Finite Group Databasebeta

← #17044404

Subgroups — #17044404

Order 4352.

866 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊C4×D17C211yesmaximal, characteristic
2#14529344C211yesmaximal, characteristic, complemented
3#14529343C211yesmaximal, characteristic, complemented
4#14529342C211yesmaximal, characteristic, complemented
5#14529341C211yesmaximal, characteristic, complemented
6#14529340C211yesmaximal, characteristic, complemented
7#14529339C211yesmaximal, characteristic, complemented
8#14529338C211yesmaximal, characteristic, complemented
9#14528851C211yesmaximal, characteristic
10#14528851C211yesmaximal, characteristic
11#14528763C211yesmaximal, characteristic
12#14528761C211yesmaximal, characteristic
13C4×#10587491C211yesmaximal, characteristic
14#14528541C211yesmaximal, characteristic
15C17×#1916C211yesmaximal, characteristic, complemented
16C2×C4⋊C4×D17C411yescharacteristic
17C2×C4⋊C4×D17C411yescharacteristic
18C2×C42×D17C2211yescharacteristic
19C2×C42×D17C2211yescharacteristic
20C2×C42×D17C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×#10587485C2211yescharacteristic
23C2×#10587485C2211yescharacteristic
24C2×#10587491C2211yescharacteristic
25C2×#10587491C2211yescharacteristic
26C2×#10587491C2211yescharacteristic
27C2×#10587491C2211yescharacteristic
28C2×C4×#3622C2211yescharacteristic
29C2×C8×Dic17C2211yescharacteristic
30C2×C8×Dic17C2211yescharacteristic
31C4⋊C4×Dic17C411yescharacteristic
32C4⋊C4×Dic17C411yescharacteristic
33C4×#63225C2211yescharacteristic
34C4×#63225C2211yescharacteristic
35C4×C4⋊Dic17C2211yescharacteristic
36#11762660C411yescharacteristic
37#11762660C411yescharacteristic
38#95×D17C411yescharacteristic
39#95×D17C411yescharacteristic
40#11762564C2211yescharacteristic
41#11762564C2211yescharacteristic
42#11762564C2211yescharacteristic
43#11762564C2211yescharacteristic
44#11762563C2211yescharacteristic
45#11762563C2211yescharacteristic
46#11762563C2211yescharacteristic
47#11762563C2211yescharacteristic
48#11762563C2211yescharacteristic
49#11762563C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×C4⋊C8C2211yescharacteristic
53C2×C4×C8×C17C2211yescharacteristic
54C4×C4⋊C4×C17C2211yescharacteristic
55C17×#336C2211yescharacteristic
56C17×#336C2211yescharacteristic
57C17×#336C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C22×C4×D17C2×C411yescharacteristic
60C22×C4×D17C2×C411yescharacteristic
61C22×C4×D17C2×C411yescharacteristic
62C22×C4×D17C2×C411yescharacteristic
63C22×C4×D17C2×C411yescharacteristic
64C22×C4×D17C2×C411yescharacteristic
65C22×C4×D17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2×C411yescharacteristic
71C2×C4⋊C4×C17C2×C411yescharacteristic
72C2×C42×C17C2311yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C4⋊Dic17C2×C411yescharacteristic
76C2×C4⋊Dic17C2×C411yescharacteristic
77C2×#63225C2×C411yescharacteristic
78C2×#63225C2×C411yescharacteristic
79C2×#63225C2×C411yescharacteristic
80C2×#63225C2×C411yescharacteristic
81C2×C4×Dic17C2×C411yescharacteristic
82C2×C4×Dic17C2×C411yescharacteristic
83C2×C4×Dic17C2×C411yescharacteristic
84C2×C4×Dic17C2×C411yescharacteristic
85C2×C4×Dic17C2×C411yescharacteristic
86C2×C4×Dic17C2×C411yescharacteristic
87C2×C4×Dic17C2311yescharacteristic
88C2×C4×Dic17C2311yescharacteristic
89C2×C4×Dic17C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C4⋊C4×D17—24no—
95C4⋊C4×D17C818yescomplemented
96C42×D17—22no—
97C42×D17—22no—
98C42×D17—41no—
99C42×D17—41no—
100C42×D17—41no—
101C42×D17—41no—
102C17×C4⋊C8—41no—
103C4×C8×C17—41no—
104C17×#95C2×C411yescharacteristic
105C17×#95C2×C411yescharacteristic
106#10587502C2×C411yescharacteristic
107#10587502C2×C411yescharacteristic
108#10587502C2×C411yescharacteristic
109#10587502C2×C411yescharacteristic
110#10587502C2×C411yescharacteristic
111#10587502C2×C411yescharacteristic
112#10587491—22no—
113#10587491—22no—
114#10587491—22no—
115#10587491—22no—
116#10587485—41no—
117#10587485—41no—
118C8×Dic17—41no—
119C8×Dic17—41no—
120C4⋊#3622—41no—
121C4×#3622—41no—
122C23×D17C22×C411yescharacteristic
123C22×C4×C17C22×C411yescharacteristic
124C22×C4×C17C22×C411yescharacteristic
125C22×C4×C17C22×C411yescharacteristic
126C22×C4×C17C22×C411yescharacteristic
127C22×C4×C17C22×C411yescharacteristic
128C22×C4×C17C22×C411yescharacteristic
129C22×C4×C17C2411yescharacteristic
130C22×Dic17C22×C411yescharacteristic
131C22×Dic17C22×C411yescharacteristic
132C22×Dic17C22×C411yescharacteristic
133C22×Dic17C22×C411yescharacteristic
134C22×Dic17C22×C411yescharacteristic
135C22×Dic17C22×C411yescharacteristic
136C22×Dic17C22×C411yescharacteristic
137C2×C4×D17—22no—
138C2×C4×D17—22no—
139C2×C4×D17—21no—
140C2×C4×D17—21no—
141C2×C4×D17—21no—
142C2×C4×D17—21no—
143C2×C4×D17—22no—
144C2×C4×D17—22no—
145C2×C4×D17—41no—
146C2×C4×D17—41no—
147C2×C4×D17—41no—
148C2×C4×D17—41no—
149C2×C4×D17—41no—
150C2×C4×D17—41no—
151C2×C4×D17—41no—
152C2×C4×D17—41no—
153C2×C4×D17—41no—
154C2×C4×D17—41no—
155C2×C4×D17C2×C814yes—
156C2×C4×D17C2×C814yes—
157C2×C4×D17C2×C814yes—
158C2×C4×D17M1614yes—
159C2×C4×D17#4112yes—
160C2×C4×D17#4112yes—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17—21no—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C4⋊C4×C17—22no—
174C4⋊C4×C17C2×C814yes—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—21no—
178C42×C17—21no—
179C42×C17—21no—
180C42×C17—21no—
181C4⋊Dic17—22no—
182C4⋊Dic17C2×C814yes—
183#63225—22no—
184#63225—22no—
185#63225C2×C814yes—
186#63225C2×C814yes—
187C4×Dic17—21no—
188C4×Dic17—21no—
189C4×Dic17—21no—
190C4×Dic17—21no—
191C4×Dic17—21no—
192C4×Dic17—21no—
193C4×Dic17—22no—
194C4×Dic17—22no—
195C4×Dic17—41no—
196C4×Dic17—41no—
197C4×Dic17—41no—
198C4×Dic17—41no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211#21912—171nomaximal, Sylow
212C23×C17C23×C411yescharacteristic
213C22×D17—21no—
214C22×D17—21no—
215C22×D17—21no—
216C22×D17—21no—
217C22×D17#11712yes—
218C22×D17C22×C812yes—
219C22×D17C2×M1612yes—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
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×C17C22×C812yes—
235C2×C4×C17C22×C812yes—
236C2×C4×C17C22×C812yes—
237C2×C4×C17C2×M1612yes—
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17H(5,𝔽2)11yescharacteristic
241C2×C4×C17H(5,𝔽2)11yescharacteristic
242C2×C4×C172−1+411yescharacteristic
243C2×C4×C172−1+411yescharacteristic
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
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#11712yes—
261C2×Dic17C22×C812yes—
262C2×Dic17C22×C812yes—
263C2×Dic17C22×C812yes—
264C2×Dic17C2×M1612yes—
265C4×D17—28no—
266C4×D17—22no—
267C4×D17—22no—
268C4×D17—22no—
269C4×D17—22no—
270C4×D17—44no—
271C4×D17—44no—
272C4×D17—44no—
273C4×D17—44no—
274C4×D17—44no—
275C8×C17—41no—
276C8×C17—41no—
277C8×C17—41no—
278C8×C17—41no—
279#3622—41no—
280#3622—41no—
281#3622—41no—
282#3622—41no—
283C2×C4×C4⋊C4—171no—
284#1916—171no—
285#1916—171no—
286#1916—171no—
287#1916—171no—
288#1916—171no—
289#1916—171no—
290#1916—171no—
291#1916D1711yescharacteristic, complemented
292#1817—171no—
293#1817—171no—
294#1797—171no—
295#1797—171no—
296C4×C22⋊C8—171no—
297C4×C22⋊C8—171no—
298C22×C17C2×#11711yescharacteristic
299C22×C17#51811yescharacteristic
300C22×C17#51911yescharacteristic
301C22×C17C23×C811yescharacteristic
302C22×C17C22×M1611yescharacteristic
303C22×C17#56811yescharacteristic
304C22×C17#56811yescharacteristic
305C2×D17—22no—
306C2×D17—22no—
307C2×D17—22no—
308C2×D17—22no—
309C2×D17—22no—
310C2×D17—22no—
311C2×D17#43114yes—
312C4×C17—22no—
313C4×C17—22no—
314C4×C17—22no—
315C4×C17—22no—
316C4×C17—41no—
317C4×C17—41no—
318C4×C17#53911yescharacteristic
319C4×C17#53911yescharacteristic
320C4×C17#53911yescharacteristic
321C4×C17#53911yescharacteristic
322Dic17—22no—
323Dic17—22no—
324Dic17—22no—
325Dic17—22no—
326Dic17—22no—
327Dic17—41no—
328Dic17—41no—
329C22×C4⋊C4—171no—
330C22×C4⋊C4—171no—
331C22×C42—171no—
332C22×C42—171no—
333C22×C42—171no—
334C2×C4⋊C8—171no—
335C2×C4⋊C8—171no—
336C2×C4⋊C8—171no—
337C2×C4⋊C8C2×D1711yescharacteristic
338C2×C22⋊C8—171no—
339C2×C22⋊C8—171no—
340C2×C22⋊C8—171no—
341C2×C22⋊C8—171no—
342C2×C4×C8—171no—
343C2×C4×C8—171no—
344C2×C4×C8—171no—
345C2×C4×C8C2×D1711yescharacteristic
346C4×C4⋊C4—171no—
347C4×C4⋊C4—171no—
348C4×C4⋊C4—171no—
349C4×C4⋊C4—171no—
350C4×C4⋊C4—171no—
351C4×C4⋊C4—171no—
352C4×C4⋊C4—171no—
353C4×C4⋊C4C2×D1711yescharacteristic
354C2×#95—171no—
355C2×#95—171no—
356#336—171no—
357#336—171no—
358#336—171no—
359#336—171no—
360#336—171no—
361#336—171no—
362#336—171no—
363#336—171no—
364#336—171no—
365#336—171no—
366#336—171no—
367#336—171no—
368#336C2×D1711yescharacteristic
369#336C2×D1711yescharacteristic
370#336C2×D1711yescharacteristic
371#336C2×D1711yescharacteristic
372C2×C17#231311yescharacteristic
373C2×C17C2×#43111yescharacteristic
374C2×C17#292111yescharacteristic
375C2×C17#297211yescharacteristic
376C2×C17#297311yescharacteristic
377C2×C17#298211yescharacteristic
378C2×C17#298311yescharacteristic
379D17—24no—
380C23×C4—171no—
381C23×C4—171no—
382C23×C4—171no—
383C23×C4—171no—
384C23×C4—171no—
385C23×C4—171no—
386C23×C4—171no—
387C22×C8—171no—
388C22×C8—171no—
389C22×C8—171no—
390C22×C8—171no—
391C22×C8C22×D1711yescharacteristic
392C22×C8C22×D1711yescharacteristic
393C22×C8C22×D1711yescharacteristic
394C22×C8C22×D1711yescharacteristic
395C2×C4⋊C4—171no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—178no—
399C2×C4⋊C4—171no—
400C2×C4⋊C4—171no—
401C2×C4⋊C4—171no—
402C2×C4⋊C4—344no—
403C2×C4⋊C4C4×D1711yescharacteristic
404C2×C4⋊C4C4×D1711yescharacteristic
405C2×C42—171no—
406C2×C42—171no—
407C2×C42—171no—
408C2×C42—171no—
409C2×C42—171no—
410C2×C42—171no—
411C2×C42—171no—
412C2×C42—171no—
413C2×C42—171no—
414C2×C42—342no—
415C2×C42—342no—
416C2×C42—681no—
417C2×C42—681no—
418C2×C42—681no—
419C2×C42—681no—
420C2×C42C22×D1711yescharacteristic
421C2×C42C22×D1711yescharacteristic
422C2×C42C22×D1711yescharacteristic
423C4⋊C8—41no—
424C4⋊C8—681no—
425C4⋊C8—681no—
426C4⋊C8—681no—
427C22⋊C8—342no—
428C22⋊C8—342no—
429C22⋊C8—342no—
430C22⋊C8—342no—
431C4×C8—41no—
432C4×C8—681no—
433C4×C8—681no—
434C4×C8—681no—
435#95—171no—
436#95—171no—
437#95—171no—
438#95—171no—
439#95—171no—
440#95—171no—
441#95C4×D1711yescharacteristic
442#95C4×D1711yescharacteristic
443C17#2191211yescharacteristic, complemented, Sylow
444C24—171no—
445C22×C4—171no—
446C22×C4—171no—
447C22×C4—171no—
448C22×C4—171no—
449C22×C4—171no—
450C22×C4—172no—
451C22×C4—172no—
452C22×C4—174no—
453C22×C4—174no—
454C22×C4—174no—
455C22×C4—174no—
456C22×C4—171no—
457C22×C4—171no—
458C22×C4—342no—
459C22×C4—342no—
460C22×C4—342no—
461C22×C4—342no—
462C22×C4—341no—
463C22×C4—341no—
464C22×C4—341no—
465C22×C4—341no—
466C22×C4—681no—
467C22×C4—681no—
468C22×C4—681no—
469C22×C4—681no—
470C22×C4—681no—
471C22×C4—681no—
472C22×C4—681no—
473C22×C4—681no—
474C22×C4—681no—
475C22×C4—681no—
476C22×C4C2×C4×D1711yescharacteristic
477C22×C4C2×C4×D1711yescharacteristic
478C22×C4C2×C4×D1711yescharacteristic
479C22×C4C2×C4×D1711yescharacteristic
480C22×C4C2×C4×D1711yescharacteristic
481C22×C4C2×C4×D1711yescharacteristic
482C22×C4C23×D1711yescharacteristic
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—341no—
496C2×C8—341no—
497C2×C8—341no—
498C2×C8—341no—
499C2×C8—341no—
500C2×C8—341no—
501C2×C8—341no—
502C2×C8—341no—
503C2×C8—341no—
504C2×C8—341no—
505C2×C8—341no—
506C2×C8—341no—
507C4⋊C4—22no—
508C4⋊C4—174no—
509C4⋊C4—174no—
510C4⋊C4—174no—
511C4⋊C4—342no—
512C4⋊C4—342no—
513C4⋊C4—342no—
514C4⋊C4C8×D1714yes—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—342no—
522C42—342no—
523C42—341no—
524C42—341no—
525C42—341no—
526C42—341no—
527C42—341no—
528C42—341no—
529C42—681no—
530C42—681no—
531C42—681no—
532C42—681no—
533C23—172no—
534C23—172no—
535C23—172no—
536C23—341no—
537C23—341no—
538C23—341no—
539C23—341no—
540C23C22×C4×D1711yescharacteristic
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—172no—
556C2×C4—172no—
557C2×C4—172no—
558C2×C4—172no—
559C2×C4—172no—
560C2×C4—348no—
561C2×C4—341no—
562C2×C4—341no—
563C2×C4—341no—
564C2×C4—341no—
565C2×C4—341no—
566C2×C4—341no—
567C2×C4—341no—
568C2×C4—341no—
569C2×C4—341no—
570C2×C4—341no—
571C2×C4—342no—
572C2×C4—342no—
573C2×C4—342no—
574C2×C4—342no—
575C2×C4—341no—
576C2×C4—341no—
577C2×C4—341no—
578C2×C4—341no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—684no—
582C2×C4—684no—
583C2×C4—684no—
584C2×C4—684no—
585C2×C4—684no—
586C2×C4C2×C8×D1712yes—
587C2×C4C2×C8×D1712yes—
588C2×C4C2×C8×D1712yes—
589C2×C4M16×D1712yes—
590C2×C4#1058769111yescharacteristic
591C2×C4#1058769411yescharacteristic
592C2×C4#41×D1711yescharacteristic
593C2×C4#41×D1711yescharacteristic
594C2×C4#1058769611yescharacteristic
595C2×C4#1058769711yescharacteristic
596C8—41no—
597C8—41no—
598C8—41no—
599C8—41no—
600C8—681no—
601C8—681no—
602C8—681no—
603C8—681no—
604C22—174no—
605C22—342no—
606C22—342no—
607C22—342no—
608C22—342no—
609C22—342no—
610C22—342no—
611C22#117×D1711yescharacteristic
612C22#1176328111yescharacteristic
613C22#1176328511yescharacteristic
614C22C22×C8×D1711yescharacteristic
615C22C2×M16×D1711yescharacteristic
616C22#1176357411yescharacteristic
617C22#1176357411yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—342no—
625C4—342no—
626C4—342no—
627C4—342no—
628C4—342no—
629C4—681no—
630C4—681no—
631C4#1176340011yescharacteristic
632C4#1176340211yescharacteristic
633C4#1176340511yescharacteristic
634C4#1176340711yescharacteristic
635C2—344no—
636C2#1453138311yescharacteristic
637C2D17×#43111yescharacteristic
638C2#1453741811yescharacteristic
639C2#1453780511yescharacteristic
640C2#1453780911yescharacteristic
641C2#1453786311yescharacteristic
642C2#1453786611yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D17
commutatorC22×C17C23×C8
FittingC17×#1916C2
socleC23×C17C23×C4
prime radicalC23×D17C22×C4
cosocleC22×C4×C17C24
hypercenter#1916D17
hypocenterC17#21912
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1916

Complements

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

NH ≅ G/Ntype
2. #14529344C2semidirect
3. #14529343C2semidirect
4. #14529342C2semidirect
5. #14529341C2semidirect
6. #14529340C2semidirect
7. #14529339C2semidirect
8. #14529338C2semidirect
15. C17×#1916C2semidirect
95. C4⋊C4×D17C8semidirect
291. #1916D17semidirect
443. C17#21912semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊C4×D172
#145293442
#145293432
#145293422
#145293412
#145293402
#145293392
#145293382
#145288512
#145287632
#145287612
C4×#105874912
#145285412
C17×#19162
#2191217