Finite Group Databasebeta

← #28340000

Subgroups — #28340000

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×#16812077C211yesmaximal, characteristic, complemented
4C2×#16812077C211yesmaximal, characteristic, complemented
5C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic, complemented
6C2×C22⋊C4×D107C211yesmaximal, characteristic, complemented
7#20292550C211yesmaximal, characteristic, complemented
8#20292543C211yesmaximal, characteristic, complemented
9#20292542C211yesmaximal, characteristic, complemented
10#20292530C211yesmaximal, characteristic, complemented
11#20292519C211yesmaximal, characteristic, complemented
12#20292507C211yesmaximal, characteristic, complemented
13#20292503C211yesmaximal, characteristic, complemented
14#20292501C211yesmaximal, characteristic, complemented
15#392×C107C211yesmaximal, characteristic, complemented
16C24×D107C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C22×C22⋊D107C2211yescharacteristic
19C22×C22⋊D107C2211yescharacteristic
20C22×C22⋊D107C2211yescharacteristic
21C23×Dic107C2211yescharacteristic, complemented
22C22×C4×D107C2211yescharacteristic
23C22×C4×D107C2211yescharacteristic
24#16812081—24no—
25#16812080—24no—
26#16812077—24no—
27#16812077—24no—
28(C22⋊D107)⋊C4—24no—
29C2×C4⋊Dic107C2211yescharacteristic, complemented
30C2×C4⋊Dic107C2211yescharacteristic, complemented
31C2×Dic107⋊C4C2211yescharacteristic, complemented
32C2×Dic107⋊C4C2211yescharacteristic, complemented
33C2×C22⋊Dic107C2211yescharacteristic
34C2×C22⋊Dic107C2211yescharacteristic
35C2×C22⋊Dic107C2211yescharacteristic
36C2×C22⋊Dic107C2211yescharacteristic, complemented
37C2×C22⋊Dic107C2211yescharacteristic, complemented
38C2×C22⋊Dic107C2211yescharacteristic, complemented
39C2×(C2×D107)⋊C4C2211yescharacteristic
40C2×(C2×D107)⋊C4C2211yescharacteristic
41C2×(C2×D107)⋊C4C2211yescharacteristic
42C2×(C2×D107)⋊C4C2211yescharacteristic
43C2×(C2×D107)⋊C4C2211yescharacteristic
44C2×(C2×D107)⋊C4C2211yescharacteristic
45C22⋊C4×D107—24no—
46C2×C4×Dic107C2211yescharacteristic, complemented
47C107⋊#95C2211yescharacteristic, complemented
48C107⋊#95C2211yescharacteristic, complemented
49C107⋊#95C2211yescharacteristic, complemented
50C22×D4×C107C2211yescharacteristic
51C22×D4×C107C2211yescharacteristic
52C22×D4×C107C2211yescharacteristic, complemented
53C2×C22⋊C4×C107C2211yescharacteristic
54C2×C22⋊C4×C107C2211yescharacteristic, complemented
55C2×C22⋊C4×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
66C2×C22⋊D107—22no—
67C2×C22⋊D107—22no—
68C2×C22⋊D107—22no—
69C2×C22⋊D107—22no—
70C2×C22⋊D107—22no—
71C2×C22⋊D107—22no—
72C2×C22⋊D107—22no—
73C2×C22⋊D107—22no—
74C2×C22⋊D107—22no—
75C2×C22⋊D107—41no—
76C2×C22⋊D107—41no—
77C2×C22⋊D107D414yescomplemented
78C22×Dic107—22no—
79C22×Dic107—22no—
80C22×Dic107—41no—
81C22×Dic107C2311yescharacteristic
82C22×Dic107C2311yescharacteristic
83C22×Dic107C2311yescharacteristic
84C22×Dic107C2311yescharacteristic
85C22×Dic107C2311yescharacteristic
86C22×Dic107C2311yescharacteristic
87C2×C4×D107—22no—
88C2×C4×D107—22no—
89C2×C4×D107—22no—
90C2×C4×D107—22no—
91C2×C4×D107—41no—
92C2×C4×D107—41no—
93C4⋊Dic107—22no—
94C4⋊Dic107—41no—
95Dic107⋊C4—22no—
96Dic107⋊C4—22no—
97C22⋊Dic107—22no—
98C22⋊Dic107—22no—
99C22⋊Dic107—22no—
100C22⋊Dic107—22no—
101C22⋊Dic107—22no—
102C22⋊Dic107—22no—
103(C2×D107)⋊C4—22no—
104(C2×D107)⋊C4—22no—
105(C2×D107)⋊C4—22no—
106(C2×D107)⋊C4—22no—
107(C2×D107)⋊C4D414yescomplemented
108(C2×D107)⋊C4D414yescomplemented
109C4×Dic107—22no—
110C24×C107C2311yescharacteristic
111C24×C107C2311yescharacteristic
112C24×C107C2311yescharacteristic
113C2×D4×C107—22no—
114C2×D4×C107—22no—
115C2×D4×C107—22no—
116C2×D4×C107—22no—
117C2×D4×C107—22no—
118C2×D4×C107—22no—
119C2×D4×C107—41no—
120C2×D4×C107—41no—
121C2×D4×C107—41no—
122C22×C4×C107C2311yescharacteristic
123C22×C4×C107C2311yescharacteristic
124C22×C4×C107C2311yescharacteristic
125C22×C4×C107C2311yescharacteristic
126C22⋊C4×C107—22no—
127C22⋊C4×C107—22no—
128C22⋊C4×C107—41no—
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×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×C107C2×D412yes—
149C23×C107#4112yes—
150C23×C107C2411yescharacteristic
151C22×D107—21no—
152C22×D107—21no—
153C22×D107—21no—
154C22×D107—21no—
155C22×D107—28no—
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×D107#4112yes—
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×C107—21no—
183C2×C4×C107—21no—
184C2×C4×C107—21no—
185C2×C4×C107C2×D412yes—
186C2×C4×C107C2×D412yes—
187C22⋊D107—28no—
188C22⋊D107—44no—
189C22⋊D107—44no—
190C22⋊D107—44no—
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×Dic107C2×D412yes—
208C2×Dic107C2×D412yes—
209C2×Dic107C2×D412yes—
210C4×D107—44no—
211C4×D107—44no—
212C22×C107—22no—
213C22×C107—22no—
214C22×C107—22no—
215C22×C107—22no—
216C22×C107—22no—
217C22×C107—22no—
218C22×C107—41no—
219C22×C107—41no—
220C22×C107—41no—
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×C107—41no—
230C22×C107—41no—
231C22×C107—41no—
232C22×C107—41no—
233C22×C107C22×D411yescharacteristic
234C22×C107C22×D411yescharacteristic
235C22×C107C22×D411yescharacteristic
236C22×C107C2×#4111yescharacteristic
237C22×C107C2×#4111yescharacteristic
238C22×C107H(5,𝔽2)11yescharacteristic
239C22×C107H(5,𝔽2)11yescharacteristic
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—22no—
249C2×D107—416no—
250C2×D107—41no—
251C2×D107—41no—
252C2×D107—41no—
253C2×D107—41no—
254C2×D107#12114yescomplemented
255C4×C107—22no—
256C4×C107—22no—
257C4×C107—41no—
258C4×C107—41no—
259Dic107—22no—
260Dic107—22no—
261Dic107—22no—
262Dic107—41no—
263Dic107—41no—
264Dic107—41no—
265C2×C107—42no—
266C2×C107—42no—
267C2×C107—81no—
268C2×C107C2×#12111yescharacteristic
269C2×C107#52511yescharacteristic
270C2×C107#53811yescharacteristic
271C2×C107D4211yescharacteristic
272C2×C107D4211yescharacteristic
273C2×C107#54611yescharacteristic
274C2×C107#54611yescharacteristic
275D107—24no—
276D107—42no—
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#392D10711yescharacteristic, complemented
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×D4C2×D10711yescharacteristic
300C22×D4C2×D10711yescharacteristic
301C22×D4C2×D10711yescharacteristic, complemented
302C23×C4—1071no—
303C23×C4—1071no—
304C23×C4—1071no—
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⋊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, complemented
329C2×C22⋊C4C2×D10711yescharacteristic, complemented
330C2×C42—1071no—
331#95—1071no—
332#95—1071no—
333#95—1071no—
334#95C2×D10711yescharacteristic, complemented
335C24—1071no—
336C24—1071no—
337C24—4281no—
338C24—4281no—
339C24—2142no—
340C24—2142no—
341C24—2142no—
342C24—2142no—
343C24—2142no—
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—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—1074no—
357C2×D4—4281no—
358C2×D4—4281no—
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—1071no—
369C22×C4—1071no—
370C22×C4—1071no—
371C22×C4—1071no—
372C22×C4—1071no—
373C22×C4—1071no—
374C22×C4—4281no—
375C22×C4—4281no—
376C22×C4—4281no—
377C22×C4—2142no—
378C22×C4—2142no—
379C22×C4—2142no—
380C22×C4—2142no—
381C22×C4—2142no—
382C22×C4—2142no—
383C22×C4C22×D10711yescharacteristic
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—41no—
394C22⋊C4—1074no—
395C22⋊C4—1074no—
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⋊C4—2142no—
406C42—2142no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
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—21no—
424C23—21no—
425C23—21no—
426C23—1072no—
427C23—1072no—
428C23—1072no—
429C23—1072no—
430C23—4284no—
431C23—4284no—
432C23—4284no—
433C23—4284no—
434C23—4284no—
435C23—4284no—
436C23—2148no—
437C23—2141no—
438C23—2141no—
439C23—2141no—
440C23—2141no—
441C23—2141no—
442C23—2141no—
443C23—2141no—
444C23—2141no—
445C23—2141no—
446C23—2141no—
447C23D4×D10712yes—
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—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—1072no—
468C2×C4—1072no—
469C2×C4—1072no—
470C2×C4—4284no—
471C2×C4—4284no—
472C2×C4—4284no—
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×C4D4×D10712yes—
489C2×C4D4×D10712yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—1074no—
512C22—42816no—
513C22—4281no—
514C22—4281no—
515C22—4281no—
516C22—4281no—
517C22—2142no—
518C22—2142no—
519C22—2142no—
520C22—2142no—
521C22—2142no—
522C22—2142no—
523C22—2142no—
524C22—2142no—
525C22—2142no—
526C22C2×D4×D10711yescharacteristic
527C22C2×D4×D10711yescharacteristic
528C22C2×D4×D10711yescharacteristic
529C22#41×D10711yescharacteristic
530C22C2×#1313381711yescharacteristic
531C22#1681215411yescharacteristic
532C22#1681215411yescharacteristic
533C4—22no—
534C4—22no—
535C4—41no—
536C4—41no—
537C4—4281no—
538C4—4281no—
539C4—4281no—
540C4—2142no—
541C4—2142no—
542C4—2142no—
543C2—42no—
544C2—42no—
545C2—81no—
546C2—4282no—
547C2—2144no—
548C2#121×D10711yescharacteristic
549C2#2029313711yescharacteristic
550C2#2029320911yescharacteristic
551C2#2029325911yescharacteristic
552C2#2029326211yescharacteristic
553C2#2029326411yescharacteristic
554C2#2029326411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#392×C107C2
hypercenter#392D107
hypocenterC107#2424
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#16812081C2semidirect
2. C2×#16812080C2semidirect
3. C2×#16812077C2semidirect
4. C2×#16812077C2semidirect
5. C2×(C22⋊D107)⋊C4C2semidirect
6. C2×C22⋊C4×D107C2semidirect
7. #20292550C2semidirect
8. #20292543C2semidirect
9. #20292542C2semidirect
10. #20292530C2semidirect
11. #20292519C2semidirect
12. #20292507C2semidirect
13. #20292503C2semidirect
14. #20292501C2semidirect
15. #392×C107C2semidirect
21. C23×Dic107C22semidirect
29. C2×C4⋊Dic107C22semidirect
30. C2×C4⋊Dic107C22semidirect
31. C2×Dic107⋊C4C22semidirect
32. C2×Dic107⋊C4C22semidirect
36. C2×C22⋊Dic107C22semidirect
37. C2×C22⋊Dic107C22semidirect
38. C2×C22⋊Dic107C22semidirect
46. C2×C4×Dic107C22semidirect
47. C107⋊#95C22semidirect
48. C107⋊#95C22semidirect
49. C107⋊#95C22semidirect
52. C22×D4×C107C22semidirect
54. C2×C22⋊C4×C107C22semidirect
55. C2×C22⋊C4×C107C22semidirect
56. #95×C107C22semidirect
77. C2×C22⋊D107D4semidirect
107. (C2×D107)⋊C4D4semidirect
108. (C2×D107)⋊C4D4semidirect
254. C2×D107#121semidirect
278. C107#2424semidirect
288. #392D107semidirect
301. C22×D4C2×D107semidirect
328. C2×C22⋊C4C2×D107semidirect
329. C2×C22⋊C4C2×D107semidirect
334. #95C2×D107semidirect

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×#168120772
C2×(C22⋊D107)⋊C42
C2×C22⋊C4×D1072
#202925502
#202925432
#202925422
#202925302
#202925192
#202925072
#202925032
#202925012
#392×C1072
#2424107