Finite Group Databasebeta

← #424×Dic59 #28586579

Subgroups — #424×Dic59 #28586579

Order 15104.

1174 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837429C211yesmaximal, characteristic
2C2×M16×Dic59C211yesmaximal, characteristic
3C22×C8×Dic59C211yesmaximal, characteristic
4#117×Dic59C211yesmaximal, characteristic
5#21423955C214yesmaximal, complemented
6C4⋊C8×Dic59C214yesmaximal, complemented
7#21423794C211yesmaximal, characteristic
8#21423792C211yesmaximal, characteristic
9C2×C59×#424C211yesmaximal, characteristic
10C22×#11745664C2211yescharacteristic
11C23×C59⋊C8C2211yescharacteristic
12C2×#14273686C2211yescharacteristic
13C22×C4×Dic59C2211yescharacteristic
14#16837429C418yescomplemented
15C2×C4⋊(C59⋊C8)C2214yes—
16M16×Dic59—24no—
17C2×#14273637C2212yes—
18C2×C8×Dic59—24no—
19C2×C8×Dic59C2212yes—
20C2×C8×Dic59C2212yes—
21C2×C8×Dic59C2212yes—
22C4⋊C4×Dic59C414yes—
23C22⋊C4×Dic59C414yes—
24#16837248C2212yes—
25C42×Dic59C2212yes—
26#16837156C2214yes—
27#16837156C2214yes—
28C22×M16×C59C2211yescharacteristic
29C23×C8×C59C2211yescharacteristic
30C2×#117×C59C2211yescharacteristic
31C59×#424C418yescomplemented
32C2×C4⋊C8×C59C2214yes—
33C23×Dic59C2×C412yes—
34C2×#11745664—22no—
35C2×#11745664—22no—
36C2×#11745664C2×C414yes—
37C22×C59⋊C8—22no—
38C22×C59⋊C8—22no—
39C22×C59⋊C8C2×C414yes—
40C22×C59⋊C8C2312yes—
41C22×C59⋊C8C2312yes—
42#14273686—22no—
43#14273686—22no—
44C2×C4⋊Dic59C2×C412yes—
45C2×C22⋊Dic59C2×C412yes—
46C2×C4×Dic59C2×C418yes—
47C2×C4×Dic59C2×C412yes—
48C2×C4×Dic59D414yes—
49C2×C4×Dic59Q814yes—
50C2×C4×Dic59C2312yes—
51C2×C4×Dic59C2311yescharacteristic
52C2×C4×Dic59C2311yescharacteristic
53C4⋊(C59⋊C8)C2×C4116yescomplemented
54#14273637—24no—
55C8×Dic59—24no—
56C8×Dic59—24no—
57C8×Dic59—24no—
58C59⋊#95C2×C418yes—
59C23×C4×C59C2311yescharacteristic
60C2×M16×C59—22no—
61C2×M16×C59—22no—
62C2×M16×C59C2×C414yes—
63C22×C8×C59—22no—
64C22×C8×C59—22no—
65C22×C8×C59C2×C414yes—
66C22×C8×C59C2312yes—
67C22×C8×C59C2312yes—
68#117×C59C2×C414yes—
69#117×C59C2×C414yes—
70C2×C4⋊C4×C59C2×C412yes—
71C2×C22⋊C4×C59C2×C412yes—
72C2×C42×C59C2312yes—
73C4⋊C8×C59C2×C4116yescomplemented
74C24×C59C22×C411yescharacteristic
75C22×C4×C59C22×C414yes—
76C22×C4×C59C22×C411yescharacteristic
77C22×C4×C59C22×C412yes—
78C22×C4×C59C22×C412yes—
79C22×C4×C59C2×D412yes—
80C22×C4×C59C2×Q812yes—
81C22×C4×C59#4112yes—
82C22×C4×C59#4112yes—
83C22×C4×C59C2411yescharacteristic
84C22×Dic59—28no—
85C22×Dic59C4⋊C418yes—
86C22×Dic59C22×C414yes—
87C22×Dic59C22×C412yes—
88C22×Dic59C22×C412yes—
89M16×C59—28no—
90C2×C8×C59—28no—
91C2×C8×C59—22no—
92C2×C8×C59—22no—
93C2×C8×C59—22no—
94C2×C8×C59—22no—
95C2×C8×C59C22×C414yes—
96C2×C8×C59C22×C414yes—
97C4⋊C4×C59C4218yes—
98C22⋊C4×C59C4218yes—
99C42×C59C22×C414yes—
100C42×C59C22×C414yes—
101C22⋊Dic59—24no—
102C4⋊Dic59—24no—
103C4×Dic59—22no—
104C4×Dic59—22no—
105C4×Dic59—216no—
106C4×Dic59C4⋊C418yes—
107C4×Dic59C2×D412yes—
108C4×Dic59C2×D412yes—
109C4×Dic59C2×Q812yes—
110C4×Dic59C2×Q812yes—
111#11745664—28no—
112C2×C59⋊C8—28no—
113C2×C59⋊C8—22no—
114C2×C59⋊C8—22no—
115C2×C59⋊C8—22no—
116C2×C59⋊C8—22no—
117C2×C59⋊C8C22×C414yes—
118C2×C59⋊C8C22×C414yes—
119C23×C59—24no—
120C23×C59C2×C4212yes—
121C23×C59C2×C4⋊C412yes—
122C23×C59#11712yes—
123C23×C59C23×C411yescharacteristic
124C2×C4×C59—28no—
125C2×C4×C59C2×C4218yes—
126C2×C4×C59C2×C4212yes—
127C2×C4×C59C2×C4⋊C412yes—
128C2×C4×C59#11712yes—
129C2×C4×C59C4×D414yes—
130C2×C4×C59C4×D414yes—
131C2×C4×C59C4×Q814yes—
132C2×C4×C59C4×Q814yes—
133C2×C4×C59C23×C411yescharacteristic
134C2×C4×C59C23×C411yescharacteristic
135C2×C4×C59C22×D411yescharacteristic
136C2×C4×C59C22×Q811yescharacteristic
137C2×C4×C59C2×#4111yescharacteristic
138C2×C4×C59C2×#4111yescharacteristic
139C2×Dic59—24no—
140C2×Dic59—216no—
141C2×Dic59—24no—
142C2×Dic59—28no—
143C2×Dic59C2×C4⋊C414yes—
144C2×Dic59C2×C4⋊C414yes—
145C2×Dic59#13118yes—
146C8×C59—24no—
147C8×C59—24no—
148C59⋊C8—24no—
149C59⋊C8—24no—
150C4×#424—591nomaximal, Sylow
151C22×C59—24no—
152C22×C59—24no—
153C22×C59—24no—
154C22×C59C4×C4⋊C414yes—
155C22×C59C22×C4211yescharacteristic
156C22×C59C22×C4⋊C411yescharacteristic
157C22×C59C2×#11711yescharacteristic
158C22×C59C2×#13114yes—
159C4×C59—28no—
160C4×C59C4×C4⋊C414yes—
161C4×C59C2×C4×D411yescharacteristic
162C4×C59C2×C4×D411yescharacteristic
163C4×C59C2×C4×Q811yescharacteristic
164C4×C59C2×C4×Q811yescharacteristic
165Dic59—24no—
166Dic59—44no—
167Dic59#42418yesdirect product factor
168C2×#424—591no—
169C2×#424D5911yescharacteristic
170C2×C4×M16—591no—
171C22×C4×C8—591no—
172C4×#117—591no—
173#1768—594no—
174C4×C4⋊C8—594no—
175#1729—591no—
176#1729—591no—
177C2×C59—24no—
178C2×C59C2×C4×C4⋊C411yescharacteristic
179C2×C59C4×#13112yes—
180C2×C59C2×#42412yes—
181C2×C59#292212yes—
182C22×M16—591no—
183C22×M16C2×D5911yescharacteristic
184C23×C8—591no—
185C23×C8C2×D5911yescharacteristic
186C2×#117—591no—
187C2×#117C2×D5911yescharacteristic
188C22×C42—591no—
189#424—598no—
190#424Dic5918yesdirect product factor
191C2×C4⋊C8—594no—
192C2×C4⋊C8C2×D5914yes—
193C4×M16—1184no—
194C2×#97—592no—
195C2×C4×C8—592no—
196C2×C4×C8—592no—
197C2×C4×C8—592no—
198C2×C4×C8—1184no—
199C4×C4⋊C4—594no—
200C4×C22⋊C4—594no—
201#376—592no—
202C43—592no—
203#336—594no—
204#336—594no—
205C59C4×#42411yescharacteristic, complemented, Sylow
206C23×C4—592no—
207C23×C4C22×D5911yescharacteristic
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—594no—
211C2×M16—1182no—
212C2×M16—1182no—
213C2×M16C2×Dic5914yes—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—594no—
217C22×C8—592no—
218C22×C8—592no—
219C22×C8—1182no—
220C22×C8—1182no—
221C22×C8C2×Dic5914yes—
222C22×C8C22×D5912yes—
223C22×C8C22×D5912yes—
224#117—1182no—
225#117—1182no—
226#117C4×D5914yes—
227#117C2×Dic5914yes—
228C2×C4⋊C4—592no—
229C2×C4⋊C4C4×D5912yes—
230C2×C22⋊C4—592no—
231C2×C22⋊C4C4×D5912yes—
232C2×C42—592no—
233C2×C42—594no—
234C2×C42—594no—
235C2×C42—591no—
236C2×C42—591no—
237C2×C42—598no—
238C2×C42—592no—
239C2×C42C22×D5912yes—
240C4⋊C8—5916no—
241C4⋊C8C2×Dic59116yescomplemented
242#97—1184no—
243C4×C8—1184no—
244C4×C8—1184no—
245C4×C8—1184no—
246#95—598no—
247C24C2×C4×D5911yescharacteristic
248C22×C4—592no—
249C22×C4—598no—
250C22×C4—592no—
251C22×C4—594no—
252C22×C4—1188no—
253C22×C4C2×C4×D5914yes—
254C22×C4C2×C4×D5911yescharacteristic
255C22×C4C2×C4×D5912yes—
256C22×C4D4×D5912yes—
257C22×C4#1174568712yes—
258C22×C4Q8×D5912yes—
259C22×C4#1174568912yes—
260C22×C4C22×Dic5912yes—
261C22×C4C23×D5911yescharacteristic
262M16—28no—
263M16—1188no—
264C2×C8—28no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—594no—
270C2×C8—594no—
271C2×C8—1188no—
272C2×C8—1182no—
273C2×C8—1182no—
274C2×C8—1182no—
275C2×C8—1182no—
276C2×C8C22×Dic5914yes—
277C2×C8C22×Dic5914yes—
278C4⋊C4—1184no—
279C4⋊C4C4×Dic5918yes—
280C22⋊C4—1184no—
281C22⋊C4C4×Dic5918yes—
282C42—598no—
283C42—592no—
284C42—592no—
285C42—592no—
286C42—592no—
287C42—11816no—
288C42—1182no—
289C42—1182no—
290C42C2×C4×D5914yes—
291C42C22×Dic5914yes—
292C23—24no—
293C23C2×C4×Dic5912yes—
294C23C4⋊C4×D5912yes—
295C23#1427368512yes—
296C23C22×C4×D5911yescharacteristic
297C2×C4—28no—
298C2×C4—598no—
299C2×C4—594no—
300C2×C4—594no—
301C2×C4—11816no—
302C2×C4—1184no—
303C2×C4—1188no—
304C2×C4—1184no—
305C2×C4C2×C4×Dic5918yes—
306C2×C4C2×C4×Dic5912yes—
307C2×C4C4⋊C4×D5912yes—
308C2×C4#1427368512yes—
309C2×C4#1427368914yes—
310C2×C4D4×Dic5914yes—
311C2×C4#1427369214yes—
312C2×C4Q8×Dic5914yes—
313C2×C4C22×C4×D5911yescharacteristic
314C2×C4C23×Dic5911yescharacteristic
315C2×C4C2×D4×D5911yescharacteristic
316C2×C4C2×Q8×D5911yescharacteristic
317C2×C4C2×#1174568711yescharacteristic
318C2×C4C2×#1174568911yescharacteristic
319C8—24no—
320C8—24no—
321C8—1184no—
322C8—1184no—
323C22—24no—
324C22—24no—
325C22—24no—
326C22C4⋊C4×Dic5914yes—
327C22C22×C4×Dic5911yescharacteristic
328C22C2×C4⋊C4×D5911yescharacteristic
329C22C2×#1427368511yescharacteristic
330C22#131×D5914yes—
331C4—28no—
332C4—598no—
333C4—1184no—
334C4—2364no—
335C4C4⋊C4×Dic5914yes—
336C4C2×#1427368911yescharacteristic
337C4C2×D4×Dic5911yescharacteristic
338C4C2×#1427369211yescharacteristic
339C4C2×Q8×Dic5911yescharacteristic
340C2—24no—
341C2C2×C4⋊C4×Dic5911yescharacteristic
342C2#131×Dic5912yes—
343C2#424×D5912yes—
344C2#2143273912yes—
subgroupG/H
center, FrattiniC22×C4C23×D59
commutatorC22×C59C22×C42
FittingC2×C59×#424C2
socleC23×C59C23×C4
prime radicalC24×C59C22×C4
cosocleC22×C4×C59C24
hypercenterC2×#424D59
hypocenterC59C4×#424
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C2×#424

Complements

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

NH ≅ G/Ntype
5. #21423955C2semidirect
6. C4⋊C8×Dic59C2semidirect
14. #16837429C4semidirect
31. C59×#424C4semidirect
53. C4⋊(C59⋊C8)C2×C4semidirect
73. C4⋊C8×C59C2×C4semidirect
167. Dic59#424direct (N × H)
190. #424Dic59direct (N × H)
205. C59C4×#424semidirect
241. C4⋊C8C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
C2×#168374292
C2×M16×Dic592
C22×C8×Dic592
#117×Dic592
#214239552
C4⋊C8×Dic592
#214237942
#214237922
C2×C59×#4242
C4×#42459