Finite Group Databasebeta

← #14535006

Subgroups — #14535006

Order 2176.

796 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587622C211yesmaximal, characteristic, complemented
2C2×#10587571C211yesmaximal, characteristic, complemented
3C2×#10587639C211yesmaximal, characteristic, complemented
4#11762725C211yesmaximal, characteristic, complemented
5#11762723C211yesmaximal, characteristic, complemented
6#11762721C211yesmaximal, characteristic, complemented
7#11762717C211yesmaximal, characteristic, complemented
8#11762717C211yesmaximal, characteristic, complemented
9#11762717C211yesmaximal, characteristic, complemented
10#11762712C211yesmaximal, characteristic, complemented
11#11762709C211yesmaximal, characteristic, complemented
12#11762702C211yesmaximal, characteristic, complemented
13#11762697C211yesmaximal, characteristic, complemented
14#11762665C211yesmaximal, characteristic, complemented
15C2×C17×#120C211yesmaximal, characteristic, complemented
16C25×C17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C22×D4×C17C2211yescharacteristic
19C22×D4×C17C2211yescharacteristic
20C22×C22⋊D17C2211yescharacteristic
21C22×C22⋊D17C2211yescharacteristic
22C22×C22⋊D17C2211yescharacteristic
23C22×C22⋊D17C2211yescharacteristic, complemented
24C22×Dic34C2211yescharacteristic, complemented
25C17×#120—24no—
26C2×C22⋊C4×C17C2211yescharacteristic
27C2×C22⋊C4×C17C2211yescharacteristic
28C2×C22⋊C4×C17C2211yescharacteristic
29#10587639—24no—
30C2×C22⋊Dic17C2211yescharacteristic
31C2×C22⋊Dic17C2211yescharacteristic
32C2×C22⋊Dic17C2211yescharacteristic
33C2×C22⋊Dic17C2211yescharacteristic
34C2×C22⋊Dic17C2211yescharacteristic
35C2×C22⋊Dic17C2211yescharacteristic
36C2×C22⋊Dic17C2211yescharacteristic
37C2×C22⋊Dic17C2211yescharacteristic
38C2×C22⋊Dic17C2211yescharacteristic
39C2×C22⋊Dic17C2211yescharacteristic, complemented
40C2×C22⋊Dic17C2211yescharacteristic, complemented
41C2×C22⋊Dic17C2211yescharacteristic, complemented
42#10587622—24no—
43C2×#63227C2211yescharacteristic, complemented
44C2×#63227C2211yescharacteristic, complemented
45C2×#63227C2211yescharacteristic, complemented
46C2×C4⋊Dic17C2211yescharacteristic, complemented
47C2×#63225C2211yescharacteristic, complemented
48C2×C4×Dic17C2211yescharacteristic, complemented
49C2×C4×Dic17C2211yescharacteristic, complemented
50#10587571—24no—
51#10587502C2211yescharacteristic, complemented
52#10587502C2211yescharacteristic, complemented
53#10587502C2211yescharacteristic, complemented
54#10587502C2211yescharacteristic, complemented
55C24×C17—22no—
56C24×C17—22no—
57C24×C17—22no—
58C24×C17—41no—
59C24×C17—41no—
60C24×C17—41no—
61C24×C17—41no—
62C24×C17C2311yescharacteristic
63C24×C17C2311yescharacteristic
64C24×C17C2311yescharacteristic
65C24×C17C2311yescharacteristic
66C23×D17C2311yescharacteristic
67C2×D4×C17—22no—
68C2×D4×C17—22no—
69C2×D4×C17—22no—
70C2×D4×C17—22no—
71C2×D4×C17—22no—
72C2×D4×C17—41no—
73C2×D4×C17—41no—
74C2×D4×C17—41no—
75C2×D4×C17—41no—
76C22×C4×C17C2311yescharacteristic
77C22×C4×C17C2311yescharacteristic
78C22×C4×C17C2311yescharacteristic
79C2×C22⋊D17—22no—
80C2×C22⋊D17—22no—
81C2×C22⋊D17—22no—
82C2×C22⋊D17—22no—
83C2×C22⋊D17—22no—
84C2×C22⋊D17—41no—
85C2×C22⋊D17—41no—
86C2×C22⋊D17—41no—
87C2×C22⋊D17—41no—
88C2×C22⋊D17—41no—
89C2×C22⋊D17—41no—
90C2×C22⋊D17—41no—
91C22×Dic17C2311yescharacteristic
92C22×Dic17C2311yescharacteristic
93C22×Dic17C2311yescharacteristic
94C22×Dic17C2311yescharacteristic
95C22×Dic17C2311yescharacteristic
96C22×Dic17C2311yescharacteristic
97C22×Dic17C2311yescharacteristic
98C2×Dic34—22no—
99C2×Dic34—22no—
100C2×Dic34—41no—
101C22⋊C4×C17—22no—
102C22⋊C4×C17—22no—
103C22⋊C4×C17D414yescomplemented
104C22⋊Dic17—22no—
105C22⋊Dic17—22no—
106C22⋊Dic17—22no—
107C22⋊Dic17—22no—
108C22⋊Dic17—22no—
109C22⋊Dic17—22no—
110C22⋊Dic17—41no—
111C22⋊Dic17—41no—
112C22⋊Dic17—41no—
113C22⋊Dic17—41no—
114C22⋊Dic17—41no—
115C22⋊Dic17D414yescomplemented
116#63227—22no—
117#63227—22no—
118#63227—41no—
119C4⋊Dic17—41no—
120#63225—41no—
121C4×Dic17—22no—
122C4×Dic17—22no—
123C23×C17—21no—
124C23×C17—21no—
125C23×C17—21no—
126C23×C17—21no—
127C23×C17—21no—
128C23×C17—21no—
129C23×C17—21no—
130C23×C17—21no—
131C23×C17—21no—
132C23×C17—21no—
133C23×C17—21no—
134C23×C17—21no—
135C23×C17—21no—
136C23×C17—21no—
137C23×C17—21no—
138C23×C17—21no—
139C23×C17—21no—
140C23×C17—21no—
141C23×C17—21no—
142C23×C17—21no—
143C23×C17—21no—
144C23×C17—21no—
145C23×C17—21no—
146C23×C17—21no—
147C23×C17—21no—
148C23×C17—44no—
149C23×C17—44no—
150C23×C17—44no—
151C23×C17—44no—
152C23×C17—44no—
153C23×C17—44no—
154C23×C17—44no—
155C23×C17C2×D412yes—
156C23×C17C2×D412yes—
157C23×C17#4112yes—
158C23×C17C2411yescharacteristic
159C22×D17—21no—
160C22×D17—21no—
161C22×D17—21no—
162C22×D17—21no—
163C22×D17—21no—
164C22×D17—21no—
165C22×D17—21no—
166D4×C17—44no—
167D4×C17—44no—
168D4×C17—44no—
169C2×C4×C17—21no—
170C2×C4×C17—21no—
171C2×C4×C17—21no—
172C2×C4×C17—21no—
173C2×C4×C17—21no—
174C2×C4×C17—21no—
175C2×C4×C17—21no—
176C2×C4×C17C2×D412yes—
177C2×C4×C17C2×D412yes—
178C22⋊D17—44no—
179C22⋊D17—44no—
180C22⋊D17—44no—
181C22⋊D17—44no—
182C2×Dic17—21no—
183C2×Dic17—21no—
184C2×Dic17—21no—
185C2×Dic17—21no—
186C2×Dic17—21no—
187C2×Dic17—21no—
188C2×Dic17—21no—
189C2×Dic17—21no—
190C2×Dic17—21no—
191C2×Dic17—21no—
192C2×Dic17—21no—
193C2×Dic17—21no—
194C2×Dic17—21no—
195C2×Dic17—21no—
196C2×Dic17—21no—
197C2×Dic17—21no—
198C2×Dic17—21no—
199C2×Dic17—21no—
200C2×Dic17—21no—
201C2×Dic17C2×D412yes—
202C2×Dic17C2×D412yes—
203Dic34—44no—
204#2670—171nomaximal, Sylow
205C22×C17—22no—
206C22×C17—22no—
207C22×C17—22no—
208C22×C17—22no—
209C22×C17—22no—
210C22×C17—22no—
211C22×C17—22no—
212C22×C17—22no—
213C22×C17—22no—
214C22×C17—416no—
215C22×C17—41no—
216C22×C17—41no—
217C22×C17—41no—
218C22×C17—41no—
219C22×C17—41no—
220C22×C17—41no—
221C22×C17—41no—
222C22×C17—41no—
223C22×C17—41no—
224C22×C17—41no—
225C22×C17—41no—
226C22×C17—41no—
227C22×C17—41no—
228C22×C17—41no—
229C22×C17—41no—
230C22×C17—41no—
231C22×C17—41no—
232C22×C17—41no—
233C22×C17—41no—
234C22×C17C22×D411yescharacteristic
235C22×C17C22×D411yescharacteristic
236C22×C17C2×#4111yescharacteristic
237C22×C17H(5,𝔽2)11yescharacteristic
238C22×C17H(5,𝔽2)11yescharacteristic
239C22×C17H(5,𝔽2)11yescharacteristic
240C22×C17H(5,𝔽2)11yescharacteristic
241C2×D17—41no—
242C2×D17—41no—
243C2×D17—41no—
244C2×D17—41no—
245C2×D17—41no—
246C2×D17—41no—
247C2×D17—41no—
248C4×C17—22no—
249C4×C17—22no—
250C4×C17—41no—
251Dic17—22no—
252Dic17—22no—
253Dic17—41no—
254Dic17—41no—
255Dic17—41no—
256Dic17—41no—
257Dic17—41no—
258C2×#124—171no—
259C2×#124—171no—
260C2×#120—171no—
261C2×#120D1711yescharacteristic, complemented
262#394—171no—
263#394—171no—
264#393—171no—
265#392—171no—
266#392—171no—
267#392—171no—
268#390—171no—
269#390—171no—
270#388—171no—
271#388—171no—
272#379—171no—
273C2×C17—42no—
274C2×C17—42no—
275C2×C17—42no—
276C2×C17—81no—
277C2×C17#53511yescharacteristic
278C2×C17#53511yescharacteristic
279C2×C17#53811yescharacteristic
280C2×C17D4211yescharacteristic
281C2×C17#54611yescharacteristic
282C2×C17#54611yescharacteristic
283C2×C17H(3,𝔽4)11yescharacteristic
284D17—81no—
285C25C2×D1711yescharacteristic
286C22×Q8—171no—
287C22×D4—171no—
288C22×D4—171no—
289C22×D4—171no—
290C22×D4—171no—
291C22×D4C2×D1711yescharacteristic
292C22×D4C2×D1711yescharacteristic
293C22×D4C2×D1711yescharacteristic
294#124—344no—
295#124—344no—
296#120—24no—
297#120—344no—
298C2×C4⋊C4—171no—
299C2×C4⋊C4—171no—
300C2×C22⋊C4—171no—
301C2×C22⋊C4—171no—
302C2×C22⋊C4—171no—
303C2×C22⋊C4—171no—
304C2×C22⋊C4—171no—
305C2×C22⋊C4—171no—
306C2×C22⋊C4—171no—
307C2×C22⋊C4—171no—
308C2×C22⋊C4—171no—
309C2×C22⋊C4—171no—
310C2×C22⋊C4—171no—
311C2×C22⋊C4—171no—
312C2×C22⋊C4—171no—
313C2×C22⋊C4—171no—
314C2×C22⋊C4—171no—
315C2×C22⋊C4C2×D1711yescharacteristic
316C2×C22⋊C4C2×D1711yescharacteristic
317C2×C22⋊C4C2×D1711yescharacteristic
318C2×C42—171no—
319C2×C42—171no—
320#95—171no—
321#95—171no—
322#95—171no—
323#95—171no—
324C17#267011yescharacteristic, complemented, Sylow
325C24—22no—
326C24—22no—
327C24—22no—
328C24—41no—
329C24—41no—
330C24—41no—
331C24—41no—
332C24—171no—
333C24C22×D1711yescharacteristic
334C24C22×D1711yescharacteristic
335C24C22×D1711yescharacteristic
336C24C22×D1711yescharacteristic
337C2×Q8—342no—
338C2×Q8—342no—
339C2×Q8—681no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—342no—
350C2×D4—342no—
351C2×D4—342no—
352C2×D4—342no—
353C2×D4—342no—
354C2×D4—681no—
355C2×D4—681no—
356C2×D4—681no—
357C2×D4—681no—
358C2×D4—681no—
359C2×D4—681no—
360C2×D4—681no—
361C22×C4—171no—
362C22×C4—171no—
363C22×C4—171no—
364C22×C4—171no—
365C22×C4—171no—
366C22×C4—171no—
367C22×C4—171no—
368C22×C4C22×D1711yescharacteristic
369C22×C4C22×D1711yescharacteristic
370C22×C4C22×D1711yescharacteristic
371C4⋊C4—681no—
372C4⋊C4—681no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—174no—
376C22⋊C4—342no—
377C22⋊C4—342no—
378C22⋊C4—342no—
379C22⋊C4—342no—
380C22⋊C4—342no—
381C22⋊C4—342no—
382C22⋊C4—342no—
383C22⋊C4—342no—
384C22⋊C4—681no—
385C22⋊C4—681no—
386C22⋊C4—681no—
387C22⋊C4—681no—
388C22⋊C4—681no—
389C22⋊C4—681no—
390C22⋊C4C22⋊D1714yescomplemented
391C42—342no—
392C42—342no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—341no—
426C23—341no—
427C23—341no—
428C23—341no—
429C23—341no—
430C23—341no—
431C23—341no—
432C23D4×D1712yes—
433C23#6325412yes—
434C23C2×C22⋊D1712yes—
435C23C23×D1711yescharacteristic
436Q8—684no—
437D4—44no—
438D4—44no—
439D4—44no—
440D4—684no—
441D4—684no—
442D4—684no—
443D4—684no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—172no—
452C2×C4—172no—
453C2×C4—341no—
454C2×C4—341no—
455C2×C4—341no—
456C2×C4—341no—
457C2×C4—341no—
458C2×C4—341no—
459C2×C4—341no—
460C2×C4—341no—
461C2×C4—341no—
462C2×C4—341no—
463C2×C4—341no—
464C2×C4—341no—
465C2×C4—341no—
466C2×C4—341no—
467C2×C4—341no—
468C2×C4—341no—
469C2×C4—341no—
470C2×C4—341no—
471C2×C4—341no—
472C2×C4C2×C22⋊D1712yes—
473C2×C4C2×C22⋊D1712yes—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—416no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—681no—
504C22—681no—
505C22—681no—
506C22—681no—
507C22—681no—
508C22—681no—
509C22—681no—
510C22C2×D4×D1711yescharacteristic
511C22C2×#6325411yescharacteristic
512C22#1058769111yescharacteristic
513C22#1058769111yescharacteristic
514C22#1058769111yescharacteristic
515C22#1058769111yescharacteristic
516C22C22×C22⋊D1711yescharacteristic
517C4—22no—
518C4—22no—
519C4—41no—
520C4—342no—
521C4—342no—
522C4—681no—
523C4—681no—
524C4—681no—
525C4—681no—
526C4—681no—
527C2—42no—
528C2—42no—
529C2—42no—
530C2—81no—
531C2—1361no—
532C2#1176337411yescharacteristic
533C2#1176337611yescharacteristic
534C2#1176339411yescharacteristic
535C2D4×C22⋊D1711yescharacteristic
536C2#1176344611yescharacteristic
537C2#1176345311yescharacteristic
538C2#1176355211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#120C2
hypercenterC2×#120D17
hypocenterC17#2670
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#10587622C2semidirect
2. C2×#10587571C2semidirect
3. C2×#10587639C2semidirect
4. #11762725C2semidirect
5. #11762723C2semidirect
6. #11762721C2semidirect
7. #11762717C2semidirect
8. #11762717C2semidirect
9. #11762717C2semidirect
10. #11762712C2semidirect
11. #11762709C2semidirect
12. #11762702C2semidirect
13. #11762697C2semidirect
14. #11762665C2semidirect
15. C2×C17×#120C2semidirect
23. C22×C22⋊D17C22semidirect
24. C22×Dic34C22semidirect
39. C2×C22⋊Dic17C22semidirect
40. C2×C22⋊Dic17C22semidirect
41. C2×C22⋊Dic17C22semidirect
43. C2×#63227C22semidirect
44. C2×#63227C22semidirect
45. C2×#63227C22semidirect
46. C2×C4⋊Dic17C22semidirect
47. C2×#63225C22semidirect
48. C2×C4×Dic17C22semidirect
49. C2×C4×Dic17C22semidirect
51. #10587502C22semidirect
52. #10587502C22semidirect
53. #10587502C22semidirect
54. #10587502C22semidirect
103. C22⋊C4×C17D4semidirect
115. C22⋊Dic17D4semidirect
261. C2×#120D17semidirect
324. C17#2670semidirect
390. C22⋊C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876222
C2×#105875712
C2×#105876392
#117627252
#117627232
#117627212
#117627172
#117627122
#117627092
#117627022
#117626972
#117626652
C2×C17×#1202
#267017