Finite Group Databasebeta

← C22×#11763674 #17968147

Subgroups — C22×#11763674 #17968147

Order 4352.

3086 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#63242C211yesmaximal, characteristic
2C23×C4×#595C211yesmaximal, characteristic
3C2×#11763674C2112yesmaximal, direct product factor
4C23×C8×D17C211yesmaximal, characteristic
5C24×#595C412yes—
6C22×#63242—24no—
7C22×#63242—24no—
8C22×#63242C416yescomplemented
9C22×#63242C2216yes—
10C23×#3624C412yescomplemented
11C22×C4×#595—24no—
12C22×C4×#595—24no—
13C22×C4×#595C416yes—
14C22×C4×#595C2216yes—
15#11763674C22116yesdirect product factor
16C23×C4×D17C2211yescharacteristic
17C23×#3622C411yescharacteristic, complemented
18C22×C8×D17—24no—
19C22×C8×D17—24no—
20C22×C8×D17C416yescomplemented
21C22×C8×D17C2216yes—
22C23×C8×C17C411yescharacteristic, complemented
23C24×D17C2×C411yescharacteristic
24C23×#595—28no—
25C23×#595—28no—
26C23×#595C8116yescomplemented
27C23×#595C2×C4112yes—
28C23×C4×C17C2×C411yescharacteristic
29C23×Dic17C2×C411yescharacteristic
30C22×C4×D17C2×C413yes—
31C22×C4×D17C2×C413yes—
32C22×C4×D17C2×C413yes—
33C22×C4×D17D414yes—
34C22×C4×D17D414yes—
35C22×C4×D17D414yes—
36C22×C4×D17Q814yes—
37C22×C4×D17C2313yes—
38C22×#3624—28no—
39C22×#3624C2×C4112yescomplemented
40C2×C4×#595—224no—
41C2×C4×#595—212no—
42C2×C4×#595—212no—
43C2×C4×#595C2×C4112yes—
44C2×C4×#595C2314yes—
45C2×#63242—224no—
46C2×#63242—212no—
47C2×#63242—212no—
48C2×#63242C2×C4112yescomplemented
49C2×#63242C2314yes—
50C22×C8×C17—24no—
51C22×C8×C17C2×C416yescomplemented
52C22×#3622—24no—
53C22×#3622C2×C416yescomplemented
54C2×C8×D17—224no—
55C2×C8×D17—212no—
56C2×C8×D17—212no—
57C2×C8×D17C2×C4112yescomplemented
58C2×C8×D17C2314yes—
59C24×C17C4211yescharacteristic
60C23×D17C4213yes—
61C23×D17C22⋊C414yes—
62C23×D17C4⋊C414yes—
63C23×D17C2×C818yes—
64C23×D17M1618yes—
65C23×D17C22×C413yes—
66C22×#595—264no—
67C22×#595—224no—
68C22×#595—224no—
69C22×#595C2×C8148yescomplemented
70C22×#595C22×C418yes—
71C22×C4×C17C4213yes—
72C22×C4×C17C22⋊C414yes—
73C22×C4×C17C4⋊C414yes—
74C22×C4×C17C22×C413yes—
75C22×Dic17C4213yes—
76C22×Dic17C22⋊C414yes—
77C22×Dic17C4⋊C414yes—
78C22×Dic17C22×C413yes—
79C2×C4×D17C4216yes—
80C2×C4×D17C22⋊C4112yes—
81C2×C4×D17C22⋊C4112yes—
82C2×C4×D17C22⋊C4112yes—
83C2×C4×D17C4⋊C4112yes—
84C2×C4×D17C4⋊C4112yes—
85C2×C4×D17C4⋊C4112yes—
86C2×C4×D17C22×C413yes—
87C2×C4×D17C22×C413yes—
88C2×C4×D17C22×C413yes—
89C2×C4×D17C2×D416yes—
90C2×C4×D17C2×D416yes—
91C2×C4×D17C2×D416yes—
92C2×C4×D17C2×Q816yes—
93C2×C4×D17C2411yescharacteristic
94C2×#3624—224no—
95C2×#3624C22×C418yescomplemented
96C4×#595—224no—
97C4×#595—24no—
98C4×#595—24no—
99#63242—224no—
100#63242—24no—
101#63242—24no—
102C2×C8×C17—212no—
103C2×C8×C17C22×C414yescomplemented
104C2×#3622—212no—
105C2×#3622C22×C414yescomplemented
106C8×D17—224no—
107C8×D17—24no—
108C8×D17—24no—
109C22×#336—171nomaximal, Sylow
110C23×C17#9514yes—
111C23×C17C4×C814yes—
112C23×C17#9714yes—
113C23×C17C2×C4213yes—
114C22×D17#95112yes—
115C22×D17C4×C8112yes—
116C22×D17#97112yes—
117C22×D17C22⋊C8132yes—
118C22×D17C4⋊C8132yes—
119C22×D17C2×C4213yes—
120C22×D17C2×C22⋊C416yes—
121C22×D17C2×C4⋊C416yes—
122C22×D17C22×C8112yes—
123C22×D17C2×M16112yes—
124C22×D17C23×C411yescharacteristic
125C2×#595—28no—
126C2×#595—28no—
127C2×#595—296no—
128C2×#595C22×C8116yescomplemented
129C2×C4×C17#95112yes—
130C2×C4×C17C2×C4213yes—
131C2×C4×C17C2×C22⋊C416yes—
132C2×C4×C17C2×C4⋊C416yes—
133C2×C4×C17C23×C411yescharacteristic
134C2×Dic17#95112yes—
135C2×Dic17C2×C4213yes—
136C2×Dic17C2×C22⋊C416yes—
137C2×Dic17C2×C4⋊C416yes—
138C2×Dic17C23×C411yescharacteristic
139C4×D17#95124yes—
140C4×D17C2×C22⋊C416yes—
141C4×D17C2×C22⋊C416yes—
142C4×D17C2×C22⋊C416yes—
143C4×D17C2×C4⋊C416yes—
144C4×D17C2×C4⋊C416yes—
145C4×D17C2×C4⋊C416yes—
146C4×D17C22×D411yescharacteristic
147C4×D17C22×D411yescharacteristic
148C4×D17C22×D411yescharacteristic
149C4×D17C22×Q811yescharacteristic
150#3624—28no—
151C8×C17—24no—
152#3622—24no—
153C24×C8—171no—
154C24×C8—171no—
155C23×C42—171no—
156C2×#336—1712no—
157C22×C17#336116yescomplemented
158C22×C17C2×#9516yes—
159C22×C17C2×C4×C816yes—
160C22×C17C2×#9716yes—
161C22×C17C22×C4211yescharacteristic
162C2×D17#336148yescomplemented
163C2×D17C2×#9516yes—
164C2×D17C2×C4×C816yes—
165C2×D17C2×#9716yes—
166C2×D17C2×C22⋊C8124yes—
167C2×D17C2×C4⋊C8124yes—
168C2×D17C22×C22⋊C411yescharacteristic
169C2×D17C22×C4⋊C411yescharacteristic
170C2×D17C23×C812yes—
171C2×D17C22×M1612yes—
172#595—216no—
173C4×C17C2×#9516yes—
174C4×C17C22×C22⋊C411yescharacteristic
175C4×C17C22×C4⋊C411yescharacteristic
176Dic17C2×#9516yes—
177Dic17C22×C22⋊C411yescharacteristic
178Dic17C22×C4⋊C411yescharacteristic
179C24×C4—171no—
180C24×C4—172no—
181C23×C8—172no—
182C23×C8—176no—
183C23×C8—176no—
184C23×C8—171no—
185C23×C8—176no—
186C23×C8—176no—
187C23×C8—344no—
188C23×C8—344no—
189C23×C8—344no—
190C23×C8—344no—
191C23×C8#59511yescharacteristic, complemented
192C22×C42—176no—
193C22×C42—176no—
194C22×C42—344no—
195C22×C42—344no—
196#336—1716no—
197C2×C17C2×#336112yescomplemented
198C2×C17C22×#9511yescharacteristic
199C2×C17C22×C4×C811yescharacteristic
200C2×C17C22×#9711yescharacteristic
201D17C2×#336112yescomplemented
202D17C22×C22⋊C812yes—
203D17C22×C4⋊C812yes—
204C25—171no—
205C23×C4—171no—
206C23×C4—173no—
207C23×C4—173no—
208C23×C4—174no—
209C23×C4—174no—
210C23×C4—174no—
211C23×C4—174no—
212C23×C4—173no—
213C23×C4—173no—
214C23×C4—1716no—
215C23×C4—1712no—
216C23×C4—348no—
217C23×C4—348no—
218C23×C4C2×#59511yescharacteristic
219C22×C8—24no—
220C22×C8—1712no—
221C22×C8—174no—
222C22×C8—1712no—
223C22×C8—176no—
224C22×C8—174no—
225C22×C8—1712no—
226C22×C8—3424no—
227C22×C8—3412no—
228C22×C8—3412no—
229C22×C8—3412no—
230C22×C8—3412no—
231C22×C8—3424no—
232C22×C8—348no—
233C22×C8—344no—
234C22×C8C2×#59516yescomplemented
235C2×C42—1712no—
236C2×C42—174no—
237C2×C42—3412no—
238C2×C42—3412no—
239C2×C42—3424no—
240C17C22×#33611yescharacteristic, complemented, Sylow
241C24—173no—
242C24—173no—
243C24—178no—
244C24—178no—
245C24—174no—
246C24—174no—
247C24C4×#59511yescharacteristic
248C22×C4—173no—
249C22×C4—173no—
250C22×C4—174no—
251C22×C4—174no—
252C22×C4—176no—
253C22×C4—173no—
254C22×C4—173no—
255C22×C4—1712no—
256C22×C4—176no—
257C22×C4—176no—
258C22×C4—1712no—
259C22×C4—176no—
260C22×C4—176no—
261C22×C4—173no—
262C22×C4—171no—
263C22×C4—1712no—
264C22×C4—1712no—
265C22×C4—1712no—
266C22×C4—1712no—
267C22×C4—1748no—
268C22×C4—178no—
269C22×C4—3424no—
270C22×C4—3424no—
271C22×C4—3464no—
272C22×C4C4×#59513yes—
273C22×C4C4⋊#59514yes—
274C22×C4C22⋊#59514yes—
275C22×C4C22×#59513yes—
276C2×C8—212no—
277C2×C8—178no—
278C2×C8—174no—
279C2×C8—344no—
280C2×C8—344no—
281C2×C8—3424no—
282C2×C8—3424no—
283C2×C8—344no—
284C2×C8—344no—
285C2×C8—3424no—
286C2×C8—3412no—
287C2×C8C22×#59514yescomplemented
288C42—3424no—
289C42—344no—
290C42—344no—
291C23—176no—
292C23—1712no—
293C23—176no—
294C23—1712no—
295C23—1712no—
296C23—1712no—
297C23—1712no—
298C23—171no—
299C23—173no—
300C23—1732no—
301C23—1732no—
302C23C8×#59514yes—
303C23#1058753214yes—
304C23(C2×#595)⋊C414yes—
305C23C2×C4×#59513yes—
306C2×C4—176no—
307C2×C4—1712no—
308C2×C4—176no—
309C2×C4—171no—
310C2×C4—173no—
311C2×C4—176no—
312C2×C4—176no—
313C2×C4—1724no—
314C2×C4—176no—
315C2×C4—176no—
316C2×C4—171no—
317C2×C4—171no—
318C2×C4—176no—
319C2×C4—171no—
320C2×C4—176no—
321C2×C4—171no—
322C2×C4—1716no—
323C2×C4—348no—
324C2×C4—3496no—
325C2×C4—348no—
326C2×C4(C2×#595)⋊C4112yes—
327C2×C4C2×C4×#59513yes—
328C2×C4C2×C4⋊#59516yes—
329C2×C4C2×C22⋊#59516yes—
330C2×C4C23×#59511yescharacteristic
331C8—24no—
332C8—348no—
333C8—344no—
334C22—171no—
335C22—171no—
336C22—176no—
337C22—1724no—
338C22—1748no—
339C22—1724no—
340C22—172no—
341C22—176no—
342C22—172no—
343C22—176no—
344C22#11763674116yesdirect product factor
345C22C2×(C2×#595)⋊C416yes—
346C22C2×C8×#59516yes—
347C22C2×#1058753216yes—
348C22C22×C4×#59511yescharacteristic
349C4—171no—
350C4—171no—
351C4—176no—
352C4—3416no—
353C4C2×(C2×#595)⋊C416yes—
354C4C22×C22⋊#59511yescharacteristic
355C4C22×C4⋊#59511yescharacteristic
356C2—172no—
357C2—172no—
358C2—1712no—
359C2C2×#11763674112yesdirect product factor
360C2C22×(C2×#595)⋊C411yescharacteristic
361C2C22×C8×#59511yescharacteristic
362C2C22×#1058753211yescharacteristic
subgroupG/H
centerC23×C4C2×#595
commutatorC2×C17C22×C4×C8
FrattiniC2×C4C23×#595
FittingC23×C8×C17C4
socleC24×C17C42
prime radicalC24×D17C2×C4
cosocleC2×C4×D17C24
hypercenterC23×C8#595
hypocenterC17C22×#336
LCSG ▷ C2×C17 ▷ C17
UCSC1 ◁ C23×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
3. C2×#11763674C2direct (N × H)
8. C22×#63242C4semidirect
10. C23×#3624C4semidirect
15. #11763674C22direct (N × H)
17. C23×#3622C4semidirect
20. C22×C8×D17C4semidirect
22. C23×C8×C17C4semidirect
26. C23×#595C8semidirect
39. C22×#3624C2×C4semidirect
48. C2×#63242C2×C4semidirect
51. C22×C8×C17C2×C4semidirect
53. C22×#3622C2×C4semidirect
57. C2×C8×D17C2×C4semidirect
69. C22×#595C2×C8semidirect
95. C2×#3624C22×C4semidirect
103. C2×C8×C17C22×C4semidirect
105. C2×#3622C22×C4semidirect
128. C2×#595C22×C8semidirect
157. C22×C17#336semidirect
162. C2×D17#336semidirect
191. C23×C8#595semidirect
197. C2×C17C2×#336semidirect
201. D17C2×#336semidirect
234. C22×C8C2×#595semidirect
240. C17C22×#336semidirect
287. C2×C8C22×#595semidirect
344. C22#11763674direct (N × H)
359. C2C2×#11763674direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#632422
C23×C4×#5952
C2×#117636742
C23×C8×D172
C22×#33617