Finite Group Databasebeta

← #17951125

Subgroups — #17951125

Order 4352.

1054 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763781C211yesmaximal, characteristic
2C2×#11763752C211yesmaximal, characteristic
3#14543657C214yesmaximal, complemented
4C2×#11762666C211yesmaximal, characteristic
5C22×#63247—21no—
6C22×C22⋊#595—21no—
7#11763781—22no—
8#11763781C414yescomplemented
9C2×#10587543—21no—
10C2×#10587543C2212yes—
11#11763752—22no—
12#11763752C414yescomplemented
13C2×(C2×#595)⋊C4—21no—
14C2×(C2×#595)⋊C4C2212yes—
15C23×C4×D17C2211yescharacteristic
16C22×C4⋊Dic17C411yescharacteristic, complemented
17C22×#63227—21no—
18#11762666—21no—
19#11762666—21no—
20#11762666C412yescomplemented
21#11762666C2212yes—
22C2×#10587502—21no—
23C22×C4⋊C4×C17C411yescharacteristic, complemented
24C24×D17D411yescharacteristic
25C23×#595—22no—
26C23×C4×C17—21no—
27C23×C4×C17C2×C411yescharacteristic
28C23×Dic17—21no—
29C23×Dic17D411yescharacteristic
30C22×C4×D17—42no—
31C22×C4×D17—42no—
32C22×C4×D17C2×C411yescharacteristic
33C22×C4×D17C2×C412yes—
34C22×C4×D17C2×C412yes—
35C22×C4×D17D411yescharacteristic
36C22×C4×D17D411yescharacteristic
37C22×C4×D17D412yes—
38C22×C4×D17Q812yes—
39C22×C4×D17C2311yescharacteristic
40C2×C22⋊#595—24no—
41C2×C22⋊#595—42no—
42C2×C22⋊#595—42no—
43C2×#63247—24no—
44C2×#63247—42no—
45C2×#63247—42no—
46C22×#3624—22no—
47C2×C4⋊C4×C17—21no—
48C2×C4⋊C4×C17—42no—
49C2×C4⋊C4×C17C2×C412yes—
50C2×#63227—22no—
51C2×#63227—22no—
52C2×#63227—44no—
53C2×C4⋊Dic17—21no—
54C2×C4⋊Dic17—42no—
55C2×C4⋊Dic17C2×C412yes—
56(C2×#595)⋊C4—24no—
57(C2×#595)⋊C4—22no—
58(C2×#595)⋊C4C2×C414yescomplemented
59#10587543—24no—
60#10587543—22no—
61#10587543C2×C414yescomplemented
62#10587502—22no—
63#10587502—22no—
64C24×C17C22⋊C411yescharacteristic
65C23×D17—22no—
66C23×D17—22no—
67C23×D17—22no—
68C23×D17—22no—
69C23×D17—41no—
70C23×D17—41no—
71C23×D17C22⋊C411yescharacteristic
72C23×D17C22⋊C412yes—
73C23×D17C4⋊C412yes—
74C23×D17C2×D411yescharacteristic
75C22×#595—24no—
76C22×#595—42no—
77C22×#595—42no—
78C22×#595—42no—
79C22×#595—42no—
80C22×#595—42no—
81C22×#595—42no—
82C22×C4×C17—22no—
83C22×C4×C17—22no—
84C22×C4×C17—22no—
85C22×C4×C17—42no—
86C22×C4×C17—42no—
87C22×C4×C17—42no—
88C22×C4×C17C4212yes—
89C22×C4×C17C22⋊C411yescharacteristic
90C22×C4×C17C4⋊C412yes—
91C22×C4×C17C22×C411yescharacteristic
92C22×Dic17—22no—
93C22×Dic17—22no—
94C22×Dic17—22no—
95C22×Dic17—42no—
96C22×Dic17—42no—
97C22×Dic17—42no—
98C22×Dic17C22⋊C411yescharacteristic
99C22×Dic17C22⋊C412yes—
100C22×Dic17C4⋊C412yes—
101C22×Dic17C2×D411yescharacteristic
102C2×C4×D17—41no—
103C2×C4×D17—41no—
104C2×C4×D17—44no—
105C2×C4×D17—44no—
106C2×C4×D17—41no—
107C2×C4×D17—41no—
108C2×C4×D17—41no—
109C2×C4×D17—41no—
110C2×C4×D17—44no—
111C2×C4×D17—44no—
112C2×C4×D17—41no—
113C2×C4×D17—41no—
114C2×C4×D17C4212yes—
115C2×C4×D17C22⋊C412yes—
116C2×C4×D17C22⋊C412yes—
117C2×C4×D17C4⋊C412yes—
118C2×C4×D17C4⋊C412yes—
119C2×C4×D17C4⋊C412yes—
120C2×C4×D17C22×C411yescharacteristic
121C2×C4×D17C22×C411yescharacteristic
122C2×C4×D17C2×D411yescharacteristic
123C2×C4×D17C2×Q811yescharacteristic
124C22⋊#595—48no—
125#63247—48no—
126C2×#3624—24no—
127C2×#3624—42no—
128C2×#3624—42no—
129C4⋊C4×C17—42no—
130C4⋊C4×C17—42no—
131#63227—44no—
132#63227—44no—
133C4⋊Dic17—42no—
134C4⋊Dic17—42no—
135#8015—171nomaximal, Sylow
136C23×C17—21no—
137C23×C17—21no—
138C23×C17—21no—
139C23×C17—21no—
140C23×C17#9512yes—
141C23×C17#9912yes—
142C23×C17#9912yes—
143C23×C17C2×C22⋊C411yescharacteristic
144C22×D17—21no—
145C22×D17—21no—
146C22×D17—21no—
147C22×D17—21no—
148C22×D17—24no—
149C22×D17—24no—
150C22×D17—21no—
151C22×D17—21no—
152C22×D17—21no—
153C22×D17—21no—
154C22×D17—21no—
155C22×D17—21no—
156C22×D17—24no—
157C22×D17—24no—
158C22×D17—42no—
159C22×D17—42no—
160C22×D17—44no—
161C22×D17—44no—
162C22×D17—44no—
163C22×D17—44no—
164C22×D17#9512yes—
165C22×D17C2×C22⋊C411yescharacteristic
166C22×D17C2×C4⋊C411yescharacteristic
167C22×D17AGL(1,ℤ/8)12yes—
168C22×D17#13712yes—
169C2×#595—44no—
170C2×#595—44no—
171C2×#595—44no—
172C2×#595—82no—
173C2×#595—82no—
174C2×#595—82no—
175C2×#595—82no—
176C2×C4×C17—24no—
177C2×C4×C17—24no—
178C2×C4×C17—21no—
179C2×C4×C17—21no—
180C2×C4×C17—21no—
181C2×C4×C17—21no—
182C2×C4×C17—42no—
183C2×C4×C17—42no—
184C2×C4×C17—42no—
185C2×C4×C17—84no—
186C2×C4×C17#9512yes—
187C2×C4×C17C2×C4211yescharacteristic
188C2×C4×C17C2×C4⋊C411yescharacteristic
189C2×Dic17—24no—
190C2×Dic17—24no—
191C2×Dic17—21no—
192C2×Dic17—21no—
193C2×Dic17—21no—
194C2×Dic17—21no—
195C2×Dic17—42no—
196C2×Dic17—42no—
197C2×Dic17—42no—
198C2×Dic17—84no—
199C2×Dic17#9512yes—
200C2×Dic17C2×C22⋊C411yescharacteristic
201C2×Dic17C2×C4⋊C411yescharacteristic
202C4×D17—42no—
203C4×D17—42no—
204C4×D17—42no—
205C4×D17—42no—
206C4×D17—42no—
207C4×D17—42no—
208C4×D17—41no—
209C4×D17—41no—
210C4×D17—41no—
211C4×D17—41no—
212#3624—44no—
213C2×#407—171no—
214C2×#381—171no—
215C2×#380—171no—
216#1294—174no—
217C22×C17—22no—
218C22×C17—22no—
219C22×C17—22no—
220C22×C17—22no—
221C22×C17—22no—
222C22×C17—22no—
223C22×C17#34214yes—
224C22×C17C2×#9511yescharacteristic
225C22×C17C2×#9911yescharacteristic
226C22×C17C2×#9911yescharacteristic
227C22×C17#41811yescharacteristic
228C22×C17#41911yescharacteristic
229C22×C17#42111yescharacteristic
230C22×C17#42111yescharacteristic
231C2×D17—21no—
232C2×D17—21no—
233C2×D17—21no—
234C2×D17—21no—
235C2×D17—22no—
236C2×D17—22no—
237C2×D17—22no—
238C2×D17—22no—
239C2×D17—22no—
240C2×D17—22no—
241C2×D17—22no—
242C2×D17—48no—
243C2×D17—48no—
244C2×D17—41no—
245C2×D17—41no—
246C2×D17—41no—
247C2×D17—41no—
248C2×D17#41712yes—
249C2×D17#42812yes—
250#595—84no—
251C4×C17—22no—
252C4×C17—21no—
253C4×C17—21no—
254C4×C17—82no—
255Dic17—21no—
256Dic17—22no—
257Dic17—21no—
258Dic17—82no—
259C24×C4—171no—
260C22×M16—341no—
261C22×C4⋊C4—171no—
262C22×C4⋊C4#59511yescharacteristic, complemented
263C22×C22⋊C4—341no—
264C22×C22⋊C4—341no—
265#407—174no—
266#407—342no—
267C2×C22⋊C8—172no—
268C2×C22⋊C8—341no—
269#381—174no—
270#381—342no—
271#380—172no—
272#380—172no—
273#380—341no—
274#380—341no—
275C2×#95—172no—
276C2×#95—341no—
277C2×#95—341no—
278C2×C17—24no—
279C2×C17#151011yescharacteristic
280C2×C17#151011yescharacteristic
281C2×C17#151111yescharacteristic
282C2×C17#151111yescharacteristic
283C2×C17#173311yescharacteristic
284C2×C17#173611yescharacteristic
285C2×C17C2×#34211yescharacteristic
286D17—22no—
287D17—22no—
288D17—42no—
289C25—171no—
290C23×C4—21no—
291C23×C4—171no—
292C23×C4—172no—
293C23×C4—172no—
294C23×C4—172no—
295C23×C4—172no—
296C23×C4—171no—
297C23×C4—171no—
298C23×C4—171no—
299C23×C4—171no—
300C23×C4—342no—
301C23×C4—341no—
302C23×C4—682no—
303C23×C4—682no—
304C23×C4C2×#59511yescharacteristic
305C2×M16—344no—
306C2×M16—682no—
307C2×M16—682no—
308C22×C8—342no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—172no—
312C2×C4⋊C4—341no—
313C2×C4⋊C4—682no—
314C2×C4⋊C4C2×#59512yes—
315C2×C22⋊C4—342no—
316C2×C22⋊C4—342no—
317C2×C22⋊C4—344no—
318C2×C22⋊C4—684no—
319C2×C22⋊C4—682no—
320C2×C22⋊C4—682no—
321C22⋊C8—174no—
322C22⋊C8—342no—
323C22⋊C8—344no—
324#95—174no—
325#95—342no—
326#95—342no—
327#95—342no—
328#95—344no—
329C17#801511yescharacteristic, complemented, Sylow
330C24—171no—
331C24—171no—
332C24—172no—
333C24—172no—
334C24—342no—
335C24—342no—
336C24—342no—
337C24—342no—
338C24—681no—
339C24—681no—
340C24C22⋊#59511yescharacteristic
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—171no—
348C22×C4—171no—
349C22×C4—172no—
350C22×C4—172no—
351C22×C4—172no—
352C22×C4—171no—
353C22×C4—171no—
354C22×C4—172no—
355C22×C4—171no—
356C22×C4—171no—
357C22×C4—172no—
358C22×C4—172no—
359C22×C4—172no—
360C22×C4—172no—
361C22×C4—342no—
362C22×C4—342no—
363C22×C4—344no—
364C22×C4—342no—
365C22×C4—681no—
366C22×C4—681no—
367C22×C4—681no—
368C22×C4—681no—
369C22×C4—681no—
370C22×C4—681no—
371C22×C4—681no—
372C22×C4—681no—
373C22×C4—684no—
374C22×C4—684no—
375C22×C4—684no—
376C22×C4—684no—
377C22×C4—682no—
378C22×C4—682no—
379C22×C4—682no—
380C22×C4—682no—
381C22×C4—682no—
382C22×C4—682no—
383C22×C4—682no—
384C22×C4—682no—
385C22×C4—682no—
386C22×C4C4×#59512yes—
387C22×C4C4⋊#59512yes—
388C22×C4C22⋊#59511yescharacteristic
389C22×C4C22×#59511yescharacteristic
390M16—688no—
391C2×C8—344no—
392C2×C8—682no—
393C2×C8—682no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—682no—
397C4⋊C4—682no—
398C22⋊C4—684no—
399C22⋊C4—684no—
400C22⋊C4—688no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—171no—
406C23—171no—
407C23—172no—
408C23—172no—
409C23—172no—
410C23—344no—
411C23—344no—
412C23—341no—
413C23—341no—
414C23—341no—
415C23—341no—
416C23—341no—
417C23—341no—
418C23—341no—
419C23—341no—
420C23—341no—
421C23—341no—
422C23—344no—
423C23—344no—
424C23—682no—
425C23—682no—
426C23—684no—
427C23—684no—
428C23—684no—
429C23—684no—
430C23#1058753912yes—
431C23(C2×#595)⋊C412yes—
432C23#1058755112yes—
433C23C2×C22⋊#59511yescharacteristic
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—84no—
444C2×C4—171no—
445C2×C4—172no—
446C2×C4—171no—
447C2×C4—341no—
448C2×C4—341no—
449C2×C4—341no—
450C2×C4—341no—
451C2×C4—344no—
452C2×C4—344no—
453C2×C4—681no—
454C2×C4—681no—
455C2×C4—682no—
456C2×C4—681no—
457C2×C4—681no—
458C2×C4—682no—
459C2×C4—682no—
460C2×C4—682no—
461C2×C4—682no—
462C2×C4—682no—
463C2×C4—684no—
464C2×C4—684no—
465C2×C4—684no—
466C2×C4—682no—
467C2×C4—682no—
468C2×C4—682no—
469C2×C4—1364no—
470C2×C4—1362no—
471C2×C4—1362no—
472C2×C4—1362no—
473C2×C4—1362no—
474C2×C4(C2×#595)⋊C412yes—
475C2×C4C2×C4×#59511yescharacteristic
476C2×C4C2×C4⋊#59511yescharacteristic
477C8—684no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—172no—
485C22—172no—
486C22—342no—
487C22—342no—
488C22—342no—
489C22—342no—
490C22—341no—
491C22—341no—
492C22—341no—
493C22—341no—
494C22—342no—
495C22—342no—
496C22—342no—
497C22—688no—
498C22—688no—
499C22—681no—
500C22—681no—
501C22—681no—
502C22—681no—
503C22#1176368714yes—
504C22C2×(C2×#595)⋊C411yescharacteristic
505C22C2×#1058755111yescharacteristic
506C22C2×#1058753911yescharacteristic
507C22#1176380111yescharacteristic
508C22#1176380211yescharacteristic
509C22#1176380711yescharacteristic
510C22#1176380811yescharacteristic
511C4—21no—
512C4—21no—
513C4—22no—
514C4—82no—
515C4—341no—
516C4—342no—
517C4—341no—
518C4—1364no—
519C4—1362no—
520C2—24no—
521C2—342no—
522C2—342no—
523C2—682no—
524C2#1454418911yescharacteristic
525C2#1454419111yescharacteristic
526C2#1454419211yescharacteristic
527C2#1454419411yescharacteristic
528C2#1454446311yescharacteristic
529C2#1454447111yescharacteristic
530C2C2×#1176368711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC24×D17D4
cosocleC22×C4×D17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#8015
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #14543657C2semidirect
8. #11763781C4semidirect
12. #11763752C4semidirect
16. C22×C4⋊Dic17C4semidirect
20. #11762666C4semidirect
23. C22×C4⋊C4×C17C4semidirect
58. (C2×#595)⋊C4C2×C4semidirect
61. #10587543C2×C4semidirect
262. C22×C4⋊C4#595semidirect
329. C17#8015semidirect

Maximal subgroups

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

subgroupindex
C2×#117637812
C2×#117637522
#145436572
C2×#117626662
#801517