Finite Group Databasebeta

← #14535015

Subgroups — #14535015

Order 2176.

682 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587569C211yesmaximal, characteristic, complemented
2C2×#10587625C211yesmaximal, characteristic, complemented
3C2×#10587624C211yesmaximal, characteristic, complemented
4#11762745C211yesmaximal, characteristic, complemented
5#11762742C211yesmaximal, characteristic, complemented
6#11762725C211yesmaximal, characteristic, complemented
7#11762723C211yesmaximal, characteristic, complemented
8#11762723C211yesmaximal, characteristic, complemented
9#11762717C211yesmaximal, characteristic, complemented
10#11762689C211yesmaximal, characteristic, complemented
11#11762688C211yesmaximal, characteristic, complemented
12#11762684C211yesmaximal, characteristic, complemented
13#11762680C211yesmaximal, characteristic, complemented
14#11762669C211yesmaximal, characteristic, complemented
15C2×C17×#121C211yesmaximal, characteristic, complemented
16C22×D4×C17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C22×D4×C17C2211yescharacteristic
19C23×C4×C17C2211yescharacteristic
20C22×C22⋊D17C2211yescharacteristic
21C22×C22⋊D17C2211yescharacteristic
22C22×C22⋊D17C2211yescharacteristic
23C23×Dic17C2211yescharacteristic
24C22×C4×D17C2211yescharacteristic
25C17×#121—24no—
26C2×C4⋊C4×C17C2211yescharacteristic
27C2×C22⋊C4×C17C2211yescharacteristic
28C2×C22⋊C4×C17C2211yescharacteristic
29C2×C22⋊Dic17C2211yescharacteristic
30C2×C22⋊Dic17C2211yescharacteristic
31C2×C22⋊Dic17C2211yescharacteristic
32C2×C22⋊Dic17C2211yescharacteristic
33C2×C22⋊Dic17C2211yescharacteristic
34C2×C22⋊Dic17C2211yescharacteristic
35C2×C22⋊Dic17C2211yescharacteristic
36C2×C22⋊Dic17C2211yescharacteristic
37#10587625—24no—
38#10587624—24no—
39C2×#63227C2211yescharacteristic
40C2×#63227C2211yescharacteristic
41C2×#63227C2211yescharacteristic
42C2×C4⋊Dic17C2211yescharacteristic
43C2×C4⋊Dic17C2211yescharacteristic
44C2×C4⋊Dic17C2211yescharacteristic
45C2×#63225C2211yescharacteristic
46C2×#63225C2211yescharacteristic
47C2×#63225C2211yescharacteristic
48#10587569—24no—
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53#10587502C2211yescharacteristic
54#10587502C2211yescharacteristic
55C24×C17C2311yescharacteristic
56C24×C17C2311yescharacteristic
57C24×C17C2311yescharacteristic
58C23×D17C2311yescharacteristic
59C2×D4×C17—22no—
60C2×D4×C17—22no—
61C2×D4×C17—22no—
62C2×D4×C17—22no—
63C2×D4×C17—22no—
64C2×D4×C17—22no—
65C2×D4×C17—41no—
66C2×D4×C17—41no—
67C2×D4×C17—41no—
68C22×C4×C17—22no—
69C22×C4×C17—22no—
70C22×C4×C17—41no—
71C22×C4×C17C2311yescharacteristic
72C22×C4×C17C2311yescharacteristic
73C22×C4×C17C2311yescharacteristic
74C22×C4×C17C2311yescharacteristic
75C2×C22⋊D17—22no—
76C2×C22⋊D17—22no—
77C2×C22⋊D17—22no—
78C2×C22⋊D17—41no—
79C2×C22⋊D17—41no—
80C2×C22⋊D17—41no—
81C2×C22⋊D17—41no—
82C2×C22⋊D17—41no—
83C2×C22⋊D17D414yescomplemented
84C22×Dic17—22no—
85C22×Dic17—41no—
86C22×Dic17—41no—
87C22×Dic17C2311yescharacteristic
88C22×Dic17C2311yescharacteristic
89C22×Dic17C2311yescharacteristic
90C22×Dic17C2311yescharacteristic
91C22×Dic17C2311yescharacteristic
92C22×Dic17C2311yescharacteristic
93C22×Dic17C2311yescharacteristic
94C2×C4×D17—22no—
95C2×C4×D17—22no—
96C2×C4×D17—41no—
97C4⋊C4×C17—22no—
98C22⋊C4×C17—22no—
99C22⋊C4×C17D414yescomplemented
100C22⋊Dic17—22no—
101C22⋊Dic17—22no—
102C22⋊Dic17—22no—
103C22⋊Dic17—22no—
104C22⋊Dic17—41no—
105C22⋊Dic17—41no—
106C22⋊Dic17—41no—
107C22⋊Dic17—41no—
108#63227—22no—
109#63227—22no—
110#63227—22no—
111C4⋊Dic17—22no—
112C4⋊Dic17—22no—
113C4⋊Dic17—41no—
114#63225—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×C17—21no—
131C23×C17—21no—
132C23×C17—21no—
133C23×C17—21no—
134C23×C17—21no—
135C23×C17—21no—
136C23×C17C2×D412yes—
137C23×C17C2×D412yes—
138C23×C17C2411yescharacteristic
139C22×D17—21no—
140C22×D17—21no—
141C22×D17—21no—
142C22×D17—21no—
143C22×D17—21no—
144C22×D17—21no—
145C22×D17C2×D412yes—
146D4×C17—44no—
147D4×C17—44no—
148D4×C17—44no—
149C2×C4×C17—21no—
150C2×C4×C17—21no—
151C2×C4×C17—21no—
152C2×C4×C17—21no—
153C2×C4×C17—21no—
154C2×C4×C17—21no—
155C2×C4×C17—21no—
156C2×C4×C17—21no—
157C2×C4×C17—21no—
158C2×C4×C17—44no—
159C2×C4×C17C2×D412yes—
160C2×C4×C17C2×D412yes—
161C2×C4×C17#4112yes—
162C22⋊D17—44no—
163C22⋊D17—44no—
164C22⋊D17—44no—
165C2×Dic17—21no—
166C2×Dic17—21no—
167C2×Dic17—21no—
168C2×Dic17—21no—
169C2×Dic17—21no—
170C2×Dic17—21no—
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—44no—
186C2×Dic17C2×D412yes—
187C4×D17—44no—
188#2671—171nomaximal, Sylow
189C22×C17—22no—
190C22×C17—22no—
191C22×C17—22no—
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×C17—41no—
207C22×C17—41no—
208C22×C17—41no—
209C22×C17—41no—
210C22×C17C22×D411yescharacteristic
211C22×C17C22×D411yescharacteristic
212C22×C17C2×#4111yescharacteristic
213C22×C17H(5,𝔽2)11yescharacteristic
214C22×C17H(5,𝔽2)11yescharacteristic
215C22×C17H(5,𝔽2)11yescharacteristic
216C22×C172−1+411yescharacteristic
217C2×D17—22no—
218C2×D17—22no—
219C2×D17—22no—
220C2×D17—41no—
221C2×D17—41no—
222C2×D17—41no—
223C2×D17—41no—
224C4×C17—22no—
225C4×C17—22no—
226C4×C17—22no—
227C4×C17—41no—
228Dic17—22no—
229Dic17—41no—
230Dic17—41no—
231Dic17—41no—
232Dic17—41no—
233Dic17—41no—
234Dic17—41no—
235C2×#123—171no—
236C2×#121—171no—
237C2×#121—171no—
238C2×#121D1711yescharacteristic, complemented
239#399—171no—
240#398—171no—
241#394—171no—
242#394—171no—
243#394—171no—
244#392—171no—
245#386—171no—
246#386—171no—
247#385—171no—
248#385—171no—
249#380—171no—
250C2×C17—42no—
251C2×C17—42no—
252C2×C17—81no—
253C2×C17#53411yescharacteristic
254C2×C17#53711yescharacteristic
255C2×C17#54011yescharacteristic
256C2×C17D4211yescharacteristic
257C2×C17#54611yescharacteristic
258C2×C17#54711yescharacteristic
259C2×C17#56211yescharacteristic
260D17—42no—
261C22×D4—171no—
262C22×D4—171no—
263C22×D4—171no—
264C22×D4C2×D1711yescharacteristic
265C22×D4C2×D1711yescharacteristic
266C22×D4C2×D1711yescharacteristic
267C23×C4—171no—
268C23×C4—171no—
269C23×C4C2×D1711yescharacteristic
270#123—344no—
271#121—24no—
272#121—344no—
273#121—344no—
274C2×C4⋊C4—171no—
275C2×C4⋊C4—171no—
276C2×C4⋊C4—171no—
277C2×C4⋊C4—171no—
278C2×C4⋊C4—171no—
279C2×C4⋊C4—171no—
280C2×C4⋊C4C2×D1711yescharacteristic
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⋊C4C2×D1711yescharacteristic
293C2×C22⋊C4C2×D1711yescharacteristic
294#95—171no—
295#95—171no—
296#95—171no—
297#95—171no—
298#95—171no—
299#95—171no—
300C17#267111yescharacteristic, complemented, Sylow
301C24—171no—
302C24C22×D1711yescharacteristic
303C24C22×D1711yescharacteristic
304C24C22×D1711yescharacteristic
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—174no—
315C2×D4—342no—
316C2×D4—342no—
317C2×D4—342no—
318C2×D4—681no—
319C2×D4—681no—
320C2×D4—681no—
321C2×D4—681no—
322C2×D4—681no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—41no—
326C22×C4—171no—
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—681no—
337C22×C4—681no—
338C22×C4—681no—
339C22×C4C22×D1711yescharacteristic
340C22×C4C22×D1711yescharacteristic
341C22×C4C22×D1711yescharacteristic
342C22×C4C22×D1711yescharacteristic
343C4⋊C4—22no—
344C4⋊C4—342no—
345C4⋊C4—342no—
346C4⋊C4—342no—
347C4⋊C4—342no—
348C4⋊C4—681no—
349C4⋊C4—681no—
350C22⋊C4—22no—
351C22⋊C4—342no—
352C22⋊C4—342no—
353C22⋊C4—342no—
354C22⋊C4—342no—
355C22⋊C4—342no—
356C22⋊C4—342no—
357C22⋊C4—342no—
358C22⋊C4—681no—
359C22⋊C4—681no—
360C22⋊C4—681no—
361C22⋊C4—681no—
362C22⋊C4C22⋊D1714yescomplemented
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
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—21no—
381C23—21no—
382C23—172no—
383C23—341no—
384C23—341no—
385C23—341no—
386C23—341no—
387C23—341no—
388C23—341no—
389C23D4×D1712yes—
390C23C2×C22⋊D1712yes—
391C23C23×D1711yescharacteristic
392D4—44no—
393D4—44no—
394D4—44no—
395D4—684no—
396D4—684no—
397D4—684no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—44no—
408C2×C4—172no—
409C2×C4—341no—
410C2×C4—341no—
411C2×C4—341no—
412C2×C4—341no—
413C2×C4—341no—
414C2×C4—341no—
415C2×C4—341no—
416C2×C4—341no—
417C2×C4—341no—
418C2×C4—341no—
419C2×C4—341no—
420C2×C4—341no—
421C2×C4—341no—
422C2×C4—341no—
423C2×C4—341no—
424C2×C4—341no—
425C2×C4—341no—
426C2×C4—341no—
427C2×C4—341no—
428C2×C4—341no—
429C2×C4—684no—
430C2×C4—684no—
431C2×C4#6325412yes—
432C2×C4C2×C22⋊D1712yes—
433C2×C4C2×C22⋊D1712yes—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—342no—
456C22—342no—
457C22—342no—
458C22—681no—
459C22—681no—
460C22—681no—
461C22—681no—
462C22C2×D4×D1711yescharacteristic
463C22C2×#6325411yescharacteristic
464C22#1058769111yescharacteristic
465C22#1058769111yescharacteristic
466C22#1058769611yescharacteristic
467C22#1058769711yescharacteristic
468C22C22×C22⋊D1711yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—41no—
473C4—342no—
474C4—681no—
475C4—681no—
476C4—681no—
477C4—681no—
478C4—681no—
479C4—681no—
480C2—42no—
481C2—42no—
482C2—81no—
483C2—682no—
484C2#1176336711yescharacteristic
485C2#1176338711yescharacteristic
486C2#1176341211yescharacteristic
487C2D4×C22⋊D1711yescharacteristic
488C2#1176345311yescharacteristic
489C2#1176345911yescharacteristic
490C2#1176355411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#121C2
hypercenterC2×#121D17
hypocenterC17#2671
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#10587569C2semidirect
2. C2×#10587625C2semidirect
3. C2×#10587624C2semidirect
4. #11762745C2semidirect
5. #11762742C2semidirect
6. #11762725C2semidirect
7. #11762723C2semidirect
8. #11762723C2semidirect
9. #11762717C2semidirect
10. #11762689C2semidirect
11. #11762688C2semidirect
12. #11762684C2semidirect
13. #11762680C2semidirect
14. #11762669C2semidirect
15. C2×C17×#121C2semidirect
83. C2×C22⋊D17D4semidirect
99. C22⋊C4×C17D4semidirect
238. C2×#121D17semidirect
300. C17#2671semidirect
362. C22⋊C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105875692
C2×#105876252
C2×#105876242
#117627452
#117627422
#117627252
#117627232
#117627172
#117626892
#117626882
#117626842
#117626802
#117626692
C2×C17×#1212
#267117