Finite Group Databasebeta

← C2×C8×#63256 #17898757

Subgroups — C2×C8×#63256 #17898757

Order 4352.

2566 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#63256C211yesmaximal, characteristic
2C8×#63256C2116yesmaximal, direct product factor
3C2×Q8×#3622C211yesmaximal, characteristic, complemented
4C2×C8×D68C213yesmaximal, complemented
5C2×#11762874C213yesmaximal, complemented
6C2×#11762861C213yesmaximal, complemented
7C2×C4×C8×D17C213yesmaximal, complemented
8C2×C8×Q8×C17C211yesmaximal, characteristic, complemented
9C22×#63256C411yescharacteristic
10C22×C8×D17C2213yes—
11C22×C8×D17C2213yes—
12C4×#63256C418yes—
13C4×#63256C2218yes—
14C2×Q8×Dic17C411yescharacteristic
15C2×C4×D68C2213yes—
16C2×#10587579C413yes—
17C2×#10587578C413yes—
18C2×C42×D17C2213yes—
19Q8×#3622C2218yescomplemented
20C8×D68C22124yescomplemented
21#11762874C22124yescomplemented
22#11762861C22124yescomplemented
23C2×C4⋊#3622C2213yes—
24C2×#10587491C2216yes—
25C2×C4×#3622C2213yes—
26C2×C8×Dic17C2213yes—
27C2×C8×Dic17C2211yescharacteristic
28C4×C8×D17C22124yescomplemented
29C2×C4×Q8×C17C2211yescharacteristic
30C8×Q8×C17C2218yescomplemented
31C2×C17×C4⋊C8C2213yes—
32C2×C4×C8×C17C2213yes—
33C22×Q8×C17C2×C411yescharacteristic
34C2×#63256C8116yesdirect product factor
35C2×#63256C2×C418yes—
36C22×D68C2×C413yes—
37C22×C4×D17C2×C413yes—
38C22×C4×D17C2×C413yes—
39C22×C4×D17C2313yes—
40C22×C8×C17C2313yes—
41C22×C8×C17C2311yescharacteristic
42C4×Q8×C17C2×C414yes—
43C4×Q8×C17C2314yes—
44C2×C4⋊C4×C17C2×C413yes—
45C2×C42×C17C2313yes—
46Q8×Dic17C2×C418yes—
47C2×#63227C2×C416yes—
48C2×C4⋊Dic17C2×C413yes—
49C2×C4×Dic17C2×C413yes—
50C2×C4×Dic17C2×C413yes—
51C2×C4×Dic17C2311yescharacteristic
52C22×#3622C2313yes—
53C22×#3622C2311yescharacteristic
54C2×C8×D17—212no—
55C2×C8×D17—212no—
56C2×C8×D17C23112yes—
57C2×C8×D17C23112yes—
58#10587579C2×C4124yes—
59#10587578C2×C4124yes—
60C4×D68C2×C4112yes—
61C4×D68C23112yes—
62C42×D17C2×C4112yes—
63C42×D17C23112yes—
64C17×C4⋊C8C23112yes—
65C4×C8×C17C23112yes—
66#10587491C23124yes—
67C8×Dic17C2314yes—
68C8×Dic17C23112yes—
69C4⋊#3622C23112yes—
70C4×#3622C23112yes—
71C23×D17C22×C413yes—
72C2×Q8×C17C2×C818yescomplemented
73C2×Q8×C17C22×C414yes—
74C22×C4×C17C22×C413yes—
75C22×C4×C17C22×C413yes—
76C22×C4×C17C2411yescharacteristic
77C22×Dic17C22×C413yes—
78C22×Dic17C22×C411yescharacteristic
79C22×Dic17C22×C411yescharacteristic
80#63256C2×C8164yesdirect product factor
81C2×D68C2×C8124yescomplemented
82C2×D68C22×C4112yes—
83C2×C4×D17—26no—
84C2×C4×D17—26no—
85C2×C4×D17—212no—
86C2×C4×D17C2×C8124yescomplemented
87C2×C4×D17C22×C4112yes—
88C2×C4×D17C22×C416yes—
89C2×C4×D17C22×C4112yes—
90C2×C4×D17C2416yes—
91C2×C8×C17—26no—
92C2×C8×C17#4114yescomplemented
93C2×C8×C17C2416yes—
94C2×C8×C17C2412yes—
95C4⋊C4×C17C22×C4112yes—
96C42×C17C22×C416yes—
97C42×C17C2416yes—
98#63227C22×C4124yes—
99C4⋊Dic17C22×C4112yes—
100C4×Dic17C22×C412yes—
101C4×Dic17C22×C4112yes—
102C4×Dic17C22×C4112yes—
103C4×Dic17C2412yes—
104C2×#3622—26no—
105C2×#3622#4114yescomplemented
106C2×#3622C2416yes—
107C2×#3622C2412yes—
108C8×D17—224no—
109C8×D17—224no—
110C2×C8×#41—171nomaximal, Sylow
111C23×C17C23×C411yescharacteristic
112C22×D17—26no—
113C22×D17—26no—
114C22×D17C22×C8112yes—
115C22×D17C23×C416yes—
116Q8×C17C22×C8116yescomplemented
117C2×C4×C17—26no—
118C2×C4×C17C22×C8112yes—
119C2×C4×C17C23×C416yes—
120C2×C4×C17C23×C411yescharacteristic
121C2×C4×C17C23×C416yes—
122C2×C4×C17C2×#4112yes—
123C2×C4×C17C2×#4112yes—
124C2×C4×C17C2511yescharacteristic
125C2×Dic17—26no—
126C2×Dic17C22×C814yes—
127C2×Dic17#13114yes—
128C2×Dic17C23×C416yes—
129C2×Dic17C23×C412yes—
130C2×Dic17C23×C412yes—
131D68C22×C8148yescomplemented
132C4×D17—212no—
133C4×D17—26no—
134C4×D17—26no—
135C4×D17—224no—
136C4×D17C22×C8148yescomplemented
137C8×C17—26no—
138C8×C17C2×#4114yescomplemented
139#3622—26no—
140#3622C2×#4114yescomplemented
141C2×C4×#41—171no—
142C8×#41—1716no—
143C2×C8×Q8—171no—
144C2×C8×Q8D1711yescharacteristic, complemented
145C2×C8×D4—173no—
146C2×C8×D4—173no—
147C2×#431—173no—
148C22×C4×C8—173no—
149C22×C17C4×#4112yes—
150C22×C17C23×C812yes—
151C22×C17C2×#13112yes—
152C22×C17C24×C411yescharacteristic
153C2×D17—26no—
154C2×D17—224no—
155C2×D17—26no—
156C2×D17C23×C8112yes—
157C4×C17—26no—
158C4×C17C4×#4112yes—
159C4×C17C23×C8112yes—
160C4×C17C22×#4111yescharacteristic
161C4×C17C22×#4111yescharacteristic
162Dic17—26no—
163Dic17C23×C814yes—
164Dic17C2×#13114yes—
165C22×#41—171no—
166C23×C8—173no—
167C23×C8—173no—
168C4×#41—178no—
169C4×#41—178no—
170C2×C4×Q8—171no—
171C2×C4×Q8C2×D1711yescharacteristic
172C2×C4×D4—173no—
173C2×C4×D4—173no—
174C2×#117—173no—
175C22×C42—173no—
176C8×Q8—178no—
177C8×Q8C2×D1718yescomplemented
178C8×D4—1724no—
179C8×D4—1724no—
180#431—1724no—
181C2×C4⋊C8—173no—
182C2×C4⋊C8C2×D1713yes—
183C2×C22⋊C8—176no—
184C2×C4×C8—173no—
185C2×C4×C8—171no—
186C2×C4×C8—173no—
187C2×C4×C8—1724no—
188C2×C4×C8C2×D1713yes—
189C2×C17C8×#4114yescomplemented
190C2×C17C2×C4×#4111yescharacteristic
191C2×C17C24×C811yescharacteristic
192C2×C17C22×#13111yescharacteristic
193D17—212no—
194C2×#41—178no—
195C2×#41—1716no—
196C22×Q8C4×D1711yescharacteristic
197C22×D4—173no—
198C23×C4—173no—
199C23×C4—173no—
200C23×C4—173no—
201C22×C8—171no—
202C22×C8—173no—
203C22×C8—1712no—
204C22×C8—1712no—
205C22×C8—3412no—
206C22×C8—3412no—
207C22×C8C22×D1713yes—
208C22×C8C22×D1711yescharacteristic
209C4×Q8—178no—
210C4×Q8C4×D1714yes—
211C4×Q8C22×D1714yes—
212C4×D4—1724no—
213C4×D4—1712no—
214C4×D4—1712no—
215#117—1724no—
216C2×C4⋊C4—173no—
217C2×C4⋊C4C4×D1713yes—
218C2×C22⋊C4—176no—
219C2×C42—173no—
220C2×C42—173no—
221C2×C42—171no—
222C2×C42—1712no—
223C2×C42—1712no—
224C2×C42C22×D1713yes—
225C4⋊C8—1712no—
226C4⋊C8C22×D17112yes—
227C22⋊C8—1724no—
228C4×C8—174no—
229C4×C8—1712no—
230C4×C8—1712no—
231C4×C8C22×D17112yes—
232C17C2×C8×#4111yescharacteristic, complemented, Sylow
233C24—173no—
234#41—1764no—
235C2×Q8C8×D1718yescomplemented
236C2×Q8C2×C4×D1714yes—
237C2×D4—1712no—
238C2×D4—1724no—
239C22×C4—171no—
240C22×C4—171no—
241C22×C4—1712no—
242C22×C4—1724no—
243C22×C4—173no—
244C22×C4—1712no—
245C22×C4—176no—
246C22×C4—176no—
247C22×C4—3412no—
248C22×C4—346no—
249C22×C4—346no—
250C22×C4C2×C4×D1713yes—
251C22×C4C2×C4×D1713yes—
252C22×C4C23×D1711yescharacteristic
253C2×C8—26no—
254C2×C8—174no—
255C2×C8—172no—
256C2×C8—176no—
257C2×C8—3424no—
258C2×C8—346no—
259C2×C8—3424no—
260C2×C8#6325614yesdirect product factor
261C2×C8C23×D1716yes—
262C2×C8C23×D1712yes—
263C4⋊C4—1712no—
264C4⋊C4C2×C4×D17112yes—
265C22⋊C4—1724no—
266C42—172no—
267C42—1712no—
268C42—1712no—
269C42—172no—
270C42C2×C4×D1716yes—
271C42C23×D1716yes—
272C23—1712no—
273C23—176no—
274C23—346no—
275C23—346no—
276C23C22×C4×D1711yescharacteristic
277Q8C2×C8×D17116yescomplemented
278D4—1748no—
279C2×C4—26no—
280C2×C4—174no—
281C2×C4—174no—
282C2×C4—172no—
283C2×C4—172no—
284C2×C4—1748no—
285C2×C4—176no—
286C2×C4—3424no—
287C2×C4—346no—
288C2×C4—3412no—
289C2×C4—346no—
290C2×C4—346no—
291C2×C4C2×C8×D17112yes—
292C2×C4C22×C4×D1716yes—
293C2×C4C22×C4×D1711yescharacteristic
294C2×C4C22×C4×D1716yes—
295C2×C4C2×#6325612yes—
296C2×C4#41×D1712yes—
297C2×C4C24×D1711yescharacteristic
298C8—26no—
299C8—174no—
300C8—346no—
301C8C2×#6325614yesdirect product factor
302C22—1712no—
303C22—3424no—
304C22—346no—
305C22—346no—
306C22C4×#6325612yes—
307C22C22×C8×D1712yes—
308C22#131×D1712yes—
309C22C23×C4×D1711yescharacteristic
310C4—26no—
311C4—174no—
312C4—174no—
313C4—346no—
314C4C4×#6325612yes—
315C4C22×C8×D17112yes—
316C4C2×#41×D1711yescharacteristic
317C4C22×#6325611yescharacteristic
318C2—3412no—
319C2C8×#6325614yesdirect product factor
320C2C2×C4×#6325611yescharacteristic
321C2C23×C8×D1711yescharacteristic
322C2C2×#131×D1711yescharacteristic
subgroupG/H
centerC22×C8C22×D17
commutatorC2×C17C24×C8
FrattiniC2×C4C24×D17
FittingC2×C8×Q8×C17C2
socleC23×C17C23×C4
prime radicalC22×#63256C4
cosocleC2×C4×C17C25
hypercenterC2×C8×Q8D17
hypocenterC17C2×C8×#41
LCSG ▷ C2×C17 ▷ C17
UCSC1 ◁ C22×C8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. C8×#63256C2direct (N × H)
3. C2×Q8×#3622C2semidirect
4. C2×C8×D68C2semidirect
5. C2×#11762874C2semidirect
6. C2×#11762861C2semidirect
7. C2×C4×C8×D17C2semidirect
8. C2×C8×Q8×C17C2semidirect
19. Q8×#3622C22semidirect
20. C8×D68C22semidirect
21. #11762874C22semidirect
22. #11762861C22semidirect
28. C4×C8×D17C22semidirect
30. C8×Q8×C17C22semidirect
34. C2×#63256C8direct (N × H)
72. C2×Q8×C17C2×C8semidirect
80. #63256C2×C8direct (N × H)
81. C2×D68C2×C8semidirect
86. C2×C4×D17C2×C8semidirect
92. C2×C8×C17#41semidirect
105. C2×#3622#41semidirect
116. Q8×C17C22×C8semidirect
131. D68C22×C8semidirect
136. C4×D17C22×C8semidirect
138. C8×C17C2×#41semidirect
140. #3622C2×#41semidirect
144. C2×C8×Q8D17semidirect
177. C8×Q8C2×D17semidirect
189. C2×C17C8×#41semidirect
232. C17C2×C8×#41semidirect
235. C2×Q8C8×D17semidirect
260. C2×C8#63256direct (N × H)
277. Q8C2×C8×D17semidirect
301. C8C2×#63256direct (N × H)
319. C2C8×#63256direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#632562
C8×#632562
C2×Q8×#36222
C2×C8×D682
C2×#117628742
C2×#117628612
C2×C4×C8×D172
C2×C8×Q8×C172
C2×C8×#4117