Finite Group Databasebeta

← #28504682

Subgroups — #28504682

Order 15104.

524 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 101 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#21426809C211yesmaximal, characteristic
2C2×#16837197C211yesmaximal, characteristic
3#21424082C212yesmaximal
4#21422220C212yesmaximal
5C2×C59×#354C211yesmaximal, characteristic
6C2×Q8×Dic59C411yescharacteristic, complemented
7C2×D4×Dic59C2211yescharacteristic
8C2×#14273642C2212yes—
9#16837309C411yescharacteristic, complemented
10#16837305C2211yescharacteristic
11#16837295—22no—
12#16837277—22no—
13#16837272C411yescharacteristic, complemented
14#16837260—22no—
15#16837248C411yescharacteristic, complemented
16#16837197—22no—
17#16837168—22no—
18C2×#124×C59C2211yescharacteristic
19C2×#99×C59C2212yes—
20C59×#354—22no—
21C23×Dic59D412yes—
22Q8×Dic59—42no—
23D4×Dic59—42no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59—22no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—42no—
29C2×C22⋊Dic59—22no—
30C2×C22⋊Dic59—22no—
31C2×C22⋊Dic59D412yes—
32C2×C4×Dic59—21no—
33C2×C4×Dic59C2×C411yescharacteristic
34C2×C4×Dic59C2×C411yescharacteristic
35#14273642—22no—
36#14273642—22no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—22no—
40C59⋊#95—42no—
41C22×Q8×C59C2×C411yescharacteristic
42C22×D4×C59C2311yescharacteristic
43#124×C59—22no—
44#124×C59D412yes—
45#124×C59Q812yes—
46C2×C22⋊C4×C59—22no—
47C2×C22⋊C4×C59—22no—
48C2×C42×C59C2×C411yescharacteristic
49#99×C59—22no—
50#99×C59—22no—
51C24×C59C2×D412yes—
52C2×Q8×C59—21no—
53C2×Q8×C59—41no—
54C2×Q8×C59—41no—
55C2×Q8×C59C4⋊C412yescomplemented
56C2×D4×C59—41no—
57C2×D4×C59—41no—
58C2×D4×C59C2×D411yescharacteristic
59C2×D4×C59C2×Q811yescharacteristic
60C2×D4×C59#4112yes—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—42no—
64C22×C4×C59C22×C411yescharacteristic
65C22×Dic59—22no—
66C22×Dic59—22no—
67C22×Dic59—21no—
68C22×Dic59—21no—
69C22×Dic59—42no—
70C22×Dic59—82no—
71C22×Dic59C22⋊C411yescharacteristic
72C22×Dic59C22⋊C412yes—
73C22×Dic59C22⋊C411yescharacteristic
74C22⋊C4×C59—24no—
75C22⋊C4×C59—24no—
76C22⋊C4×C59—42no—
77C22⋊C4×C59—42no—
78C42×C59—21no—
79C42×C59C4⋊C412yescomplemented
80C22⋊Dic59—42no—
81C22⋊Dic59—42no—
82C22⋊Dic59—42no—
83C22⋊Dic59—82no—
84C4⋊Dic59—41no—
85C4⋊Dic59—42no—
86C4⋊Dic59—41no—
87C4⋊Dic59—82no—
88Dic59⋊C4—84no—
89C4×Dic59—22no—
90C4×Dic59—42no—
91C4×Dic59—41no—
92C23×C59—22no—
93C23×C59—22no—
94C23×C59—42no—
95C23×C59—42no—
96C23×C59C2×C22⋊C411yescharacteristic
97C23×C59#12112yes—
98C23×C59C22⋊Q812yes—
99Q8×C59—42no—
100Q8×C59—42no—
101D4×C59—41no—
102D4×C59—41no—
103D4×C59—42no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—22no—
107C2×C4×C59—22no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—82no—
113C2×C4×C59—82no—
114C2×C4×C59—82no—
115C2×C4×C59C2×C4⋊C411yescharacteristic
116C2×C4×C59#11711yescharacteristic
117C2×Dic59—21no—
118C2×Dic59—22no—
119C2×Dic59—41no—
120C2×Dic59—44no—
121C2×Dic59—42no—
122C2×Dic59—41no—
123C2×Dic59—42no—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—82no—
129C2×Dic59—82no—
130C2×Dic59—82no—
131C2×Dic59—82no—
132C2×Dic59#9912yes—
133#12800—591nomaximal, Sylow
134C22×C59—21no—
135C22×C59—21no—
136C22×C59—22no—
137C22×C59—22no—
138C22×C59—42no—
139C22×C59—42no—
140C22×C59—42no—
141C22×C59—42no—
142C22×C59—82no—
143C22×C59#38011yescharacteristic
144C22×C59C2×#9911yescharacteristic
145C22×C59#41011yescharacteristic
146C4×C59—21no—
147C4×C59—21no—
148C4×C59—42no—
149C4×C59—42no—
150C4×C59—162no—
151Dic59—22no—
152Dic59—42no—
153Dic59—41no—
154Dic59—81no—
155Dic59—81no—
156Dic59—162no—
157#2326—591no—
158C2×#354—591no—
159C2×#354D5911yescharacteristic
160#1799—592no—
161#1393—592no—
162C2×C59—21no—
163C2×C59—21no—
164C2×C59—42no—
165C2×C59—42no—
166C2×C59#179911yescharacteristic
167C2×C59#212711yescharacteristic
168C2×C59#212911yescharacteristic
169C2×#124C2×D5911yescharacteristic
170C2×C4×Q8—591no—
171C2×C4×D4—591no—
172C2×#99—592no—
173C2×#99C2×D5912yes—
174#391—591no—
175#390—591no—
176#388—1182no—
177#385—1182no—
178#384—591no—
179#380—1182no—
180#376—591no—
181#354—22no—
182#354—1182no—
183#342—1182no—
184C59#1280011yescharacteristic, complemented, Sylow
185C22×Q8C4×D5911yescharacteristic
186C22×D4C22×D5911yescharacteristic
187C23×C4—592no—
188#124—22no—
189#124Dic11812yes—
190#124C22⋊D5912yes—
191C4×Q8—2362no—
192C4×D4—2362no—
193C2×C4⋊C4—591no—
194C2×C4⋊C4—591no—
195C2×C4⋊C4—1182no—
196C2×C4⋊C4—1181no—
197C2×C4⋊C4—2362no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—592no—
201C2×C22⋊C4—1182no—
202C2×C22⋊C4—1182no—
203C2×C42—591no—
204C2×C42—591no—
205C2×C42—1181no—
206C2×C42C4×D5911yescharacteristic
207#99—22no—
208#99—22no—
209#99—1182no—
210#99—1182no—
211#95—1181no—
212#95—1181no—
213#95—1182no—
214#95—2362no—
215C24D4×D5912yes—
216C2×Q8—21no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8Dic59⋊C412yescomplemented
220C2×D4—41no—
221C2×D4—41no—
222C2×D4C2×Dic11811yescharacteristic
223C2×D4#1174568712yes—
224C2×D4C2×C22⋊D5911yescharacteristic
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—591no—
229C22×C4—591no—
230C22×C4—592no—
231C22×C4—1182no—
232C22×C4—1182no—
233C22×C4—1181no—
234C22×C4—1181no—
235C22×C4—4722no—
236C22×C4—2362no—
237C22×C4C2×C4×D5911yescharacteristic
238C4⋊C4—4722no—
239C4⋊C4—4724no—
240C4⋊C4—2361no—
241C4⋊C4—2361no—
242C4⋊C4—2362no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—4722no—
248C22⋊C4—2362no—
249C22⋊C4—2362no—
250C22⋊C4—2362no—
251C42—21no—
252C42—1182no—
253C42—2361no—
254C42—2362no—
255C42Dic59⋊C412yescomplemented
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23C22⋊C4×D5911yescharacteristic
261C23#1427369912yes—
262C23#1427370412yes—
263Q8—42no—
264Q8—42no—
265D4—41no—
266D4—42no—
267D4—41no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—592no—
280C2×C4—1181no—
281C2×C4—1182no—
282C2×C4—4722no—
283C2×C4—4722no—
284C2×C4—4722no—
285C2×C4—4722no—
286C2×C4—2361no—
287C2×C4—2364no—
288C2×C4—2361no—
289C2×C4—2362no—
290C2×C4—2361no—
291C2×C4—2361no—
292C2×C4—2361no—
293C2×C4—2362no—
294C2×C4—2361no—
295C2×C4C2×Dic59⋊C411yescharacteristic
296C2×C4#1427368311yescharacteristic
297C22—21no—
298C22—21no—
299C22—22no—
300C22—22no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22#1683726011yescharacteristic
307C22#99×D5911yescharacteristic
308C22#1683737411yescharacteristic
309C4—21no—
310C4—21no—
311C4—42no—
312C4—42no—
313C4—162no—
314C4—1182no—
315C4—9442no—
316C4—4721no—
317C4—4721no—
318C4—2362no—
319C4—2361no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—42no—
324C2#2142408211yescharacteristic
325C2#2142576411yescharacteristic
326C2#2142576911yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC2×C59×#354C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenterC2×#354D59
hypocenterC59#12800
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
6. C2×Q8×Dic59C4semidirect
9. #16837309C4semidirect
13. #16837272C4semidirect
15. #16837248C4semidirect
55. C2×Q8×C59C4⋊C4semidirect
79. C42×C59C4⋊C4semidirect
184. C59#12800semidirect
219. C2×Q8Dic59⋊C4semidirect
255. C42Dic59⋊C4semidirect

Maximal subgroups

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

subgroupindex
#214268092
C2×#168371972
#214240822
#214222202
C2×C59×#3542
#1280059