Finite Group Databasebeta

← #28339991

Subgroups — #28339991

Order 13696.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812083C211yesmaximal, characteristic, complemented
2C2×#16812081C211yesmaximal, characteristic, complemented
3C2×#16812079C211yesmaximal, characteristic, complemented
4C2×#16812077C211yesmaximal, characteristic, complemented
5C22×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
6#20292549C211yesmaximal, characteristic, complemented
7#20292544C211yesmaximal, characteristic, complemented
8#20292543C211yesmaximal, characteristic, complemented
9#20292536C211yesmaximal, characteristic, complemented
10#20292529C211yesmaximal, characteristic, complemented
11#20292518C211yesmaximal, characteristic, complemented
12#20292506C211yesmaximal, characteristic, complemented
13#20292506C211yesmaximal, characteristic, complemented
14#20292502C211yesmaximal, characteristic, complemented
15C2×C4×D4×C107C211yesmaximal, characteristic, complemented
16C24×D107C2211yescharacteristic
17C22×D428C2211yescharacteristic
18C22×D428C2211yescharacteristic
19C22×C22⋊D107C2211yescharacteristic
20C22×C22⋊D107C2211yescharacteristic
21C22×C22⋊D107C2211yescharacteristic
22C22×C22⋊D107C2211yescharacteristic, complemented
23C23×Dic107C2211yescharacteristic
24#16812083—24no—
25#16812081—24no—
26#16812079—24no—
27#16812077—24no—
28C2×C4⋊Dic107C2211yescharacteristic, complemented
29C2×C4⋊Dic107C2211yescharacteristic, complemented
30C2×Dic107⋊C4C2211yescharacteristic, complemented
31C2×C22⋊Dic107C2211yescharacteristic
32C2×C22⋊Dic107C2211yescharacteristic, complemented
33C2×(C2×D107)⋊C4—24no—
34C2×(C2×D107)⋊C4C2211yescharacteristic
35C2×(C2×D107)⋊C4C2211yescharacteristic
36C2×(C2×D107)⋊C4C2211yescharacteristic
37C2×(C2×D107)⋊C4C2211yescharacteristic
38C2×(C2×D107)⋊C4C2211yescharacteristic
39C2×(C2×D107)⋊C4C2211yescharacteristic
40C2×(C2×D107)⋊C4C2211yescharacteristic
41C2×(C2×D107)⋊C4C2211yescharacteristic
42C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
45C107⋊#95C2211yescharacteristic, complemented
46C107⋊#95C2211yescharacteristic, complemented
47C107⋊#95C2211yescharacteristic, complemented
48C107⋊#95C2211yescharacteristic, complemented
49C22×D4×C107C2211yescharacteristic
50C23×C4×C107C2211yescharacteristic
51C23×C4×C107C2211yescharacteristic, complemented
52C4×D4×C107—24no—
53C2×C4⋊C4×C107C2211yescharacteristic, complemented
54C2×C22⋊C4×C107C2211yescharacteristic
55C2×C22⋊C4×C107C2211yescharacteristic, complemented
56C2×C42×C107C2211yescharacteristic, complemented
57C23×D107—22no—
58C23×D107—22no—
59C23×D107—22no—
60C23×D107—22no—
61C23×D107—22no—
62C23×D107—41no—
63C23×D107—41no—
64C23×D107C2311yescharacteristic
65C23×D107C2311yescharacteristic
66C23×D107C2311yescharacteristic
67C2×D428—22no—
68C2×D428—22no—
69C2×D428—22no—
70C2×D428—22no—
71C2×D428—41no—
72C2×D428—41no—
73C2×C22⋊D107—22no—
74C2×C22⋊D107—22no—
75C2×C22⋊D107—22no—
76C2×C22⋊D107—22no—
77C2×C22⋊D107—22no—
78C2×C22⋊D107—22no—
79C2×C22⋊D107—22no—
80C2×C22⋊D107—22no—
81C2×C22⋊D107—22no—
82C2×C22⋊D107—41no—
83C2×C22⋊D107—41no—
84C2×C22⋊D107—41no—
85C22×Dic107—22no—
86C22×Dic107—22no—
87C22×Dic107—41no—
88C22×Dic107C2311yescharacteristic
89C22×Dic107C2311yescharacteristic
90C22×Dic107C2311yescharacteristic
91C22×Dic107C2311yescharacteristic
92C22×Dic107C2311yescharacteristic
93C4⋊Dic107—22no—
94C4⋊Dic107—41no—
95Dic107⋊C4—22no—
96C22⋊Dic107—22no—
97C22⋊Dic107—22no—
98(C2×D107)⋊C4—22no—
99(C2×D107)⋊C4—22no—
100(C2×D107)⋊C4—22no—
101(C2×D107)⋊C4—22no—
102(C2×D107)⋊C4—22no—
103(C2×D107)⋊C4—22no—
104(C2×D107)⋊C4—22no—
105(C2×D107)⋊C4—22no—
106(C2×D107)⋊C4—22no—
107(C2×D107)⋊C4—41no—
108(C2×D107)⋊C4D414yescomplemented
109C24×C107C2311yescharacteristic
110C24×C107C2311yescharacteristic
111C2×D4×C107—22no—
112C2×D4×C107—22no—
113C2×D4×C107D414yescomplemented
114C22×C4×C107—22no—
115C22×C4×C107—22no—
116C22×C4×C107—22no—
117C22×C4×C107—22no—
118C22×C4×C107—41no—
119C22×C4×C107—41no—
120C22×C4×C107C2311yescharacteristic
121C22×C4×C107C2311yescharacteristic
122C22×C4×C107C2311yescharacteristic
123C22×C4×C107C2311yescharacteristic
124C22×C4×C107C2311yescharacteristic
125C4⋊C4×C107—22no—
126C22⋊C4×C107—22no—
127C22⋊C4×C107D414yescomplemented
128C42×C107—22no—
129C23×C107—21no—
130C23×C107—21no—
131C23×C107—21no—
132C23×C107—21no—
133C23×C107—21no—
134C23×C107—21no—
135C23×C107—21no—
136C23×C107—21no—
137C23×C107—21no—
138C23×C107—21no—
139C23×C107C2×D412yes—
140C23×C107C2×D412yes—
141C23×C107C2×D412yes—
142C23×C107#4112yes—
143C23×C107C2411yescharacteristic
144C22×D107—28no—
145C22×D107—21no—
146C22×D107—21no—
147C22×D107—21no—
148C22×D107—21no—
149C22×D107—21no—
150C22×D107—21no—
151C22×D107—21no—
152C22×D107—21no—
153C22×D107—21no—
154C22×D107—21no—
155C22×D107—21no—
156C22×D107—21no—
157C22×D107—21no—
158C22×D107—21no—
159C22×D107—21no—
160C22×D107—21no—
161C22×D107—21no—
162C22×D107—21no—
163C22×D107—21no—
164C22×D107—44no—
165C22×D107—44no—
166C22×D107—44no—
167C22×D107—44no—
168C22×D107—44no—
169C22×D107—44no—
170C22×D107C2×D412yes—
171C22×D107#4112yes—
172D4×C107—28no—
173C2×C4×C107—21no—
174C2×C4×C107—21no—
175C2×C4×C107—21no—
176C2×C4×C107—21no—
177C2×C4×C107—21no—
178C2×C4×C107—21no—
179C2×C4×C107—21no—
180C2×C4×C107—21no—
181C2×C4×C107—21no—
182C2×C4×C107—21no—
183C2×C4×C107—21no—
184C2×C4×C107—44no—
185C2×C4×C107—44no—
186C2×C4×C107C2×D412yes—
187C2×C4×C107C2×D412yes—
188C2×C4×C107C2×D412yes—
189C2×C4×C107C2×D412yes—
190C22⋊D107—44no—
191C22⋊D107—44no—
192C22⋊D107—44no—
193C22⋊D107—44no—
194C2×Dic107—21no—
195C2×Dic107—21no—
196C2×Dic107—21no—
197C2×Dic107—21no—
198C2×Dic107—21no—
199C2×Dic107—21no—
200C2×Dic107—21no—
201C2×Dic107—21no—
202C2×Dic107—21no—
203C2×Dic107—21no—
204C2×Dic107—21no—
205C2×Dic107—21no—
206C2×Dic107—21no—
207C2×Dic107—21no—
208C2×Dic107—44no—
209C2×Dic107C2×D412yes—
210D428—44no—
211D428—44no—
212C22×C107—22no—
213C22×C107—22no—
214C22×C107—22no—
215C22×C107—22no—
216C22×C107—22no—
217C22×C107—22no—
218C22×C107—22no—
219C22×C107—22no—
220C22×C107—22no—
221C22×C107—41no—
222C22×C107—41no—
223C22×C107—41no—
224C22×C107—41no—
225C22×C107#12114yescomplemented
226C22×C107C22×D411yescharacteristic
227C22×C107C22×D411yescharacteristic
228C22×C107C22×D411yescharacteristic
229C22×C107C2×#4111yescharacteristic
230C22×C107C2×#4111yescharacteristic
231C22×C107H(5,𝔽2)11yescharacteristic
232C22×C107H(5,𝔽2)11yescharacteristic
233C2×D107—22no—
234C2×D107—22no—
235C2×D107—22no—
236C2×D107—22no—
237C2×D107—22no—
238C2×D107—22no—
239C2×D107—416no—
240C2×D107—41no—
241C2×D107—41no—
242C2×D107—41no—
243C2×D107—41no—
244C2×D107—41no—
245C2×D107—41no—
246C2×D107—41no—
247C2×D107—41no—
248C2×D107—41no—
249C2×D107—41no—
250C2×D107—41no—
251C2×D107—41no—
252C2×D107—41no—
253C2×D107—41no—
254C2×D107—41no—
255C4×C107—22no—
256C4×C107—22no—
257C4×C107—22no—
258C4×C107—22no—
259C4×C107—41no—
260Dic107—22no—
261Dic107—41no—
262Dic107—41no—
263Dic107—41no—
264Dic107—41no—
265C2×C107—24no—
266C2×C107—42no—
267C2×C107C2×#12111yescharacteristic
268C2×C107#52511yescharacteristic
269C2×C107#53811yescharacteristic
270C2×C107D4211yescharacteristic
271C2×C107D4211yescharacteristic
272C2×C107#54611yescharacteristic
273C2×C107#54611yescharacteristic
274D107—42no—
275D107—42no—
276D107—81no—
277#2424—1071nomaximal, Sylow
278C107#242411yescharacteristic, complemented, Sylow
279C2×#121—1071no—
280C2×#121—1071no—
281C2×#120—1071no—
282C2×#120—1071no—
283C2×C4×D4D10711yescharacteristic, complemented
284C22×C22⋊C4—1071no—
285#396—1071no—
286#394—1071no—
287#394—1071no—
288#392—1071no—
289#390—1071no—
290#388—1071no—
291#386—1071no—
292#386—1071no—
293#385—1071no—
294C25—1071no—
295C22×D4—1071no—
296C22×D4—1071no—
297C22×D4—1071no—
298C22×D4—1071no—
299C22×D4—1071no—
300C22×D4—1071no—
301C22×D4C2×D10711yescharacteristic
302C23×C4—1071no—
303C23×C4C2×D10711yescharacteristic
304C23×C4C2×D10711yescharacteristic, complemented
305#121—2144no—
306#121—2144no—
307#120—2144no—
308#120—2144no—
309C4×D4—24no—
310C2×C4⋊C4—1071no—
311C2×C4⋊C4—1071no—
312C2×C4⋊C4—1071no—
313C2×C4⋊C4C2×D10711yescharacteristic, complemented
314C2×C22⋊C4—1071no—
315C2×C22⋊C4—1071no—
316C2×C22⋊C4—1071no—
317C2×C22⋊C4—1071no—
318C2×C22⋊C4—1071no—
319C2×C22⋊C4—1071no—
320C2×C22⋊C4—1071no—
321C2×C22⋊C4—1071no—
322C2×C22⋊C4—1071no—
323C2×C22⋊C4—1071no—
324C2×C22⋊C4—1071no—
325C2×C22⋊C4—1071no—
326C2×C22⋊C4—1071no—
327C2×C22⋊C4—2144no—
328C2×C22⋊C4C2×D10711yescharacteristic
329C2×C22⋊C4C2×D10711yescharacteristic, complemented
330C2×C42C2×D10711yescharacteristic, complemented
331#95—1071no—
332#95—1071no—
333#95—1071no—
334#95—1071no—
335C24—1071no—
336C24—1071no—
337C24—1071no—
338C24—4281no—
339C24—4281no—
340C24—2142no—
341C24—2142no—
342C24—2142no—
343C24—2142no—
344C24—2142no—
345C24C22×D10711yescharacteristic
346C24C22×D10711yescharacteristic
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—4281no—
350C2×D4—4281no—
351C2×D4—4281no—
352C2×D4—4281no—
353C2×D4—4281no—
354C2×D4—2142no—
355C2×D4—2142no—
356C2×D4—2142no—
357C2×D4—2142no—
358C2×D4—2142no—
359C2×D4—2142no—
360C2×D4—2142no—
361C2×D4—2142no—
362C2×D4—2142no—
363C2×D4—2142no—
364C2×D4—2142no—
365C2×D4—2142no—
366C2×D4—2142no—
367C2×D4D42814yescomplemented
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—1071no—
375C22×C4—1071no—
376C22×C4—1071no—
377C22×C4—1071no—
378C22×C4—1071no—
379C22×C4—4281no—
380C22×C4—2142no—
381C22×C4—2142no—
382C22×C4C22×D10711yescharacteristic
383C22×C4C22×D10711yescharacteristic
384C22×C4C22×D10711yescharacteristic
385C22×C4C22×D10711yescharacteristic
386C22×C4C22×D10711yescharacteristic
387C4⋊C4—22no—
388C4⋊C4—4281no—
389C4⋊C4—2142no—
390C4⋊C4—2142no—
391C22⋊C4—22no—
392C22⋊C4—1074no—
393C22⋊C4—4281no—
394C22⋊C4—2142no—
395C22⋊C4—2142no—
396C22⋊C4—2142no—
397C22⋊C4—2142no—
398C22⋊C4—2142no—
399C22⋊C4—2142no—
400C22⋊C4—2142no—
401C22⋊C4—2142no—
402C22⋊C4—2142no—
403C22⋊C4—2142no—
404C22⋊C4—2142no—
405C22⋊C4C22⋊D10714yescomplemented
406C42—22no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—1072no—
418C23—1072no—
419C23—4284no—
420C23—4284no—
421C23—4284no—
422C23—4284no—
423C23—4284no—
424C23—4284no—
425C23—2148no—
426C23—2141no—
427C23—2141no—
428C23—2141no—
429C23—2141no—
430C23—2141no—
431C23—2141no—
432C23—2141no—
433C23—2141no—
434C23—2141no—
435C23—2141no—
436C23—2141no—
437C23—2141no—
438C23—2141no—
439C23—2141no—
440C23—2141no—
441C23—2141no—
442C23—2141no—
443C23—2141no—
444C23—2141no—
445C23C2×C22⋊D10712yes—
446C23C2×D42812yes—
447C23C2×D42812yes—
448C23#1313381812yes—
449C23C23×D10711yescharacteristic
450D4—28no—
451D4—4284no—
452D4—4284no—
453D4—4284no—
454D4—4284no—
455D4—4284no—
456D4—4284no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—1072no—
471C2×C4—4284no—
472C2×C4—2141no—
473C2×C4—2141no—
474C2×C4—2141no—
475C2×C4—2141no—
476C2×C4—2141no—
477C2×C4—2141no—
478C2×C4—2141no—
479C2×C4—2141no—
480C2×C4—2141no—
481C2×C4—2141no—
482C2×C4—2141no—
483C2×C4—2141no—
484C2×C4—2141no—
485C2×C4—2141no—
486C2×C4D4×D10712yes—
487C2×C4C2×C22⋊D10712yes—
488C2×C4C2×C22⋊D10712yes—
489C2×C4C2×D42812yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—42816no—
504C22—4281no—
505C22—4281no—
506C22—4281no—
507C22—4281no—
508C22—4281no—
509C22—4281no—
510C22—4281no—
511C22—4281no—
512C22—4281no—
513C22—4281no—
514C22—4281no—
515C22—4281no—
516C22—4281no—
517C22—4281no—
518C22—4281no—
519C22—2142no—
520C22—2142no—
521C22—2142no—
522C22—2142no—
523C22—2142no—
524C22—2142no—
525C22#1681208314yescomplemented
526C22C2×D4×D10711yescharacteristic
527C22C22×C22⋊D10711yescharacteristic
528C22C22×D42811yescharacteristic
529C22#41×D10711yescharacteristic
530C22C2×#1313381811yescharacteristic
531C22#1681215411yescharacteristic
532C22#1681215511yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—41no—
538C4—4281no—
539C4—4281no—
540C4—4281no—
541C4—4281no—
542C4—2142no—
543C2—24no—
544C2—42no—
545C2—8561no—
546C2—4282no—
547C2—4282no—
548C2C2×#1681208311yescharacteristic
549C2#2029313611yescharacteristic
550C2#2029321011yescharacteristic
551C2D4×C22⋊D10711yescharacteristic
552C2D4×D42811yescharacteristic
553C2#2029326511yescharacteristic
554C2#2029327211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C4×D4×C107C2
hypercenterC2×C4×D4D107
hypocenterC107#2424
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16812083C2semidirect
2. C2×#16812081C2semidirect
3. C2×#16812079C2semidirect
4. C2×#16812077C2semidirect
5. C22×(C2×D107)⋊C4C2semidirect
6. #20292549C2semidirect
7. #20292544C2semidirect
8. #20292543C2semidirect
9. #20292536C2semidirect
10. #20292529C2semidirect
11. #20292518C2semidirect
12. #20292506C2semidirect
13. #20292506C2semidirect
14. #20292502C2semidirect
15. C2×C4×D4×C107C2semidirect
22. C22×C22⋊D107C22semidirect
28. C2×C4⋊Dic107C22semidirect
29. C2×C4⋊Dic107C22semidirect
30. C2×Dic107⋊C4C22semidirect
32. C2×C22⋊Dic107C22semidirect
42. C2×(C2×D107)⋊C4C22semidirect
43. C2×(C2×D107)⋊C4C22semidirect
44. C2×(C2×D107)⋊C4C22semidirect
45. C107⋊#95C22semidirect
46. C107⋊#95C22semidirect
47. C107⋊#95C22semidirect
48. C107⋊#95C22semidirect
51. C23×C4×C107C22semidirect
53. C2×C4⋊C4×C107C22semidirect
55. C2×C22⋊C4×C107C22semidirect
56. C2×C42×C107C22semidirect
108. (C2×D107)⋊C4D4semidirect
113. C2×D4×C107D4semidirect
127. C22⋊C4×C107D4semidirect
225. C22×C107#121semidirect
278. C107#2424semidirect
283. C2×C4×D4D107semidirect
304. C23×C4C2×D107semidirect
313. C2×C4⋊C4C2×D107semidirect
329. C2×C22⋊C4C2×D107semidirect
330. C2×C42C2×D107semidirect
367. C2×D4D428semidirect
405. C22⋊C4C22⋊D107semidirect
525. C22#16812083semidirect

Maximal subgroups

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

subgroupindex
C2×#168120832
C2×#168120812
C2×#168120792
C2×#168120772
C22×(C2×D107)⋊C42
#202925492
#202925442
#202925432
#202925362
#202925292
#202925182
#202925062
#202925022
C2×C4×D4×C1072
#2424107