D59C2semidirect
3. #21422030C2semidirect
4. #21422027C2semidirect
5. C59×#1319C2semidirect
22. C2×#11745689C8semidirect
51. C2×Q8×C59C2×C8semidirect
67. C2×D236C2×C8semidirect
160. #1319D59semidirect
182. C59#11432semidirect
217. C2×Q8C8×D59semidirect

Maximal subgroups

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

subgroupindex
#214266872
#324×D592
#214220302
#214220272
C59×#13192
#1143259