Finite Group Databasebeta

← #519×#6157 #28328637

Subgroups — #519×#6157 #28328637

Order 13568.

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×#16811691C211yesmaximal, characteristic
2C22×Q8×#6157C211yesmaximal, characteristic
3#20291155C218yesmaximal, complemented
4#20291152C216yesmaximal
5C4×Q8×#6157C218yesmaximal, complemented
6#117×#6157C216yesmaximal
7#519×D53C211yesmaximal, characteristic
8#16811691C4116yescomplemented
9C2×Q8×#6157C4116yescomplemented
10C2×Q8×#6157C2214yes—
11C2×Q8×#6157C2214yes—
12C2×Q8×#6157C2218yes—
13C2×#13133296C2216yes—
14C22×C4⋊#6157C2213yes—
15C22×C4×#6157C2213yes—
16C4⋊C4×#6157C22112yes—
17C4⋊C4×#6157C22112yes—
18C4⋊C4×#6157C22124yes—
19C4×C4⋊#6157C22124yes—
20C22⋊C4×#6157C2214yes—
21C22⋊C4×#6157C22112yes—
22#16811559C22112yes—
23C42×#6157C22112yes—
24C22×Q8×D53C2211yescharacteristic
25#16811101C411yescharacteristic, complemented
26#16811100C416yescomplemented
27#16811099C418yescomplemented
28C4×Q8×D53C2218yes—
29#117×D53C2216yes—
30C53×#519C411yescharacteristic, complemented
31C23×#6157C2311yescharacteristic
32C23×#6157C2313yes—
33Q8×#6157C2×C4132yes—
34Q8×#6157C2×C4132yes—
35Q8×#6157C2×C4164yescomplemented
36#13133296C2×C4148yes—
37C2×C4⋊#6157C2×C4124yes—
38C2×C4⋊#6157C23112yes—
39C2×C4⋊#6157C23112yes—
40C2×C22⋊#6157C2314yes—
41C2×C4×#6157C2×C4124yes—
42C2×C4×#6157C2318yes—
43C2×C4×#6157C23124yes—
44C2×C4×#6157C2316yes—
45C2×C4×#6157C23112yes—
46C2×C4×#6157C2316yes—
47(C2×#6157)⋊C4C2318yes—
48(C2×#6157)⋊C4C23124yes—
49C22×Dic106C2×C413yes—
50C2×Q8×D53C2×C418yes—
51C2×Q8×D53C2×C418yes—
52C2×Q8×D53C2×C414yes—
53C2×Q8×D53C2314yes—
54C22×C4×D53C2313yes—
55Q8×Dic53C2×C418yescomplemented
56C4×Dic106C2×C4124yescomplemented
57#13133182C2×C4124yescomplemented
58#13133176C2×C416yes—
59#13133175C2×C4112yes—
60#13133174C2×C4112yes—
61#13133173C2×C4112yes—
62C4⋊C4×D53C23112yes—
63C22⋊C4×D53C2314yes—
64C42×D53C23112yes—
65C22×Q8×C53C2×C411yescharacteristic
66C4×Q8×C53C2×C418yescomplemented
67#117×C53C2×C416yes—
68C23×D53C2411yescharacteristic
69C22×#6157—24no—
70C22×#6157—46no—
71C22×#6157C22×C414yes—
72C22×#6157C22×C4112yes—
73C22×#6157C2411yescharacteristic
74C22×#6157C2418yes—
75C22×#6157C2416yes—
76C22×#6157C2411yescharacteristic
77C2×Q8×C53C4218yescomplemented
78C2×Q8×C53C22×C414yes—
79C22×C4×C53C22×C413yes—
80C22×Dic53C22×C413yes—
81Q8×D53C42132yescomplemented
82Q8×D53C22×C4116yes—
83Q8×D53C22×C4116yes—
84C2×C4×D53C22×C416yes—
85C2×C4×D53C22×C4112yes—
86C2×C4×D53C22×C4112yes—
87C2×C4×D53C2418yes—
88C2×C4×D53C2416yes—
89C2×Dic106C42124yescomplemented
90C2×Dic106C22×C4112yes—
91C22⋊#6157C22×C4116yes—
92C4⋊#6157C22×C4148yes—
93C4⋊#6157C22×C4148yes—
94C4×#6157—216no—
95C4×#6157—424no—
96C4×#6157C22×C4124yes—
97C4×#6157C22×C4124yes—
98C4×#6157C22×C4148yes—
99C4⋊C4×C53C22×C4112yes—
100C22⋊C4×C53C22×C414yes—
101C42×C53C22×C4112yes—
102C22⋊Dic53C22×C414yes—
103#11719921C22×C418yes—
104C4⋊Dic53C22×C4112yes—
105#11719919C22×C4124yes—
106C4×Dic53C22×C4124yes—
107C4×Dic53C22×C4112yes—
108C23×C53C23×C411yescharacteristic
109C22×D53—22no—
110C22×D53—22no—
111C22×D53C23×C412yes—
112C22×D53C23×C412yes—
113C22×D53C23×C411yescharacteristic
114C22×D53C2511yescharacteristic
115C2×#6157—28no—
116C2×#6157—22no—
117C2×#6157—28no—
118C2×#6157—212no—
119C2×#6157—28no—
120C2×#6157—412no—
121C2×#6157C23×C412yes—
122C2×#6157C23×C4116yes—
123C2×#6157C23×C4112yes—
124C2×#6157C23×C412yes—
125C2×#61572−1+414yes—
126Q8×C53C2×C42116yes—
127C2×C4×C53C2×C42112yes—
128C2×C4×C53C23×C418yes—
129C2×C4×C53C23×C416yes—
130C2×Dic53C2×C42112yes—
131C2×Dic53C23×C418yes—
132C2×Dic53C23×C416yes—
133C4×D53—28no—
134C4×D53—28no—
135C4×D53C2×C42124yes—
136C4×D53C23×C4112yes—
137C4×D53C23×C4112yes—
138Dic106C2×C42148yes—
139C4×#519—531nomaximal, Sylow
140C22×C53—22no—
141C22×C53C22×C4212yes—
142C22×C53C24×C411yescharacteristic
143C2×D53—22no—
144C2×D53—22no—
145C2×D53—22no—
146C2×D53—24no—
147C2×D53C22×C4212yes—
148C2×D53C24×C411yescharacteristic
149C2×D53C24×C411yescharacteristic
150C2×D53C2×2−1+412yes—
151C2×D53#58512yes—
152#6157—22no—
153#6157—212no—
154#6157—216no—
155#6157#51914yesdirect product factor
156C4×C53—28no—
157C4×C53C22×C42112yes—
158Dic53—28no—
159Dic53C22×C42112yes—
160C2×#519—531no—
161C2×#519—531no—
162C22×C4×Q8—531no—
163#2276—538no—
164#2274—536no—
165C42×Q8—538no—
166C4×#117—536no—
167C2×C53—22no—
168C2×C53C23×C4211yescharacteristic
169C2×C53C4×2−1+412yes—
170D53—22no—
171D53C2×#51911yescharacteristic
172D53#342811yescharacteristic
173D53C4×2−1+412yes—
174C23×Q8—531no—
175#519—5316no—
176#519—531no—
177#519—538no—
178#519—536no—
179#519#615711yescharacteristic, direct product factor
180C2×C4×Q8—5316no—
181C2×C4×Q8—534no—
182C2×C4×Q8—534no—
183C2×C4×Q8—538no—
184C2×C4×Q8—538no—
185C2×#117—536no—
186C2×#117—536no—
187C22×C4⋊C4—533no—
188C22×C42—533no—
189C4×C4⋊C4—5324no—
190C4×C4⋊C4—5312no—
191C4×C4⋊C4—5312no—
192C4×C4⋊C4—5324no—
193C4×C22⋊C4—534no—
194C4×C22⋊C4—5312no—
195#376—5312no—
196C43—5312no—
197C53C4×#51911yescharacteristic, complemented, Sylow
198C22×Q8—538no—
199C22×Q8—538no—
200C22×Q8—534no—
201C22×Q8—534no—
202C22×Q8—533no—
203C22×Q8C2×#615711yescharacteristic
204C23×C4—533no—
205C23×C4—531no—
206C23×C4—533no—
207C4×Q8—5332no—
208C4×Q8—5332no—
209C4×Q8—5364no—
210C4×Q8—538no—
211C4×Q8—5324no—
212C4×Q8—5324no—
213C4×Q8C2×#615718yescomplemented
214#117—5348no—
215#117—536no—
216#117—5312no—
217#117—5312no—
218#117—5312no—
219#117C2×#615716yes—
220C2×C4⋊C4—5312no—
221C2×C4⋊C4—5324no—
222C2×C4⋊C4—5312no—
223C2×C4⋊C4—5312no—
224C2×C22⋊C4—534no—
225C2×C22⋊C4—534no—
226C2×C42—5312no—
227C2×C42—536no—
228C2×C42—536no—
229C2×C42—5324no—
230C2×C42—5312no—
231C2×C42—538no—
232C2×C42—5324no—
233#95—538no—
234#95—5324no—
235C24—531no—
236C2×Q8—5332no—
237C2×Q8—5316no—
238C2×Q8—5316no—
239C2×Q8—5324no—
240C2×Q8—5312no—
241C2×Q8C4×#615718yescomplemented
242C2×Q8C22×#615714yes—
243C22×C4—531no—
244C22×C4—533no—
245C22×C4—5312no—
246C22×C4—5312no—
247C22×C4—536no—
248C22×C4—536no—
249C22×C4—534no—
250C22×C4—531no—
251C22×C4—538no—
252C22×C4—536no—
253C22×C4—5312no—
254C22×C4—538no—
255C22×C4—1064no—
256C22×C4—2126no—
257C22×C4C22×#615713yes—
258C4⋊C4—5348no—
259C4⋊C4—5348no—
260C4⋊C4—5324no—
261C4⋊C4—5312no—
262C4⋊C4C22×#6157112yes—
263C22⋊C4—5316no—
264C22⋊C4—534no—
265C22⋊C4—538no—
266C22⋊C4C22×#615714yes—
267C42—5324no—
268C42—5324no—
269C42—5348no—
270C42—5324no—
271C42—5312no—
272C42—10616no—
273C42—21224no—
274C42C22×#6157112yes—
275C23—531no—
276C23—532no—
277C23—532no—
278C23—531no—
279C23—1062no—
280C23—1062no—
281C23C23×#615711yescharacteristic
282Q8—5348no—
283Q8C2×C4×#6157116yes—
284C2×C4—532no—
285C2×C4—534no—
286C2×C4—5312no—
287C2×C4—536no—
288C2×C4—5324no—
289C2×C4—5312no—
290C2×C4—5312no—
291C2×C4—532no—
292C2×C4—5316no—
293C2×C4—5312no—
294C2×C4—538no—
295C2×C4—1068no—
296C2×C4—1068no—
297C2×C4—1068no—
298C2×C4—1068no—
299C2×C4—1068no—
300C2×C4—10612no—
301C2×C4—1062no—
302C2×C4—21212no—
303C2×C4C2×C4×#6157112yes—
304C2×C4C23×#615718yes—
305C2×C4C23×#615716yes—
306C22—22no—
307C22—531no—
308C22—531no—
309C22—532no—
310C22—532no—
311C22—532no—
312C22—1062no—
313C22—1062no—
314C22—1064no—
315C22—1062no—
316C22C22×C4×#615712yes—
317C22C24×#615711yescharacteristic
318C4—28no—
319C4—534no—
320C4—5312no—
321C4—1062no—
322C4—10612no—
323C4—10616no—
324C4—1068no—
325C4C22×C4×#6157112yes—
326C2—22no—
327C2—531no—
328C2—531no—
329C2—532no—
330C2—1062no—
331C2C23×C4×#615711yescharacteristic
332C22−1+4×#615712yes—
subgroupG/H
center, FrattiniC22C24×#6157
commutatorC2×C53C23×C42
FittingC53×#519C4
socleC22×C53C24×C4
prime radicalC23×D53C24
cosocleC22×D53C25
hypercenter#519#6157
hypocenterC53C4×#519
LCSG ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
3. #20291155C2semidirect
5. C4×Q8×#6157C2semidirect
8. #16811691C4semidirect
9. C2×Q8×#6157C4semidirect
25. #16811101C4semidirect
26. #16811100C4semidirect
27. #16811099C4semidirect
30. C53×#519C4semidirect
35. Q8×#6157C2×C4semidirect
55. Q8×Dic53C2×C4semidirect
56. C4×Dic106C2×C4semidirect
57. #13133182C2×C4semidirect
66. C4×Q8×C53C2×C4semidirect
77. C2×Q8×C53C42semidirect
81. Q8×D53C42semidirect
89. C2×Dic106C42semidirect
155. #6157#519direct (N × H)
179. #519#6157direct (N × H)
197. C53C4×#519semidirect
213. C4×Q8C2×#6157semidirect
241. C2×Q8C4×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168116912
C22×Q8×#61572
#202911552
#202911522
C4×Q8×#61572
#117×#61572
#519×D532
C4×#51953