Finite Group Databasebeta

← C2×#20293216 #28349803

Subgroups — C2×#20293216 #28349803

Order 13696.

1336 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293216C2116yesmaximal, direct product factor
2C2×#16812104C211yesmaximal, characteristic, complemented
3C2×#16812097C211yesmaximal, characteristic, complemented
4C2×#16812095C211yesmaximal, characteristic, complemented
5C2×#16812094C211yesmaximal, characteristic, complemented
6C2×#16812091C211yesmaximal, characteristic, complemented
7C2×#16812087C211yesmaximal, characteristic, complemented
8C2×#16812084C211yesmaximal, characteristic, complemented
9C2×#16812084C211yesmaximal, characteristic, complemented
10C2×#16812082C211yesmaximal, characteristic, complemented
11C2×#16812079C211yesmaximal, characteristic, complemented
12C2×#16812079C211yesmaximal, characteristic, complemented
13C2×C4×D428C211yesmaximal, characteristic, complemented
14C2×#16812071C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×D107C211yesmaximal, characteristic, complemented
16C2×#126×C107C211yesmaximal, characteristic, complemented
17C24×D107C2211yescharacteristic
18C22×Dic214C2211yescharacteristic, complemented
19C22×D428C2211yescharacteristic
20C22×D428C2211yescharacteristic
21C22×D428C2211yescharacteristic
22C22×D428C2211yescharacteristic
23C22×D428C2211yescharacteristic, complemented
24C22×C22⋊D107C2211yescharacteristic
25C22×C22⋊D107C2211yescharacteristic
26C22×C4×D107C2211yescharacteristic
27C22×C4×D107C2211yescharacteristic
28C22×C4×D107C2211yescharacteristic, complemented
29C22×C4×D107C2211yescharacteristic, complemented
30#16812104C2218yescomplemented
31#16812097C2218yescomplemented
32#16812095C2218yescomplemented
33#16812094C2218yescomplemented
34#16812091C2218yescomplemented
35#16812087C2218yescomplemented
36#16812084C2218yescomplemented
37#16812084C2218yescomplemented
38#16812082C2218yescomplemented
39#16812079C2218yescomplemented
40#16812079C2218yescomplemented
41C4×D428C2218yescomplemented
42#16812071C2218yescomplemented
43C2×C4⋊Dic107C2211yescharacteristic, complemented
44C2×Dic107⋊C4C2211yescharacteristic, complemented
45C2×Dic107⋊C4C2211yescharacteristic, complemented
46C2×C22⋊Dic107C2211yescharacteristic
47C2×(C2×D107)⋊C4C2211yescharacteristic
48C2×(C2×D107)⋊C4C2211yescharacteristic
49C2×(C2×D107)⋊C4C2211yescharacteristic
50C2×(C2×D107)⋊C4C2211yescharacteristic
51C2×(C2×D107)⋊C4C2211yescharacteristic
52C2×(C2×D107)⋊C4C2211yescharacteristic
53C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
54C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
55C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
56C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
57C22⋊C4×D107C2218yescomplemented
58C2×C4×Dic107C2211yescharacteristic, complemented
59#126×C107C2218yescomplemented
60C2×C4⋊C4×C107C2211yescharacteristic, complemented
61C2×C4⋊C4×C107C2211yescharacteristic, complemented
62C2×C4⋊C4×C107C2211yescharacteristic, complemented
63C2×C22⋊C4×C107C2211yescharacteristic
64C2×C22⋊C4×C107C2211yescharacteristic
65C2×C22⋊C4×C107C2211yescharacteristic
66C2×C42×C107C2211yescharacteristic, complemented
67C23×D107—24no—
68C23×D107—42no—
69C23×D107—42no—
70C23×D107C2311yescharacteristic
71C23×D107C2311yescharacteristic
72C23×D107C2311yescharacteristic
73C23×D107C2311yescharacteristic
74C23×D107C2314yes—
75C2×Dic214—42no—
76C2×Dic214C2314yescomplemented
77C2×D428—24no—
78C2×D428—24no—
79C2×D428—42no—
80C2×D428—42no—
81C2×D428—42no—
82C2×D428C2314yes—
83C2×D428C2314yes—
84C2×D428C2314yes—
85C2×D428C2314yes—
86C2×D428C2314yescomplemented
87C2×C22⋊D107—42no—
88C2×C22⋊D107—42no—
89C2×C22⋊D107C2314yes—
90C2×C22⋊D107C2314yes—
91C22×Dic107C2311yescharacteristic
92C22×Dic107C2311yescharacteristic
93C22×Dic107C2311yescharacteristic
94C22×Dic107C2311yescharacteristic
95C2×C4×D107—42no—
96C2×C4×D107—42no—
97C2×C4×D107—42no—
98C2×C4×D107—42no—
99C2×C4×D107C2314yes—
100C2×C4×D107C2314yes—
101C2×C4×D107C2314yescomplemented
102C2×C4×D107C2314yescomplemented
103C4⋊Dic107C2314yescomplemented
104Dic107⋊C4C2314yescomplemented
105Dic107⋊C4C2314yescomplemented
106C22⋊Dic107C2314yes—
107(C2×D107)⋊C4C2314yes—
108(C2×D107)⋊C4C2314yes—
109(C2×D107)⋊C4C2314yes—
110(C2×D107)⋊C4C2314yes—
111(C2×D107)⋊C4C2314yes—
112(C2×D107)⋊C4C2314yes—
113(C2×D107)⋊C4C2314yescomplemented
114(C2×D107)⋊C4C2314yescomplemented
115(C2×D107)⋊C4C2314yescomplemented
116(C2×D107)⋊C4C2314yescomplemented
117C4×Dic107C2314yescomplemented
118C24×C107C2311yescharacteristic
119C22×C4×C107C2311yescharacteristic
120C22×C4×C107C2311yescharacteristic
121C22×C4×C107C2311yescharacteristic
122C22×C4×C107C2311yescharacteristic
123C22×C4×C107C2311yescharacteristic
124C22×C4×C107C2311yescharacteristic
125C4⋊C4×C107C2314yescomplemented
126C4⋊C4×C107C2314yescomplemented
127C4⋊C4×C107C2314yescomplemented
128C22⋊C4×C107C2314yes—
129C22⋊C4×C107C2314yes—
130C22⋊C4×C107C2314yes—
131C42×C107C2314yescomplemented
132C23×C107—22no—
133C23×C107—22no—
134C23×C107—22no—
135C23×C107C2412yes—
136C23×C107C2411yescharacteristic
137C22×D107—22no—
138C22×D107—22no—
139C22×D107—22no—
140C22×D107—22no—
141C22×D107—22no—
142C22×D107—22no—
143C22×D107—22no—
144C22×D107—22no—
145C22×D107—22no—
146C22×D107—22no—
147C22×D107—22no—
148C22×D107—28no—
149C22×D107—44no—
150C22×D107—44no—
151C22×D107—416no—
152C22×D107#4114yes—
153C22×D107C2412yes—
154C22×D107C2412yes—
155C22×D107C2412yes—
156C22×D107C2412yes—
157C2×C4×C107—22no—
158C2×C4×C107—22no—
159C2×C4×C107—22no—
160C2×C4×C107—22no—
161C2×C4×C107—22no—
162C2×C4×C107—22no—
163C2×C4×C107C2412yes—
164C2×C4×C107C2412yes—
165C2×C4×C107C2412yes—
166C2×C4×C107C2412yes—
167C2×C4×C107C2412yes—
168C2×C4×C107C2412yes—
169C22⋊D107—44no—
170C22⋊D107—44no—
171C2×Dic107—22no—
172C2×Dic107—22no—
173C2×Dic107—22no—
174C2×Dic107—22no—
175C2×Dic107C2412yes—
176C2×Dic107C2412yes—
177C2×Dic107C2412yes—
178C2×Dic107C2412yes—
179D428—28no—
180D428—28no—
181D428—44no—
182D428—44no—
183D428—44no—
184C4×D107—44no—
185C4×D107—44no—
186C4×D107—44no—
187C4×D107—44no—
188Dic214—44no—
189C22×C107—22no—
190C22×C107—22no—
191C22×C107—22no—
192C22×C107—44no—
193C22×C107C2×#4112yes—
194C22×C107H(5,𝔽2)12yes—
195C22×C107H(5,𝔽2)12yes—
196C22×C107C2511yescharacteristic
197C2×D107—28no—
198C2×D107—22no—
199C2×D107—22no—
200C2×D107—22no—
201C2×D107—22no—
202C2×D107—22no—
203C2×D107—22no—
204C2×D107—22no—
205C2×D107—22no—
206C2×D107—22no—
207C2×D107—22no—
208C2×D107—22no—
209C2×D107—416no—
210C2×D107—44no—
211C2×D107—44no—
212C2×D107—44no—
213C2×D107C2×#4114yes—
214C4×C107—22no—
215C4×C107—22no—
216C4×C107—22no—
217C4×C107—22no—
218C4×C107—22no—
219C4×C107—22no—
220Dic107—22no—
221Dic107—22no—
222Dic107—22no—
223Dic107—22no—
224C2×C107—42no—
225C2×C107#53814yescomplemented
226C2×C107C22×#4111yescharacteristic
227C2×C107C2×H(5,𝔽2)11yescharacteristic
228C2×C107C2×H(5,𝔽2)11yescharacteristic
229D107—24no—
230D107—42no—
231D107—42no—
232D107—42no—
233C2×#538—1071nomaximal, Sylow
234C107C2×#53811yescharacteristic, complemented, Sylow
235#538—10716no—
236C2×#126—1071no—
237C2×#126D10711yescharacteristic, complemented
238C2×#124—1071no—
239C2×#124—1071no—
240C2×#123—1071no—
241C2×#123—1071no—
242C2×C22⋊Q8—1071no—
243C2×#121—1071no—
244C2×#121—1071no—
245C2×#121—1071no—
246C2×#120—1071no—
247C2×#120—1071no—
248C2×C4×D4—1071no—
249C2×C4×D4—1071no—
250C22×C22⋊C4—1071no—
251C25—1071no—
252C22×Q8—1071no—
253C22×D4—1071no—
254C22×D4—1071no—
255C22×D4—1071no—
256C22×D4—1071no—
257C22×D4—1071no—
258C22×D4—1071no—
259C22×D4—1071no—
260C23×C4—1071no—
261C23×C4—1071no—
262C23×C4—1071no—
263C23×C4—1071no—
264#126—1078no—
265#126C2×D10718yescomplemented
266#124—1078no—
267#124—1078no—
268#123—1078no—
269#123—1078no—
270C22⋊Q8—1078no—
271#121—1078no—
272#121—1078no—
273#121—1078no—
274#120—1078no—
275#120—1078no—
276C4×D4—1078no—
277C4×D4—1078no—
278C2×C4⋊C4—1071no—
279C2×C4⋊C4—1071no—
280C2×C4⋊C4—1071no—
281C2×C4⋊C4C2×D10711yescharacteristic, complemented
282C2×C4⋊C4C2×D10711yescharacteristic, complemented
283C2×C4⋊C4C2×D10711yescharacteristic, complemented
284C2×C22⋊C4—1071no—
285C2×C22⋊C4—1071no—
286C2×C22⋊C4—1071no—
287C2×C22⋊C4—1071no—
288C2×C22⋊C4—1071no—
289C2×C22⋊C4—1071no—
290C2×C22⋊C4—1071no—
291C2×C22⋊C4—1071no—
292C2×C22⋊C4—1071no—
293C2×C22⋊C4—1071no—
294C2×C22⋊C4—1071no—
295C2×C22⋊C4—1078no—
296C2×C22⋊C4C2×D10711yescharacteristic
297C2×C22⋊C4C2×D10711yescharacteristic
298C2×C22⋊C4C2×D10711yescharacteristic
299C2×C42—1071no—
300C2×C42C2×D10711yescharacteristic, complemented
301C24—1071no—
302C24—1074no—
303C24—1071no—
304C24—1071no—
305C24—1071no—
306C24—4282no—
307C24—4282no—
308C24—2144no—
309C24C22×D10711yescharacteristic
310C2×Q8—1074no—
311C2×Q8—4282no—
312C2×D4—1074no—
313C2×D4—1074no—
314C2×D4—1074no—
315C2×D4—1074no—
316C2×D4—1074no—
317C2×D4—1074no—
318C2×D4—1074no—
319C2×D4—4282no—
320C2×D4—4282no—
321C2×D4—4282no—
322C2×D4—4282no—
323C2×D4—4282no—
324C2×D4—2144no—
325C2×D4—2144no—
326C22×C4—1071no—
327C22×C4—1071no—
328C22×C4—1071no—
329C22×C4—1071no—
330C22×C4—1074no—
331C22×C4—1074no—
332C22×C4—1074no—
333C22×C4—1074no—
334C22×C4—4282no—
335C22×C4—4282no—
336C22×C4—4282no—
337C22×C4—4282no—
338C22×C4C22×D10711yescharacteristic
339C22×C4C22×D10711yescharacteristic
340C22×C4C22×D10711yescharacteristic
341C22×C4C22×D10711yescharacteristic
342C22×C4C22×D10711yescharacteristic
343C22×C4C22×D10711yescharacteristic
344C4⋊C4—1074no—
345C4⋊C4—1074no—
346C4⋊C4—1074no—
347C4⋊C4C22×D10714yescomplemented
348C4⋊C4C22×D10714yescomplemented
349C4⋊C4C22×D10714yescomplemented
350C22⋊C4—1074no—
351C22⋊C4—1074no—
352C22⋊C4—1074no—
353C22⋊C4—1074no—
354C22⋊C4—1074no—
355C22⋊C4—1074no—
356C22⋊C4—1074no—
357C22⋊C4—1074no—
358C22⋊C4—1074no—
359C22⋊C4—1074no—
360C22⋊C4—1074no—
361C22⋊C4C22×D10714yes—
362C22⋊C4C22×D10714yes—
363C22⋊C4C22×D10714yes—
364C42—1074no—
365C42C22×D10714yescomplemented
366C23—22no—
367C23—22no—
368C23—22no—
369C23—1074no—
370C23—1072no—
371C23—1072no—
372C23—1072no—
373C23—1072no—
374C23—4284no—
375C23—4284no—
376C23—42816no—
377C23—2148no—
378C23—2142no—
379C23—2142no—
380C23—2142no—
381C23—2142no—
382C23—2142no—
383C23—2142no—
384C23—2142no—
385C23—2142no—
386C23—2142no—
387C23—2142no—
388C23—2142no—
389C23C23×D10712yes—
390C23C23×D10711yescharacteristic
391Q8—4284no—
392D4—4284no—
393D4—4284no—
394D4—4284no—
395D4—4284no—
396D4—4284no—
397D4—2148no—
398D4—2148no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—1072no—
406C2×C4—1072no—
407C2×C4—1072no—
408C2×C4—1072no—
409C2×C4—4284no—
410C2×C4—4284no—
411C2×C4—4284no—
412C2×C4—4284no—
413C2×C4—2142no—
414C2×C4—2142no—
415C2×C4—2142no—
416C2×C4—2142no—
417C2×C4C23×D10712yes—
418C2×C4C23×D10712yes—
419C2×C4C23×D10712yes—
420C2×C4C23×D10712yes—
421C2×C4C23×D10712yes—
422C2×C4C23×D10712yes—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—44no—
427C22—1074no—
428C22—42816no—
429C22—4284no—
430C22—4284no—
431C22—4284no—
432C22—2142no—
433C22—2142no—
434C22—2142no—
435C22—2142no—
436C22—2142no—
437C22—2142no—
438C22—2142no—
439C22—2142no—
440C22—2142no—
441C22—2142no—
442C22—2142no—
443C22—2148no—
444C22#41×D10712yes—
445C22#1681215512yes—
446C22#1681215512yes—
447C22C24×D10711yescharacteristic
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—22no—
453C4—22no—
454C4—2142no—
455C4—2142no—
456C4—2142no—
457C4—2142no—
458C2—42no—
459C2—4282no—
460C2—4282no—
461C2—4282no—
462C2—2144no—
463C2#2029321614yesdirect product factor
464C2C2×#41×D10711yescharacteristic
465C2C2×#1681215511yescharacteristic
466C2C2×#1681215511yescharacteristic
subgroupG/H
centerC23C23×D107
commutator, cosocleC22×C107C25
FrattiniC22C24×D107
FittingC2×#126×C107C2
socleC23×C107C24
hypercenterC2×#126D107
hypocenterC107C2×#538
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #20293216C2direct (N × H)
2. C2×#16812104C2semidirect
3. C2×#16812097C2semidirect
4. C2×#16812095C2semidirect
5. C2×#16812094C2semidirect
6. C2×#16812091C2semidirect
7. C2×#16812087C2semidirect
8. C2×#16812084C2semidirect
9. C2×#16812084C2semidirect
10. C2×#16812082C2semidirect
11. C2×#16812079C2semidirect
12. C2×#16812079C2semidirect
13. C2×C4×D428C2semidirect
14. C2×#16812071C2semidirect
15. C2×C22⋊C4×D107C2semidirect
16. C2×#126×C107C2semidirect
18. C22×Dic214C22semidirect
23. C22×D428C22semidirect
28. C22×C4×D107C22semidirect
29. C22×C4×D107C22semidirect
30. #16812104C22semidirect
31. #16812097C22semidirect
32. #16812095C22semidirect
33. #16812094C22semidirect
34. #16812091C22semidirect
35. #16812087C22semidirect
36. #16812084C22semidirect
37. #16812084C22semidirect
38. #16812082C22semidirect
39. #16812079C22semidirect
40. #16812079C22semidirect
41. C4×D428C22semidirect
42. #16812071C22semidirect
43. C2×C4⋊Dic107C22semidirect
44. C2×Dic107⋊C4C22semidirect
45. C2×Dic107⋊C4C22semidirect
53. C2×(C2×D107)⋊C4C22semidirect
54. C2×(C2×D107)⋊C4C22semidirect
55. C2×(C2×D107)⋊C4C22semidirect
56. C2×(C2×D107)⋊C4C22semidirect
57. C22⋊C4×D107C22semidirect
58. C2×C4×Dic107C22semidirect
59. #126×C107C22semidirect
60. C2×C4⋊C4×C107C22semidirect
61. C2×C4⋊C4×C107C22semidirect
62. C2×C4⋊C4×C107C22semidirect
66. C2×C42×C107C22semidirect
76. C2×Dic214C23semidirect
86. C2×D428C23semidirect
101. C2×C4×D107C23semidirect
102. C2×C4×D107C23semidirect
103. C4⋊Dic107C23semidirect
104. Dic107⋊C4C23semidirect
105. Dic107⋊C4C23semidirect
113. (C2×D107)⋊C4C23semidirect
114. (C2×D107)⋊C4C23semidirect
115. (C2×D107)⋊C4C23semidirect
116. (C2×D107)⋊C4C23semidirect
117. C4×Dic107C23semidirect
125. C4⋊C4×C107C23semidirect
126. C4⋊C4×C107C23semidirect
127. C4⋊C4×C107C23semidirect
131. C42×C107C23semidirect
225. C2×C107#538semidirect
234. C107C2×#538semidirect
237. C2×#126D107semidirect
265. #126C2×D107semidirect
281. C2×C4⋊C4C2×D107semidirect
282. C2×C4⋊C4C2×D107semidirect
283. C2×C4⋊C4C2×D107semidirect
300. C2×C42C2×D107semidirect
347. C4⋊C4C22×D107semidirect
348. C4⋊C4C22×D107semidirect
349. C4⋊C4C22×D107semidirect
365. C42C22×D107semidirect
463. C2#20293216direct (N × H)

Maximal subgroups

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

subgroupindex
#202932162
C2×#168121042
C2×#168120972
C2×#168120952
C2×#168120942
C2×#168120912
C2×#168120872
C2×#168120842
C2×#168120822
C2×#168120792
C2×C4×D4282
C2×#168120712
C2×C22⋊C4×D1072
C2×#126×C1072
C2×#538107