Finite Group Databasebeta

← #17051878

Subgroups — #17051878

Order 4352.

924 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762865C211yesmaximal, characteristic
2C2×#11762864C211yesmaximal, characteristic
3C4×C4⋊C4×D17C211yesmaximal, characteristic
4#14529501C211yesmaximal, characteristic, complemented
5#14529499C211yesmaximal, characteristic, complemented
6#14529350C211yesmaximal, characteristic, complemented
7#14529349C211yesmaximal, characteristic, complemented
8#14529347C211yesmaximal, characteristic, complemented
9#14528851C211yesmaximal, characteristic
10#14528851C211yesmaximal, characteristic
11#14528763C211yesmaximal, characteristic
12#14528652C211yesmaximal, characteristic, complemented
13#14528650C211yesmaximal, characteristic, complemented
14#14528541C211yesmaximal, characteristic
15C17×#1917C211yesmaximal, characteristic, complemented
16C2×C4⋊C4×D17C411yescharacteristic
17C2×C4⋊C4×D17C411yescharacteristic
18C2×C42×D17C2211yescharacteristic
19C2×C42×D17C2211yescharacteristic
20C2×C42×D17C2211yescharacteristic
21#11762865—24no—
22#11762864—24no—
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587485C2211yescharacteristic
27C2×#10587485C2211yescharacteristic
28C2×#10587491C2211yescharacteristic
29C2×#10587491C2211yescharacteristic
30C2×#10587491C2211yescharacteristic
31C2×#10587491C2211yescharacteristic
32C2×#10587474C2211yescharacteristic
33C2×#10587486C2211yescharacteristic
34C2×#10587486C2211yescharacteristic
35C2×C8×Dic17C2211yescharacteristic
36C2×C8×Dic17C2211yescharacteristic
37C4⋊C4×Dic17C411yescharacteristic
38C4⋊C4×Dic17C411yescharacteristic
39C4×#63225C2211yescharacteristic
40C4×#63225C2211yescharacteristic
41C4×C4⋊Dic17C2211yescharacteristic
42#11762660C411yescharacteristic
43#11762660C411yescharacteristic
44#95×D17C411yescharacteristic
45#95×D17C411yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53#11762563C2211yescharacteristic
54C2×C17×C4⋊C8C2211yescharacteristic
55C2×C17×#97C2211yescharacteristic
56C4×C4⋊C4×C17C2211yescharacteristic
57C17×#336C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C17×#336C2211yescharacteristic
61C22×C4×D17C2×C411yescharacteristic
62C22×C4×D17C2×C411yescharacteristic
63C22×C4×D17C2×C411yescharacteristic
64C22×C4×D17C2×C411yescharacteristic
65C22×C4×D17C2×C411yescharacteristic
66C22×C4×D17C2×C411yescharacteristic
67C22×C4×D17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C22×C8×C17C2311yescharacteristic
71C22×C8×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2×C411yescharacteristic
73C2×C4⋊C4×C17C2×C411yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C2×C42×C17C2311yescharacteristic
77C2×C4⋊Dic17C2×C411yescharacteristic
78C2×C4⋊Dic17C2×C411yescharacteristic
79C2×#63225C2×C411yescharacteristic
80C2×#63225C2×C411yescharacteristic
81C2×#63225C2×C411yescharacteristic
82C2×#63225C2×C411yescharacteristic
83C2×C4×Dic17C2×C411yescharacteristic
84C2×C4×Dic17C2×C411yescharacteristic
85C2×C4×Dic17C2×C411yescharacteristic
86C2×C4×Dic17C2×C411yescharacteristic
87C2×C4×Dic17C2×C411yescharacteristic
88C2×C4×Dic17C2×C411yescharacteristic
89C2×C4×Dic17C2311yescharacteristic
90C2×C4×Dic17C2311yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C4⋊C4×D17—24no—
97C4⋊C4×D17—24no—
98C42×D17—22no—
99C42×D17—22no—
100C42×D17—22no—
101C42×D17—22no—
102C42×D17—22no—
103C42×D17—22no—
104C17×C4⋊C8—22no—
105C17×#97—22no—
106C17×#95C2×C411yescharacteristic
107C17×#95C2×C411yescharacteristic
108#10587502C2×C411yescharacteristic
109#10587502C2×C411yescharacteristic
110#10587502C2×C411yescharacteristic
111#10587502C2×C411yescharacteristic
112#10587502C2×C411yescharacteristic
113#10587502C2×C411yescharacteristic
114#10587491—22no—
115#10587491—22no—
116#10587491—22no—
117#10587491D414yes—
118#10587486—22no—
119#10587486—22no—
120#10587485—22no—
121#10587485—22no—
122#10587485—22no—
123#10587485—22no—
124C8×Dic17—22no—
125C8×Dic17—22no—
126C4⋊#3622—22no—
127#10587474—22no—
128C23×D17C22×C411yescharacteristic
129C22×C4×C17C22×C411yescharacteristic
130C22×C4×C17C22×C411yescharacteristic
131C22×C4×C17C22×C411yescharacteristic
132C22×C4×C17C22×C411yescharacteristic
133C22×C4×C17C22×C411yescharacteristic
134C22×C4×C17C22×C411yescharacteristic
135C22×C4×C17C2411yescharacteristic
136C22×Dic17C22×C411yescharacteristic
137C22×Dic17C22×C411yescharacteristic
138C22×Dic17C22×C411yescharacteristic
139C22×Dic17C22×C411yescharacteristic
140C22×Dic17C22×C411yescharacteristic
141C22×Dic17C22×C411yescharacteristic
142C22×Dic17C22×C411yescharacteristic
143C2×C4×D17—21no—
144C2×C4×D17—21no—
145C2×C4×D17—21no—
146C2×C4×D17—21no—
147C2×C4×D17—22no—
148C2×C4×D17—22no—
149C2×C4×D17—22no—
150C2×C4×D17—22no—
151C2×C4×D17—22no—
152C2×C4×D17—22no—
153C2×C4×D17—22no—
154C2×C4×D17—22no—
155C2×C4×D17—22no—
156C2×C4×D17—22no—
157C2×C4×D17—22no—
158C2×C4×D17—22no—
159C2×C4×D17—22no—
160C2×C4×D17—22no—
161C2×C4×D17—41no—
162C2×C4×D17—41no—
163C2×C4×D17—41no—
164C2×C4×D17M1614yes—
165C2×C4×D17C2×D412yes—
166C2×C4×D17#4112yes—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17—21no—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17C2×D412yes—
179C4⋊C4×C17—22no—
180C4⋊C4×C17—22no—
181C42×C17—21no—
182C42×C17—21no—
183C42×C17—21no—
184C42×C17—21no—
185C42×C17—21no—
186C42×C17—21no—
187C4⋊Dic17—22no—
188C4⋊Dic17—22no—
189#63225—22no—
190#63225—22no—
191#63225—22no—
192#63225M1614yes—
193C4×Dic17—21no—
194C4×Dic17—21no—
195C4×Dic17—21no—
196C4×Dic17—21no—
197C4×Dic17—22no—
198C4×Dic17—22no—
199C4×Dic17—22no—
200C4×Dic17—22no—
201C4×Dic17—22no—
202C4×Dic17—22no—
203C4×Dic17#4112yes—
204C4×Dic17#4112yes—
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×#3622—21no—
215C2×#3622—21no—
216C2×#3622C2×D412yes—
217#22603—171nomaximal, Sylow
218C23×C17C23×C411yescharacteristic
219C22×D17—21no—
220C22×D17—21no—
221C22×D17—21no—
222C22×D17—21no—
223C22×D17C4×D412yes—
224C22×D17C2×M1612yes—
225C22×D17#13112yes—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17C2×M1612yes—
243C2×C4×C17C2×M1612yes—
244C2×C4×C17C22×D411yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17C2×#4111yescharacteristic
247C2×C4×C17C2×#4111yescharacteristic
248C2×C4×C172−1+411yescharacteristic
249C2×C4×C172−1+411yescharacteristic
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17#11712yes—
266C2×Dic17#11712yes—
267C2×Dic17C4×D412yes—
268C2×Dic17C2×M1612yes—
269C2×Dic17C2×M1612yes—
270C2×Dic17C2×M1612yes—
271C4×D17—22no—
272C4×D17—22no—
273C4×D17—22no—
274C4×D17—22no—
275C4×D17—28no—
276C4×D17—44no—
277C4×D17—44no—
278C4×D17—44no—
279C4×D17—44no—
280C4×D17—44no—
281C8×C17—22no—
282C8×C17—41no—
283C8×C17—41no—
284C8×C17—41no—
285#3622—22no—
286#3622—41no—
287#3622—41no—
288#3622—41no—
289C2×#432—171no—
290C2×#432—171no—
291C2×C4×C4⋊C4—171no—
292#1939—171no—
293#1939—171no—
294#1917—171no—
295#1917—171no—
296#1917—171no—
297#1917D1711yescharacteristic, complemented
298#1817—171no—
299#1817—171no—
300#1797—171no—
301#1770—171no—
302#1770—171no—
303C4×C22⋊C8—171no—
304C22×C17C2×#11711yescharacteristic
305C22×C17C2×C4×D411yescharacteristic
306C22×C17#51911yescharacteristic
307C22×C17C22×M1611yescharacteristic
308C22×C17C22×M1611yescharacteristic
309C22×C17C2×#13111yescharacteristic
310C22×C17#56811yescharacteristic
311C2×D17—22no—
312C2×D17—22no—
313C2×D17—22no—
314C2×D17—22no—
315C2×D17—22no—
316C2×D17—22no—
317C2×D17#43614yes—
318C4×C17—22no—
319C4×C17—22no—
320C4×C17—41no—
321C4×C17—41no—
322C4×C17—41no—
323C4×C17—41no—
324C4×C17#54711yescharacteristic
325C4×C17#54711yescharacteristic
326C4×C17#55211yescharacteristic
327C4×C17#55211yescharacteristic
328Dic17—22no—
329Dic17—22no—
330Dic17—22no—
331Dic17—41no—
332Dic17—41no—
333Dic17—41no—
334Dic17#43214yescomplemented
335C22×C4⋊C4—171no—
336C22×C4⋊C4—171no—
337C22×C42—171no—
338C22×C42—171no—
339C22×C42—171no—
340#432—344no—
341#432—344no—
342C2×C4⋊C8—171no—
343C2×C4⋊C8—171no—
344C2×C4⋊C8—171no—
345C2×C4⋊C8—171no—
346C2×C4⋊C8—171no—
347C2×C4⋊C8C2×D1711yescharacteristic
348C2×C22⋊C8—171no—
349C2×C22⋊C8—171no—
350C2×C22⋊C8—171no—
351C2×C22⋊C8—171no—
352C2×#97—171no—
353C2×#97—171no—
354C2×#97—171no—
355C2×#97C2×D1711yescharacteristic
356C2×C4×C8—171no—
357C2×C4×C8—171no—
358C4×C4⋊C4—171no—
359C4×C4⋊C4—171no—
360C4×C4⋊C4—171no—
361C4×C4⋊C4—171no—
362C4×C4⋊C4—171no—
363C4×C4⋊C4—171no—
364C4×C4⋊C4—171no—
365C4×C4⋊C4C2×D1711yescharacteristic
366C2×#95—171no—
367C2×#95—171no—
368#336—171no—
369#336—171no—
370#336—171no—
371#336—171no—
372#336—171no—
373#336—171no—
374#336—171no—
375#336—171no—
376#336C2×D1711yescharacteristic
377#336C2×D1711yescharacteristic
378#336C2×D1711yescharacteristic
379#336C2×D1711yescharacteristic
380C2×C17#234411yescharacteristic
381C2×C17C2×#43211yescharacteristic
382C2×C17#292411yescharacteristic
383C2×C17C2×#43611yescharacteristic
384C2×C17D4×M1611yescharacteristic
385C2×C17#298311yescharacteristic
386C2×C17#298411yescharacteristic
387D17—24no—
388C23×C4—171no—
389C23×C4—171no—
390C23×C4—171no—
391C23×C4—171no—
392C23×C4—171no—
393C23×C4—171no—
394C23×C4—171no—
395C22×C8—171no—
396C22×C8—171no—
397C22×C8—171no—
398C22×C8—171no—
399C22×C8C22×D1711yescharacteristic
400C22×C8C22×D1711yescharacteristic
401C22×C8C22×D1711yescharacteristic
402C22×C8C22×D1711yescharacteristic
403C2×C4⋊C4—171no—
404C2×C4⋊C4—171no—
405C2×C4⋊C4—171no—
406C2×C4⋊C4—171no—
407C2×C4⋊C4—171no—
408C2×C4⋊C4—171no—
409C2×C4⋊C4—344no—
410C2×C4⋊C4—344no—
411C2×C4⋊C4C4×D1711yescharacteristic
412C2×C4⋊C4C4×D1711yescharacteristic
413C2×C42—171no—
414C2×C42—171no—
415C2×C42—171no—
416C2×C42—171no—
417C2×C42—171no—
418C2×C42—171no—
419C2×C42—171no—
420C2×C42—171no—
421C2×C42—171no—
422C2×C42—342no—
423C2×C42—342no—
424C2×C42—342no—
425C2×C42—342no—
426C2×C42—342no—
427C2×C42—342no—
428C2×C42C22×D1711yescharacteristic
429C2×C42C22×D1711yescharacteristic
430C2×C42C22×D1711yescharacteristic
431C4⋊C8—22no—
432C4⋊C8—342no—
433C4⋊C8—342no—
434C4⋊C8—342no—
435C4⋊C8—342no—
436C4⋊C8—342no—
437C22⋊C8—174no—
438C22⋊C8—342no—
439C22⋊C8—342no—
440C22⋊C8—342no—
441#97—22no—
442#97—342no—
443#97—342no—
444#97—342no—
445C4×C8—342no—
446C4×C8—342no—
447#95—171no—
448#95—171no—
449#95—171no—
450#95—171no—
451#95—171no—
452#95—171no—
453#95C4×D1711yescharacteristic
454#95C4×D1711yescharacteristic
455C17#2260311yescharacteristic, complemented, Sylow
456C24—171no—
457C22×C4—171no—
458C22×C4—171no—
459C22×C4—171no—
460C22×C4—171no—
461C22×C4—172no—
462C22×C4—172no—
463C22×C4—174no—
464C22×C4—171no—
465C22×C4—171no—
466C22×C4—171no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—342no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4—341no—
483C22×C4—341no—
484C22×C4—342no—
485C22×C4—681no—
486C22×C4—681no—
487C22×C4—681no—
488C22×C4C2×C4×D1711yescharacteristic
489C22×C4C2×C4×D1711yescharacteristic
490C22×C4C2×C4×D1711yescharacteristic
491C22×C4C2×C4×D1711yescharacteristic
492C22×C4C2×C4×D1711yescharacteristic
493C22×C4C2×C4×D1711yescharacteristic
494C22×C4C23×D1711yescharacteristic
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—172no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8D4×D1712yes—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—174no—
522C4⋊C4—342no—
523C4⋊C4—342no—
524C4⋊C4—342no—
525C4⋊C4—342no—
526C4⋊C4—342no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—172no—
534C42—172no—
535C42—342no—
536C42—342no—
537C42—342no—
538C42—342no—
539C42—342no—
540C42—342no—
541C42—341no—
542C42—341no—
543C42—341no—
544C42—341no—
545C23—172no—
546C23—172no—
547C23—172no—
548C23—341no—
549C23—341no—
550C23—341no—
551C23—341no—
552C23C22×C4×D1711yescharacteristic
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—172no—
570C2×C4—172no—
571C2×C4—172no—
572C2×C4—172no—
573C2×C4—172no—
574C2×C4—172no—
575C2×C4—348no—
576C2×C4—341no—
577C2×C4—341no—
578C2×C4—341no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—341no—
582C2×C4—342no—
583C2×C4—342no—
584C2×C4—342no—
585C2×C4—342no—
586C2×C4—341no—
587C2×C4—341no—
588C2×C4—341no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—341no—
595C2×C4—684no—
596C2×C4—684no—
597C2×C4—684no—
598C2×C4—684no—
599C2×C4—684no—
600C2×C4M16×D1712yes—
601C2×C4M16×D1712yes—
602C2×C4C2×D4×D1711yescharacteristic
603C2×C4#1058769411yescharacteristic
604C2×C4#41×D1711yescharacteristic
605C2×C4#41×D1711yescharacteristic
606C2×C4#41×D1711yescharacteristic
607C2×C4#1058769711yescharacteristic
608C8—22no—
609C8—41no—
610C8—41no—
611C8—41no—
612C8—342no—
613C8—681no—
614C8—681no—
615C8—681no—
616C22—174no—
617C22—342no—
618C22—342no—
619C22—342no—
620C22—342no—
621C22—342no—
622C22—342no—
623C22#117×D1711yescharacteristic
624C22C4×D4×D1711yescharacteristic
625C22#1176328511yescharacteristic
626C22C2×M16×D1711yescharacteristic
627C22C2×M16×D1711yescharacteristic
628C22#131×D1711yescharacteristic
629C22#1176357411yescharacteristic
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—174no—
637C4—342no—
638C4—342no—
639C4—342no—
640C4—681no—
641C4—681no—
642C4—681no—
643C4#1176345711yescharacteristic
644C4#1176346111yescharacteristic
645C4#1176349111yescharacteristic
646C4#1176349511yescharacteristic
647C2—344no—
648C2#1453165011yescharacteristic
649C2D17×#43211yescharacteristic
650C2#1453744011yescharacteristic
651C2D17×#43611yescharacteristic
652C2#1453751711yescharacteristic
653C2#1453786611yescharacteristic
654C2#1453787011yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1917C2
prime radicalC23×D17C22×C4
cosocleC22×C4×C17C24
hypercenter#1917D17
hypocenterC17#22603
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1917

Complements

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

NH ≅ G/Ntype
4. #14529501C2semidirect
5. #14529499C2semidirect
6. #14529350C2semidirect
7. #14529349C2semidirect
8. #14529347C2semidirect
12. #14528652C2semidirect
13. #14528650C2semidirect
15. C17×#1917C2semidirect
297. #1917D17semidirect
334. Dic17#432semidirect
455. C17#22603semidirect

Maximal subgroups

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

subgroupindex
C2×#117628652
C2×#117628642
C4×C4⋊C4×D172
#145295012
#145294992
#145293502
#145293492
#145293472
#145288512
#145287632
#145286522
#145286502
#145285412
C17×#19172
#2260317