Finite Group Databasebeta

← C2×#137×D97 #27156472

Subgroups — C2×#137×D97 #27156472

Order 12416.

1464 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D97C211yesmaximal, characteristic
2C22×Q8×D97C211yesmaximal, characteristic, complemented
3C2×#15690843C211yesmaximal, characteristic, complemented
4C2×#15690842C212yesmaximal, complemented
5C2×#15690841C211yesmaximal, characteristic, complemented
6C2×#15690840C211yesmaximal, characteristic, complemented
7C2×#15690839C212yesmaximal, complemented
8#137×D97C2116yesmaximal, direct product factor
9C2×Q16×D97C212yesmaximal, complemented
10C2×SD16×D97C212yesmaximal, complemented
11C2×M16×D97C211yesmaximal, characteristic, complemented
12C2×#137×C97C211yesmaximal, characteristic, complemented
13C2×#13095512C2211yescharacteristic
14C2×#13095511—21no—
15C2×#13095511C2211yescharacteristic
16C2×#13095510—21no—
17C2×#13095510C2211yescharacteristic
18#41×D97—24no—
19#41×D97C2218yes—
20C22×Dic194—21no—
21C22×Dic194C2211yescharacteristic, complemented
22C2×Q8×D97—21no—
23C2×Q8×D97—24no—
24C2×Q8×D97—21no—
25C2×Q8×D97C2212yes—
26C2×Q8×D97C2211yescharacteristic
27C2×Q8×D97C2212yescomplemented
28C2×Q8×D97C2218yescomplemented
29C2×D4×D97—21no—
30C2×D4×D97C2211yescharacteristic
31C22×C4×D97—21no—
32C22×C4×D97—21no—
33C22×C4×D97C2211yescharacteristic
34#15690843C2218yescomplemented
35#15690842C22116yescomplemented
36#15690841C2218yescomplemented
37#15690840C2218yescomplemented
38#15690839C22116yescomplemented
39C2×Dic388C2212yescomplemented
40C2×#13095501C2212yescomplemented
41C2×#13095501C2212yescomplemented
42Q16×D97C22116yescomplemented
43C2×#13095500C2212yescomplemented
44C2×#13095499C2212yescomplemented
45C2×#13095498C2212yescomplemented
46SD16×D97C22116yescomplemented
47C2×#13095495C2211yescharacteristic, complemented
48C2×#13095494C2212yescomplemented
49M16×D97C2218yescomplemented
50C2×C8×D97C2212yes—
51C2×#41×C97C2211yescharacteristic
52C22×Q8×C97C2211yescharacteristic, complemented
53#137×C97C2218yescomplemented
54C2×Q16×C97C2212yescomplemented
55C2×SD16×C97C2212yescomplemented
56C2×M16×C97C2211yescharacteristic, complemented
57C23×D97—21no—
58C23×D97D411yescharacteristic
59#13095512—22no—
60#13095512C2314yes—
61#13095511—22no—
62#13095511—28no—
63#13095511C2314yes—
64#13095510—28no—
65#13095510—22no—
66#13095510C2314yes—
67C2×Dic194—22no—
68C2×Dic194—21no—
69C2×Dic194—22no—
70C2×Dic194—28no—
71C2×Dic194—22no—
72C2×Dic194—21no—
73C2×Dic194C2312yes—
74C2×Dic194C2311yescharacteristic
75C2×Dic194C2314yescomplemented
76Q8×D97—24no—
77Q8×D97—24no—
78Q8×D97—28no—
79Q8×D97—28no—
80Q8×D97—22no—
81Q8×D97C2318yes—
82Q8×D97C2314yes—
83Q8×D97C2318yescomplemented
84C2×D388—21no—
85C2×D388C2311yescharacteristic
86C2×C22⋊D97—21no—
87C2×C22⋊D97—21no—
88C2×C22⋊D97—21no—
89D4×D97—28no—
90D4×D97—22no—
91D4×D97C2314yes—
92C22×Dic97—21no—
93C22×Dic97—21no—
94C22×Dic97D411yescharacteristic
95C2×C4×D97—21no—
96C2×C4×D97—22no—
97C2×C4×D97—28no—
98C2×C4×D97—22no—
99C2×C4×D97—21no—
100C2×C4×D97—21no—
101C2×C4×D97—28no—
102C2×C4×D97D411yescharacteristic
103C2×C4×D97D414yes—
104C2×C4×D97D411yescharacteristic
105C2×C4×D97C2311yescharacteristic
106C2×C4×D97C2314yes—
107C2×C4×D97C2311yescharacteristic
108Dic388C2318yescomplemented
109#13095501C2318yescomplemented
110#13095501C2318yescomplemented
111#13095500C2318yescomplemented
112#13095499C2318yescomplemented
113#13095498C2318yescomplemented
114#13095495C2314yescomplemented
115#13095494C2318yescomplemented
116C2×#11715895C2312yes—
117C8×D97C2318yes—
118#41×C97—22no—
119#41×C97C2314yes—
120C2×Q8×C97—21no—
121C2×Q8×C97—22no—
122C2×Q8×C97C2312yes—
123C2×Q8×C97C2311yescharacteristic
124C2×Q8×C97C2314yescomplemented
125C2×D4×C97—21no—
126C2×D4×C97C2311yescharacteristic
127C22×C4×C97—21no—
128C22×C4×C97—21no—
129C22×C4×C97C2311yescharacteristic
130Q16×C97C2318yescomplemented
131SD16×C97C2318yescomplemented
132M16×C97C2314yescomplemented
133C2×C8×C97C2312yes—
134C23×C97—21no—
135C23×C97C2×D411yescharacteristic
136C22×D97—24no—
137C22×D97—21no—
138C22×D97—24no—
139C22×D97—44no—
140C22×D97C2×D414yes—
141C22×D97C2×D411yescharacteristic
142C22×D97C2×D411yescharacteristic
143Q8×C97—22no—
144Q8×C97—24no—
145Q8×C97—21no—
146Q8×C97C2414yes—
147Q8×C97C2412yes—
148D4×C97—24no—
149D4×C97—21no—
150D4×C97C2412yes—
151C2×C4×C97—21no—
152C2×C4×C97—22no—
153C2×C4×C97—24no—
154C2×C4×C97—24no—
155C2×C4×C97C2×D411yescharacteristic
156C2×C4×C97C2×D412yes—
157C2×C4×C97C2411yescharacteristic
158C2×C4×C97C2412yes—
159C22⋊D97—24no—
160C22⋊D97—24no—
161C22⋊D97—24no—
162C2×Dic97—21no—
163C2×Dic97—22no—
164C2×Dic97—24no—
165C2×Dic97—24no—
166C2×Dic97C2×D411yescharacteristic
167C2×Dic97C2×D411yescharacteristic
168C2×Dic97C2×D414yes—
169D388—24no—
170D388—21no—
171D388C2412yes—
172C4×D97—24no—
173C4×D97—28no—
174C4×D97—28no—
175C4×D97—24no—
176C4×D97—24no—
177C4×D97C2×D414yes—
178C4×D97C2×D412yes—
179C4×D97C2×D412yes—
180C4×D97C2×D414yes—
181C4×D97C2412yes—
182C4×D97C2412yes—
183Dic194—22no—
184Dic194—24no—
185Dic194—21no—
186Dic194—28no—
187Dic194—28no—
188Dic194—24no—
189Dic194C2414yes—
190Dic194C2412yes—
191C8×C97C2414yes—
192#11715895C2414yes—
193C22×C97—22no—
194C22×C97—22no—
195C22×C97—42no—
196C22×C97C22×D412yes—
197C22×C97C22×D411yescharacteristic
198C2×D97—22no—
199C2×D97—28no—
200C2×D97—22no—
201C2×D97—42no—
202C2×D97—48no—
203C2×D97#13714yesdirect product factor
204C2×D97C22×D412yes—
205C2×D97C22×D412yes—
206C4×C97—24no—
207C4×C97—22no—
208C4×C97C22×D411yescharacteristic
209C4×C97C22×D412yes—
210C4×C97C2511yescharacteristic
211Dic97—24no—
212Dic97—22no—
213Dic97C22×D412yes—
214Dic97C22×D412yes—
215C2×C97—22no—
216C2×C97—42no—
217C2×C97C2×#13712yescomplemented
218C2×C97C23×D411yescharacteristic
219D97—22no—
220D97—42no—
221D97C2×#13714yesdirect product factor
222C22×#137—971nomaximal, Sylow
223C97C22×#13711yescharacteristic, complemented, Sylow
224C22×#41—971no—
225C23×Q8—971no—
226C2×#137—971no—
227C2×#137—971no—
228C2×#137—9716no—
229C2×#137—972no—
230C2×#137—972no—
231C2×#137—971no—
232C2×#137D9711yescharacteristic, direct product factor
233C22×Q16—972no—
234C22×SD16—972no—
235C22×M16—971no—
236C2×#41—978no—
237C2×#41—971no—
238C2×#41—971no—
239C2×#41—971no—
240C2×#41—1941no—
241C2×#41—1941no—
242C2×#41—1944no—
243C2×#41C2×D9711yescharacteristic
244C22×Q8—972no—
245C22×Q8—971no—
246C22×Q8—972no—
247C22×Q8—978no—
248C22×Q8—971no—
249C22×Q8—1941no—
250C22×Q8—1941no—
251C22×Q8—1944no—
252C22×Q8—1941no—
253C22×Q8C2×D9711yescharacteristic, complemented
254C22×D4—971no—
255C22×D4—1941no—
256C23×C4—971no—
257C23×C4—1941no—
258C23×C4—1941no—
259#137—978no—
260#137—978no—
261#137—9716no—
262#137—9716no—
263#137—978no—
264#137C2×D9718yesdirect product factor
265C2×Q16—972no—
266C2×Q16—972no—
267C2×Q16—9716no—
268C2×Q16—972no—
269C2×Q16C2×D9712yescomplemented
270C2×SD16—972no—
271C2×SD16—972no—
272C2×SD16—9716no—
273C2×SD16—972no—
274C2×SD16C2×D9712yescomplemented
275C2×M16—971no—
276C2×M16—978no—
277C2×M16—972no—
278C2×M16C2×D9711yescharacteristic, complemented
279C22×C8—972no—
280C24—971no—
281C24—1941no—
282#41—22no—
283#41—974no—
284#41—974no—
285#41—974no—
286#41—1948no—
287#41—1948no—
288#41—1942no—
289#41—1942no—
290#41—1942no—
291#41C22×D9714yes—
292C2×Q8—22no—
293C2×Q8—21no—
294C2×Q8—978no—
295C2×Q8—974no—
296C2×Q8—978no—
297C2×Q8—972no—
298C2×Q8—971no—
299C2×Q8—974no—
300C2×Q8—1948no—
301C2×Q8—1942no—
302C2×Q8—1942no—
303C2×Q8—1941no—
304C2×Q8—1948no—
305C2×Q8—1944no—
306C2×Q8—1944no—
307C2×Q8—1948no—
308C2×Q8—1942no—
309C2×Q8—1942no—
310C2×Q8—1941no—
311C2×Q8C22×D9712yes—
312C2×Q8C22×D9711yescharacteristic
313C2×Q8C22×D9714yescomplemented
314C2×D4—21no—
315C2×D4—974no—
316C2×D4—971no—
317C2×D4—1941no—
318C2×D4—1942no—
319C2×D4—1941no—
320C2×D4—1948no—
321C2×D4—1941no—
322C2×D4—1941no—
323C2×D4C22×D9711yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—971no—
327C22×C4—971no—
328C22×C4—974no—
329C22×C4—971no—
330C22×C4—971no—
331C22×C4—974no—
332C22×C4—971no—
333C22×C4—1941no—
334C22×C4—1948no—
335C22×C4—1941no—
336C22×C4—1948no—
337C22×C4—1942no—
338C22×C4—1942no—
339C22×C4—1941no—
340C22×C4—1941no—
341C22×C4—1941no—
342C22×C4C22×D9711yescharacteristic
343Q16—978no—
344Q16—978no—
345Q16—978no—
346Q16C22×D9718yescomplemented
347SD16—978no—
348SD16—978no—
349SD16—978no—
350SD16C22×D9718yescomplemented
351M16—974no—
352M16—978no—
353M16C22×D9714yescomplemented
354C2×C8—972no—
355C2×C8—978no—
356C2×C8C22×D9712yes—
357C23—21no—
358C23—971no—
359C23—974no—
360C23—971no—
361C23—3884no—
362C23—1944no—
363C23—1944no—
364C23—1941no—
365C23D4×D9711yescharacteristic
366Q8—22no—
367Q8—24no—
368Q8—21no—
369Q8—974no—
370Q8—972no—
371Q8—1948no—
372Q8—1948no—
373Q8—1944no—
374Q8—1944no—
375Q8—1942no—
376Q8—1941no—
377Q8C23×D9714yes—
378Q8C23×D9712yes—
379D4—24no—
380D4—21no—
381D4—972no—
382D4—1944no—
383D4—1944no—
384D4—1944no—
385D4—1944no—
386D4—1941no—
387D4C23×D9712yes—
388C2×C4—24no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—974no—
393C2×C4—971no—
394C2×C4—971no—
395C2×C4—974no—
396C2×C4—972no—
397C2×C4—972no—
398C2×C4—974no—
399C2×C4—972no—
400C2×C4—972no—
401C2×C4—1944no—
402C2×C4—1944no—
403C2×C4—1948no—
404C2×C4—1948no—
405C2×C4—1944no—
406C2×C4—1944no—
407C2×C4—1941no—
408C2×C4—1942no—
409C2×C4—1944no—
410C2×C4D4×D9712yes—
411C2×C4D4×D9711yescharacteristic
412C2×C4C23×D9712yes—
413C2×C4C23×D9711yescharacteristic
414C8—974no—
415C8C23×D9714yes—
416C22—22no—
417C22—22no—
418C22—42no—
419C22—974no—
420C22—972no—
421C22—972no—
422C22—3888no—
423C22—3882no—
424C22—1948no—
425C22—1942no—
426C22—1942no—
427C22C2×D4×D9712yes—
428C22C2×D4×D9711yescharacteristic
429C4—22no—
430C4—24no—
431C4—972no—
432C4—972no—
433C4—1942no—
434C4—1944no—
435C4C2×D4×D9712yes—
436C4C2×D4×D9711yescharacteristic
437C4C24×D9711yescharacteristic
438C2—22no—
439C2—42no—
440C2—974no—
441C2—3882no—
442C2—1942no—
443C2#137×D9712yesdirect product factor
444C2C22×D4×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
FittingC2×#137×C97C2
socleC22×C97C22×D4
prime radicalC2×#41×D97C2
hypercenterC2×#137D97
hypocenterC97C22×#137
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×Q8×D97C2semidirect
3. C2×#15690843C2semidirect
4. C2×#15690842C2semidirect
5. C2×#15690841C2semidirect
6. C2×#15690840C2semidirect
7. C2×#15690839C2semidirect
8. #137×D97C2direct (N × H)
9. C2×Q16×D97C2semidirect
10. C2×SD16×D97C2semidirect
11. C2×M16×D97C2semidirect
12. C2×#137×C97C2semidirect
21. C22×Dic194C22semidirect
27. C2×Q8×D97C22semidirect
28. C2×Q8×D97C22semidirect
34. #15690843C22semidirect
35. #15690842C22semidirect
36. #15690841C22semidirect
37. #15690840C22semidirect
38. #15690839C22semidirect
39. C2×Dic388C22semidirect
40. C2×#13095501C22semidirect
41. C2×#13095501C22semidirect
42. Q16×D97C22semidirect
43. C2×#13095500C22semidirect
44. C2×#13095499C22semidirect
45. C2×#13095498C22semidirect
46. SD16×D97C22semidirect
47. C2×#13095495C22semidirect
48. C2×#13095494C22semidirect
49. M16×D97C22semidirect
52. C22×Q8×C97C22semidirect
53. #137×C97C22semidirect
54. C2×Q16×C97C22semidirect
55. C2×SD16×C97C22semidirect
56. C2×M16×C97C22semidirect
75. C2×Dic194C23semidirect
83. Q8×D97C23semidirect
108. Dic388C23semidirect
109. #13095501C23semidirect
110. #13095501C23semidirect
111. #13095500C23semidirect
112. #13095499C23semidirect
113. #13095498C23semidirect
114. #13095495C23semidirect
115. #13095494C23semidirect
124. C2×Q8×C97C23semidirect
130. Q16×C97C23semidirect
131. SD16×C97C23semidirect
132. M16×C97C23semidirect
203. C2×D97#137direct (N × H)
217. C2×C97C2×#137semidirect
221. D97C2×#137direct (N × H)
223. C97C22×#137semidirect
232. C2×#137D97direct (N × H)
253. C22×Q8C2×D97semidirect
264. #137C2×D97direct (N × H)
269. C2×Q16C2×D97semidirect
274. C2×SD16C2×D97semidirect
278. C2×M16C2×D97semidirect
313. C2×Q8C22×D97semidirect
346. Q16C22×D97semidirect
350. SD16C22×D97semidirect
353. M16C22×D97semidirect
443. C2#137×D97direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#41×D972
C22×Q8×D972
C2×#156908432
C2×#156908422
C2×#156908412
C2×#156908402
C2×#156908392
#137×D972
C2×Q16×D972
C2×SD16×D972
C2×M16×D972
C2×#137×C972
C22×#13797