Finite Group Databasebeta

← #14535112

Subgroups — #14535112

Order 2176.

802 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×#10587615C211yesmaximal, characteristic, complemented
2C2×#10587623C211yesmaximal, characteristic, complemented
3C2×#10587623C211yesmaximal, characteristic, complemented
4#11762737C211yesmaximal, characteristic, complemented
5#11762731C211yesmaximal, characteristic, complemented
6#11762725C211yesmaximal, characteristic, complemented
7#11762723C211yesmaximal, characteristic, complemented
8#11762717C211yesmaximal, characteristic, complemented
9#11762717C211yesmaximal, characteristic, complemented
10#11762712C211yesmaximal, characteristic, complemented
11#11762709C211yesmaximal, characteristic, complemented
12#11762699C211yesmaximal, characteristic, complemented
13#11762688C211yesmaximal, characteristic, complemented
14#11762664C211yesmaximal, characteristic, complemented
15C2×C17×#121C211yesmaximal, characteristic, complemented
16C24×D17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C22×D4×C17C2211yescharacteristic
19C22×D4×C17C2211yescharacteristic, complemented
20C23×C4×C17C2211yescharacteristic, complemented
21C22×C22⋊D17C2211yescharacteristic
22C22×C22⋊D17C2211yescharacteristic
23C22×C22⋊D17C2211yescharacteristic
24C22×C22⋊D17C2211yescharacteristic
25C17×#121—24no—
26C2×C4⋊C4×C17C2211yescharacteristic, complemented
27C2×C22⋊C4×C17C2211yescharacteristic
28C2×C22⋊C4×C17C2211yescharacteristic, complemented
29C2×C22⋊Dic17C2211yescharacteristic
30C2×C22⋊Dic17C2211yescharacteristic
31C2×C22⋊Dic17C2211yescharacteristic
32C2×C22⋊Dic17C2211yescharacteristic, complemented
33C2×C22⋊Dic17C2211yescharacteristic, complemented
34C2×C22⋊Dic17C2211yescharacteristic, complemented
35C2×C22⋊Dic17C2211yescharacteristic, complemented
36#10587623—24no—
37#10587623—24no—
38#10587615—24no—
39C2×#63227C2211yescharacteristic
40C2×#63227C2211yescharacteristic
41C2×#63227C2211yescharacteristic
42C2×#63227C2211yescharacteristic
43C2×#63227C2211yescharacteristic
44C2×#63227C2211yescharacteristic
45C2×#63227C2211yescharacteristic
46C2×#63227C2211yescharacteristic
47C2×C4⋊Dic17C2211yescharacteristic, complemented
48C2×#63225C2211yescharacteristic, complemented
49C2×#63225C2211yescharacteristic, complemented
50C2×C4×Dic17C2211yescharacteristic, complemented
51#10587502C2211yescharacteristic, complemented
52#10587502C2211yescharacteristic, complemented
53#10587502C2211yescharacteristic, complemented
54#10587502C2211yescharacteristic, complemented
55C24×C17C2311yescharacteristic
56C24×C17C2311yescharacteristic
57C24×C17C2311yescharacteristic
58C23×D17—22no—
59C23×D17—22no—
60C23×D17—22no—
61C23×D17—41no—
62C23×D17—41no—
63C23×D17—41no—
64C23×D17—41no—
65C23×D17C2311yescharacteristic
66C23×D17C2311yescharacteristic
67C2×D4×C17—22no—
68C2×D4×C17—22no—
69C2×D4×C17—22no—
70C2×D4×C17—41no—
71C2×D4×C17—41no—
72C2×D4×C17—41no—
73C2×D4×C17—41no—
74C2×D4×C17—41no—
75C2×D4×C17D414yescomplemented
76C22×C4×C17—22no—
77C22×C4×C17—22no—
78C22×C4×C17—41no—
79C22×C4×C17C2311yescharacteristic
80C22×C4×C17C2311yescharacteristic
81C22×C4×C17C2311yescharacteristic
82C22×C4×C17C2311yescharacteristic
83C2×C22⋊D17—22no—
84C2×C22⋊D17—22no—
85C2×C22⋊D17—22no—
86C2×C22⋊D17—22no—
87C2×C22⋊D17—22no—
88C2×C22⋊D17—22no—
89C2×C22⋊D17—22no—
90C2×C22⋊D17—41no—
91C2×C22⋊D17—41no—
92C2×C22⋊D17—41no—
93C2×C22⋊D17—41no—
94C2×C22⋊D17—41no—
95C22×Dic17C2311yescharacteristic
96C22×Dic17C2311yescharacteristic
97C22×Dic17C2311yescharacteristic
98C22×Dic17C2311yescharacteristic
99C22×Dic17C2311yescharacteristic
100C22×Dic17C2311yescharacteristic
101C4⋊C4×C17—22no—
102C22⋊C4×C17—22no—
103C22⋊C4×C17—22no—
104C22⋊Dic17—22no—
105C22⋊Dic17—22no—
106C22⋊Dic17—22no—
107C22⋊Dic17—41no—
108C22⋊Dic17—41no—
109C22⋊Dic17—41no—
110C22⋊Dic17—41no—
111#63227—22no—
112#63227—22no—
113#63227—22no—
114#63227—22no—
115#63227—41no—
116#63227—41no—
117#63227—41no—
118#63227D414yescomplemented
119C4⋊Dic17—41no—
120#63225—22no—
121#63225—22no—
122C4×Dic17—41no—
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×C17C2×D412yes—
143C23×C17C2×D412yes—
144C23×C17C2411yescharacteristic
145C22×D17—21no—
146C22×D17—21no—
147C22×D17—21no—
148C22×D17—21no—
149C22×D17—21no—
150C22×D17—21no—
151C22×D17—21no—
152C22×D17—21no—
153C22×D17—21no—
154C22×D17—21no—
155C22×D17—21no—
156C22×D17—21no—
157C22×D17—44no—
158C22×D17—44no—
159C22×D17—44no—
160C22×D17—44no—
161C22×D17—44no—
162C22×D17—44no—
163C22×D17—44no—
164C22×D17C2×D412yes—
165C22×D17#4112yes—
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×C17—21no—
177C2×C4×C17—21no—
178C2×C4×C17—21no—
179C2×C4×C17—44no—
180C2×C4×C17C2×D412yes—
181C2×C4×C17C2×D412yes—
182C22⋊D17—44no—
183C22⋊D17—44no—
184C22⋊D17—44no—
185C22⋊D17—44no—
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×Dic17—21no—
202C2×Dic17—21no—
203C2×Dic17C2×D412yes—
204#2678—171nomaximal, Sylow
205C22×C17—22no—
206C22×C17—22no—
207C22×C17—22no—
208C22×C17—22no—
209C22×C17—22no—
210C22×C17—22no—
211C22×C17—41no—
212C22×C17—41no—
213C22×C17—41no—
214C22×C17—41no—
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×C17C22×D411yescharacteristic
227C22×C17C22×D411yescharacteristic
228C22×C17C2×#4111yescharacteristic
229C22×C17H(5,𝔽2)11yescharacteristic
230C22×C17H(5,𝔽2)11yescharacteristic
231C22×C17H(5,𝔽2)11yescharacteristic
232C22×C17H(5,𝔽2)11yescharacteristic
233C2×D17—22no—
234C2×D17—22no—
235C2×D17—22no—
236C2×D17—22no—
237C2×D17—22no—
238C2×D17—22no—
239C2×D17—416no—
240C2×D17—41no—
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—
251C4×C17—41no—
252Dic17—22no—
253Dic17—41no—
254Dic17—41no—
255Dic17—41no—
256Dic17—41no—
257Dic17—41no—
258C2×#123—171no—
259C2×#121D1711yescharacteristic, complemented
260C2×#120—171no—
261C2×#120—171no—
262#397—171no—
263#396—171no—
264#394—171no—
265#394—171no—
266#392—171no—
267#392—171no—
268#390—171no—
269#390—171no—
270#388—171no—
271#386—171no—
272#379—171no—
273C2×C17—42no—
274C2×C17—42no—
275C2×C17—81no—
276C2×C17#53411yescharacteristic
277C2×C17#53511yescharacteristic
278C2×C17#53811yescharacteristic
279C2×C17D4211yescharacteristic
280C2×C17#54611yescharacteristic
281C2×C17#54611yescharacteristic
282C2×C17#56011yescharacteristic
283D17—42no—
284D17—42no—
285C25—171no—
286C22×D4—171no—
287C22×D4—171no—
288C22×D4—171no—
289C22×D4—171no—
290C22×D4C2×D1711yescharacteristic
291C22×D4C2×D1711yescharacteristic
292C22×D4C2×D1711yescharacteristic, complemented
293C23×C4C2×D1711yescharacteristic, complemented
294#123—344no—
295#121—24no—
296#120—344no—
297#120—344no—
298C2×C4⋊C4—171no—
299C2×C4⋊C4—171no—
300C2×C4⋊C4—171no—
301C2×C4⋊C4C2×D1711yescharacteristic, complemented
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⋊C4—171no—
316C2×C22⋊C4—171no—
317C2×C22⋊C4C2×D1711yescharacteristic
318C2×C22⋊C4C2×D1711yescharacteristic, complemented
319C2×C42—171no—
320#95—171no—
321#95—171no—
322#95—171no—
323#95—171no—
324C17#267811yescharacteristic, complemented, Sylow
325C24—171no—
326C24—171no—
327C24—342no—
328C24—342no—
329C24—342no—
330C24—681no—
331C24—681no—
332C24—681no—
333C24—681no—
334C24C22×D1711yescharacteristic
335C24C22×D1711yescharacteristic
336C24C22×D1711yescharacteristic
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—342no—
346C2×D4—342no—
347C2×D4—342no—
348C2×D4—342no—
349C2×D4—342no—
350C2×D4—342no—
351C2×D4—342no—
352C2×D4—681no—
353C2×D4—681no—
354C2×D4—681no—
355C2×D4—681no—
356C2×D4—681no—
357C2×D4C22⋊D1714yescomplemented
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—41no—
361C22×C4—171no—
362C22×C4—171no—
363C22×C4—171no—
364C22×C4—171no—
365C22×C4—171no—
366C22×C4—171no—
367C22×C4C22×D1711yescharacteristic
368C22×C4C22×D1711yescharacteristic
369C22×C4C22×D1711yescharacteristic
370C22×C4C22×D1711yescharacteristic
371C4⋊C4—22no—
372C4⋊C4—342no—
373C4⋊C4—342no—
374C4⋊C4—681no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—174no—
378C22⋊C4—342no—
379C22⋊C4—342no—
380C22⋊C4—342no—
381C22⋊C4—342no—
382C22⋊C4—342no—
383C22⋊C4—342no—
384C22⋊C4—342no—
385C22⋊C4—681no—
386C22⋊C4—681no—
387C22⋊C4—681no—
388C22⋊C4—681no—
389C22⋊C4—681no—
390C22⋊C4—681no—
391C22⋊C4—681no—
392C42—681no—
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—172no—
413C23—172no—
414C23—341no—
415C23—341no—
416C23—341no—
417C23—341no—
418C23—341no—
419C23—341no—
420C23—341no—
421C23—341no—
422C23—341no—
423C23—341no—
424C23—341no—
425C23—341no—
426C23—684no—
427C23—684no—
428C23—684no—
429C23—684no—
430C23—684no—
431C23—684no—
432C23—684no—
433C23C2×C22⋊D1712yes—
434C23C2×C22⋊D1712yes—
435C23C23×D1711yescharacteristic
436D4—44no—
437D4—44no—
438D4—44no—
439D4—684no—
440D4—684no—
441D4—684no—
442D4—684no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—44no—
454C2×C4—172no—
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×C4D4×D1712yes—
473C2×C4C2×C22⋊D1712yes—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—342no—
496C22—342no—
497C22—342no—
498C22—342no—
499C22—342no—
500C22—342no—
501C22—6816no—
502C22—681no—
503C22—681no—
504C22—681no—
505C22—681no—
506C22—681no—
507C22—681no—
508C22—681no—
509C22—681no—
510C22C2×D4×D1711yescharacteristic
511C22#1058769111yescharacteristic
512C22#1058769111yescharacteristic
513C22#1058769111yescharacteristic
514C22#41×D1711yescharacteristic
515C22#1058769611yescharacteristic
516C22C22×C22⋊D1711yescharacteristic
517C4—22no—
518C4—22no—
519C4—41no—
520C4—41no—
521C4—342no—
522C4—681no—
523C4—681no—
524C4—681no—
525C4—681no—
526C4—681no—
527C2—42no—
528C2—42no—
529C2—81no—
530C2—682no—
531C2—682no—
532C2#1176336811yescharacteristic
533C2#1176337511yescharacteristic
534C2#1176339111yescharacteristic
535C2D4×C22⋊D1711yescharacteristic
536C2#1176344711yescharacteristic
537C2#1176345611yescharacteristic
538C2#1176354711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#121C2
hypercenterC2×#121D17
hypocenterC17#2678
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#10587615C2semidirect
2. C2×#10587623C2semidirect
3. C2×#10587623C2semidirect
4. #11762737C2semidirect
5. #11762731C2semidirect
6. #11762725C2semidirect
7. #11762723C2semidirect
8. #11762717C2semidirect
9. #11762717C2semidirect
10. #11762712C2semidirect
11. #11762709C2semidirect
12. #11762699C2semidirect
13. #11762688C2semidirect
14. #11762664C2semidirect
15. C2×C17×#121C2semidirect
19. C22×D4×C17C22semidirect
20. C23×C4×C17C22semidirect
26. C2×C4⋊C4×C17C22semidirect
28. C2×C22⋊C4×C17C22semidirect
32. C2×C22⋊Dic17C22semidirect
33. C2×C22⋊Dic17C22semidirect
34. C2×C22⋊Dic17C22semidirect
35. C2×C22⋊Dic17C22semidirect
47. C2×C4⋊Dic17C22semidirect
48. C2×#63225C22semidirect
49. C2×#63225C22semidirect
50. C2×C4×Dic17C22semidirect
51. #10587502C22semidirect
52. #10587502C22semidirect
53. #10587502C22semidirect
54. #10587502C22semidirect
75. C2×D4×C17D4semidirect
118. #63227D4semidirect
259. C2×#121D17semidirect
292. C22×D4C2×D17semidirect
293. C23×C4C2×D17semidirect
301. C2×C4⋊C4C2×D17semidirect
318. C2×C22⋊C4C2×D17semidirect
324. C17#2678semidirect
357. C2×D4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876152
C2×#105876232
#117627372
#117627312
#117627252
#117627232
#117627172
#117627122
#117627092
#117626992
#117626882
#117626642
C2×C17×#1212
#267817