Finite Group Databasebeta

← #28501738

Subgroups — #28501738

Order 15104.

378 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 63 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#21425449C211yesmaximal, characteristic
2#21423120C211yesmaximal, characteristic
3#21423120C211yesmaximal, characteristic
4#21422120C211yesmaximal, characteristic
5#21422120C211yesmaximal, characteristic
6#21422044C211yesmaximal, characteristic
7C59×#2807C211yesmaximal, characteristic
8#16838028C2211yescharacteristic
9#16837737—21no—
10#16837737—21no—
11#16837275—21no—
12#16837169C2211yescharacteristic
13#16837168—21no—
14#16837138—21no—
15#16837138—21no—
16#16837126C2211yescharacteristic
17#16837126C2211yescharacteristic
18C59×#400—21no—
19C59×#399C2211yescharacteristic
20C59×#399C2211yescharacteristic
21C59×#398—21no—
22C59×#389—21no—
23C59×#389—21no—
24C59×#388—21no—
25C59×#385—21no—
26C59×#385C2211yescharacteristic
27#14273706—21no—
28#14273706—21no—
29#14273704—41no—
30#14273704—41no—
31C2×Dic59⋊C4—41no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34#14273660—41no—
35#14273660—41no—
36#14273650—41no—
37#14273650—41no—
38C22⋊(C59⋊C8)—21no—
39C22⋊(C59⋊C8)—21no—
40C59⋊#95—41no—
41C23×C4×C59D411yescharacteristic
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59—21no—
46C2×C4⋊C4×C59—21no—
47C2×C4⋊C4×C59—21no—
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59D411yescharacteristic
53C2×C22⋊C4×C59C2311yescharacteristic
54C2×C42×C59—21no—
55#95×C59—21no—
56#95×C59—21no—
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60#95×C59D411yescharacteristic
61#95×C59D411yescharacteristic
62C24×C59C2×D411yescharacteristic
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—41no—
70C22×C4×C59—81no—
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—41no—
76C2×Dic118—21no—
77C2×Dic118—21no—
78C4⋊C4×C59—42no—
79C4⋊C4×C59—42no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—81no—
83C4⋊C4×C59—81no—
84C4⋊C4×C59—81no—
85C4⋊C4×C59—81no—
86C22⋊C4×C59—21no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—42no—
89C22⋊C4×C59—81no—
90C22⋊C4×C59Q1612yes—
91C42×C59—81no—
92C22⋊Dic59—22no—
93C22⋊Dic59—41no—
94C22⋊Dic59—41no—
95C22⋊Dic59—41no—
96C4⋊Dic59—21no—
97C4⋊Dic59—21no—
98Dic59⋊C4—41no—
99Dic59⋊C4—41no—
100Dic59⋊C4—82no—
101C2×C59⋊C8—41no—
102C2×C59⋊C8—41no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59#12011yescharacteristic
108C23×C59C2×Q1611yescharacteristic
109C23×C59#13711yescharacteristic
110C2×C4×C59—22no—
111C2×C4×C59—22no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—42no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—81no—
134C2×Dic59—42no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—81no—
138C2×Dic59—81no—
139C2×Dic59—81no—
140C2×Dic59—81no—
141Dic118—81no—
142Dic118—81no—
143C59⋊C8—81no—
144C59⋊C8—81no—
145#12255—591nomaximal, Sylow
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—41no—
154C22×C59—81no—
155C22×C59#45111yescharacteristic
156C22×C59#45611yescharacteristic
157C22×C59B(4,𝔽2)11yescharacteristic
158C4×C59—42no—
159C4×C59—42no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—81no—
163C4×C59—81no—
164C4×C59—81no—
165C4×C59—81no—
166Dic59—82no—
167Dic59—81no—
168Dic59—81no—
169#2807D5911yescharacteristic
170#2062—591no—
171#1614—591no—
172#1614—591no—
173#1352—591no—
174#1352—591no—
175#1326—591no—
176C2×C59—21no—
177C2×C59—21no—
178C2×C59—41no—
179C2×C59—41no—
180C2×C59#160511yescharacteristic
181C2×C59#219811yescharacteristic
182C2×C59#220011yescharacteristic
183#543—591no—
184#484—1181no—
185#484—1181no—
186#400—21no—
187#399C2×D5911yescharacteristic
188#399C2×D5911yescharacteristic
189#398—21no—
190#389—21no—
191#389—21no—
192#388—21no—
193#385—21no—
194#385—1181no—
195#385C2×D5911yescharacteristic
196#342—591no—
197#342—1181no—
198#328—1181no—
199#328—1181no—
200#323—591no—
201#323—591no—
202C59#1225511yescharacteristic, complemented, Sylow
203C23×C4C22⋊D5911yescharacteristic
204C22⋊Q8—1181no—
205C22⋊Q8—1181no—
206C22⋊Q8—2361no—
207C22⋊Q8—2361no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—2361no—
215C2×C4⋊C4C22⋊D5911yescharacteristic
216C2×C4⋊C4C22⋊D5911yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—1181no—
220C2×C22⋊C4—1181no—
221C2×C22⋊C4C22⋊D5911yescharacteristic
222C2×C22⋊C4C22×D5911yescharacteristic
223C2×C42—21no—
224#107—2361no—
225#107—2361no—
226#103—2361no—
227#103—2361no—
228C22⋊C8—1181no—
229C22⋊C8—1181no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—2361no—
236#95C22⋊D5911yescharacteristic
237#95C22⋊D5911yescharacteristic
238C24C2×C22⋊D5911yescharacteristic
239C2×Q8—1181no—
240C2×Q8—1181no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4—81no—
249C22×C4—1181no—
250C22×C4—1181no—
251C22×C4—2361no—
252C22×C4C2×C22⋊D5911yescharacteristic
253C22×C4C2×C22⋊D5911yescharacteristic
254C2×C8—2361no—
255C2×C8—2361no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—81no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C4⋊C4—1181no—
265C4⋊C4—1181no—
266C4⋊C4—4722no—
267C4⋊C4—2361no—
268C4⋊C4—2361no—
269C22⋊C4—21no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—81no—
273C22⋊C4—1182no—
274C22⋊C4—2361no—
275C22⋊C4—2361no—
276C22⋊C4—2361no—
277C22⋊C4#1174567212yes—
278C42—81no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23#1427369611yescharacteristic
284C23C2×#1174567211yescharacteristic
285C23#1427375811yescharacteristic
286Q8—4721no—
287Q8—4721no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—4721no—
313C2×C4—4721no—
314C2×C4—4721no—
315C2×C4—4721no—
316C2×C4—2362no—
317C2×C4—2361no—
318C2×C4—2361no—
319C8—4721no—
320C8—4721no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22#1683755811yescharacteristic
331C22#1683758511yescharacteristic
332C22#1683758911yescharacteristic
333C4—42no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—4722no—
342C4—4721no—
343C4—4721no—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2#2142307011yescharacteristic
349C2#2142609211yescharacteristic
350C2#2142610311yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C22⋊C4×C59C23
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2807C2
socleC22×C59B(4,𝔽2)
prime radicalC24×C59C2×D4
hypercenter#2807D59
hypocenterC59#12255
LCSG ▷ C2×C22⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2807

Complements

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

NH ≅ G/Ntype
202. C59#12255semidirect

Maximal subgroups

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

subgroupindex
#214254492
#214231202
#214221202
#214220442
C59×#28072
#1225559