Finite Group Databasebeta

← #557×#595 #17979036

Subgroups — #557×#595 #17979036

Order 4352.

2220 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14545318C216yesmaximal
2#14545313C213yesmaximal
3#14545303C213yesmaximal
4#14545299C212yesmaximal
5Q8×C4⋊#595C211yesmaximal, characteristic
6C4⋊Q8×#595C213yesmaximal
7#125×#595C216yesmaximal
8C4×Q8×#595C212yesmaximal
9C4×Q8×#595C213yesmaximal
10C4×Q8×#595C211yesmaximal, characteristic
11D17×#557C211yesmaximal, characteristic
12C2×Q8×#595C2213yes—
13C2×Q8×#595C2212yes—
14C2×Q8×#595C2211yescharacteristic
15C2×Q8×#595C2213yes—
16C2×Q8×#595C2211yescharacteristic
17(C2×#595)⋊Q8C2216yes—
18#11763760C2216yes—
19#11763760C2216yes—
20#11763760C2216yes—
21#11763760C2216yes—
22#11763758C22112yes—
23#11763758C22112yes—
24#11763755C2213yes—
25#11763753C2216yes—
26C4⋊C4×#595C2211yescharacteristic
27C4⋊C4×#595C2216yes—
28C4⋊C4×#595C2213yes—
29C4⋊C4×#595C2216yes—
30C4⋊C4×#595C2213yes—
31C4⋊C4×#595C2216yes—
32C4⋊C4×#595C2213yes—
33C4⋊C4×#595C2216yes—
34C4⋊C4×#595C2216yes—
35C4×C4⋊#595C2213yes—
36C4×C4⋊#595C2216yes—
37C4×C4⋊#595C2213yes—
38C4×C4⋊#595C2216yes—
39C42×#595C2216yes—
40C42×#595C2213yes—
41#11763538C411yescharacteristic, complemented
42#11763537C413yescomplemented
43#11763536C416yescomplemented
44#11763535C412yescomplemented
45#11763534C413yescomplemented
46C4⋊Q8×D17C2213yes—
47#125×D17C2216yes—
48C4×Q8×D17C2211yescharacteristic
49C4×Q8×D17C2212yes—
50C4×Q8×D17C2213yes—
51C17×#557C411yescharacteristic, complemented
52C2×Q8×D17C2313yes—
53C2×Q8×D17C2311yescharacteristic
54Q8×#595—28no—
55Q8×#595—24no—
56Q8×#595—212no—
57Q8×#595—212no—
58Q8×#595Q818yescomplemented
59C2×C4⋊#595C2313yes—
60C2×C4⋊#595C2316yes—
61C2×C4⋊#595C2313yes—
62C2×C4⋊#595C2316yes—
63C2×C4⋊#595C2316yes—
64C2×C4⋊#595C2316yes—
65C2×C4⋊#595C2313yes—
66C2×C4⋊#595C2311yescharacteristic
67C2×C4×#595C2312yes—
68C2×C4×#595C2316yes—
69C2×C4×#595C2313yes—
70C2×C4×#595C2313yes—
71C2×C4×#595C2313yes—
72C2×C4×#595C2316yes—
73C2×C4×#595C2316yes—
74C2×C4×#595C2316yes—
75C2×C4×#595C2313yes—
76C2×C4×#595C2313yes—
77C2×C4×#595C2311yescharacteristic
78C17×C4⋊Q8C2×C413yes—
79C17×#125C2×C416yes—
80C4×Q8×C17C2×C412yes—
81C4×Q8×C17C2×C413yes—
82C4×Q8×C17C2×C411yescharacteristic
83Q8×Dic17C2×C413yes—
84Q8×Dic17C2×C411yescharacteristic
85Q8×Dic17C2×C412yes—
86#10587630C2×C416yes—
87C4⋊C4×D17C2311yescharacteristic
88C4⋊C4×D17C2316yes—
89C4⋊C4×D17C2313yes—
90C4⋊C4×D17C2316yes—
91C4⋊C4×D17C2316yes—
92#10587576C2×C416yes—
93#10587576C2×C416yes—
94#10587575C2×C4112yes—
95#10587575C2×C4112yes—
96#10587574C2×C416yes—
97#10587574C2×C416yes—
98#10587573C2×C416yes—
99#10587573C2×C416yes—
100#10587573C2×C413yes—
101#10587573C2×C413yes—
102C42×D17C2316yes—
103C42×D17C2313yes—
104#10587556C2×C416yes—
105#10587555C2×C413yes—
106C4×Dic34C2×C413yes—
107C4×Dic34C2×C413yes—
108C4×Dic34C2×C416yes—
109C4×Dic34C2×C416yes—
110(C2×#595)⋊C4C2312yes—
111(C2×#595)⋊C4C2316yes—
112(C2×#595)⋊C4C2316yes—
113(C2×#595)⋊C4C2316yes—
114(C2×#595)⋊C4C23112yes—
115(C2×#595)⋊C4C23112yes—
116C22×#595C2416yes—
117C22×#595C2413yes—
118C22×#595C2413yes—
119C22×#595C2411yescharacteristic
120C22×#595C2412yes—
121C22×#595C2411yescharacteristic
122C2×Q8×C17C22×C413yes—
123C2×Q8×C17C22×C411yescharacteristic
124Q8×D17—26no—
125Q8×D17—26no—
126Q8×D17C2×Q814yes—
127Q8×D17#4114yes—
128C2×C4×D17C2412yes—
129C2×C4×D17C2416yes—
130C2×C4×D17C2413yes—
131C2×C4×D17C2413yes—
132C2×C4×D17C2411yescharacteristic
133C2×Dic34C22×C413yes—
134C2×Dic34C22×C416yes—
135C2×Dic34C22×C413yes—
136C4⋊#595—212no—
137C4⋊#595—212no—
138C4⋊#595—26no—
139C4⋊#595—212no—
140C4⋊#595—26no—
141C4⋊#595—46no—
142C4⋊#595C2×Q8112yes—
143C4⋊#595#4114yes—
144C4×#595—212no—
145C4×#595—212no—
146C4×#595—24no—
147C4×#595—26no—
148C4×#595—26no—
149C4×#595—212no—
150C4×#595—26no—
151C4×#595—46no—
152C4×#595—43no—
153C4×#595C2×Q8112yes—
154C4×#595#4114yes—
155C4⋊C4×C17C22×C416yes—
156C4⋊C4×C17C22×C416yes—
157C4⋊C4×C17C22×C413yes—
158C4⋊C4×C17C22×C416yes—
159C4⋊C4×C17C22×C411yescharacteristic
160C42×C17C22×C416yes—
161C42×C17C22×C413yes—
162C4⋊Dic17C22×C411yescharacteristic
163C4⋊Dic17C22×C416yes—
164C4⋊Dic17C22×C413yes—
165C4⋊Dic17C22×C416yes—
166C4⋊Dic17C22×C416yes—
167#63225C22×C412yes—
168#63225C22×C416yes—
169#63225C22×C4112yes—
170#63225C22×C416yes—
171#63225C22×C416yes—
172#63225C22×C4112yes—
173C4×Dic17C22×C416yes—
174C4×Dic17C22×C413yes—
175C4×Dic17C22×C416yes—
176C4×Dic17C22×C413yes—
177C4×Dic17C22×C413yes—
178C4×Dic17C22×C416yes—
179C4×#557—171nomaximal, Sylow
180C22×D17C2511yescharacteristic
181C2×#595—23no—
182C2×#595—21no—
183C2×#595—22no—
184C2×#595—22no—
185C2×#595—23no—
186C2×#595—21no—
187C2×#595—26no—
188C2×#595—26no—
189C2×#595—23no—
190C2×#595—23no—
191C2×#595—23no—
192C2×#595—26no—
193C2×#595C22×Q816yes—
194C2×#595C22×Q812yes—
195C2×#595C2×#4114yes—
196C2×#595C2×#4112yes—
197C2×#595C2×#4112yes—
198C2×#5952−1+412yes—
199Q8×C17—26no—
200Q8×C17C4×Q814yescomplemented
201C2×C4×C17C23×C412yes—
202C2×C4×C17C23×C413yes—
203C2×C4×C17C23×C416yes—
204C2×C4×C17C23×C413yes—
205C2×C4×C17C23×C411yescharacteristic
206C2×Dic17C23×C416yes—
207C2×Dic17C23×C412yes—
208C2×Dic17C23×C413yes—
209C2×Dic17C23×C413yes—
210C2×Dic17C23×C411yescharacteristic
211C4×D17—26no—
212C4×D17—26no—
213C4×D17—22no—
214C4×D17—23no—
215C4×D17—22no—
216C4×D17—23no—
217C4×D17C22×Q816yes—
218C4×D17C2×#4116yes—
219C4×D17C2×#4112yes—
220C4×D17C2×#4112yes—
221Dic34—26no—
222Dic34—212no—
223Dic34C4×Q8112yescomplemented
224C2×#557—171no—
225#2370—176no—
226#2365—173no—
227#2355—173no—
228#2351—172no—
229Q8×C4⋊C4—171no—
230C4×C4⋊Q8—173no—
231C4×#125—176no—
232C42×Q8—171no—
233C42×Q8—172no—
234C42×Q8—173no—
235C22×C17C24×C411yescharacteristic
236C2×D17C23×Q811yescharacteristic
237C2×D17C22×#4111yescharacteristic
238C2×D17C22×#4111yescharacteristic
239C2×D17C22×#4111yescharacteristic
240C2×D17C2×2−1+411yescharacteristic
241C2×D17#58511yescharacteristic
242#595—26no—
243#595—22no—
244#595—24no—
245#595—46no—
246#595—43no—
247#595#55714yesdirect product factor
248C4×C17—26no—
249C4×C17—22no—
250C4×C17—23no—
251C4×C17C2×C4×Q816yes—
252C4×C17C4×#4112yes—
253Dic17—26no—
254Dic17—22no—
255Dic17—23no—
256Dic17C2×C4×Q816yes—
257Dic17C4×#4112yes—
258#557—173no—
259#557—176no—
260#557—172no—
261#557—173no—
262#557—171no—
263#557#59511yescharacteristic, direct product factor
264C2×C4⋊Q8—173no—
265C2×#125—176no—
266C2×C4×Q8—171no—
267C2×C4×Q8—171no—
268C2×C4×Q8—172no—
269C2×C4×Q8—173no—
270C2×C4×Q8—173no—
271C2×C4×Q8—172no—
272C2×C4×Q8—171no—
273C2×C4×Q8—173no—
274#391—176no—
275#389—176no—
276#389—176no—
277#389—176no—
278#389—176no—
279#387—1712no—
280#387—1712no—
281#384—173no—
282#382—176no—
283C4×C4⋊C4—176no—
284C4×C4⋊C4—176no—
285C4×C4⋊C4—173no—
286C4×C4⋊C4—176no—
287C4×C4⋊C4—173no—
288C4×C4⋊C4—176no—
289C4×C4⋊C4—173no—
290C4×C4⋊C4—176no—
291C4×C4⋊C4—173no—
292C4×C4⋊C4—176no—
293C4×C4⋊C4—176no—
294C4×C4⋊C4—173no—
295C4×C4⋊C4—171no—
296C43—176no—
297C43—173no—
298C2×C17C22×C4×Q811yescharacteristic
299C2×C17C2×C4×#4111yescharacteristic
300C2×C17C4×2−1+411yescharacteristic
301D17C2×#55711yescharacteristic
302D17Q8×#4111yescharacteristic
303D17#348011yescharacteristic
304D17#348211yescharacteristic
305C22×Q8—171no—
306C22×Q8—173no—
307C4⋊Q8—176no—
308C4⋊Q8—176no—
309C4⋊Q8—176no—
310C4⋊Q8—173no—
311C4⋊Q8C2×#59513yes—
312#125—1712no—
313#125—1712no—
314#125—176no—
315#125—176no—
316#125—176no—
317#125C2×#59516yes—
318C4×Q8—178no—
319C4×Q8—172no—
320C4×Q8—173no—
321C4×Q8—171no—
322C4×Q8—176no—
323C4×Q8—176no—
324C4×Q8—173no—
325C4×Q8—173no—
326C4×Q8—176no—
327C4×Q8—173no—
328C4×Q8—176no—
329C4×Q8—173no—
330C4×Q8—3412no—
331C4×Q8—3412no—
332C4×Q8—348no—
333C4×Q8—344no—
334C4×Q8C2×#59512yes—
335C4×Q8C2×#59513yes—
336C4×Q8C2×#59511yescharacteristic
337C2×C4⋊C4—171no—
338C2×C4⋊C4—173no—
339C2×C4⋊C4—171no—
340C2×C4⋊C4—176no—
341C2×C4⋊C4—173no—
342C2×C4⋊C4—176no—
343C2×C4⋊C4—176no—
344C2×C4⋊C4—173no—
345C2×C4⋊C4—176no—
346C2×C4⋊C4—173no—
347C2×C4⋊C4—176no—
348C2×C4⋊C4—176no—
349C2×C4⋊C4—176no—
350C2×C42—171no—
351C2×C42—173no—
352C2×C42—176no—
353C2×C42—173no—
354C2×C42—176no—
355C2×C42—176no—
356C2×C42—176no—
357C2×C42—173no—
358C2×C42—173no—
359C2×C42—172no—
360C2×C42—173no—
361C2×C42—176no—
362C2×C42—173no—
363#95—172no—
364#95—176no—
365#95—176no—
366#95—176no—
367#95—1712no—
368#95—1712no—
369C17C4×#55711yescharacteristic, complemented, Sylow
370C2×Q8—174no—
371C2×Q8—174no—
372C2×Q8—173no—
373C2×Q8—176no—
374C2×Q8—173no—
375C2×Q8—346no—
376C2×Q8—346no—
377C2×Q8C22×#59513yes—
378C2×Q8C22×#59511yescharacteristic
379C22×C4—171no—
380C22×C4—173no—
381C22×C4—171no—
382C22×C4—173no—
383C22×C4—172no—
384C22×C4—171no—
385C22×C4—172no—
386C22×C4—173no—
387C22×C4—176no—
388C22×C4—173no—
389C22×C4—176no—
390C4⋊C4—1712no—
391C4⋊C4—174no—
392C4⋊C4—172no—
393C4⋊C4—171no—
394C4⋊C4—176no—
395C4⋊C4—176no—
396C4⋊C4—176no—
397C4⋊C4—176no—
398C4⋊C4—173no—
399C4⋊C4—1712no—
400C4⋊C4—176no—
401C4⋊C4—1712no—
402C4⋊C4—176no—
403C4⋊C4—346no—
404C4⋊C4—3412no—
405C4⋊C4—346no—
406C4⋊C4—3412no—
407C4⋊C4—3412no—
408C4⋊C4—686no—
409C4⋊C4C22×#59511yescharacteristic
410C4⋊C4C22×#59516yes—
411C4⋊C4C22×#59513yes—
412C4⋊C4C22×#59516yes—
413C4⋊C4C22×#59516yes—
414C42—1712no—
415C42—174no—
416C42—176no—
417C42—176no—
418C42—176no—
419C42—173no—
420C42—173no—
421C42—173no—
422C42—344no—
423C42—346no—
424C42—3412no—
425C42—3412no—
426C42—346no—
427C42—346no—
428C42—3412no—
429C42—686no—
430C42—683no—
431C42C22×#59516yes—
432C42C22×#59513yes—
433C23—171no—
434Q8—26no—
435Q8—1712no—
436Q8—3412no—
437Q8—346no—
438Q8Q8×#59514yescomplemented
439C2×C4—172no—
440C2×C4—172no—
441C2×C4—172no—
442C2×C4—173no—
443C2×C4—171no—
444C2×C4—176no—
445C2×C4—176no—
446C2×C4—172no—
447C2×C4—172no—
448C2×C4—176no—
449C2×C4—172no—
450C2×C4—174no—
451C2×C4—172no—
452C2×C4—173no—
453C2×C4—176no—
454C2×C4—346no—
455C2×C4—346no—
456C2×C4—342no—
457C2×C4—343no—
458C2×C4—342no—
459C2×C4—343no—
460C2×C4—342no—
461C2×C4—341no—
462C2×C4—343no—
463C2×C4—342no—
464C2×C4—343no—
465C2×C4—341no—
466C2×C4—343no—
467C2×C4—346no—
468C2×C4—343no—
469C2×C4—346no—
470C2×C4—343no—
471C2×C4—346no—
472C2×C4C23×#59512yes—
473C2×C4C23×#59513yes—
474C2×C4C23×#59516yes—
475C2×C4C23×#59513yes—
476C2×C4C23×#59511yescharacteristic
477C22—171no—
478C22—171no—
479C22—171no—
480C22—171no—
481C22—171no—
482C22—171no—
483C22C24×#59511yescharacteristic
484C4—26no—
485C4—22no—
486C4—23no—
487C4—174no—
488C4—176no—
489C4—172no—
490C4—346no—
491C4—344no—
492C4—342no—
493C4—346no—
494C4—342no—
495C4—343no—
496C4—683no—
497C4—686no—
498C4C2×Q8×#59516yes—
499C4#41×#59512yes—
500C2—171no—
501C2—171no—
502C2—171no—
503C2—171no—
504C2C22×Q8×#59511yescharacteristic
505C2C2×#41×#59511yescharacteristic
506C22−1+4×#59511yescharacteristic
subgroupG/H
center, FrattiniC22C24×#595
commutator, socleC22×C17C24×C4
FittingC17×#557C4
prime radical, cosocleC22×D17C25
hypercenter#557#595
hypocenterC17C4×#557
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
41. #11763538C4semidirect
42. #11763537C4semidirect
43. #11763536C4semidirect
44. #11763535C4semidirect
45. #11763534C4semidirect
51. C17×#557C4semidirect
58. Q8×#595Q8semidirect
200. Q8×C17C4×Q8semidirect
223. Dic34C4×Q8semidirect
247. #595#557direct (N × H)
263. #557#595direct (N × H)
369. C17C4×#557semidirect
438. Q8Q8×#595semidirect

Maximal subgroups

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

subgroupindex
#145453182
#145453132
#145453032
#145452992
Q8×C4⋊#5952
C4⋊Q8×#5952
#125×#5952
C4×Q8×#5952
D17×#5572
C4×#55717