Finite Group Databasebeta

← #14533857

Subgroups — #14533857

Order 2176.

696 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×#10587622C211yesmaximal, characteristic, complemented
2C2×#10587571C211yesmaximal, characteristic, complemented
3C2×#10587625C211yesmaximal, characteristic, complemented
4#11762729C211yesmaximal, characteristic, complemented
5#11762723C211yesmaximal, characteristic, complemented
6#11762717C211yesmaximal, characteristic, complemented
7#11762717C211yesmaximal, characteristic, complemented
8#11762709C211yesmaximal, characteristic, complemented
9#11762699C211yesmaximal, characteristic, complemented
10#11762697C211yesmaximal, characteristic, complemented
11#11762689C211yesmaximal, characteristic, complemented
12#11762688C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic17C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic17C211yesmaximal, 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×Dic34C2211yescharacteristic
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#10587622—24no—
39C2×#63227C2211yescharacteristic
40C2×#63227C2211yescharacteristic
41C2×#63227C2211yescharacteristic
42C2×#63227C2211yescharacteristic
43C2×#63225C2211yescharacteristic
44C2×#63225C2211yescharacteristic
45C2×C4×Dic17C2211yescharacteristic
46C2×C4×Dic17C2211yescharacteristic
47C2×C4×Dic17C2211yescharacteristic
48#10587571—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—22no—
79C2×C22⋊D17—22no—
80C2×C22⋊D17—41no—
81C2×C22⋊D17—41no—
82C2×C22⋊D17—41no—
83C2×C22⋊D17—41no—
84C22×Dic17—22no—
85C22×Dic17—22no—
86C22×Dic17—41no—
87C22×Dic17C2311yescharacteristic
88C22×Dic17C2311yescharacteristic
89C22×Dic17C2311yescharacteristic
90C22×Dic17C2311yescharacteristic
91C22×Dic17C2311yescharacteristic
92C22×Dic17C2311yescharacteristic
93C22×Dic17C2311yescharacteristic
94C2×Dic34—22no—
95C2×Dic34—22no—
96C2×Dic34—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—22no—
105C22⋊Dic17—22no—
106C22⋊Dic17—22no—
107C22⋊Dic17D414yescomplemented
108#63227—22no—
109#63227—22no—
110#63227—22no—
111#63227—22no—
112#63225—22no—
113#63225—41no—
114C4×Dic17—22no—
115C4×Dic17—22no—
116C4×Dic17—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×D17—21no—
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—44no—
183C2×Dic17C2×D412yes—
184C2×Dic17C2×D412yes—
185C2×Dic17#4112yes—
186C2×Dic17#4112yes—
187Dic34—44no—
188#2555—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×C17C2×#4111yescharacteristic
214C22×C17C2×#4111yescharacteristic
215C22×C17H(5,𝔽2)11yescharacteristic
216C22×C17H(5,𝔽2)11yescharacteristic
217C2×D17—41no—
218C2×D17—41no—
219C2×D17—41no—
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—22no—
230Dic17—22no—
231Dic17—22no—
232Dic17—41no—
233Dic17—41no—
234Dic17—41no—
235C2×#124—171no—
236C2×#124—171no—
237C2×#121—171no—
238C2×#121D1711yescharacteristic, complemented
239#395—171no—
240#394—171no—
241#392—171no—
242#392—171no—
243#390—171no—
244#388—171no—
245#388—171no—
246#386—171no—
247#386—171no—
248C4×C22⋊C4—171no—
249C4×C22⋊C4—171no—
250C2×C17—42no—
251C2×C17—42no—
252C2×C17—81no—
253C2×C17#53211yescharacteristic
254C2×C17#53211yescharacteristic
255C2×C17#53811yescharacteristic
256C2×C17D4211yescharacteristic
257C2×C17#54611yescharacteristic
258C2×C17#54611yescharacteristic
259C2×C17#55511yescharacteristic
260D17—81no—
261C22×Q8—171no—
262C22×D4—171no—
263C22×D4—171no—
264C22×D4—171no—
265C22×D4C2×D1711yescharacteristic
266C22×D4C2×D1711yescharacteristic
267C22×D4C2×D1711yescharacteristic
268C23×C4—171no—
269C23×C4C2×D1711yescharacteristic
270#124—344no—
271#124—344no—
272#121—24no—
273#121—344no—
274C2×C4⋊C4—171no—
275C2×C4⋊C4—171no—
276C2×C4⋊C4C2×D1711yescharacteristic
277C2×C22⋊C4—171no—
278C2×C22⋊C4—171no—
279C2×C22⋊C4—171no—
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⋊C4C2×D1711yescharacteristic
290C2×C22⋊C4C2×D1711yescharacteristic
291C2×C42—171no—
292C2×C42—171no—
293C2×C42—171no—
294#95—171no—
295#95—171no—
296#95—171no—
297#95—171no—
298#95—171no—
299#95—171no—
300C17#255511yescharacteristic, complemented, Sylow
301C24—171no—
302C24C22×D1711yescharacteristic
303C24C22×D1711yescharacteristic
304C24C22×D1711yescharacteristic
305C2×Q8—342no—
306C2×Q8—342no—
307C2×Q8—681no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—342no—
318C2×D4—342no—
319C2×D4—342no—
320C2×D4—342no—
321C2×D4—342no—
322C2×D4—681no—
323C2×D4—681no—
324C2×D4—681no—
325C2×D4—681no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—171no—
330C22×C4—171no—
331C22×C4—171no—
332C22×C4—171no—
333C22×C4—171no—
334C22×C4—171no—
335C22×C4—171no—
336C22×C4—342no—
337C22×C4—342no—
338C22×C4—681no—
339C22×C4C22×D1711yescharacteristic
340C22×C4C22×D1711yescharacteristic
341C22×C4C22×D1711yescharacteristic
342C22×C4C22×D1711yescharacteristic
343C4⋊C4—22no—
344C4⋊C4—342no—
345C4⋊C4—681no—
346C22⋊C4—22no—
347C22⋊C4—174no—
348C22⋊C4—342no—
349C22⋊C4—342no—
350C22⋊C4—342no—
351C22⋊C4—342no—
352C22⋊C4—342no—
353C22⋊C4—342no—
354C22⋊C4—342no—
355C22⋊C4—342no—
356C22⋊C4—342no—
357C22⋊C4—342no—
358C22⋊C4—342no—
359C22⋊C4C22⋊D1714yescomplemented
360C42—342no—
361C42—342no—
362C42—681no—
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—341no—
383C23—341no—
384C23—341no—
385C23—341no—
386C23—341no—
387C23—341no—
388C23—341no—
389C23D4×D1712yes—
390C23C2×C22⋊D1712yes—
391C23C23×D1711yescharacteristic
392Q8—684no—
393D4—44no—
394D4—44no—
395D4—44no—
396D4—684no—
397D4—684no—
398D4—684no—
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—21no—
408C2×C4—44no—
409C2×C4—172no—
410C2×C4—172no—
411C2×C4—172no—
412C2×C4—172no—
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—341no—
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—681no—
456C22—681no—
457C22—681no—
458C22—681no—
459C22—681no—
460C22—681no—
461C22—681no—
462C22C2×D4×D1711yescharacteristic
463C22C2×#6325411yescharacteristic
464C22#1058769111yescharacteristic
465C22#1058769111yescharacteristic
466C22#41×D1711yescharacteristic
467C22#41×D1711yescharacteristic
468C22C22×C22⋊D1711yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—41no—
473C4—342no—
474C4—342no—
475C4—342no—
476C4—342no—
477C4—681no—
478C4—681no—
479C4—681no—
480C2—42no—
481C2—42no—
482C2—81no—
483C2—1361no—
484C2#1176335411yescharacteristic
485C2#1176335511yescharacteristic
486C2#1176339111yescharacteristic
487C2D4×C22⋊D1711yescharacteristic
488C2#1176344811yescharacteristic
489C2#1176345311yescharacteristic
490C2#1176352311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#121C2
hypercenterC2×#121D17
hypocenterC17#2555
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#10587622C2semidirect
2. C2×#10587571C2semidirect
3. C2×#10587625C2semidirect
4. #11762729C2semidirect
5. #11762723C2semidirect
6. #11762717C2semidirect
7. #11762717C2semidirect
8. #11762709C2semidirect
9. #11762699C2semidirect
10. #11762697C2semidirect
11. #11762689C2semidirect
12. #11762688C2semidirect
13. C22⋊C4×Dic17C2semidirect
14. C4×C22⋊Dic17C2semidirect
15. C2×C17×#121C2semidirect
99. C22⋊C4×C17D4semidirect
107. C22⋊Dic17D4semidirect
238. C2×#121D17semidirect
300. C17#2555semidirect
359. C22⋊C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876222
C2×#105875712
C2×#105876252
#117627292
#117627232
#117627172
#117627092
#117626992
#117626972
#117626892
#117626882
C22⋊C4×Dic172
C4×C22⋊Dic172
C2×C17×#1212
#255517