Finite Group Databasebeta

← #28563589

Subgroups — #28563589

Order 15104.

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

normal subgroups only

This group has 906 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#21435237C212yesmaximal
2#21429081C211yesmaximal, characteristic
3#21423104C218yesmaximal
4#21422595C212yesmaximal
5#21422420C211yesmaximal, characteristic
6C59×#3502C211yesmaximal, characteristic
7C2×#14273727C2211yescharacteristic
8C2×#14273725—21no—
9#16837798—22no—
10#16837752—22no—
11#16837730C2218yes—
12#16837687—24no—
13#16837455C2212yes—
14#16837332C2214yes—
15#16837307—21no—
16#16837300—22no—
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—
32#14273725—44no—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59—21no—
35C2×C4⋊Dic59C2312yes—
36C2×Dic59⋊C4—24no—
37C2×C4×Dic59—21no—
38#14273658—28no—
39C4⋊(C59⋊C8)—22no—
40#14273651—28no—
41C22⋊(C59⋊C8)C2314yes—
42C4×C59⋊C8—22no—
43C59⋊#95—24no—
44C59⋊#95C2312yes—
45C22×Q8×C59D411yescharacteristic
46C4⋊Q8×C59—22no—
47C4⋊Q8×C59—22no—
48C4⋊Q8×C59—24no—
49C4⋊Q8×C59—28no—
50C4⋊Q8×C59D412yes—
51C4⋊Q8×C59D412yes—
52#125×C59—24no—
53#125×C59—24no—
54#125×C59—28no—
55#125×C59—22no—
56C22⋊Q8×C59—22no—
57C22⋊Q8×C59—24no—
58C22⋊Q8×C59C2314yes—
59C4×Q8×C59—24no—
60C4×Q8×C59—24no—
61C4×Q8×C59—24no—
62C4×Q8×C59—24no—
63C4×Q8×C59—22no—
64C4×Q8×C59—22no—
65C4×Q8×C59—22no—
66C4×Q8×C59D414yes—
67#117×C59—24no—
68#117×C59—22no—
69C2×C4⋊C4×C59D411yescharacteristic
70C2×C4⋊C4×C59C2312yes—
71C2×C42×C59D412yes—
72C2×C42×C59C2311yescharacteristic
73C2×Q8×C59—24no—
74C2×Q8×C59—22no—
75C2×Q8×C59—22no—
76C2×Q8×C59—22no—
77C2×Q8×C59SD1614yes—
78C2×Q8×C59C2×D414yes—
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D412yes—
81C22×C4×C59C2×D412yes—
82C22×C4×C59C2×D411yescharacteristic
83C22×C4×C59C2411yescharacteristic
84C22×Dic59—22no—
85C22×Dic59—22no—
86C2×Dic118—42no—
87C2×Dic118—42no—
88C2×Dic118—42no—
89C4⋊C4×C59—24no—
90C4⋊C4×C59—24no—
91C4⋊C4×C59—24no—
92C4⋊C4×C59—28no—
93C4⋊C4×C59—22no—
94C4⋊C4×C59—22no—
95C4⋊C4×C59—22no—
96C4⋊C4×C59—22no—
97C4⋊C4×C59—24no—
98C4⋊C4×C59—24no—
99C4⋊C4×C59—24no—
100C4⋊C4×C59C2×D414yes—
101C22⋊C4×C59—24no—
102C42×C59—28no—
103C42×C59—22no—
104C42×C59—22no—
105C42×C59—24no—
106C42×C59C2×D412yes—
107C42×C59C2×D412yes—
108C4⋊Dic59—24no—
109C4⋊Dic59—24no—
110C4⋊Dic59—41no—
111C4⋊Dic59—41no—
112Dic59⋊C4—48no—
113C4×Dic59—42no—
114C2×C59⋊C8—24no—
115C23×C59C22×D411yescharacteristic
116Q8×C59—28no—
117Q8×C59—44no—
118Q8×C59—44no—
119Q8×C59—84no—
120C2×C4×C59—22no—
121C2×C4×C59—22no—
122C2×C4×C59—22no—
123C2×C4×C59—24no—
124C2×C4×C59—24no—
125C2×C4×C59—22no—
126C2×C4×C59—41no—
127C2×C4×C59—42no—
128C2×C4×C59#12011yescharacteristic
129C2×C4×C59#12012yes—
130C2×C4×C59#12011yescharacteristic
131C2×C4×C59C2×SD1614yes—
132C2×C4×C59C2×SD1612yes—
133C2×C4×C59#13712yes—
134C2×C4×C59C22×D412yes—
135C2×Dic59—24no—
136C2×Dic59—42no—
137C2×Dic59—42no—
138C2×Dic59—42no—
139C2×Dic59—42no—
140C2×Dic59—42no—
141Dic118—84no—
142C59⋊C8—44no—
143#18410—591nomaximal, Sylow
144C22×C59—21no—
145C22×C59—21no—
146C22×C59C2×#12011yescharacteristic
147C22×C59C22×SD1611yescharacteristic
148C22×C59C2×#13711yescharacteristic
149C4×C59—24no—
150C4×C59—21no—
151C4×C59—22no—
152C4×C59—44no—
153C4×C59—42no—
154C4×C59—44no—
155C4×C59#44814yes—
156C4×C59#45612yes—
157Dic59—44no—
158Dic59—82no—
159#3502D5911yescharacteristic
160#3229—592no—
161#2546—591no—
162#1611—598no—
163#1503—592no—
164#1461—591no—
165C2×C59—21no—
166C2×C59—21no—
167C2×C59C2×#44811yescharacteristic
168C2×C59C2×#45611yescharacteristic
169C2×C59#302811yescharacteristic
170Q82—22no—
171#557—22no—
172#557—24no—
173#544—21no—
174#544C2×D5914yes—
175#533—22no—
176#533C2×D5912yes—
177C2×C4⋊Q8—591no—
178C2×C4⋊Q8—1181no—
179#519—21no—
180C2×C4×Q8C2×D5911yescharacteristic
181#498—1182no—
182#487—1182no—
183#483—598no—
184#475—1184no—
185#431—592no—
186#398—594no—
187#391—1181no—
188#389—1182no—
189C4×C4⋊C4—591no—
190#328—598no—
191#324—594no—
192C59#1841011yescharacteristic, complemented, Sylow
193C22×Q8—1181no—
194C22×Q8C22⋊D5911yescharacteristic
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8—24no—
198C4⋊Q8—28no—
199C4⋊Q8—1182no—
200C4⋊Q8—1182no—
201C4⋊Q8—2364no—
202C4⋊Q8C22⋊D5912yes—
203C4⋊Q8C22⋊D5912yes—
204#125—24no—
205#125—24no—
206#125—28no—
207#125—22no—
208C22⋊Q8—22no—
209C22⋊Q8—24no—
210C22⋊Q8C22×D5914yes—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8C22⋊D5914yes—
219#117—24no—
220#117—22no—
221C2×C4⋊C4—592no—
222C2×C4⋊C4—1181no—
223C2×C4⋊C4—1181no—
224C2×C4⋊C4—1184no—
225C2×C4⋊C4C22⋊D5911yescharacteristic
226C2×C4⋊C4C22×D5912yes—
227C2×C42—1181no—
228C2×C42C22⋊D5912yes—
229C2×C42C22×D5911yescharacteristic
230#106—1188no—
231C4⋊C8—1182no—
232#103—1188no—
233C22⋊C8—594no—
234C4×C8—1182no—
235#95—592no—
236#95—1184no—
237C2×Q8—24no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—2362no—
242C2×Q8—2362no—
243C2×Q8—2362no—
244C2×Q8#1174567114yes—
245C2×Q8C2×C22⋊D5914yes—
246C22×C4—1182no—
247C22×C4—1182no—
248C22×C4C2×C22⋊D5911yescharacteristic
249C22×C4C2×C22⋊D5912yes—
250C22×C4C2×C22⋊D5912yes—
251C22×C4C2×C22⋊D5911yescharacteristic
252C22×C4C23×D5911yescharacteristic
253C2×C8—1184no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—28no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—1184no—
266C4⋊C4—1184no—
267C4⋊C4—2361no—
268C4⋊C4—2361no—
269C4⋊C4—2368no—
270C4⋊C4C2×C22⋊D5914yes—
271C22⋊C4—24no—
272C42—28no—
273C42—22no—
274C42—22no—
275C42—24no—
276C42—2362no—
277C42C2×C22⋊D5912yes—
278C42C2×C22⋊D5912yes—
279C23C22×C22⋊D5911yescharacteristic
280Q8—28no—
281Q8—44no—
282Q8—44no—
283Q8—84no—
284Q8—4724no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—1184no—
294C2×C4—2362no—
295C2×C4—2362no—
296C2×C4—2362no—
297C2×C4—2362no—
298C2×C4—2362no—
299C2×C4#1427369611yescharacteristic
300C2×C4#1427369612yes—
301C2×C4#1427369611yescharacteristic
302C2×C4C2×#1174567114yes—
303C2×C4C2×#1174567112yes—
304C2×C4#1427375812yes—
305C2×C4C22×C22⋊D5912yes—
306C8—2364no—
307C22—21no—
308C22—21no—
309C22C2×#1427369611yescharacteristic
310C22C22×#1174567111yescharacteristic
311C22C2×#1427375811yescharacteristic
312C4—21no—
313C4—22no—
314C4—24no—
315C4—44no—
316C4—42no—
317C4—44no—
318C4—4722no—
319C4—2364no—
320C4#1683754114yes—
321C4#1683758512yes—
322C2—21no—
323C2—21no—
324C2C2×#1683754111yescharacteristic
325C2C2×#1683758511yescharacteristic
326C2#2143350211yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#3502C2
socleC22×C59C2×#120
prime radicalC23×C59C22×D4
hypercenter#3502D59
hypocenterC59#18410
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
192. C59#18410semidirect

Maximal subgroups

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

subgroupindex
#214352372
#214290812
#214231042
#214225952
#214224202
C59×#35022
#1841059