Finite Group Databasebeta

← #14532175

Subgroups — #14532175

Order 2176.

814 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587569C211yesmaximal, characteristic, complemented
2C2×#10587623C211yesmaximal, characteristic
3C2×C22⋊C4×D17C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D17C211yesmaximal, characteristic, complemented
5#11762737C211yesmaximal, characteristic, complemented
6#11762734C211yesmaximal, characteristic, complemented
7#11762723C211yesmaximal, characteristic, complemented
8#11762685C211yesmaximal, characteristic, complemented
9#11762685C211yesmaximal, characteristic, complemented
10#11762685C211yesmaximal, characteristic, complemented
11#11762680C211yesmaximal, characteristic, complemented
12#11762671C211yesmaximal, characteristic, complemented
13#11762668C211yesmaximal, characteristic, complemented
14#11762666C211yesmaximal, characteristic, complemented
15C17×#394C211yesmaximal, characteristic, complemented
16C24×D17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C22×C22⋊D17C2211yescharacteristic
19C22×C22⋊D17C2211yescharacteristic
20C22×C4×D17C2211yescharacteristic
21C22×C4×D17C2211yescharacteristic
22C22×C4×D17C2211yescharacteristic
23C22×C4×D17C2211yescharacteristic
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
32C2×C22⋊Dic17C2211yescharacteristic, complemented
33C2×C22⋊Dic17C2211yescharacteristic, complemented
34#10587623—24no—
35C2×#63227C2211yescharacteristic
36C2×#63227C2211yescharacteristic
37C2×#63227C2211yescharacteristic
38C2×#63227C2211yescharacteristic
39C2×#63227C2211yescharacteristic
40C2×#63227C2211yescharacteristic
41C2×C4⋊Dic17C2211yescharacteristic, complemented
42C2×#63225C2211yescharacteristic, complemented
43C2×#63225C2211yescharacteristic, complemented
44#10587569—24no—
45C22⋊C4×D17—24no—
46C22⋊C4×D17—24no—
47C17×#95C2211yescharacteristic, complemented
48#10587502C2211yescharacteristic, complemented
49#10587502C2211yescharacteristic, complemented
50#10587502C2211yescharacteristic, complemented
51#10587502C2211yescharacteristic, complemented
52#10587502C2211yescharacteristic, complemented
53#10587502C2211yescharacteristic, complemented
54#10587502C2211yescharacteristic, complemented
55C24×C17C2311yescharacteristic
56C24×C17C2311yescharacteristic
57C23×D17—22no—
58C23×D17—22no—
59C23×D17—22no—
60C23×D17—22no—
61C23×D17—41no—
62C23×D17—41no—
63C23×D17D414yes—
64C23×D17C2311yescharacteristic
65C23×D17C2311yescharacteristic
66C2×D4×C17—22no—
67C2×D4×C17—41no—
68C2×D4×C17—41no—
69C22×C4×C17C2311yescharacteristic
70C22×C4×C17C2311yescharacteristic
71C22×C4×C17C2311yescharacteristic
72C22×C4×C17C2311yescharacteristic
73C22×C4×C17C2311yescharacteristic
74C2×C22⋊D17—22no—
75C2×C22⋊D17—22no—
76C2×C22⋊D17—22no—
77C2×C22⋊D17—22no—
78C2×C22⋊D17—22no—
79C2×C22⋊D17—22no—
80C22×Dic17C2311yescharacteristic
81C22×Dic17C2311yescharacteristic
82C22×Dic17C2311yescharacteristic
83C22×Dic17C2311yescharacteristic
84C22×Dic17C2311yescharacteristic
85C22×Dic17C2311yescharacteristic
86C2×C4×D17—22no—
87C2×C4×D17—22no—
88C2×C4×D17—22no—
89C2×C4×D17—22no—
90C2×C4×D17—22no—
91C2×C4×D17—22no—
92C2×C4×D17—22no—
93C2×C4×D17—22no—
94C2×C4×D17—41no—
95C2×C4×D17—41no—
96C2×C4×D17—41no—
97C2×C4×D17—41no—
98C4⋊C4×C17—41no—
99C22⋊C4×C17—22no—
100C22⋊C4×C17—22no—
101C22⋊C4×C17—22no—
102C22⋊C4×C17—41no—
103C22⋊Dic17—22no—
104C22⋊Dic17—22no—
105C22⋊Dic17—22no—
106C22⋊Dic17—22no—
107C22⋊Dic17—41no—
108#63227—22no—
109#63227—22no—
110#63227—22no—
111#63227—22no—
112#63227—22no—
113#63227D414yescomplemented
114C4⋊Dic17—22no—
115#63225—22no—
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—28no—
141C22×D17—28no—
142C22×D17—44no—
143C22×D17—44no—
144C22×D17—44no—
145C22×D17—44no—
146C22×D17—44no—
147C22×D17C2×D412yes—
148C22×D17C2×D412yes—
149C22×D17C2×D412yes—
150C22×D17#4112yes—
151C22×D17#4112yes—
152C22×D17#4112yes—
153D4×C17—44no—
154C2×C4×C17—21no—
155C2×C4×C17—21no—
156C2×C4×C17—21no—
157C2×C4×C17—21no—
158C2×C4×C17—21no—
159C2×C4×C17—21no—
160C2×C4×C17—21no—
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×C17C2×D412yes—
169C22⋊D17—44no—
170C22⋊D17—44no—
171C2×Dic17—21no—
172C2×Dic17—21no—
173C2×Dic17—21no—
174C2×Dic17—21no—
175C2×Dic17—21no—
176C2×Dic17—21no—
177C2×Dic17—21no—
178C2×Dic17—21no—
179C2×Dic17—21no—
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×Dic17C2×D412yes—
189C4×D17—44no—
190C4×D17—44no—
191C4×D17—44no—
192C4×D17—44no—
193#2412—171nomaximal, Sylow
194C22×C17—22no—
195C22×C17—22no—
196C22×C17—22no—
197C22×C17—41no—
198C22×C17—41no—
199C22×C17—41no—
200C22×C17—41no—
201C22×C17—41no—
202C22×C17—41no—
203C22×C17—41no—
204C22×C17—41no—
205C22×C17—41no—
206C22×C17—41no—
207C22×C17—41no—
208C22×C17C22×D411yescharacteristic
209C22×C17C22×D411yescharacteristic
210C22×C17C2×#4111yescharacteristic
211C22×C17C2×#4111yescharacteristic
212C22×C17C2×#4111yescharacteristic
213C22×C17H(5,𝔽2)11yescharacteristic
214C22×C17H(5,𝔽2)11yescharacteristic
215C2×D17—22no—
216C2×D17—22no—
217C2×D17—22no—
218C2×D17—22no—
219C2×D17—22no—
220C2×D17—22no—
221C2×D17—22no—
222C2×D17—22no—
223C2×D17—22no—
224C2×D17—22no—
225C2×D17—22no—
226C2×D17—22no—
227C2×D17—416no—
228C2×D17#12114yes—
229C2×D17#12314yescomplemented
230C4×C17—22no—
231C4×C17—41no—
232C4×C17—41no—
233C4×C17—41no—
234C4×C17—41no—
235Dic17—22no—
236Dic17—41no—
237Dic17—41no—
238Dic17—41no—
239Dic17—41no—
240Dic17—41no—
241C2×#123—171no—
242C2×#120—171no—
243C22×C22⋊C4—171no—
244C22×C22⋊C4—171no—
245#397—171no—
246#397—171no—
247#394—171no—
248#394D1711yescharacteristic, complemented
249#386—171no—
250#386—171no—
251#386—171no—
252#385—171no—
253#381—171no—
254#380—171no—
255#380—171no—
256C2×C17—42no—
257C2×C17—81no—
258C2×C17C2×#12111yescharacteristic
259C2×C17C2×#12311yescharacteristic
260C2×C17#53411yescharacteristic
261C2×C17#54611yescharacteristic
262C2×C17#54611yescharacteristic
263C2×C17#55111yescharacteristic
264C2×C17#55111yescharacteristic
265D17—24no—
266D17—24no—
267C25—171no—
268C22×D4—171no—
269C22×D4—171no—
270C22×D4C2×D1711yescharacteristic
271C23×C4—171no—
272C23×C4—171no—
273C23×C4—171no—
274C23×C4—171no—
275#123—344no—
276#120—344no—
277C2×C4⋊C4—171no—
278C2×C4⋊C4—171no—
279C2×C4⋊C4—171no—
280C2×C4⋊C4C2×D1711yescharacteristic, complemented
281C2×C22⋊C4—171no—
282C2×C22⋊C4—171no—
283C2×C22⋊C4—171no—
284C2×C22⋊C4—171no—
285C2×C22⋊C4—171no—
286C2×C22⋊C4—171no—
287C2×C22⋊C4—171no—
288C2×C22⋊C4—171no—
289C2×C22⋊C4—171no—
290C2×C22⋊C4—171no—
291C2×C22⋊C4—171no—
292C2×C22⋊C4—344no—
293C2×C22⋊C4—344no—
294C2×C22⋊C4C2×D1711yescharacteristic
295C2×C22⋊C4C2×D1711yescharacteristic
296C2×C22⋊C4C2×D1711yescharacteristic, complemented
297C2×C22⋊C4C2×D1711yescharacteristic, complemented
298#95—171no—
299#95—171no—
300#95—171no—
301#95—171no—
302#95—171no—
303#95—171no—
304#95—171no—
305#95C2×D1711yescharacteristic, complemented
306C17#241211yescharacteristic, complemented, Sylow
307C24—171no—
308C24—171no—
309C24—174no—
310C24—342no—
311C24—342no—
312C24—342no—
313C24—342no—
314C24—681no—
315C24—681no—
316C24C22×D1711yescharacteristic
317C24C22×D1711yescharacteristic
318C2×D4—22no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—342no—
322C2×D4—342no—
323C2×D4—342no—
324C2×D4—342no—
325C2×D4—342no—
326C2×D4—342no—
327C22×C4—171no—
328C22×C4—171no—
329C22×C4—171no—
330C22×C4—171no—
331C22×C4—171no—
332C22×C4—171no—
333C22×C4—342no—
334C22×C4—342no—
335C22×C4—342no—
336C22×C4—342no—
337C22×C4—342no—
338C22×C4—342no—
339C22×C4—342no—
340C22×C4—342no—
341C22×C4—681no—
342C22×C4—681no—
343C22×C4—681no—
344C22×C4—681no—
345C22×C4C22×D1711yescharacteristic
346C22×C4C22×D1711yescharacteristic
347C22×C4C22×D1711yescharacteristic
348C22×C4C22×D1711yescharacteristic
349C22×C4C22×D1711yescharacteristic
350C4⋊C4—41no—
351C4⋊C4—342no—
352C4⋊C4—342no—
353C4⋊C4—681no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—41no—
358C22⋊C4—174no—
359C22⋊C4—342no—
360C22⋊C4—342no—
361C22⋊C4—342no—
362C22⋊C4—342no—
363C22⋊C4—342no—
364C22⋊C4—342no—
365C22⋊C4—342no—
366C22⋊C4—342no—
367C22⋊C4—342no—
368C22⋊C4—681no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—172no—
383C23—172no—
384C23—172no—
385C23—172no—
386C23—172no—
387C23—172no—
388C23—348no—
389C23—348no—
390C23—341no—
391C23—341no—
392C23—341no—
393C23—341no—
394C23—341no—
395C23—341no—
396C23—341no—
397C23—341no—
398C23—684no—
399C23—684no—
400C23—684no—
401C23—684no—
402C23—684no—
403C23D4×D1712yes—
404C23C23×D1711yescharacteristic
405D4—44no—
406D4—684no—
407D4—684no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—172no—
423C2×C4—341no—
424C2×C4—341no—
425C2×C4—341no—
426C2×C4—341no—
427C2×C4—341no—
428C2×C4—341no—
429C2×C4—341no—
430C2×C4—341no—
431C2×C4—341no—
432C2×C4—341no—
433C2×C4—341no—
434C2×C4—341no—
435C2×C4—341no—
436C2×C4—341no—
437C2×C4—341no—
438C2×C4—341no—
439C2×C4—341no—
440C2×C4—684no—
441C2×C4—684no—
442C2×C4—684no—
443C2×C4—684no—
444C2×C4D4×D1712yes—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—174no—
460C22—174no—
461C22—342no—
462C22—342no—
463C22—342no—
464C22—342no—
465C22—342no—
466C22—342no—
467C22—342no—
468C22—342no—
469C22—342no—
470C22—342no—
471C22—342no—
472C22—342no—
473C22—6816no—
474C22C2×D4×D1711yescharacteristic
475C22C2×D4×D1711yescharacteristic
476C22#1058769111yescharacteristic
477C22#1058769111yescharacteristic
478C22#41×D1711yescharacteristic
479C22#41×D1711yescharacteristic
480C22#41×D1711yescharacteristic
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—342no—
487C4—681no—
488C4—681no—
489C4—681no—
490C4—681no—
491C4—681no—
492C2—42no—
493C2—81no—
494C2—344no—
495C2—344no—
496C2#121×D1711yescharacteristic
497C2#123×D1711yescharacteristic
498C2#1176336911yescharacteristic
499C2#1176344811yescharacteristic
500C2#1176345611yescharacteristic
501C2#1176348711yescharacteristic
502C2#1176348911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC2×#10587623C2
hypercenter#394D17
hypocenterC17#2412
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#10587569C2semidirect
3. C2×C22⋊C4×D17C2semidirect
4. C2×C22⋊C4×D17C2semidirect
5. #11762737C2semidirect
6. #11762734C2semidirect
7. #11762723C2semidirect
8. #11762685C2semidirect
9. #11762685C2semidirect
10. #11762685C2semidirect
11. #11762680C2semidirect
12. #11762671C2semidirect
13. #11762668C2semidirect
14. #11762666C2semidirect
15. C17×#394C2semidirect
24. C2×C4⋊C4×C17C22semidirect
27. C2×C22⋊C4×C17C22semidirect
28. C2×C22⋊C4×C17C22semidirect
32. C2×C22⋊Dic17C22semidirect
33. C2×C22⋊Dic17C22semidirect
41. C2×C4⋊Dic17C22semidirect
42. C2×#63225C22semidirect
43. C2×#63225C22semidirect
47. C17×#95C22semidirect
48. #10587502C22semidirect
49. #10587502C22semidirect
50. #10587502C22semidirect
51. #10587502C22semidirect
52. #10587502C22semidirect
53. #10587502C22semidirect
54. #10587502C22semidirect
113. #63227D4semidirect
229. C2×D17#123semidirect
248. #394D17semidirect
280. C2×C4⋊C4C2×D17semidirect
296. C2×C22⋊C4C2×D17semidirect
297. C2×C22⋊C4C2×D17semidirect
305. #95C2×D17semidirect
306. C17#2412semidirect

Maximal subgroups

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

subgroupindex
C2×#105875692
C2×#105876232
C2×C22⋊C4×D172
#117627372
#117627342
#117627232
#117626852
#117626802
#117626712
#117626682
#117626662
C17×#3942
#241217