Finite Group Databasebeta

← C2×#22572847 #31735391

Subgroups — C2×#22572847 #31735391

Order 17152.

2058 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×D268C211yesmaximal, characteristic
2#22572847C2116yesmaximal, direct product factor
3C2×#16863746C213yesmaximal, complemented
4C2×#16863681C213yesmaximal, complemented
5C2×#16863598C213yesmaximal
6C2×Q8×C67⋊C8C211yesmaximal, characteristic, complemented
7C2×C4×#11761615C213yesmaximal
8C2×#557×C67C211yesmaximal, characteristic, complemented
9C22×Q8×D67—21no—
10C22×#11761615C2213yes—
11Q8×D268—24no—
12Q8×D268C2218yes—
13C2×#14524043C2213yes—
14C2×#14524023—23no—
15C2×C4×D268C2213yes—
16#16863746C22124yescomplemented
17#16863681C22124yescomplemented
18#16863598C22124yes—
19Q8×C67⋊C8C2218yescomplemented
20C4×#11761615C22124yes—
21C2×#14523974C2213yes—
22C2×C4⋊(C67⋊C8)C2213yes—
23C2×#14523967C2213yes—
24C2×#14523967C2216yes—
25C2×#14523963C2211yescharacteristic
26C2×C4×C67⋊C8C2213yes—
27#557×C67—24no—
28#557×C67C2218yescomplemented
29C2×C4⋊Q8×C67C2213yes—
30C2×#125×C67—23no—
31C2×C4×Q8×C67—21no—
32C2×C4×Q8×C67C2213yes—
33C2×C4×Q8×C67C2211yescharacteristic
34C22×Dic134—23no—
35C2×Q8×D67—28no—
36C2×Q8×D67—48no—
37C22×D268C2311yescharacteristic
38C22×C4×D67—23no—
39C2×#11761615—212no—
40C2×#11761615C23112yes—
41C22×C67⋊C8C2311yescharacteristic
42C22×C67⋊C8C2313yes—
43#14524043—26no—
44#14524043C23112yes—
45#14524023—224no—
46C4×D268—26no—
47C4×D268C23112yes—
48C2×C4⋊Dic67—23no—
49C2×C4⋊Dic67C2313yes—
50C2×(C2×D67)⋊C4—23no—
51#14523974C23112yes—
52C4⋊(C67⋊C8)C23112yes—
53#14523967C23112yes—
54#14523967C23124yes—
55#14523963C2314yes—
56C4×C67⋊C8C23112yes—
57C22×Q8×C67D411yescharacteristic
58C22×Q8×C67C2313yes—
59C4⋊Q8×C67—26no—
60C4⋊Q8×C67C23112yes—
61#125×C67—224no—
62C4×Q8×C67—28no—
63C4×Q8×C67—26no—
64C4×Q8×C67D414yes—
65C4×Q8×C67C23112yes—
66C4×Q8×C67C2314yes—
67C2×C4⋊C4×C67—23no—
68C2×C4⋊C4×C67—23no—
69C2×C4⋊C4×C67—23no—
70C2×C4⋊C4×C67D413yes—
71C2×C4⋊C4×C67C2311yescharacteristic
72C2×C42×C67—23no—
73C2×C42×C67C2313yes—
74C23×D67—21no—
75C2×Dic134—212no—
76C2×Dic134—412no—
77Q8×D67—432no—
78C2×D268—21no—
79C2×D268—22no—
80C2×D268#4114yes—
81C2×D268C2412yes—
82C22×Dic67—23no—
83C2×C4×D67—212no—
84C2×C4×D67—412no—
85#11761615—224no—
86C2×C67⋊C8—26no—
87C2×C67⋊C8—22no—
88C2×C67⋊C8C2412yes—
89C2×C67⋊C8C2416yes—
90C4⋊Dic67—212no—
91C4⋊Dic67—23no—
92C4⋊Dic67C2416yes—
93(C2×D67)⋊C4—212no—
94C2×Q8×C67—23no—
95C2×Q8×C67—26no—
96C2×Q8×C67—26no—
97C2×Q8×C67SD1618yescomplemented
98C2×Q8×C67C2×D414yes—
99C2×Q8×C67C2416yes—
100C22×C4×C67—21no—
101C22×C4×C67—23no—
102C22×C4×C67C2×D413yes—
103C22×C4×C67C2×D413yes—
104C22×C4×C67C2411yescharacteristic
105C4⋊C4×C67—212no—
106C4⋊C4×C67—212no—
107C4⋊C4×C67—212no—
108C4⋊C4×C67—21no—
109C4⋊C4×C67C2×D4112yes—
110C4⋊C4×C67C2412yes—
111C42×C67—212no—
112C42×C67C2×D416yes—
113C42×C67C2416yes—
114C23×C67C22×D411yescharacteristic
115C22×D67—24no—
116C22×D67—22no—
117C22×D67—42no—
118C22×D67—42no—
119Q8×C67—26no—
120Q8×C67—26no—
121Q8×C67—46no—
122Q8×C67C2×SD16116yescomplemented
123C2×C4×C67—26no—
124C2×C4×C67—22no—
125C2×C4×C67—26no—
126C2×C4×C67—46no—
127C2×C4×C67—42no—
128C2×C4×C67C2×SD16112yes—
129C2×C4×C67C22×D416yes—
130C2×C4×C67C22×D411yescharacteristic
131C2×C4×C67C22×D416yes—
132C2×C4×C67C2×#4112yes—
133C2×C4×C672−1+412yes—
134C2×C4×C67C2511yescharacteristic
135C2×Dic67—26no—
136C2×Dic67—46no—
137D268—24no—
138D268—22no—
139D268C2×#4114yes—
140C4×D67—424no—
141Dic134—424no—
142C67⋊C8—26no—
143C67⋊C8—22no—
144C22×C67#54812yes—
145C22×C67C22×SD1612yes—
146C22×C67#57812yes—
147C22×C67C23×D411yescharacteristic
148C2×D67—24no—
149C2×D67—48no—
150C2×D67—42no—
151C2×D67—42no—
152C4×C67—26no—
153C4×C67—46no—
154C4×C67—42no—
155C4×C67#54812yes—
156C4×C67C22×SD16112yes—
157C4×C67C22×#4111yescharacteristic
158C4×C67C2×2−1+411yescharacteristic
159Dic67—46no—
160C2×#3359—671nomaximal, Sylow
161C2×C67#335914yescomplemented
162C2×C67C2×#54811yescharacteristic
163C2×C67C23×SD1611yescharacteristic
164C2×C67C2×#57811yescharacteristic
165D67—44no—
166C2×#557D6711yescharacteristic, complemented
167C2×D4×Q8—671no—
168#3359—6716no—
169C2×#487—673no—
170C2×#475—673no—
171C2×#461—673no—
172C2×C8×Q8—671no—
173C2×C4×SD16—673no—
174C67C2×#335911yescharacteristic, complemented, Sylow
175C23×Q8—1341no—
176C22×SD16—673no—
177#557—24no—
178#557C2×D6718yescomplemented
179D4×Q8—678no—
180D4×Q8—1344no—
181C2×C4⋊Q8—673no—
182C2×C4⋊Q8C2×D6713yes—
183C2×#125—23no—
184C2×C22⋊Q8—1343no—
185C2×C4×Q8—21no—
186C2×C4×Q8C2×D6713yes—
187C2×C4×Q8C2×D6711yescharacteristic
188C2×C4×D4—673no—
189#487—6724no—
190#475—6724no—
191#461—6724no—
192C8×Q8—678no—
193C4×SD16—6724no—
194C2×#106—673no—
195C2×C4⋊C8—673no—
196C2×#103—676no—
197C2×#103—673no—
198C2×#102—671no—
199C2×C4×C8—673no—
200C22×Q8—2688no—
201C22×Q8—1348no—
202C22×Q8—1343no—
203C22×Q8C22⋊D6711yescharacteristic
204C22×Q8C22×D6713yes—
205C22×D4—671no—
206C23×C4—1343no—
207C2×SD16—6712no—
208C2×SD16—13412no—
209C22×C8—673no—
210C22×C8—671no—
211C4⋊Q8—26no—
212C4⋊Q8—6712no—
213C4⋊Q8—1346no—
214C4⋊Q8C22×D67112yes—
215#125—224no—
216C22⋊Q8—13424no—
217C4×Q8—28no—
218C4×Q8—26no—
219C4×Q8C22⋊D6714yes—
220C4×Q8C22×D67112yes—
221C4×Q8C22×D6714yes—
222C4×D4—6712no—
223C4×D4—1346no—
224C2×C4⋊C4—23no—
225C2×C4⋊C4—23no—
226C2×C4⋊C4—23no—
227C2×C4⋊C4—673no—
228C2×C4⋊C4—1343no—
229C2×C4⋊C4C22⋊D6713yes—
230C2×C4⋊C4C22×D6711yescharacteristic
231C2×C22⋊C4—1343no—
232C2×C42—23no—
233C2×C42C22×D6713yes—
234#106—6712no—
235C4⋊C8—6712no—
236#103—6712no—
237#103—6724no—
238#102—674no—
239C4×C8—6712no—
240C24—1341no—
241C2×Q8—23no—
242C2×Q8—26no—
243C2×Q8—26no—
244C2×Q8—26832no—
245C2×Q8—26812no—
246C2×Q8—13412no—
247C2×Q8#1176161518yescomplemented
248C2×Q8C2×C22⋊D6714yes—
249C2×Q8C23×D6716yes—
250C2×D4—674no—
251C2×D4—672no—
252C2×D4—1341no—
253C2×D4—1342no—
254C22×C4—21no—
255C22×C4—23no—
256C22×C4—26812no—
257C22×C4—1343no—
258C22×C4—13412no—
259C22×C4C2×C22⋊D6713yes—
260C22×C4C2×C22⋊D6713yes—
261C22×C4C23×D6711yescharacteristic
262SD16—13424no—
263C2×C8—672no—
264C2×C8—676no—
265C2×C8—1346no—
266C2×C8—1342no—
267C4⋊C4—212no—
268C4⋊C4—212no—
269C4⋊C4—212no—
270C4⋊C4—21no—
271C4⋊C4—676no—
272C4⋊C4—1343no—
273C4⋊C4—13412no—
274C4⋊C4C2×C22⋊D67112yes—
275C4⋊C4C23×D6712yes—
276C22⋊C4—13412no—
277C42—212no—
278C42C2×C22⋊D6716yes—
279C42C23×D6716yes—
280C23—2682no—
281C23—2682no—
282C23—1344no—
283C23—1342no—
284C23C22×C22⋊D6711yescharacteristic
285Q8—26no—
286Q8—26no—
287Q8—46no—
288Q8—26824no—
289Q8C2×#11761615116yescomplemented
290D4—674no—
291D4—1344no—
292D4—1342no—
293C2×C4—26no—
294C2×C4—22no—
295C2×C4—26no—
296C2×C4—46no—
297C2×C4—42no—
298C2×C4—2686no—
299C2×C4—26824no—
300C2×C4—1346no—
301C2×C4C2×#11761615112yes—
302C2×C4C22×C22⋊D6716yes—
303C2×C4C22×C22⋊D6711yescharacteristic
304C2×C4C22×C22⋊D6716yes—
305C2×C4#41×D6712yes—
306C2×C4#1452409012yes—
307C2×C4C24×D6711yescharacteristic
308C8—1346no—
309C8—1342no—
310C22—2688no—
311C22—2682no—
312C22—2682no—
313C22—1344no—
314C22#1686405312yes—
315C22C22×#1176161512yes—
316C22#1686421112yes—
317C22C23×C22⋊D6711yescharacteristic
318C4—26no—
319C4—46no—
320C4—42no—
321C4—2686no—
322C4#1686405312yes—
323C4C22×#11761615112yes—
324C4C2×#41×D6711yescharacteristic
325C4C2×#1452409011yescharacteristic
326C2—2684no—
327C2#2257284714yesdirect product factor
328C2C2×#1686405311yescharacteristic
329C2C23×#1176161511yescharacteristic
330C2C2×#1686421111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D67
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC2×#557×C67C2
socleC23×C67C22×D4
prime radicalC22×D268C23
hypercenterC2×#557D67
hypocenterC67C2×#3359
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
2. #22572847C2direct (N × H)
3. C2×#16863746C2semidirect
4. C2×#16863681C2semidirect
6. C2×Q8×C67⋊C8C2semidirect
8. C2×#557×C67C2semidirect
16. #16863746C22semidirect
17. #16863681C22semidirect
19. Q8×C67⋊C8C22semidirect
28. #557×C67C22semidirect
97. C2×Q8×C67SD16semidirect
122. Q8×C67C2×SD16semidirect
161. C2×C67#3359semidirect
166. C2×#557D67semidirect
174. C67C2×#3359semidirect
178. #557C2×D67semidirect
247. C2×Q8#11761615semidirect
289. Q8C2×#11761615semidirect
327. C2#22572847direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×D2682
#225728472
C2×#168637462
C2×#168636812
C2×#168635982
C2×Q8×C67⋊C82
C2×C4×#117616152
C2×#557×C672
C2×#335967