Finite Group Databasebeta

← #14535374

Subgroups — #14535374

Order 2176.

612 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587625C211yesmaximal, characteristic, complemented
2C2×#10587624C211yesmaximal, characteristic, complemented
3#11762743C211yesmaximal, characteristic, complemented
4#11762724C211yesmaximal, characteristic, complemented
5#11762723C211yesmaximal, characteristic, complemented
6#11762723C211yesmaximal, characteristic, complemented
7#11762717C211yesmaximal, characteristic, complemented
8#11762709C211yesmaximal, characteristic, complemented
9#11762702C211yesmaximal, characteristic, complemented
10#11762697C211yesmaximal, characteristic, complemented
11#11762689C211yesmaximal, characteristic, complemented
12#11762685C211yesmaximal, characteristic, complemented
13#11762671C211yesmaximal, characteristic, complemented
14#11762670C211yesmaximal, characteristic, complemented
15C17×#392C211yesmaximal, characteristic, complemented
16C22×D4×C17C2211yescharacteristic
17C22×D4×C17C2211yescharacteristic
18C22×D4×C17C2211yescharacteristic
19C22×C22⋊D17C2211yescharacteristic
20C22×C22⋊D17C2211yescharacteristic
21C22×C22⋊D17C2211yescharacteristic
22C23×Dic17C2211yescharacteristic
23C22×C4×D17C2211yescharacteristic
24C2×C22⋊C4×C17C2211yescharacteristic
25C2×C22⋊C4×C17C2211yescharacteristic
26C2×C22⋊C4×C17C2211yescharacteristic
27C2×C22⋊Dic17C2211yescharacteristic
28C2×C22⋊Dic17C2211yescharacteristic
29C2×C22⋊Dic17C2211yescharacteristic
30C2×C22⋊Dic17C2211yescharacteristic
31C2×C22⋊Dic17C2211yescharacteristic
32C2×C22⋊Dic17C2211yescharacteristic
33C2×C22⋊Dic17C2211yescharacteristic
34C2×C22⋊Dic17C2211yescharacteristic
35#10587625—24no—
36#10587624—24no—
37C2×#63227C2211yescharacteristic
38C2×#63227C2211yescharacteristic
39C2×#63227C2211yescharacteristic
40C2×C4⋊Dic17C2211yescharacteristic
41C2×#63225C2211yescharacteristic
42C2×#63225C2211yescharacteristic
43C2×#63225C2211yescharacteristic
44C2×C4×Dic17C2211yescharacteristic
45C17×#95C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53C24×C17C2311yescharacteristic
54C24×C17C2311yescharacteristic
55C24×C17C2311yescharacteristic
56C23×D17C2311yescharacteristic
57C2×D4×C17—22no—
58C2×D4×C17—22no—
59C2×D4×C17—22no—
60C2×D4×C17—41no—
61C2×D4×C17—41no—
62C2×D4×C17—41no—
63C2×D4×C17—41no—
64C2×D4×C17—41no—
65C2×D4×C17—41no—
66C22×C4×C17C2311yescharacteristic
67C22×C4×C17C2311yescharacteristic
68C22×C4×C17C2311yescharacteristic
69C22×C4×C17C2311yescharacteristic
70C2×C22⋊D17—22no—
71C2×C22⋊D17—22no—
72C2×C22⋊D17—22no—
73C2×C22⋊D17—41no—
74C2×C22⋊D17—41no—
75C2×C22⋊D17—41no—
76C2×C22⋊D17—41no—
77C2×C22⋊D17—41no—
78C2×C22⋊D17D414yescomplemented
79C22×Dic17—22no—
80C22×Dic17—22no—
81C22×Dic17—41no—
82C22×Dic17C2311yescharacteristic
83C22×Dic17C2311yescharacteristic
84C22×Dic17C2311yescharacteristic
85C22×Dic17C2311yescharacteristic
86C22×Dic17C2311yescharacteristic
87C22×Dic17C2311yescharacteristic
88C22×Dic17C2311yescharacteristic
89C2×C4×D17—22no—
90C2×C4×D17—22no—
91C2×C4×D17—41no—
92C22⋊C4×C17—22no—
93C22⋊C4×C17—41no—
94C22⋊C4×C17—41no—
95C22⋊Dic17—22no—
96C22⋊Dic17—22no—
97C22⋊Dic17—22no—
98C22⋊Dic17—22no—
99C22⋊Dic17—41no—
100C22⋊Dic17—41no—
101C22⋊Dic17—41no—
102C22⋊Dic17—41no—
103#63227—22no—
104#63227—41no—
105#63227—41no—
106C4⋊Dic17—22no—
107#63225—22no—
108#63225—41no—
109#63225—41no—
110C4×Dic17—41no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
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×C17C2×D412yes—
132C23×C17C2411yescharacteristic
133C22×D17—21no—
134C22×D17—21no—
135C22×D17—21no—
136C22×D17—21no—
137C22×D17—21no—
138C22×D17—21no—
139C22×D17C2×D412yes—
140D4×C17—44no—
141D4×C17—44no—
142D4×C17—44no—
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×C17—21no—
152C2×C4×C17—21no—
153C2×C4×C17—21no—
154C2×C4×C17#4112yes—
155C22⋊D17—44no—
156C22⋊D17—44no—
157C22⋊D17—44no—
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×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—44no—
178C2×Dic17C2×D412yes—
179C2×Dic17#4112yes—
180C4×D17—44no—
181#2699—171nomaximal, Sylow
182C22×C17—22no—
183C22×C17—22no—
184C22×C17—22no—
185C22×C17—41no—
186C22×C17—41no—
187C22×C17—41no—
188C22×C17—41no—
189C22×C17—41no—
190C22×C17—41no—
191C22×C17—41no—
192C22×C17—41no—
193C22×C17—41no—
194C22×C17—41no—
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×C17C22×D411yescharacteristic
204C22×C17C2×#4111yescharacteristic
205C22×C17C2×#4111yescharacteristic
206C22×C17H(5,𝔽2)11yescharacteristic
207C22×C17H(5,𝔽2)11yescharacteristic
208C22×C17H(5,𝔽2)11yescharacteristic
209C22×C17H(5,𝔽2)11yescharacteristic
210C2×D17—22no—
211C2×D17—22no—
212C2×D17—22no—
213C2×D17—41no—
214C2×D17—41no—
215C2×D17—41no—
216C2×D17—41no—
217C4×C17—22no—
218C4×C17—41no—
219C4×C17—41no—
220C4×C17—41no—
221Dic17—22no—
222Dic17—22no—
223Dic17—41no—
224Dic17—41no—
225Dic17—41no—
226Dic17—41no—
227Dic17—41no—
228C2×#121—171no—
229C2×#121—171no—
230#398—171no—
231#394—171no—
232#394—171no—
233#394—171no—
234#392—171no—
235#392D1711yescharacteristic, complemented
236#390—171no—
237#388—171no—
238#388—171no—
239#386—171no—
240#386—171no—
241#381—171no—
242#381—171no—
243C2×C17—42no—
244C2×C17—81no—
245C2×C17—81no—
246C2×C17#53411yescharacteristic
247C2×C17#54011yescharacteristic
248C2×C17#54011yescharacteristic
249C2×C17#54611yescharacteristic
250C2×C17#54611yescharacteristic
251C2×C17#55511yescharacteristic
252C2×C17#56011yescharacteristic
253D17—42no—
254C22×D4—171no—
255C22×D4—171no—
256C22×D4—171no—
257C22×D4C2×D1711yescharacteristic
258C22×D4C2×D1711yescharacteristic
259C22×D4C2×D1711yescharacteristic
260C23×C4—171no—
261C23×C4—171no—
262#121—344no—
263#121—344no—
264C2×C4⋊C4—171no—
265C2×C4⋊C4—171no—
266C2×C4⋊C4—171no—
267C2×C4⋊C4—171no—
268C2×C22⋊C4—171no—
269C2×C22⋊C4—171no—
270C2×C22⋊C4—171no—
271C2×C22⋊C4—171no—
272C2×C22⋊C4—171no—
273C2×C22⋊C4—171no—
274C2×C22⋊C4—171no—
275C2×C22⋊C4—171no—
276C2×C22⋊C4—171no—
277C2×C22⋊C4—171no—
278C2×C22⋊C4—171no—
279C2×C22⋊C4C2×D1711yescharacteristic
280C2×C22⋊C4C2×D1711yescharacteristic
281C2×C22⋊C4C2×D1711yescharacteristic
282C2×C42—171no—
283#95—171no—
284#95—171no—
285#95—171no—
286#95—171no—
287#95—171no—
288#95—171no—
289#95—171no—
290#95C2×D1711yescharacteristic
291C17#269911yescharacteristic, complemented, Sylow
292C24—171no—
293C24C22×D1711yescharacteristic
294C24C22×D1711yescharacteristic
295C24C22×D1711yescharacteristic
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—174no—
306C2×D4—342no—
307C2×D4—342no—
308C2×D4—342no—
309C2×D4—681no—
310C2×D4—681no—
311C2×D4—681no—
312C2×D4—681no—
313C2×D4—681no—
314C22×C4—171no—
315C22×C4—171no—
316C22×C4—171no—
317C22×C4—171no—
318C22×C4—171no—
319C22×C4—171no—
320C22×C4—171no—
321C22×C4—342no—
322C22×C4—342no—
323C22×C4—342no—
324C22×C4—342no—
325C22×C4—681no—
326C22×C4—681no—
327C22×C4C22×D1711yescharacteristic
328C22×C4C22×D1711yescharacteristic
329C22×C4C22×D1711yescharacteristic
330C22×C4C22×D1711yescharacteristic
331C4⋊C4—342no—
332C4⋊C4—342no—
333C4⋊C4—681no—
334C4⋊C4—681no—
335C22⋊C4—22no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—342no—
339C22⋊C4—342no—
340C22⋊C4—342no—
341C22⋊C4—342no—
342C22⋊C4—342no—
343C22⋊C4—681no—
344C22⋊C4—681no—
345C22⋊C4—681no—
346C22⋊C4—681no—
347C22⋊C4—681no—
348C22⋊C4—681no—
349C42—681no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—172no—
371C23—341no—
372C23—341no—
373C23—341no—
374C23—341no—
375C23—341no—
376C23—341no—
377C23D4×D1712yes—
378C23C23×D1711yescharacteristic
379D4—44no—
380D4—44no—
381D4—44no—
382D4—684no—
383D4—684no—
384D4—684no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—172no—
397C2×C4—172no—
398C2×C4—341no—
399C2×C4—341no—
400C2×C4—341no—
401C2×C4—341no—
402C2×C4—341no—
403C2×C4—341no—
404C2×C4—341no—
405C2×C4—341no—
406C2×C4—341no—
407C2×C4—341no—
408C2×C4—341no—
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—684no—
418C2×C4—684no—
419C2×C4#6325412yes—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—342no—
442C22—342no—
443C22—342no—
444C22—681no—
445C22—681no—
446C22—681no—
447C22—681no—
448C22C2×D4×D1711yescharacteristic
449C22C2×#6325411yescharacteristic
450C22#1058769111yescharacteristic
451C22#1058769111yescharacteristic
452C22#1058769111yescharacteristic
453C22#1058769111yescharacteristic
454C22#41×D1711yescharacteristic
455C4—22no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—342no—
460C4—342no—
461C4—681no—
462C4—681no—
463C4—681no—
464C4—681no—
465C4—681no—
466C2—42no—
467C2—81no—
468C2—81no—
469C2—682no—
470C2#1176336911yescharacteristic
471C2#1176340911yescharacteristic
472C2#1176341311yescharacteristic
473C2#1176344611yescharacteristic
474C2#1176344811yescharacteristic
475C2#1176352311yescharacteristic
476C2#1176354811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#392C2
hypercenter#392D17
hypocenterC17#2699
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#10587625C2semidirect
2. C2×#10587624C2semidirect
3. #11762743C2semidirect
4. #11762724C2semidirect
5. #11762723C2semidirect
6. #11762723C2semidirect
7. #11762717C2semidirect
8. #11762709C2semidirect
9. #11762702C2semidirect
10. #11762697C2semidirect
11. #11762689C2semidirect
12. #11762685C2semidirect
13. #11762671C2semidirect
14. #11762670C2semidirect
15. C17×#392C2semidirect
78. C2×C22⋊D17D4semidirect
235. #392D17semidirect
291. C17#2699semidirect

Maximal subgroups

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

subgroupindex
C2×#105876252
C2×#105876242
#117627432
#117627242
#117627232
#117627172
#117627092
#117627022
#117626972
#117626892
#117626852
#117626712
#117626702
C17×#3922
#269917