Finite Group Databasebeta

← C2×#137×D101 #27185868

Subgroups — C2×#137×D101 #27185868

Order 12928.

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×D101C211yesmaximal, characteristic
2C22×Q8×D101C211yesmaximal, characteristic, complemented
3C2×#15692288C211yesmaximal, characteristic, complemented
4C2×#15692287C212yesmaximal, complemented
5C2×#15692286C211yesmaximal, characteristic, complemented
6C2×#15692285C211yesmaximal, characteristic, complemented
7C2×#15692284C212yesmaximal, complemented
8#137×D101C2116yesmaximal, direct product factor
9C2×Q16×D101C212yesmaximal, complemented
10C2×SD16×D101C212yesmaximal, complemented
11C2×M16×D101C211yesmaximal, characteristic, complemented
12C2×#137×C101C211yesmaximal, characteristic, complemented
13C2×#13108075C2211yescharacteristic
14C2×#13108074—21no—
15C2×#13108074C2211yescharacteristic
16C2×#13108073—21no—
17C2×#13108073C2211yescharacteristic
18#41×D101—24no—
19#41×D101C2218yes—
20C22×Dic202—21no—
21C22×Dic202C2211yescharacteristic, complemented
22C2×Q8×D101—21no—
23C2×Q8×D101—24no—
24C2×Q8×D101—21no—
25C2×Q8×D101C2212yes—
26C2×Q8×D101C2211yescharacteristic
27C2×Q8×D101C2212yescomplemented
28C2×Q8×D101C2218yescomplemented
29C2×D4×D101—21no—
30C2×D4×D101C2211yescharacteristic
31C22×C4×D101—21no—
32C22×C4×D101—21no—
33C22×C4×D101C2211yescharacteristic
34#15692288C2218yescomplemented
35#15692287C22116yescomplemented
36#15692286C2218yescomplemented
37#15692285C2218yescomplemented
38#15692284C22116yescomplemented
39C2×Dic404C2212yescomplemented
40C2×#13108064C2212yescomplemented
41C2×#13108064C2212yescomplemented
42Q16×D101C22116yescomplemented
43C2×#13108063C2212yescomplemented
44C2×#13108062C2212yescomplemented
45C2×#13108061C2212yescomplemented
46SD16×D101C22116yescomplemented
47C2×#13108058C2211yescharacteristic, complemented
48C2×#13108057C2212yescomplemented
49M16×D101C2218yescomplemented
50C2×C8×D101C2212yes—
51C2×#41×C101C2211yescharacteristic
52C22×Q8×C101C2211yescharacteristic, complemented
53#137×C101C2218yescomplemented
54C2×Q16×C101C2212yescomplemented
55C2×SD16×C101C2212yescomplemented
56C2×M16×C101C2211yescharacteristic, complemented
57C23×D101—21no—
58C23×D101D411yescharacteristic
59#13108075—22no—
60#13108075C2314yes—
61#13108074—22no—
62#13108074—28no—
63#13108074C2314yes—
64#13108073—28no—
65#13108073—22no—
66#13108073C2314yes—
67C2×Dic202—22no—
68C2×Dic202—21no—
69C2×Dic202—22no—
70C2×Dic202—28no—
71C2×Dic202—22no—
72C2×Dic202—21no—
73C2×Dic202C2312yes—
74C2×Dic202C2311yescharacteristic
75C2×Dic202C2314yescomplemented
76Q8×D101—24no—
77Q8×D101—24no—
78Q8×D101—28no—
79Q8×D101—28no—
80Q8×D101—22no—
81Q8×D101C2318yes—
82Q8×D101C2314yes—
83Q8×D101C2318yescomplemented
84C2×D404—21no—
85C2×D404C2311yescharacteristic
86C2×C22⋊D101—21no—
87C2×C22⋊D101—21no—
88C2×C22⋊D101—21no—
89D4×D101—28no—
90D4×D101—22no—
91D4×D101C2314yes—
92C22×Dic101—21no—
93C22×Dic101—21no—
94C22×Dic101D411yescharacteristic
95C2×C4×D101—21no—
96C2×C4×D101—22no—
97C2×C4×D101—28no—
98C2×C4×D101—22no—
99C2×C4×D101—21no—
100C2×C4×D101—21no—
101C2×C4×D101—28no—
102C2×C4×D101D411yescharacteristic
103C2×C4×D101D414yes—
104C2×C4×D101D411yescharacteristic
105C2×C4×D101C2311yescharacteristic
106C2×C4×D101C2314yes—
107C2×C4×D101C2311yescharacteristic
108Dic404C2318yescomplemented
109#13108064C2318yescomplemented
110#13108064C2318yescomplemented
111#13108063C2318yescomplemented
112#13108062C2318yescomplemented
113#13108061C2318yescomplemented
114#13108058C2314yescomplemented
115#13108057C2318yescomplemented
116C2×#11717579C2312yes—
117C8×D101C2318yes—
118#41×C101—22no—
119#41×C101C2314yes—
120C2×Q8×C101—21no—
121C2×Q8×C101—22no—
122C2×Q8×C101C2312yes—
123C2×Q8×C101C2311yescharacteristic
124C2×Q8×C101C2314yescomplemented
125C2×D4×C101—21no—
126C2×D4×C101C2311yescharacteristic
127C22×C4×C101—21no—
128C22×C4×C101—21no—
129C22×C4×C101C2311yescharacteristic
130Q16×C101C2318yescomplemented
131SD16×C101C2318yescomplemented
132M16×C101C2314yescomplemented
133C2×C8×C101C2312yes—
134C23×C101—21no—
135C23×C101C2×D411yescharacteristic
136C22×D101—24no—
137C22×D101—21no—
138C22×D101—24no—
139C22×D101—44no—
140C22×D101C2×D414yes—
141C22×D101C2×D411yescharacteristic
142C22×D101C2×D411yescharacteristic
143Q8×C101—22no—
144Q8×C101—24no—
145Q8×C101—21no—
146Q8×C101C2414yes—
147Q8×C101C2412yes—
148D4×C101—24no—
149D4×C101—21no—
150D4×C101C2412yes—
151C2×C4×C101—21no—
152C2×C4×C101—22no—
153C2×C4×C101—24no—
154C2×C4×C101—24no—
155C2×C4×C101C2×D411yescharacteristic
156C2×C4×C101C2×D412yes—
157C2×C4×C101C2411yescharacteristic
158C2×C4×C101C2412yes—
159C22⋊D101—24no—
160C22⋊D101—24no—
161C22⋊D101—24no—
162C2×Dic101—21no—
163C2×Dic101—22no—
164C2×Dic101—24no—
165C2×Dic101—24no—
166C2×Dic101C2×D411yescharacteristic
167C2×Dic101C2×D411yescharacteristic
168C2×Dic101C2×D414yes—
169D404—24no—
170D404—21no—
171D404C2412yes—
172C4×D101—24no—
173C4×D101—28no—
174C4×D101—28no—
175C4×D101—24no—
176C4×D101—24no—
177C4×D101C2×D414yes—
178C4×D101C2×D412yes—
179C4×D101C2×D412yes—
180C4×D101C2×D414yes—
181C4×D101C2412yes—
182C4×D101C2412yes—
183Dic202—22no—
184Dic202—24no—
185Dic202—21no—
186Dic202—28no—
187Dic202—28no—
188Dic202—24no—
189Dic202C2414yes—
190Dic202C2412yes—
191C8×C101C2414yes—
192#11717579C2414yes—
193C22×C101—22no—
194C22×C101—22no—
195C22×C101—42no—
196C22×C101C22×D412yes—
197C22×C101C22×D411yescharacteristic
198C2×D101—22no—
199C2×D101—28no—
200C2×D101—22no—
201C2×D101—42no—
202C2×D101—48no—
203C2×D101#13714yesdirect product factor
204C2×D101C22×D412yes—
205C2×D101C22×D412yes—
206C4×C101—24no—
207C4×C101—22no—
208C4×C101C22×D411yescharacteristic
209C4×C101C22×D412yes—
210C4×C101C2511yescharacteristic
211Dic101—24no—
212Dic101—22no—
213Dic101C22×D412yes—
214Dic101C22×D412yes—
215C2×C101—22no—
216C2×C101—42no—
217C2×C101C2×#13712yescomplemented
218C2×C101C23×D411yescharacteristic
219D101—22no—
220D101—42no—
221D101C2×#13714yesdirect product factor
222C22×#137—1011nomaximal, Sylow
223C101C22×#13711yescharacteristic, complemented, Sylow
224C22×#41—1011no—
225C23×Q8—1011no—
226C2×#137—1011no—
227C2×#137—1011no—
228C2×#137—10116no—
229C2×#137—1012no—
230C2×#137—1012no—
231C2×#137—1011no—
232C2×#137D10111yescharacteristic, direct product factor
233C22×Q16—1012no—
234C22×SD16—1012no—
235C22×M16—1011no—
236C2×#41—1018no—
237C2×#41—1011no—
238C2×#41—1011no—
239C2×#41—1011no—
240C2×#41—2021no—
241C2×#41—2021no—
242C2×#41—2024no—
243C2×#41C2×D10111yescharacteristic
244C22×Q8—1012no—
245C22×Q8—1011no—
246C22×Q8—1012no—
247C22×Q8—1018no—
248C22×Q8—1011no—
249C22×Q8—2021no—
250C22×Q8—2021no—
251C22×Q8—2024no—
252C22×Q8—2021no—
253C22×Q8C2×D10111yescharacteristic, complemented
254C22×D4—1011no—
255C22×D4—2021no—
256C23×C4—1011no—
257C23×C4—2021no—
258C23×C4—2021no—
259#137—1018no—
260#137—1018no—
261#137—10116no—
262#137—10116no—
263#137—1018no—
264#137C2×D10118yesdirect product factor
265C2×Q16—1012no—
266C2×Q16—1012no—
267C2×Q16—10116no—
268C2×Q16—1012no—
269C2×Q16C2×D10112yescomplemented
270C2×SD16—1012no—
271C2×SD16—1012no—
272C2×SD16—10116no—
273C2×SD16—1012no—
274C2×SD16C2×D10112yescomplemented
275C2×M16—1011no—
276C2×M16—1018no—
277C2×M16—1012no—
278C2×M16C2×D10111yescharacteristic, complemented
279C22×C8—1012no—
280C24—1011no—
281C24—2021no—
282#41—22no—
283#41—1014no—
284#41—1014no—
285#41—1014no—
286#41—2028no—
287#41—2028no—
288#41—2022no—
289#41—2022no—
290#41—2022no—
291#41C22×D10114yes—
292C2×Q8—22no—
293C2×Q8—21no—
294C2×Q8—1018no—
295C2×Q8—1014no—
296C2×Q8—1018no—
297C2×Q8—1012no—
298C2×Q8—1011no—
299C2×Q8—1014no—
300C2×Q8—2028no—
301C2×Q8—2022no—
302C2×Q8—2022no—
303C2×Q8—2021no—
304C2×Q8—2028no—
305C2×Q8—2024no—
306C2×Q8—2024no—
307C2×Q8—2028no—
308C2×Q8—2022no—
309C2×Q8—2022no—
310C2×Q8—2021no—
311C2×Q8C22×D10112yes—
312C2×Q8C22×D10111yescharacteristic
313C2×Q8C22×D10114yescomplemented
314C2×D4—21no—
315C2×D4—1014no—
316C2×D4—1011no—
317C2×D4—2021no—
318C2×D4—2022no—
319C2×D4—2021no—
320C2×D4—2028no—
321C2×D4—2021no—
322C2×D4—2021no—
323C2×D4C22×D10111yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—1011no—
327C22×C4—1011no—
328C22×C4—1014no—
329C22×C4—1011no—
330C22×C4—1011no—
331C22×C4—1014no—
332C22×C4—1011no—
333C22×C4—2021no—
334C22×C4—2028no—
335C22×C4—2021no—
336C22×C4—2028no—
337C22×C4—2022no—
338C22×C4—2022no—
339C22×C4—2021no—
340C22×C4—2021no—
341C22×C4—2021no—
342C22×C4C22×D10111yescharacteristic
343Q16—1018no—
344Q16—1018no—
345Q16—1018no—
346Q16C22×D10118yescomplemented
347SD16—1018no—
348SD16—1018no—
349SD16—1018no—
350SD16C22×D10118yescomplemented
351M16—1014no—
352M16—1018no—
353M16C22×D10114yescomplemented
354C2×C8—1012no—
355C2×C8—1018no—
356C2×C8C22×D10112yes—
357C23—21no—
358C23—1011no—
359C23—1014no—
360C23—1011no—
361C23—4044no—
362C23—2024no—
363C23—2024no—
364C23—2021no—
365C23D4×D10111yescharacteristic
366Q8—22no—
367Q8—24no—
368Q8—21no—
369Q8—1014no—
370Q8—1012no—
371Q8—2028no—
372Q8—2028no—
373Q8—2024no—
374Q8—2024no—
375Q8—2022no—
376Q8—2021no—
377Q8C23×D10114yes—
378Q8C23×D10112yes—
379D4—24no—
380D4—21no—
381D4—1012no—
382D4—2024no—
383D4—2024no—
384D4—2024no—
385D4—2024no—
386D4—2021no—
387D4C23×D10112yes—
388C2×C4—24no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—1014no—
393C2×C4—1011no—
394C2×C4—1011no—
395C2×C4—1014no—
396C2×C4—1012no—
397C2×C4—1012no—
398C2×C4—1014no—
399C2×C4—1012no—
400C2×C4—1012no—
401C2×C4—2024no—
402C2×C4—2024no—
403C2×C4—2028no—
404C2×C4—2028no—
405C2×C4—2024no—
406C2×C4—2024no—
407C2×C4—2021no—
408C2×C4—2022no—
409C2×C4—2024no—
410C2×C4D4×D10112yes—
411C2×C4D4×D10111yescharacteristic
412C2×C4C23×D10112yes—
413C2×C4C23×D10111yescharacteristic
414C8—1014no—
415C8C23×D10114yes—
416C22—22no—
417C22—22no—
418C22—42no—
419C22—1012no—
420C22—1014no—
421C22—1012no—
422C22—4048no—
423C22—4042no—
424C22—2028no—
425C22—2022no—
426C22—2022no—
427C22C2×D4×D10112yes—
428C22C2×D4×D10111yescharacteristic
429C4—22no—
430C4—24no—
431C4—1012no—
432C4—1012no—
433C4—2022no—
434C4—2024no—
435C4C2×D4×D10112yes—
436C4C2×D4×D10111yescharacteristic
437C4C24×D10111yescharacteristic
438C2—22no—
439C2—42no—
440C2—1014no—
441C2—4042no—
442C2—2022no—
443C2#137×D10112yesdirect product factor
444C2C22×D4×D10111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC4×C101C25
FrattiniC4C24×D101
FittingC2×#137×C101C2
socleC22×C101C22×D4
prime radicalC2×#41×D101C2
hypercenterC2×#137D101
hypocenterC101C22×#137
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×Q8×D101C2semidirect
3. C2×#15692288C2semidirect
4. C2×#15692287C2semidirect
5. C2×#15692286C2semidirect
6. C2×#15692285C2semidirect
7. C2×#15692284C2semidirect
8. #137×D101C2direct (N × H)
9. C2×Q16×D101C2semidirect
10. C2×SD16×D101C2semidirect
11. C2×M16×D101C2semidirect
12. C2×#137×C101C2semidirect
21. C22×Dic202C22semidirect
27. C2×Q8×D101C22semidirect
28. C2×Q8×D101C22semidirect
34. #15692288C22semidirect
35. #15692287C22semidirect
36. #15692286C22semidirect
37. #15692285C22semidirect
38. #15692284C22semidirect
39. C2×Dic404C22semidirect
40. C2×#13108064C22semidirect
41. C2×#13108064C22semidirect
42. Q16×D101C22semidirect
43. C2×#13108063C22semidirect
44. C2×#13108062C22semidirect
45. C2×#13108061C22semidirect
46. SD16×D101C22semidirect
47. C2×#13108058C22semidirect
48. C2×#13108057C22semidirect
49. M16×D101C22semidirect
52. C22×Q8×C101C22semidirect
53. #137×C101C22semidirect
54. C2×Q16×C101C22semidirect
55. C2×SD16×C101C22semidirect
56. C2×M16×C101C22semidirect
75. C2×Dic202C23semidirect
83. Q8×D101C23semidirect
108. Dic404C23semidirect
109. #13108064C23semidirect
110. #13108064C23semidirect
111. #13108063C23semidirect
112. #13108062C23semidirect
113. #13108061C23semidirect
114. #13108058C23semidirect
115. #13108057C23semidirect
124. C2×Q8×C101C23semidirect
130. Q16×C101C23semidirect
131. SD16×C101C23semidirect
132. M16×C101C23semidirect
203. C2×D101#137direct (N × H)
217. C2×C101C2×#137semidirect
221. D101C2×#137direct (N × H)
223. C101C22×#137semidirect
232. C2×#137D101direct (N × H)
253. C22×Q8C2×D101semidirect
264. #137C2×D101direct (N × H)
269. C2×Q16C2×D101semidirect
274. C2×SD16C2×D101semidirect
278. C2×M16C2×D101semidirect
313. C2×Q8C22×D101semidirect
346. Q16C22×D101semidirect
350. SD16C22×D101semidirect
353. M16C22×D101semidirect
443. C2#137×D101direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#41×D1012
C22×Q8×D1012
C2×#156922882
C2×#156922872
C2×#156922862
C2×#156922852
C2×#156922842
#137×D1012
C2×Q16×D1012
C2×SD16×D1012
C2×M16×D1012
C2×#137×C1012
C22×#137101