Finite Group Databasebeta

← #28501731

Subgroups — #28501731

Order 15104.

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

normal subgroups only

This group has 388 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#21423052C211yesmaximal, characteristic
4#21422120C211yesmaximal, characteristic
5#21422117C211yesmaximal, characteristic
6#21422044C211yesmaximal, characteristic
7C59×#2742C211yesmaximal, characteristic
8#16838028C2211yescharacteristic
9#16837737—21no—
10#16837558—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×#395—21no—
21C59×#393C2211yescharacteristic
22C59×#389—21no—
23C59×#388—21no—
24C59×#387—21no—
25C59×#385C2211yescharacteristic
26C59×#380—21no—
27C2×#11745672—41no—
28#14273706—21no—
29#14273706—21no—
30#14273704—41no—
31#14273704—41no—
32C2×Dic59⋊C4—41no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35#14273660—41no—
36#14273651—41no—
37#14273650—41no—
38C22⋊(C59⋊C8)—21no—
39C22⋊(C59⋊C8)—21no—
40C59⋊#95—41no—
41C22×Q8×C59D411yescharacteristic
42C23×C4×C59D411yescharacteristic
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×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59D411yescharacteristic
52C2×C22⋊C4×C59C2311yescharacteristic
53C2×C42×C59—21no—
54#95×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
63C2×Q8×C59—41no—
64C2×Q8×C59—81no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—42no—
72C22×C4×C59—41no—
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—41no—
78C2×Dic118—21no—
79C2×Dic118—21no—
80C4⋊C4×C59—42no—
81C4⋊C4×C59—42no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—81no—
84C4⋊C4×C59—81no—
85C4⋊C4×C59—81no—
86C22⋊C4×C59—21no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—81no—
89C22⋊C4×C59—81no—
90C22⋊C4×C59Q1612yescomplemented
91C42×C59—81no—
92C22⋊Dic59—22no—
93C22⋊Dic59—41no—
94C22⋊Dic59—41no—
95C22⋊Dic59—41no—
96#11745672—82no—
97C4⋊Dic59—21no—
98C4⋊Dic59—21no—
99Dic59⋊C4—41no—
100Dic59⋊C4—41no—
101Dic59⋊C4—82no—
102C2×C59⋊C8—41no—
103C2×C59⋊C8—41no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59#12011yescharacteristic
109C23×C59C2×Q1611yescharacteristic
110C23×C59#13711yescharacteristic
111Q8×C59—42no—
112Q8×C59—81no—
113C2×C4×C59—22no—
114C2×C4×C59—22no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
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—42no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—81no—
137C2×Dic59—42no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—81no—
141C2×Dic59—81no—
142C2×Dic59—81no—
143C2×Dic59—81no—
144Dic118—81no—
145Dic118—81no—
146C59⋊C8—81no—
147C59⋊C8—81no—
148#12254—591nomaximal, Sylow
149C22×C59—21no—
150C22×C59—21no—
151C22×C59—21no—
152C22×C59—21no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—81no—
158C22×C59#45111yescharacteristic
159C22×C59#45611yescharacteristic
160C22×C59B(4,𝔽2)11yescharacteristic
161C4×C59—42no—
162C4×C59—41no—
163C4×C59—42no—
164C4×C59—41no—
165C4×C59—81no—
166C4×C59—81no—
167C4×C59—81no—
168C4×C59—81no—
169Dic59—82no—
170Dic59—81no—
171Dic59—81no—
172#2742D5911yescharacteristic
173#2062—591no—
174#1614—591no—
175#1602—591no—
176#1352—591no—
177#1351—591no—
178#1326—591no—
179C2×C59—21no—
180C2×C59—21no—
181C2×C59—41no—
182C2×C59—41no—
183C2×C59#160511yescharacteristic
184C2×C59#219711yescharacteristic
185C2×C59#220011yescharacteristic
186#543—591no—
187#484—1181no—
188#451—1181no—
189#400—21no—
190#399C2×D5911yescharacteristic
191#395—21no—
192#393C2×D5911yescharacteristic
193#389—21no—
194#388—21no—
195#387—21no—
196#385—1181no—
197#385C2×D5911yescharacteristic
198#380—21no—
199#342—591no—
200#342—1181no—
201#328—1181no—
202#328—1181no—
203#323—591no—
204#323—591no—
205C59#1225411yescharacteristic, complemented, Sylow
206C22×Q8C22⋊D5911yescharacteristic
207C23×C4C22⋊D5911yescharacteristic
208C2×Q16—2361no—
209C22⋊Q8—1181no—
210C22⋊Q8—1181no—
211C22⋊Q8—2361no—
212C22⋊Q8—2361no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—2361no—
219C2×C4⋊C4C22⋊D5911yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—1181no—
223C2×C22⋊C4—1181no—
224C2×C22⋊C4C22⋊D5911yescharacteristic
225C2×C22⋊C4C22×D5911yescharacteristic
226C2×C42—21no—
227#107—2361no—
228#103—2361no—
229#103—2361no—
230C22⋊C8—1181no—
231C22⋊C8—1181no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—2361no—
239#95C22⋊D5911yescharacteristic
240#95C22⋊D5911yescharacteristic
241C24C2×C22⋊D5911yescharacteristic
242C2×Q8—41no—
243C2×Q8—81no—
244C2×Q8—1181no—
245C2×Q8—1181no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—1181no—
255C22×C4—1181no—
256C22×C4—2361no—
257C22×C4C2×C22⋊D5911yescharacteristic
258C22×C4C2×C22⋊D5911yescharacteristic
259Q16—4722no—
260C2×C8—2361no—
261C2×C8—2361no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—81no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C4⋊C4—1181no—
269C4⋊C4—1181no—
270C4⋊C4—4722no—
271C4⋊C4—2361no—
272C4⋊C4—2361no—
273C22⋊C4—21no—
274C22⋊C4—41no—
275C22⋊C4—81no—
276C22⋊C4—81no—
277C22⋊C4—1182no—
278C22⋊C4—2361no—
279C22⋊C4—2361no—
280C22⋊C4—2361no—
281C22⋊C4#1174567212yescomplemented
282C42—81no—
283C23—21no—
284C23—21no—
285C23—41no—
286C23—41no—
287C23#1427369611yescharacteristic
288C23C2×#1174567211yescharacteristic
289C23#1427375811yescharacteristic
290Q8—42no—
291Q8—81no—
292Q8—4721no—
293Q8—4721no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—4721no—
319C2×C4—4721no—
320C2×C4—4721no—
321C2×C4—4721no—
322C2×C4—2362no—
323C2×C4—2361no—
324C2×C4—2361no—
325C8—4721no—
326C8—4721no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22#1683755811yescharacteristic
337C22#1683758511yescharacteristic
338C22#1683758911yescharacteristic
339C4—42no—
340C4—41no—
341C4—42no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—4722no—
348C4—4721no—
349C4—4721no—
350C2—21no—
351C2—21no—
352C2—41no—
353C2—41no—
354C2#2142307011yescharacteristic
355C2#2142608611yescharacteristic
356C2#2142610311yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C22⋊C4×C59C23
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2742C2
socleC22×C59B(4,𝔽2)
prime radicalC24×C59C2×D4
hypercenter#2742D59
hypocenterC59#12254
LCSG ▷ C2×C22⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2742

Complements

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

NH ≅ G/Ntype
90. C22⋊C4×C59Q16semidirect
205. C59#12254semidirect
281. C22⋊C4#11745672semidirect

Maximal subgroups

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

subgroupindex
#214254492
#214231202
#214230522
#214221202
#214221172
#214220442
C59×#27422
#1225459