Finite Group Databasebeta

← #16936648

Subgroups — #16936648

Order 4352.

644 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763094C211yesmaximal, characteristic, complemented
2C2×#11763015C211yesmaximal, characteristic
3#14533702C211yesmaximal, characteristic
4#14529745C211yesmaximal, characteristic, complemented
5C2×#11762538C211yesmaximal, characteristic
6#14526668C211yesmaximal, characteristic, complemented
7C2×C17×Q8⋊C8C211yesmaximal, characteristic, complemented
8C22×#63219—21no—
9C2×#10587555C2211yescharacteristic
10C2×#10587632—21no—
11C2×C4×D68C2211yescharacteristic
12#11763094—22no—
13#11763094—22no—
14#11763015—22no—
15#11763015—22no—
16C2×#10587479—21no—
17C2×C4⋊#3622C2211yescharacteristic
18C2×#10587487—21no—
19C2×#10587506—21no—
20#11762731—22no—
21#11762711—21no—
22#11762707—21no—
23#11762679C2211yescharacteristic
24#11762666—21no—
25#11762572—22no—
26#11762538—22no—
27C2×C4×Q8×C17C2211yescharacteristic
28C2×C17×C4⋊C8C2211yescharacteristic
29C2×C4×C8×C17C2211yescharacteristic
30C17×Q8⋊C8—22no—
31C22×Q8×C17D411yescharacteristic
32C22×D68D411yescharacteristic
33C22×C4×D17—21no—
34C22×Dic34—21no—
35C22×C8×C17—21no—
36C22×C8×C17—21no—
37C4×Q8×C17—21no—
38C4×Q8×C17—21no—
39C4×Q8×C17—21no—
40C4×Q8×C17—21no—
41C2×C4⋊C4×C17—21no—
42C2×C4⋊C4×C17D411yescharacteristic
43C2×C42×C17—21no—
44C2×C42×C17C2311yescharacteristic
45#10587632—42no—
46#10587632—42no—
47C2×#63227—22no—
48C2×#63227—22no—
49C2×#63227—21no—
50C2×C4⋊Dic17—21no—
51C2×C4⋊Dic17—21no—
52C2×C4⋊Dic17—21no—
53C2×C4⋊Dic17—21no—
54C2×C4⋊Dic17D411yescharacteristic
55C2×C4⋊Dic17D412yes—
56C2×#63225—22no—
57C22×#3622—21no—
58C2×#63219—24no—
59C2×#63219—42no—
60C2×#63219—42no—
61C4×D68—21no—
62C4×D68—21no—
63C4×D68—21no—
64C4×D68—21no—
65#10587555—21no—
66#10587555—21no—
67#10587555—21no—
68#10587555D412yes—
69C17×C4⋊C8—21no—
70C17×C4⋊C8D412yes—
71C4×C8×C17—21no—
72C4×C8×C17—21no—
73#10587506—24no—
74#10587502—22no—
75#10587487—24no—
76#10587479—24no—
77C4⋊#3622—21no—
78C4⋊#3622D412yes—
79C23×D17—21no—
80C2×Q8×C17—21no—
81C2×Q8×C17—21no—
82C2×Q8×C17—21no—
83C2×Q8×C17—21no—
84C2×Q8×C17—21no—
85C2×Q8×C17SD1612yescomplemented
86C22×C4×C17—21no—
87C22×C4×C17—21no—
88C22×C4×C17C2×D411yescharacteristic
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×Dic17—21no—
92C22×Dic17—22no—
93C22×Dic17—41no—
94C2×D68—21no—
95C2×D68—21no—
96C2×D68—21no—
97C2×D68—21no—
98C2×D68—21no—
99C2×D68SD1612yes—
100C2×C4×D17—41no—
101C2×C4×D17—41no—
102C2×C4×D17—41no—
103C2×C4×D17—41no—
104C2×C4×D17—41no—
105C2×C4×D17—41no—
106C2×Dic34—22no—
107C2×Dic34—41no—
108C2×Dic34—41no—
109C2×Dic34—41no—
110C2×Dic34—41no—
111C2×Dic34—41no—
112C2×C8×C17—22no—
113C2×C8×C17—22no—
114C2×C8×C17—41no—
115C2×C8×C17—41no—
116C2×C8×C17—41no—
117C2×C8×C17—41no—
118C4⋊C4×C17—21no—
119C4⋊C4×C17—21no—
120C4⋊C4×C17—41no—
121C4⋊C4×C17—41no—
122C42×C17—41no—
123C42×C17—41no—
124C42×C17C2×D411yescharacteristic
125C42×C17C2×D411yescharacteristic
126C42×C17C2×D411yescharacteristic
127C42×C17#4111yescharacteristic
128#63227—44no—
129#63227—41no—
130#63227—41no—
131#63227—82no—
132C4⋊Dic17—22no—
133C4⋊Dic17—21no—
134C4⋊Dic17—22no—
135C4⋊Dic17—21no—
136C4⋊Dic17—41no—
137C4⋊Dic17—41no—
138C4⋊Dic17—41no—
139C4⋊Dic17—41no—
140C4⋊Dic17—41no—
141C4⋊Dic17—41no—
142C4⋊Dic17—42no—
143#63225—44no—
144C2×#3622—22no—
145C2×#3622—41no—
146C2×#3622—41no—
147#63219—48no—
148#10232—171nomaximal, Sylow
149C23×C17#12011yescharacteristic
150C22×D17—22no—
151C22×D17—41no—
152C22×D17—41no—
153C22×D17—41no—
154C22×D17—41no—
155C22×D17—41no—
156C22×D17—41no—
157Q8×C17—22no—
158Q8×C17—22no—
159Q8×C17—41no—
160Q8×C17—41no—
161C2×C4×C17—21no—
162C2×C4×C17—21no—
163C2×C4×C17—22no—
164C2×C4×C17—41no—
165C2×C4×C17—41no—
166C2×C4×C17—41no—
167C2×C4×C17—41no—
168C2×C4×C17—41no—
169C2×C4×C17#12011yescharacteristic
170C2×C4×C17#12011yescharacteristic
171C2×C4×C17#12111yescharacteristic
172C2×C4×C17#12111yescharacteristic
173C2×C4×C17#12111yescharacteristic
174C2×C4×C17#12711yescharacteristic
175C2×C4×C17C2×SD1611yescharacteristic
176C2×C4×C17C2×SD1611yescharacteristic
177C2×C4×C17#13511yescharacteristic
178C2×C4×C17#13511yescharacteristic
179C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C17#13711yescharacteristic
182C2×C4×C17#13711yescharacteristic
183C2×Dic17—42no—
184C2×Dic17—41no—
185C2×Dic17—42no—
186C2×Dic17—41no—
187C2×Dic17—42no—
188C2×Dic17—41no—
189C2×Dic17—42no—
190C2×Dic17—81no—
191C2×Dic17—81no—
192D68—22no—
193D68—22no—
194D68—41no—
195D68—41no—
196C4×D17—81no—
197C4×D17—81no—
198C4×D17—81no—
199C4×D17—81no—
200Dic34—42no—
201Dic34—42no—
202Dic34—81no—
203Dic34—81no—
204C8×C17—42no—
205C8×C17—42no—
206#3622—42no—
207C2×#475—171no—
208C2×#461—171no—
209#2538—171no—
210#1990—171no—
211C2×Q8⋊C8D1711yescharacteristic, complemented
212C2×D4⋊C8—171no—
213#1296—171no—
214C22×C17#39211yescharacteristic
215C22×C17#44811yescharacteristic
216C22×C17#44911yescharacteristic
217C22×C17#45011yescharacteristic
218C22×C17#45211yescharacteristic
219C22×C17#45511yescharacteristic
220C22×C17#45611yescharacteristic
221C2×D17—42no—
222C2×D17—42no—
223C2×D17—42no—
224C2×D17—81no—
225C2×D17—81no—
226C2×D17—81no—
227C2×D17—81no—
228C4×C17—21no—
229C4×C17—21no—
230C4×C17—42no—
231C4×C17—81no—
232C4×C17#44811yescharacteristic
233C4×C17#44911yescharacteristic
234C4×C17#45011yescharacteristic
235C4×C17#45211yescharacteristic
236C4×C17#46511yescharacteristic
237C4×C17#46511yescharacteristic
238C4×C17#46811yescharacteristic
239C4×C17#46811yescharacteristic
240Dic17—82no—
241Dic17—82no—
242Dic17—81no—
243C22×SD16—341no—
244C2×C4⋊Q8—171no—
245C2×C22⋊Q8—341no—
246C2×C4×Q8C2×D1711yescharacteristic
247C2×C4×D4—171no—
248#475—342no—
249#475—342no—
250#461—342no—
251#461—342no—
252C2×#106—341no—
253C2×C4⋊C8—171no—
254C2×C4⋊C8C2×D1711yescharacteristic
255C2×#103—341no—
256C2×#103—341no—
257C2×C4×C8C2×D1711yescharacteristic
258#396—342no—
259#390—341no—
260#389—341no—
261#384—171no—
262#380—341no—
263#340—342no—
264Q8⋊C8—22no—
265D4⋊C8—342no—
266C2×C17#164211yescharacteristic
267C2×C17#164311yescharacteristic
268C2×C17#165011yescharacteristic
269C2×C17#165111yescharacteristic
270C2×C17#200011yescharacteristic
271C2×C17#200011yescharacteristic
272C2×C17#200711yescharacteristic
273D17—82no—
274C22×Q8—341no—
275C22×Q8D6811yescharacteristic
276C22×D4—171no—
277C23×C4—341no—
278C2×SD16—344no—
279C2×SD16—682no—
280C2×SD16—682no—
281C22×C8—21no—
282C22×C8—21no—
283C22×C8—341no—
284C4⋊Q8—172no—
285C4⋊Q8—341no—
286C4⋊Q8—341no—
287C4⋊Q8—341no—
288C22⋊Q8—682no—
289C22⋊Q8—682no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×D4—341no—
295C4×D4—341no—
296C4×D4—341no—
297C4×D4—341no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—171no—
300C2×C4⋊C4—172no—
301C2×C4⋊C4—341no—
302C2×C4⋊C4—341no—
303C2×C4⋊C4—342no—
304C2×C4⋊C4—341no—
305C2×C4⋊C4—341no—
306C2×C4⋊C4D6811yescharacteristic
307C2×C22⋊C4—342no—
308C2×C22⋊C4—341no—
309C2×C22⋊C4—342no—
310C2×C42—21no—
311C2×C42C22×D1711yescharacteristic
312#106—344no—
313C4⋊C8—21no—
314C4⋊C8—172no—
315C4⋊C8—341no—
316C4⋊C8C22⋊D1712yes—
317#103—344no—
318#103—344no—
319C4×C8—21no—
320C4×C8—21no—
321#95—342no—
322C17#1023211yescharacteristic, complemented, Sylow
323C24—341no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—342no—
330C2×Q8—681no—
331C2×Q8—681no—
332C2×Q8—681no—
333C2×Q8—681no—
334C2×Q8—681no—
335C2×Q8#6321912yescomplemented
336C2×D4—172no—
337C2×D4—341no—
338C2×D4—341no—
339C2×D4—341no—
340C2×D4—341no—
341C2×D4—341no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—341no—
345C22×C4—342no—
346C22×C4—681no—
347C22×C4—681no—
348C22×C4—681no—
349C22×C4—681no—
350C22×C4—681no—
351C22×C4—681no—
352C22×C4—681no—
353C22×C4C2×D6811yescharacteristic
354C22×C4D4×D1711yescharacteristic
355C22×C4D4×D1711yescharacteristic
356SD16—688no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—342no—
364C2×C8—681no—
365C2×C8—681no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—342no—
371C4⋊C4—341no—
372C4⋊C4—342no—
373C4⋊C4—341no—
374C4⋊C4—682no—
375C4⋊C4—684no—
376C4⋊C4—681no—
377C4⋊C4—681no—
378C4⋊C4—681no—
379C4⋊C4—681no—
380C4⋊C4—681no—
381C4⋊C4—681no—
382C22⋊C4—684no—
383C22⋊C4—681no—
384C22⋊C4—681no—
385C22⋊C4—1362no—
386C42—41no—
387C42—41no—
388C42#6325211yescharacteristic
389C42D4×D1711yescharacteristic
390C42D4×D1711yescharacteristic
391C42C2×C22⋊D1711yescharacteristic
392C23—342no—
393C23—681no—
394C23—681no—
395C23—681no—
396C23—681no—
397C23—681no—
398C23—681no—
399C23#1058756811yescharacteristic
400Q8—22no—
401Q8—22no—
402Q8—41no—
403Q8—41no—
404Q8—682no—
405Q8—682no—
406Q8—1361no—
407Q8—1361no—
408D4—342no—
409D4—342no—
410D4—681no—
411D4—681no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—682no—
421C2×C4—682no—
422C2×C4—681no—
423C2×C4—681no—
424C2×C4—681no—
425C2×C4—682no—
426C2×C4—682no—
427C2×C4—1361no—
428C2×C4—1361no—
429C2×C4—1361no—
430C2×C4—1361no—
431C2×C4—1361no—
432C2×C4—1361no—
433C2×C4#1058756811yescharacteristic
434C2×C4#1058757011yescharacteristic
435C2×C4#1058757011yescharacteristic
436C2×C4C2×#6321911yescharacteristic
437C2×C4#1058759011yescharacteristic
438C2×C4#1058759411yescharacteristic
439C2×C4#1058759511yescharacteristic
440C2×C4SD16×D1711yescharacteristic
441C2×C4#1058760011yescharacteristic
442C2×C4#1058760411yescharacteristic
443C2×C4#1058760511yescharacteristic
444C2×C4#1058761611yescharacteristic
445C2×C4#1058762311yescharacteristic
446C2×C4#1058762611yescharacteristic
447C8—42no—
448C8—42no—
449C8—682no—
450C22—682no—
451C22—682no—
452C22—682no—
453C22—1361no—
454C22—1361no—
455C22—1361no—
456C22—1361no—
457C22#1176271811yescharacteristic
458C22#1176294911yescharacteristic
459C22#1176295711yescharacteristic
460C22#1176296211yescharacteristic
461C22#1176297311yescharacteristic
462C22#1176298811yescharacteristic
463C22#1176299311yescharacteristic
464C4—21no—
465C4—21no—
466C4—42no—
467C4—81no—
468C4—1362no—
469C4—1362no—
470C4—1361no—
471C4#1176295011yescharacteristic
472C4#1176295611yescharacteristic
473C4#1176295811yescharacteristic
474C4#1176297011yescharacteristic
475C4#1176303811yescharacteristic
476C4#1176304111yescharacteristic
477C4#1176305511yescharacteristic
478C4#1176305811yescharacteristic
479C2—1362no—
480C2#1452798711yescharacteristic
481C2#1452799411yescharacteristic
482C2#1452804311yescharacteristic
483C2#1452805011yescharacteristic
484C2#1452978611yescharacteristic
485C2#1452978911yescharacteristic
486C2#1452983211yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×Q8⋊C8C2
socleC23×C17#120
prime radicalC22×D68D4
hypercenterC2×Q8⋊C8D17
hypocenterC17#10232
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#11763094C2semidirect
4. #14529745C2semidirect
6. #14526668C2semidirect
7. C2×C17×Q8⋊C8C2semidirect
85. C2×Q8×C17SD16semidirect
211. C2×Q8⋊C8D17semidirect
322. C17#10232semidirect
335. C2×Q8#63219semidirect

Maximal subgroups

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

subgroupindex
C2×#117630942
C2×#117630152
#145337022
#145297452
C2×#117625382
#145266682
C2×C17×Q8⋊C82
#1023217