Finite Group Databasebeta

← #28507381

Subgroups — #28507381

Order 15104.

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

normal subgroups only

This group has 396 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#21429283C211yesmaximal, characteristic
2#21424092C211yesmaximal, characteristic
3#21422222C212yesmaximal
4#21422222C212yesmaximal
5C4×#101×C59C211yesmaximal, characteristic
6C2×#14273719C411yescharacteristic
7#16837426—21no—
8C2×#14273646C2212yes—
9#16837335—21no—
10#16837335—21no—
11#16837309C2211yescharacteristic
12#16837309C2211yescharacteristic
13#16837286—21no—
14#16837286—21no—
15#16837271C411yescharacteristic
16#16837267C411yescharacteristic
17C4×C4⋊Dic59C411yescharacteristic
18#16837171—21no—
19C2×C4×Q8×C59C2211yescharacteristic
20C2×#101×C59C2212yes—
21C4×M16×C59—21no—
22C59×#343—21no—
23C2×#11745664—22no—
24#14273719—41no—
25#14273719—41no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×C4×Dic59C2×C411yescharacteristic
34C2×C4×Dic59C2×C411yescharacteristic
35C4⋊(C59⋊C8)—41no—
36C4⋊(C59⋊C8)—41no—
37#14273646—24no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C22×Q8×C59C2311yescharacteristic
46C2×M16×C59—22no—
47C4×Q8×C59—41no—
48C4×Q8×C59—41no—
49C2×C4⋊C4×C59C2×C411yescharacteristic
50C2×C4⋊C4×C59D411yescharacteristic
51C2×C4⋊C4×C59D411yescharacteristic
52C2×C42×C59C2×C411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C2×C42×C59D411yescharacteristic
55#101×C59—22no—
56#101×C59Q814yes—
57#97×C59—41no—
58C4×C8×C59—41no—
59C2×Q8×C59—41no—
60C2×Q8×C59—41no—
61C2×Q8×C59C2×Q812yes—
62C2×Q8×C59#4112yes—
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22×C411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×Dic59—21no—
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74M16×C59—24no—
75M16×C59—42no—
76C2×C8×C59—42no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C42×C59—21no—
81C42×C59—21no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59—21no—
86C4⋊Dic59—42no—
87C4⋊Dic59—41no—
88C4⋊Dic59—41no—
89Dic59⋊C4—42no—
90Dic59⋊C4—42no—
91Dic59⋊C4—42no—
92Dic59⋊C4—41no—
93Dic59⋊C4—41no—
94C4×Dic59—41no—
95C4×Dic59—41no—
96#11745664—42no—
97#11745664—42no—
98C2×C59⋊C8—42no—
99C23×C59C2×C22⋊C411yescharacteristic
100Q8×C59—42no—
101Q8×C59—42no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59#9911yescharacteristic
116C2×C4×C59#9911yescharacteristic
117C2×C4×C59C4×Q812yes—
118C2×C4×C59C22⋊Q812yes—
119C2×C4×C59#12411yescharacteristic
120C2×C4×C59C4⋊Q811yescharacteristic
121C2×Dic59—22no—
122C2×Dic59—41no—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—42no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C8×C59—44no—
133C59⋊C8—82no—
134#13303—591nomaximal, Sylow
135C22×C59—21no—
136C22×C59—21no—
137C22×C59#39111yescharacteristic
138C22×C59C2×#9911yescharacteristic
139C22×C59#41011yescharacteristic
140C4×C59—21no—
141C4×C59—21no—
142C4×C59—21no—
143C4×C59—22no—
144C4×C59—22no—
145C4×C59—21no—
146C4×C59—41no—
147C4×C59—41no—
148C4×C59—41no—
149Dic59—42no—
150Dic59—81no—
151Dic59—81no—
152Dic59—81no—
153#2565—591no—
154#1801—591no—
155C4×#101D5911yescharacteristic
156#1394—592no—
157#1394—592no—
158C2×C59—21no—
159C2×C59—21no—
160C2×C59#199611yescharacteristic
161C2×C59#213211yescharacteristic
162C2×C59#213311yescharacteristic
163C2×#125—591no—
164C2×C4×Q8C2×D5911yescharacteristic
165#423—1181no—
166C2×#101—592no—
167C2×#101C2×D5912yes—
168C4×M16—21no—
169#398—1181no—
170#398—1181no—
171#391—591no—
172#391—591no—
173#387—1181no—
174#387—1181no—
175#384—591no—
176#382—591no—
177C4×C4⋊C4—591no—
178#343—21no—
179#343—1181no—
180C59#1330311yescharacteristic, complemented, Sylow
181C22×Q8C22×D5911yescharacteristic
182C2×M16—22no—
183C2×M16—1182no—
184#125—2361no—
185#125—2361no—
186C4×Q8—41no—
187C4×Q8—41no—
188C2×C4⋊C4—591no—
189C2×C4⋊C4—591no—
190C2×C4⋊C4—1181no—
191C2×C4⋊C4—1181no—
192C2×C4⋊C4—1181no—
193C2×C4⋊C4—1181no—
194C2×C4⋊C4—1181no—
195C2×C4⋊C4C4×D5911yescharacteristic
196C2×C4⋊C4D23611yescharacteristic
197C2×C4⋊C4C22⋊D5911yescharacteristic
198C2×C42—591no—
199C2×C42—591no—
200C2×C42C4×D5911yescharacteristic
201C2×C42D23611yescharacteristic
202C2×C42C22⋊D5911yescharacteristic
203C4⋊C8—2361no—
204C4⋊C8—2361no—
205#101—22no—
206#101—1184no—
207#101Dic11814yes—
208#97—41no—
209C4×C8—41no—
210#95—1181no—
211#95—1181no—
212#95—1181no—
213#95—1181no—
214#95—1181no—
215#95—1181no—
216#95—1181no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8C2×Dic11812yes—
220C2×Q8#1174568512yes—
221C22×C4—1181no—
222C22×C4—1181no—
223C22×C4—1181no—
224C22×C4—1181no—
225C22×C4(C2×D59)⋊C411yescharacteristic
226C22×C4(C2×D59)⋊C411yescharacteristic
227C22×C4(C2×D59)⋊C411yescharacteristic
228C22×C4(C2×D59)⋊C411yescharacteristic
229C22×C4C2×C4×D5911yescharacteristic
230C22×C4C2×D23611yescharacteristic
231C22×C4C2×C22⋊D5911yescharacteristic
232M16—24no—
233M16—42no—
234M16—2362no—
235M16—2362no—
236C2×C8—42no—
237C2×C8—2362no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—2361no—
242C4⋊C4—2361no—
243C4⋊C4—2362no—
244C4⋊C4—2362no—
245C4⋊C4—2362no—
246C4⋊C4—2362no—
247C4⋊C4—2361no—
248C4⋊C4—2361no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—2361no—
256C42—2361no—
257C23C2×(C2×D59)⋊C411yescharacteristic
258Q8—42no—
259Q8—42no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—1182no—
274C2×C4—2361no—
275C2×C4—2361no—
276C2×C4—2361no—
277C2×C4—2361no—
278C2×C4—2362no—
279C2×C4—2361no—
280C2×C4—2361no—
281C2×C4—2361no—
282C2×C4—2361no—
283C2×C4—2361no—
284C2×C4#1427364111yescharacteristic
285C2×C4#1427364111yescharacteristic
286C2×C4C4×Dic11812yes—
287C2×C4#1427370612yes—
288C2×C4#1427371511yescharacteristic
289C2×C4#1427372711yescharacteristic
290C8—44no—
291C8—4722no—
292C22—21no—
293C22—21no—
294C22#1683730811yescharacteristic
295C22C2×#1427364111yescharacteristic
296C22#1683737511yescharacteristic
297C4—21no—
298C4—21no—
299C4—22no—
300C4—21no—
301C4—22no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—4721no—
307C4—4721no—
308C4—4721no—
309C4—2362no—
310C2—21no—
311C2—21no—
312C2#2142508111yescharacteristic
313C2#2142577911yescharacteristic
314C2#2142578311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC22×Q8C22×D59
FittingC4×#101×C59C2
socleC22×C59C2×#99
prime radicalC23×C59C2×C22⋊C4
cosocleC22×Q8×C59C23
hypercenterC4×#101D59
hypocenterC59#13303
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
180. C59#13303semidirect

Maximal subgroups

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

subgroupindex
#214292832
#214240922
#214222222
C4×#101×C592
#1330359