Finite Group Databasebeta

← #28744118

Subgroups — #28744118

Order 15104.

918 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 179 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
1C2×#16837839C214yesmaximal, complemented
2#21434525C211yesmaximal, characteristic, complemented
3#493×D59C211yesmaximal, characteristic, complemented
4#21426361C214yesmaximal, complemented
5#21426353C211yesmaximal, characteristic, complemented
6#21426346C211yesmaximal, characteristic, complemented
7#21426345C211yesmaximal, characteristic, complemented
8C4×#14273666C211yesmaximal, characteristic, complemented
9C59×#2248C211yesmaximal, characteristic, complemented
10C2×#14273750—21no—
11C2×#14273750C2212yes—
12C2×D8×D59—21no—
13C2×D8×D59C2212yes—
14#16837953—21no—
15#127×D59—21no—
16#16837839—216no—
17C2×#14273670C2214yescomplemented
18C2×D944C2214yescomplemented
19C2×#14273668C2214yescomplemented
20C2×#14273666C2212yes—
21#16837787—21no—
22#16837787C2212yescomplemented
23#16837779C2211yescharacteristic
24#16837779C2211yescharacteristic, complemented
25#16837778—21no—
26#16837777—21no—
27#16837777C2212yes—
28#16837776—21no—
29C4×C8×D59C2211yescharacteristic
30#16837177C2212yescomplemented
31C4×C59⋊C16C2211yescharacteristic, complemented
32C2×SD32×C59C2214yescomplemented
33C59×#494C2211yescharacteristic, complemented
34C59×#493C2211yescharacteristic
35C4×C16×C59C2211yescharacteristic, complemented
36C2×#11745689—22no—
37C2×D4×D59—22no—
38C2×D4×D59—42no—
39#14273750—28no—
40#14273750—44no—
41C2×#11745671—22no—
42C2×#11745671—22no—
43C2×D472—21no—
44C2×D472—21no—
45C2×D472C2312yes—
46C2×D472C2312yes—
47C2×#11745669—22no—
48C2×#11745669—22no—
49D8×D59—28no—
50D8×D59—44no—
51C2×C8×D59D411yescharacteristic
52C2×C8×D59C2311yescharacteristic
53#14273724—21no—
54#14273724—21no—
55#14273723—22no—
56#14273723—42no—
57#14273716—22no—
58#14273702—42no—
59#14273689—42no—
60C42×D59D411yescharacteristic
61#14273670—28no—
62D944—28no—
63#14273668—28no—
64#14273666D418yescomplemented
65C2×C59⋊C16C2312yes—
66C4×C59⋊C8D411yescharacteristic
67C8×Dic59D411yescharacteristic
68C8×Dic59C2311yescharacteristic
69C2×Q16×C59—21no—
70C2×Q16×C59C2312yes—
71C2×D8×C59—21no—
72C2×D8×C59C2312yes—
73C4⋊Q8×C59—21no—
74#127×C59—21no—
75SD32×C59—28no—
76C2×C16×C59C2312yes—
77C4×C8×C59C2311yescharacteristic
78C23×D59—42no—
79C2×Q8×C59—22no—
80C2×D4×C59—22no—
81C2×D4×C59—42no—
82C2×C22⋊D59—44no—
83C2×C22⋊D59—42no—
84#11745689—44no—
85#11745689—44no—
86D4×D59—44no—
87D4×D59—44no—
88D4×D59—88no—
89C2×D236—22no—
90C2×D236—22no—
91C2×D236—42no—
92C2×D236—42no—
93C2×D236—42no—
94C2×C4×D59—42no—
95C2×C4×D59D812yes—
96C2×C4×D59C2×D411yescharacteristic
97Q16×C59—24no—
98Q16×C59—42no—
99D8×C59—24no—
100D8×C59—42no—
101C2×C8×C59C2×D411yescharacteristic
102C2×C8×C59C2411yescharacteristic
103C4⋊C4×C59—42no—
104C42×C59C2×D411yescharacteristic
105#11745671—44no—
106#11745671—44no—
107#11745669—44no—
108#11745669—44no—
109(C2×D59)⋊C4—44no—
110C4×Dic59D812yes—
111C4×Dic59C2×D411yescharacteristic
112C2×C59⋊C8C2×D411yescharacteristic
113C2×C59⋊C8C2×D411yescharacteristic
114D472—24no—
115D472—24no—
116D472—42no—
117D472—42no—
118C8×D59D814yes—
119C8×D59C2×D414yes—
120C16×C59C2×D414yescomplemented
121C59⋊C16C2×D414yescomplemented
122C23×C59—42no—
123C22×D59—42no—
124C22×D59—42no—
125C22×D59—88no—
126C22×D59—84no—
127C22×D59C2×D811yescharacteristic
128Q8×C59—42no—
129Q8×C59—42no—
130D4×C59—42no—
131D4×C59—42no—
132D4×C59—84no—
133C2×C4×C59—42no—
134C2×C4×C59C2×D812yes—
135C2×C4×C59C22×D411yescharacteristic
136C22⋊D59—88no—
137C22⋊D59—84no—
138C2×Dic59C2×D812yes—
139C2×Dic59C2×D811yescharacteristic
140D236—42no—
141D236—42no—
142D236—42no—
143D236—42no—
144D236—84no—
145D236—84no—
146D236—84no—
147C4×D59—24no—
148C4×D59—84no—
149C4×D59#12712yes—
150C4×D59C2×D812yes—
151C8×C59C2×D812yes—
152C8×C59C22×D412yes—
153C59⋊C8#12712yes—
154C59⋊C8C2×D812yes—
155#33717—591nomaximal, Sylow
156C22×C59—82no—
157C22×C59—84no—
158C22×C59C22×D811yescharacteristic
159C2×D59—22no—
160C2×D59—82no—
161C2×D59—82no—
162C2×D59—84no—
163C2×D59—84no—
164C2×D59—168no—
165C2×D59#49312yes—
166C4×C59—82no—
167C4×C59#50914yescomplemented
168C4×C59C2×#12711yescharacteristic
169C4×C59C22×D811yescharacteristic
170Dic59—22no—
171Dic59#49312yes—
172C2×#509—594no—
173#3147—591no—
174C2×#493—591no—
175#2250—594no—
176#2248—591no—
177#2248D5911yescharacteristic, complemented
178#2246—591no—
179#2246—591no—
180C4×M32—591no—
181C2×C59—162no—
182C2×C59C2×#49311yescharacteristic
183C2×C59C2×#50912yes—
184D59—22no—
185D59—162no—
186D59—162no—
187C2×#135—592no—
188C2×#135—1181no—
189C22×D8—592no—
190C22×D8—1181no—
191#532—1181no—
192C2×#127—1181no—
193#509—11816no—
194C2×SD32—594no—
195C2×SD32C2×D5914yescomplemented
196C2×D16—594no—
197C2×D16—594no—
198C2×M32—592no—
199#495—592no—
200#495—1181no—
201#494C2×D5911yescharacteristic, complemented
202#493—592no—
203#493—591no—
204#493—591no—
205#493—1181no—
206#493—1181no—
207#493C2×D5911yescharacteristic
208#492—1181no—
209C2×C4×C8—591no—
210#346—592no—
211C4×C16—591no—
212C4×C16C2×D5911yescharacteristic, complemented
213C59#3371711yescharacteristic, complemented, Sylow
214C2×#41—1182no—
215C22×D4—1182no—
216C22×D4—2362no—
217#135—1188no—
218#135—2364no—
219C2×Q16—21no—
220C2×Q16C22×D5912yes—
221C2×SD16—1182no—
222C2×SD16—1182no—
223C2×D8—21no—
224C2×D8—592no—
225C2×D8—592no—
226C2×D8—1188no—
227C2×D8—1182no—
228C2×D8—1182no—
229C2×D8—1181no—
230C2×D8—1181no—
231C2×D8—2364no—
232C2×D8C22×D5912yes—
233C22×C8—591no—
234C22×C8—591no—
235C4⋊Q8—21no—
236#127—21no—
237#127—1182no—
238#127—1181no—
239#127—1181no—
240#127—2362no—
241#124—1182no—
242#121—2362no—
243C4×D4—2362no—
244C2×C42—591no—
245SD32—28no—
246SD32—1188no—
247D16—1188no—
248D16—1188no—
249M32—598no—
250C2×C16—592no—
251C2×C16C22×D5912yes—
252C4×C8—591no—
253C4×C8—591no—
254C4×C8—591no—
255C4×C8C22×D5911yescharacteristic
256C24—2362no—
257#41—2364no—
258#41—2364no—
259C2×Q8—22no—
260C2×D4—22no—
261C2×D4—42no—
262C2×D4—1182no—
263C2×D4—1182no—
264C2×D4—4728no—
265C2×D4—2362no—
266C2×D4—2364no—
267C2×D4—2364no—
268C2×D4—2364no—
269C2×D4—2362no—
270C2×D4—2362no—
271C2×D4—2362no—
272C22×C4—591no—
273C22×C4—592no—
274C22×C4—2362no—
275Q16—24no—
276Q16—42no—
277SD16—2364no—
278SD16—2364no—
279D8—24no—
280D8—42no—
281D8—1184no—
282D8—1184no—
283D8—2364no—
284D8—2364no—
285D8—2362no—
286D8—2362no—
287C2×C8—591no—
288C2×C8—591no—
289C2×C8—594no—
290C2×C8—594no—
291C2×C8D4×D5911yescharacteristic
292C2×C8C23×D5911yescharacteristic
293C4⋊C4—42no—
294C22⋊C4—2364no—
295C42—592no—
296C42—591no—
297C42D4×D5911yescharacteristic
298C16—594no—
299C16D4×D5914yescomplemented
300C23—42no—
301C23—591no—
302C23—4724no—
303C23—4728no—
304C23—2362no—
305C23—2362no—
306Q8—42no—
307Q8—42no—
308D4—42no—
309D4—42no—
310D4—84no—
311D4—4724no—
312D4—4724no—
313D4—4728no—
314D4—4724no—
315D4—4724no—
316D4—2362no—
317D4—2362no—
318D4—2362no—
319D4—2362no—
320C2×C4—42no—
321C2×C4—591no—
322C2×C4—592no—
323C2×C4—592no—
324C2×C4—592no—
325C2×C4—1184no—
326C2×C4—4724no—
327C2×C4D8×D5912yes—
328C2×C4C2×D4×D5911yescharacteristic
329C8—592no—
330C8—592no—
331C8D8×D5912yes—
332C8C2×D4×D5912yes—
333C22—84no—
334C22—82no—
335C22—592no—
336C22—1182no—
337C22—9448no—
338C22—4724no—
339C22—4724no—
340C22—4722no—
341C22—4722no—
342C22C2×D8×D5911yescharacteristic
343C4—82no—
344C4—592no—
345C4—1182no—
346C4#1683783914yescomplemented
347C4#127×D5911yescharacteristic
348C4C2×D8×D5911yescharacteristic
349C2—162no—
350C2—1182no—
351C2—9442no—
352C2—9442no—
353C2#493×D5911yescharacteristic
354C2C2×#1683783912yes—
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2248C2
socleC22×C59C22×D8
hypercenter#2248D59
hypocenterC59#33717
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
1. C2×#16837839C2semidirect
2. #21434525C2semidirect
3. #493×D59C2semidirect
4. #21426361C2semidirect
5. #21426353C2semidirect
6. #21426346C2semidirect
7. #21426345C2semidirect
8. C4×#14273666C2semidirect
9. C59×#2248C2semidirect
17. C2×#14273670C22semidirect
18. C2×D944C22semidirect
19. C2×#14273668C22semidirect
22. #16837787C22semidirect
24. #16837779C22semidirect
30. #16837177C22semidirect
31. C4×C59⋊C16C22semidirect
32. C2×SD32×C59C22semidirect
33. C59×#494C22semidirect
35. C4×C16×C59C22semidirect
64. #14273666D4semidirect
120. C16×C59C2×D4semidirect
121. C59⋊C16C2×D4semidirect
167. C4×C59#509semidirect
177. #2248D59semidirect
195. C2×SD32C2×D59semidirect
201. #494C2×D59semidirect
212. C4×C16C2×D59semidirect
213. C59#33717semidirect
299. C16D4×D59semidirect
346. C4#16837839semidirect

Maximal subgroups

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

subgroupindex
C2×#168378392
#214345252
#493×D592
#214263612
#214263532
#214263462
#214263452
C4×#142736662
C59×#22482
#3371759