Finite Group Databasebeta

← #28507359

Subgroups — #28507359

Order 15104.

556 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 107 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#21429049C211yesmaximal, characteristic
2#21424084C211yesmaximal, characteristic
3#21422220C214yesmaximal
4C4×#99×C59C211yesmaximal, characteristic
5C2×#14273727C411yescharacteristic, complemented
6C2×#14273706—21no—
7C2×#14273706—21no—
8C2×#14273642C2212yes—
9#16837338—21no—
10#16837338—21no—
11#16837305C2212yes—
12#16837271C412yescomplemented
13#16837262—21no—
14C4×C4⋊Dic59C411yescharacteristic, complemented
15#16837169—21no—
16C2×C4×D4×C59C2211yescharacteristic
17C2×#99×C59C2212yes—
18C4×C22⋊C4×C59—21no—
19C59×#342—21no—
20C22×Dic118—21no—
21#14273727—41no—
22#14273727—41no—
23#14273706—44no—
24#14273706—44no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59—41no—
28C2×C4⋊Dic59C2×C412yes—
29C2×Dic59⋊C4—22no—
30C2×Dic59⋊C4—22no—
31C2×C22⋊Dic59—22no—
32C2×C22⋊Dic59—22no—
33C2×C22⋊Dic59—22no—
34C2×C4×Dic59C2×C412yes—
35#14273642—24no—
36C59⋊#95—21no—
37C59⋊#95—41no—
38C22×D4×C59C2311yescharacteristic
39C23×C4×C59D411yescharacteristic
40C23×C4×C59D411yescharacteristic
41C4×D4×C59—41no—
42C4×D4×C59—41no—
43C2×C4⋊C4×C59C2×C411yescharacteristic
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59D411yescharacteristic
46C2×C22⋊C4×C59D411yescharacteristic
47C2×C42×C59—41no—
48C2×C42×C59C2×C411yescharacteristic
49#99×C59—22no—
50#99×C59Q814yescomplemented
51#95×C59—41no—
52C24×C59C2×D411yescharacteristic
53C24×C59C2×D411yescharacteristic
54C2×D4×C59—41no—
55C2×D4×C59—41no—
56C2×D4×C59C2×Q812yes—
57C2×D4×C59#4112yes—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—42no—
63C22×C4×C59—42no—
64C22×C4×C59—42no—
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22×C411yescharacteristic
70C22×Dic59—22no—
71C22×Dic59—22no—
72C22×Dic59—42no—
73C2×Dic118—42no—
74C2×Dic118—42no—
75C2×Dic118—41no—
76C2×Dic118—41no—
77C4⋊C4×C59—41no—
78C22⋊C4×C59—24no—
79C22⋊C4×C59—41no—
80C22⋊C4×C59—41no—
81C22⋊C4×C59—42no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59—82no—
85C22⋊Dic59—44no—
86C22⋊Dic59—44no—
87C22⋊Dic59—42no—
88C22⋊Dic59—42no—
89C4⋊Dic59—24no—
90C4⋊Dic59—42no—
91C4⋊Dic59—42no—
92C4⋊Dic59—82no—
93Dic59⋊C4—44no—
94Dic59⋊C4—44no—
95C4×Dic59—42no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—41no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59C2×C22⋊C411yescharacteristic
105C23×C59C22⋊Q812yes—
106C23×C59#12411yescharacteristic
107C23×C59C4⋊Q811yescharacteristic
108D4×C59—42no—
109D4×C59—42no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—22no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—44no—
122C2×C4×C59—81no—
123C2×C4×C59—81no—
124C2×C4×C59—82no—
125C2×C4×C59—82no—
126C2×C4×C59#9911yescharacteristic
127C2×C4×C59#9911yescharacteristic
128C2×C4×C59C4×Q812yes—
129C2×Dic59—24no—
130C2×Dic59—42no—
131C2×Dic59—42no—
132C2×Dic59—42no—
133C2×Dic59—42no—
134C2×Dic59—42no—
135C2×Dic59—82no—
136C2×Dic59—82no—
137C2×Dic59—82no—
138Dic118—82no—
139Dic118—82no—
140#13298—591nomaximal, Sylow
141C22×C59—21no—
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—22no—
145C22×C59—21no—
146C22×C59—42no—
147C22×C59—41no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—81no—
154C22×C59—81no—
155C22×C59#39111yescharacteristic
156C22×C59C2×#9911yescharacteristic
157C22×C59#41011yescharacteristic
158C4×C59—21no—
159C4×C59—22no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—41no—
163C4×C59—84no—
164C4×C59#35312yes—
165Dic59—44no—
166Dic59—82no—
167Dic59—162no—
168#2542—591no—
169#1799—591no—
170C4×#99D5911yescharacteristic
171#1393—594no—
172C2×C59—21no—
173C2×C59—21no—
174C2×C59—41no—
175C2×C59—42no—
176C2×C59—41no—
177C2×C59#199611yescharacteristic
178C2×C59C2×#35311yescharacteristic
179C2×C59#212911yescharacteristic
180C2×C4⋊Q8—591no—
181C2×C22⋊Q8—1181no—
182C2×C22⋊Q8—1181no—
183C2×C4×D4C2×D5911yescharacteristic
184C2×#99—592no—
185C2×#99C2×D5912yes—
186#399—1181no—
187#399—1181no—
188#390—592no—
189#384—592no—
190#380—1181no—
191C4×C4⋊C4—591no—
192C4×C22⋊C4—21no—
193#342—21no—
194#342—1181no—
195C59#1329811yescharacteristic, complemented, Sylow
196C22×Q8—1181no—
197C22×D4C22×D5911yescharacteristic
198C23×C4D23611yescharacteristic
199C23×C4C22⋊D5911yescharacteristic
200C4⋊Q8—2361no—
201C4⋊Q8—2361no—
202C22⋊Q8—2364no—
203C22⋊Q8—2364no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—592no—
207C2×C4⋊C4—1181no—
208C2×C4⋊C4—1182no—
209C2×C4⋊C4—1182no—
210C2×C4⋊C4—1181no—
211C2×C4⋊C4—2361no—
212C2×C4⋊C4C4×D5911yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4—1182no—
215C2×C22⋊C4—1182no—
216C2×C22⋊C4—1182no—
217C2×C22⋊C4D23611yescharacteristic
218C2×C22⋊C4C22⋊D5911yescharacteristic
219C2×C42—41no—
220C2×C42—592no—
221C2×C42C4×D5911yescharacteristic
222#99—22no—
223#99—1184no—
224#99Dic11814yescomplemented
225#95—41no—
226#95—1181no—
227#95—2361no—
228C24C2×D23611yescharacteristic
229C24C2×C22⋊D5911yescharacteristic
230C2×Q8—2362no—
231C2×Q8—2362no—
232C2×Q8—2361no—
233C2×Q8—2361no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4C2×Dic11812yes—
237C2×D4#1174568512yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—1182no—
246C22×C4—1182no—
247C22×C4—2362no—
248C22×C4(C2×D59)⋊C411yescharacteristic
249C22×C4(C2×D59)⋊C411yescharacteristic
250C22×C4(C2×D59)⋊C411yescharacteristic
251C22×C4(C2×D59)⋊C411yescharacteristic
252C22×C4C2×C4×D5911yescharacteristic
253C4⋊C4—41no—
254C4⋊C4—1184no—
255C4⋊C4—4722no—
256C4⋊C4—2362no—
257C4⋊C4—2362no—
258C4⋊C4—2364no—
259C4⋊C4—2364no—
260C22⋊C4—24no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—42no—
264C22⋊C4—2364no—
265C22⋊C4—2364no—
266C22⋊C4—2362no—
267C22⋊C4—2362no—
268C42—21no—
269C42—21no—
270C42—82no—
271C42—2362no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×(C2×D59)⋊C411yescharacteristic
281C23#1427370612yes—
282C23#1427371511yescharacteristic
283C23#1427372711yescharacteristic
284Q8—4722no—
285Q8—4722no—
286D4—42no—
287D4—42no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—1184no—
305C2×C4—4722no—
306C2×C4—4722no—
307C2×C4—4722no—
308C2×C4—2362no—
309C2×C4—2362no—
310C2×C4—2362no—
311C2×C4—2362no—
312C2×C4—2362no—
313C2×C4#1427364111yescharacteristic
314C2×C4#1427364111yescharacteristic
315C2×C4C4×Dic11812yes—
316C22—21no—
317C22—21no—
318C22—22no—
319C22—21no—
320C22—21no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22—81no—
330C22#1683730811yescharacteristic
331C22C2×#1427364111yescharacteristic
332C22#1683737511yescharacteristic
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—84no—
339C4—9442no—
340C4—4722no—
341C4—2364no—
342C4#1683719412yes—
343C2—21no—
344C2—21no—
345C2—42no—
346C2—41no—
347C2—41no—
348C2#2142508111yescharacteristic
349C2C2×#1683719411yescharacteristic
350C2#2142577011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC4×#99×C59C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenterC4×#99D59
hypocenterC59#13298
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
5. C2×#14273727C4semidirect
12. #16837271C4semidirect
14. C4×C4⋊Dic59C4semidirect
50. #99×C59Q8semidirect
195. C59#13298semidirect
224. #99Dic118semidirect

Maximal subgroups

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

subgroupindex
#214290492
#214240842
#214222202
C4×#99×C592
#1329859