Finite Group Databasebeta

← #28497803

Subgroups — #28497803

Order 15104.

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

normal subgroups only

This group has 526 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#21426679C211yesmaximal, characteristic
2#21422409C212yesmaximal
3#21422030C211yesmaximal, characteristic, complemented
4#21422029C212yesmaximal, complemented
5C59×#1319C211yesmaximal, characteristic, complemented
6C22×Q8×D59C411yescharacteristic
7C2×#14273685C2211yescharacteristic
8C2×#14273684C411yescharacteristic
9#16837452—22no—
10#16837309C411yescharacteristic
11#16837308C411yescharacteristic
12#16837307C2211yescharacteristic
13#16837306—22no—
14#16837264—22no—
15#16837128C2212yes—
16#16837125—22no—
17C59×#391C2211yescharacteristic
18C59×#324C2212yes—
19C22×Dic118—21no—
20C22×Dic118C2×C411yescharacteristic
21C2×Q8×D59—22no—
22C2×Q8×D59—42no—
23C2×Q8×D59—42no—
24C2×Q8×D59C814yescomplemented
25C22×C4×D59—21no—
26C22×C4×D59C2×C411yescharacteristic
27#14273685—42no—
28#14273684—42no—
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×Dic59⋊C4—21no—
31C2×(C2×D59)⋊C4—21no—
32C2×(C2×D59)⋊C4D412yes—
33C2×C4×Dic59C2×C411yescharacteristic
34C2×C4×Dic59D412yes—
35Dic59⋊C8—42no—
36C22⋊(C59⋊C8)—22no—
37C8×Dic59—42no—
38C59⋊#95—22no—
39C59⋊#95—22no—
40C59⋊#95—22no—
41C22×Q8×C59C2×C411yescharacteristic
42C2×C4⋊C4×C59C2311yescharacteristic
43C2×C42×C59C2×C411yescharacteristic
44C22⋊C8×C59—22no—
45#95×C59—22no—
46C23×D59C22⋊C411yescharacteristic
47C2×Q8×C59—21no—
48C2×Q8×C59—41no—
49C2×Q8×C59—41no—
50C2×Q8×C59C2×C812yes—
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—22no—
63C2×C4×D59—24no—
64C2×C4×D59—42no—
65C2×C4×D59—42no—
66C2×C4×D59C2×C812yes—
67C2×C4×D59M1612yes—
68C2×Dic118—21no—
69C2×Dic118—24no—
70C2×Dic118—42no—
71C2×Dic118—42no—
72C2×Dic118—41no—
73C2×Dic118—41no—
74C2×Dic118C2×C812yes—
75C2×C8×C59—42no—
76C4⋊C4×C59—41no—
77C42×C59—41no—
78(C2×D59)⋊C4—42no—
79(C2×D59)⋊C4—42no—
80C4⋊Dic59—41no—
81Dic59⋊C4—42no—
82C4×Dic59—22no—
83C4×Dic59—22no—
84C4×Dic59—41no—
85C2×C59⋊C8—42no—
86C23×C59C2×C22⋊C411yescharacteristic
87C22×D59—21no—
88C22×D59—21no—
89C22×D59—41no—
90C22×D59—41no—
91C22×D59C22⋊C812yes—
92Q8×C59—42no—
93Q8×C59—42no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—42no—
101C2×C4×C59—41no—
102C2×C4×C59—41no—
103C2×C4×C59—41no—
104C2×C4×C59C22×C811yescharacteristic
105C2×C4×C59C2×M1611yescharacteristic
106C2×Dic59—22no—
107C2×Dic59—22no—
108C2×Dic59—41no—
109C2×Dic59—41no—
110C2×Dic59—42no—
111C2×Dic59—41no—
112C2×Dic59—41no—
113C2×Dic59—41no—
114C2×Dic59C22⋊C812yes—
115C2×Dic59#9912yes—
116C2×Dic59#10012yes—
117C4×D59—24no—
118C4×D59—24no—
119C4×D59—48no—
120Dic118—42no—
121Dic118—48no—
122Dic118—42no—
123C8×C59—82no—
124C59⋊C8—82no—
125#11434—591nomaximal, Sylow
126C22×C59—21no—
127C22×C59—21no—
128C22×C59C2×C22⋊C811yescharacteristic
129C22×C59C2×#9911yescharacteristic
130C22×C59C2×#10011yescharacteristic
131C2×D59—22no—
132C2×D59—42no—
133C2×D59—41no—
134C2×D59—41no—
135C2×D59—42no—
136C4×C59—21no—
137C4×C59—21no—
138C4×C59—42no—
139C4×C59—42no—
140C4×C59—81no—
141Dic59—42no—
142Dic59—42no—
143Dic59—81no—
144Dic59#32314yes—
145#2310—591no—
146#1458—592no—
147#1319—591no—
148#1319—592no—
149#1319D5911yescharacteristic, complemented
150C2×C59—21no—
151C2×C59—21no—
152C2×C59C2×#32311yescharacteristic
153C2×C59#212911yescharacteristic
154C2×C59#213511yescharacteristic
155D59—42no—
156C23×Q8—591no—
157C2×#117—591no—
158C2×#117—591no—
159#431—1182no—
160#391—591no—
161#391—591no—
162#391—591no—
163#391—1182no—
164#391C2×D5911yescharacteristic
165#381—1182no—
166#324—592no—
167#324C2×D5912yes—
168#323—1182no—
169C59#1143411yescharacteristic, complemented, Sylow
170C22×Q8—594no—
171C22×Q8—591no—
172C22×Q8—1181no—
173C22×Q8—1182no—
174C22×Q8—2362no—
175C22×Q8—2362no—
176C22×Q8C4×D5911yescharacteristic
177C23×C4—591no—
178C23×C4—1181no—
179#117—2362no—
180#117—2362no—
181C2×C4⋊C4—591no—
182C2×C4⋊C4—1181no—
183C2×C4⋊C4C22×D5911yescharacteristic
184C2×C22⋊C4—592no—
185C2×C22⋊C4—1181no—
186C2×C42—592no—
187C2×C42—591no—
188C2×C42C4×D5911yescharacteristic
189C4⋊C8—2362no—
190C22⋊C8—22no—
191C22⋊C8—1182no—
192C4×C8—2362no—
193#95—22no—
194#95—1182no—
195#95—1182no—
196#95—1182no—
197C24—591no—
198C2×Q8—21no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—592no—
202C2×Q8—1184no—
203C2×Q8—1181no—
204C2×Q8—2361no—
205C2×Q8—2361no—
206C2×Q8—2362no—
207C2×Q8—2362no—
208C2×Q8—2368no—
209C2×Q8—2368no—
210C2×Q8C8×D5912yes—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—591no—
214C22×C4—592no—
215C22×C4—592no—
216C22×C4—592no—
217C22×C4—1182no—
218C22×C4—1182no—
219C22×C4—1184no—
220C22×C4—1181no—
221C22×C4—1181no—
222C22×C4—2362no—
223C22×C4—2362no—
224C22×C4C2×C4×D5911yescharacteristic
225C22×C4D4×D5912yes—
226C2×C8—42no—
227C2×C8—2362no—
228C4⋊C4—41no—
229C4⋊C4—2361no—
230C4⋊C4—2362no—
231C22⋊C4—2362no—
232C22⋊C4—2362no—
233C42—41no—
234C42—1182no—
235C42—1182no—
236C42—2361no—
237C23—592no—
238C23—1181no—
239C23—1181no—
240C23—2361no—
241C23—2361no—
242C23C22⋊C4×D5911yescharacteristic
243Q8—42no—
244Q8—42no—
245Q8—2362no—
246Q8—2362no—
247Q8—2368no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—592no—
259C2×C4—592no—
260C2×C4—592no—
261C2×C4—1184no—
262C2×C4—1184no—
263C2×C4—1182no—
264C2×C4—1182no—
265C2×C4—2368no—
266C2×C4—2361no—
267C2×C4—2361no—
268C2×C4—2361no—
269C2×C4—2361no—
270C2×C4—2361no—
271C2×C4—2362no—
272C2×C4C2×C8×D5911yescharacteristic
273C2×C4M16×D5911yescharacteristic
274C8—82no—
275C8—4722no—
276C22—21no—
277C22—21no—
278C22—1182no—
279C22—2362no—
280C22—2362no—
281C22—2361no—
282C22—2361no—
283C22C22⋊C8×D5911yescharacteristic
284C22#99×D5911yescharacteristic
285C22#100×D5911yescharacteristic
286C4—21no—
287C4—21no—
288C4—42no—
289C4—42no—
290C4—81no—
291C4—594no—
292C4—4721no—
293C4—2362no—
294C4—2362no—
295C2—21no—
296C2—21no—
297C2—2362no—
298C2#323×D5911yescharacteristic
299C2#2142576911yescharacteristic
300C2#2142578911yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC2×C4×C59C22×C8
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1319C2
socleC22×C59C2×#99
prime radicalC23×D59C22⋊C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1319D59
hypocenterC59#11434
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
3. #21422030C2semidirect
4. #21422029C2semidirect
5. C59×#1319C2semidirect
24. C2×Q8×D59C8semidirect
149. #1319D59semidirect
169. C59#11434semidirect

Maximal subgroups

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

subgroupindex
#214266792
#214224092
#214220302
#214220292
C59×#13192
#1143459