Finite Group Databasebeta

← #28339997

Subgroups — #28339997

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×#16812083C211yesmaximal, characteristic, complemented
2C2×#16812082C211yesmaximal, characteristic, complemented
3C2×#16812078C211yesmaximal, characteristic, complemented
4C2×#16812078C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊D107C211yesmaximal, characteristic, complemented
6#20292550C211yesmaximal, characteristic, complemented
7#20292542C211yesmaximal, characteristic, complemented
8#20292542C211yesmaximal, characteristic, complemented
9#20292536C211yesmaximal, characteristic, complemented
10#20292527C211yesmaximal, characteristic, complemented
11#20292520C211yesmaximal, characteristic, complemented
12#20292506C211yesmaximal, characteristic, complemented
13#20292503C211yesmaximal, characteristic, complemented
14#20292500C211yesmaximal, characteristic, complemented
15C22×C22⋊C4×C107C211yesmaximal, characteristic, complemented
16C22×D428C2211yescharacteristic, complemented
17C22×C22⋊D107C2211yescharacteristic
18C22×C22⋊D107C2211yescharacteristic
19C22×C22⋊D107C2211yescharacteristic
20C22×C22⋊D107C2211yescharacteristic
21C22×C22⋊D107C2211yescharacteristic
22C22×C22⋊D107C2211yescharacteristic
23C22×C4×D107C2211yescharacteristic, complemented
24#16812083—24no—
25#16812082—24no—
26#16812078—24no—
27#16812078—24no—
28C4×C22⋊D107—24no—
29C2×C4⋊Dic107C2211yescharacteristic, complemented
30C2×Dic107⋊C4C2211yescharacteristic, complemented
31C2×Dic107⋊C4C2211yescharacteristic, complemented
32C2×Dic107⋊C4C2211yescharacteristic, complemented
33C2×C22⋊Dic107C2211yescharacteristic
34C2×C22⋊Dic107C2211yescharacteristic
35C2×C22⋊Dic107C2211yescharacteristic
36C2×C22⋊Dic107C2211yescharacteristic
37C2×C22⋊Dic107C2211yescharacteristic
38C2×C22⋊Dic107C2211yescharacteristic
39C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D107)⋊C4C2211yescharacteristic, 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
50C23×C4×C107C2211yescharacteristic
51C23×C4×C107C2211yescharacteristic
52C2×C22⋊C4×C107—24no—
53C2×C22⋊C4×C107C2211yescharacteristic
54C2×C22⋊C4×C107C2211yescharacteristic
55C2×C22⋊C4×C107C2211yescharacteristic
56C2×C22⋊C4×C107C2211yescharacteristic
57C23×D107C2311yescharacteristic
58C23×D107C2311yescharacteristic
59C2×D428—22no—
60C2×D428—22no—
61C2×D428—41no—
62C2×C22⋊D107—22no—
63C2×C22⋊D107—22no—
64C2×C22⋊D107—22no—
65C2×C22⋊D107—22no—
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⋊D107—41no—
78C2×C22⋊D107—41no—
79C2×C22⋊D107D414yescomplemented
80C22×Dic107C2311yescharacteristic
81C22×Dic107C2311yescharacteristic
82C22×Dic107C2311yescharacteristic
83C22×Dic107C2311yescharacteristic
84C22×Dic107C2311yescharacteristic
85C22×Dic107C2311yescharacteristic
86C2×C4×D107—22no—
87C2×C4×D107—22no—
88C2×C4×D107—41no—
89C4⋊Dic107—22no—
90Dic107⋊C4—22no—
91Dic107⋊C4—22no—
92Dic107⋊C4—41no—
93C22⋊Dic107—22no—
94C22⋊Dic107—22no—
95C22⋊Dic107—22no—
96C22⋊Dic107—22no—
97C22⋊Dic107—22no—
98C22⋊Dic107D414yescomplemented
99(C2×D107)⋊C4—22no—
100(C2×D107)⋊C4—22no—
101(C2×D107)⋊C4—22no—
102(C2×D107)⋊C4—22no—
103(C2×D107)⋊C4—41no—
104C4×Dic107—22no—
105C24×C107—22no—
106C24×C107—22no—
107C24×C107—22no—
108C24×C107—22no—
109C24×C107—22no—
110C24×C107—41no—
111C24×C107—41no—
112C24×C107C2311yescharacteristic
113C24×C107C2311yescharacteristic
114C24×C107C2311yescharacteristic
115C22×C4×C107—22no—
116C22×C4×C107—22no—
117C22×C4×C107—22no—
118C22×C4×C107—22no—
119C22×C4×C107—41no—
120C22×C4×C107—41no—
121C22×C4×C107C2311yescharacteristic
122C22×C4×C107C2311yescharacteristic
123C22×C4×C107C2311yescharacteristic
124C22×C4×C107C2311yescharacteristic
125C22⋊C4×C107—22no—
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—44no—
147C23×C107—44no—
148C23×C107—44no—
149C23×C107—44no—
150C23×C107—44no—
151C23×C107—44no—
152C23×C107C2×D412yes—
153C23×C107C2×D412yes—
154C23×C107C2×D412yes—
155C23×C107#4112yes—
156C23×C107#4112yes—
157C23×C107C2411yescharacteristic
158C22×D107—21no—
159C22×D107—21no—
160C22×D107—21no—
161C22×D107—21no—
162C22×D107—21no—
163C22×D107—21no—
164C22×D107—21no—
165C22×D107—21no—
166C22×D107—21no—
167C22×D107—21no—
168C22×D107—21no—
169C22×D107—21no—
170C22×D107—21no—
171C22×D107C2×D412yes—
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—44no—
183C2×C4×C107—44no—
184C2×C4×C107C2×D412yes—
185C2×C4×C107C2×D412yes—
186C22⋊D107—28no—
187C22⋊D107—44no—
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×Dic107—21no—
205C2×Dic107—21no—
206C2×Dic107—21no—
207C2×Dic107C2×D412yes—
208C2×Dic107C2×D412yes—
209C2×Dic107C2×D412yes—
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—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#12114yescomplemented
234C22×C107C22×D411yescharacteristic
235C22×C107C22×D411yescharacteristic
236C22×C107C22×D411yescharacteristic
237C22×C107C2×#4111yescharacteristic
238C22×C107C2×#4111yescharacteristic
239C22×C107H(5,𝔽2)11yescharacteristic
240C22×C107H(5,𝔽2)11yescharacteristic
241C2×D107—22no—
242C2×D107—22no—
243C2×D107—22no—
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—41no—
258C4×C107—41no—
259Dic107—22no—
260Dic107—22no—
261Dic107—22no—
262Dic107—41no—
263Dic107—41no—
264Dic107—41no—
265C2×C107—24no—
266C2×C107—42no—
267C2×C107—42no—
268C2×C107C2×#12111yescharacteristic
269C2×C107#52511yescharacteristic
270C2×C107#53811yescharacteristic
271C2×C107D4211yescharacteristic
272C2×C107D4211yescharacteristic
273C2×C107#54611yescharacteristic
274C2×C107#54611yescharacteristic
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⋊C4D10711yescharacteristic, complemented
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×D4—1071no—
300C22×D4—1071no—
301C22×D4—1071no—
302C23×C4—1071no—
303C23×C4C2×D10711yescharacteristic
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⋊C4—1071no—
313C2×C4⋊C4—1071no—
314C2×C22⋊C4—24no—
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⋊C4C2×D10711yescharacteristic
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—
343C24—1071no—
344C24C22×D10711yescharacteristic
345C24C22×D10711yescharacteristic
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—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—1071no—
375C22×C4—1071no—
376C22×C4—1071no—
377C22×C4—1071no—
378C22×C4—1071no—
379C22×C4—1071no—
380C22×C4—4281no—
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—22no—
394C22⋊C4—1074no—
395C22⋊C4—4281no—
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—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—1072no—
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—
444C23D4×D10712yes—
445C23D4×D10712yes—
446C23C2×C22⋊D10712yes—
447C23#1313381812yes—
448C23#1313381812yes—
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—44no—
468C2×C4—44no—
469C2×C4—1072no—
470C2×C4—1072no—
471C2×C4—1072no—
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×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—4281no—
512C22—4281no—
513C22—4281no—
514C22—4281no—
515C22—4281no—
516C22—4281no—
517C22—4281no—
518C22—4281no—
519C22—4281no—
520C22—4281no—
521C22—4281no—
522C22—2142no—
523C22—2142no—
524C22—2142no—
525C22#1681208214yescomplemented
526C22C2×D4×D10711yescharacteristic
527C22C2×D4×D10711yescharacteristic
528C22C22×C22⋊D10711yescharacteristic
529C22C2×#1313381811yescharacteristic
530C22C2×#1313381811yescharacteristic
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—24no—
544C2—42no—
545C2—42no—
546C2—8561no—
547C2—4282no—
548C2C2×#1681208211yescharacteristic
549C2#2029313811yescharacteristic
550C2#2029321511yescharacteristic
551C2D4×C22⋊D10711yescharacteristic
552C2D4×C22⋊D10711yescharacteristic
553C2#2029326711yescharacteristic
554C2#2029326711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC22×C22⋊C4×C107C2
hypercenterC22×C22⋊C4D107
hypocenterC107#2424
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16812083C2semidirect
2. C2×#16812082C2semidirect
3. C2×#16812078C2semidirect
4. C2×#16812078C2semidirect
5. C2×C4×C22⋊D107C2semidirect
6. #20292550C2semidirect
7. #20292542C2semidirect
8. #20292542C2semidirect
9. #20292536C2semidirect
10. #20292527C2semidirect
11. #20292520C2semidirect
12. #20292506C2semidirect
13. #20292503C2semidirect
14. #20292500C2semidirect
15. C22×C22⋊C4×C107C2semidirect
16. C22×D428C22semidirect
23. C22×C4×D107C22semidirect
29. C2×C4⋊Dic107C22semidirect
30. C2×Dic107⋊C4C22semidirect
31. C2×Dic107⋊C4C22semidirect
32. C2×Dic107⋊C4C22semidirect
39. C2×(C2×D107)⋊C4C22semidirect
40. C2×(C2×D107)⋊C4C22semidirect
41. C2×(C2×D107)⋊C4C22semidirect
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
79. C2×C22⋊D107D4semidirect
98. C22⋊Dic107D4semidirect
128. C22⋊C4×C107D4semidirect
233. C22×C107#121semidirect
278. C107#2424semidirect
284. C22×C22⋊C4D107semidirect
405. C22⋊C4C22⋊D107semidirect
525. C22#16812082semidirect

Maximal subgroups

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

subgroupindex
C2×#168120832
C2×#168120822
C2×#168120782
C2×C4×C22⋊D1072
#202925502
#202925422
#202925362
#202925272
#202925202
#202925062
#202925032
#202925002
C22×C22⋊C4×C1072
#2424107