Finite Group Databasebeta

← #28503870

Subgroups — #28503870

Order 15104.

500 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 115 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#21426665C211yesmaximal, characteristic
2C2×#16837200C212yesmaximal
3#101×Dic59C212yesmaximal
4#21422223C211yesmaximal, characteristic
5C59×#1394C211yesmaximal, characteristic
6C2×#14273725C2211yescharacteristic
7C2×Q8×Dic59C2211yescharacteristic
8C2×#14273692C411yescharacteristic
9C2×#14273646C2212yes—
10M16×Dic59—22no—
11#16837308C411yescharacteristic
12#16837299—22no—
13#16837271C411yescharacteristic
14#16837253—22no—
15C42×Dic59C411yescharacteristic
16#16837200C418yescomplemented
17#16837170—22no—
18C2×#101×C59C2212yes—
19C59×#391C2211yescharacteristic
20C22×Dic118C2×C411yescharacteristic
21C2×#11745664—22no—
22#14273725—21no—
23#14273725—21no—
24#14273725C2×C414yes—
25Q8×Dic59—41no—
26Q8×Dic59—41no—
27#14273692—41no—
28#14273692—41no—
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×C4⋊Dic59D412yes—
31C2×Dic59⋊C4—22no—
32C2×Dic59⋊C4—22no—
33C2×Dic59⋊C4—21no—
34C2×C4×Dic59—21no—
35C2×C4×Dic59—21no—
36C2×C4×Dic59—21no—
37C2×C4×Dic59C2×C411yescharacteristic
38C2×C4×Dic59C2×C411yescharacteristic
39C2×C4×Dic59D412yes—
40#14273646—22no—
41#14273646C2×C414yes—
42#14273637—42no—
43C8×Dic59—42no—
44C59⋊#95—22no—
45C22×Q8×C59C2311yescharacteristic
46C2×M16×C59—22no—
47C2×C4⋊C4×C59C2×C411yescharacteristic
48C2×C42×C59C2×C411yescharacteristic
49#101×C59—22no—
50#101×C59C2×C414yes—
51#95×C59—22no—
52C2×Q8×C59—41no—
53C2×Q8×C59—41no—
54C2×Q8×C59C22×C412yes—
55C2×Q8×C59#4111yescharacteristic
56C2×Q8×C59#4111yescharacteristic
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59C22×C411yescharacteristic
60C22×C4×C59C2×D412yes—
61C22×Dic59—21no—
62C22×Dic59—21no—
63C22×Dic59C22⋊C411yescharacteristic
64C22×Dic59C22⋊C412yes—
65C22×Dic59C22⋊C411yescharacteristic
66C2×Dic118—21no—
67C2×Dic118—41no—
68C2×Dic118—41no—
69C2×Dic118C4212yes—
70M16×C59—24no—
71M16×C59—42no—
72C2×C8×C59—42no—
73C4⋊C4×C59—41no—
74C42×C59—41no—
75C4⋊Dic59—41no—
76C4⋊Dic59—42no—
77Dic59⋊C4—24no—
78Dic59⋊C4—24no—
79Dic59⋊C4—44no—
80Dic59⋊C4—42no—
81C4×Dic59—24no—
82C4×Dic59—21no—
83C4×Dic59—22no—
84C4×Dic59—22no—
85C4×Dic59—21no—
86C4×Dic59—21no—
87C4×Dic59—42no—
88C4×Dic59—41no—
89C4×Dic59—41no—
90C4×Dic59C4212yes—
91#11745664—24no—
92#11745664—42no—
93C2×C59⋊C8—42no—
94C23×C59C2×C22⋊C411yescharacteristic
95Q8×C59—42no—
96Q8×C59—41no—
97Q8×C59—41no—
98C2×C4×C59—22no—
99C2×C4×C59—21no—
100C2×C4×C59—22no—
101C2×C4×C59—21no—
102C2×C4×C59—22no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59—42no—
106C2×C4×C59—41no—
107C2×C4×C59C2×C4211yescharacteristic
108C2×C4×C59#11711yescharacteristic
109C2×C4×C59C4×D412yes—
110C2×C4×C59C4×D412yes—
111C2×Dic59—22no—
112C2×Dic59—22no—
113C2×Dic59—22no—
114C2×Dic59—22no—
115C2×Dic59—22no—
116C2×Dic59—21no—
117C2×Dic59—22no—
118C2×Dic59—41no—
119C2×Dic59—41no—
120C2×Dic59—41no—
121C2×Dic59—42no—
122C2×Dic59#9911yescharacteristic
123C2×Dic59#9911yescharacteristic
124Dic118—42no—
125Dic118—42no—
126C8×C59—44no—
127C59⋊C8—44no—
128C4×#356—591nomaximal, Sylow
129C22×C59—21no—
130C22×C59—21no—
131C22×C59C4×C22⋊C411yescharacteristic
132C22×C59C2×#9911yescharacteristic
133C22×C59#41011yescharacteristic
134C4×C59—22no—
135C4×C59—22no—
136C4×C59—21no—
137C4×C59—21no—
138C4×C59—42no—
139C4×C59—81no—
140Dic59—24no—
141Dic59—21no—
142Dic59—41no—
143Dic59—42no—
144Dic59—42no—
145Dic59#35612yescomplemented
146C4×C4⋊Q8—591no—
147C2×#356—592no—
148C4×#101—592no—
149#1394—591no—
150#1394D5911yescharacteristic
151C2×C59—21no—
152C2×C59—21no—
153C2×C59C4×#9911yescharacteristic
154C2×C59C2×#35611yescharacteristic
155C2×C59#213211yescharacteristic
156C2×C4⋊Q8—591no—
157C2×C4×Q8—591no—
158C2×C4×Q8—591no—
159C2×#101—592no—
160C2×#101C2×D5912yes—
161C4×M16—1182no—
162#391—591no—
163#391C2×D5911yescharacteristic
164#389—1182no—
165#384—591no—
166C4×C4⋊C4—1182no—
167C43—591no—
168#356—598no—
169#343—1182no—
170C59C4×#35611yescharacteristic, complemented, Sylow
171C22×Q8—591no—
172C22×Q8C22×D5911yescharacteristic
173C2×M16—22no—
174C2×M16—1182no—
175C4⋊Q8—594no—
176C4⋊Q8—1181no—
177C4⋊Q8—1181no—
178C4×Q8—2361no—
179C4×Q8—2361no—
180C4×Q8—2361no—
181C4×Q8—2361no—
182C2×C4⋊C4—591no—
183C2×C4⋊C4—592no—
184C2×C4⋊C4—1182no—
185C2×C4⋊C4—1181no—
186C2×C4⋊C4—1182no—
187C2×C4⋊C4C4×D5911yescharacteristic
188C2×C42—591no—
189C2×C42—591no—
190C2×C42—592no—
191C2×C42—1181no—
192C2×C42—1181no—
193C2×C42—1181no—
194C2×C42C4×D5911yescharacteristic
195#101—22no—
196#101—594no—
197#101—1182no—
198#101C4×D5914yes—
199#97—2362no—
200C4×C8—2362no—
201#95—22no—
202#95—1182no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—592no—
206C2×Q8—1181no—
207C2×Q8—2361no—
208C2×Q8—2361no—
209C2×Q8C2×C4×D5912yes—
210C2×Q8#1174568711yescharacteristic
211C2×Q8#1174568911yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—591no—
215C22×C4—592no—
216C22×C4—591no—
217C22×C4—1181no—
218C22×C4—1181no—
219C22×C4C2×C4×D5911yescharacteristic
220C22×C4D4×D5912yes—
221M16—24no—
222M16—42no—
223M16—1184no—
224M16—2362no—
225C2×C8—42no—
226C2×C8—2362no—
227C4⋊C4—41no—
228C4⋊C4—1184no—
229C4⋊C4—1184no—
230C4⋊C4—2361no—
231C4⋊C4—2362no—
232C4⋊C4—2362no—
233C4⋊C4—2364no—
234C42—41no—
235C42—592no—
236C42—1182no—
237C42—1182no—
238C42—1181no—
239C42—1181no—
240C42—1184no—
241C42—1181no—
242C42—2361no—
243C42—2361no—
244C42—2362no—
245C23C22⋊C4×D5911yescharacteristic
246Q8—42no—
247Q8—41no—
248Q8—41no—
249Q8—2362no—
250Q8—2362no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—591no—
261C2×C4—591no—
262C2×C4—1182no—
263C2×C4—1182no—
264C2×C4—1182no—
265C2×C4—1182no—
266C2×C4—1182no—
267C2×C4—1181no—
268C2×C4—1182no—
269C2×C4—2361no—
270C2×C4—2361no—
271C2×C4—2361no—
272C2×C4—2362no—
273C2×C4C42×D5911yescharacteristic
274C2×C4#1427368511yescharacteristic
275C2×C4(C22⋊D59)⋊C412yes—
276C2×C4#1427368912yes—
277C8—44no—
278C8—2364no—
279C22—21no—
280C22—21no—
281C22#1683724911yescharacteristic
282C22#99×D5911yescharacteristic
283C22#1683737411yescharacteristic
284C4—22no—
285C4—22no—
286C4—21no—
287C4—21no—
288C4—42no—
289C4—81no—
290C4—592no—
291C4—1184no—
292C4—1181no—
293C4—2362no—
294C4—2361no—
295C4—2362no—
296C2—21no—
297C2—21no—
298C2#2142388311yescharacteristic
299C2#356×D5911yescharacteristic
300C2#2142577811yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC2×C4×C59C2×C42
FrattiniC22×Q8C22×D59
FittingC59×#1394C2
socleC22×C59C2×#99
prime radicalC23×C59C2×C22⋊C4
cosocleC22×Q8×C59C23
hypercenter#1394D59
hypocenterC59C4×#356
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
16. #16837200C4semidirect
145. Dic59#356semidirect
170. C59C4×#356semidirect

Maximal subgroups

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

subgroupindex
#214266652
C2×#168372002
#101×Dic592
#214222232
C59×#13942
C4×#35659