Finite Group Databasebeta

← #28339990

Subgroups — #28339990

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×#16812082C211yesmaximal, characteristic, complemented
2C2×#16812080C211yesmaximal, characteristic, complemented
3C2×#16812079C211yesmaximal, characteristic, complemented
4C2×#16812077C211yesmaximal, characteristic, complemented
5C2×#16812071C211yesmaximal, characteristic, complemented
6C2×C22⋊C4×D107C211yesmaximal, characteristic, complemented
7#20292549C211yesmaximal, characteristic, complemented
8#20292542C211yesmaximal, characteristic, complemented
9#20292536C211yesmaximal, characteristic, complemented
10#20292527C211yesmaximal, characteristic, complemented
11#20292521C211yesmaximal, characteristic, complemented
12#20292505C211yesmaximal, characteristic, complemented
13#20292503C211yesmaximal, characteristic, complemented
14#20292501C211yesmaximal, characteristic, complemented
15#394×C107C211yesmaximal, characteristic, complemented
16C24×D107C2211yescharacteristic
17C22×D428C2211yescharacteristic
18C22×D428C2211yescharacteristic
19C22×C22⋊D107C2211yescharacteristic
20C22×C22⋊D107C2211yescharacteristic
21C22×C22⋊D107C2211yescharacteristic
22C22×C22⋊D107C2211yescharacteristic, complemented
23C22×C4×D107C2211yescharacteristic
24C22×C4×D107C2211yescharacteristic
25C22×C4×D107C2211yescharacteristic, complemented
26#16812082—24no—
27#16812080—24no—
28#16812079—24no—
29#16812077—24no—
30#16812071—24no—
31C2×C4⋊Dic107C2211yescharacteristic, complemented
32C2×Dic107⋊C4C2211yescharacteristic, complemented
33C2×Dic107⋊C4C2211yescharacteristic, complemented
34C2×C22⋊Dic107C2211yescharacteristic
35C2×C22⋊Dic107C2211yescharacteristic
36C2×C22⋊Dic107C2211yescharacteristic, complemented
37C2×(C2×D107)⋊C4C2211yescharacteristic
38C2×(C2×D107)⋊C4C2211yescharacteristic
39C2×(C2×D107)⋊C4C2211yescharacteristic
40C2×(C2×D107)⋊C4C2211yescharacteristic
41C2×(C2×D107)⋊C4C2211yescharacteristic
42C2×(C2×D107)⋊C4C2211yescharacteristic
43C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
45C22⋊C4×D107—24no—
46C2×C4×Dic107C2211yescharacteristic, complemented
47C107⋊#95C2211yescharacteristic, complemented
48C107⋊#95C2211yescharacteristic, complemented
49C107⋊#95C2211yescharacteristic, complemented
50C22×D4×C107C2211yescharacteristic
51C2×C4⋊C4×C107C2211yescharacteristic, complemented
52C2×C22⋊C4×C107C2211yescharacteristic
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
66C23×D107C2311yescharacteristic
67C2×D428—22no—
68C2×D428—22no—
69C2×D428—22no—
70C2×D428—22no—
71C2×D428—41no—
72C2×D428D414yescomplemented
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×Dic107C2311yescharacteristic
86C22×Dic107C2311yescharacteristic
87C22×Dic107C2311yescharacteristic
88C22×Dic107C2311yescharacteristic
89C22×Dic107C2311yescharacteristic
90C2×C4×D107—22no—
91C2×C4×D107—22no—
92C2×C4×D107—22no—
93C2×C4×D107—22no—
94C2×C4×D107—22no—
95C2×C4×D107—22no—
96C2×C4×D107—41no—
97C2×C4×D107—41no—
98C2×C4×D107—41no—
99C4⋊Dic107—22no—
100Dic107⋊C4—22no—
101Dic107⋊C4—41no—
102C22⋊Dic107—22no—
103C22⋊Dic107—22no—
104C22⋊Dic107D414yescomplemented
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)⋊C4D414yescomplemented
113C4×Dic107—22no—
114C24×C107C2311yescharacteristic
115C24×C107C2311yescharacteristic
116C2×D4×C107—22no—
117C2×D4×C107—22no—
118C2×D4×C107—41no—
119C22×C4×C107C2311yescharacteristic
120C22×C4×C107C2311yescharacteristic
121C22×C4×C107C2311yescharacteristic
122C22×C4×C107C2311yescharacteristic
123C22×C4×C107C2311yescharacteristic
124C4⋊C4×C107—22no—
125C22⋊C4×C107—22no—
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×C107C2×D412yes—
143C23×C107C2411yescharacteristic
144C22×D107—21no—
145C22×D107—21no—
146C22×D107—21no—
147C22×D107—21no—
148C22×D107—28no—
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—44no—
162C22×D107—44no—
163C22×D107—44no—
164C22×D107—44no—
165C22×D107—44no—
166C22×D107—44no—
167C22×D107C2×D412yes—
168C22×D107C2×D412yes—
169C22×D107C2×D412yes—
170C22×D107#4112yes—
171C22×D107#4112yes—
172D4×C107—44no—
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×C107C2×D412yes—
187C2×C4×C107C2×D412yes—
188C22⋊D107—44no—
189C22⋊D107—44no—
190C22⋊D107—44no—
191C22⋊D107—44no—
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×Dic107C2×D412yes—
205C2×Dic107C2×D412yes—
206C2×Dic107C2×D412yes—
207D428—28no—
208D428—44no—
209C4×D107—44no—
210C4×D107—44no—
211C4×D107—44no—
212C22×C107—22no—
213C22×C107—22no—
214C22×C107—22no—
215C22×C107—41no—
216C22×C107—41no—
217C22×C107—41no—
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×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—22no—
240C2×D107—22no—
241C2×D107—22no—
242C2×D107—22no—
243C2×D107—22no—
244C2×D107—22no—
245C2×D107—416no—
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#12114yescomplemented
255C4×C107—22no—
256C4×C107—22no—
257C4×C107—41no—
258C4×C107—41no—
259C4×C107—41no—
260Dic107—22no—
261Dic107—22no—
262Dic107—22no—
263Dic107—41no—
264Dic107—41no—
265C2×C107—42no—
266C2×C107—81no—
267C2×C107C2×#12111yescharacteristic
268C2×C107#52511yescharacteristic
269C2×C107#53811yescharacteristic
270C2×C107D4211yescharacteristic
271C2×C107D4211yescharacteristic
272C2×C107#54611yescharacteristic
273C2×C107#54611yescharacteristic
274D107—24no—
275D107—42no—
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#394D10711yescharacteristic, complemented
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×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⋊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—2144no—
326C2×C22⋊C4C2×D10711yescharacteristic
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—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—41no—
350C2×D4—1074no—
351C2×D4—4281no—
352C2×D4—4281no—
353C2×D4—4281no—
354C2×D4—4281no—
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—1071no—
369C22×C4—1071no—
370C22×C4—1071no—
371C22×C4—1071no—
372C22×C4—1071no—
373C22×C4—4281no—
374C22×C4—4281no—
375C22×C4—4281no—
376C22×C4—2142no—
377C22×C4—2142no—
378C22×C4—2142no—
379C22×C4—2142no—
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—22no—
393C22⋊C4—22no—
394C22⋊C4—41no—
395C22⋊C4—1074no—
396C22⋊C4—1074no—
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—1072no—
421C23—1072no—
422C23—1072no—
423C23—1072no—
424C23—1072no—
425C23—4284no—
426C23—4284no—
427C23—4284no—
428C23—4284no—
429C23—4284no—
430C23—4284no—
431C23—2148no—
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—
446C23—2141no—
447C23—2141no—
448C23D4×D10712yes—
449C23C23×D10711yescharacteristic
450D4—44no—
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—1072no—
471C2×C4—1072no—
472C2×C4—1072no—
473C2×C4—4284no—
474C2×C4—4284no—
475C2×C4—4284no—
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—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—1074no—
505C22—42816no—
506C22—4281no—
507C22—4281no—
508C22—4281no—
509C22—4281no—
510C22—4281no—
511C22—4281no—
512C22—4281no—
513C22—4281no—
514C22—2142no—
515C22—2142no—
516C22—2142no—
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
530C22#41×D10711yescharacteristic
531C22#1681215411yescharacteristic
532C22#1681215511yescharacteristic
533C4—22no—
534C4—22no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—4281no—
539C4—4281no—
540C4—2142no—
541C4—2142no—
542C4—2142no—
543C2—42no—
544C2—81no—
545C2—4282no—
546C2—4282no—
547C2—2144no—
548C2#121×D10711yescharacteristic
549C2#2029313511yescharacteristic
550C2#2029321311yescharacteristic
551C2#2029325911yescharacteristic
552C2#2029326111yescharacteristic
553C2#2029327011yescharacteristic
554C2#2029327411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#394×C107C2
hypercenter#394D107
hypocenterC107#2424
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#16812082C2semidirect
2. C2×#16812080C2semidirect
3. C2×#16812079C2semidirect
4. C2×#16812077C2semidirect
5. C2×#16812071C2semidirect
6. C2×C22⋊C4×D107C2semidirect
7. #20292549C2semidirect
8. #20292542C2semidirect
9. #20292536C2semidirect
10. #20292527C2semidirect
11. #20292521C2semidirect
12. #20292505C2semidirect
13. #20292503C2semidirect
14. #20292501C2semidirect
15. #394×C107C2semidirect
22. C22×C22⋊D107C22semidirect
25. C22×C4×D107C22semidirect
31. C2×C4⋊Dic107C22semidirect
32. C2×Dic107⋊C4C22semidirect
33. C2×Dic107⋊C4C22semidirect
36. C2×C22⋊Dic107C22semidirect
43. C2×(C2×D107)⋊C4C22semidirect
44. C2×(C2×D107)⋊C4C22semidirect
46. C2×C4×Dic107C22semidirect
47. C107⋊#95C22semidirect
48. C107⋊#95C22semidirect
49. C107⋊#95C22semidirect
51. C2×C4⋊C4×C107C22semidirect
54. C2×C22⋊C4×C107C22semidirect
55. C2×C22⋊C4×C107C22semidirect
56. #95×C107C22semidirect
72. C2×D428D4semidirect
104. C22⋊Dic107D4semidirect
112. (C2×D107)⋊C4D4semidirect
254. C2×D107#121semidirect
278. C107#2424semidirect
287. #394D107semidirect
313. C2×C4⋊C4C2×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×#168120822
C2×#168120802
C2×#168120792
C2×#168120772
C2×#168120712
C2×C22⋊C4×D1072
#202925492
#202925422
#202925362
#202925272
#202925212
#202925052
#202925032
#202925012
#394×C1072
#2424107