Finite Group Databasebeta

← #14533031

Subgroups — #14533031

Order 2176.

664 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587584C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D17C211yesmaximal, characteristic
3#11762738C211yesmaximal, characteristic, complemented
4#11762736C211yesmaximal, characteristic, complemented
5#11762732C211yesmaximal, characteristic, complemented
6#11762702C211yesmaximal, characteristic, complemented
7#11762702C211yesmaximal, characteristic, complemented
8#11762701C211yesmaximal, characteristic, complemented
9#11762683C211yesmaximal, characteristic, complemented
10#11762683C211yesmaximal, characteristic, complemented
11#11762666C211yesmaximal, characteristic, complemented
12#11762664C211yesmaximal, characteristic
13#11762664C211yesmaximal, characteristic
14#11762656C211yesmaximal, characteristic, complemented
15C17×#397C211yesmaximal, characteristic, complemented
16C24×D17C2211yescharacteristic
17C22×C4×D17C2211yescharacteristic
18C22×C4×D17C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic, complemented
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊Dic17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C2×#63227C2211yescharacteristic
27C2×#63227C2211yescharacteristic
28C2×#63227C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×#63227C2211yescharacteristic
33C2×#63227C2211yescharacteristic
34C2×#63227C2211yescharacteristic
35C2×#63227C2211yescharacteristic
36C2×C4⋊Dic17C2211yescharacteristic, complemented
37C2×C4⋊Dic17C2211yescharacteristic, complemented
38C2×C4⋊Dic17C2211yescharacteristic, complemented
39C2×#63225C2211yescharacteristic, complemented
40C2×#63225C2211yescharacteristic, complemented
41C2×C4×Dic17C2211yescharacteristic, complemented
42C2×C4×Dic17C2211yescharacteristic, complemented
43#10587584—24no—
44C22⋊C4×D17—24no—
45C17×#95C2211yescharacteristic, complemented
46C17×#95C2211yescharacteristic, complemented
47C17×#95C2211yescharacteristic, complemented
48#10587502C2211yescharacteristic, complemented
49#10587502C2211yescharacteristic, complemented
50#10587502C2211yescharacteristic, complemented
51#10587502C2211yescharacteristic, complemented
52#10587502C2211yescharacteristic, complemented
53C24×C17C2311yescharacteristic
54C23×D17—22no—
55C23×D17—22no—
56C23×D17—22no—
57C23×D17—22no—
58C23×D17—22no—
59C23×D17—41no—
60C23×D17—41no—
61C23×D17C2311yescharacteristic
62C23×D17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C22×C4×C17C2311yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C22×C4×C17C2311yescharacteristic
67C22×C4×C17C2311yescharacteristic
68C22×C4×C17C2311yescharacteristic
69C22×Dic17C2311yescharacteristic
70C22×Dic17C2311yescharacteristic
71C22×Dic17C2311yescharacteristic
72C22×Dic17C2311yescharacteristic
73C22×Dic17C2311yescharacteristic
74C22×Dic17C2311yescharacteristic
75C2×C4×D17—22no—
76C2×C4×D17—22no—
77C2×C4×D17—22no—
78C2×C4×D17—22no—
79C2×C4×D17—41no—
80C2×C4×D17—41no—
81C4⋊C4×C17—22no—
82C22⋊C4×C17—22no—
83C22⋊C4×C17—22no—
84C22⋊C4×C17—22no—
85C22⋊Dic17—22no—
86C22⋊Dic17—41no—
87C22⋊Dic17—41no—
88#63227—22no—
89#63227—22no—
90#63227—22no—
91#63227—22no—
92#63227—22no—
93#63227—22no—
94#63227—22no—
95#63227—41no—
96#63227—41no—
97#63227D414yes—
98C4⋊Dic17—22no—
99C4⋊Dic17—41no—
100C4⋊Dic17—41no—
101#63225—22no—
102#63225—41no—
103C4×Dic17—22no—
104C4×Dic17—41no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17#4112yes—
112C23×C17C2411yescharacteristic
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—28no—
118C22×D17—21no—
119C22×D17—21no—
120C22×D17—21no—
121C22×D17—21no—
122C22×D17—21no—
123C22×D17—21no—
124C22×D17—44no—
125C22×D17—44no—
126C22×D17—44no—
127C22×D17—44no—
128C22×D17—44no—
129C22×D17—44no—
130C22×D17C2×D412yes—
131C22×D17#4112yes—
132C22×D17#4112yes—
133C22×D17#4112yes—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—21no—
146C2×C4×C17—21no—
147C2×C4×C17—21no—
148C2×C4×C17—21no—
149C2×C4×C17—21no—
150C2×C4×C17—21no—
151C2×C4×C17C2×D412yes—
152C2×Dic17—21no—
153C2×Dic17—21no—
154C2×Dic17—21no—
155C2×Dic17—21no—
156C2×Dic17—21no—
157C2×Dic17—21no—
158C2×Dic17—21no—
159C2×Dic17—21no—
160C2×Dic17—21no—
161C2×Dic17—21no—
162C2×Dic17—21no—
163C2×Dic17—21no—
164C2×Dic17—21no—
165C2×Dic17—21no—
166C2×Dic17—21no—
167C2×Dic17—21no—
168C2×Dic17—21no—
169C2×Dic17C2×D412yes—
170C4×D17—44no—
171C4×D17—44no—
172#2480—171nomaximal, Sylow
173C22×C17—22no—
174C22×C17—22no—
175C22×C17—22no—
176C22×C17—41no—
177C22×C17—41no—
178C22×C17—41no—
179C22×C17—41no—
180C22×C17C22×D411yescharacteristic
181C22×C17C2×#4111yescharacteristic
182C22×C17C2×#4111yescharacteristic
183C22×C17C2×#4111yescharacteristic
184C22×C17C2×#4111yescharacteristic
185C22×C17H(5,𝔽2)11yescharacteristic
186C22×C17H(5,𝔽2)11yescharacteristic
187C2×D17—22no—
188C2×D17—22no—
189C2×D17—22no—
190C2×D17—22no—
191C2×D17—22no—
192C2×D17—22no—
193C2×D17—22no—
194C2×D17—22no—
195C2×D17—22no—
196C2×D17—416no—
197C2×D17—41no—
198C2×D17—41no—
199C2×D17—41no—
200C2×D17—41no—
201C2×D17#12614yescomplemented
202C4×C17—22no—
203C4×C17—41no—
204C4×C17—41no—
205C4×C17—41no—
206C4×C17—41no—
207C4×C17—41no—
208Dic17—22no—
209Dic17—41no—
210Dic17—41no—
211Dic17—41no—
212Dic17—41no—
213Dic17—41no—
214C2×#126—171no—
215C22×C22⋊C4—171no—
216#397—171no—
217#397—171no—
218#397D1711yescharacteristic, complemented
219#396—171no—
220#388—171no—
221#388—171no—
222#388—171no—
223#385—171no—
224#385—171no—
225#380—171no—
226#379—171no—
227#379—171no—
228C4×C22⋊C4—171no—
229C2×C17—42no—
230C2×C17#52511yescharacteristic
231C2×C17C2×#12611yescharacteristic
232C2×C17#53811yescharacteristic
233C2×C17#54611yescharacteristic
234C2×C17#54611yescharacteristic
235C2×C17#55111yescharacteristic
236C2×C17#55111yescharacteristic
237D17—24no—
238D17—42no—
239C25—171no—
240C23×C4—171no—
241C23×C4—171no—
242#126—344no—
243C2×C4⋊C4—171no—
244C2×C4⋊C4—171no—
245C2×C4⋊C4—171no—
246C2×C4⋊C4—171no—
247C2×C4⋊C4—171no—
248C2×C4⋊C4C2×D1711yescharacteristic, complemented
249C2×C22⋊C4—171no—
250C2×C22⋊C4—171no—
251C2×C22⋊C4—171no—
252C2×C22⋊C4—171no—
253C2×C22⋊C4—171no—
254C2×C22⋊C4—171no—
255C2×C22⋊C4—171no—
256C2×C22⋊C4—171no—
257C2×C22⋊C4—171no—
258C2×C22⋊C4—171no—
259C2×C22⋊C4—171no—
260C2×C22⋊C4—171no—
261C2×C22⋊C4—171no—
262C2×C22⋊C4—344no—
263C2×C22⋊C4C2×D1711yescharacteristic
264C2×C22⋊C4C2×D1711yescharacteristic
265C2×C22⋊C4C2×D1711yescharacteristic
266C2×C42—171no—
267C2×C42—171no—
268#95—171no—
269#95—171no—
270#95—171no—
271#95—171no—
272#95—171no—
273#95C2×D1711yescharacteristic, complemented
274#95C2×D1711yescharacteristic, complemented
275#95C2×D1711yescharacteristic, complemented
276C17#248011yescharacteristic, complemented, Sylow
277C24—171no—
278C24—171no—
279C24—342no—
280C24—342no—
281C24—342no—
282C24—342no—
283C24—342no—
284C24—681no—
285C24—681no—
286C24C22×D1711yescharacteristic
287C22×C4—171no—
288C22×C4—171no—
289C22×C4—171no—
290C22×C4—171no—
291C22×C4—171no—
292C22×C4—171no—
293C22×C4—342no—
294C22×C4—342no—
295C22×C4—342no—
296C22×C4—342no—
297C22×C4—681no—
298C22×C4—681no—
299C22×C4C22×D1711yescharacteristic
300C22×C4C22×D1711yescharacteristic
301C22×C4C22×D1711yescharacteristic
302C22×C4C22×D1711yescharacteristic
303C22×C4C22×D1711yescharacteristic
304C22×C4C22×D1711yescharacteristic
305C4⋊C4—22no—
306C4⋊C4—342no—
307C4⋊C4—342no—
308C4⋊C4—681no—
309C4⋊C4—681no—
310C4⋊C4—681no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—174no—
315C22⋊C4—342no—
316C22⋊C4—342no—
317C22⋊C4—342no—
318C22⋊C4—342no—
319C22⋊C4—342no—
320C22⋊C4—342no—
321C22⋊C4—342no—
322C22⋊C4—342no—
323C22⋊C4—681no—
324C22⋊C4—681no—
325C22⋊C4—681no—
326C22⋊C4—681no—
327C42—342no—
328C42—681no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—172no—
336C23—172no—
337C23—172no—
338C23—172no—
339C23—348no—
340C23—341no—
341C23—341no—
342C23—341no—
343C23—341no—
344C23—341no—
345C23—341no—
346C23—341no—
347C23—341no—
348C23—341no—
349C23—341no—
350C23—684no—
351C23—684no—
352C23—684no—
353C23—684no—
354C23—684no—
355C23—684no—
356C23#6325412yes—
357C23C23×D1711yescharacteristic
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—172no—
376C2×C4—341no—
377C2×C4—341no—
378C2×C4—341no—
379C2×C4—341no—
380C2×C4—341no—
381C2×C4—341no—
382C2×C4—341no—
383C2×C4—341no—
384C2×C4—341no—
385C2×C4—341no—
386C2×C4—341no—
387C2×C4—341no—
388C2×C4—341no—
389C2×C4—341no—
390C2×C4—341no—
391C2×C4—341no—
392C2×C4—341no—
393C2×C4—684no—
394C2×C4—684no—
395C2×C4D4×D1712yes—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—174no—
404C22—342no—
405C22—342no—
406C22—342no—
407C22—342no—
408C22—342no—
409C22—342no—
410C22—342no—
411C22—342no—
412C22—342no—
413C22—6816no—
414C22—681no—
415C22—681no—
416C22—681no—
417C22—681no—
418C22C2×D4×D1711yescharacteristic
419C22C2×#6325411yescharacteristic
420C22#41×D1711yescharacteristic
421C22#41×D1711yescharacteristic
422C22#41×D1711yescharacteristic
423C22#1058769611yescharacteristic
424C22#1058769611yescharacteristic
425C4—22no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—342no—
432C4—681no—
433C4—681no—
434C4—681no—
435C4—681no—
436C4—681no—
437C2—42no—
438C2—344no—
439C2—682no—
440C2#1176331911yescharacteristic
441C2#126×D1711yescharacteristic
442C2#1176339811yescharacteristic
443C2#1176345111yescharacteristic
444C2#1176345211yescharacteristic
445C2#1176348211yescharacteristic
446C2#1176348211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#397C2
prime radicalC24×D17C22
hypercenter#397D17
hypocenterC17#2480
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#10587584C2semidirect
3. #11762738C2semidirect
4. #11762736C2semidirect
5. #11762732C2semidirect
6. #11762702C2semidirect
7. #11762702C2semidirect
8. #11762701C2semidirect
9. #11762683C2semidirect
10. #11762683C2semidirect
11. #11762666C2semidirect
14. #11762656C2semidirect
15. C17×#397C2semidirect
19. C2×C4⋊C4×C17C22semidirect
36. C2×C4⋊Dic17C22semidirect
37. C2×C4⋊Dic17C22semidirect
38. C2×C4⋊Dic17C22semidirect
39. C2×#63225C22semidirect
40. C2×#63225C22semidirect
41. C2×C4×Dic17C22semidirect
42. C2×C4×Dic17C22semidirect
45. C17×#95C22semidirect
46. C17×#95C22semidirect
47. C17×#95C22semidirect
48. #10587502C22semidirect
49. #10587502C22semidirect
50. #10587502C22semidirect
51. #10587502C22semidirect
52. #10587502C22semidirect
201. C2×D17#126semidirect
218. #397D17semidirect
248. C2×C4⋊C4C2×D17semidirect
273. #95C2×D17semidirect
274. #95C2×D17semidirect
275. #95C2×D17semidirect
276. C17#2480semidirect

Maximal subgroups

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

subgroupindex
C2×#105875842
C2×C22⋊C4×D172
#117627382
#117627362
#117627322
#117627022
#117627012
#117626832
#117626662
#117626642
#117626562
C17×#3972
#248017