Finite Group Databasebeta

← #519×#595 #17978869

Subgroups — #519×#595 #17978869

Order 4352.

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×#11763877C211yesmaximal, characteristic
2C22×Q8×#595C211yesmaximal, characteristic
3#14545224C218yesmaximal, complemented
4#14545221C216yesmaximal
5C4×Q8×#595C218yesmaximal, complemented
6#117×#595C216yesmaximal
7D17×#519C211yesmaximal, characteristic
8#11763877C4116yescomplemented
9C2×Q8×#595C4116yescomplemented
10C2×Q8×#595C2214yes—
11C2×Q8×#595C2214yes—
12C2×Q8×#595C2218yes—
13C2×#10587677C2216yes—
14C22×C4⋊#595C2213yes—
15C22×C4×#595C2213yes—
16C4⋊C4×#595C22112yes—
17C4⋊C4×#595C22112yes—
18C4⋊C4×#595C22124yes—
19C4×C4⋊#595C22124yes—
20C22⋊C4×#595C2214yes—
21C22⋊C4×#595C22112yes—
22#11763745C22112yes—
23C42×#595C22112yes—
24C22×Q8×D17C2211yescharacteristic
25#11763287C411yescharacteristic, complemented
26#11763286C416yescomplemented
27#11763285C418yescomplemented
28C4×Q8×D17C2218yes—
29#117×D17C2216yes—
30C17×#519C411yescharacteristic, complemented
31C23×#595C2311yescharacteristic
32C23×#595C2313yes—
33C22×Q8×C17C2×C411yescharacteristic
34C2×Q8×D17C2×C418yes—
35C2×Q8×D17C2×C418yes—
36C2×Q8×D17C2×C414yes—
37C2×Q8×D17C2314yes—
38C22×C4×D17C2313yes—
39C22×Dic34C2×C413yes—
40C2×C22⋊#595C2314yes—
41Q8×#595C2×C4132yes—
42Q8×#595C2×C4132yes—
43Q8×#595C2×C4164yescomplemented
44#10587677C2×C4148yes—
45C2×C4⋊#595C2×C4124yes—
46C2×C4⋊#595C23112yes—
47C2×C4⋊#595C23112yes—
48C2×C4×#595C2×C4124yes—
49C2×C4×#595C2318yes—
50C2×C4×#595C23124yes—
51C2×C4×#595C2316yes—
52C2×C4×#595C23112yes—
53C2×C4×#595C2316yes—
54C4×Q8×C17C2×C418yescomplemented
55C17×#117C2×C416yes—
56Q8×Dic17C2×C418yescomplemented
57#10587612C2×C416yes—
58#10587578C2×C4112yes—
59C4⋊C4×D17C23112yes—
60#10587573C2×C4124yescomplemented
61C22⋊C4×D17C2314yes—
62#10587563C2×C4112yes—
63#10587558C2×C4112yes—
64C42×D17C23112yes—
65C4×Dic34C2×C4124yescomplemented
66(C2×#595)⋊C4C2318yes—
67(C2×#595)⋊C4C23124yes—
68C23×D17C2411yescharacteristic
69C22×#595—24no—
70C22×#595—46no—
71C22×#595C22×C414yes—
72C22×#595C22×C4112yes—
73C22×#595C2418yes—
74C22×#595C2416yes—
75C22×#595C2411yescharacteristic
76C22×#595C2411yescharacteristic
77C2×Q8×C17C4218yescomplemented
78C2×Q8×C17C22×C414yes—
79C22×C4×C17C22×C413yes—
80C22×Dic17C22×C413yes—
81Q8×D17C42132yescomplemented
82Q8×D17C22×C4116yes—
83Q8×D17C22×C4116yes—
84C2×C4×D17C22×C416yes—
85C2×C4×D17C22×C4112yes—
86C2×C4×D17C22×C4112yes—
87C2×C4×D17C2418yes—
88C2×C4×D17C2416yes—
89C2×Dic34C42124yescomplemented
90C2×Dic34C22×C4112yes—
91C22⋊#595C22×C4116yes—
92C4⋊#595C22×C4148yes—
93C4⋊#595C22×C4148yes—
94C4×#595—216no—
95C4×#595—424no—
96C4×#595C22×C4124yes—
97C4×#595C22×C4124yes—
98C4×#595C22×C4148yes—
99C4⋊C4×C17C22×C4112yes—
100C22⋊C4×C17C22×C414yes—
101C42×C17C22×C4112yes—
102C22⋊Dic17C22×C414yes—
103#63227C22×C418yes—
104C4⋊Dic17C22×C4112yes—
105#63225C22×C4124yes—
106C4×Dic17C22×C4124yes—
107C4×Dic17C22×C4112yes—
108C4×#519—171nomaximal, Sylow
109C23×C17C23×C411yescharacteristic
110C22×D17—22no—
111C22×D17—22no—
112C22×D17C23×C412yes—
113C22×D17C23×C412yes—
114C22×D17C23×C411yescharacteristic
115C22×D17C2511yescharacteristic
116C2×#595—28no—
117C2×#595—22no—
118C2×#595—28no—
119C2×#595—212no—
120C2×#595—28no—
121C2×#595—412no—
122C2×#595C23×C412yes—
123C2×#595C23×C4116yes—
124C2×#595C23×C4112yes—
125C2×#595C23×C412yes—
126C2×#5952−1+414yes—
127Q8×C17C2×C42116yes—
128C2×C4×C17C2×C42112yes—
129C2×C4×C17C23×C418yes—
130C2×C4×C17C23×C416yes—
131C2×Dic17C2×C42112yes—
132C2×Dic17C23×C418yes—
133C2×Dic17C23×C416yes—
134C4×D17—28no—
135C4×D17—28no—
136C4×D17C2×C42124yes—
137C4×D17C23×C4112yes—
138C4×D17C23×C4112yes—
139Dic34C2×C42148yes—
140C2×#519—171no—
141C2×#519—171no—
142C22×C4×Q8—171no—
143#2276—178no—
144#2274—176no—
145C42×Q8—178no—
146C4×#117—176no—
147C22×C17—22no—
148C22×C17C22×C4212yes—
149C22×C17C24×C411yescharacteristic
150C2×D17—22no—
151C2×D17—22no—
152C2×D17—22no—
153C2×D17—24no—
154C2×D17C22×C4212yes—
155C2×D17C24×C411yescharacteristic
156C2×D17C24×C411yescharacteristic
157C2×D17C2×2−1+412yes—
158C2×D17#58512yes—
159#595—22no—
160#595—212no—
161#595—216no—
162#595#51914yesdirect product factor
163C4×C17—28no—
164C4×C17C22×C42112yes—
165Dic17—28no—
166Dic17C22×C42112yes—
167C23×Q8—171no—
168#519—1716no—
169#519—171no—
170#519—178no—
171#519—176no—
172#519#59511yescharacteristic, direct product factor
173C2×C4×Q8—1716no—
174C2×C4×Q8—174no—
175C2×C4×Q8—174no—
176C2×C4×Q8—178no—
177C2×C4×Q8—178no—
178C2×#117—176no—
179C2×#117—176no—
180C22×C4⋊C4—173no—
181C22×C42—173no—
182C4×C4⋊C4—1724no—
183C4×C4⋊C4—1712no—
184C4×C4⋊C4—1712no—
185C4×C4⋊C4—1724no—
186C4×C22⋊C4—174no—
187C4×C22⋊C4—1712no—
188#376—1712no—
189C43—1712no—
190C2×C17—22no—
191C2×C17C23×C4211yescharacteristic
192C2×C17C4×2−1+412yes—
193D17—22no—
194D17C2×#51911yescharacteristic
195D17#342811yescharacteristic
196D17C4×2−1+412yes—
197C22×Q8—178no—
198C22×Q8—178no—
199C22×Q8—174no—
200C22×Q8—174no—
201C22×Q8—173no—
202C22×Q8C2×#59511yescharacteristic
203C23×C4—173no—
204C23×C4—171no—
205C23×C4—173no—
206C4×Q8—1732no—
207C4×Q8—1732no—
208C4×Q8—1764no—
209C4×Q8—178no—
210C4×Q8—1724no—
211C4×Q8—1724no—
212C4×Q8C2×#59518yescomplemented
213#117—1748no—
214#117—176no—
215#117—1712no—
216#117—1712no—
217#117—1712no—
218#117C2×#59516yes—
219C2×C4⋊C4—1724no—
220C2×C4⋊C4—1712no—
221C2×C4⋊C4—1712no—
222C2×C4⋊C4—1712no—
223C2×C22⋊C4—174no—
224C2×C22⋊C4—174no—
225C2×C42—1712no—
226C2×C42—1724no—
227C2×C42—176no—
228C2×C42—176no—
229C2×C42—1712no—
230C2×C42—178no—
231C2×C42—1724no—
232#95—178no—
233#95—1724no—
234C17C4×#51911yescharacteristic, complemented, Sylow
235C24—171no—
236C2×Q8—1732no—
237C2×Q8—1716no—
238C2×Q8—1716no—
239C2×Q8—1724no—
240C2×Q8—1712no—
241C2×Q8C4×#59518yescomplemented
242C2×Q8C22×#59514yes—
243C22×C4—171no—
244C22×C4—173no—
245C22×C4—1712no—
246C22×C4—1712no—
247C22×C4—176no—
248C22×C4—176no—
249C22×C4—174no—
250C22×C4—171no—
251C22×C4—178no—
252C22×C4—176no—
253C22×C4—1712no—
254C22×C4—178no—
255C22×C4—344no—
256C22×C4—686no—
257C22×C4C22×#59513yes—
258C4⋊C4—1748no—
259C4⋊C4—1748no—
260C4⋊C4—1724no—
261C4⋊C4—1712no—
262C4⋊C4C22×#595112yes—
263C22⋊C4—1716no—
264C22⋊C4—174no—
265C22⋊C4—178no—
266C22⋊C4C22×#59514yes—
267C42—1724no—
268C42—1724no—
269C42—1748no—
270C42—1724no—
271C42—1712no—
272C42—3416no—
273C42—6824no—
274C42C22×#595112yes—
275C23—171no—
276C23—172no—
277C23—172no—
278C23—171no—
279C23—342no—
280C23—342no—
281C23C23×#59511yescharacteristic
282Q8—1748no—
283Q8C2×C4×#595116yes—
284C2×C4—172no—
285C2×C4—174no—
286C2×C4—1712no—
287C2×C4—176no—
288C2×C4—1712no—
289C2×C4—1712no—
290C2×C4—1724no—
291C2×C4—172no—
292C2×C4—1716no—
293C2×C4—1712no—
294C2×C4—178no—
295C2×C4—348no—
296C2×C4—348no—
297C2×C4—348no—
298C2×C4—348no—
299C2×C4—348no—
300C2×C4—3412no—
301C2×C4—342no—
302C2×C4—6812no—
303C2×C4C2×C4×#595112yes—
304C2×C4C23×#59518yes—
305C2×C4C23×#59516yes—
306C22—22no—
307C22—171no—
308C22—171no—
309C22—172no—
310C22—172no—
311C22—172no—
312C22—342no—
313C22—342no—
314C22—344no—
315C22—342no—
316C22C22×C4×#59512yes—
317C22C24×#59511yescharacteristic
318C4—28no—
319C4—174no—
320C4—1712no—
321C4—342no—
322C4—3412no—
323C4—3416no—
324C4—348no—
325C4C22×C4×#595112yes—
326C2—22no—
327C2—171no—
328C2—171no—
329C2—172no—
330C2—342no—
331C2C23×C4×#59511yescharacteristic
332C22−1+4×#59512yes—
subgroupG/H
center, FrattiniC22C24×#595
commutatorC2×C17C23×C42
FittingC17×#519C4
socleC22×C17C24×C4
prime radicalC23×D17C24
cosocleC22×D17C25
hypercenter#519#595
hypocenterC17C4×#519
LCSG ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
3. #14545224C2semidirect
5. C4×Q8×#595C2semidirect
8. #11763877C4semidirect
9. C2×Q8×#595C4semidirect
25. #11763287C4semidirect
26. #11763286C4semidirect
27. #11763285C4semidirect
30. C17×#519C4semidirect
43. Q8×#595C2×C4semidirect
54. C4×Q8×C17C2×C4semidirect
56. Q8×Dic17C2×C4semidirect
60. #10587573C2×C4semidirect
65. C4×Dic34C2×C4semidirect
77. C2×Q8×C17C42semidirect
81. Q8×D17C42semidirect
89. C2×Dic34C42semidirect
162. #595#519direct (N × H)
172. #519#595direct (N × H)
212. C4×Q8C2×#595semidirect
234. C17C4×#519semidirect
241. C2×Q8C4×#595semidirect

Maximal subgroups

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

subgroupindex
C2×#117638772
C22×Q8×#5952
#145452242
#145452212
C4×Q8×#5952
#117×#5952
D17×#5192
C4×#51917