Finite Group Databasebeta

← #28339778

Subgroups — #28339778

Order 13696.

1056 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133818C211yesmaximal, characteristic, complemented
2C2×#16812079C211yesmaximal, characteristic, complemented
3C2×#16812077C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D107C211yesmaximal, characteristic, complemented
5#20292540C211yesmaximal, characteristic, complemented
6#20292539C211yesmaximal, characteristic, complemented
7#20292536C211yesmaximal, characteristic, complemented
8#20292535C211yesmaximal, characteristic, complemented
9#20292506C211yesmaximal, characteristic, complemented
10#20292505C211yesmaximal, characteristic, complemented
11#20292505C211yesmaximal, characteristic, complemented
12#20292503C211yesmaximal, characteristic, complemented
13#20292503C211yesmaximal, characteristic, complemented
14#20292489C211yesmaximal, characteristic, complemented
15#386×C107C211yesmaximal, characteristic, complemented
16C24×D107C2211yescharacteristic
17C2×#13133818—24no—
18C2×#13133818—24no—
19C2×#13133818—24no—
20C22×Dic214C2211yescharacteristic, complemented
21C22×D428C2211yescharacteristic
22C22×D428C2211yescharacteristic
23C22×C22⋊D107C2211yescharacteristic
24C22×C22⋊D107C2211yescharacteristic
25C22×C22⋊D107C2211yescharacteristic
26C22×C22⋊D107C2211yescharacteristic, complemented
27C22×C4×D107C2211yescharacteristic
28C22×C4×D107C2211yescharacteristic
29C22×C4×D107C2211yescharacteristic, complemented
30#16812079—24no—
31#16812077—24no—
32C2×C22⋊Dic107C2211yescharacteristic
33C2×C22⋊Dic107C2211yescharacteristic
34C2×C22⋊Dic107C2211yescharacteristic, complemented
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, complemented
42C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
43C22⋊C4×D107—24no—
44C107⋊#95C2211yescharacteristic, complemented
45C107⋊#95C2211yescharacteristic, complemented
46C107⋊#95C2211yescharacteristic, complemented
47C107⋊#95C2211yescharacteristic, complemented
48C107⋊#95C2211yescharacteristic, complemented
49C107⋊#95C2211yescharacteristic, complemented
50C22×D4×C107C2211yescharacteristic
51C23×C4×C107C2211yescharacteristic, complemented
52C2×C22⋊C4×C107C2211yescharacteristic
53C2×C22⋊C4×C107C2211yescharacteristic
54C2×C22⋊C4×C107C2211yescharacteristic, complemented
55#95×C107C2211yescharacteristic, complemented
56#95×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
67#13133818—232no—
68C2×Dic214—22no—
69C2×Dic214—22no—
70C2×Dic214—22no—
71C2×D428—22no—
72C2×D428—22no—
73C2×D428—22no—
74C2×D428—22no—
75C2×D428—41no—
76C2×D428D414yescomplemented
77C2×C22⋊D107—22no—
78C2×C22⋊D107—22no—
79C2×C22⋊D107—22no—
80C2×C22⋊D107—22no—
81C2×C22⋊D107—22no—
82C2×C22⋊D107—22no—
83C2×C22⋊D107—22no—
84C2×C22⋊D107—22no—
85C2×C22⋊D107—22no—
86C2×C22⋊D107—41no—
87C2×C22⋊D107—41no—
88C2×C22⋊D107D414yescomplemented
89C22×Dic107C2311yescharacteristic
90C22×Dic107C2311yescharacteristic
91C22×Dic107C2311yescharacteristic
92C22×Dic107C2311yescharacteristic
93C22×Dic107C2311yescharacteristic
94C2×C4×D107—22no—
95C2×C4×D107—22no—
96C2×C4×D107—22no—
97C2×C4×D107—22no—
98C2×C4×D107—22no—
99C2×C4×D107—22no—
100C2×C4×D107—41no—
101C2×C4×D107—41no—
102C2×C4×D107D414yescomplemented
103C22⋊Dic107—22no—
104C22⋊Dic107—22no—
105C22⋊Dic107—41no—
106(C2×D107)⋊C4—22no—
107(C2×D107)⋊C4—22no—
108(C2×D107)⋊C4—22no—
109(C2×D107)⋊C4—22no—
110(C2×D107)⋊C4—22no—
111(C2×D107)⋊C4—22no—
112(C2×D107)⋊C4—41no—
113(C2×D107)⋊C4—41no—
114C24×C107C2311yescharacteristic
115C24×C107C2311yescharacteristic
116C2×D4×C107—22no—
117C2×D4×C107—22no—
118C2×D4×C107—41no—
119C22×C4×C107—22no—
120C22×C4×C107—22no—
121C22×C4×C107—22no—
122C22×C4×C107C2311yescharacteristic
123C22×C4×C107C2311yescharacteristic
124C22×C4×C107C2311yescharacteristic
125C22×C4×C107C2311yescharacteristic
126C22×C4×C107C2311yescharacteristic
127C22⋊C4×C107—22no—
128C22⋊C4×C107—22no—
129C22⋊C4×C107—41no—
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×C107—21no—
140C23×C107—21no—
141C23×C107—21no—
142C23×C107C2×D412yes—
143C23×C107#4112yescomplemented
144C23×C107C2411yescharacteristic
145C22×D107—21no—
146C22×D107—21no—
147C22×D107—21no—
148C22×D107—21no—
149C22×D107—28no—
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—44no—
163C22×D107—44no—
164C22×D107—44no—
165C22×D107—44no—
166C22×D107—44no—
167C22×D107—44no—
168C22×D107C2×D412yes—
169C22×D107C2×D412yes—
170C22×D107C2×D412yes—
171C22×D107C2×D412yes—
172C22×D107#4112yescomplemented
173D4×C107—44no—
174C2×C4×C107—28no—
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—21no—
185C2×C4×C107—21no—
186C2×C4×C107—21no—
187C2×C4×C107—21no—
188C2×C4×C107C2×D412yes—
189C2×C4×C107C2×D412yes—
190C22⋊D107—28no—
191C22⋊D107—28no—
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×Dic107C2×D412yes—
208C2×Dic107C2×D412yes—
209D428—28no—
210D428—44no—
211C4×D107—28no—
212C4×D107—28no—
213C4×D107—44no—
214Dic214—28no—
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—41no—
226C22×C107—41no—
227C22×C107—41no—
228C22×C107—41no—
229C22×C107C22×D411yescharacteristic
230C22×C107C22×D411yescharacteristic
231C22×C107C22×D411yescharacteristic
232C22×C107C2×#4111yescharacteristic
233C22×C107C2×#4111yescharacteristic
234C22×C107H(5,𝔽2)11yescharacteristic
235C22×C107H(5,𝔽2)11yescharacteristic
236C2×D107—22no—
237C2×D107—22no—
238C2×D107—22no—
239C2×D107—22no—
240C2×D107—22no—
241C2×D107—22no—
242C2×D107—22no—
243C2×D107—22no—
244C2×D107—22no—
245C2×D107—22no—
246C2×D107—22no—
247C2×D107—22no—
248C2×D107—416no—
249C2×D107—41no—
250C2×D107—41no—
251C2×D107—41no—
252C2×D107—41no—
253C2×D107—41no—
254C2×D107—41no—
255C2×D107—41no—
256C2×D107—41no—
257C2×D107#12014yes—
258C4×C107—22no—
259C4×C107—22no—
260C4×C107—41no—
261C4×C107—41no—
262C4×C107—41no—
263Dic107—22no—
264Dic107—22no—
265Dic107—41no—
266Dic107—41no—
267Dic107—41no—
268C2×C107—42no—
269C2×C107—42no—
270C2×C107C2×#12011yescharacteristic
271C2×C107#52511yescharacteristic
272C2×C107#53511yescharacteristic
273C2×C107#54611yescharacteristic
274C2×C107#54611yescharacteristic
275C2×C107#54611yescharacteristic
276C2×C107#54611yescharacteristic
277D107—24no—
278D107—42no—
279D107—42no—
280#2404—1071nomaximal, Sylow
281C107#240411yescharacteristic, complemented, Sylow
282C22×#41—1071no—
283C2×#120—1071no—
284C2×#120—1071no—
285C22×C22⋊C4—1071no—
286#393—1071no—
287#393—1071no—
288#392—1071no—
289#392—1071no—
290#386—1071no—
291#386—1071no—
292#386—1071no—
293#386—1071no—
294#386—1071no—
295#386D10711yescharacteristic, complemented
296#381—1071no—
297C25—1071no—
298C2×#41—2144no—
299C2×#41—2144no—
300C2×#41—2144no—
301C22×Q8—1071no—
302C22×D4—1071no—
303C22×D4—1071no—
304C22×D4—1071no—
305C22×D4—1071no—
306C22×D4—1071no—
307C22×D4—1071no—
308C22×D4C2×D10711yescharacteristic
309C23×C4—1071no—
310C23×C4—1071no—
311C23×C4—1071no—
312C23×C4C2×D10711yescharacteristic, complemented
313#120—2144no—
314#120—2144no—
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—2144no—
327C2×C22⋊C4C2×D10711yescharacteristic
328C2×C22⋊C4C2×D10711yescharacteristic
329C2×C22⋊C4C2×D10711yescharacteristic, complemented
330#95—1071no—
331#95—1071no—
332#95—1071no—
333#95—1071no—
334#95—1071no—
335#95—1071no—
336#95C2×D10711yescharacteristic, complemented
337#95C2×D10711yescharacteristic, complemented
338C24—1071no—
339C24—1071no—
340C24—1071no—
341C24—4281no—
342C24—4281no—
343C24—2142no—
344C24—2142no—
345C24—2142no—
346C24—2142no—
347C24—2142no—
348C24C22×D10711yescharacteristic
349C24C22×D10711yescharacteristic
350#41—21432no—
351C2×Q8—2142no—
352C2×Q8—2142no—
353C2×Q8—2142no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—41no—
357C2×D4—1074no—
358C2×D4—1074no—
359C2×D4—4281no—
360C2×D4—4281no—
361C2×D4—4281no—
362C2×D4—2142no—
363C2×D4—2142no—
364C2×D4—2142no—
365C2×D4—2142no—
366C2×D4—2142no—
367C2×D4—2142no—
368C2×D4—2142no—
369C2×D4—2142no—
370C2×D4—2142no—
371C2×D4—2142no—
372C2×D4—2142no—
373C2×D4—2142no—
374C2×D4—2142no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—1071no—
379C22×C4—1071no—
380C22×C4—1074no—
381C22×C4—1071no—
382C22×C4—1071no—
383C22×C4—1071no—
384C22×C4—4281no—
385C22×C4—4281no—
386C22×C4—2142no—
387C22×C4—2142no—
388C22×C4—2142no—
389C22×C4—2142no—
390C22×C4—2142no—
391C22×C4—2142no—
392C22×C4C22×D10711yescharacteristic
393C22×C4C22×D10711yescharacteristic
394C22×C4C22×D10711yescharacteristic
395C22×C4C22×D10711yescharacteristic
396C22×C4C22×D10711yescharacteristic
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—41no—
400C22⋊C4—4281no—
401C22⋊C4—4281no—
402C22⋊C4—4281no—
403C22⋊C4—2142no—
404C22⋊C4—2142no—
405C22⋊C4—2142no—
406C22⋊C4—2142no—
407C22⋊C4—2142no—
408C22⋊C4—2142no—
409C22⋊C4—2142no—
410C22⋊C4—2142no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—1072no—
424C23—1072no—
425C23—1072no—
426C23—1072no—
427C23—1072no—
428C23—4284no—
429C23—4284no—
430C23—4284no—
431C23—4284no—
432C23—4284no—
433C23—4284no—
434C23—2148no—
435C23—2141no—
436C23—2141no—
437C23—2141no—
438C23—2141no—
439C23—2141no—
440C23—2141no—
441C23—2141no—
442C23—2141no—
443C23—2141no—
444C23—2141no—
445C23—2141no—
446C23—2141no—
447C23—2141no—
448C23—2141no—
449C23—2141no—
450C23—2141no—
451C23D4×D10712yes—
452C23#1313381812yescomplemented
453C23C23×D10711yescharacteristic
454Q8—2148no—
455D4—44no—
456D4—4284no—
457D4—4284no—
458D4—4284no—
459D4—2148no—
460D4—2148no—
461D4—2148no—
462C2×C4—28no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—1072no—
477C2×C4—1072no—
478C2×C4—4284no—
479C2×C4—2148no—
480C2×C4—2148no—
481C2×C4—2141no—
482C2×C4—2141no—
483C2×C4—2141no—
484C2×C4—2141no—
485C2×C4—2141no—
486C2×C4—2141no—
487C2×C4—2141no—
488C2×C4—2141no—
489C2×C4—2141no—
490C2×C4—2141no—
491C2×C4—2141no—
492C2×C4—2141no—
493C2×C4—2141no—
494C2×C4D4×D10712yes—
495C2×C4D4×D10712yes—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—1074no—
511C22—42816no—
512C22—4281no—
513C22—4281no—
514C22—4281no—
515C22—4281no—
516C22—4281no—
517C22—4281no—
518C22—4281no—
519C22—4281no—
520C22—2142no—
521C22—2142no—
522C22—2142no—
523C22—2142no—
524C22—2142no—
525C22—2142no—
526C22—2142no—
527C22—2142no—
528C22—2142no—
529C22—2142no—
530C22—2142no—
531C22—2142no—
532C22C2×D4×D10711yescharacteristic
533C22C2×D4×D10711yescharacteristic
534C22C2×D4×D10711yescharacteristic
535C22#41×D10711yescharacteristic
536C22C2×#1313381811yescharacteristic
537C22#1681215411yescharacteristic
538C22#1681215511yescharacteristic
539C4—22no—
540C4—22no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—4281no—
545C4—4281no—
546C4—4281no—
547C4—2142no—
548C4—2142no—
549C2—42no—
550C2—42no—
551C2—4282no—
552C2—4282no—
553C2—2144no—
554C2#120×D10711yescharacteristic
555C2#2029313611yescharacteristic
556C2#2029319311yescharacteristic
557C2#2029326711yescharacteristic
558C2#2029326911yescharacteristic
559C2#2029327311yescharacteristic
560C2#2029327411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#386×C107C2
hypercenter#386D107
hypocenterC107#2404
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×#13133818C2semidirect
2. C2×#16812079C2semidirect
3. C2×#16812077C2semidirect
4. C2×C22⋊C4×D107C2semidirect
5. #20292540C2semidirect
6. #20292539C2semidirect
7. #20292536C2semidirect
8. #20292535C2semidirect
9. #20292506C2semidirect
10. #20292505C2semidirect
11. #20292505C2semidirect
12. #20292503C2semidirect
13. #20292503C2semidirect
14. #20292489C2semidirect
15. #386×C107C2semidirect
20. C22×Dic214C22semidirect
26. C22×C22⋊D107C22semidirect
29. C22×C4×D107C22semidirect
34. C2×C22⋊Dic107C22semidirect
41. C2×(C2×D107)⋊C4C22semidirect
42. C2×(C2×D107)⋊C4C22semidirect
44. C107⋊#95C22semidirect
45. C107⋊#95C22semidirect
46. C107⋊#95C22semidirect
47. C107⋊#95C22semidirect
48. C107⋊#95C22semidirect
49. C107⋊#95C22semidirect
51. C23×C4×C107C22semidirect
54. C2×C22⋊C4×C107C22semidirect
55. #95×C107C22semidirect
56. #95×C107C22semidirect
76. C2×D428D4semidirect
88. C2×C22⋊D107D4semidirect
102. C2×C4×D107D4semidirect
143. C23×C107#41semidirect
172. C22×D107#41semidirect
281. C107#2404semidirect
295. #386D107semidirect
312. C23×C4C2×D107semidirect
329. C2×C22⋊C4C2×D107semidirect
336. #95C2×D107semidirect
337. #95C2×D107semidirect
452. C23#13133818semidirect

Maximal subgroups

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

subgroupindex
C22×#131338182
C2×#168120792
C2×#168120772
C2×C22⋊C4×D1072
#202925402
#202925392
#202925362
#202925352
#202925062
#202925052
#202925032
#202924892
#386×C1072
#2404107