Finite Group Databasebeta

← C67×#25901 #30705152

Subgroups — C67×#25901 #30705152

Order 17152.

1406 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3496C211yesmaximal, characteristic
2C67×#2591C212yesmaximal, complemented
3C67×#2579C211yesmaximal, characteristic, complemented
4C67×#2044C212yesmaximal, complemented
5C67×#2044C212yesmaximal, complemented
6C67×#1964C212yesmaximal, complemented
7C67×#1790C212yesmaximal, complemented
8C67×#1790C212yesmaximal, complemented
9C4×#99×C67C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)×C67C2212yes—
11#547×C67—22no—
12#546×C67—24no—
13D42×C67—22no—
14#537×C67—21no—
15#534×C67—21no—
16#534×C67C2212yes—
17#532×C67—21no—
18C2×#121×C67—21no—
19C2×#121×C67C2212yes—
20C2×C4×D4×C67C2211yescharacteristic
21C22×C4⋊C4×C67C2211yescharacteristic, complemented
22C2×#99×C67C2212yes—
23C2×#99×C67C2214yescomplemented
24C2×#99×C67C2214yescomplemented
25#400×C67—22no—
26#398×C67—21no—
27#396×C67—21no—
28#394×C67—22no—
29#388×C67—24no—
30#387×C67—21no—
31#386×C67C2211yescharacteristic
32#386×C67C2211yescharacteristic
33#386×C67C2214yescomplemented
34#386×C67C2214yescomplemented
35#385×C67—21no—
36#385×C67C2211yescharacteristic, complemented
37#385×C67C2211yescharacteristic, complemented
38#380×C67C2211yescharacteristic, complemented
39C4×C22⋊C4×C67C2212yescomplemented
40#342×C67C2212yescomplemented
41H(5,𝔽2)×C67—24no—
42H(5,𝔽2)×C67—24no—
43H(5,𝔽2)×C67—44no—
44C2×#41×C67—22no—
45C2×#41×C67—24no—
46C22×D4×C67—22no—
47C22×D4×C67—24no—
48C22×D4×C67—21no—
49C22×D4×C67D412yes—
50C22×D4×C67C2314yes—
51C22×D4×C67C2311yescharacteristic
52C23×C4×C67D411yescharacteristic
53C23×C4×C67C2311yescharacteristic
54C23×C4×C67C2311yescharacteristic
55C23×C4×C67C2311yescharacteristic
56C4⋊Q8×C67—21no—
57#127×C67—21no—
58#124×C67—22no—
59#123×C67—24no—
60#123×C67—24no—
61C22⋊Q8×C67—24no—
62#121×C67—22no—
63#121×C67—22no—
64#121×C67—24no—
65#121×C67—22no—
66#121×C67—22no—
67#121×C67—22no—
68#121×C67—22no—
69#121×C67—22no—
70#121×C67—22no—
71#120×C67—24no—
72#120×C67—24no—
73C4×D4×C67—22no—
74C4×D4×C67—22no—
75C4×D4×C67—22no—
76C4×D4×C67—22no—
77C2×C4⋊C4×C67—21no—
78C2×C4⋊C4×C67—21no—
79C2×C4⋊C4×C67—22no—
80C2×C4⋊C4×C67—22no—
81C2×C4⋊C4×C67—21no—
82C2×C4⋊C4×C67—21no—
83C2×C4⋊C4×C67—21no—
84C2×C4⋊C4×C67—21no—
85C2×C4⋊C4×C67D411yescharacteristic
86C2×C4⋊C4×C67D412yes—
87C2×C22⋊C4×C67—22no—
88C2×C22⋊C4×C67—21no—
89C2×C22⋊C4×C67—24no—
90C2×C22⋊C4×C67—21no—
91C2×C22⋊C4×C67—24no—
92C2×C22⋊C4×C67D411yescharacteristic
93C2×C22⋊C4×C67D414yes—
94C2×C22⋊C4×C67C2314yes—
95C2×C22⋊C4×C67C2311yescharacteristic
96C2×C22⋊C4×C67C2311yescharacteristic
97C2×C22⋊C4×C67C2311yescharacteristic
98C2×C42×C67—22no—
99C2×C42×C67D411yescharacteristic
100#99×C67—24no—
101#99×C67—44no—
102#99×C67—44no—
103#95×C67—22no—
104#95×C67—21no—
105#95×C67—21no—
106#95×C67—21no—
107#95×C67—21no—
108#95×C67—21no—
109#95×C67—24no—
110#95×C67—24no—
111C24×C67—22no—
112C24×C67C2×D414yes—
113C24×C67C2×D411yescharacteristic
114C24×C67C2411yescharacteristic
115#41×C67—48no—
116#41×C67—48no—
117#41×C67—44no—
118#41×C67—44no—
119C2×Q8×C67—22no—
120C2×D4×C67—22no—
121C2×D4×C67—22no—
122C2×D4×C67—24no—
123C2×D4×C67—22no—
124C2×D4×C67—24no—
125C2×D4×C67—24no—
126C2×D4×C67—24no—
127C2×D4×C67—24no—
128C2×D4×C67—24no—
129C2×D4×C67—24no—
130C2×D4×C67—22no—
131C2×D4×C67—22no—
132C2×D4×C67—22no—
133C2×D4×C67—22no—
134C2×D4×C67—48no—
135C2×D4×C67—48no—
136C2×D4×C67—44no—
137C2×D4×C67—44no—
138C2×D4×C67—82no—
139C2×D4×C67—82no—
140C2×D4×C67#4114yes—
141C22×C4×C67—22no—
142C22×C4×C67—24no—
143C22×C4×C67—22no—
144C22×C4×C67—21no—
145C22×C4×C67—21no—
146C22×C4×C67—21no—
147C22×C4×C67—24no—
148C22×C4×C67—22no—
149C22×C4×C67—21no—
150C22×C4×C67—21no—
151C22×C4×C67—21no—
152C22×C4×C67—42no—
153C22×C4×C67—81no—
154C22×C4×C67—81no—
155C22×C4×C67—81no—
156C22×C4×C67C2×D414yes—
157C22×C4×C67C2×D411yescharacteristic
158C22×C4×C67C2×D411yescharacteristic
159C22×C4×C67C2×D411yescharacteristic
160C22×C4×C67C2×D412yes—
161C22×C4×C67C2×D411yescharacteristic
162C22×C4×C67C2×D411yescharacteristic
163C22×C4×C67C2×D412yes—
164C4⋊C4×C67—22no—
165C4⋊C4×C67—22no—
166C4⋊C4×C67—22no—
167C4⋊C4×C67—22no—
168C4⋊C4×C67—22no—
169C4⋊C4×C67—42no—
170C4⋊C4×C67—41no—
171C4⋊C4×C67—41no—
172C4⋊C4×C67—42no—
173C4⋊C4×C67—81no—
174C4⋊C4×C67—81no—
175C22⋊C4×C67—24no—
176C22⋊C4×C67—24no—
177C22⋊C4×C67—22no—
178C22⋊C4×C67—24no—
179C22⋊C4×C67—22no—
180C22⋊C4×C67—24no—
181C22⋊C4×C67—24no—
182C22⋊C4×C67—22no—
183C22⋊C4×C67—22no—
184C22⋊C4×C67—24no—
185C22⋊C4×C67—48no—
186C22⋊C4×C67—42no—
187C22⋊C4×C67—84no—
188C22⋊C4×C67—81no—
189C42×C67—21no—
190C42×C67—21no—
191C42×C67—82no—
192C23×C67—21no—
193C23×C67—21no—
194C23×C67—24no—
195C23×C67—24no—
196C23×C67—24no—
197C23×C67—41no—
198C23×C67—41no—
199C23×C67—41no—
200C23×C67—41no—
201C23×C67—44no—
202C23×C67—44no—
203C23×C67—44no—
204C23×C67—82no—
205C23×C67—82no—
206C23×C67#12412yes—
207C23×C67C22×D411yescharacteristic
208C23×C67C22×D412yes—
209C23×C67C2×#4112yes—
210C23×C67H(5,𝔽2)12yes—
211Q8×C67—44no—
212D4×C67—28no—
213D4×C67—44no—
214D4×C67—44no—
215D4×C67—48no—
216D4×C67—44no—
217D4×C67—44no—
218D4×C67—44no—
219D4×C67—44no—
220D4×C67—82no—
221D4×C67—82no—
222D4×C67—82no—
223C2×C4×C67—21no—
224C2×C4×C67—21no—
225C2×C4×C67—24no—
226C2×C4×C67—22no—
227C2×C4×C67—21no—
228C2×C4×C67—21no—
229C2×C4×C67—24no—
230C2×C4×C67—24no—
231C2×C4×C67—24no—
232C2×C4×C67—42no—
233C2×C4×C67—41no—
234C2×C4×C67—41no—
235C2×C4×C67—42no—
236C2×C4×C67—42no—
237C2×C4×C67—48no—
238C2×C4×C67—41no—
239C2×C4×C67—42no—
240C2×C4×C67—44no—
241C2×C4×C67—41no—
242C2×C4×C67—41no—
243C2×C4×C67—41no—
244C2×C4×C67—44no—
245C2×C4×C67—41no—
246C2×C4×C67—44no—
247C2×C4×C67—41no—
248C2×C4×C67—41no—
249C2×C4×C67—44no—
250C2×C4×C67—41no—
251C2×C4×C67—41no—
252C2×C4×C67—41no—
253C2×C4×C67—41no—
254C2×C4×C67—41no—
255C2×C4×C67—41no—
256C2×C4×C67—44no—
257C2×C4×C67—81no—
258C2×C4×C67#12011yescharacteristic
259C2×C4×C67#12011yescharacteristic
260C2×C4×C67#12012yes—
261C2×C4×C67#12412yes—
262C22×C67—21no—
263C22×C67—22no—
264C22×C67—21no—
265C22×C67—22no—
266C22×C67—22no—
267C22×C67—42no—
268C22×C67—42no—
269C22×C67—42no—
270C22×C67—41no—
271C22×C67—41no—
272C22×C67—44no—
273C22×C67—44no—
274C22×C67—44no—
275C22×C67—81no—
276C22×C67—84no—
277C22×C67—84no—
278C22×C67—164no—
279C22×C67C2×#12011yescharacteristic
280C22×C67C2×#12411yescharacteristic
281C22×C67#53511yescharacteristic
282C22×C67#54614yes—
283C4×C67—22no—
284C4×C67—21no—
285C4×C67—21no—
286C4×C67—22no—
287C4×C67—44no—
288C4×C67—42no—
289C4×C67—41no—
290C4×C67—41no—
291C4×C67—81no—
292C4×C67—81no—
293C4×C67—84no—
294C4×C67—81no—
295#25901C6711yesmaximal, characteristic, direct product factor, Sylow
296C2×C67—21no—
297C2×C67—24no—
298C2×C67—21no—
299C2×C67—42no—
300C2×C67—84no—
301C2×C67#243511yescharacteristic
302C2×C67#302511yescharacteristic
303C2×C67#302611yescharacteristic
304#3496C2×C6711yescharacteristic
305#2591C2×C6712yescomplemented
306#2579C2×C6711yescharacteristic, complemented
307#2044C2×C6712yescomplemented
308#2044C2×C6712yescomplemented
309#1964C2×C6712yescomplemented
310#1790C2×C6712yescomplemented
311#1790C2×C6712yescomplemented
312C4×#99C2×C6711yescharacteristic, complemented
313C67#2590111yescharacteristic, direct product factor, Sylow
314C2×H(5,𝔽2)C22×C6712yes—
315#547—22no—
316#546—24no—
317D42—22no—
318#537—21no—
319#534—21no—
320#534C22×C6712yes—
321#532—21no—
322C2×#121—21no—
323C2×#121C22×C6712yes—
324C2×C4×D4C22×C6711yescharacteristic
325C22×C4⋊C4C22×C6711yescharacteristic, complemented
326C2×#99C22×C6712yes—
327C2×#99C22×C6714yescomplemented
328C2×#99C22×C6714yescomplemented
329#400—22no—
330#398—21no—
331#396—21no—
332#394—22no—
333#388—24no—
334#387—21no—
335#386C22×C6711yescharacteristic
336#386C22×C6711yescharacteristic
337#386C22×C6714yescomplemented
338#386C22×C6714yescomplemented
339#385—21no—
340#385C22×C6711yescharacteristic, complemented
341#385C22×C6711yescharacteristic, complemented
342#380C22×C6711yescharacteristic, complemented
343C4×C22⋊C4C22×C6712yescomplemented
344#342C22×C6712yescomplemented
345H(5,𝔽2)—24no—
346H(5,𝔽2)—24no—
347H(5,𝔽2)—44no—
348C2×#41—22no—
349C2×#41—24no—
350C22×D4—22no—
351C22×D4—24no—
352C22×D4—21no—
353C22×D4D4×C6712yes—
354C22×D4C23×C6714yes—
355C22×D4C23×C6711yescharacteristic
356C23×C4D4×C6711yescharacteristic
357C23×C4C23×C6711yescharacteristic
358C23×C4C23×C6711yescharacteristic
359C23×C4C23×C6711yescharacteristic
360C4⋊Q8—21no—
361#127—21no—
362#124—22no—
363#123—24no—
364#123—24no—
365C22⋊Q8—24no—
366#121—22no—
367#121—22no—
368#121—24no—
369#121—22no—
370#121—22no—
371#121—22no—
372#121—22no—
373#121—22no—
374#121—22no—
375#120—24no—
376#120—24no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4D4×C6711yescharacteristic
390C2×C4⋊C4D4×C6712yes—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4D4×C6711yescharacteristic
397C2×C22⋊C4D4×C6714yes—
398C2×C22⋊C4C23×C6714yes—
399C2×C22⋊C4C23×C6711yescharacteristic
400C2×C22⋊C4C23×C6711yescharacteristic
401C2×C22⋊C4C23×C6711yescharacteristic
402C2×C42—22no—
403C2×C42D4×C6711yescharacteristic
404#99—24no—
405#99—44no—
406#99—44no—
407#95—22no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—24no—
414#95—24no—
415C24—22no—
416C24C2×D4×C6714yes—
417C24C2×D4×C6711yescharacteristic
418C24C24×C6711yescharacteristic
419#41—48no—
420#41—48no—
421#41—44no—
422#41—44no—
423C2×Q8—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—48no—
439C2×D4—48no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4#41×C6714yes—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—42no—
457C22×C4—81no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4C2×D4×C6714yes—
461C22×C4C2×D4×C6711yescharacteristic
462C22×C4C2×D4×C6711yescharacteristic
463C22×C4C2×D4×C6711yescharacteristic
464C22×C4C2×D4×C6711yescharacteristic
465C22×C4C2×D4×C6712yes—
466C22×C4C2×D4×C6712yes—
467C22×C4C2×D4×C6711yescharacteristic
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C4⋊C4—81no—
478C4⋊C4—81no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—22no—
482C22⋊C4—24no—
483C22⋊C4—22no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—24no—
489C22⋊C4—48no—
490C22⋊C4—42no—
491C22⋊C4—84no—
492C22⋊C4—81no—
493C42—21no—
494C42—21no—
495C42—82no—
496C23—21no—
497C23—21no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—82no—
509C23—82no—
510C23#124×C6712yes—
511C23C22×D4×C6711yescharacteristic
512C23C22×D4×C6712yes—
513C23C2×#41×C6712yes—
514C23H(5,𝔽2)×C6712yes—
515Q8—44no—
516D4—28no—
517D4—44no—
518D4—44no—
519D4—48no—
520D4—44no—
521D4—44no—
522D4—44no—
523D4—44no—
524D4—82no—
525D4—82no—
526D4—82no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—48no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—81no—
562C2×C4#120×C6711yescharacteristic
563C2×C4#120×C6711yescharacteristic
564C2×C4#120×C6712yes—
565C2×C4#124×C6712yes—
566C22—21no—
567C22—22no—
568C22—21no—
569C22—22no—
570C22—22no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—81no—
580C22—84no—
581C22—84no—
582C22—164no—
583C22C2×#120×C6711yescharacteristic
584C22C2×#124×C6711yescharacteristic
585C22#535×C6711yescharacteristic
586C22#546×C6714yes—
587C4—21no—
588C4—21no—
589C4—22no—
590C4—22no—
591C4—42no—
592C4—41no—
593C4—41no—
594C4—44no—
595C4—81no—
596C4—81no—
597C4—84no—
598C4—81no—
599C2—21no—
600C2—24no—
601C2—21no—
602C2—42no—
603C2—84no—
604C2C67×#243511yescharacteristic
605C2C67×#302511yescharacteristic
606C2C67×#302611yescharacteristic
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC24C24×C67
prime radicalC67×#3496C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2591C2semidirect
3. C67×#2579C2semidirect
4. C67×#2044C2semidirect
5. C67×#2044C2semidirect
6. C67×#1964C2semidirect
7. C67×#1790C2semidirect
8. C67×#1790C2semidirect
9. C4×#99×C67C2semidirect
21. C22×C4⋊C4×C67C22semidirect
23. C2×#99×C67C22semidirect
24. C2×#99×C67C22semidirect
33. #386×C67C22semidirect
34. #386×C67C22semidirect
36. #385×C67C22semidirect
37. #385×C67C22semidirect
38. #380×C67C22semidirect
39. C4×C22⋊C4×C67C22semidirect
40. #342×C67C22semidirect
295. #25901C67direct (N × H)
305. #2591C2×C67semidirect
306. #2579C2×C67semidirect
307. #2044C2×C67semidirect
308. #2044C2×C67semidirect
309. #1964C2×C67semidirect
310. #1790C2×C67semidirect
311. #1790C2×C67semidirect
312. C4×#99C2×C67semidirect
313. C67#25901direct (N × H)
325. C22×C4⋊C4C22×C67semidirect
327. C2×#99C22×C67semidirect
328. C2×#99C22×C67semidirect
337. #386C22×C67semidirect
338. #386C22×C67semidirect
340. #385C22×C67semidirect
341. #385C22×C67semidirect
342. #380C22×C67semidirect
343. C4×C22⋊C4C22×C67semidirect
344. #342C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34962
C67×#25912
C67×#25792
C67×#20442
C67×#19642
C67×#17902
C4×#99×C672
#2590167