Finite Group Databasebeta

← #14536780

Subgroups — #14536780

Order 2176.

712 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587568C211yesmaximal, characteristic, complemented
2C2×#10587623C211yesmaximal, characteristic, complemented
3#11762737C211yesmaximal, characteristic, complemented
4#11762736C211yesmaximal, characteristic, complemented
5#11762735C211yesmaximal, characteristic, complemented
6#11762727C211yesmaximal, characteristic, complemented
7#11762726C211yesmaximal, characteristic, complemented
8#11762725C211yesmaximal, characteristic, complemented
9#11762725C211yesmaximal, characteristic, complemented
10#11762724C211yesmaximal, characteristic, complemented
11#11762718C211yesmaximal, characteristic, complemented
12#11762718C211yesmaximal, characteristic, complemented
13#11762717C211yesmaximal, characteristic, complemented
14#11762664C211yesmaximal, characteristic, complemented
15C17×#394C211yesmaximal, characteristic, complemented
16C24×D17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C22×C22⋊D17C2211yescharacteristic
19C22×C22⋊D17C2211yescharacteristic
20C22×C22⋊D17C2211yescharacteristic
21C22×C22⋊D17C2211yescharacteristic, complemented
22C22×D68C2211yescharacteristic
23C22×D68C2211yescharacteristic
24C2×C4⋊C4×C17C2211yescharacteristic, complemented
25C2×C22⋊C4×C17C2211yescharacteristic
26C2×C22⋊C4×C17C2211yescharacteristic
27C2×C22⋊C4×C17C2211yescharacteristic, complemented
28C2×C22⋊C4×C17C2211yescharacteristic, complemented
29C2×C22⋊Dic17C2211yescharacteristic
30C2×C22⋊Dic17C2211yescharacteristic
31C2×C22⋊Dic17C2211yescharacteristic, complemented
32#10587623—24no—
33C2×#63227C2211yescharacteristic
34C2×#63227C2211yescharacteristic
35C2×#63227C2211yescharacteristic
36C2×#63227C2211yescharacteristic
37C2×#63227C2211yescharacteristic
38C2×#63227C2211yescharacteristic
39C2×#63227C2211yescharacteristic
40C2×#63227C2211yescharacteristic
41C2×#63227C2211yescharacteristic, complemented
42C2×#63227C2211yescharacteristic, complemented
43C2×#63227C2211yescharacteristic, complemented
44C2×C4⋊Dic17C2211yescharacteristic, complemented
45C2×#63225C2211yescharacteristic, complemented
46#10587568—24no—
47C17×#95C2211yescharacteristic, complemented
48#10587502C2211yescharacteristic, complemented
49#10587502C2211yescharacteristic, complemented
50#10587502C2211yescharacteristic, complemented
51#10587502C2211yescharacteristic, complemented
52#10587502C2211yescharacteristic, complemented
53C24×C17C2311yescharacteristic
54C24×C17C2311yescharacteristic
55C23×D17—41no—
56C23×D17—41no—
57C23×D17—41no—
58C23×D17—41no—
59C23×D17—41no—
60C23×D17—41no—
61C23×D17D414yescomplemented
62C23×D17C2311yescharacteristic
63C23×D17C2311yescharacteristic
64C23×D17C2311yescharacteristic
65C2×D4×C17—22no—
66C2×D4×C17—41no—
67C2×D4×C17—41no—
68C22×C4×C17C2311yescharacteristic
69C22×C4×C17C2311yescharacteristic
70C22×C4×C17C2311yescharacteristic
71C22×C4×C17C2311yescharacteristic
72C22×C4×C17C2311yescharacteristic
73C2×C22⋊D17—22no—
74C2×C22⋊D17—22no—
75C2×C22⋊D17—22no—
76C2×C22⋊D17—41no—
77C2×C22⋊D17—41no—
78C2×C22⋊D17—41no—
79C2×C22⋊D17—41no—
80C2×C22⋊D17—41no—
81C2×C22⋊D17—41no—
82C2×C22⋊D17—41no—
83C2×C22⋊D17—41no—
84C2×C22⋊D17—41no—
85C22×Dic17C2311yescharacteristic
86C22×Dic17C2311yescharacteristic
87C22×Dic17C2311yescharacteristic
88C22×Dic17C2311yescharacteristic
89C22×Dic17C2311yescharacteristic
90C2×D68—22no—
91C2×D68—22no—
92C2×D68—41no—
93C2×D68—41no—
94C2×D68—41no—
95C2×D68—41no—
96C4⋊C4×C17—41no—
97C22⋊C4×C17—22no—
98C22⋊C4×C17—41no—
99C22⋊C4×C17—41no—
100C22⋊C4×C17—41no—
101C22⋊Dic17—22no—
102C22⋊Dic17—41no—
103C22⋊Dic17—41no—
104#63227—22no—
105#63227—22no—
106#63227—22no—
107#63227—22no—
108#63227—41no—
109#63227—41no—
110#63227—41no—
111#63227—41no—
112#63227—41no—
113#63227—41no—
114#63227—41no—
115C4⋊Dic17—41no—
116#63225—41no—
117C23×C17—21no—
118C23×C17—21no—
119C23×C17—21no—
120C23×C17—21no—
121C23×C17—21no—
122C23×C17—21no—
123C23×C17—21no—
124C23×C17—21no—
125C23×C17—21no—
126C23×C17—21no—
127C23×C17—21no—
128C23×C17—21no—
129C23×C17—21no—
130C23×C17C2×D412yes—
131C23×C17C2411yescharacteristic
132C22×D17—21no—
133C22×D17—21no—
134C22×D17—21no—
135C22×D17—21no—
136C22×D17—21no—
137C22×D17—21no—
138C22×D17—21no—
139C22×D17—21no—
140C22×D17—21no—
141C22×D17—21no—
142C22×D17—21no—
143C22×D17—21no—
144C22×D17—21no—
145C22×D17—21no—
146C22×D17—21no—
147C22×D17—21no—
148C22×D17—21no—
149C22×D17—21no—
150C22×D17—21no—
151C22×D17—44no—
152C22×D17—44no—
153C22×D17—44no—
154C22×D17—44no—
155C22×D17—44no—
156C22×D17—44no—
157C22×D17—44no—
158C22×D17C2×D412yes—
159C22×D17C2×D412yes—
160D4×C17—44no—
161C2×C4×C17—21no—
162C2×C4×C17—21no—
163C2×C4×C17—21no—
164C2×C4×C17—21no—
165C2×C4×C17—21no—
166C2×C4×C17—21no—
167C2×C4×C17—21no—
168C2×C4×C17—21no—
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—
176C22⋊D17—44no—
177C22⋊D17—44no—
178C22⋊D17—44no—
179C22⋊D17—44no—
180C2×Dic17—21no—
181C2×Dic17—21no—
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—
195D68—44no—
196D68—44no—
197#2812—171nomaximal, Sylow
198C22×C17—22no—
199C22×C17—22no—
200C22×C17—22no—
201C22×C17—41no—
202C22×C17—41no—
203C22×C17—41no—
204C22×C17—41no—
205C22×C17—41no—
206C22×C17—41no—
207C22×C17—41no—
208C22×C17—41no—
209C22×C17—41no—
210C22×C17—41no—
211C22×C17—41no—
212C22×C17C22×D411yescharacteristic
213C22×C17H(5,𝔽2)11yescharacteristic
214C22×C17H(5,𝔽2)11yescharacteristic
215C22×C17H(5,𝔽2)11yescharacteristic
216C22×C17H(5,𝔽2)11yescharacteristic
217C22×C17H(5,𝔽2)11yescharacteristic
218C22×C17H(5,𝔽2)11yescharacteristic
219C2×D17—22no—
220C2×D17—22no—
221C2×D17—22no—
222C2×D17—22no—
223C2×D17—22no—
224C2×D17—22no—
225C2×D17—416no—
226C2×D17—41no—
227C2×D17—41no—
228C2×D17—41no—
229C2×D17—41no—
230C2×D17—41no—
231C2×D17—41no—
232C2×D17—41no—
233C2×D17—41no—
234C2×D17—41no—
235C2×D17—41no—
236C2×D17—41no—
237C2×D17—41no—
238C2×D17—41no—
239C2×D17—41no—
240C2×D17—41no—
241C4×C17—41no—
242C4×C17—41no—
243C4×C17—41no—
244C4×C17—41no—
245C4×C17—41no—
246Dic17—41no—
247Dic17—41no—
248Dic17—41no—
249Dic17—41no—
250Dic17—41no—
251C2×#120—171no—
252C2×#120—171no—
253#397—171no—
254#397—171no—
255#397—171no—
256#394—171no—
257#394—171no—
258#394—171no—
259#394—171no—
260#394—171no—
261#394D1711yescharacteristic, complemented
262#392—171no—
263#392—171no—
264#392—171no—
265#379—171no—
266C2×C17—42no—
267C2×C17—81no—
268C2×C17#53411yescharacteristic
269C2×C17#53411yescharacteristic
270C2×C17#53411yescharacteristic
271C2×C17#56011yescharacteristic
272C2×C17#56011yescharacteristic
273C2×C17#56011yescharacteristic
274C2×C17H(3,𝔽4)11yescharacteristic
275D17—42no—
276D17—42no—
277D17—81no—
278C25—171no—
279C22×D4—171no—
280C22×D4—171no—
281C22×D4—171no—
282C22×D4—171no—
283C22×D4—171no—
284C22×D4—171no—
285C22×D4C2×D1711yescharacteristic
286#120—344no—
287#120—344no—
288C2×C4⋊C4—171no—
289C2×C4⋊C4—171no—
290C2×C4⋊C4C2×D1711yescharacteristic, complemented
291C2×C22⋊C4—171no—
292C2×C22⋊C4—171no—
293C2×C22⋊C4—171no—
294C2×C22⋊C4—171no—
295C2×C22⋊C4—171no—
296C2×C22⋊C4—171no—
297C2×C22⋊C4—171no—
298C2×C22⋊C4—171no—
299C2×C22⋊C4—171no—
300C2×C22⋊C4—171no—
301C2×C22⋊C4—171no—
302C2×C22⋊C4—171no—
303C2×C22⋊C4—171no—
304C2×C22⋊C4—171no—
305C2×C22⋊C4C2×D1711yescharacteristic
306C2×C22⋊C4C2×D1711yescharacteristic
307C2×C22⋊C4C2×D1711yescharacteristic, complemented
308C2×C22⋊C4C2×D1711yescharacteristic, complemented
309#95—171no—
310#95—171no—
311#95—171no—
312#95—171no—
313#95—171no—
314#95C2×D1711yescharacteristic, complemented
315C17#281211yescharacteristic, complemented, Sylow
316C24—171no—
317C24—171no—
318C24—174no—
319C24—171no—
320C24—681no—
321C24—681no—
322C24—681no—
323C24—681no—
324C24—681no—
325C24—681no—
326C24C22×D1711yescharacteristic
327C24C22×D1711yescharacteristic
328C2×D4—22no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—342no—
332C2×D4—342no—
333C2×D4—342no—
334C2×D4—342no—
335C2×D4—342no—
336C2×D4—681no—
337C2×D4—681no—
338C2×D4—681no—
339C2×D4—681no—
340C2×D4—681no—
341C2×D4—681no—
342C2×D4—681no—
343C2×D4—681no—
344C2×D4—681no—
345C2×D4—681no—
346C2×D4—681no—
347C2×D4—681no—
348C2×D4—681no—
349C22×C4—171no—
350C22×C4—171no—
351C22×C4—171no—
352C22×C4—171no—
353C22×C4—171no—
354C22×C4C22×D1711yescharacteristic
355C22×C4C22×D1711yescharacteristic
356C22×C4C22×D1711yescharacteristic
357C22×C4C22×D1711yescharacteristic
358C22×C4C22×D1711yescharacteristic
359C4⋊C4—41no—
360C4⋊C4—681no—
361C4⋊C4—681no—
362C22⋊C4—22no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—342no—
367C22⋊C4—342no—
368C22⋊C4—342no—
369C22⋊C4—342no—
370C22⋊C4—342no—
371C22⋊C4—681no—
372C22⋊C4—681no—
373C22⋊C4—681no—
374C22⋊C4—681no—
375C22⋊C4—681no—
376C22⋊C4—681no—
377C22⋊C4—681no—
378C22⋊C4—681no—
379C22⋊C4—681no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—172no—
394C23—172no—
395C23—341no—
396C23—341no—
397C23—341no—
398C23—341no—
399C23—341no—
400C23—341no—
401C23—341no—
402C23—341no—
403C23—341no—
404C23—341no—
405C23—341no—
406C23—341no—
407C23—341no—
408C23—341no—
409C23—341no—
410C23—341no—
411C23—341no—
412C23—341no—
413C23—341no—
414C23—684no—
415C23—684no—
416C23—684no—
417C23—684no—
418C23—684no—
419C23—684no—
420C23—684no—
421C23D4×D1712yes—
422C23C23×D1711yescharacteristic
423D4—44no—
424D4—684no—
425D4—684no—
426D4—684no—
427D4—684no—
428D4—684no—
429D4—684no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—341no—
446C2×C4—341no—
447C2×C4—341no—
448C2×C4—341no—
449C2×C4—341no—
450C2×C4—341no—
451C2×C4—341no—
452C2×C4—341no—
453C2×C4—341no—
454C2×C4—341no—
455C2×C4—341no—
456C2×C4—341no—
457C2×C4—341no—
458C2×C4—341no—
459C2×C4—341no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—342no—
475C22—342no—
476C22—342no—
477C22—342no—
478C22—342no—
479C22—342no—
480C22—6816no—
481C22—681no—
482C22—681no—
483C22—681no—
484C22—681no—
485C22—681no—
486C22—681no—
487C22—681no—
488C22—681no—
489C22—681no—
490C22—681no—
491C22—681no—
492C22—681no—
493C22—681no—
494C22—681no—
495C22—681no—
496C22C2×D4×D1711yescharacteristic
497C22#1058769111yescharacteristic
498C22#1058769111yescharacteristic
499C22#1058769111yescharacteristic
500C22#1058769611yescharacteristic
501C22#1058769611yescharacteristic
502C22#1058769611yescharacteristic
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—681no—
509C4—681no—
510C4—681no—
511C4—681no—
512C4—681no—
513C2—42no—
514C2—81no—
515C2—682no—
516C2—682no—
517C2—1361no—
518C2#1176336711yescharacteristic
519C2#1176336911yescharacteristic
520C2#1176337011yescharacteristic
521C2#1176354711yescharacteristic
522C2#1176354711yescharacteristic
523C2#1176354911yescharacteristic
524C2#1176355111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
hypercenter#394D17
hypocenterC17#2812
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#10587568C2semidirect
2. C2×#10587623C2semidirect
3. #11762737C2semidirect
4. #11762736C2semidirect
5. #11762735C2semidirect
6. #11762727C2semidirect
7. #11762726C2semidirect
8. #11762725C2semidirect
9. #11762725C2semidirect
10. #11762724C2semidirect
11. #11762718C2semidirect
12. #11762718C2semidirect
13. #11762717C2semidirect
14. #11762664C2semidirect
15. C17×#394C2semidirect
21. C22×C22⋊D17C22semidirect
24. C2×C4⋊C4×C17C22semidirect
27. C2×C22⋊C4×C17C22semidirect
28. C2×C22⋊C4×C17C22semidirect
31. C2×C22⋊Dic17C22semidirect
41. C2×#63227C22semidirect
42. C2×#63227C22semidirect
43. C2×#63227C22semidirect
44. C2×C4⋊Dic17C22semidirect
45. C2×#63225C22semidirect
47. C17×#95C22semidirect
48. #10587502C22semidirect
49. #10587502C22semidirect
50. #10587502C22semidirect
51. #10587502C22semidirect
52. #10587502C22semidirect
61. C23×D17D4semidirect
261. #394D17semidirect
290. C2×C4⋊C4C2×D17semidirect
307. C2×C22⋊C4C2×D17semidirect
308. C2×C22⋊C4C2×D17semidirect
314. #95C2×D17semidirect
315. C17#2812semidirect

Maximal subgroups

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

subgroupindex
C2×#105875682
C2×#105876232
#117627372
#117627362
#117627352
#117627272
#117627262
#117627252
#117627242
#117627182
#117627172
#117626642
C17×#3942
#281217