Finite Group Databasebeta

← #28339993

Subgroups — #28339993

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×#16812081C211yesmaximal, characteristic, complemented
2C2×#16812080C211yesmaximal, characteristic, complemented
3C2×#16812078C211yesmaximal, characteristic, complemented
4C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic, complemented
5C22×C22⋊Dic107C211yesmaximal, characteristic, complemented
6#20292550C211yesmaximal, characteristic, complemented
7#20292543C211yesmaximal, characteristic, complemented
8#20292541C211yesmaximal, characteristic, complemented
9#20292535C211yesmaximal, characteristic, complemented
10#20292530C211yesmaximal, characteristic, complemented
11#20292515C211yesmaximal, characteristic, complemented
12#20292507C211yesmaximal, characteristic, complemented
13#20292503C211yesmaximal, characteristic, complemented
14#20292502C211yesmaximal, characteristic, complemented
15C2×#120×C107C211yesmaximal, characteristic, complemented
16C22×C22⋊D107C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C22×C22⋊D107C2211yescharacteristic
19C22×C22⋊D107C2211yescharacteristic, complemented
20C23×Dic107C2211yescharacteristic
21C23×Dic107C2211yescharacteristic
22C22×C4×D107C2211yescharacteristic, complemented
23#16812081—24no—
24#16812080—24no—
25#16812078—24no—
26(C22⋊D107)⋊C4—24no—
27C2×C4⋊Dic107C2211yescharacteristic, complemented
28C2×C4⋊Dic107C2211yescharacteristic, complemented
29C2×Dic107⋊C4C2211yescharacteristic, complemented
30C2×Dic107⋊C4C2211yescharacteristic, complemented
31C2×C22⋊Dic107—24no—
32C2×C22⋊Dic107C2211yescharacteristic
33C2×C22⋊Dic107C2211yescharacteristic
34C2×C22⋊Dic107C2211yescharacteristic
35C2×C22⋊Dic107C2211yescharacteristic
36C2×C22⋊Dic107C2211yescharacteristic
37C2×C22⋊Dic107C2211yescharacteristic
38C2×C22⋊Dic107C2211yescharacteristic
39C2×C22⋊Dic107C2211yescharacteristic, complemented
40C2×C22⋊Dic107C2211yescharacteristic, complemented
41C2×C22⋊Dic107C2211yescharacteristic, complemented
42C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
44C2×C4×Dic107C2211yescharacteristic, complemented
45C107⋊#95C2211yescharacteristic, complemented
46C107⋊#95C2211yescharacteristic, complemented
47C107⋊#95C2211yescharacteristic, complemented
48C107⋊#95C2211yescharacteristic, complemented
49C25×C107C2211yescharacteristic
50C22×D4×C107C2211yescharacteristic
51C22×D4×C107C2211yescharacteristic
52C22×D4×C107C2211yescharacteristic
53#120×C107—24no—
54C2×C22⋊C4×C107C2211yescharacteristic
55C2×C22⋊C4×C107C2211yescharacteristic
56C2×C22⋊C4×C107C2211yescharacteristic
57C23×D107C2311yescharacteristic
58C2×C22⋊D107—22no—
59C2×C22⋊D107—22no—
60C2×C22⋊D107—22no—
61C2×C22⋊D107—22no—
62C2×C22⋊D107—22no—
63C2×C22⋊D107—22no—
64C2×C22⋊D107—22no—
65C2×C22⋊D107—22no—
66C2×C22⋊D107—41no—
67C2×C22⋊D107—41no—
68C2×C22⋊D107—41no—
69C2×C22⋊D107D414yescomplemented
70C22×Dic107—22no—
71C22×Dic107—22no—
72C22×Dic107—22no—
73C22×Dic107—22no—
74C22×Dic107—41no—
75C22×Dic107—41no—
76C22×Dic107C2311yescharacteristic
77C22×Dic107C2311yescharacteristic
78C22×Dic107C2311yescharacteristic
79C22×Dic107C2311yescharacteristic
80C22×Dic107C2311yescharacteristic
81C22×Dic107C2311yescharacteristic
82C22×Dic107C2311yescharacteristic
83C2×C4×D107—22no—
84C2×C4×D107—22no—
85C2×C4×D107—41no—
86C4⋊Dic107—22no—
87C4⋊Dic107—41no—
88Dic107⋊C4—22no—
89Dic107⋊C4—22no—
90C22⋊Dic107—22no—
91C22⋊Dic107—22no—
92C22⋊Dic107—22no—
93C22⋊Dic107—22no—
94C22⋊Dic107—22no—
95C22⋊Dic107—22no—
96C22⋊Dic107—22no—
97C22⋊Dic107—22no—
98C22⋊Dic107—41no—
99C22⋊Dic107D414yescomplemented
100(C2×D107)⋊C4—22no—
101(C2×D107)⋊C4—22no—
102C4×Dic107—22no—
103C24×C107—22no—
104C24×C107—22no—
105C24×C107—22no—
106C24×C107—22no—
107C24×C107—22no—
108C24×C107—41no—
109C24×C107—41no—
110C24×C107C2311yescharacteristic
111C24×C107C2311yescharacteristic
112C24×C107C2311yescharacteristic
113C24×C107C2311yescharacteristic
114C2×D4×C107—22no—
115C2×D4×C107—22no—
116C2×D4×C107—22no—
117C2×D4×C107—22no—
118C2×D4×C107—22no—
119C2×D4×C107—22no—
120C2×D4×C107—22no—
121C2×D4×C107—41no—
122C2×D4×C107—41no—
123C22×C4×C107C2311yescharacteristic
124C22×C4×C107C2311yescharacteristic
125C22×C4×C107C2311yescharacteristic
126C22⋊C4×C107—22no—
127C22⋊C4×C107—22no—
128C22⋊C4×C107D414yescomplemented
129C23×C107—21no—
130C23×C107—21no—
131C23×C107—21no—
132C23×C107—21no—
133C23×C107—28no—
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×C107—21no—
143C23×C107—21no—
144C23×C107—21no—
145C23×C107—21no—
146C23×C107—21no—
147C23×C107—21no—
148C23×C107—21no—
149C23×C107—21no—
150C23×C107—21no—
151C23×C107—21no—
152C23×C107—21no—
153C23×C107—44no—
154C23×C107—44no—
155C23×C107—44no—
156C23×C107—44no—
157C23×C107—44no—
158C23×C107—44no—
159C23×C107C2×D412yes—
160C23×C107C2×D412yes—
161C23×C107C2×D412yes—
162C23×C107#4112yes—
163C23×C107#4112yes—
164C23×C107C2411yescharacteristic
165C22×D107—21no—
166C22×D107—21no—
167C22×D107—21no—
168C22×D107—21no—
169C22×D107—21no—
170C22×D107—21no—
171C22×D107C2×D412yes—
172D4×C107—44no—
173D4×C107—44no—
174D4×C107—44no—
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×C107C2×D412yes—
183C2×C4×C107C2×D412yes—
184C22⋊D107—28no—
185C22⋊D107—44no—
186C22⋊D107—44no—
187C22⋊D107—44no—
188C2×Dic107—21no—
189C2×Dic107—21no—
190C2×Dic107—21no—
191C2×Dic107—21no—
192C2×Dic107—21no—
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—21no—
204C2×Dic107—21no—
205C2×Dic107—21no—
206C2×Dic107—44no—
207C2×Dic107—44no—
208C2×Dic107C2×D412yes—
209C2×Dic107C2×D412yes—
210C2×Dic107C2×D412yes—
211C4×D107—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—22no—
222C22×C107—22no—
223C22×C107—22no—
224C22×C107—416no—
225C22×C107—41no—
226C22×C107—41no—
227C22×C107—41no—
228C22×C107—41no—
229C22×C107—41no—
230C22×C107—41no—
231C22×C107—41no—
232C22×C107—41no—
233C22×C107—41no—
234C22×C107—41no—
235C22×C107—41no—
236C22×C107—41no—
237C22×C107—41no—
238C22×C107—41no—
239C22×C107—41no—
240C22×C107#12114yescomplemented
241C22×C107C22×D411yescharacteristic
242C22×C107C22×D411yescharacteristic
243C22×C107C22×D411yescharacteristic
244C22×C107C2×#4111yescharacteristic
245C22×C107C2×#4111yescharacteristic
246C22×C107H(5,𝔽2)11yescharacteristic
247C22×C107H(5,𝔽2)11yescharacteristic
248C2×D107—22no—
249C2×D107—22no—
250C2×D107—22no—
251C2×D107—41no—
252C2×D107—41no—
253C2×D107—41no—
254C2×D107—41no—
255C4×C107—22no—
256C4×C107—22no—
257C4×C107—41no—
258Dic107—22no—
259Dic107—22no—
260Dic107—22no—
261Dic107—41no—
262Dic107—41no—
263Dic107—41no—
264Dic107—41no—
265C2×C107—24no—
266C2×C107—42no—
267C2×C107—42no—
268C2×C107—81no—
269C2×C107C2×#12111yescharacteristic
270C2×C107#52511yescharacteristic
271C2×C107#53811yescharacteristic
272C2×C107D4211yescharacteristic
273C2×C107D4211yescharacteristic
274C2×C107#54611yescharacteristic
275C2×C107#54611yescharacteristic
276D107—42no—
277#2424—1071nomaximal, Sylow
278C107#242411yescharacteristic, complemented, Sylow
279C2×#121—1071no—
280C2×#121—1071no—
281C2×#120—1071no—
282C2×#120D10711yescharacteristic, complemented
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#385—1071no—
294C25C2×D10711yescharacteristic
295C22×D4—1071no—
296C22×D4—1071no—
297C22×D4—1071no—
298C22×D4—1071no—
299C22×D4C2×D10711yescharacteristic
300C22×D4C2×D10711yescharacteristic
301C22×D4C2×D10711yescharacteristic
302C23×C4—1071no—
303C23×C4—1071no—
304C23×C4—1071no—
305#121—2144no—
306#121—2144no—
307#120—24no—
308#120—2144no—
309C4×D4—2144no—
310C2×C4⋊C4—1071no—
311C2×C4⋊C4—1071no—
312C2×C4⋊C4—1071no—
313C2×C4⋊C4—1071no—
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—2144no—
327C2×C22⋊C4C2×D10711yescharacteristic
328C2×C22⋊C4C2×D10711yescharacteristic
329C2×C22⋊C4C2×D10711yescharacteristic
330C2×C42—1071no—
331#95—1071no—
332#95—1071no—
333#95—1071no—
334#95—1071no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—41no—
341C24—41no—
342C24—1071no—
343C24C22×D10711yescharacteristic
344C24C22×D10711yescharacteristic
345C24C22×D10711yescharacteristic
346C24C22×D10711yescharacteristic
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—1074no—
357C2×D4—4281no—
358C2×D4—4281no—
359C2×D4—4281no—
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—1071no—
369C22×C4—1071no—
370C22×C4—1071no—
371C22×C4—1071no—
372C22×C4—1071no—
373C22×C4—1071no—
374C22×C4—1071no—
375C22×C4—4281no—
376C22×C4—4281no—
377C22×C4—4281no—
378C22×C4—2142no—
379C22×C4—2142no—
380C22×C4—2142no—
381C22×C4—2142no—
382C22×C4—2142no—
383C22×C4—2142no—
384C22×C4C22×D10711yescharacteristic
385C22×C4C22×D10711yescharacteristic
386C22×C4C22×D10711yescharacteristic
387C4⋊C4—4281no—
388C4⋊C4—2142no—
389C4⋊C4—2142no—
390C4⋊C4—2142no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—1074no—
394C22⋊C4—4281no—
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—2142no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—28no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—1072no—
438C23—2141no—
439C23—2141no—
440C23—2141no—
441C23—2141no—
442C23—2141no—
443C23—2141no—
444C23D4×D10712yes—
445C23D4×D10712yes—
446C23C2×C22⋊D10712yes—
447C23#1313381712yes—
448C23#1313381712yes—
449C23C23×D10711yescharacteristic
450D4—44no—
451D4—44no—
452D4—44no—
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—1072no—
465C2×C4—1072no—
466C2×C4—1072no—
467C2×C4—4284no—
468C2×C4—4284no—
469C2×C4—4284no—
470C2×C4—2141no—
471C2×C4—2141no—
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×C4—2141no—
487C2×C4—2141no—
488C2×C4C2×C22⋊D10712yes—
489C2×C4C2×C22⋊D10712yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—416no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—4281no—
519C22—4281no—
520C22—4281no—
521C22—4281no—
522C22—2142no—
523C22—2142no—
524C22—2142no—
525C22#1681208014yescomplemented
526C22C2×D4×D10711yescharacteristic
527C22C2×D4×D10711yescharacteristic
528C22C22×C22⋊D10711yescharacteristic
529C22C2×#1313381711yescharacteristic
530C22C2×#1313381711yescharacteristic
531C22#1681215411yescharacteristic
532C22#1681215411yescharacteristic
533C4—22no—
534C4—22no—
535C4—41no—
536C4—4281no—
537C4—4281no—
538C4—4281no—
539C4—4281no—
540C4—2142no—
541C4—2142no—
542C4—2142no—
543C2—24no—
544C2—42no—
545C2—42no—
546C2—81no—
547C2—4282no—
548C2C2×#1681208011yescharacteristic
549C2#2029313311yescharacteristic
550C2#2029321211yescharacteristic
551C2#2029325911yescharacteristic
552C2D4×C22⋊D10711yescharacteristic
553C2#2029326411yescharacteristic
554C2#2029327111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#120×C107C2
hypercenterC2×#120D107
hypocenterC107#2424
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#16812081C2semidirect
2. C2×#16812080C2semidirect
3. C2×#16812078C2semidirect
4. C2×(C22⋊D107)⋊C4C2semidirect
5. C22×C22⋊Dic107C2semidirect
6. #20292550C2semidirect
7. #20292543C2semidirect
8. #20292541C2semidirect
9. #20292535C2semidirect
10. #20292530C2semidirect
11. #20292515C2semidirect
12. #20292507C2semidirect
13. #20292503C2semidirect
14. #20292502C2semidirect
15. C2×#120×C107C2semidirect
19. C22×C22⋊D107C22semidirect
22. C22×C4×D107C22semidirect
27. C2×C4⋊Dic107C22semidirect
28. C2×C4⋊Dic107C22semidirect
29. C2×Dic107⋊C4C22semidirect
30. C2×Dic107⋊C4C22semidirect
39. C2×C22⋊Dic107C22semidirect
40. C2×C22⋊Dic107C22semidirect
41. C2×C22⋊Dic107C22semidirect
42. C2×(C2×D107)⋊C4C22semidirect
43. C2×(C2×D107)⋊C4C22semidirect
44. C2×C4×Dic107C22semidirect
45. C107⋊#95C22semidirect
46. C107⋊#95C22semidirect
47. C107⋊#95C22semidirect
48. C107⋊#95C22semidirect
69. C2×C22⋊D107D4semidirect
99. C22⋊Dic107D4semidirect
128. C22⋊C4×C107D4semidirect
240. C22×C107#121semidirect
278. C107#2424semidirect
282. C2×#120D107semidirect
405. C22⋊C4C22⋊D107semidirect
525. C22#16812080semidirect

Maximal subgroups

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

subgroupindex
C2×#168120812
C2×#168120802
C2×#168120782
C2×(C22⋊D107)⋊C42
C22×C22⋊Dic1072
#202925502
#202925432
#202925412
#202925352
#202925302
#202925152
#202925072
#202925032
#202925022
C2×#120×C1072
#2424107