Finite Group Databasebeta

← C2×#22572881 #31735428

Subgroups — C2×#22572881 #31735428

Order 17152.

2174 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic134C211yesmaximal, characteristic, complemented
2C2×#16864068C211yesmaximal, characteristic
3#22572881C2116yesmaximal, direct product factor
4C2×#16863743C213yesmaximal, complemented
5C2×#16863687C213yesmaximal, complemented
6C2×#16863597C213yesmaximal, complemented
7C2×C4×#11761617C213yesmaximal, complemented
8C2×C8×Q8×C67C211yesmaximal, characteristic, complemented
9C22×#11761629—21no—
10C22×#11761617C2213yes—
11Q8×Dic134—24no—
12Q8×Dic134C2218yescomplemented
13#16864068—24no—
14#16864068C2218yes—
15C2×#14524043C2213yes—
16C2×#14524042—23no—
17C2×#14524040C2213yes—
18C2×#14524018—23no—
19C2×Q8×Dic67—21no—
20C2×C4×Dic134C2213yes—
21C2×C4×D268C2213yes—
22#16863743C22124yescomplemented
23#16863687C22124yescomplemented
24#16863597C22124yescomplemented
25C4×#11761617C22124yescomplemented
26C2×#14523975C2213yes—
27C2×#14523968C2211yescharacteristic
28C2×#14523965C2213yes—
29C2×#14523965C2216yes—
30C2×C4×Q8×C67C2211yescharacteristic
31C8×Q8×C67C2218yescomplemented
32C2×C4⋊C8×C67C2213yes—
33C2×C4×C8×C67C2213yes—
34C2×#11761629—28no—
35C2×#11761629—48no—
36C22×Dic134—23no—
37C22×Dic134C2311yescharacteristic
38C22×D268—23no—
39C22×D268—23no—
40C22×D268C2313yes—
41C22×C4×D67—23no—
42C2×#11761617—212no—
43C2×#11761617C23112yes—
44#14524043—26no—
45#14524043C23112yes—
46#14524042—224no—
47#14524040—26no—
48#14524040C23112yes—
49#14524018—224no—
50Q8×Dic67—28no—
51C4×Dic134—26no—
52C4×Dic134C23112yes—
53C4×D268—26no—
54C4×D268C23112yes—
55C2×C4⋊Dic67—23no—
56C2×C4⋊Dic67C2311yescharacteristic
57C2×C4⋊Dic67C2313yes—
58C2×Dic67⋊C4—23no—
59C2×(C2×D67)⋊C4—23no—
60C2×C4×Dic67—23no—
61#14523975C23112yes—
62#14523968C2314yes—
63#14523965C23124yes—
64#14523965C23112yes—
65C22×Q8×C67D411yescharacteristic
66C22×C8×C67C2311yescharacteristic
67C22×C8×C67C2313yes—
68C4×Q8×C67D414yes—
69C4×Q8×C67C2314yes—
70C2×C4⋊C4×C67D413yes—
71C2×C42×C67C2313yes—
72C4⋊C8×C67C23112yes—
73C4×C8×C67C23112yes—
74C23×D67—23no—
75#11761629—432no—
76C2×Dic134—21no—
77C2×Dic134—22no—
78C2×Dic134—212no—
79C2×Dic134—86no—
80C2×Dic134#4114yescomplemented
81C2×Dic134C2412yes—
82C2×D268—26no—
83C2×D268—26no—
84C2×D268—23no—
85C2×D268—212no—
86C2×D268—212no—
87C2×D268—412no—
88C2×D268—86no—
89C2×D268C2416yes—
90C22×Dic67—21no—
91C22×Dic67—23no—
92C22×Dic67—21no—
93C2×C4×D67—212no—
94C2×C4×D67—412no—
95#11761617—224no—
96C4⋊Dic67—212no—
97C4⋊Dic67—21no—
98C4⋊Dic67—23no—
99C4⋊Dic67C2416yes—
100C4⋊Dic67C2412yes—
101Dic67⋊C4—212no—
102(C2×D67)⋊C4—212no—
103C4×Dic67—212no—
104C2×Q8×C67SD1618yescomplemented
105C2×Q8×C67C2×D414yes—
106C22×C4×C67C2×D413yes—
107C22×C4×C67C2×D413yes—
108C22×C4×C67C2411yescharacteristic
109C2×C8×C67—26no—
110C2×C8×C67—22no—
111C2×C8×C67C2416yes—
112C2×C8×C67C2412yes—
113C4⋊C4×C67C2×D4112yes—
114C42×C67C2×D416yes—
115C42×C67C2416yes—
116C23×C67C22×D411yescharacteristic
117C22×D67—26no—
118C22×D67—46no—
119C22×D67—46no—
120C22×D67—46no—
121Q8×C67C2×SD16116yescomplemented
122C2×C4×C67—26no—
123C2×C4×C67C2×SD16112yes—
124C2×C4×C67C22×D416yes—
125C2×C4×C67C22×D411yescharacteristic
126C2×C4×C67C22×D416yes—
127C2×C4×C67C2×#4112yes—
128C2×C4×C672−1+412yes—
129C2×C4×C67C2511yescharacteristic
130C2×Dic67—24no—
131C2×Dic67—22no—
132C2×Dic67—26no—
133C2×Dic67—22no—
134C2×Dic67—42no—
135C2×Dic67—46no—
136D268—26no—
137D268—26no—
138D268—46no—
139D268—424no—
140D268—812no—
141C4×D67—424no—
142Dic134—24no—
143Dic134—22no—
144Dic134—812no—
145Dic134C2×#4114yescomplemented
146C8×C67—26no—
147C8×C67—22no—
148C22×C67#54812yes—
149C22×C67C22×SD1612yes—
150C22×C67#57612yes—
151C22×C67C23×D411yescharacteristic
152C2×D67—46no—
153C2×D67—46no—
154C2×D67—46no—
155C2×D67—812no—
156C4×C67—26no—
157C4×C67#54812yes—
158C4×C67C22×SD16112yes—
159C4×C67C22×#4111yescharacteristic
160C4×C67C2×2−1+411yescharacteristic
161Dic67—24no—
162Dic67—42no—
163Dic67—46no—
164C2×#3362—671nomaximal, Sylow
165C2×C67#336214yescomplemented
166C2×C67C2×#54811yescharacteristic
167C2×C67C23×SD1611yescharacteristic
168C2×C67C2×#57611yescharacteristic
169D67—86no—
170C2×Q82—671no—
171C2×#550—671no—
172#3362—6716no—
173C2×#486—673no—
174C2×#476—673no—
175C2×#460—673no—
176C2×C8×Q8D6711yescharacteristic, complemented
177C2×C4×SD16—673no—
178C67C2×#336211yescharacteristic, complemented, Sylow
179C22×#41—1341no—
180C22×SD16—673no—
181Q82—678no—
182Q82—1344no—
183#550—678no—
184#550—1344no—
185C2×C4⋊Q8—673no—
186C2×C4⋊Q8—1343no—
187C2×#127—673no—
188C2×#121—1343no—
189C2×C4×Q8—673no—
190C2×C4×Q8—1341no—
191C2×C4×Q8C2×D6711yescharacteristic
192C2×C4×D4—673no—
193#486—6724no—
194#476—6724no—
195#460—6724no—
196C8×Q8C2×D6718yescomplemented
197C4×SD16—6724no—
198C2×#106—673no—
199C2×C4⋊C8C2×D6713yes—
200C2×#103—671no—
201C2×#102—673no—
202C2×#102—676no—
203C2×C4×C8C2×D6713yes—
204C2×#41—2688no—
205C2×#41—1348no—
206C22×Q8—671no—
207C22×Q8—1343no—
208C22×Q8D26811yescharacteristic
209C22×D4—673no—
210C22×D4—1343no—
211C22×D4—1343no—
212C23×C4—1343no—
213C2×SD16—6712no—
214C2×SD16—13412no—
215C22×C8C22×D6713yes—
216C22×C8C22×D6711yescharacteristic
217C4⋊Q8—6712no—
218C4⋊Q8—13424no—
219C4⋊Q8—1346no—
220#127—6712no—
221#127—1346no—
222#121—13424no—
223C4×Q8—6712no—
224C4×Q8—1348no—
225C4×Q8—1346no—
226C4×Q8D26814yes—
227C4×Q8C22×D6714yes—
228C4×D4—6712no—
229C4×D4—1346no—
230C2×C4⋊C4—671no—
231C2×C4⋊C4—673no—
232C2×C4⋊C4—1343no—
233C2×C4⋊C4—1343no—
234C2×C4⋊C4D26813yes—
235C2×C22⋊C4—1343no—
236C2×C42—1343no—
237C2×C42C22×D6713yes—
238#106—6712no—
239C4⋊C8C22×D67112yes—
240#103—674no—
241#102—6712no—
242#102—6724no—
243C4×C8C22×D67112yes—
244C24—1343no—
245#41—26832no—
246C2×Q8—5366no—
247C2×Q8—674no—
248C2×Q8—672no—
249C2×Q8—13412no—
250C2×Q8—1341no—
251C2×Q8—1342no—
252C2×Q8#1176161718yescomplemented
253C2×Q8C2×D26814yes—
254C2×D4—26812no—
255C2×D4—5366no—
256C2×D4—676no—
257C2×D4—1343no—
258C2×D4—13412no—
259C2×D4—13412no—
260C2×D4—1346no—
261C2×D4—1346no—
262C22×C4—26812no—
263C22×C4—1343no—
264C22×C4—1341no—
265C22×C4—13412no—
266C22×C4—1341no—
267C22×C4C2×D26813yes—
268C22×C4C2×D26813yes—
269C22×C4C23×D6711yescharacteristic
270SD16—13424no—
271C2×C8—26no—
272C2×C8—22no—
273C2×C8C23×D6712yes—
274C2×C8C23×D6716yes—
275C4⋊C4—676no—
276C4⋊C4—672no—
277C4⋊C4—1341no—
278C4⋊C4—1343no—
279C4⋊C4—13412no—
280C4⋊C4—13412no—
281C4⋊C4C2×D268112yes—
282C22⋊C4—13412no—
283C42—13412no—
284C42C2×D26816yes—
285C42C23×D6716yes—
286C23—2686no—
287C23—2686no—
288C23—2686no—
289C23—1346no—
290C23C22×D26811yescharacteristic
291Q8—53612no—
292Q8—674no—
293Q8—1344no—
294Q8—1342no—
295Q8C2×#11761617116yescomplemented
296D4—2686no—
297D4—26824no—
298D4—53612no—
299D4—1346no—
300D4—1346no—
301C2×C4—26no—
302C2×C4—2682no—
303C2×C4—2686no—
304C2×C4—26824no—
305C2×C4—1346no—
306C2×C4—1342no—
307C2×C4—1344no—
308C2×C4—1342no—
309C2×C4C2×#11761617112yes—
310C2×C4C22×D26816yes—
311C2×C4C22×D26811yescharacteristic
312C2×C4C22×D26816yes—
313C2×C4#41×D6712yes—
314C2×C4#1452409112yes—
315C2×C4C24×D6711yescharacteristic
316C8—26no—
317C8—22no—
318C22—2686no—
319C22—2686no—
320C22—2686no—
321C22—53612no—
322C22#1686405812yes—
323C22C22×#1176161712yes—
324C22#1686420112yes—
325C22C23×D26811yescharacteristic
326C4—26no—
327C4—2682no—
328C4—2686no—
329C4—1344no—
330C4#1686405812yes—
331C4C22×#11761617112yes—
332C4C2×#41×D6711yescharacteristic
333C4C2×#1452409111yescharacteristic
334C2—5366no—
335C2#2257288114yesdirect product factor
336C2C2×#1686405811yescharacteristic
337C2C23×#1176161711yescharacteristic
338C2C2×#1686420111yescharacteristic
subgroupG/H
centerC23C22×D268
commutator, cosocleC2×C4×C67C25
FrattiniC2×C4C24×D67
FittingC2×C8×Q8×C67C2
socleC23×C67C22×D4
prime radicalC2×#16864068C2
hypercenterC2×C8×Q8D67
hypocenterC67C2×#3362
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic134C2semidirect
3. #22572881C2direct (N × H)
4. C2×#16863743C2semidirect
5. C2×#16863687C2semidirect
6. C2×#16863597C2semidirect
7. C2×C4×#11761617C2semidirect
8. C2×C8×Q8×C67C2semidirect
12. Q8×Dic134C22semidirect
22. #16863743C22semidirect
23. #16863687C22semidirect
24. #16863597C22semidirect
25. C4×#11761617C22semidirect
31. C8×Q8×C67C22semidirect
80. C2×Dic134#41semidirect
104. C2×Q8×C67SD16semidirect
121. Q8×C67C2×SD16semidirect
145. Dic134C2×#41semidirect
165. C2×C67#3362semidirect
176. C2×C8×Q8D67semidirect
178. C67C2×#3362semidirect
196. C8×Q8C2×D67semidirect
252. C2×Q8#11761617semidirect
295. Q8C2×#11761617semidirect
335. C2#22572881direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1342
C2×#168640682
#225728812
C2×#168637432
C2×#168636872
C2×#168635972
C2×C4×#117616172
C2×C8×Q8×C672
C2×#336267