Finite Group Databasebeta

← C2×#22560520 #30857036

Subgroups — C2×#22560520 #30857036

Order 17152.

1966 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14524023C211yesmaximal, characteristic
2C22×#14524001C211yesmaximal, characteristic
3C22×#14523968C211yesmaximal, characteristic, complemented
4C22×#14523966C211yesmaximal, characteristic, complemented
5C22×(C2×D67)⋊C8C211yesmaximal, characteristic
6#22560520C218yesmaximal, direct product factor
7C2×#16863158C211yesmaximal, characteristic, complemented
8C2×#340×C67C211yesmaximal, characteristic, complemented
9C23×Dic134C2211yescharacteristic
10C23×C4×D67C2211yescharacteristic
11C23×C67⋊C8C2211yescharacteristic
12C2×#14524023—24no—
13C2×#14524023—22no—
14C2×#14524023C418yescomplemented
15C2×#14524023C2214yes—
16C2×#14524001—22no—
17C2×#14524001—24no—
18C2×#14524001—24no—
19C2×#14524001C2214yes—
20C22×C4⋊Dic67—21no—
21C22×C4⋊Dic67C2211yescharacteristic
22C22×(C2×D67)⋊C4—21no—
23C22×(C2×D67)⋊C4—21no—
24C22×C4×Dic67—21no—
25C2×#14523968—24no—
26C2×#14523968C2214yescomplemented
27C2×#14523966—24no—
28C2×#14523966C2214yescomplemented
29C2×(C2×D67)⋊C8—24no—
30C2×(C2×D67)⋊C8C2214yes—
31#16863158C2214yescomplemented
32C23×C8×C67C2211yescharacteristic
33C22×C4⋊C4×C67C2211yescharacteristic
34C22×C4⋊C4×C67C2211yescharacteristic
35#340×C67C2214yescomplemented
36C24×D67D411yescharacteristic
37C22×Dic134—21no—
38C22×Dic134—22no—
39C22×Dic134—22no—
40C22×Dic134—22no—
41C22×Dic134—22no—
42C22×Dic134—22no—
43C22×Dic134C2×C414yes—
44C22×Dic134C2312yes—
45C23×Dic67—21no—
46C23×Dic67—21no—
47C23×Dic67D411yescharacteristic
48C22×C4×D67—22no—
49C22×C4×D67—22no—
50C22×C4×D67—22no—
51C22×C4×D67—22no—
52C22×C4×D67C2×C414yes—
53C22×C4×D67D412yes—
54C22×C4×D67D414yes—
55C22×C4×D67C2312yes—
56C22×C67⋊C8—22no—
57C22×C67⋊C8—22no—
58C22×C67⋊C8—22no—
59C22×C67⋊C8C2312yes—
60#14524023—216no—
61#14524023—28no—
62#14524023C2×C4116yescomplemented
63#14524001—216no—
64#14524001—28no—
65#14524001—28no—
66C2×C4⋊Dic67—28no—
67C2×C4⋊Dic67—24no—
68C2×C4⋊Dic67—21no—
69C2×C4⋊Dic67—22no—
70C2×C4⋊Dic67D414yes—
71C2×C4⋊Dic67C2312yes—
72C2×(C2×D67)⋊C4—28no—
73C2×(C2×D67)⋊C4—28no—
74C2×(C2×D67)⋊C4—24no—
75C2×(C2×D67)⋊C4—24no—
76C2×C4×Dic67—28no—
77C2×C4×Dic67—24no—
78#14523968—28no—
79#14523966—28no—
80(C2×D67)⋊C8—28no—
81C23×C4×C67—21no—
82C23×C4×C67—21no—
83C23×C4×C67C2311yescharacteristic
84C22×C8×C67—22no—
85C22×C8×C67—22no—
86C22×C8×C67—22no—
87C22×C8×C67C2312yes—
88C2×C4⋊C4×C67—21no—
89C2×C4⋊C4×C67—22no—
90C2×C4⋊C4×C67—21no—
91C2×C4⋊C4×C67—22no—
92C2×C4⋊C4×C67C2×C414yes—
93C2×C4⋊C4×C67D414yes—
94C2×C4⋊C4×C67C2312yes—
95C2×C4⋊C4×C67C2312yes—
96C23×D67—22no—
97C23×D67—22no—
98C23×D67—22no—
99C23×D67—22no—
100C23×D67—22no—
101C23×D67—22no—
102C23×D67C22⋊C414yes—
103C23×D67C2×D412yes—
104C2×Dic134—28no—
105C2×Dic134—28no—
106C2×Dic134—22no—
107C2×Dic134—22no—
108C2×Dic134—22no—
109C2×Dic134—22no—
110C2×Dic134—24no—
111C2×Dic134—22no—
112C2×Dic134—42no—
113C2×Dic134—44no—
114C2×Dic134—44no—
115C2×Dic134—42no—
116C2×Dic134C22×C414yes—
117C22×Dic67—22no—
118C22×Dic67—22no—
119C22×Dic67—24no—
120C22×Dic67—24no—
121C22×Dic67—22no—
122C22×Dic67—22no—
123C22×Dic67—42no—
124C22×Dic67—42no—
125C22×Dic67—42no—
126C22×Dic67—42no—
127C22×Dic67C22⋊C414yes—
128C22×Dic67C2×D412yes—
129C2×C4×D67—28no—
130C2×C4×D67—28no—
131C2×C4×D67—24no—
132C2×C4×D67—28no—
133C2×C4×D67—28no—
134C2×C4×D67—24no—
135C2×C4×D67—22no—
136C2×C4×D67—22no—
137C2×C4×D67—22no—
138C2×C4×D67—22no—
139C2×C4×D67C22⋊C418yes—
140C2×C4×D67C22×C414yes—
141C2×C4×D67C2×D414yes—
142C2×C67⋊C8—22no—
143C2×C67⋊C8—22no—
144C2×C67⋊C8—22no—
145C2×C67⋊C8—44no—
146C4⋊Dic67—24no—
147C4⋊Dic67—22no—
148C4⋊Dic67—216no—
149C4⋊Dic67C2×D414yes—
150(C2×D67)⋊C4—216no—
151(C2×D67)⋊C4—216no—
152C4×Dic67—216no—
153C24×C67C2×D411yescharacteristic
154C22×C4×C67—24no—
155C22×C4×C67—24no—
156C22×C4×C67—22no—
157C22×C4×C67—22no—
158C22×C4×C67—42no—
159C22×C4×C67—42no—
160C22×C4×C67—42no—
161C22×C4×C67—42no—
162C22×C4×C67C22×C412yes—
163C22×C4×C67C2×D411yescharacteristic
164C22×C4×C67C2×D412yes—
165C22×C4×C67C2×D412yes—
166C22×C4×C67C2×D412yes—
167C22×C4×C67#4112yes—
168C22×C4×C67#4112yes—
169C22×C4×C67C2411yescharacteristic
170C2×C8×C67—22no—
171C2×C8×C67—22no—
172C2×C8×C67—22no—
173C2×C8×C67—44no—
174C4⋊C4×C67—24no—
175C4⋊C4×C67—24no—
176C4⋊C4×C67—22no—
177C4⋊C4×C67—22no—
178C4⋊C4×C67C22×C414yes—
179C4⋊C4×C67C2×D414yes—
180C23×C67C2×C22⋊C412yes—
181C23×C67#12012yes—
182C23×C67#12312yes—
183C23×C67#13512yes—
184C23×C67#13512yes—
185C23×C67#13712yes—
186C23×C67#13712yes—
187C23×C67C22×D411yescharacteristic
188C22×D67—22no—
189C22×D67—22no—
190C22×D67—22no—
191C22×D67—22no—
192C22×D67—22no—
193C22×D67—22no—
194C22×D67—28no—
195C22×D67—28no—
196C22×D67—28no—
197C22×D67—44no—
198C22×D67—44no—
199C22×D67—44no—
200C22×D67—44no—
201C22×D67C2×C22⋊C414yes—
202C2×C4×C67—24no—
203C2×C4×C67—24no—
204C2×C4×C67—42no—
205C2×C4×C67—42no—
206C2×C4×C67—42no—
207C2×C4×C67—42no—
208C2×C4×C67—48no—
209C2×C4×C67—48no—
210C2×C4×C67C2×C22⋊C412yes—
211C2×C4×C67C4×D414yes—
212C2×C4×C67C4×D414yes—
213C2×C4×C67#12012yes—
214C2×C4×C67#12114yes—
215C2×C4×C67#12114yes—
216C2×C4×C67#12312yes—
217C2×C4×C67C23×C411yescharacteristic
218C2×C4×C67C22×D411yescharacteristic
219C2×C4×C67C22×D411yescharacteristic
220C2×C4×C67C22×D411yescharacteristic
221C2×C4×C67C2×#4111yescharacteristic
222C2×C4×C67C2×#4111yescharacteristic
223C2×Dic67—22no—
224C2×Dic67—22no—
225C2×Dic67—28no—
226C2×Dic67—24no—
227C2×Dic67—24no—
228C2×Dic67—42no—
229C2×Dic67—42no—
230C2×Dic67—42no—
231C2×Dic67—42no—
232C2×Dic67—48no—
233C2×Dic67—48no—
234C2×Dic67C2×C22⋊C414yes—
235C4×D67—24no—
236C4×D67—24no—
237C4×D67—24no—
238C4×D67—24no—
239C4×D67—216no—
240Dic134—24no—
241Dic134—24no—
242Dic134—42no—
243Dic134—42no—
244Dic134—48no—
245C8×C67—42no—
246C67⋊C8—42no—
247C22×C67#38614yes—
248C22×C67#41614yes—
249C22×C67#41914yes—
250C22×C67#45214yes—
251C22×C67#45214yes—
252C22×C67#48514yes—
253C22×C67#48514yes—
254C22×C67C22×C22⋊C411yescharacteristic
255C22×C67C2×#12011yescharacteristic
256C22×C67C2×#12311yescharacteristic
257C22×C67C2×#13511yescharacteristic
258C22×C67C2×#13511yescharacteristic
259C22×C67C2×#13711yescharacteristic
260C22×C67C2×#13711yescharacteristic
261C2×D67—24no—
262C2×D67—24no—
263C2×D67—24no—
264C2×D67—42no—
265C2×D67—42no—
266C2×D67—416no—
267C2×D67—42no—
268C2×D67—42no—
269C4×C67—44no—
270C4×C67—44no—
271C4×C67#38614yes—
272C4×C67C2×C4×D411yescharacteristic
273C4×C67C2×C4×D411yescharacteristic
274C4×C67C2×#12111yescharacteristic
275C4×C67C2×#12111yescharacteristic
276Dic67—24no—
277Dic67—44no—
278Dic67—44no—
279C2×#1873—671nomaximal, Sylow
280C2×C67#187318yescomplemented
281C2×C67C2×#38611yescharacteristic
282C2×C67C2×#41611yescharacteristic
283C2×C67C2×#41911yescharacteristic
284C2×C67C2×#45211yescharacteristic
285C2×C67C2×#45211yescharacteristic
286C2×C67C2×#48511yescharacteristic
287C2×C67C2×#48511yescharacteristic
288D67—44no—
289C22×C22⋊Q8—671no—
290C22×#117—671no—
291C22×#103—671no—
292C22×#103—671no—
293C22×C22⋊C8—671no—
294#1873—678no—
295C2×#340—671no—
296C2×#340D6711yescharacteristic, complemented
297C67C2×#187311yescharacteristic, complemented, Sylow
298C23×Q8—671no—
299C24×C4—671no—
300C23×C8—671no—
301C23×C8C2×D6711yescharacteristic
302C2×C22⋊Q8—678no—
303C2×C22⋊Q8—674no—
304C2×C22⋊Q8—1342no—
305C2×C22⋊Q8—1344no—
306C2×#117—674no—
307C2×#117—1344no—
308C2×#117—1342no—
309C2×#117—1344no—
310C22×C4⋊C4—671no—
311C22×C4⋊C4—1341no—
312C22×C4⋊C4C2×D6711yescharacteristic
313C22×C4⋊C4C2×D6711yescharacteristic
314C22×C22⋊C4—1341no—
315C22×C22⋊C4—1341no—
316C22×C42—1341no—
317C2×#103—674no—
318C2×#103—674no—
319C2×#103—1344no—
320C2×#103—1344no—
321C2×C22⋊C8—674no—
322C2×C22⋊C8—1344no—
323#340—674no—
324#340C2×D6714yescomplemented
325C25—671no—
326C22×Q8—674no—
327C22×Q8—672no—
328C22×Q8—1342no—
329C22×Q8—1342no—
330C22×Q8—1342no—
331C22×Q8—1342no—
332C22×Q8—1342no—
333C22×Q8—1341no—
334C23×C4—21no—
335C23×C4—21no—
336C23×C4—671no—
337C23×C4—674no—
338C23×C4—672no—
339C23×C4—674no—
340C23×C4—672no—
341C23×C4—1341no—
342C23×C4—1341no—
343C23×C4—1342no—
344C23×C4—1342no—
345C23×C4—1342no—
346C23×C4—1342no—
347C23×C4C22×D6711yescharacteristic
348C22×C8—22no—
349C22×C8—22no—
350C22×C8—22no—
351C22×C8—672no—
352C22×C8—1342no—
353C22×C8—1342no—
354C22×C8—1342no—
355C22×C8C22×D6712yes—
356C22⋊Q8—6716no—
357C22⋊Q8—13416no—
358C22⋊Q8—1348no—
359#117—1348no—
360#117—13416no—
361#117—1348no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—674no—
367C2×C4⋊C4—672no—
368C2×C4⋊C4—1348no—
369C2×C4⋊C4—1344no—
370C2×C4⋊C4—1341no—
371C2×C4⋊C4—1342no—
372C2×C4⋊C4C4×D6714yes—
373C2×C4⋊C4D26814yes—
374C2×C4⋊C4C22×D6712yes—
375C2×C4⋊C4C22×D6712yes—
376C2×C22⋊C4—1344no—
377C2×C22⋊C4—1348no—
378C2×C22⋊C4—1344no—
379C2×C22⋊C4—1348no—
380C2×C42—1344no—
381C2×C42—1348no—
382#103—1348no—
383#103—1348no—
384C22⋊C8—1348no—
385C24—674no—
386C24—672no—
387C24—1342no—
388C24—1342no—
389C24—1342no—
390C24—1342no—
391C24—1342no—
392C24—1342no—
393C24D4×D6711yescharacteristic
394C2×Q8—2682no—
395C2×Q8—2684no—
396C2×Q8—2684no—
397C2×Q8—2682no—
398C2×Q8—674no—
399C2×Q8—1348no—
400C2×Q8—1348no—
401C2×Q8—1342no—
402C2×Q8—1342no—
403C2×Q8—1342no—
404C2×Q8—1342no—
405C2×Q8—1344no—
406C2×Q8—1342no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—2682no—
416C22×C4—2682no—
417C22×C4—2682no—
418C22×C4—2682no—
419C22×C4—674no—
420C22×C4—672no—
421C22×C4—678no—
422C22×C4—674no—
423C22×C4—674no—
424C22×C4—1344no—
425C22×C4—1344no—
426C22×C4—1342no—
427C22×C4—1342no—
428C22×C4—1342no—
429C22×C4—1342no—
430C22×C4—1348no—
431C22×C4—1348no—
432C22×C4—1344no—
433C22×C4—1348no—
434C22×C4—1348no—
435C22×C4—1344no—
436C22×C4—1342no—
437C22×C4—1342no—
438C22×C4—1342no—
439C22×C4—1342no—
440C22×C4C2×C4×D6712yes—
441C22×C4D4×D6711yescharacteristic
442C22×C4D4×D6712yes—
443C22×C4D4×D6712yes—
444C22×C4C2×D26812yes—
445C22×C4#1176162812yes—
446C22×C4#1176162912yes—
447C22×C4C23×D6711yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—44no—
452C2×C8—2684no—
453C2×C8—1342no—
454C2×C8—1342no—
455C2×C8—1342no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—674no—
461C4⋊C4—13416no—
462C4⋊C4—1344no—
463C4⋊C4—1342no—
464C4⋊C4C2×C4×D6714yes—
465C4⋊C4C2×D26814yes—
466C22⋊C4—13416no—
467C22⋊C4—13416no—
468C42—13416no—
469C23—2684no—
470C23—2684no—
471C23—2684no—
472C23—2684no—
473C23—674no—
474C23—1348no—
475C23—1342no—
476C23—1348no—
477C23—1342no—
478C23—1348no—
479C23—1342no—
480C23—1342no—
481C23—1342no—
482C23—1342no—
483C23C22⋊C4×D6712yes—
484C23#1452401312yes—
485C23#1452402812yes—
486C23#1452406212yes—
487C23#1452406412yes—
488C23#1452407212yes—
489C23#1452407512yes—
490C23C2×D4×D6711yescharacteristic
491Q8—2688no—
492Q8—2682no—
493Q8—2682no—
494Q8—1344no—
495Q8—1344no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—48no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—2688no—
505C2×C4—2688no—
506C2×C4—2682no—
507C2×C4—2682no—
508C2×C4—2682no—
509C2×C4—2682no—
510C2×C4—674no—
511C2×C4—1344no—
512C2×C4—1344no—
513C2×C4—1348no—
514C2×C4—1342no—
515C2×C4—1342no—
516C2×C4—13416no—
517C2×C4—1344no—
518C2×C4—1344no—
519C2×C4—1344no—
520C2×C4—1344no—
521C2×C4C22⋊C4×D6712yes—
522C2×C4#1452400514yes—
523C2×C4C4×D26814yes—
524C2×C4#1452401312yes—
525C2×C4#1452401614yes—
526C2×C4#1452401814yes—
527C2×C4#1452402812yes—
528C2×C4C22×C4×D6711yescharacteristic
529C2×C4C2×D4×D6711yescharacteristic
530C2×C4C2×D4×D6711yescharacteristic
531C2×C4C22×D26811yescharacteristic
532C2×C4C2×#1176162811yescharacteristic
533C2×C4C2×#1176162911yescharacteristic
534C8—42no—
535C8—2682no—
536C22—26816no—
537C22—2682no—
538C22—2682no—
539C22—2682no—
540C22—2682no—
541C22—1344no—
542C22—1344no—
543C22—1344no—
544C22#1686327414yes—
545C22#1686338914yes—
546C22#1686340314yes—
547C22#1686355914yes—
548C22#1686356114yes—
549C22#1686373914yes—
550C22#1686374014yes—
551C22C2×C22⋊C4×D6711yescharacteristic
552C22C2×#1452401311yescharacteristic
553C22C2×#1452402811yescharacteristic
554C22C2×#1452406211yescharacteristic
555C22C2×#1452406411yescharacteristic
556C22C2×#1452407211yescharacteristic
557C22C2×#1452407511yescharacteristic
558C4—44no—
559C4—44no—
560C4—2684no—
561C4—2684no—
562C4—1344no—
563C4#1686327414yes—
564C4C2×#1452400511yescharacteristic
565C4C2×C4×D26811yescharacteristic
566C4C2×#1452401611yescharacteristic
567C4C2×#1452401811yescharacteristic
568C2—2684no—
569C2#2256052018yesdirect product factor
570C2C2×#1686327411yescharacteristic
571C2C2×#1686338911yescharacteristic
572C2C2×#1686340311yescharacteristic
573C2C2×#1686355911yescharacteristic
574C2C2×#1686356111yescharacteristic
575C2C2×#1686373911yescharacteristic
576C2C2×#1686374011yescharacteristic
subgroupG/H
centerC24D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC2×#340×C67C2
socleC24×C67C2×D4
prime radicalC24×D67D4
cosocleC22×C4×C67C24
hypercenterC2×#340D67
hypocenterC67C2×#1873
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. C22×#14523968C2semidirect
4. C22×#14523966C2semidirect
6. #22560520C2direct (N × H)
7. C2×#16863158C2semidirect
8. C2×#340×C67C2semidirect
14. C2×#14524023C4semidirect
26. C2×#14523968C22semidirect
28. C2×#14523966C22semidirect
31. #16863158C22semidirect
35. #340×C67C22semidirect
62. #14524023C2×C4semidirect
280. C2×C67#1873semidirect
296. C2×#340D67semidirect
297. C67C2×#1873semidirect
324. #340C2×D67semidirect
569. C2#22560520direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240232
C22×#145240012
C22×#145239682
C22×#145239662
C22×(C2×D67)⋊C82
#225605202
C2×#168631582
C2×#340×C672
C2×#187367