Finite Group Databasebeta

← #14532166

Subgroups — #14532166

Order 2176.

786 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587569C211yesmaximal, characteristic, complemented
2C2×#10587623C211yesmaximal, characteristic
3C2×#10587558C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D17C211yesmaximal, characteristic, complemented
5#11762732C211yesmaximal, characteristic, complemented
6#11762731C211yesmaximal, characteristic, complemented
7#11762724C211yesmaximal, characteristic, complemented
8#11762723C211yesmaximal, characteristic, complemented
9#11762709C211yesmaximal, characteristic, complemented
10#11762703C211yesmaximal, characteristic, complemented
11#11762700C211yesmaximal, characteristic, complemented
12#11762685C211yesmaximal, characteristic, complemented
13#11762680C211yesmaximal, characteristic, complemented
14#11762664C211yesmaximal, characteristic, complemented
15C17×#390C211yesmaximal, characteristic, complemented
16C24×D17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C22×C22⋊D17C2211yescharacteristic
19C22×C22⋊D17C2211yescharacteristic
20C22×C4×D17C2211yescharacteristic
21C22×C4×D17C2211yescharacteristic
22C2×C4⋊C4×C17C2211yescharacteristic, complemented
23C2×C22⋊C4×C17C2211yescharacteristic
24C2×C22⋊C4×C17C2211yescharacteristic
25C2×C22⋊C4×C17C2211yescharacteristic, complemented
26C2×C22⋊C4×C17C2211yescharacteristic, complemented
27C2×C42×C17C2211yescharacteristic, complemented
28C2×C22⋊Dic17C2211yescharacteristic
29C2×C22⋊Dic17C2211yescharacteristic
30C2×C22⋊Dic17C2211yescharacteristic
31C2×C22⋊Dic17C2211yescharacteristic, complemented
32C2×C22⋊Dic17C2211yescharacteristic, complemented
33#10587623—24no—
34C2×#63227C2211yescharacteristic
35C2×#63227C2211yescharacteristic
36C2×#63227C2211yescharacteristic
37C2×#63227C2211yescharacteristic
38C2×#63227C2211yescharacteristic
39C2×#63227C2211yescharacteristic
40C2×#63227C2211yescharacteristic
41C2×#63227C2211yescharacteristic
42C2×C4⋊Dic17C2211yescharacteristic, complemented
43C2×#63225C2211yescharacteristic, complemented
44C2×#63225C2211yescharacteristic, complemented
45C2×#63225C2211yescharacteristic, complemented
46C2×#63225C2211yescharacteristic, complemented
47C2×C4×Dic17C2211yescharacteristic, complemented
48#10587569—24no—
49C22⋊C4×D17—24no—
50#10587558—24no—
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—22no—
62C23×D17—41no—
63C23×D17—41no—
64C23×D17C2311yescharacteristic
65C23×D17C2311yescharacteristic
66C2×D4×C17—22no—
67C2×D4×C17—22no—
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—41no—
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—41no—
90C2×C4×D17—41no—
91C2×C4×D17D414yescomplemented
92C4⋊C4×C17—41no—
93C22⋊C4×C17—22no—
94C22⋊C4×C17—22no—
95C22⋊C4×C17—41no—
96C22⋊C4×C17—41no—
97C42×C17—22no—
98C22⋊Dic17—22no—
99C22⋊Dic17—22no—
100C22⋊Dic17—22no—
101C22⋊Dic17—22no—
102C22⋊Dic17—41no—
103#63227—22no—
104#63227—22no—
105#63227—22no—
106#63227—22no—
107#63227—22no—
108#63227—22no—
109#63227—41no—
110#63227D414yes—
111C4⋊Dic17—22no—
112#63225—22no—
113#63225—22no—
114#63225—22no—
115#63225—41no—
116C4×Dic17—22no—
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×C17#4112yes—
131C23×C17C2411yescharacteristic
132C22×D17—21no—
133C22×D17—21no—
134C22×D17—21no—
135C22×D17—21no—
136C22×D17—28no—
137C22×D17—21no—
138C22×D17—21no—
139C22×D17—21no—
140C22×D17—21no—
141C22×D17—21no—
142C22×D17—21no—
143C22×D17—44no—
144C22×D17—44no—
145C22×D17—44no—
146C22×D17—44no—
147C22×D17—44no—
148C22×D17—44no—
149C22×D17C2×D412yes—
150C22×D17C2×D412yes—
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×C17C2×D412yes—
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×Dic17C2×D412yes—
188C2×Dic17C2×D412yes—
189C4×D17—28no—
190C4×D17—44no—
191#2411—171nomaximal, Sylow
192C22×C17—22no—
193C22×C17—22no—
194C22×C17—22no—
195C22×C17—41no—
196C22×C17—41no—
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×C17C22×D411yescharacteristic
207C22×C17C22×D411yescharacteristic
208C22×C17C2×#4111yescharacteristic
209C22×C17C2×#4111yescharacteristic
210C22×C17C2×#4111yescharacteristic
211C22×C17H(5,𝔽2)11yescharacteristic
212C22×C17H(5,𝔽2)11yescharacteristic
213C2×D17—22no—
214C2×D17—22no—
215C2×D17—22no—
216C2×D17—22no—
217C2×D17—22no—
218C2×D17—22no—
219C2×D17—22no—
220C2×D17—22no—
221C2×D17—22no—
222C2×D17—416no—
223C2×D17—41no—
224C2×D17—41no—
225C2×D17—41no—
226C2×D17—41no—
227C2×D17#12114yes—
228C4×C17—22no—
229C4×C17—22no—
230C4×C17—41no—
231C4×C17—41no—
232C4×C17—41no—
233Dic17—22no—
234Dic17—22no—
235Dic17—41no—
236Dic17—41no—
237Dic17—41no—
238Dic17—41no—
239C2×#123—171no—
240C2×#120—171no—
241C2×#117—171no—
242C22×C22⋊C4—171no—
243#396—171no—
244#396—171no—
245#394—171no—
246#394—171no—
247#390—171no—
248#390D1711yescharacteristic, complemented
249#388—171no—
250#388—171no—
251#386—171no—
252#385—171no—
253#379—171no—
254C2×C17—42no—
255C2×C17—81no—
256C2×C17C2×#12111yescharacteristic
257C2×C17#52511yescharacteristic
258C2×C17#53511yescharacteristic
259C2×C17#54611yescharacteristic
260C2×C17#54611yescharacteristic
261C2×C17#55111yescharacteristic
262C2×C17#55111yescharacteristic
263D17—24no—
264D17—42no—
265C25—171no—
266C22×D4—171no—
267C22×D4—171no—
268C22×D4C2×D1711yescharacteristic
269C23×C4—171no—
270C23×C4—171no—
271#123—344no—
272#120—344no—
273#117—344no—
274C2×C4⋊C4—171no—
275C2×C4⋊C4—171no—
276C2×C4⋊C4—171no—
277C2×C4⋊C4—171no—
278C2×C4⋊C4—171no—
279C2×C4⋊C4C2×D1711yescharacteristic, complemented
280C2×C22⋊C4—171no—
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—171no—
293C2×C22⋊C4—344no—
294C2×C22⋊C4C2×D1711yescharacteristic
295C2×C22⋊C4C2×D1711yescharacteristic
296C2×C22⋊C4C2×D1711yescharacteristic, complemented
297C2×C22⋊C4C2×D1711yescharacteristic, complemented
298C2×C42—171no—
299C2×C42C2×D1711yescharacteristic, complemented
300#95—171no—
301#95—171no—
302#95—171no—
303#95—171no—
304C17#241111yescharacteristic, complemented, Sylow
305C24—171no—
306C24—171no—
307C24—342no—
308C24—342no—
309C24—342no—
310C24—342no—
311C24—342no—
312C24—681no—
313C24—681no—
314C24C22×D1711yescharacteristic
315C24C22×D1711yescharacteristic
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—41no—
319C2×D4—342no—
320C2×D4—342no—
321C2×D4—342no—
322C2×D4—342no—
323C2×D4—342no—
324C2×D4—681no—
325C22×C4—171no—
326C22×C4—171no—
327C22×C4—174no—
328C22×C4—171no—
329C22×C4—171no—
330C22×C4—171no—
331C22×C4—171no—
332C22×C4—342no—
333C22×C4—342no—
334C22×C4—342no—
335C22×C4—681no—
336C22×C4—681no—
337C22×C4C22×D1711yescharacteristic
338C22×C4C22×D1711yescharacteristic
339C22×C4C22×D1711yescharacteristic
340C22×C4C22×D1711yescharacteristic
341C22×C4C22×D1711yescharacteristic
342C4⋊C4—41no—
343C4⋊C4—342no—
344C4⋊C4—342no—
345C4⋊C4—342no—
346C4⋊C4—342no—
347C4⋊C4—681no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—174no—
353C22⋊C4—342no—
354C22⋊C4—342no—
355C22⋊C4—342no—
356C22⋊C4—342no—
357C22⋊C4—342no—
358C22⋊C4—342no—
359C22⋊C4—342no—
360C22⋊C4—342no—
361C22⋊C4—342no—
362C22⋊C4—342no—
363C22⋊C4—681no—
364C22⋊C4—681no—
365C42—22no—
366C42—342no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—172no—
381C23—172no—
382C23—172no—
383C23—172no—
384C23—348no—
385C23—341no—
386C23—341no—
387C23—341no—
388C23—341no—
389C23—341no—
390C23—341no—
391C23—341no—
392C23—341no—
393C23—341no—
394C23—341no—
395C23—684no—
396C23—684no—
397C23—684no—
398C23—684no—
399C23—684no—
400C23—684no—
401C23#6325212yes—
402C23C23×D1711yescharacteristic
403D4—44no—
404D4—684no—
405D4—684no—
406C2×C4—21no—
407C2×C4—21no—
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—172no—
420C2×C4—172no—
421C2×C4—348no—
422C2×C4—341no—
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—684no—
439C2×C4D4×D1712yes—
440C2×C4D4×D1712yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—174no—
456C22—342no—
457C22—342no—
458C22—342no—
459C22—342no—
460C22—342no—
461C22—342no—
462C22—342no—
463C22—342no—
464C22—342no—
465C22—6816no—
466C22—681no—
467C22—681no—
468C22—681no—
469C22—681no—
470C22C2×#6325211yescharacteristic
471C22C2×D4×D1711yescharacteristic
472C22C2×D4×D1711yescharacteristic
473C22#1058769111yescharacteristic
474C22#1058769111yescharacteristic
475C22#41×D1711yescharacteristic
476C22#41×D1711yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—342no—
483C4—342no—
484C4—681no—
485C4—681no—
486C4—681no—
487C4—681no—
488C2—42no—
489C2—81no—
490C2—344no—
491C2—682no—
492C2#121×D1711yescharacteristic
493C2#1176331811yescharacteristic
494C2#1176337211yescharacteristic
495C2#1176344911yescharacteristic
496C2#1176345611yescharacteristic
497C2#1176348811yescharacteristic
498C2#1176348911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#390C2
prime radicalC2×#10587623C2
hypercenter#390D17
hypocenterC17#2411
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#10587569C2semidirect
3. C2×#10587558C2semidirect
4. C2×C22⋊C4×D17C2semidirect
5. #11762732C2semidirect
6. #11762731C2semidirect
7. #11762724C2semidirect
8. #11762723C2semidirect
9. #11762709C2semidirect
10. #11762703C2semidirect
11. #11762700C2semidirect
12. #11762685C2semidirect
13. #11762680C2semidirect
14. #11762664C2semidirect
15. C17×#390C2semidirect
22. C2×C4⋊C4×C17C22semidirect
25. C2×C22⋊C4×C17C22semidirect
26. C2×C22⋊C4×C17C22semidirect
27. C2×C42×C17C22semidirect
31. C2×C22⋊Dic17C22semidirect
32. C2×C22⋊Dic17C22semidirect
42. C2×C4⋊Dic17C22semidirect
43. C2×#63225C22semidirect
44. C2×#63225C22semidirect
45. C2×#63225C22semidirect
46. C2×#63225C22semidirect
47. C2×C4×Dic17C22semidirect
51. #10587502C22semidirect
52. #10587502C22semidirect
53. #10587502C22semidirect
54. #10587502C22semidirect
91. C2×C4×D17D4semidirect
248. #390D17semidirect
279. C2×C4⋊C4C2×D17semidirect
296. C2×C22⋊C4C2×D17semidirect
297. C2×C22⋊C4C2×D17semidirect
299. C2×C42C2×D17semidirect
304. C17#2411semidirect

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×#105875582
C2×C22⋊C4×D172
#117627322
#117627312
#117627242
#117627232
#117627092
#117627032
#117627002
#117626852
#117626802
#117626642
C17×#3902
#241117