Finite Group Databasebeta

← #28498976

Subgroups — #28498976

Order 15104.

448 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 83 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#21429916C211yesmaximal, characteristic
2#21423827C211yesmaximal, characteristic
3#21422407C211yesmaximal, characteristic
4#21422118C211yesmaximal, characteristic
5#21422117C211yesmaximal, characteristic
6#21422116C211yesmaximal, characteristic
7C59×#1351C211yesmaximal, characteristic
8C2×#14273725C411yescharacteristic, complemented
9C2×#14273683—21no—
10C2×#14273683C2211yescharacteristic
11#16837452—21no—
12#16837343—21no—
13#16837338—21no—
14#16837323C411yescharacteristic, complemented
15#16837321C411yescharacteristic, complemented
16#16837314C2211yescharacteristic
17#16837299—21no—
18#16837266C411yescharacteristic, complemented
19#16837260—21no—
20#16837169C2211yescharacteristic
21#16837168—21no—
22#16837127—21no—
23#16837126C2211yescharacteristic
24C59×#393C2211yescharacteristic
25C59×#342C2211yescharacteristic
26C59×#323C2211yescharacteristic
27C22×Dic118C2×C411yescharacteristic
28C23×Dic59D411yescharacteristic
29#14273725—41no—
30#14273725—41no—
31#14273683—24no—
32#14273683—41no—
33#14273683—41no—
34#14273683—42no—
35C2×C4⋊Dic59—21no—
36C2×Dic59⋊C4—21no—
37C2×Dic59⋊C4—21no—
38C2×Dic59⋊C4—21no—
39C2×Dic59⋊C4—21no—
40C2×Dic59⋊C4—41no—
41C2×Dic59⋊C4C2×C411yescharacteristic
42C2×Dic59⋊C4D411yescharacteristic
43C2×C22⋊Dic59—21no—
44C2×C22⋊Dic59—21no—
45C2×C22⋊Dic59D411yescharacteristic
46C2×C4×Dic59—41no—
47C2×C4×Dic59C2×C411yescharacteristic
48C2×C4×Dic59D411yescharacteristic
49Dic59⋊C8—41no—
50C22⋊(C59⋊C8)—21no—
51C8×Dic59—41no—
52C59⋊#95—21no—
53C59⋊#95—21no—
54C59⋊#95—21no—
55C59⋊#95—21no—
56C59⋊#95C2×C411yescharacteristic
57C22×Q8×C59C2×C411yescharacteristic
58C2×C22⋊C4×C59—21no—
59C2×C22⋊C4×C59—21no—
60C2×C22⋊C4×C59C2311yescharacteristic
61C22⋊C8×C59—21no—
62#95×C59—21no—
63#95×C59C2×C411yescharacteristic
64C24×C59C2×D411yescharacteristic
65C2×Q8×C59—41no—
66C2×Q8×C59—81no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—41no—
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×Dic59—21no—
73C22×Dic59—21no—
74C22×Dic59—22no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—41no—
78C22×Dic59C22⋊C411yescharacteristic
79C22×Dic59C22⋊C411yescharacteristic
80C22×Dic59C22⋊C411yescharacteristic
81C22×Dic59C22⋊C411yescharacteristic
82C22×Dic59SD1612yes—
83C22×Dic59Q1612yes—
84C2×Dic118—41no—
85C2×Dic118—81no—
86C2×C8×C59—41no—
87C22⋊C4×C59—22no—
88C22⋊C4×C59—41no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—81no—
91C22⋊Dic59—22no—
92C22⋊Dic59—41no—
93C22⋊Dic59—41no—
94C22⋊Dic59—81no—
95C4⋊Dic59—81no—
96Dic59⋊C4—41no—
97Dic59⋊C4—41no—
98Dic59⋊C4—42no—
99Dic59⋊C4—44no—
100Dic59⋊C4—41no—
101Dic59⋊C4—41no—
102Dic59⋊C4—42no—
103Dic59⋊C4—81no—
104C4×Dic59—21no—
105C4×Dic59—21no—
106C4×Dic59—21no—
107C4×Dic59—21no—
108C4×Dic59—44no—
109C2×C59⋊C8—41no—
110C23×C59—21no—
111C23×C59—21no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59C2×C22⋊C411yescharacteristic
115C23×C59C2×SD1611yescharacteristic
116C23×C59C2×Q1611yescharacteristic
117Q8×C59—82no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—42no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—81no—
130C2×C4×C59—81no—
131C2×Dic59—21no—
132C2×Dic59—21no—
133C2×Dic59—22no—
134C2×Dic59—22no—
135C2×Dic59—22no—
136C2×Dic59—22no—
137C2×Dic59—21no—
138C2×Dic59—21no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—42no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—81no—
149C2×Dic59—81no—
150C2×Dic59#9911yescharacteristic
151C2×Dic59#9911yescharacteristic
152C2×Dic59#10312yes—
153C2×Dic59#10312yes—
154C2×Dic59#10412yes—
155Dic118—82no—
156C8×C59—81no—
157C59⋊C8—81no—
158#11717—591nomaximal, Sylow
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—81no—
168C22×C59C2×#9911yescharacteristic
169C22×C59C2×#10311yescharacteristic
170C22×C59C2×#10411yescharacteristic
171C4×C59—41no—
172C4×C59—41no—
173C4×C59—81no—
174C4×C59—82no—
175C4×C59—81no—
176Dic59—22no—
177Dic59—41no—
178Dic59—41no—
179Dic59—81no—
180Dic59—82no—
181Dic59—81no—
182Dic59#32812yes—
183Dic59#32812yes—
184#2628—591no—
185#1740—591no—
186#1458—591no—
187#1351—591no—
188#1351—591no—
189#1351—591no—
190#1351D5911yescharacteristic
191C2×C59—21no—
192C2×C59—21no—
193C2×C59—41no—
194C2×C59—41no—
195C2×C59C2×#32811yescharacteristic
196C2×C59#212211yescharacteristic
197C2×C59#212911yescharacteristic
198C2×C4⋊Q8—591no—
199C2×#117—591no—
200C2×#117—1181no—
201#431—1181no—
202#400—1181no—
203#399—1181no—
204#395—591no—
205#395—591no—
206#393—591no—
207#393C2×D5911yescharacteristic
208#389—1181no—
209#382—591no—
210#380—1181no—
211#342—591no—
212#342—1181no—
213#342C2×D5911yescharacteristic
214#324—1181no—
215#323—591no—
216#323C2×D5911yescharacteristic
217C59#1171711yescharacteristic, complemented, Sylow
218C22×Q8—591no—
219C22×Q8C4×D5911yescharacteristic
220C23×C4—591no—
221C4⋊Q8—2361no—
222C4⋊Q8—2361no—
223#117—1184no—
224#117—2361no—
225#117—2361no—
226#117—2362no—
227C2×C4⋊C4—591no—
228C2×C4⋊C4—591no—
229C2×C4⋊C4—1181no—
230C2×C4⋊C4—1181no—
231C2×C4⋊C4—1181no—
232C2×C4⋊C4—1181no—
233C2×C4⋊C4—1181no—
234C2×C4⋊C4—2361no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—591no—
238C2×C22⋊C4—1181no—
239C2×C22⋊C4—1181no—
240C2×C22⋊C4C22×D5911yescharacteristic
241C2×C42—591no—
242C2×C42—591no—
243C2×C42—2361no—
244C4⋊C8—2361no—
245C22⋊C8—21no—
246C22⋊C8—1181no—
247C4×C8—2361no—
248#95—21no—
249#95—591no—
250#95—1181no—
251#95—1181no—
252#95—1181no—
253#95—1181no—
254#95C4×D5911yescharacteristic
255C24D4×D5911yescharacteristic
256C2×Q8—41no—
257C2×Q8—81no—
258C2×Q8—4721no—
259C2×Q8—2361no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—591no—
264C22×C4—591no—
265C22×C4—592no—
266C22×C4—592no—
267C22×C4—591no—
268C22×C4—591no—
269C22×C4—1181no—
270C22×C4—1181no—
271C22×C4—1182no—
272C22×C4—1181no—
273C22×C4—1181no—
274C22×C4—2361no—
275C22×C4C2×C4×D5911yescharacteristic
276C22×C4D4×D5911yescharacteristic
277C2×C8—41no—
278C2×C8—2361no—
279C4⋊C4—4721no—
280C4⋊C4—4721no—
281C4⋊C4—2361no—
282C4⋊C4—2361no—
283C4⋊C4—2361no—
284C4⋊C4—2361no—
285C4⋊C4—2362no—
286C4⋊C4—2362no—
287C4⋊C4—2364no—
288C22⋊C4—22no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—81no—
292C22⋊C4—1182no—
293C22⋊C4—4721no—
294C22⋊C4—2361no—
295C22⋊C4—2361no—
296C42—1181no—
297C42—1181no—
298C42—1181no—
299C42—1181no—
300C42—2364no—
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23C22⋊C4×D5911yescharacteristic
306C23SD16×D5911yescharacteristic
307C23Q16×D5911yescharacteristic
308Q8—82no—
309Q8—4722no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—592no—
324C2×C4—592no—
325C2×C4—591no—
326C2×C4—591no—
327C2×C4—592no—
328C2×C4—1182no—
329C2×C4—1182no—
330C2×C4—1182no—
331C2×C4—1182no—
332C2×C4—1181no—
333C2×C4—1181no—
334C2×C4—1181no—
335C2×C4—1181no—
336C2×C4—4721no—
337C2×C4—4721no—
338C2×C4—2362no—
339C2×C4—2361no—
340C2×C4—2361no—
341C2×C4—2361no—
342C2×C4—2361no—
343C2×C4—2361no—
344C2×C4—2361no—
345C2×C4—2361no—
346C2×C4—2361no—
347C8—81no—
348C8—4721no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—81no—
358C22#99×D5911yescharacteristic
359C22#103×D5911yescharacteristic
360C22#104×D5911yescharacteristic
361C4—41no—
362C4—41no—
363C4—81no—
364C4—82no—
365C4—81no—
366C4—592no—
367C4—592no—
368C4—1182no—
369C4—4721no—
370C4—4722no—
371C4—4721no—
372C4—2361no—
373C4—2361no—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#328×D5911yescharacteristic
379C2#2142574411yescharacteristic
380C2#2142576911yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1351C2
socleC22×C59C2×#99
prime radicalC24×C59C2×D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1351D59
hypocenterC59#11717
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1351

Complements

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

NH ≅ G/Ntype
8. C2×#14273725C4semidirect
14. #16837323C4semidirect
15. #16837321C4semidirect
18. #16837266C4semidirect
217. C59#11717semidirect

Maximal subgroups

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

subgroupindex
#214299162
#214238272
#214224072
#214221182
#214221172
#214221162
C59×#13512
#1171759