Finite Group Databasebeta

← #17968197

Subgroups — #17968197

Order 4352.

2158 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763878C212yesmaximal
2#14544472C216yesmaximal, complemented
3#14544470C213yesmaximal, complemented
4C2×#11763683C213yesmaximal, complemented
5C2×#131×D17C211yesmaximal, characteristic
6#11763878—28no—
7#11763878C4116yescomplemented
8C2×Q8×#595—22no—
9C2×D4×#595—26no—
10C2×#10587677C2216yes—
11C22×C4⋊#595C2216yes—
12#11763683C2213yes—
13#11763683C2216yes—
14#11763683C2211yescharacteristic
15#11763683C2216yes—
16C2×#41×D17C2211yescharacteristic
17C2×#10587593C413yescomplemented
18C2×#10587634C411yescharacteristic, complemented
19C2×#10587587C413yescomplemented
20#131×D17—28no—
21C2×M16×D17C2213yes—
22C22×C8×D17C2213yes—
23C2×C17×#131C411yescharacteristic, complemented
24C23×#595—26no—
25C2×#41×C17C2×C411yescharacteristic
26#41×D17C2×C418yes—
27#41×D17D418yes—
28C2×#63256Q811yescharacteristic
29C2×Q8×D17D411yescharacteristic
30C2×#63254Q813yes—
31C2×D4×D17D413yes—
32C2×#63252C2×C413yes—
33C22×C4×D17C2313yes—
34C2×C22⋊#595—26no—
35Q8×#595—216no—
36D4×#595—248no—
37#10587677—212no—
38#10587677C2×C4124yescomplemented
39C2×C4⋊#595—26no—
40C2×C4⋊#595—212no—
41C2×C4⋊#595C2×C4124yescomplemented
42C2×C4⋊#595C2316yes—
43C2×C4⋊#595C2312yes—
44C2×C4×#595—26no—
45C17×#131—24no—
46C2×M16×C17C2×C413yescomplemented
47C22×C8×C17C2×C413yescomplemented
48#10587634—24no—
49C2×#63223C2×C413yescomplemented
50C22×#3622C2×C413yescomplemented
51#10587593—212no—
52M16×D17—212no—
53#10587587—212no—
54C2×#63218C2×C416yescomplemented
55C2×C8×D17—212no—
56C2×C8×D17C2×C413yescomplemented
57C2×C8×D17C2×C413yescomplemented
58C2×C8×D17C2311yescharacteristic
59C2×C8×D17C2313yes—
60C23×D17C2×D413yes—
61C22×#595—22no—
62C22×#595—26no—
63C22×#595—224no—
64C22×#595—86no—
65C22×#595—86no—
66#41×C17C4214yes—
67#41×C17C22⋊C414yes—
68C2×Q8×C17C4⋊C411yescharacteristic
69C2×D4×C17C4⋊C413yes—
70C22×C4×C17C22×C413yes—
71C2×C22⋊D17C4⋊C416yes—
72C22×Dic17C2×Q813yes—
73#63256C4⋊C418yes—
74Q8×D17C22⋊C418yes—
75#63254C4⋊C4124yes—
76D4×D17C22⋊C4124yes—
77#63252C42112yes—
78#63252C22⋊C4112yes—
79C2×D68C4⋊C413yes—
80C2×C4×D17C22×C413yes—
81C2×C4×D17C22×C4112yes—
82C2×C4×D17C2×D413yes—
83C2×C4×D17C2×D4112yes—
84C2×C4×D17C2×Q813yes—
85C2×C4×D17C2411yescharacteristic
86C2×Dic34C4⋊C413yes—
87C22⋊#595—224no—
88C4⋊#595—224no—
89C4⋊#595—26no—
90C4⋊#595—22no—
91C4⋊#595C22×C4112yes—
92C4⋊#595C22×C414yes—
93C4×#595—224no—
94M16×C17—26no—
95C2×C8×C17—26no—
96C2×C8×C17C22×C411yescharacteristic
97C2×C8×C17C22×C413yes—
98#63223—26no—
99C2×#3622—26no—
100C2×#3622C22×C411yescharacteristic
101C2×#3622C22×C413yes—
102#63218—212no—
103C8×D17—26no—
104C8×D17—26no—
105C8×D17—22no—
106C8×D17—26no—
107#20348—171nomaximal, Sylow
108C23×C17C2×C4⋊C413yes—
109C22×D17—23no—
110C22×D17—26no—
111C22×D17—23no—
112C22×D17C2×C22⋊C4112yes—
113C22×D17C2×C4⋊C413yes—
114C22×D17C22×D411yescharacteristic
115C2×#595—24no—
116C2×#595—212no—
117C2×#595—46no—
118C2×#595—42no—
119C2×#595—46no—
120C2×#595—42no—
121C2×#595—824no—
122Q8×C17#9514yes—
123D4×C17#95112yes—
124C2×C4×C17C2×C4216yes—
125C2×C4×C17C2×C22⋊C416yes—
126C2×C4×C17C2×C4⋊C413yes—
127C2×C4×C17C23×C411yescharacteristic
128C22⋊D17#95124yes—
129C2×Dic17C2×C4⋊C4112yes—
130C2×Dic17C2×C4⋊C413yes—
131C2×Dic17C22×Q811yescharacteristic
132D68#95112yes—
133C4×D17C2×C4216yes—
134C4×D17C2×C22⋊C416yes—
135C4×D17C2×C22⋊C4112yes—
136C4×D17C2×C4⋊C4112yes—
137C4×D17C23×C412yes—
138C4×D17C22×D412yes—
139Dic34#95112yes—
140C8×C17—21no—
141C8×C17—23no—
142#3622—21no—
143#3622—23no—
144C22×#131—171no—
145C2×#520—172no—
146#1736—176no—
147#1735—173no—
148C2×#340—173no—
149C22×C17—23no—
150C22×C17—23no—
151C22×C17C2×#9516yes—
152C22×C17C22×C4⋊C411yescharacteristic
153C2×D17—23no—
154C2×D17—23no—
155C2×D17—212no—
156C2×D17—212no—
157C2×D17C2×#9516yes—
158C2×D17C22×C22⋊C412yes—
159C2×D17#57611yescharacteristic
160C2×D17#57712yes—
161C2×D17#57811yescharacteristic
162#595—412no—
163#595—44no—
164C4×C17C2×#9516yes—
165C4×C17C22×C4211yescharacteristic
166C4×C17C22×C22⋊C411yescharacteristic
167Dic17C2×#9516yes—
168Dic17C22×C4⋊C412yes—
169C22×#41—171no—
170C2×#131—171no—
171C2×#131—173no—
172C2×#131—173no—
173C2×#131—348no—
174C2×#131#59511yescharacteristic, complemented
175C22×M16—173no—
176C23×C8—173no—
177#520—1716no—
178#520—348no—
179C2×C4×Q8—342no—
180C2×C4×D4—346no—
181C2×#117—176no—
182C22×C4⋊C4—176no—
183#340—173no—
184#340—176no—
185#340—171no—
186#340—176no—
187C2×C17—26no—
188C2×C17C22×#9511yescharacteristic
189C2×C17#291211yescharacteristic
190C2×C17#291311yescharacteristic
191D17—26no—
192D17#289712yes—
193D17#289812yes—
194C2×#41—178no—
195C2×#41—178no—
196C2×#41—171no—
197C2×#41—173no—
198C2×#41—173no—
199C2×#41C2×#59511yescharacteristic
200C22×Q8—171no—
201C22×D4—173no—
202C23×C4—173no—
203C23×C4—346no—
204#131—24no—
205#131—3412no—
206#131—3412no—
207#131—344no—
208C2×M16—173no—
209C2×M16—176no—
210C2×M16—3412no—
211C2×M16C2×#59513yescomplemented
212C22×C8—173no—
213C22×C8—171no—
214C22×C8—173no—
215C22×C8—173no—
216C22×C8—173no—
217C22×C8—3412no—
218C22×C8C2×#59513yescomplemented
219C4×Q8—3416no—
220C4×D4—3448no—
221#117—1724no—
222#117—3412no—
223C2×C4⋊C4—176no—
224C2×C4⋊C4—1724no—
225C2×C4⋊C4—172no—
226C2×C4⋊C4—346no—
227C2×C4⋊C4—3412no—
228C2×C22⋊C4—346no—
229C2×C42—346no—
230C17#2034811yescharacteristic, complemented, Sylow
231C24—173no—
232#41—178no—
233#41—1724no—
234#41—1712no—
235#41—1712no—
236#41C4×#59514yes—
237#41C22⋊#59514yes—
238C2×Q8—178no—
239C2×Q8—173no—
240C2×Q8C4⋊#59511yescharacteristic
241C2×D4—176no—
242C2×D4—1724no—
243C2×D4—173no—
244C2×D4C4⋊#59513yes—
245C22×C4—173no—
246C22×C4—1712no—
247C22×C4—173no—
248C22×C4—173no—
249C22×C4—1712no—
250C22×C4—173no—
251C22×C4—171no—
252C22×C4—342no—
253C22×C4—346no—
254C22×C4—3424no—
255C22×C4—1366no—
256C22×C4—1366no—
257C22×C4C22×#59513yes—
258M16—26no—
259M16—3412no—
260M16—346no—
261C2×C8—26no—
262C2×C8—171no—
263C2×C8—173no—
264C2×C8—342no—
265C2×C8—346no—
266C2×C8—346no—
267C2×C8—346no—
268C2×C8—346no—
269C2×C8C22×#59511yescharacteristic
270C2×C8C22×#59513yes—
271C4⋊C4—1712no—
272C4⋊C4—174no—
273C4⋊C4—346no—
274C4⋊C4—3424no—
275C4⋊C4—342no—
276C22⋊C4—3424no—
277C42—3424no—
278C23—171no—
279C23—1712no—
280C23—173no—
281C23—343no—
282C23—346no—
283C23—343no—
284C23C2×C4⋊#59513yes—
285Q8—1712no—
286Q8(C2×#595)⋊C414yes—
287D4—1724no—
288D4—1712no—
289D4(C2×#595)⋊C4112yes—
290C2×C4—1712no—
291C2×C4—171no—
292C2×C4—173no—
293C2×C4—1712no—
294C2×C4—1712no—
295C2×C4—172no—
296C2×C4—176no—
297C2×C4—172no—
298C2×C4—176no—
299C2×C4—344no—
300C2×C4—3412no—
301C2×C4—686no—
302C2×C4—686no—
303C2×C4—682no—
304C2×C4—682no—
305C2×C4—13624no—
306C2×C4C2×C4×#59516yes—
307C2×C4C2×C4⋊#59513yes—
308C2×C4C2×C22⋊#59516yes—
309C2×C4C23×#59511yescharacteristic
310C8—21no—
311C8—23no—
312C8—341no—
313C8—343no—
314C22—23no—
315C22—23no—
316C22—172no—
317C22—171no—
318C22—171no—
319C22—172no—
320C22—176no—
321C22—3412no—
322C22—3412no—
323C22—343no—
324C22—343no—
325C22C2×(C2×#595)⋊C416yes—
326C22C22×C4⋊#59511yescharacteristic
327C4—172no—
328C4—176no—
329C4—6812no—
330C4—684no—
331C4C2×(C2×#595)⋊C416yes—
332C4C22×C4×#59511yescharacteristic
333C4C22×C22⋊#59511yescharacteristic
334C2—26no—
335C2—172no—
336C2—172no—
337C2—346no—
338C2C22×(C2×#595)⋊C411yescharacteristic
339C2#1454543811yescharacteristic
340C2#1454543911yescharacteristic
subgroupG/H
centerC22C22×C4⋊#595
commutatorC4×C17C22×C42
FrattiniC2×C4C23×#595
FittingC2×C17×#131C4
socleC22×C17C22×C4⋊C4
prime radicalC2×#41×D17C22
cosocleC2×C4×D17C24
hypercenterC2×#131#595
hypocenterC17#20348
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
2. #14544472C2semidirect
3. #14544470C2semidirect
4. C2×#11763683C2semidirect
7. #11763878C4semidirect
17. C2×#10587593C4semidirect
18. C2×#10587634C4semidirect
19. C2×#10587587C4semidirect
23. C2×C17×#131C4semidirect
38. #10587677C2×C4semidirect
41. C2×C4⋊#595C2×C4semidirect
46. C2×M16×C17C2×C4semidirect
47. C22×C8×C17C2×C4semidirect
49. C2×#63223C2×C4semidirect
50. C22×#3622C2×C4semidirect
54. C2×#63218C2×C4semidirect
56. C2×C8×D17C2×C4semidirect
57. C2×C8×D17C2×C4semidirect
174. C2×#131#595semidirect
211. C2×M16C2×#595semidirect
218. C22×C8C2×#595semidirect
230. C17#20348semidirect

Maximal subgroups

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

subgroupindex
C2×#117638782
#145444722
#145444702
C2×#117636832
C2×#131×D172
#2034817