Finite Group Databasebeta

← #28486014

Subgroups — #28486014

Order 15104.

548 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 115 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#21429272C211yesmaximal, characteristic
2#21424084C211yesmaximal, characteristic
3#21422669C212yesmaximal, complemented
4#21421976C212yesmaximal, complemented
5C59×#2973C211yesmaximal, characteristic
6C2×#14273706C2211yescharacteristic
7C2×#14273642—21no—
8#16837341C411yescharacteristic, complemented
9#16837338C411yescharacteristic, complemented
10#16837322C411yescharacteristic, complemented
11#16837314C411yescharacteristic, complemented
12#16837286—21no—
13#16837282—21no—
14#16837275—21no—
15#16837262C2212yes—
16#16837260—21no—
17C2×C59⋊#95C2211yescharacteristic
18#16837169—21no—
19#16837140—24no—
20C2×C4×D4×C59C2211yescharacteristic
21C59×#435—24no—
22C59×#432—21no—
23C59×#407—21no—
24C2×C22⋊C8×C59C2212yes—
25C22×Dic118C2×C411yescharacteristic
26C23×Dic59—21no—
27#14273706—22no—
28#14273706—22no—
29C2×C4⋊Dic59—22no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—22no—
37C2×C22⋊Dic59—21no—
38#14273642—42no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—22no—
45C59⋊#95C2×C411yescharacteristic
46C59⋊#95C2×C411yescharacteristic
47C22×D4×C59D411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C23×C4×C59C2311yescharacteristic
50C2×M16×C59—22no—
51C22×C8×C59—22no—
52C4×D4×C59—22no—
53C4×D4×C59—22no—
54C2×C4⋊C4×C59C2×C411yescharacteristic
55C2×C22⋊C4×C59C2×C411yescharacteristic
56C2×C22⋊C4×C59D411yescharacteristic
57C2×C42×C59D411yescharacteristic
58C4⋊C8×C59—22no—
59C22⋊C8×C59—22no—
60C22⋊C8×C59—22no—
61C22⋊C8×C59D414yes—
62#97×C59—22no—
63C24×C59C22⋊C411yescharacteristic
64C24×C59C2×D411yescharacteristic
65C2×D4×C59—22no—
66C2×D4×C59—21no—
67C2×D4×C59—21no—
68C2×D4×C59—22no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—81no—
72C22×C4×C59—81no—
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22×C411yescharacteristic
77C22×C4×C59C2×D412yes—
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59#4112yes—
80C22×Dic59—21no—
81C22×Dic59—21no—
82C22×Dic59—21no—
83C22×Dic59—21no—
84C22×Dic59—42no—
85C22×Dic59—41no—
86C22×Dic59—41no—
87C22×Dic59—42no—
88C22×Dic59—81no—
89C2×Dic118—22no—
90C2×Dic118—81no—
91M16×C59—82no—
92C2×C8×C59—24no—
93C2×C8×C59—82no—
94C4⋊C4×C59—22no—
95C22⋊C4×C59—22no—
96C22⋊C4×C59—22no—
97C42×C59—21no—
98C42×C59—21no—
99C22⋊Dic59—42no—
100C22⋊Dic59—42no—
101C22⋊Dic59—42no—
102C22⋊Dic59—42no—
103C22⋊Dic59—81no—
104C4⋊Dic59—22no—
105C4⋊Dic59—82no—
106Dic59⋊C4—42no—
107Dic59⋊C4—41no—
108Dic59⋊C4—41no—
109Dic59⋊C4—81no—
110C23×C59—21no—
111C23×C59—21no—
112C23×C59—21no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59C2×C22⋊C411yescharacteristic
118C23×C59#12112yes—
119C23×C59AGL(1,ℤ/8)11yescharacteristic
120C23×C59#13711yescharacteristic
121C23×C59#13712yes—
122D4×C59—42no—
123D4×C59—42no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—42no—
130C2×C4×C59—42no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—42no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—81no—
137C2×C4×C59C4×D412yes—
138C2×C4×C59#12011yescharacteristic
139C2×C4×C59#12311yescharacteristic
140C2×Dic59—22no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—82no—
153C2×Dic59—82no—
154C2×Dic59—82no—
155C2×Dic59—82no—
156C2×Dic59—82no—
157Dic118—82no—
158C8×C59—44no—
159#9385—591nomaximal, Sylow
160C22×C59—21no—
161C22×C59—22no—
162C22×C59—21no—
163C22×C59—22no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—42no—
167C22×C59—42no—
168C22×C59—41no—
169C22×C59—41no—
170C22×C59—41no—
171C22×C59—81no—
172C22×C59#38611yescharacteristic
173C22×C59#41711yescharacteristic
174C22×C59#41911yescharacteristic
175C22×C59#46812yes—
176C22×C59#47012yes—
177C4×C59—21no—
178C4×C59—22no—
179C4×C59—21no—
180C4×C59—41no—
181C4×C59—41no—
182C4×C59—41no—
183Dic59—42no—
184Dic59—81no—
185Dic59—81no—
186Dic59—81no—
187Dic59—162no—
188#2973D5911yescharacteristic
189#2564—591no—
190#1799—591no—
191#1520—592no—
192#1294—592no—
193C2×C59—21no—
194C2×C59—22no—
195C2×C59—21no—
196C2×C59—22no—
197C2×C59—81no—
198C2×C59#188411yescharacteristic
199C2×C59#189411yescharacteristic
200C2×C59#191311yescharacteristic
201C2×C22⋊Q8—591no—
202C2×C4×D4C2×D5911yescharacteristic
203#435—24no—
204#432—21no—
205C2×#99—1181no—
206#407—21no—
207C2×C22⋊C8C2×D5912yes—
208#400—591no—
209#399—591no—
210#395—591no—
211#393—591no—
212#387—1181no—
213#386—1181no—
214#385—1181no—
215#380—592no—
216#380—1181no—
217C2×#95—591no—
218#342—1181no—
219#328—1184no—
220C59#938511yescharacteristic, complemented, Sylow
221C22×Q8—591no—
222C22×D4D23611yescharacteristic
223C23×C4—1181no—
224C23×C4C22⋊D5911yescharacteristic
225C23×C4C22×D5911yescharacteristic
226C2×M16—22no—
227C22×C8—22no—
228C22⋊Q8—1182no—
229C22⋊Q8—1182no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—591no—
233C2×C4⋊C4—1181no—
234C2×C4⋊C4—1182no—
235C2×C4⋊C4—1181no—
236C2×C4⋊C4—1181no—
237C2×C4⋊C4C4×D5911yescharacteristic
238C2×C22⋊C4—1181no—
239C2×C22⋊C4—1181no—
240C2×C22⋊C4—1181no—
241C2×C22⋊C4—1182no—
242C2×C22⋊C4C4×D5911yescharacteristic
243C2×C22⋊C4D23611yescharacteristic
244C2×C42C22⋊D5911yescharacteristic
245C4⋊C8—22no—
246#99—2362no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8C22⋊D5914yes—
250#97—22no—
251#95—591no—
252#95—591no—
253#95—1181no—
254#95—1181no—
255#95—1181no—
256#95—1181no—
257#95—1181no—
258#95—1182no—
259C24(C2×D59)⋊C411yescharacteristic
260C24C2×D23611yescharacteristic
261C2×Q8—1182no—
262C2×Q8—4721no—
263C2×D4—22no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—1181no—
272C22×C4—1181no—
273C22×C4—1181no—
274C22×C4—1181no—
275C22×C4—4721no—
276C22×C4—2362no—
277C22×C4—2362no—
278C22×C4—2361no—
279C22×C4—2361no—
280C22×C4(C2×D59)⋊C411yescharacteristic
281C22×C4(C2×D59)⋊C411yescharacteristic
282C22×C4(C2×D59)⋊C411yescharacteristic
283C22×C4C2×C4×D5911yescharacteristic
284C22×C4#1174568512yes—
285C22×C4C2×C22⋊D5912yes—
286C22×C4C2×C22⋊D5911yescharacteristic
287M16—82no—
288C2×C8—24no—
289C2×C8—82no—
290C4⋊C4—22no—
291C4⋊C4—1182no—
292C4⋊C4—4722no—
293C4⋊C4—4721no—
294C4⋊C4—2362no—
295C4⋊C4—2361no—
296C4⋊C4—2361no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—4721no—
300C22⋊C4—2362no—
301C22⋊C4—2362no—
302C22⋊C4—2362no—
303C22⋊C4—2362no—
304C42—21no—
305C42—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23C2×(C2×D59)⋊C411yescharacteristic
314C23#1427370112yes—
315C23#1427375511yescharacteristic
316C23#1427376012yes—
317C23#1427376011yescharacteristic
318Q8—4722no—
319D4—42no—
320D4—42no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—1182no—
335C2×C4—4722no—
336C2×C4—4722no—
337C2×C4—4722no—
338C2×C4—4722no—
339C2×C4—4722no—
340C2×C4—2361no—
341C2×C4—2361no—
342C2×C4—2361no—
343C2×C4—2361no—
344C2×C4—2361no—
345C2×C4—2361no—
346C2×C4—2361no—
347C2×C4—2361no—
348C2×C4—2361no—
349C2×C4—2361no—
350C2×C4—2361no—
351C2×C4C4×C22⋊D5912yes—
352C2×C4#1427369611yescharacteristic
353C2×C4#1427371011yescharacteristic
354C8—44no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—81no—
367C22#1683728111yescharacteristic
368C22#1683740411yescharacteristic
369C22#1683741211yescharacteristic
370C22#1683764812yes—
371C22#1683766012yes—
372C4—21no—
373C4—21no—
374C4—22no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—9442no—
379C4—4721no—
380C4—4721no—
381C4—4721no—
382C4—2362no—
383C2—22no—
384C2—21no—
385C2—21no—
386C2—22no—
387C2—81no—
388C2#2142447911yescharacteristic
389C2#2142452911yescharacteristic
390C2#2142463811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2973C2
socleC22×C59#386
prime radicalC22×D4×C59D4
cosocleC23×C4×C59C23
hypercenter#2973D59
hypocenterC59#9385
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
3. #21422669C2semidirect
4. #21421976C2semidirect
8. #16837341C4semidirect
9. #16837338C4semidirect
10. #16837322C4semidirect
11. #16837314C4semidirect
220. C59#9385semidirect

Maximal subgroups

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

subgroupindex
#214292722
#214240842
#214226692
#214219762
C59×#29732
#938559