Finite Group Databasebeta

← #28563594

Subgroups — #28563594

Order 15104.

870 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 191 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#21437906C211yesmaximal, characteristic
2#21435247C212yesmaximal
3#21423137C218yesmaximal
4#21422594C212yesmaximal
5#21422420C211yesmaximal, characteristic
6C59×#2568C211yesmaximal, characteristic
7Q8×Dic118—22no—
8#16838131—22no—
9#16838128—24no—
10#16838033—21no—
11#16838033C2214yes—
12#16837959—22no—
13#16837959C2212yes—
14#16837880—21no—
15#16837805—22no—
16#16837754—22no—
17#16837737C2218yes—
18#16837694—24no—
19#16837455C2212yes—
20#16837138C2218yes—
21#16837128C2214yes—
22C2×C4⋊Q8×C59C2211yescharacteristic
23C2×C4×Q8×C59C2211yescharacteristic
24C59×#400C2214yes—
25C59×#391—21no—
26C59×#391—22no—
27C59×#382—21no—
28C4×C4⋊C4×C59C2211yescharacteristic
29#14273727—22no—
30#14273727—22no—
31#14273727D412yes—
32#14273727D412yes—
33#14273726—24no—
34#14273726—28no—
35#14273719—24no—
36#14273719—22no—
37#14273718—24no—
38#14273718—28no—
39#14273706—22no—
40#14273706—24no—
41#14273706C2314yes—
42Q8×Dic59—24no—
43Q8×Dic59—24no—
44C4×Dic118—24no—
45C4×Dic118—24no—
46C4×Dic118—22no—
47C4×Dic118D414yes—
48#14273686—24no—
49#14273686—22no—
50#14273660—28no—
51C4⋊(C59⋊C8)—22no—
52#14273650—28no—
53C22⋊(C59⋊C8)C2314yes—
54C4×C59⋊C8—22no—
55C22×Q8×C59—21no—
56C22×Q8×C59D411yescharacteristic
57C4⋊Q8×C59—22no—
58C4⋊Q8×C59—22no—
59C4×Q8×C59—22no—
60C4×Q8×C59—22no—
61C2×C4⋊C4×C59—21no—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59D411yescharacteristic
64C2×C4⋊C4×C59C2312yes—
65C2×C4⋊C4×C59C2312yes—
66C2×C42×C59—21no—
67C2×C42×C59D412yes—
68C2×C42×C59C2311yescharacteristic
69#95×C59—24no—
70#95×C59—24no—
71#95×C59C2312yes—
72C2×Q8×C59—22no—
73C2×Q8×C59—22no—
74C2×Q8×C59—44no—
75C2×Q8×C59—41no—
76C2×Q8×C59—41no—
77C2×Q8×C59Q1614yes—
78C22×C4×C59—22no—
79C22×C4×C59—22no—
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59C2×D412yes—
82C22×C4×C59C2×D412yes—
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59C2411yescharacteristic
85C2×Dic118—22no—
86C2×Dic118—24no—
87C2×Dic118C2×D414yes—
88C4⋊C4×C59—22no—
89C4⋊C4×C59—24no—
90C4⋊C4×C59—24no—
91C4⋊C4×C59—42no—
92C4⋊C4×C59—42no—
93C42×C59—22no—
94C42×C59—22no—
95C42×C59—42no—
96C42×C59C2×D412yes—
97C42×C59C2×D412yes—
98C22⋊Dic59—24no—
99C4⋊Dic59—24no—
100C4⋊Dic59—24no—
101C4⋊Dic59—24no—
102C4⋊Dic59—22no—
103C4⋊Dic59—22no—
104C4⋊Dic59—22no—
105C4⋊Dic59C2×D414yes—
106Dic59⋊C4—28no—
107Dic59⋊C4—24no—
108Dic59⋊C4—24no—
109C4×Dic59—28no—
110C4×Dic59—24no—
111C2×C59⋊C8—24no—
112C23×C59C22×D411yescharacteristic
113Q8×C59—28no—
114Q8×C59—82no—
115Q8×C59—82no—
116C2×C4×C59—22no—
117C2×C4×C59—22no—
118C2×C4×C59—22no—
119C2×C4×C59—22no—
120C2×C4×C59—41no—
121C2×C4×C59—42no—
122C2×C4×C59—42no—
123C2×C4×C59—42no—
124C2×C4×C59—44no—
125C2×C4×C59—42no—
126C2×C4×C59—42no—
127C2×C4×C59#12011yescharacteristic
128C2×C4×C59#12012yes—
129C2×C4×C59#12011yescharacteristic
130C2×C4×C59C2×Q1614yes—
131C2×C4×C59C2×Q1612yes—
132C2×C4×C59#13712yes—
133C2×C4×C59C22×D412yes—
134C2×Dic59—24no—
135C2×Dic59—24no—
136Dic118—44no—
137Dic118—44no—
138Dic118—84no—
139C59⋊C8—44no—
140#18411—591nomaximal, Sylow
141C22×C59—21no—
142C22×C59—21no—
143C22×C59C2×#12011yescharacteristic
144C22×C59C22×Q1611yescharacteristic
145C22×C59C2×#13711yescharacteristic
146C4×C59—24no—
147C4×C59—21no—
148C4×C59—22no—
149C4×C59—42no—
150C4×C59—82no—
151C4×C59—82no—
152C4×C59#45114yes—
153C4×C59#45412yes—
154Dic59—44no—
155Dic59—44no—
156#3502—591no—
157#3230—592no—
158#2568D5911yescharacteristic
159#1617—598no—
160#1503—592no—
161#1461—591no—
162C2×C59—21no—
163C2×C59—21no—
164C2×C59C2×#45111yescharacteristic
165C2×C59C2×#45411yescharacteristic
166C2×C59#302811yescharacteristic
167Q82—1182no—
168#557—1184no—
169#557—1182no—
170#544—594no—
171#544—1181no—
172#533—592no—
173#533—1182no—
174C2×C4⋊Q8C2×D5911yescharacteristic
175#519—1181no—
176C2×C4×Q8C2×D5911yescharacteristic
177#500—1182no—
178#487—1182no—
179#484—598no—
180#477—1184no—
181#431—592no—
182#400C2×D5914yes—
183#391—21no—
184#391—22no—
185#382—21no—
186C4×C4⋊C4C2×D5911yescharacteristic
187#328—598no—
188#324—594no—
189C59#1841111yescharacteristic, complemented, Sylow
190C22×Q8—21no—
191C22×Q8C22⋊D5911yescharacteristic
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—592no—
195C4⋊Q8—592no—
196C4⋊Q8—1182no—
197C4⋊Q8—1182no—
198C4⋊Q8—1188no—
199C4⋊Q8—1184no—
200#125—1184no—
201#125—1188no—
202#125—1184no—
203#125—1182no—
204C22⋊Q8—594no—
205C22⋊Q8—1182no—
206C22⋊Q8—1184no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—594no—
210C4×Q8—1184no—
211C4×Q8—1184no—
212C4×Q8—1184no—
213C4×Q8—1184no—
214C4×Q8—1182no—
215#117—1182no—
216#117—1184no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C22⋊D5911yescharacteristic
220C2×C4⋊C4C22×D5912yes—
221C2×C4⋊C4C22×D5912yes—
222C2×C42—21no—
223C2×C42C22⋊D5912yes—
224C2×C42C22×D5911yescharacteristic
225#107—1188no—
226C4⋊C8—1182no—
227#103—1188no—
228C22⋊C8—594no—
229C4×C8—1182no—
230#95—24no—
231#95—24no—
232#95C22×D5912yes—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—594no—
239C2×Q8—1184no—
240C2×Q8—1182no—
241C2×Q8#1174567214yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4C2×C22⋊D5911yescharacteristic
245C22×C4C2×C22⋊D5912yes—
246C22×C4C2×C22⋊D5912yes—
247C22×C4C2×C22⋊D5911yescharacteristic
248C22×C4C23×D5911yescharacteristic
249C2×C8—1184no—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—594no—
256C4⋊C4—1184no—
257C4⋊C4—1184no—
258C4⋊C4—1184no—
259C4⋊C4—1188no—
260C4⋊C4—1182no—
261C4⋊C4—1182no—
262C4⋊C4—1184no—
263C4⋊C4—1182no—
264C4⋊C4—1184no—
265C22⋊C4—1184no—
266C42—22no—
267C42—22no—
268C42—42no—
269C42—1188no—
270C42—1184no—
271C42D4×D5912yes—
272C42D4×D5912yes—
273C23C22×C22⋊D5911yescharacteristic
274Q8—28no—
275Q8—82no—
276Q8—82no—
277Q8—4724no—
278Q8—2364no—
279Q8—2364no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—1184no—
292C2×C4—1184no—
293C2×C4#1427369511yescharacteristic
294C2×C4#1427369512yes—
295C2×C4#1427369511yescharacteristic
296C2×C4C2×#1174567214yes—
297C2×C4C2×#1174567212yes—
298C2×C4#1427375712yes—
299C2×C4C2×D4×D5912yes—
300C8—2364no—
301C22—21no—
302C22—21no—
303C22C2×#1427369511yescharacteristic
304C22C22×#1174567211yescharacteristic
305C22C2×#1427375711yescharacteristic
306C4—21no—
307C4—22no—
308C4—24no—
309C4—42no—
310C4—82no—
311C4—82no—
312C4—2364no—
313C4—2364no—
314C4#1683755714yes—
315C4#1683757412yes—
316C2—21no—
317C2—21no—
318C2C2×#1683755711yescharacteristic
319C2C2×#1683757411yescharacteristic
320C2#2143350111yescharacteristic
subgroupG/H
centerC22C2×#14273695
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2568C2
socleC22×C59C2×#120
prime radicalC23×C59C22×D4
hypercenter#2568D59
hypocenterC59#18411
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2568

Complements

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

NH ≅ G/Ntype
189. C59#18411semidirect

Maximal subgroups

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

subgroupindex
#214379062
#214352472
#214231372
#214225942
#214224202
C59×#25682
#1841159