Finite Group Databasebeta

← #28339995

Subgroups — #28339995

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×#16812084C211yesmaximal, characteristic, complemented
2C2×#16812083C211yesmaximal, characteristic, complemented
3C2×#16812079C211yesmaximal, characteristic, complemented
4C2×#16812079C211yesmaximal, characteristic, complemented
5C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic, complemented
6C22×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
7#20292549C211yesmaximal, characteristic, complemented
8#20292545C211yesmaximal, characteristic, complemented
9#20292544C211yesmaximal, characteristic, complemented
10#20292537C211yesmaximal, characteristic, complemented
11#20292528C211yesmaximal, characteristic, complemented
12#20292519C211yesmaximal, characteristic, complemented
13#20292505C211yesmaximal, characteristic, complemented
14#20292504C211yesmaximal, characteristic, complemented
15#385×C107C211yesmaximal, characteristic, complemented
16C24×D107C2211yescharacteristic
17C22×D428C2211yescharacteristic
18C22×D428C2211yescharacteristic
19C22×D428C2211yescharacteristic
20C22×D428C2211yescharacteristic
21C22×D428C2211yescharacteristic, complemented
22C22×C22⋊D107C2211yescharacteristic
23C22×C22⋊D107C2211yescharacteristic
24C23×Dic107C2211yescharacteristic
25C22×C4×D107C2211yescharacteristic, complemented
26#16812084—24no—
27#16812083—24no—
28#16812079—24no—
29#16812079—24no—
30(C22⋊D107)⋊C4—24no—
31C2×C4⋊Dic107C2211yescharacteristic, complemented
32C2×Dic107⋊C4C2211yescharacteristic, 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
45C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
47C2×C4×Dic107C2211yescharacteristic, complemented
48C107⋊#95C2211yescharacteristic, complemented
49C107⋊#95C2211yescharacteristic, complemented
50C23×C4×C107C2211yescharacteristic
51C2×C4⋊C4×C107C2211yescharacteristic, complemented
52C2×C4⋊C4×C107C2211yescharacteristic, complemented
53C2×C22⋊C4×C107C2211yescharacteristic
54C2×C22⋊C4×C107C2211yescharacteristic
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
67C23×D107C2311yescharacteristic
68C2×D428—22no—
69C2×D428—22no—
70C2×D428—22no—
71C2×D428—22no—
72C2×D428—22no—
73C2×D428—22no—
74C2×D428—22no—
75C2×D428—22no—
76C2×D428—22no—
77C2×D428—22no—
78C2×D428—41no—
79C2×D428—41no—
80C2×D428—41no—
81C2×D428—41no—
82C2×D428—41no—
83C2×C22⋊D107—22no—
84C2×C22⋊D107—22no—
85C2×C22⋊D107—22no—
86C2×C22⋊D107—22no—
87C2×C22⋊D107—22no—
88C2×C22⋊D107D414yescomplemented
89C22×Dic107—22no—
90C22×Dic107—22no—
91C22×Dic107—41no—
92C22×Dic107C2311yescharacteristic
93C22×Dic107C2311yescharacteristic
94C22×Dic107C2311yescharacteristic
95C22×Dic107C2311yescharacteristic
96C2×C4×D107—22no—
97C2×C4×D107—22no—
98C2×C4×D107—41no—
99C4⋊Dic107—22no—
100Dic107⋊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—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)⋊C4D414yescomplemented
114C4×Dic107—22no—
115C24×C107C2311yescharacteristic
116C22×C4×C107—22no—
117C22×C4×C107—22no—
118C22×C4×C107—41no—
119C22×C4×C107C2311yescharacteristic
120C22×C4×C107C2311yescharacteristic
121C22×C4×C107C2311yescharacteristic
122C22×C4×C107C2311yescharacteristic
123C22×C4×C107C2311yescharacteristic
124C22×C4×C107C2311yescharacteristic
125C4⋊C4×C107—22no—
126C4⋊C4×C107—41no—
127C22⋊C4×C107—22no—
128C22⋊C4×C107D414yescomplemented
129C23×C107—21no—
130C23×C107—21no—
131C23×C107—21no—
132C23×C107—21no—
133C23×C107C2×D412yes—
134C23×C107C2×D412yes—
135C23×C107#4112yes—
136C23×C107C2411yescharacteristic
137C22×D107—28no—
138C22×D107—21no—
139C22×D107—21no—
140C22×D107—21no—
141C22×D107—21no—
142C22×D107—21no—
143C22×D107—21no—
144C22×D107—21no—
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—44no—
164C22×D107—44no—
165C22×D107—44no—
166C22×D107—44no—
167C22×D107—44no—
168C22×D107—44no—
169C22×D107C2×D412yes—
170C22×D107C2×D412yes—
171C22×D107#4112yes—
172C2×C4×C107—21no—
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—21no—
185C2×C4×C107—21no—
186C2×C4×C107—21no—
187C2×C4×C107—44no—
188C2×C4×C107C2×D412yes—
189C2×C4×C107C2×D412yes—
190C2×C4×C107C2×D412yes—
191C22⋊D107—28no—
192C22⋊D107—44no—
193C2×Dic107—21no—
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—44no—
204C2×Dic107C2×D412yes—
205C2×Dic107C2×D412yes—
206D428—44no—
207D428—44no—
208D428—44no—
209D428—44no—
210D428—44no—
211C4×D107—44no—
212C22×C107—22no—
213C22×C107—22no—
214C22×C107—22no—
215C22×C107—22no—
216C22×C107—22no—
217C22×C107—22no—
218C22×C107#12114yescomplemented
219C22×C107C22×D411yescharacteristic
220C22×C107C22×D411yescharacteristic
221C22×C107C22×D411yescharacteristic
222C22×C107C2×#4111yescharacteristic
223C22×C107C2×#4111yescharacteristic
224C22×C107H(5,𝔽2)11yescharacteristic
225C22×C107H(5,𝔽2)11yescharacteristic
226C2×D107—22no—
227C2×D107—22no—
228C2×D107—22no—
229C2×D107—22no—
230C2×D107—22no—
231C2×D107—22no—
232C2×D107—22no—
233C2×D107—22no—
234C2×D107—22no—
235C2×D107—416no—
236C2×D107—41no—
237C2×D107—41no—
238C2×D107—41no—
239C2×D107—41no—
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—41no—
259C4×C107—41no—
260C4×C107—41no—
261Dic107—22no—
262Dic107—22no—
263Dic107—41no—
264Dic107—41no—
265C2×C107—24no—
266C2×C107C2×#12111yescharacteristic
267C2×C107#52511yescharacteristic
268C2×C107#53811yescharacteristic
269C2×C107D4211yescharacteristic
270C2×C107D4211yescharacteristic
271C2×C107#54611yescharacteristic
272C2×C107#54611yescharacteristic
273D107—42no—
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×D4—1071no—
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#385D10711yescharacteristic, complemented
294C25—1071no—
295C22×D4—1071no—
296C22×D4—1071no—
297C22×D4—1071no—
298C22×D4—1071no—
299C22×D4—1071no—
300C22×D4—1071no—
301C22×D4—1071no—
302C23×C4—1071no—
303C23×C4—1071no—
304C23×C4C2×D10711yescharacteristic
305#121—2144no—
306#121—2144no—
307#120—2144no—
308#120—2144no—
309C4×D4—2144no—
310C2×C4⋊C4—1071no—
311C2×C4⋊C4—1071no—
312C2×C4⋊C4C2×D10711yescharacteristic, complemented
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
330C2×C42—1071no—
331#95—1071no—
332#95—1071no—
333#95C2×D10711yescharacteristic, complemented
334#95C2×D10711yescharacteristic, complemented
335C24—1071no—
336C24—1071no—
337C24—1071no—
338C24—1071no—
339C24—4281no—
340C24—4281no—
341C24—2142no—
342C24—2142no—
343C24—2142no—
344C24—2142no—
345C24—2142no—
346C24C22×D10711yescharacteristic
347C2×D4—1074no—
348C2×D4—4281no—
349C2×D4—4281no—
350C2×D4—4281no—
351C2×D4—4281no—
352C2×D4—4281no—
353C2×D4—2142no—
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×D4—2142no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—41no—
371C22×C4—1071no—
372C22×C4—1071no—
373C22×C4—1071no—
374C22×C4—1071no—
375C22×C4—4281no—
376C22×C4—4281no—
377C22×C4—2142no—
378C22×C4—2142no—
379C22×C4—2142no—
380C22×C4—2142no—
381C22×C4C22×D10711yescharacteristic
382C22×C4C22×D10711yescharacteristic
383C22×C4C22×D10711yescharacteristic
384C22×C4C22×D10711yescharacteristic
385C22×C4C22×D10711yescharacteristic
386C22×C4C22×D10711yescharacteristic
387C4⋊C4—22no—
388C4⋊C4—41no—
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⋊C4D42814yescomplemented
406C42—2142no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—1072no—
412C23—1072no—
413C23—1072no—
414C23—4284no—
415C23—4284no—
416C23—4284no—
417C23—4284no—
418C23—4284no—
419C23—4284no—
420C23—2148no—
421C23—2141no—
422C23—2141no—
423C23—2141no—
424C23—2141no—
425C23—2141no—
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—
445C23—2141no—
446C23D4×D10712yes—
447C23C2×D42812yes—
448C23#1313381912yes—
449C23C23×D10711yescharacteristic
450D4—4284no—
451D4—4284no—
452D4—4284no—
453D4—4284no—
454D4—4284no—
455D4—4284no—
456D4—2148no—
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—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—44no—
473C2×C4—1072no—
474C2×C4—1072no—
475C2×C4—4284no—
476C2×C4—4284no—
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×C4—2141no—
487C2×C4D4×D10712yes—
488C2×C4C2×D42812yes—
489C2×C4C2×D42812yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—42816no—
497C22—4281no—
498C22—4281no—
499C22—4281no—
500C22—4281no—
501C22—4281no—
502C22—4281no—
503C22—4281no—
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—2142no—
517C22—2142no—
518C22—2142no—
519C22—2142no—
520C22—2142no—
521C22—2142no—
522C22—2142no—
523C22—2142no—
524C22—2142no—
525C22#1681208414yescomplemented
526C22C2×D4×D10711yescharacteristic
527C22C2×D4×D10711yescharacteristic
528C22C22×D42811yescharacteristic
529C22#41×D10711yescharacteristic
530C22C2×#1313381911yescharacteristic
531C22#1681215411yescharacteristic
532C22#1681215511yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—4281no—
540C4—4281no—
541C4—2142no—
542C4—2142no—
543C2—24no—
544C2—8561no—
545C2—4282no—
546C2—4282no—
547C2—4282no—
548C2C2×#1681208411yescharacteristic
549C2#2029314011yescharacteristic
550C2#2029321711yescharacteristic
551C2#2029325911yescharacteristic
552C2D4×D42811yescharacteristic
553C2#2029326911yescharacteristic
554C2#2029327211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#385×C107C2
hypercenter#385D107
hypocenterC107#2424
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#16812084C2semidirect
2. C2×#16812083C2semidirect
3. C2×#16812079C2semidirect
4. C2×#16812079C2semidirect
5. C2×(C22⋊D107)⋊C4C2semidirect
6. C22×(C2×D107)⋊C4C2semidirect
7. #20292549C2semidirect
8. #20292545C2semidirect
9. #20292544C2semidirect
10. #20292537C2semidirect
11. #20292528C2semidirect
12. #20292519C2semidirect
13. #20292505C2semidirect
14. #20292504C2semidirect
15. #385×C107C2semidirect
21. C22×D428C22semidirect
25. C22×C4×D107C22semidirect
31. C2×C4⋊Dic107C22semidirect
32. C2×Dic107⋊C4C22semidirect
42. C2×(C2×D107)⋊C4C22semidirect
43. C2×(C2×D107)⋊C4C22semidirect
44. C2×(C2×D107)⋊C4C22semidirect
45. C2×(C2×D107)⋊C4C22semidirect
46. C2×(C2×D107)⋊C4C22semidirect
47. C2×C4×Dic107C22semidirect
48. C107⋊#95C22semidirect
49. C107⋊#95C22semidirect
51. C2×C4⋊C4×C107C22semidirect
52. C2×C4⋊C4×C107C22semidirect
55. #95×C107C22semidirect
56. #95×C107C22semidirect
88. C2×C22⋊D107D4semidirect
113. (C2×D107)⋊C4D4semidirect
128. C22⋊C4×C107D4semidirect
218. C22×C107#121semidirect
278. C107#2424semidirect
293. #385D107semidirect
312. C2×C4⋊C4C2×D107semidirect
313. C2×C4⋊C4C2×D107semidirect
333. #95C2×D107semidirect
334. #95C2×D107semidirect
405. C22⋊C4D428semidirect
525. C22#16812084semidirect

Maximal subgroups

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

subgroupindex
C2×#168120842
C2×#168120832
C2×#168120792
C2×(C22⋊D107)⋊C42
C22×(C2×D107)⋊C42
#202925492
#202925452
#202925442
#202925372
#202925282
#202925192
#202925052
#202925042
#385×C1072
#2424107