Finite Group Databasebeta

← #537×D97 #27155056

Subgroups — #537×D97 #27155056

Order 12416.

1640 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D97C212yesmaximal, complemented
2#20261316C211yesmaximal, characteristic, complemented
3#20261315C214yesmaximal, complemented
4#20261314C214yesmaximal, complemented
5#20261313C212yesmaximal, complemented
6#20261312C214yesmaximal, complemented
7C22⋊Q8×D97C214yesmaximal, complemented
8#121×D97C214yesmaximal, complemented
9#121×D97C214yesmaximal, complemented
10C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
11#537×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2212yescomplemented
13C2×#13095511C2212yescomplemented
14C2×#13095511C2214yescomplemented
15C2×#13095510C2214yescomplemented
16C2×#13095510C2212yescomplemented
17#41×D97—216no—
18C2×Q8×D97C2212yes—
19C2×D4×D97C2212yes—
20C2×D4×D97C2214yes—
21C2×D4×D97C2214yes—
22C22×C4×D97C2214yes—
23C22×C4×D97C2212yes—
24C22×C4×D97C2211yescharacteristic
25#15690790C2214yescomplemented
26#15690789C2214yescomplemented
27#15690788C2218yescomplemented
28#15690787C2218yescomplemented
29#15690786C2214yescomplemented
30#15690785C2214yescomplemented
31#15690785C2214yescomplemented
32#15690784C2214yescomplemented
33#15690784C2214yescomplemented
34#15690783C2218yescomplemented
35#15690783C2218yescomplemented
36#15690782C2218yescomplemented
37#15690782C2218yescomplemented
38#15690781C2214yescomplemented
39#15690781C2214yescomplemented
40C2×C4⋊Dic97C2211yescharacteristic, complemented
41C2×#13095490C2212yescomplemented
42C4⋊C4×D97C2214yes—
43C4⋊C4×D97C2214yes—
44C4⋊C4×D97C2214yescomplemented
45C22⋊C4×D97C2218yes—
46C2×#41×C97C2212yescomplemented
47C22⋊Q8×C97C2214yescomplemented
48#121×C97C2214yescomplemented
49#121×C97C2214yescomplemented
50C2×C4⋊C4×C97C2211yescharacteristic, complemented
51C23×D97C2314yes—
52C23×D97C2311yescharacteristic
53#13095512—28no—
54#13095511—28no—
55#13095511—216no—
56#13095510—216no—
57#13095510—28no—
58C2×Dic194C2314yes—
59C2×Dic194C2312yes—
60Q8×D97—28no—
61C2×D388C2314yes—
62C2×D388C2312yes—
63C2×D388C2314yes—
64C2×C22⋊D97C2314yes—
65C2×C22⋊D97C2314yes—
66C2×C22⋊D97C2314yes—
67C2×C22⋊D97C2318yes—
68D4×D97—216no—
69D4×D97—28no—
70D4×D97—48no—
71C22×Dic97C2314yes—
72C22×Dic97C2312yes—
73C22×Dic97C2311yescharacteristic
74C2×C4×D97—216no—
75C2×C4×D97—44no—
76C2×C4×D97D418yescomplemented
77C2×C4×D97C2314yes—
78C2×C4×D97C2314yes—
79C2×C4×D97C2314yes—
80C2×C4×D97C2314yes—
81C2×C4×D97C2314yes—
82C2×C4×D97C2312yes—
83C2×C4×D97C2312yes—
84C4⋊Dic97C2314yes—
85C4⋊Dic97C2314yes—
86#13095490C2318yes—
87#13095490C2318yes—
88C22⋊Dic97C2318yes—
89#13095488C2318yes—
90#13095488C2318yes—
91#41×C97—28no—
92C2×Q8×C97C2312yes—
93C2×D4×C97C2312yes—
94C2×D4×C97C2314yes—
95C2×D4×C97C2314yes—
96C22×C4×C97C2314yes—
97C22×C4×C97C2312yes—
98C22×C4×C97C2311yescharacteristic
99C4⋊C4×C97C2314yes—
100C4⋊C4×C97C2314yes—
101C22⋊C4×C97C2318yes—
102C23×C97C2414yes—
103C23×C97C2411yescharacteristic
104C22×D97—22no—
105C22×D97—22no—
106C22×D97—28no—
107C22×D97—28no—
108C22×D97—28no—
109C22×D97C2×D414yes—
110C22×D97C2414yes—
111C22×D97C2411yescharacteristic
112C22×D97C2411yescharacteristic
113Q8×C97—24no—
114D4×C97—28no—
115D4×C97—24no—
116D4×C97—44no—
117C2×C4×C97—28no—
118C2×C4×C97—42no—
119C2×C4×C97C2×D414yescomplemented
120C2×C4×C97C2414yes—
121C2×C4×C97C2414yes—
122C2×C4×C97C2412yes—
123C22⋊D97—28no—
124C22⋊D97—28no—
125C22⋊D97—28no—
126C22⋊D97—48no—
127C2×Dic97—28no—
128C2×Dic97—42no—
129C2×Dic97C2×D414yescomplemented
130C2×Dic97C2414yes—
131C2×Dic97C2414yes—
132C2×Dic97C2412yes—
133D388—28no—
134D388—24no—
135D388—44no—
136C4×D97—28no—
137C4×D97—28no—
138C4×D97—28no—
139C4×D97—28no—
140C4×D97—44no—
141C4×D97C2×D418yes—
142C4×D97C2×D418yescomplemented
143Dic194—24no—
144Dic194—28no—
145C22×C97—21no—
146C22×C97—21no—
147C22×C97—24no—
148C22×C97—24no—
149C22×C97—24no—
150C22×C97C22×D412yes—
151C22×C97C2511yescharacteristic
152C2×D97—21no—
153C2×D97—21no—
154C2×D97—28no—
155C2×D97—24no—
156C2×D97—24no—
157C2×D97—24no—
158C2×D97—416no—
159C2×D97C22×D412yes—
160C2×D97C22×D412yes—
161C2×D97H(5,𝔽2)12yes—
162C2×D972−1+412yes—
163C4×C97—24no—
164C4×C97—24no—
165C4×C97C22×D414yes—
166Dic97—24no—
167Dic97—24no—
168Dic97C22×D414yes—
169C2×C97—22no—
170C2×C97—44no—
171C2×C97C23×D411yescharacteristic
172C2×C97C2×H(5,𝔽2)11yescharacteristic
173C2×C97C2×2−1+411yescharacteristic
174D97—22no—
175D97—44no—
176D97#53714yesdirect product factor
177C2×#537—971nomaximal, Sylow
178C97C2×#53711yescharacteristic, complemented, Sylow
179C22×#41—972no—
180#537—972no—
181#537—974no—
182#537—974no—
183#537—974no—
184#537—971no—
185#537D9711yescharacteristic, direct product factor
186C2×C22⋊Q8—974no—
187C2×#121—974no—
188C2×#121—974no—
189C22×C4⋊C4—971no—
190C2×#41—972no—
191C2×#41—974no—
192C2×#41—972no—
193C2×#41—974no—
194C2×#41—972no—
195C2×#41—19416no—
196C2×#41C2×D9712yescomplemented
197C22×Q8—972no—
198C22×D4—974no—
199C22×D4—972no—
200C22×D4—974no—
201C23×C4—971no—
202C23×C4—972no—
203C23×C4—974no—
204C22⋊Q8—974no—
205C22⋊Q8—974no—
206C22⋊Q8—978no—
207C22⋊Q8—974no—
208C22⋊Q8—978no—
209C22⋊Q8C2×D9714yescomplemented
210#121—974no—
211#121—974no—
212#121—978no—
213#121—978no—
214#121—974no—
215#121—974no—
216#121—974no—
217#121—974no—
218#121—978no—
219#121—978no—
220#121C2×D9714yescomplemented
221#121C2×D9714yescomplemented
222C2×C4⋊C4—972no—
223C2×C4⋊C4—971no—
224C2×C4⋊C4—974no—
225C2×C4⋊C4—974no—
226C2×C4⋊C4—974no—
227C2×C4⋊C4C2×D9711yescharacteristic, complemented
228C2×C22⋊C4—978no—
229C24—971no—
230C24—974no—
231#41—28no—
232#41—19416no—
233#41—19416no—
234#41—1948no—
235#41—1948no—
236#41—1948no—
237C2×Q8—974no—
238C2×Q8—972no—
239C2×Q8—1948no—
240C2×Q8C22×D9712yes—
241C2×D4—978no—
242C2×D4—974no—
243C2×D4—974no—
244C2×D4—974no—
245C2×D4—974no—
246C2×D4—974no—
247C2×D4—972no—
248C2×D4—3888no—
249C2×D4—1948no—
250C2×D4—19416no—
251C2×D4C22×D9714yes—
252C2×D4C22×D9712yes—
253C2×D4C22×D9714yes—
254C22×C4—971no—
255C22×C4—978no—
256C22×C4—972no—
257C22×C4—972no—
258C22×C4—972no—
259C22×C4—974no—
260C22×C4—974no—
261C22×C4—974no—
262C22×C4—974no—
263C22×C4—974no—
264C22×C4—974no—
265C22×C4—3884no—
266C22×C4—19416no—
267C22×C4C22×D9714yes—
268C22×C4C22×D9712yes—
269C22×C4C22×D9711yescharacteristic
270C4⋊C4—978no—
271C4⋊C4—978no—
272C4⋊C4—974no—
273C4⋊C4—974no—
274C4⋊C4C22×D9714yes—
275C4⋊C4C22×D9714yes—
276C22⋊C4—978no—
277C22⋊C4—978no—
278C22⋊C4—978no—
279C22⋊C4C22×D9718yes—
280C23—971no—
281C23—974no—
282C23—971no—
283C23—974no—
284C23—1948no—
285C23—1948no—
286C23—1948no—
287C23—1942no—
288C23—1942no—
289C23C23×D9714yes—
290C23C23×D9711yescharacteristic
291Q8—24no—
292Q8—1948no—
293Q8—1944no—
294D4—28no—
295D4—24no—
296D4—44no—
297D4—3884no—
298D4—3888no—
299D4—1948no—
300D4—1948no—
301D4—1948no—
302D4—1948no—
303D4—1944no—
304C2×C4—28no—
305C2×C4—42no—
306C2×C4—974no—
307C2×C4—972no—
308C2×C4—978no—
309C2×C4—978no—
310C2×C4—974no—
311C2×C4—974no—
312C2×C4—3884no—
313C2×C4—3882no—
314C2×C4—1948no—
315C2×C4—1948no—
316C2×C4—1948no—
317C2×C4—1948no—
318C2×C4—1948no—
319C2×C4D4×D9714yescomplemented
320C2×C4C23×D9714yes—
321C2×C4C23×D9714yes—
322C2×C4C23×D9712yes—
323C22—21no—
324C22—21no—
325C22—24no—
326C22—24no—
327C22—24no—
328C22—972no—
329C22—972no—
330C22—972no—
331C22—972no—
332C22—38816no—
333C22—1948no—
334C22—1944no—
335C22—1944no—
336C22—1944no—
337C22—1941no—
338C22—1941no—
339C22C2×D4×D9712yes—
340C22C24×D9711yescharacteristic
341C4—24no—
342C4—24no—
343C4—974no—
344C4—1944no—
345C4—1944no—
346C4C2×D4×D9714yes—
347C2—22no—
348C2—44no—
349C2—974no—
350C2—3884no—
351C2—1942no—
352C2C22×D4×D9711yescharacteristic
353C2H(5,𝔽2)×D9711yescharacteristic
354C22−1+4×D9711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#537×C97C2
hypercenter#537D97
hypocenterC97C2×#537
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#41×D97C2semidirect
2. #20261316C2semidirect
3. #20261315C2semidirect
4. #20261314C2semidirect
5. #20261313C2semidirect
6. #20261312C2semidirect
7. C22⋊Q8×D97C2semidirect
8. #121×D97C2semidirect
9. #121×D97C2semidirect
10. C2×C4⋊C4×D97C2semidirect
11. #537×C97C2semidirect
12. C2×#13095512C22semidirect
13. C2×#13095511C22semidirect
14. C2×#13095511C22semidirect
15. C2×#13095510C22semidirect
16. C2×#13095510C22semidirect
25. #15690790C22semidirect
26. #15690789C22semidirect
27. #15690788C22semidirect
28. #15690787C22semidirect
29. #15690786C22semidirect
30. #15690785C22semidirect
31. #15690785C22semidirect
32. #15690784C22semidirect
33. #15690784C22semidirect
34. #15690783C22semidirect
35. #15690783C22semidirect
36. #15690782C22semidirect
37. #15690782C22semidirect
38. #15690781C22semidirect
39. #15690781C22semidirect
40. C2×C4⋊Dic97C22semidirect
41. C2×#13095490C22semidirect
44. C4⋊C4×D97C22semidirect
46. C2×#41×C97C22semidirect
47. C22⋊Q8×C97C22semidirect
48. #121×C97C22semidirect
49. #121×C97C22semidirect
50. C2×C4⋊C4×C97C22semidirect
76. C2×C4×D97D4semidirect
119. C2×C4×C97C2×D4semidirect
129. C2×Dic97C2×D4semidirect
142. C4×D97C2×D4semidirect
176. D97#537direct (N × H)
178. C97C2×#537semidirect
185. #537D97direct (N × H)
196. C2×#41C2×D97semidirect
209. C22⋊Q8C2×D97semidirect
220. #121C2×D97semidirect
221. #121C2×D97semidirect
227. C2×C4⋊C4C2×D97semidirect
319. C2×C4D4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D972
#202613162
#202613152
#202613142
#202613132
#202613122
C22⋊Q8×D972
#121×D972
C2×C4⋊C4×D972
#537×C972
C2×#53797