Finite Group Databasebeta

← #28563595

Subgroups — #28563595

Order 15104.

870 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 207 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#21435244C212yesmaximal
2#21429308C211yesmaximal, characteristic
3#21423138C218yesmaximal
4#21422595C212yesmaximal
5#21422420C211yesmaximal, characteristic
6C59×#3502C211yesmaximal, characteristic
7C2×#14273727C2211yescharacteristic
8#16837805—22no—
9#16837752—22no—
10#16837735C2218yes—
11#16837697—24no—
12#16837455C2212yes—
13#16837341C2214yes—
14#16837309—21no—
15#16837308—22no—
16#16837267—21no—
17C4×C4⋊Dic59C2211yescharacteristic
18#16837139C2218yes—
19#16837128C2214yes—
20Q82×C59—22no—
21C59×#557—22no—
22C59×#557—24no—
23C59×#544—21no—
24C59×#544C2214yes—
25C59×#533—22no—
26C59×#533C2212yes—
27C59×#519—21no—
28C2×C4×Q8×C59C2211yescharacteristic
29C22×Dic118—21no—
30#14273727—22no—
31#14273727—22no—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59C2312yes—
35C2×C4×Dic59—21no—
36#14273660—28no—
37C4⋊(C59⋊C8)—22no—
38#14273651—28no—
39C22⋊(C59⋊C8)C2314yes—
40C4×C59⋊C8—22no—
41C59⋊#95—24no—
42C59⋊#95—24no—
43C59⋊#95C2312yes—
44C22×Q8×C59D411yescharacteristic
45C4⋊Q8×C59—22no—
46C4⋊Q8×C59—22no—
47C4⋊Q8×C59—24no—
48C4⋊Q8×C59—28no—
49C4⋊Q8×C59D412yes—
50C4⋊Q8×C59D412yes—
51#125×C59—24no—
52#125×C59—24no—
53#125×C59—28no—
54#125×C59—22no—
55C22⋊Q8×C59—22no—
56C22⋊Q8×C59—24no—
57C22⋊Q8×C59C2314yes—
58C4×Q8×C59—24no—
59C4×Q8×C59—24no—
60C4×Q8×C59—24no—
61C4×Q8×C59—24no—
62C4×Q8×C59—22no—
63C4×Q8×C59—22no—
64C4×Q8×C59—22no—
65C4×Q8×C59D414yes—
66#117×C59—24no—
67#117×C59—22no—
68C2×C4⋊C4×C59D411yescharacteristic
69C2×C4⋊C4×C59C2312yes—
70C2×C42×C59D412yes—
71C2×C42×C59C2311yescharacteristic
72C2×Q8×C59—24no—
73C2×Q8×C59—22no—
74C2×Q8×C59—22no—
75C2×Q8×C59—22no—
76C2×Q8×C59Q1614yes—
77C2×Q8×C59C2×D414yes—
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D412yes—
80C22×C4×C59C2×D412yes—
81C22×C4×C59C2×D411yescharacteristic
82C22×C4×C59C2411yescharacteristic
83C22×Dic59—22no—
84C22×Dic59—22no—
85C2×Dic118—44no—
86C2×Dic118—41no—
87C2×Dic118—41no—
88C4⋊C4×C59—24no—
89C4⋊C4×C59—24no—
90C4⋊C4×C59—24no—
91C4⋊C4×C59—28no—
92C4⋊C4×C59—22no—
93C4⋊C4×C59—22no—
94C4⋊C4×C59—22no—
95C4⋊C4×C59—22no—
96C4⋊C4×C59—24no—
97C4⋊C4×C59—24no—
98C4⋊C4×C59—24no—
99C4⋊C4×C59C2×D414yes—
100C22⋊C4×C59—24no—
101C42×C59—28no—
102C42×C59—22no—
103C42×C59—22no—
104C42×C59—24no—
105C42×C59C2×D412yes—
106C42×C59C2×D412yes—
107C4⋊Dic59—24no—
108C4⋊Dic59—42no—
109C4⋊Dic59—42no—
110C4×Dic59—42no—
111C2×C59⋊C8—24no—
112C23×C59C22×D411yescharacteristic
113Q8×C59—28no—
114Q8×C59—44no—
115Q8×C59—44no—
116Q8×C59—84no—
117C2×C4×C59—22no—
118C2×C4×C59—22no—
119C2×C4×C59—22no—
120C2×C4×C59—24no—
121C2×C4×C59—24no—
122C2×C4×C59—22no—
123C2×C4×C59—41no—
124C2×C4×C59—42no—
125C2×C4×C59#12011yescharacteristic
126C2×C4×C59#12012yes—
127C2×C4×C59#12011yescharacteristic
128C2×C4×C59C2×Q1614yes—
129C2×C4×C59C2×Q1612yes—
130C2×C4×C59#13712yes—
131C2×C4×C59C22×D412yes—
132C2×Dic59—42no—
133C2×Dic59—44no—
134C2×Dic59—42no—
135C2×Dic59—42no—
136C2×Dic59—42no—
137Dic118—82no—
138Dic118—82no—
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—44no—
150C4×C59—42no—
151C4×C59—44no—
152C4×C59#45114yes—
153C4×C59#45412yes—
154Dic59—82no—
155Dic59—82no—
156#3502D5911yescharacteristic
157#3230—592no—
158#2568—591no—
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—22no—
168#557—22no—
169#557—24no—
170#544—21no—
171#544C2×D5914yes—
172#533—22no—
173#533C2×D5912yes—
174C2×C4⋊Q8—591no—
175#519—21no—
176C2×C4×Q8C2×D5911yescharacteristic
177#500—1182no—
178#487—1182no—
179#484—598no—
180#477—1184no—
181#431—592no—
182#400—594no—
183#391—1181no—
184#391—1182no—
185#382—1181no—
186C4×C4⋊C4—591no—
187#328—598no—
188#324—594no—
189C59#1841111yescharacteristic, complemented, Sylow
190C22×Q8—1181no—
191C22×Q8C22⋊D5911yescharacteristic
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—24no—
195C4⋊Q8—28no—
196C4⋊Q8—1182no—
197C4⋊Q8—1182no—
198C4⋊Q8C22⋊D5912yes—
199C4⋊Q8C22⋊D5912yes—
200#125—24no—
201#125—24no—
202#125—28no—
203#125—22no—
204C22⋊Q8—22no—
205C22⋊Q8—24no—
206C22⋊Q8C22×D5914yes—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8C22⋊D5914yes—
215#117—24no—
216#117—22no—
217C2×C4⋊C4—592no—
218C2×C4⋊C4—1181no—
219C2×C4⋊C4—1181no—
220C2×C4⋊C4C22⋊D5911yescharacteristic
221C2×C4⋊C4C22×D5912yes—
222C2×C42—1181no—
223C2×C42C22⋊D5912yes—
224C2×C42C22×D5911yescharacteristic
225#107—1188no—
226C4⋊C8—1182no—
227#103—1188no—
228C22⋊C8—594no—
229C4×C8—1182no—
230#95—592no—
231#95—1184no—
232#95—1184no—
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—2364no—
238C2×Q8—2361no—
239C2×Q8—2361no—
240C2×Q8#1174567214yes—
241C2×Q8C2×C22⋊D5914yes—
242C22×C4—1182no—
243C22×C4—1182no—
244C22×C4C2×C22⋊D5911yescharacteristic
245C22×C4C2×C22⋊D5912yes—
246C22×C4C2×C22⋊D5912yes—
247C22×C4C2×C22⋊D5911yescharacteristic
248C22×C4C23×D5911yescharacteristic
249C2×C8—1184no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—28no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—1184no—
262C4⋊C4—2362no—
263C4⋊C4—2362no—
264C4⋊C4C2×C22⋊D5914yes—
265C22⋊C4—24no—
266C42—28no—
267C42—22no—
268C42—22no—
269C42—24no—
270C42—2362no—
271C42C2×C22⋊D5912yes—
272C42C2×C22⋊D5912yes—
273C23C22×C22⋊D5911yescharacteristic
274Q8—28no—
275Q8—44no—
276Q8—44no—
277Q8—84no—
278Q8—4722no—
279Q8—4722no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—2362no—
289C2×C4—2362no—
290C2×C4—2364no—
291C2×C4—2362no—
292C2×C4—2362no—
293C2×C4#1427369611yescharacteristic
294C2×C4#1427369612yes—
295C2×C4#1427369611yescharacteristic
296C2×C4C2×#1174567214yes—
297C2×C4C2×#1174567212yes—
298C2×C4#1427375812yes—
299C2×C4C22×C22⋊D5912yes—
300C8—2364no—
301C22—21no—
302C22—21no—
303C22C2×#1427369611yescharacteristic
304C22C22×#1174567211yescharacteristic
305C22C2×#1427375811yescharacteristic
306C4—21no—
307C4—22no—
308C4—24no—
309C4—44no—
310C4—42no—
311C4—44no—
312C4—4722no—
313C4—4722no—
314C4#1683755814yes—
315C4#1683757512yes—
316C2—21no—
317C2—21no—
318C2C2×#1683755811yescharacteristic
319C2C2×#1683757511yescharacteristic
320C2#2143350211yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#3502C2
socleC22×C59C2×#120
prime radicalC23×C59C22×D4
hypercenter#3502D59
hypocenterC59#18411
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

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
#214352442
#214293082
#214231382
#214225952
#214224202
C59×#35022
#1841159