Finite Group Databasebeta

← #30855651

Subgroups — #30855651

Order 17152.

1374 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863363C211yesmaximal, characteristic
2#22562801C211yesmaximal, characteristic
3#22560304C212yesmaximal, complemented
4#22560304C212yesmaximal, complemented
5#22560299C212yesmaximal, complemented
6#22560295C212yesmaximal, complemented
7#22559930C211yesmaximal, characteristic, complemented
8#22559928C211yesmaximal, characteristic, complemented
9#22559925C211yesmaximal, characteristic, complemented
10C2×#16863158C211yesmaximal, characteristic, complemented
11C2×#520×C67C211yesmaximal, characteristic
12C22×#11761612C2211yescharacteristic
13C23×C67⋊C8C2211yescharacteristic
14C2×#14524002C2211yescharacteristic
15C22×C4⋊Dic67C2211yescharacteristic
16#16863363—24no—
17C2×C22⋊(C67⋊C8)C2212yes—
18C2×C22⋊(C67⋊C8)C2212yes—
19#16863303—21no—
20#16863299—21no—
21#16863276—22no—
22#16863256C2212yes—
23#16863256C2212yes—
24#16863158C2211yescharacteristic
25#16863158C2212yes—
26#16863158C2212yes—
27#16863158C2212yes—
28#16863158C2211yescharacteristic
29#16863158C2212yes—
30#16863158C2212yes—
31#16863158C2211yescharacteristic
32#16863158C2211yescharacteristic
33#16863158C2212yes—
34C22×#41×C67C2211yescharacteristic
35#520×C67—24no—
36#520×C67C418yescomplemented
37C2×C4×Q8×C67—21no—
38C2×C4×D4×C67—22no—
39C2×C4×D4×C67—21no—
40C2×#117×C67C2211yescharacteristic
41C2×#117×C67C2212yes—
42C22×C4⋊C4×C67C2211yescharacteristic
43C22×C4⋊C4×C67C2212yes—
44C23×Dic67—21no—
45C2×#11761612—22no—
46C2×#11761612—22no—
47C2×#11761612—22no—
48C22×C67⋊C8—22no—
49C22×C67⋊C8—22no—
50C22×C67⋊C8—22no—
51C22×C67⋊C8C2311yescharacteristic
52C22×C67⋊C8C2311yescharacteristic
53C22×C67⋊C8C2312yes—
54#14524002—22no—
55#14524002—22no—
56C2×C4⋊Dic67—21no—
57C2×C4⋊Dic67—22no—
58C2×C4⋊Dic67—22no—
59C2×C4⋊Dic67C2312yes—
60C2×C4⋊Dic67C2311yescharacteristic
61C2×C4⋊Dic67C2311yescharacteristic
62C2×C22⋊Dic67—21no—
63C2×C22⋊Dic67—22no—
64C2×C22⋊Dic67—22no—
65C2×C4×Dic67—21no—
66C22⋊(C67⋊C8)—24no—
67C22⋊(C67⋊C8)—24no—
68C67⋊#95—22no—
69C67⋊#95—22no—
70C2×#41×C67—22no—
71C2×#41×C67—22no—
72C2×#41×C67—22no—
73C2×#41×C67—22no—
74C2×#41×C67C2×C414yes—
75C2×#41×C67D414yes—
76C22×Q8×C67D411yescharacteristic
77C22×D4×C67D412yes—
78C22×D4×C67D411yescharacteristic
79C23×C4×C67—21no—
80C23×C4×C67—22no—
81C23×C4×C67C2312yes—
82C23×C4×C67C2311yescharacteristic
83C4×Q8×C67—28no—
84C4×D4×C67—28no—
85C4×D4×C67—216no—
86#117×C67—22no—
87#117×C67—24no—
88#117×C67C2×C414yes—
89#117×C67C2×C418yescomplemented
90C2×C4⋊C4×C67—21no—
91C2×C4⋊C4×C67—22no—
92C2×C4⋊C4×C67—22no—
93C2×C4⋊C4×C67—24no—
94C2×C4⋊C4×C67C2×C414yes—
95C2×C4⋊C4×C67C2×C418yescomplemented
96C2×C4⋊C4×C67C2312yes—
97C2×C4⋊C4×C67C2311yescharacteristic
98C2×C4⋊C4×C67C2311yescharacteristic
99C2×C22⋊C4×C67—21no—
100C2×C22⋊C4×C67—22no—
101C2×C42×C67—21no—
102C2×C42×C67—22no—
103C22×Dic67—21no—
104C22×Dic67—21no—
105C22×Dic67—22no—
106C22×Dic67—42no—
107C22×Dic67—81no—
108C22×Dic67—81no—
109#11761612—44no—
110C2×C67⋊C8—21no—
111C2×C67⋊C8—21no—
112C2×C67⋊C8—22no—
113C2×C67⋊C8—21no—
114C2×C67⋊C8—21no—
115C2×C67⋊C8—21no—
116C2×C67⋊C8—21no—
117C2×C67⋊C8—22no—
118C2×C67⋊C8—22no—
119C2×C67⋊C8—44no—
120C4⋊Dic67—22no—
121C4⋊Dic67—22no—
122C4⋊Dic67—21no—
123C4⋊Dic67—21no—
124C4⋊Dic67—21no—
125C4⋊Dic67—21no—
126C4⋊Dic67—42no—
127C22⋊Dic67—42no—
128C22⋊Dic67—82no—
129C22⋊Dic67—82no—
130C4×Dic67—42no—
131C24×C67C2×D412yes—
132C24×C67C2×D411yescharacteristic
133#41×C67—28no—
134#41×C67—28no—
135#41×C67—28no—
136#41×C67—28no—
137C2×Q8×C67—22no—
138C2×Q8×C67—22no—
139C2×Q8×C67C22⋊C414yes—
140C2×D4×C67—24no—
141C2×D4×C67—24no—
142C2×D4×C67—22no—
143C2×D4×C67—22no—
144C2×D4×C67C22⋊C418yes—
145C2×D4×C67C22⋊C414yes—
146C22×C4×C67—22no—
147C22×C4×C67—22no—
148C22×C4×C67—22no—
149C22×C4×C67—22no—
150C22×C4×C67—21no—
151C22×C4×C67—21no—
152C22×C4×C67—22no—
153C22×C4×C67—24no—
154C22×C4×C67—28no—
155C22×C4×C67—81no—
156C22×C4×C67—82no—
157C22×C4×C67—81no—
158C22×C4×C67—82no—
159C22×C4×C67C22×C414yes—
160C22×C4×C67C22×C412yes—
161C22×C4×C67C2×D412yes—
162C22×C4×C67C2×D414yes—
163C22×C4×C67C2×D411yescharacteristic
164C22×C4×C67C2×D412yes—
165C22×C4×C67#4112yes—
166C22×C4×C67#4112yes—
167C22×C4×C67#4112yes—
168C22×C4×C67#4112yes—
169C22×C4×C67C2411yescharacteristic
170C4⋊C4×C67—24no—
171C4⋊C4×C67—28no—
172C4⋊C4×C67—21no—
173C4⋊C4×C67—22no—
174C4⋊C4×C67—21no—
175C4⋊C4×C67C22×C414yes—
176C4⋊C4×C67C22×C412yes—
177C4⋊C4×C67C22×C412yes—
178C22⋊C4×C67—24no—
179C22⋊C4×C67—28no—
180C42×C67—24no—
181C42×C67—28no—
182C23×C67—21no—
183C23×C67—21no—
184C23×C67—21no—
185C23×C67—21no—
186C23×C67—22no—
187C23×C67—22no—
188C23×C67—22no—
189C23×C67—22no—
190C23×C67—22no—
191C23×C67—22no—
192C23×C67C2×C22⋊C414yes—
193C23×C67C2×C22⋊C412yes—
194C23×C67#12312yes—
195C23×C67#12312yes—
196C23×C67C22×D411yescharacteristic
197Q8×C67—28no—
198D4×C67—216no—
199D4×C67—28no—
200C2×C4×C67—22no—
201C2×C4×C67—22no—
202C2×C4×C67—24no—
203C2×C4×C67—24no—
204C2×C4×C67—24no—
205C2×C4×C67—24no—
206C2×C4×C67—22no—
207C2×C4×C67—22no—
208C2×C4×C67—24no—
209C2×C4×C67—42no—
210C2×C4×C67—41no—
211C2×C4×C67—41no—
212C2×C4×C67—42no—
213C2×C4×C67—41no—
214C2×C4×C67—41no—
215C2×C4×C67—84no—
216C2×C4×C67—88no—
217C2×C4×C67C2×C22⋊C414yes—
218C2×C4×C67C2×C22⋊C412yes—
219C2×C4×C67#11714yes—
220C2×C4×C67#11714yes—
221C2×C4×C67#12314yes—
222C2×C4×C67#12314yes—
223C2×C4×C67#12312yes—
224C2×C4×C67#12312yes—
225C2×C4×C67C23×C411yescharacteristic
226C2×C4×C67C22×D411yescharacteristic
227C2×C4×C67C2×#4111yescharacteristic
228C2×C4×C67C2×#4111yescharacteristic
229C2×C4×C67C2×#4111yescharacteristic
230C2×C4×C67C2×#4111yescharacteristic
231C2×Dic67—41no—
232C2×Dic67—41no—
233C2×Dic67—42no—
234C2×Dic67—41no—
235C2×Dic67—42no—
236C2×Dic67—41no—
237C2×Dic67—42no—
238C2×Dic67—41no—
239C2×Dic67—41no—
240C2×Dic67—84no—
241C67⋊C8—41no—
242C67⋊C8—41no—
243C67⋊C8—42no—
244C22×C67—22no—
245C22×C67—22no—
246C22×C67—22no—
247C22×C67—22no—
248C22×C67—22no—
249C22×C67—22no—
250C22×C67—24no—
251C22×C67—24no—
252C22×C67—24no—
253C22×C67—42no—
254C22×C67—42no—
255C22×C67—42no—
256C22×C67—42no—
257C22×C67#38114yes—
258C22×C67C22×C22⋊C411yescharacteristic
259C22×C67C2×#12311yescharacteristic
260C22×C67C2×#12311yescharacteristic
261C22×C67#57611yescharacteristic
262C22×C67#57711yescharacteristic
263C22×C67#57711yescharacteristic
264C22×C67#57811yescharacteristic
265C4×C67—24no—
266C4×C67—42no—
267C4×C67—44no—
268C4×C67—42no—
269C4×C67#38114yes—
270C4×C67C2×#11711yescharacteristic
271C4×C67C2×#11711yescharacteristic
272C4×C67C2×#12311yescharacteristic
273C4×C67C2×#12311yescharacteristic
274Dic67—82no—
275Dic67—81no—
276Dic67—81no—
277#20703—671nomaximal, Sylow
278C2×C67—24no—
279C2×C67—44no—
280C2×C67C2×#38111yescharacteristic
281C2×C67#289711yescharacteristic
282C2×C67#289811yescharacteristic
283C2×C67#309611yescharacteristic
284C2×C67#309711yescharacteristic
285C2×C67#309811yescharacteristic
286C2×C67#309911yescharacteristic
287C2×#520D6711yescharacteristic
288C2×#408—671no—
289#2311—671no—
290#1825—672no—
291#1825—672no—
292#1824—672no—
293#1823—672no—
294#1736—671no—
295#1736—671no—
296#1735—671no—
297C2×#340—671no—
298C67#2070311yescharacteristic, complemented, Sylow
299C22×#41C2×D6711yescharacteristic
300C22×M16—671no—
301C23×C8—671no—
302#520—24no—
303#520Dic6718yescomplemented
304C2×C4×Q8—21no—
305C2×C4×D4—22no—
306C2×C4×D4—21no—
307C2×#117—671no—
308C2×#117C2×D6711yescharacteristic
309C2×#117C2×D6712yes—
310C22×C4⋊C4—671no—
311C22×C4⋊C4C2×D6711yescharacteristic
312C22×C4⋊C4C2×D6712yes—
313#408—1344no—
314C2×C22⋊C8—672no—
315C2×C22⋊C8—672no—
316#391—1341no—
317#390—1341no—
318#386—1342no—
319#380—672no—
320#380—672no—
321#340—671no—
322#340—672no—
323#340—672no—
324#340—672no—
325#340—671no—
326#340—672no—
327#340—672no—
328#340—671no—
329#340—671no—
330#340—672no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41C2×Dic6714yes—
336C2×#41C22⋊D6714yes—
337C22×Q8C22⋊D6711yescharacteristic
338C22×D4C22⋊D6712yes—
339C22×D4C22⋊D6711yescharacteristic
340C23×C4—21no—
341C23×C4—22no—
342C23×C4—1341no—
343C23×C4C22×D6712yes—
344C23×C4C22×D6711yescharacteristic
345C2×M16—1342no—
346C2×M16—1342no—
347C2×M16—1342no—
348C22×C8—671no—
349C22×C8—671no—
350C22×C8—672no—
351C22×C8—1342no—
352C22×C8—1342no—
353C22×C8—1342no—
354C4×Q8—28no—
355C4×D4—28no—
356C4×D4—216no—
357#117—22no—
358#117—24no—
359#117—1342no—
360#117—1342no—
361#117C2×Dic6714yes—
362#117C2×Dic6718yescomplemented
363C2×C4⋊C4—21no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—672no—
368C2×C4⋊C4—671no—
369C2×C4⋊C4—671no—
370C2×C4⋊C4—1341no—
371C2×C4⋊C4—1342no—
372C2×C4⋊C4—1342no—
373C2×C4⋊C4C2×Dic6714yes—
374C2×C4⋊C4C2×Dic6718yescomplemented
375C2×C4⋊C4C22×D6712yes—
376C2×C4⋊C4C22×D6711yescharacteristic
377C2×C4⋊C4C22×D6711yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—1341no—
381C2×C22⋊C4—1342no—
382C2×C22⋊C4—1342no—
383C2×C42—21no—
384C2×C42—22no—
385C2×C42—1341no—
386C22⋊C8—1344no—
387C22⋊C8—1344no—
388#95—1342no—
389#95—1342no—
390C24C2×C22⋊D6712yes—
391C24C2×C22⋊D6711yescharacteristic
392#41—28no—
393#41—28no—
394#41—28no—
395#41—28no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8C22⋊Dic6714yes—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4C22⋊Dic6718yes—
404C2×D4C22⋊Dic6714yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—81no—
415C22×C4—82no—
416C22×C4—81no—
417C22×C4—82no—
418C22×C4—2682no—
419C22×C4—5361no—
420C22×C4—5361no—
421C22×C4—1341no—
422C22×C4—1341no—
423C22×C4—1342no—
424C22×C4C22×Dic6714yes—
425C22×C4C22×Dic6712yes—
426C22×C4C2×C22⋊D6712yes—
427C22×C4C2×C22⋊D6714yes—
428C22×C4C2×C22⋊D6711yescharacteristic
429C22×C4C2×C22⋊D6712yes—
430C22×C4#1176162812yes—
431C22×C4#1176162812yes—
432C22×C4#1176162812yes—
433C22×C4#1176162812yes—
434C22×C4C23×D6711yescharacteristic
435M16—2684no—
436C2×C8—2684no—
437C2×C8—1341no—
438C2×C8—1341no—
439C2×C8—1342no—
440C2×C8—1341no—
441C2×C8—1341no—
442C2×C8—1341no—
443C2×C8—1341no—
444C2×C8—1342no—
445C2×C8—1342no—
446C4⋊C4—24no—
447C4⋊C4—28no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—2682no—
452C4⋊C4—1342no—
453C4⋊C4—1342no—
454C4⋊C4—1341no—
455C4⋊C4—1341no—
456C4⋊C4—1341no—
457C4⋊C4—1341no—
458C4⋊C4C22×Dic6714yes—
459C4⋊C4C22×Dic6712yes—
460C4⋊C4C22×Dic6712yes—
461C22⋊C4—24no—
462C22⋊C4—28no—
463C22⋊C4—2682no—
464C22⋊C4—5362no—
465C22⋊C4—5362no—
466C42—24no—
467C42—28no—
468C42—2682no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23C2×C22⋊Dic6714yes—
480C23C2×C22⋊Dic6712yes—
481C23#1452402612yes—
482C23#1452402612yes—
483C23C22×C22⋊D6711yescharacteristic
484Q8—28no—
485D4—216no—
486D4—28no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—84no—
503C2×C4—88no—
504C2×C4—2681no—
505C2×C4—2681no—
506C2×C4—2682no—
507C2×C4—2681no—
508C2×C4—2682no—
509C2×C4—2681no—
510C2×C4—2682no—
511C2×C4—2681no—
512C2×C4—2681no—
513C2×C4—5364no—
514C2×C4C2×C22⋊Dic6714yes—
515C2×C4C2×C22⋊Dic6712yes—
516C2×C4#1452400214yes—
517C2×C4#1452400214yes—
518C2×C4#1452402612yes—
519C2×C4#1452402612yes—
520C2×C4#1452402614yes—
521C2×C4#1452402614yes—
522C2×C4C23×Dic6711yescharacteristic
523C2×C4C22×C22⋊D6711yescharacteristic
524C2×C4C2×#1176162811yescharacteristic
525C2×C4C2×#1176162811yescharacteristic
526C2×C4C2×#1176162811yescharacteristic
527C2×C4C2×#1176162811yescharacteristic
528C8—2681no—
529C8—2681no—
530C8—2682no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22#1686325914yes—
545C22C22×C22⋊Dic6711yescharacteristic
546C22C2×#1452402611yescharacteristic
547C22C2×#1452402611yescharacteristic
548C22#1686419911yescharacteristic
549C22#1686420411yescharacteristic
550C22#1686420711yescharacteristic
551C22#1686421111yescharacteristic
552C4—24no—
553C4—42no—
554C4—44no—
555C4—42no—
556C4—5362no—
557C4—5361no—
558C4—5361no—
559C4#1686325914yes—
560C4C2×#1452400211yescharacteristic
561C4C2×#1452400211yescharacteristic
562C4C2×#1452402611yescharacteristic
563C4C2×#1452402611yescharacteristic
564C2—24no—
565C2—44no—
566C2C2×#1686325911yescharacteristic
567C2#2256873511yescharacteristic
568C2#2256874211yescharacteristic
569C2#2257016711yescharacteristic
570C2#2257017811yescharacteristic
571C2#2257018711yescharacteristic
572C2#2257019811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC2×#520×C67C2
socleC23×C67C22×D4
prime radicalC22×#41×C67C22
cosocleC22×C4×C67C24
hypercenterC2×#520D67
hypocenterC67#20703
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
3. #22560304C2semidirect
4. #22560304C2semidirect
5. #22560299C2semidirect
6. #22560295C2semidirect
7. #22559930C2semidirect
8. #22559928C2semidirect
9. #22559925C2semidirect
10. C2×#16863158C2semidirect
36. #520×C67C4semidirect
89. #117×C67C2×C4semidirect
95. C2×C4⋊C4×C67C2×C4semidirect
298. C67#20703semidirect
303. #520Dic67semidirect
362. #117C2×Dic67semidirect
374. C2×C4⋊C4C2×Dic67semidirect

Maximal subgroups

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

subgroupindex
C2×#168633632
#225628012
#225603042
#225602992
#225602952
#225599302
#225599282
#225599252
C2×#168631582
C2×#520×C672
#2070367