Finite Group Databasebeta

← #354×Dic59 #28503857

Subgroups — #354×Dic59 #28503857

Order 15104.

584 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 127 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#124×Dic59C211yesmaximal, characteristic
2C2×#16837197C211yesmaximal, characteristic
3#99×Dic59C212yesmaximal
4#21422220C212yesmaximal
5C2×C59×#354C211yesmaximal, characteristic
6C2×Q8×Dic59C411yescharacteristic, complemented
7C2×D4×Dic59C2211yescharacteristic
8C2×#14273642C2212yes—
9#16837309C411yescharacteristic, complemented
10#16837305C2211yescharacteristic
11#16837295—22no—
12#16837268C411yescharacteristic, complemented
13C22⋊C4×Dic59—22no—
14C22⋊C4×Dic59—22no—
15C42×Dic59C411yescharacteristic, complemented
16#16837197C414yescomplemented
17#16837168—22no—
18C2×#124×C59C2211yescharacteristic
19C2×#99×C59C2212yes—
20C59×#354C414yescomplemented
21C23×Dic59D412yes—
22Q8×Dic59—41no—
23Q8×Dic59—41no—
24D4×Dic59—41no—
25D4×Dic59—41no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×C22⋊Dic59—22no—
30C2×C22⋊Dic59—22no—
31C2×C22⋊Dic59D412yes—
32C2×C4×Dic59—21no—
33C2×C4×Dic59—21no—
34C2×C4×Dic59—21no—
35C2×C4×Dic59—42no—
36C2×C4×Dic59C2×C411yescharacteristic
37C2×C4×Dic59C2×C411yescharacteristic
38#14273642—22no—
39#14273642C2×C414yes—
40C59⋊#95—22no—
41C59⋊#95—22no—
42C59⋊#95—42no—
43C22×Q8×C59C2×C411yescharacteristic
44C22×D4×C59C2311yescharacteristic
45#124×C59—22no—
46#124×C59C2×C412yes—
47#124×C59C2×C412yes—
48C2×C22⋊C4×C59—22no—
49C2×C22⋊C4×C59—22no—
50C2×C42×C59C2×C411yescharacteristic
51#99×C59—22no—
52#99×C59C2×C414yes—
53C24×C59C2×D412yes—
54C2×Q8×C59—21no—
55C2×Q8×C59—41no—
56C2×Q8×C59—41no—
57C2×Q8×C59C4212yescomplemented
58C2×D4×C59—41no—
59C2×D4×C59—41no—
60C2×D4×C59C22×C411yescharacteristic
61C2×D4×C59C22×C411yescharacteristic
62C2×D4×C59#4112yes—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—42no—
66C22×C4×C59C22×C411yescharacteristic
67C22×Dic59—21no—
68C22×Dic59—22no—
69C22×Dic59—22no—
70C22×Dic59—21no—
71C22×Dic59—44no—
72C22×Dic59—42no—
73C22×Dic59C22⋊C411yescharacteristic
74C22×Dic59C22⋊C412yes—
75C22×Dic59C22⋊C411yescharacteristic
76C22⋊C4×C59—24no—
77C22⋊C4×C59—24no—
78C22⋊C4×C59—24no—
79C22⋊C4×C59—42no—
80C42×C59—21no—
81C42×C59C4212yescomplemented
82C22⋊Dic59—24no—
83C22⋊Dic59—42no—
84C22⋊Dic59—44no—
85C22⋊Dic59—42no—
86C4⋊Dic59—41no—
87C4⋊Dic59—42no—
88C4⋊Dic59—41no—
89C4×Dic59—21no—
90C4×Dic59—21no—
91C4×Dic59—42no—
92C4×Dic59—42no—
93C4×Dic59—41no—
94C4×Dic59—41no—
95C4×Dic59—41no—
96C4×Dic59—84no—
97C23×C59—22no—
98C23×C59—22no—
99C23×C59—42no—
100C23×C59—42no—
101C23×C59C2×C22⋊C411yescharacteristic
102C23×C59C4×D412yes—
103C23×C59C4×D412yes—
104Q8×C59—42no—
105Q8×C59—42no—
106D4×C59—41no—
107D4×C59—41no—
108D4×C59—42no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—44no—
118C2×C4×C59—82no—
119C2×C4×C59—82no—
120C2×C4×C59C2×C4211yescharacteristic
121C2×C4×C59#11711yescharacteristic
122C2×Dic59—22no—
123C2×Dic59—22no—
124C2×Dic59—21no—
125C2×Dic59—22no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—44no—
129C2×Dic59—41no—
130C2×Dic59—42no—
131C2×Dic59—44no—
132C2×Dic59—42no—
133C2×Dic59—41no—
134C2×Dic59—82no—
135C2×Dic59—82no—
136C2×Dic59—82no—
137C2×Dic59#9911yescharacteristic
138C2×Dic59#9911yescharacteristic
139C4×#354—591nomaximal, Sylow
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—22no—
143C22×C59—22no—
144C22×C59—42no—
145C22×C59—42no—
146C22×C59—42no—
147C22×C59—42no—
148C22×C59—82no—
149C22×C59C4×C22⋊C411yescharacteristic
150C22×C59C2×#9911yescharacteristic
151C22×C59#41011yescharacteristic
152C4×C59—21no—
153C4×C59—21no—
154C4×C59—42no—
155C4×C59—42no—
156C4×C59—84no—
157Dic59—21no—
158Dic59—42no—
159Dic59—42no—
160Dic59—41no—
161Dic59—84no—
162Dic59—81no—
163Dic59#35412yesdirect product factor
164C4×#124—591no—
165C2×#354—591no—
166C2×#354D5911yescharacteristic
167C4×#99—592no—
168#1393—592no—
169C2×C59—21no—
170C2×C59—21no—
171C2×C59—42no—
172C2×C59—42no—
173C2×C59C4×#9911yescharacteristic
174C2×C59C2×#35411yescharacteristic
175C2×C59#212611yescharacteristic
176C2×#124C2×D5911yescharacteristic
177C2×C4×Q8—591no—
178C2×C4×D4—591no—
179C2×#99—592no—
180C2×#99C2×D5912yes—
181#391—591no—
182#390—591no—
183#388—1182no—
184#382—591no—
185C4×C22⋊C4—1182no—
186C4×C22⋊C4—1182no—
187C43—591no—
188#354—594no—
189#354Dic5914yesdirect product factor
190#342—1182no—
191C59C4×#35411yescharacteristic, complemented, Sylow
192C22×Q8C4×D5911yescharacteristic
193C22×D4C22×D5911yescharacteristic
194C23×C4—592no—
195#124—22no—
196#124C4×D5912yes—
197#124C2×Dic5912yes—
198C4×Q8—2361no—
199C4×Q8—2361no—
200C4×D4—2361no—
201C4×D4—2361no—
202C2×C4⋊C4—591no—
203C2×C4⋊C4—591no—
204C2×C4⋊C4—1181no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—592no—
208C2×C22⋊C4—1182no—
209C2×C22⋊C4—1182no—
210C2×C42—591no—
211C2×C42—591no—
212C2×C42—1181no—
213C2×C42—1181no—
214C2×C42—1181no—
215C2×C42—2362no—
216C2×C42C4×D5911yescharacteristic
217#99—22no—
218#99—594no—
219#99—1182no—
220#99C2×Dic5914yes—
221#95—1182no—
222#95—1182no—
223#95—2362no—
224C24D4×D5912yes—
225C2×Q8—21no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8C4×Dic5912yescomplemented
229C2×D4—41no—
230C2×D4—41no—
231C2×D4C2×C4×D5911yescharacteristic
232C2×D4#1174568712yes—
233C2×D4C22×Dic5911yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4—591no—
238C22×C4—592no—
239C22×C4—591no—
240C22×C4—1182no—
241C22×C4—1182no—
242C22×C4—1181no—
243C22×C4—1181no—
244C22×C4—2364no—
245C22×C4—2362no—
246C22×C4C2×C4×D5911yescharacteristic
247C4⋊C4—2361no—
248C4⋊C4—2361no—
249C4⋊C4—2362no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—42no—
254C22⋊C4—1184no—
255C22⋊C4—2362no—
256C22⋊C4—2364no—
257C22⋊C4—2362no—
258C42—21no—
259C42—1181no—
260C42—1181no—
261C42—4724no—
262C42—2361no—
263C42—2361no—
264C42—2361no—
265C42—2362no—
266C42—2362no—
267C42C4×Dic5912yescomplemented
268C23—22no—
269C23—22no—
270C23—42no—
271C23—42no—
272C23C22⋊C4×D5911yescharacteristic
273C23(C22⋊D59)⋊C412yes—
274C23D4×Dic5912yes—
275Q8—42no—
276Q8—42no—
277D4—41no—
278D4—42no—
279D4—41no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—44no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—591no—
292C2×C4—591no—
293C2×C4—1182no—
294C2×C4—1182no—
295C2×C4—1181no—
296C2×C4—1182no—
297C2×C4—4722no—
298C2×C4—4722no—
299C2×C4—4722no—
300C2×C4—2364no—
301C2×C4—2364no—
302C2×C4—2362no—
303C2×C4—2361no—
304C2×C4—2361no—
305C2×C4—2361no—
306C2×C4—2361no—
307C2×C4—2362no—
308C2×C4C2×C4×Dic5911yescharacteristic
309C2×C4#1427368311yescharacteristic
310C22—21no—
311C22—21no—
312C22—22no—
313C22—22no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22C22⋊C4×Dic5911yescharacteristic
320C22#99×D5911yescharacteristic
321C22#1683737411yescharacteristic
322C4—21no—
323C4—21no—
324C4—42no—
325C4—42no—
326C4—84no—
327C4—592no—
328C4—1181no—
329C4—4724no—
330C4—4721no—
331C4—2362no—
332C4—2362no—
333C4—2361no—
334C2—21no—
335C2—21no—
336C2—42no—
337C2—42no—
338C2#99×Dic5911yescharacteristic
339C2#354×D5911yescharacteristic
340C2#2142576011yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC2×C4×C59C2×C42
FrattiniC22×D4C22×D59
FittingC2×C59×#354C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenterC2×#354D59
hypocenterC59C4×#354
LCSG ▷ C2×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
12. #16837268C4semidirect
15. C42×Dic59C4semidirect
16. #16837197C4semidirect
20. C59×#354C4semidirect
57. C2×Q8×C59C42semidirect
81. C42×C59C42semidirect
163. Dic59#354direct (N × H)
189. #354Dic59direct (N × H)
191. C59C4×#354semidirect
228. C2×Q8C4×Dic59semidirect
267. C42C4×Dic59semidirect

Maximal subgroups

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

subgroupindex
#124×Dic592
C2×#168371972
#99×Dic592
#214222202
C2×C59×#3542
C4×#35459