Finite Group Databasebeta

← #519×#6923 #30624558

Subgroups — #519×#6923 #30624558

Order 15616.

3402 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840830C211yesmaximal, characteristic
2C22×Q8×#6923C211yesmaximal, characteristic
3#21462259C218yesmaximal, complemented
4#21462256C216yesmaximal
5C4×Q8×#6923C218yesmaximal, complemented
6#117×#6923C216yesmaximal
7#519×D61C211yesmaximal, characteristic
8#16840830C4116yescomplemented
9C2×Q8×#6923C4116yescomplemented
10C2×Q8×#6923C2214yes—
11C2×Q8×#6923C2214yes—
12C2×Q8×#6923C2218yes—
13C2×#14519856C2216yes—
14C22×C4⋊#6923C2213yes—
15C22×C4×#6923C2213yes—
16C4⋊C4×#6923C22112yes—
17C4⋊C4×#6923C22112yes—
18C4⋊C4×#6923C22124yes—
19C4×C4⋊#6923C22124yes—
20C22⋊C4×#6923C2214yes—
21C22⋊C4×#6923C22112yes—
22#16840698C22112yes—
23C42×#6923C22112yes—
24C22×Q8×D61C2211yescharacteristic
25#16840240C411yescharacteristic, complemented
26#16840239C416yescomplemented
27#16840238C418yescomplemented
28C4×Q8×D61C2218yes—
29#117×D61C2216yes—
30C61×#519C411yescharacteristic, complemented
31C23×#6923C2311yescharacteristic
32C23×#6923C2313yes—
33Q8×#6923C2×C4132yes—
34Q8×#6923C2×C4132yes—
35Q8×#6923C2×C4164yescomplemented
36#14519856C2×C4148yes—
37C2×C4⋊#6923C2×C4124yes—
38C2×C4⋊#6923C23112yes—
39C2×C4⋊#6923C23112yes—
40C2×C22⋊#6923C2314yes—
41C2×C4×#6923C2×C4124yes—
42C2×C4×#6923C2318yes—
43C2×C4×#6923C23124yes—
44C2×C4×#6923C2316yes—
45C2×C4×#6923C2316yes—
46C2×C4×#6923C23112yes—
47(C2×#6923)⋊C4C2318yes—
48(C2×#6923)⋊C4C23124yes—
49C22×Dic122C2×C413yes—
50C2×Q8×D61C2×C418yes—
51C2×Q8×D61C2×C418yes—
52C2×Q8×D61C2×C414yes—
53C2×Q8×D61C2314yes—
54C22×C4×D61C2313yes—
55Q8×Dic61C2×C418yescomplemented
56C4×Dic122C2×C4124yescomplemented
57#14519742C2×C4124yescomplemented
58#14519736C2×C416yes—
59#14519735C2×C4112yes—
60#14519734C2×C4112yes—
61#14519733C2×C4112yes—
62C4⋊C4×D61C23112yes—
63C22⋊C4×D61C2314yes—
64C42×D61C23112yes—
65C22×Q8×C61C2×C411yescharacteristic
66C4×Q8×C61C2×C418yescomplemented
67#117×C61C2×C416yes—
68C23×D61C2411yescharacteristic
69C22×#6923—24no—
70C22×#6923—46no—
71C22×#6923C22×C414yes—
72C22×#6923C22×C4112yes—
73C22×#6923C2411yescharacteristic
74C22×#6923C2418yes—
75C22×#6923C2416yes—
76C22×#6923C2411yescharacteristic
77C2×Q8×C61C4218yescomplemented
78C2×Q8×C61C22×C414yes—
79C22×C4×C61C22×C413yes—
80C22×Dic61C22×C413yes—
81Q8×D61C42132yescomplemented
82Q8×D61C22×C4116yes—
83Q8×D61C22×C4116yes—
84C2×C4×D61C22×C416yes—
85C2×C4×D61C22×C4112yes—
86C2×C4×D61C22×C4112yes—
87C2×C4×D61C2418yes—
88C2×C4×D61C2416yes—
89C2×Dic122C42124yescomplemented
90C2×Dic122C22×C4112yes—
91C22⋊#6923C22×C4116yes—
92C4⋊#6923C22×C4148yes—
93C4⋊#6923C22×C4148yes—
94C4×#6923—216no—
95C4×#6923—424no—
96C4×#6923C22×C4124yes—
97C4×#6923C22×C4124yes—
98C4×#6923C22×C4148yes—
99C4⋊C4×C61C22×C4112yes—
100C22⋊C4×C61C22×C414yes—
101C42×C61C22×C4112yes—
102C22⋊Dic61C22×C414yes—
103#11758198C22×C418yes—
104C4⋊Dic61C22×C4112yes—
105#11758196C22×C4124yes—
106C4×Dic61C22×C4124yes—
107C4×Dic61C22×C4112yes—
108C23×C61C23×C411yescharacteristic
109C22×D61—22no—
110C22×D61—22no—
111C22×D61C23×C412yes—
112C22×D61C23×C412yes—
113C22×D61C23×C411yescharacteristic
114C22×D61C2511yescharacteristic
115C2×#6923—28no—
116C2×#6923—22no—
117C2×#6923—28no—
118C2×#6923—212no—
119C2×#6923—28no—
120C2×#6923—412no—
121C2×#6923C23×C412yes—
122C2×#6923C23×C4116yes—
123C2×#6923C23×C4112yes—
124C2×#6923C23×C412yes—
125C2×#69232−1+414yes—
126Q8×C61C2×C42116yes—
127C2×C4×C61C2×C42112yes—
128C2×C4×C61C23×C418yes—
129C2×C4×C61C23×C416yes—
130C2×Dic61C2×C42112yes—
131C2×Dic61C23×C418yes—
132C2×Dic61C23×C416yes—
133C4×D61—28no—
134C4×D61—28no—
135C4×D61C2×C42124yes—
136C4×D61C23×C4112yes—
137C4×D61C23×C4112yes—
138Dic122C2×C42148yes—
139C4×#519—611nomaximal, Sylow
140C22×C61—22no—
141C22×C61C22×C4212yes—
142C22×C61C24×C411yescharacteristic
143C2×D61—22no—
144C2×D61—22no—
145C2×D61—22no—
146C2×D61—24no—
147C2×D61C22×C4212yes—
148C2×D61C24×C411yescharacteristic
149C2×D61C24×C411yescharacteristic
150C2×D61C2×2−1+412yes—
151C2×D61#58512yes—
152#6923—22no—
153#6923—212no—
154#6923—216no—
155#6923#51914yesdirect product factor
156C4×C61—28no—
157C4×C61C22×C42112yes—
158Dic61—28no—
159Dic61C22×C42112yes—
160C2×#519—611no—
161C2×#519—611no—
162C22×C4×Q8—611no—
163#2276—618no—
164#2274—616no—
165C42×Q8—618no—
166C4×#117—616no—
167C2×C61—22no—
168C2×C61C23×C4211yescharacteristic
169C2×C61C4×2−1+412yes—
170D61—22no—
171D61C2×#51911yescharacteristic
172D61#342811yescharacteristic
173D61C4×2−1+412yes—
174C23×Q8—611no—
175#519—6116no—
176#519—611no—
177#519—618no—
178#519—616no—
179#519#692311yescharacteristic, direct product factor
180C2×C4×Q8—6116no—
181C2×C4×Q8—614no—
182C2×C4×Q8—614no—
183C2×C4×Q8—618no—
184C2×C4×Q8—618no—
185C2×#117—616no—
186C2×#117—616no—
187C22×C4⋊C4—613no—
188C22×C42—613no—
189C4×C4⋊C4—6112no—
190C4×C4⋊C4—6112no—
191C4×C4⋊C4—6124no—
192C4×C4⋊C4—6124no—
193C4×C22⋊C4—614no—
194C4×C22⋊C4—6112no—
195#376—6112no—
196C43—6112no—
197C61C4×#51911yescharacteristic, complemented, Sylow
198C22×Q8—618no—
199C22×Q8—618no—
200C22×Q8—614no—
201C22×Q8—614no—
202C22×Q8—613no—
203C22×Q8C2×#692311yescharacteristic
204C23×C4—613no—
205C23×C4—611no—
206C23×C4—613no—
207C4×Q8—6132no—
208C4×Q8—6132no—
209C4×Q8—6164no—
210C4×Q8—618no—
211C4×Q8—6124no—
212C4×Q8—6124no—
213C4×Q8C2×#692318yescomplemented
214#117—6148no—
215#117—616no—
216#117—6112no—
217#117—6112no—
218#117—6112no—
219#117C2×#692316yes—
220C2×C4⋊C4—6124no—
221C2×C4⋊C4—6112no—
222C2×C4⋊C4—6112no—
223C2×C4⋊C4—6112no—
224C2×C22⋊C4—614no—
225C2×C22⋊C4—614no—
226C2×C42—6112no—
227C2×C42—6124no—
228C2×C42—616no—
229C2×C42—616no—
230C2×C42—6112no—
231C2×C42—618no—
232C2×C42—6124no—
233#95—618no—
234#95—6124no—
235C24—611no—
236C2×Q8—6116no—
237C2×Q8—6116no—
238C2×Q8—6132no—
239C2×Q8—6124no—
240C2×Q8—6112no—
241C2×Q8C4×#692318yescomplemented
242C2×Q8C22×#692314yes—
243C22×C4—611no—
244C22×C4—613no—
245C22×C4—6112no—
246C22×C4—6112no—
247C22×C4—616no—
248C22×C4—616no—
249C22×C4—611no—
250C22×C4—618no—
251C22×C4—616no—
252C22×C4—614no—
253C22×C4—6112no—
254C22×C4—618no—
255C22×C4—1224no—
256C22×C4—2446no—
257C22×C4C22×#692313yes—
258C4⋊C4—6148no—
259C4⋊C4—6148no—
260C4⋊C4—6124no—
261C4⋊C4—6112no—
262C4⋊C4C22×#6923112yes—
263C22⋊C4—6116no—
264C22⋊C4—614no—
265C22⋊C4—618no—
266C22⋊C4C22×#692314yes—
267C42—6124no—
268C42—6124no—
269C42—6148no—
270C42—6124no—
271C42—6112no—
272C42—12216no—
273C42—24424no—
274C42C22×#6923112yes—
275C23—611no—
276C23—612no—
277C23—612no—
278C23—611no—
279C23—1222no—
280C23—1222no—
281C23C23×#692311yescharacteristic
282Q8—6148no—
283Q8C2×C4×#6923116yes—
284C2×C4—612no—
285C2×C4—614no—
286C2×C4—6112no—
287C2×C4—616no—
288C2×C4—6124no—
289C2×C4—6112no—
290C2×C4—6112no—
291C2×C4—612no—
292C2×C4—6116no—
293C2×C4—6112no—
294C2×C4—618no—
295C2×C4—1228no—
296C2×C4—1228no—
297C2×C4—1228no—
298C2×C4—1228no—
299C2×C4—1228no—
300C2×C4—12212no—
301C2×C4—1222no—
302C2×C4—24412no—
303C2×C4C2×C4×#6923112yes—
304C2×C4C23×#692318yes—
305C2×C4C23×#692316yes—
306C22—22no—
307C22—611no—
308C22—611no—
309C22—612no—
310C22—612no—
311C22—612no—
312C22—1222no—
313C22—1222no—
314C22—1224no—
315C22—1222no—
316C22C22×C4×#692312yes—
317C22C24×#692311yescharacteristic
318C4—28no—
319C4—614no—
320C4—6112no—
321C4—1222no—
322C4—12212no—
323C4—12216no—
324C4—1228no—
325C4C22×C4×#6923112yes—
326C2—22no—
327C2—611no—
328C2—611no—
329C2—612no—
330C2—1222no—
331C2C23×C4×#692311yescharacteristic
332C22−1+4×#692312yes—
subgroupG/H
center, FrattiniC22C24×#6923
commutatorC2×C61C23×C42
FittingC61×#519C4
socleC22×C61C24×C4
prime radicalC23×D61C24
cosocleC22×D61C25
hypercenter#519#6923
hypocenterC61C4×#519
LCSG ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
3. #21462259C2semidirect
5. C4×Q8×#6923C2semidirect
8. #16840830C4semidirect
9. C2×Q8×#6923C4semidirect
25. #16840240C4semidirect
26. #16840239C4semidirect
27. #16840238C4semidirect
30. C61×#519C4semidirect
35. Q8×#6923C2×C4semidirect
55. Q8×Dic61C2×C4semidirect
56. C4×Dic122C2×C4semidirect
57. #14519742C2×C4semidirect
66. C4×Q8×C61C2×C4semidirect
77. C2×Q8×C61C42semidirect
81. Q8×D61C42semidirect
89. C2×Dic122C42semidirect
155. #6923#519direct (N × H)
179. #519#6923direct (N × H)
197. C61C4×#519semidirect
213. C4×Q8C2×#6923semidirect
241. C2×Q8C4×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168408302
C22×Q8×#69232
#214622592
#214622562
C4×Q8×#69232
#117×#69232
#519×D612
C4×#51961