Finite Group Databasebeta

← #28501814

Subgroups — #28501814

Order 15104.

414 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 59 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#21425263C211yesmaximal, characteristic, complemented
2#21423120C211yesmaximal, characteristic, complemented
3#21423018C211yesmaximal, characteristic
4#21422120C211yesmaximal, characteristic, complemented
5#21422093C211yesmaximal, characteristic
6#21422026C211yesmaximal, characteristic, complemented
7C59×#2752C211yesmaximal, characteristic
8#16838028C2211yescharacteristic
9#16837737—21no—
10#16837553—21no—
11#16837265—21no—
12#16837169C2211yescharacteristic
13#16837168—21no—
14#16837138—21no—
15#16837138—21no—
16#16837126C2211yescharacteristic
17#16837126C2211yescharacteristic
18C59×#399C2211yescharacteristic
19C59×#397—21no—
20C59×#394—21no—
21C59×#392C2211yescharacteristic
22C59×#390—21no—
23C59×#388—21no—
24C59×#387—21no—
25C59×#386—21no—
26C59×#381C2211yescharacteristic
27C2×#11745670—41no—
28#14273706—21no—
29#14273706—21no—
30#14273704—41no—
31#14273704—41no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34#14273660—41no—
35#14273650—41no—
36#14273648—41no—
37C22⋊(C59⋊C8)—21no—
38C22⋊(C59⋊C8)—21no—
39C59⋊#95—41no—
40C59⋊#95—41no—
41C22×D4×C59—21no—
42C22×D4×C59D411yescharacteristic
43C23×C4×C59D411yescharacteristic
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59—21no—
46C2×C4⋊C4×C59D411yescharacteristic
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59—21no—
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59C2311yescharacteristic
55C2×C42×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×C59—21no—
63C24×C59C2×D411yescharacteristic
64C2×D4×C59—41no—
65C2×D4×C59—42no—
66C2×D4×C59—41no—
67C2×D4×C59—41no—
68C2×D4×C59—81no—
69C2×D4×C59—81no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—41no—
78C22×C4×C59—41no—
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C22×Dic59—21no—
82C22×Dic59—21no—
83C22×Dic59—41no—
84C2×Dic118—21no—
85C2×Dic118—21no—
86C4⋊C4×C59—42no—
87C4⋊C4×C59—41no—
88C4⋊C4×C59—81no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—42no—
91C22⋊C4×C59—41no—
92C22⋊C4×C59—81no—
93C22⋊C4×C59—81no—
94C22⋊C4×C59—81no—
95C22⋊C4×C59—81no—
96C22⋊C4×C59—81no—
97C42×C59—81no—
98C22⋊Dic59—41no—
99C22⋊Dic59—41no—
100C22⋊Dic59—41no—
101C22⋊Dic59—41no—
102#11745670—82no—
103C4⋊Dic59—21no—
104C4⋊Dic59—21no—
105Dic59⋊C4—41no—
106Dic59⋊C4—41no—
107C2×C59⋊C8—41no—
108C2×C59⋊C8—41no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59#12011yescharacteristic
121C23×C59#13511yescharacteristic
122C23×C59#13711yescharacteristic
123D4×C59—42no—
124D4×C59—82no—
125D4×C59—82no—
126D4×C59—81no—
127C2×C4×C59—22no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
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—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—42no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—81no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—81no—
151C2×Dic59—81no—
152C2×Dic59—81no—
153C2×Dic59—81no—
154C2×Dic59—81no—
155Dic118—81no—
156Dic118—81no—
157C59⋊C8—81no—
158C59⋊C8—81no—
159#12267—591nomaximal, Sylow
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—41no—
168C22×C59—81no—
169C22×C59—81no—
170C22×C59—81no—
171C22×C59—81no—
172C22×C59—81no—
173C22×C59—81no—
174C22×C59—81no—
175C22×C59—81no—
176C22×C59#45211yescharacteristic
177C22×C59#45511yescharacteristic
178C22×C59B(4,𝔽2)11yescharacteristic
179C4×C59—42no—
180C4×C59—41no—
181C4×C59—41no—
182C4×C59—81no—
183C4×C59—81no—
184C4×C59—81no—
185C4×C59—81no—
186Dic59—81no—
187Dic59—81no—
188Dic59—161no—
189#2752D5911yescharacteristic
190#2030—591no—
191#1614—591no—
192#1596—591no—
193#1352—591no—
194#1343—591no—
195#1317—591no—
196C2×C59—21no—
197C2×C59—21no—
198C2×C59—41no—
199C2×C59—41no—
200C2×C59—161no—
201C2×C59#160611yescharacteristic
202C2×C59#219811yescharacteristic
203C2×C59#220011yescharacteristic
204#543—591no—
205#484—1181no—
206#450—1181no—
207#399C2×D5911yescharacteristic
208#397—21no—
209#394—21no—
210#392C2×D5911yescharacteristic
211#390—21no—
212#388—21no—
213#387—21no—
214#386—21no—
215#381—1181no—
216#381C2×D5911yescharacteristic
217#342—591no—
218#342—1181no—
219#328—1181no—
220#328—1181no—
221#323—591no—
222#323—591no—
223C59#1226711yescharacteristic, complemented, Sylow
224C22×D4—21no—
225C22×D4C22⋊D5911yescharacteristic
226C23×C4C22⋊D5911yescharacteristic
227C2×SD16—2361no—
228C22⋊Q8—1181no—
229C22⋊Q8—1181no—
230C22⋊Q8—2361no—
231C22⋊Q8—2361no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C22⋊D5911yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—1181no—
242C2×C22⋊C4—1181no—
243C2×C22⋊C4C22⋊D5911yescharacteristic
244C2×C22⋊C4C22×D5911yescharacteristic
245C2×C42—21no—
246#107—2361no—
247#103—2361no—
248#102—2361no—
249C22⋊C8—1181no—
250C22⋊C8—1181no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—2361no—
256#95—2361no—
257#95C22⋊D5911yescharacteristic
258#95C22⋊D5911yescharacteristic
259C24—21no—
260C24C2×C22⋊D5911yescharacteristic
261C2×Q8—1181no—
262C2×Q8—1181no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—81no—
268C2×D4—81no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—1181no—
279C22×C4—1181no—
280C22×C4—2361no—
281C22×C4C2×C22⋊D5911yescharacteristic
282C22×C4C2×C22⋊D5911yescharacteristic
283SD16—4722no—
284C2×C8—2361no—
285C2×C8—2361no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—81no—
289C4⋊C4—1181no—
290C4⋊C4—1181no—
291C4⋊C4—2361no—
292C4⋊C4—2361no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C22⋊C4—81no—
301C22⋊C4—2361no—
302C22⋊C4—2361no—
303C22⋊C4—2361no—
304C22⋊C4—2361no—
305C42—81no—
306C23—21no—
307C23—21no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23#1427369611yescharacteristic
318C23#1427374711yescharacteristic
319C23#1427375711yescharacteristic
320Q8—4721no—
321Q8—4721no—
322D4—42no—
323D4—82no—
324D4—82no—
325D4—81no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—4721no—
348C2×C4—4721no—
349C2×C4—4721no—
350C2×C4—4721no—
351C2×C4—4721no—
352C2×C4—2361no—
353C2×C4—2361no—
354C8—4721no—
355C8—4721no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#1683756411yescharacteristic
373C22#1683758011yescharacteristic
374C22#1683758911yescharacteristic
375C4—42no—
376C4—41no—
377C4—41no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—9441no—
383C4—4721no—
384C4—4721no—
385C2—21no—
386C2—21no—
387C2—41no—
388C2—41no—
389C2—161no—
390C2#2142307611yescharacteristic
391C2#2142609211yescharacteristic
392C2#2142610311yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C22⋊C4×C59C23
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2752C2
socleC22×C59B(4,𝔽2)
prime radicalC59×#392C22
hypercenter#2752D59
hypocenterC59#12267
LCSG ▷ C2×C22⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2752

Complements

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

NH ≅ G/Ntype
1. #21425263C2semidirect
2. #21423120C2semidirect
4. #21422120C2semidirect
6. #21422026C2semidirect
223. C59#12267semidirect

Maximal subgroups

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

subgroupindex
#214252632
#214231202
#214230182
#214221202
#214220932
#214220262
C59×#27522
#1226759