Finite Group Databasebeta

← #17088493

Subgroups — #17088493

Order 4352.

1028 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763168C211yesmaximal, characteristic
2C2×#11763094C211yesmaximal, characteristic, complemented
3#14533989C211yesmaximal, characteristic, complemented
4#14530027C212yesmaximal, complemented
5#14530020C212yesmaximal
6#14529725C211yesmaximal, characteristic, complemented
7#14529717C211yesmaximal, characteristic
8#14529052C212yesmaximal, complemented
9#14529045C212yesmaximal
10#14528614C211yesmaximal, characteristic, complemented
11C2×C17×#556C211yesmaximal, characteristic
12C2×#10587555C2211yescharacteristic
13#11763168—24no—
14#11763094—24no—
15C2×#10587479C2212yes—
16C2×C4⋊#3622C2211yescharacteristic
17C2×#10587506C2212yes—
18C2×#10587506C2212yes—
19C2×#10587506C2212yes—
20C2×#10587506C2212yes—
21C2×#10587503C2212yes—
22C2×#10587503C2212yes—
23C2×#10587474C2211yescharacteristic
24#11762729—21no—
25#11762729—21no—
26#11762716—21no—
27#11762707C2212yes—
28#11762707C2212yes—
29#11762678—21no—
30C4×C4⋊Dic17C2211yescharacteristic
31#11762571C2212yes—
32#11762564C2212yes—
33#11762564C2212yes—
34C17×#556—24no—
35C17×#556—24no—
36C2×C17×C4⋊Q8C2211yescharacteristic
37C2×C17×#126—21no—
38C2×C17×#126—21no—
39C2×C17×#124C2211yescharacteristic
40C2×C17×#124C2211yescharacteristic
41C2×C17×C22⋊Q8—21no—
42C2×C4×Q8×C17C2211yescharacteristic
43C2×C4×Q8×C17C2211yescharacteristic
44C2×C4×Q8×C17C2211yescharacteristic
45C2×C4×D4×C17C2211yescharacteristic
46C2×C17×#117—21no—
47C22×Q8×C17D411yescharacteristic
48C22×Q8×C17C2311yescharacteristic
49C22×Q8×C17C2311yescharacteristic
50C22×D4×C17C2311yescharacteristic
51C23×C4×C17—21no—
52C22×Dic34—21no—
53C17×C4⋊Q8—22no—
54C17×C4⋊Q8—22no—
55C17×#126—28no—
56C17×#126—28no—
57C17×#124—22no—
58C17×#124—22no—
59C17×#124—22no—
60C17×#124D414yes—
61C17×C22⋊Q8—28no—
62C4×Q8×C17—22no—
63C4×Q8×C17—22no—
64C4×Q8×C17—22no—
65C4×Q8×C17—22no—
66C4×Q8×C17—22no—
67C4×Q8×C17D414yes—
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C17×#117—28no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17D411yescharacteristic
76C2×C4⋊C4×C17D411yescharacteristic
77C2×C4⋊C4×C17D411yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17—21no—
83C2×C22⋊C4×C17—21no—
84C2×C42×C17—21no—
85C2×C42×C17—21no—
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×C42×C17C2311yescharacteristic
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17C2312yes—
91C2×C4⋊Dic17C2312yes—
92C2×#63225—22no—
93C2×#63225—22no—
94C2×C4×Dic17—21no—
95C22×#3622C2312yes—
96C22×#3622C2312yes—
97#10587555—22no—
98#10587555—22no—
99#10587506—24no—
100#10587506—24no—
101#10587506—24no—
102#10587506—24no—
103#10587503—24no—
104#10587503—24no—
105#10587502—22no—
106#10587502—22no—
107#10587502—21no—
108#10587479—24no—
109C4⋊#3622—22no—
110#10587474—22no—
111C24×C17—21no—
112C2×Q8×C17—22no—
113C2×Q8×C17—22no—
114C2×Q8×C17—21no—
115C2×Q8×C17—21no—
116C2×Q8×C17—22no—
117C2×Q8×C17—22no—
118C2×Q8×C17—22no—
119C2×Q8×C17—22no—
120C2×Q8×C17—41no—
121C2×Q8×C17C2×D412yes—
122C2×Q8×C17C2×D412yes—
123C2×D4×C17—21no—
124C2×D4×C17—22no—
125C2×D4×C17—22no—
126C2×D4×C17C2×D412yes—
127C22×C4×C17—24no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—42no—
132C22×C4×C17—42no—
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2411yescharacteristic
140C22×Dic17—22no—
141C22×Dic17—22no—
142C2×Dic34—42no—
143C2×Dic34—82no—
144C4⋊C4×C17—24no—
145C4⋊C4×C17—24no—
146C4⋊C4×C17—24no—
147C4⋊C4×C17—24no—
148C4⋊C4×C17—22no—
149C4⋊C4×C17—22no—
150C4⋊C4×C17—22no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17C2×D412yes—
153C22⋊C4×C17—24no—
154C22⋊C4×C17—24no—
155C22⋊C4×C17—24no—
156C22⋊C4×C17—24no—
157C22⋊C4×C17—24no—
158C42×C17—24no—
159C42×C17—24no—
160C42×C17—21no—
161C42×C17—21no—
162C42×C17—21no—
163C42×C17—21no—
164C42×C17C2×D412yes—
165C42×C17C2×D412yes—
166C4⋊Dic17—22no—
167C4⋊Dic17—22no—
168C4⋊Dic17—22no—
169C4⋊Dic17—42no—
170C4⋊Dic17#4114yes—
171#63225—82no—
172#63225—82no—
173C4×Dic17—42no—
174C2×#3622—22no—
175C2×#3622—22no—
176C2×#3622—22no—
177C2×#3622—22no—
178C2×#3622—22no—
179C2×#3622—22no—
180#25662—171nomaximal, Sylow
181C23×C17—22no—
182C23×C17—42no—
183C23×C17—41no—
184C23×C17—41no—
185C23×C17—42no—
186C23×C17C22×D411yescharacteristic
187Q8×C17—22no—
188Q8×C17—22no—
189Q8×C17—24no—
190Q8×C17—44no—
191Q8×C17—41no—
192Q8×C17—41no—
193Q8×C17—42no—
194D4×C17—24no—
195D4×C17—41no—
196D4×C17—41no—
197C2×C4×C17—21no—
198C2×C4×C17—21no—
199C2×C4×C17—21no—
200C2×C4×C17—21no—
201C2×C4×C17—22no—
202C2×C4×C17—21no—
203C2×C4×C17—22no—
204C2×C4×C17—21no—
205C2×C4×C17—22no—
206C2×C4×C17—22no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—22no—
211C2×C4×C17—22no—
212C2×C4×C17—48no—
213C2×C4×C17—42no—
214C2×C4×C17—42no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17#12012yes—
218C2×C4×C17#12012yes—
219C2×C4×C17#13712yes—
220C2×C4×C17C22×D411yescharacteristic
221C2×C4×C17C22×D411yescharacteristic
222C2×C4×C17C2×#4112yes—
223C2×C4×C172−1+412yes—
224C2×Dic17—24no—
225C2×Dic17—42no—
226C2×Dic17—42no—
227C2×Dic17—42no—
228C2×Dic17—42no—
229C2×Dic17—42no—
230Dic34—84no—
231#3622—42no—
232#3622—42no—
233C2×#556D1711yescharacteristic
234C2×#489—171no—
235C2×#475—171no—
236#2567—171no—
237#2043—172no—
238#2042—172no—
239#1985—171no—
240#1983—171no—
241#1865—172no—
242#1864—172no—
243#1763—171no—
244C22×C17—42no—
245C22×C17—42no—
246C22×C17—42no—
247C22×C17—82no—
248C22×C17—82no—
249C22×C17C2×#12011yescharacteristic
250C22×C17#54812yes—
251C22×C17C2×#13711yescharacteristic
252C22×C17#57511yescharacteristic
253C22×C17#57712yes—
254C4×C17—22no—
255C4×C17—22no—
256C4×C17—22no—
257C4×C17—42no—
258C4×C17—42no—
259C4×C17—41no—
260C4×C17—42no—
261C4×C17—41no—
262C4×C17—41no—
263C4×C17#53211yescharacteristic
264C4×C17#53611yescharacteristic
265C4×C17#54712yes—
266Dic17—44no—
267Dic17—82no—
268#556—24no—
269#556—24no—
270C2×C4⋊Q8—171no—
271C2×C4⋊Q8C2×D1711yescharacteristic
272C2×#126—21no—
273C2×#126—21no—
274C2×#124C2×D1711yescharacteristic
275C2×#124C2×D1711yescharacteristic
276C2×C22⋊Q8—21no—
277C2×C4×Q8C2×D1711yescharacteristic
278C2×C4×Q8C2×D1711yescharacteristic
279C2×C4×Q8C2×D1711yescharacteristic
280C2×C4×D4C2×D1711yescharacteristic
281C2×#117—21no—
282#489—344no—
283#475—344no—
284C2×#106—172no—
285C2×C4⋊C8—171no—
286C2×#103—172no—
287C2×#103—172no—
288C2×#103—172no—
289C2×#103—172no—
290C2×#102—172no—
291C2×#102—172no—
292C2×#97—171no—
293#395—341no—
294#395—341no—
295#391—341no—
296#389—172no—
297#389—172no—
298#384—341no—
299C4×C4⋊C4—171no—
300#340—172no—
301#336—172no—
302#336—172no—
303C2×C17—82no—
304C2×C17#243211yescharacteristic
305C2×C17#300711yescharacteristic
306C2×C17#301011yescharacteristic
307C2×C17#336912yes—
308C2×C17#337612yes—
309C22×Q8—341no—
310C22×Q8C22⋊D1711yescharacteristic
311C22×Q8C22×D1711yescharacteristic
312C22×Q8C22×D1711yescharacteristic
313C22×D4C22×D1711yescharacteristic
314C23×C4—21no—
315C22×C8—172no—
316C22×C8—172no—
317C4⋊Q8—22no—
318C4⋊Q8—22no—
319C4⋊Q8—342no—
320C4⋊Q8—342no—
321#126—28no—
322#126—28no—
323#124—22no—
324#124—22no—
325#124—22no—
326#124C22⋊D1714yes—
327C22⋊Q8—28no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8—22no—
332C4×Q8—22no—
333C4×Q8C22⋊D1714yes—
334C4×D4—22no—
335C4×D4—22no—
336#117—28no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—172no—
342C2×C4⋊C4—172no—
343C2×C4⋊C4—341no—
344C2×C4⋊C4—342no—
345C2×C4⋊C4—342no—
346C2×C4⋊C4C22⋊D1711yescharacteristic
347C2×C4⋊C4C22⋊D1711yescharacteristic
348C2×C4⋊C4C22⋊D1711yescharacteristic
349C2×C4⋊C4C22×D1711yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—341no—
358C2×C42C22×D1711yescharacteristic
359C2×C42C22×D1711yescharacteristic
360C2×C42C22×D1711yescharacteristic
361#106—344no—
362C4⋊C8—342no—
363#103—344no—
364#103—344no—
365#103—344no—
366#103—344no—
367#102—344no—
368#102—344no—
369#97—342no—
370#95—341no—
371#95—342no—
372#95—342no—
373C17#2566211yescharacteristic, complemented, Sylow
374C24—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—41no—
384C2×Q8—682no—
385C2×Q8—1362no—
386C2×Q8C2×C22⋊D1712yes—
387C2×Q8C2×C22⋊D1712yes—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4C2×C22⋊D1712yes—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—342no—
399C22×C4—342no—
400C22×C4C2×C22⋊D1711yescharacteristic
401C22×C4C2×C22⋊D1711yescharacteristic
402C22×C4C2×C22⋊D1711yescharacteristic
403C22×C4C2×C22⋊D1711yescharacteristic
404C22×C4C2×C22⋊D1711yescharacteristic
405C22×C4C2×C22⋊D1711yescharacteristic
406C22×C4C23×D1711yescharacteristic
407C2×C8—342no—
408C2×C8—342no—
409C2×C8—342no—
410C2×C8—342no—
411C2×C8—342no—
412C2×C8—342no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—21no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—174no—
422C4⋊C4—342no—
423C4⋊C4—342no—
424C4⋊C4—342no—
425C4⋊C4—682no—
426C4⋊C4—1362no—
427C4⋊C4—1362no—
428C4⋊C4C2×C22⋊D1712yes—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C42—24no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—24no—
440C42—682no—
441C42C2×C22⋊D1712yes—
442C42C2×C22⋊D1712yes—
443C23—22no—
444C23—42no—
445C23—41no—
446C23—41no—
447C23—42no—
448C23C22×C22⋊D1711yescharacteristic
449Q8—24no—
450Q8—22no—
451Q8—22no—
452Q8—44no—
453Q8—42no—
454Q8—41no—
455Q8—41no—
456Q8—1364no—
457D4—24no—
458D4—41no—
459D4—41no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—344no—
481C2×C4—682no—
482C2×C4—682no—
483C2×C4—682no—
484C2×C4—682no—
485C2×C4—682no—
486C2×C4#1058763712yes—
487C2×C4#1058763912yes—
488C2×C4#1058763912yes—
489C2×C4#41×D1712yes—
490C2×C4#1058769712yes—
491C2×C4C22×C22⋊D1711yescharacteristic
492C2×C4C22×C22⋊D1711yescharacteristic
493C8—682no—
494C8—682no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—82no—
499C22—82no—
500C22C2×#1058763911yescharacteristic
501C22#1176346612yes—
502C22C2×#1058763711yescharacteristic
503C22#1176360711yescharacteristic
504C22#1176361712yes—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—42no—
509C4—42no—
510C4—41no—
511C4—42no—
512C4—41no—
513C4—41no—
514C4—684no—
515C4—1362no—
516C4#1176335511yescharacteristic
517C4#1176338011yescharacteristic
518C4#1176345812yes—
519C2—82no—
520C2#1453240811yescharacteristic
521C2#1453804411yescharacteristic
522C2#1453807011yescharacteristic
523C2#1454152012yes—
524C2#1454160512yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#556C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenterC2×#556D17
hypocenterC17#25662
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. C2×#11763094C2semidirect
3. #14533989C2semidirect
4. #14530027C2semidirect
6. #14529725C2semidirect
8. #14529052C2semidirect
10. #14528614C2semidirect
373. C17#25662semidirect

Maximal subgroups

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

subgroupindex
C2×#117631682
C2×#117630942
#145339892
#145300272
#145300202
#145297252
#145297172
#145290522
#145290452
#145286142
C2×C17×#5562
#2566217