Finite Group Databasebeta

← #17097241

Subgroups — #17097241

Order 4352.

896 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763106C211yesmaximal, characteristic, complemented
2C2×#11762852C211yesmaximal, characteristic, complemented
3#14531798C211yesmaximal, characteristic
4#14530318C211yesmaximal, characteristic, complemented
5#14530285C211yesmaximal, characteristic, complemented
6#14530169C211yesmaximal, characteristic, complemented
7#14530119C211yesmaximal, characteristic, complemented
8#14529545C211yesmaximal, characteristic, complemented
9#14529352C211yesmaximal, characteristic, complemented
10#14529220C211yesmaximal, characteristic
11#14529106C211yesmaximal, characteristic
12#14529044C211yesmaximal, characteristic, complemented
13#14528976C211yesmaximal, characteristic, complemented
14#14528969C211yesmaximal, characteristic, complemented
15C17×#2441C211yesmaximal, characteristic, complemented
16C22×#63223C2211yescharacteristic
17C2×#10587616C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×#10587567—21no—
20C22×C4⋊Dic17C2211yescharacteristic
21#11763106—24no—
22#11762852—24no—
23C2×#10587478C2211yescharacteristic
24C2×#10587478C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28C2×#10587479C2211yescharacteristic
29C2×C4⋊#3622C2211yescharacteristic
30C2×C4⋊#3622C2211yescharacteristic
31C2×#10587480C2211yescharacteristic
32C2×#10587480C2211yescharacteristic
33C2×#10587480C2211yescharacteristic
34C2×#10587480C2211yescharacteristic
35C2×C22⋊#3622C2211yescharacteristic
36#11762711C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762701—21no—
39#11762686—21no—
40#11762683—21no—
41C4×C4⋊Dic17C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C17×#125C2211yescharacteristic
51C2×C17×#117C2211yescharacteristic
52C17×#399—21no—
53C17×#396—21no—
54C17×#389—21no—
55C17×#389C2211yescharacteristic
56C17×#389C2211yescharacteristic
57C17×#387—21no—
58C17×#384C2211yescharacteristic
59C17×#380C2211yescharacteristic
60C4×C22⋊C4×C17C2211yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C22×C22⋊D17—21no—
63C23×Dic17—21no—
64C22×D68C2311yescharacteristic
65C22×C4×D17—21no—
66C17×#125—22no—
67C17×#125—41no—
68C17×#117—22no—
69C17×#117—41no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17D411yescharacteristic
83C2×C22⋊C4×C17D411yescharacteristic
84C2×C42×C17—21no—
85C2×C42×C17C2311yescharacteristic
86C2×C42×C17C2311yescharacteristic
87#10587616—22no—
88#10587616Q814yes—
89C2×#63227—21no—
90C2×#63227—21no—
91C2×#63227—21no—
92C2×#63227—21no—
93C2×C4⋊Dic17—21no—
94C2×C4⋊Dic17—22no—
95C2×C4⋊Dic17—41no—
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99C2×#63225—21no—
100C2×C4×Dic17—21no—
101C2×#63223—22no—
102C2×#63223—22no—
103C2×#63223—22no—
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108#10587567—44no—
109C4×D68—22no—
110C4×D68—41no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95—21no—
114C17×#95D411yescharacteristic
115C17×#95D411yescharacteristic
116#10587502—21no—
117#10587502—21no—
118C22⋊#3622—22no—
119#10587480—22no—
120#10587480—22no—
121#10587480—22no—
122#10587480—41no—
123#10587479—22no—
124#10587479—22no—
125#10587479—22no—
126#10587479—41no—
127#10587478—22no—
128#10587478D414yescomplemented
129C4⋊#3622—22no—
130C4⋊#3622—22no—
131C24×C17C2×D411yescharacteristic
132C23×D17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×Q812yes—
147C22×C4×C17#4112yes—
148C22×C4×C17C2411yescharacteristic
149C2×C22⋊D17—24no—
150C2×C22⋊D17—42no—
151C2×C22⋊D17—42no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—42no—
156C22×Dic17—42no—
157C22×Dic17—81no—
158C2×D68—21no—
159C2×D68—21no—
160C2×D68—21no—
161C2×D68—21no—
162C2×D68—21no—
163C2×D68C2×Q812yes—
164C2×C4×D17—42no—
165C2×C4×D17—42no—
166C2×C4×D17—81no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—42no—
175C4⋊C4×C17—42no—
176C4⋊C4×C17—42no—
177C4⋊C4×C17—42no—
178C4⋊C4×C17—81no—
179C4⋊C4×C17—81no—
180C4⋊C4×C17C2×D412yes—
181C22⋊C4×C17—22no—
182C22⋊C4×C17—41no—
183C22⋊C4×C17—42no—
184C22⋊C4×C17—81no—
185C42×C17—21no—
186C42×C17—21no—
187C42×C17—21no—
188C42×C17—21no—
189C42×C17—42no—
190#63227—24no—
191#63227—42no—
192#63227—42no—
193#63227—81no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—42no—
199C4⋊Dic17C2×D412yes—
200C4⋊Dic17C2×Q812yes—
201#63225—42no—
202C4×Dic17—42no—
203#63223—28no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622C2×D412yes—
215C2×#3622#4112yes—
216#26366—171nomaximal, Sylow
217C23×C17—21no—
218C23×C17—21no—
219C23×C17—21no—
220C23×C17—21no—
221C23×C17—21no—
222C23×C17—21no—
223C23×C17C22⋊Q812yes—
224C23×C17C22×D411yescharacteristic
225C22×D17—22no—
226C22×D17—41no—
227C22×D17—41no—
228C22×D17—41no—
229C22×D17—41no—
230C22×D17—41no—
231C22×D17—41no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—22no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17C22⋊Q812yes—
262C2×C4×C17AGL(1,ℤ/8)12yes—
263C2×C4×C17C22×D411yescharacteristic
264C2×C4×C17C22×Q811yescharacteristic
265C2×C4×C17C2×#4111yescharacteristic
266C2×C4×C17C2×#4111yescharacteristic
267C2×C4×C17H(5,𝔽2)11yescharacteristic
268C2×C4×C172−1+411yescharacteristic
269C22⋊D17—48no—
270C2×Dic17—22no—
271C2×Dic17—22no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—84no—
280D68—22no—
281D68—22no—
282D68—41no—
283D68—41no—
284C4×D17—84no—
285#3622—22no—
286#3622—22no—
287#3622—41no—
288#3622—41no—
289C2×#478—171no—
290C2×#428—171no—
291#2441D1711yescharacteristic, complemented
292#2361—171no—
293#2094—171no—
294#2088—171no—
295#2067—171no—
296#2059—171no—
297#1948—171no—
298#1918—171no—
299#1895—171no—
300#1875—171no—
301#1864—171no—
302#1853—171no—
303#1851—171no—
304C22×C17—22no—
305C22×C17—22no—
306C22×C17—22no—
307C22×C17—41no—
308C22×C17—41no—
309C22×C17—41no—
310C22×C17—41no—
311C22×C17C2×C22⋊Q811yescharacteristic
312C22×C17D4211yescharacteristic
313C22×C17#54711yescharacteristic
314C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
315C22×C17#57511yescharacteristic
316C22×C17#57711yescharacteristic
317C22×C17#57711yescharacteristic
318C2×D17—42no—
319C2×D17—42no—
320C2×D17—42no—
321C2×D17—81no—
322C2×D17—81no—
323C2×D17—81no—
324C2×D17—81no—
325C4×C17—22no—
326C4×C17—22no—
327C4×C17—41no—
328C4×C17—41no—
329C4×C17—41no—
330C4×C17—42no—
331C4×C17—81no—
332C4×C17—81no—
333C4×C17—81no—
334C4×C17#52511yescharacteristic
335C4×C17#53911yescharacteristic
336C4×C17D4×Q811yescharacteristic
337C4×C17#55411yescharacteristic
338Dic17—42no—
339Dic17—42no—
340Dic17—81no—
341C22×M16—171no—
342C2×#125C2×D1711yescharacteristic
343C2×#121—171no—
344C2×C4×D4—171no—
345C2×C4×D4—341no—
346C2×#117C2×D1711yescharacteristic
347C22×C4⋊C4—171no—
348#478—344no—
349#428—344no—
350C2×#107—171no—
351C2×#107—171no—
352C2×#106—171no—
353C2×#106—171no—
354C2×#106—171no—
355C2×#106—171no—
356C2×C4⋊C8—171no—
357C2×C4⋊C8—171no—
358C2×#102—171no—
359C2×#102—171no—
360C2×#102—171no—
361C2×#102—171no—
362C2×C22⋊C8—171no—
363#399—21no—
364#396—21no—
365#390—171no—
366#389—21no—
367#389—171no—
368#389C2×D1711yescharacteristic
369#389C2×D1711yescharacteristic
370#388—341no—
371#387—21no—
372#386—341no—
373#385—341no—
374#384C2×D1711yescharacteristic
375#380C2×D1711yescharacteristic
376C4×C4⋊C4—171no—
377C4×C22⋊C4C2×D1711yescharacteristic
378#340—171no—
379#340—171no—
380#340—171no—
381#340—171no—
382#340—171no—
383#340—171no—
384#336—171no—
385#336—171no—
386C2×C17—42no—
387C2×C17#244911yescharacteristic
388C2×C17#307711yescharacteristic
389C2×C17#308411yescharacteristic
390C2×C17#327111yescharacteristic
391C2×C17#327211yescharacteristic
392C2×C17#334511yescharacteristic
393C2×C17#335111yescharacteristic
394D17—82no—
395C22×D4—171no—
396C22×D4—341no—
397C23×C4—341no—
398C23×C4—341no—
399C23×C4C22×D1711yescharacteristic
400C2×M16—342no—
401C2×M16—342no—
402C2×M16—342no—
403C22×C8—171no—
404C22×C8—171no—
405C22×C8—171no—
406C22×C8—171no—
407#125—22no—
408#125—41no—
409#121—174no—
410#121—342no—
411C4×D4—342no—
412C4×D4—681no—
413C4×D4—684no—
414#117—22no—
415#117—41no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—342no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4—681no—
429C2×C4⋊C4C22×D1711yescharacteristic
430C2×C4⋊C4C22×D1711yescharacteristic
431C2×C4⋊C4C22×D1711yescharacteristic
432C2×C4⋊C4C22×D1711yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—341no—
436C2×C22⋊C4—341no—
437C2×C22⋊C4—341no—
438C2×C22⋊C4—341no—
439C2×C22⋊C4C22⋊D1711yescharacteristic
440C2×C22⋊C4C22⋊D1711yescharacteristic
441C2×C42—21no—
442C2×C42—341no—
443C2×C42C22×D1711yescharacteristic
444C2×C42C22×D1711yescharacteristic
445#107—174no—
446#107—342no—
447#106—342no—
448#106—342no—
449#106—342no—
450#106—681no—
451C4⋊C8—342no—
452C4⋊C8—342no—
453#102—342no—
454#102—342no—
455#102—342no—
456#102—681no—
457C22⋊C8—342no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—341no—
462#95—341no—
463#95C22⋊D1711yescharacteristic
464#95C22⋊D1711yescharacteristic
465C17#2636611yescharacteristic, complemented, Sylow
466C24—341no—
467C24C2×C22⋊D1711yescharacteristic
468C2×D4—172no—
469C2×D4—344no—
470C2×D4—341no—
471C2×D4—341no—
472C2×D4—341no—
473C2×D4—341no—
474C2×D4—341no—
475C2×D4—682no—
476C2×D4—682no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—341no—
488C22×C4—682no—
489C22×C4—682no—
490C22×C4—682no—
491C22×C4—682no—
492C22×C4—1361no—
493C22×C4—1361no—
494C22×C4Q8×D1712yes—
495C22×C4#6325612yes—
496C22×C4C2×C22⋊D1711yescharacteristic
497C22×C4C2×C22⋊D1711yescharacteristic
498C22×C4C2×C22⋊D1711yescharacteristic
499C22×C4C2×C22⋊D1711yescharacteristic
500C22×C4C2×C22⋊D1711yescharacteristic
501C22×C4C23×D1711yescharacteristic
502M16—348no—
503C2×C8—172no—
504C2×C8—172no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—81no—
527C4⋊C4—81no—
528C4⋊C4—172no—
529C4⋊C4—172no—
530C4⋊C4—341no—
531C4⋊C4—341no—
532C4⋊C4—341no—
533C4⋊C4—341no—
534C4⋊C4—682no—
535C4⋊C4—682no—
536C4⋊C4D4×D1712yes—
537C22⋊C4—22no—
538C22⋊C4—41no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C22⋊C4—344no—
542C22⋊C4—682no—
543C22⋊C4—682no—
544C22⋊C4—1361no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—42no—
550C42—682no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—342no—
558C23—681no—
559C23—681no—
560C23—681no—
561C23—681no—
562C23—681no—
563C23—681no—
564C23#1058763212yes—
565C23C22×C22⋊D1711yescharacteristic
566D4—342no—
567D4—342no—
568D4—688no—
569D4—681no—
570D4—681no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—342no—
601C2×C4—342no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—1364no—
610C2×C4—1364no—
611C2×C4#1058763212yes—
612C2×C4#1058763512yes—
613C2×C4C2×D4×D1711yescharacteristic
614C2×C4#1058769111yescharacteristic
615C2×C4C2×Q8×D1711yescharacteristic
616C2×C4C2×#6325611yescharacteristic
617C2×C4#41×D1711yescharacteristic
618C2×C4#1058769711yescharacteristic
619C8—342no—
620C8—342no—
621C8—681no—
622C8—681no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—682no—
631C22—682no—
632C22—682no—
633C22—1361no—
634C22—1361no—
635C22—1361no—
636C22—1361no—
637C22C2×#1058763211yescharacteristic
638C22D4×C22⋊D1711yescharacteristic
639C22#1176345811yescharacteristic
640C22C2×#1058763511yescharacteristic
641C22#1176360711yescharacteristic
642C22#1176361711yescharacteristic
643C22#1176362011yescharacteristic
644C4—22no—
645C4—22no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—42no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—682no—
654C4—682no—
655C4—1361no—
656C4#1176332211yescharacteristic
657C4#1176340811yescharacteristic
658C4#1176347211yescharacteristic
659C4#1176351311yescharacteristic
660C2—42no—
661C2—1362no—
662C2#1453258911yescharacteristic
663C2#1453857711yescharacteristic
664C2#1453864211yescharacteristic
665C2#1454036511yescharacteristic
666C2#1454038411yescharacteristic
667C2#1454124811yescharacteristic
668C2#1454134411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2441C2
socleC23×C17C22×D4
prime radicalC2×#10587616C22
hypercenter#2441D17
hypocenterC17#26366
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
1. C2×#11763106C2semidirect
2. C2×#11762852C2semidirect
4. #14530318C2semidirect
5. #14530285C2semidirect
6. #14530169C2semidirect
7. #14530119C2semidirect
8. #14529545C2semidirect
9. #14529352C2semidirect
12. #14529044C2semidirect
13. #14528976C2semidirect
14. #14528969C2semidirect
15. C17×#2441C2semidirect
128. #10587478D4semidirect
291. #2441D17semidirect
465. C17#26366semidirect

Maximal subgroups

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

subgroupindex
C2×#117631062
C2×#117628522
#145317982
#145303182
#145302852
#145301692
#145301192
#145295452
#145293522
#145292202
#145291062
#145290442
#145289762
#145289692
C17×#24412
#2636617