Finite Group Databasebeta

← C67×#33086 #30712337

Subgroups — C67×#33086 #30712337

Order 17152.

804 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2644C211yesmaximal, characteristic, complemented
2C67×#2275C211yesmaximal, characteristic, complemented
3C67×#1926C211yesmaximal, characteristic, complemented
4C67×#1926C211yesmaximal, characteristic, complemented
5C67×#1925C211yesmaximal, characteristic, complemented
6C67×#1920C211yesmaximal, characteristic, complemented
7C67×#1919C211yesmaximal, characteristic, complemented
8C67×#1918C211yesmaximal, characteristic, complemented
9C67×#1879C211yesmaximal, characteristic
10C67×#1879C211yesmaximal, characteristic
11C67×#1879C211yesmaximal, characteristic
12C67×#1864C211yesmaximal, characteristic, complemented
13C67×#1864C211yesmaximal, characteristic, complemented
14C67×#1862C211yesmaximal, characteristic, complemented
15C67×#1840C211yesmaximal, characteristic, complemented
16C22×D8×C67C2211yescharacteristic
17C2×C4×D4×C67C2211yescharacteristic
18C2×C4×D4×C67C2211yescharacteristic
19C2×C4×D4×C67C2211yescharacteristic
20C2×#107×C67C2211yescharacteristic
21C2×#106×C67C2211yescharacteristic
22C2×#106×C67C2211yescharacteristic
23C2×#102×C67C2211yescharacteristic
24C2×#102×C67C2211yescharacteristic
25C2×#102×C67C2211yescharacteristic
26C2×#102×C67C2211yescharacteristic
27C2×#102×C67C2211yescharacteristic
28C2×#102×C67C2211yescharacteristic
29#397×C67—21no—
30#397×C67—21no—
31#397×C67—21no—
32#396×C67—21no—
33#390×C67C2211yescharacteristic
34#390×C67C2211yescharacteristic
35#390×C67C2211yescharacteristic
36#389×C67C2211yescharacteristic
37#389×C67C2211yescharacteristic
38#389×C67C2211yescharacteristic
39C4×C4⋊C4×C67C2211yescharacteristic
40C4×C4⋊C4×C67C2211yescharacteristic
41C4×C4⋊C4×C67C2211yescharacteristic
42C4×C22⋊C4×C67—21no—
43C4×C22⋊C4×C67—21no—
44C4×C22⋊C4×C67—21no—
45#376×C67C2211yescharacteristic
46C2×#95×C67—21no—
47#340×C67C2211yescharacteristic
48#340×C67C2211yescharacteristic
49#340×C67C2211yescharacteristic
50#340×C67C2211yescharacteristic
51#340×C67C2211yescharacteristic
52#340×C67C2211yescharacteristic
53#336×C67C2211yescharacteristic
54#336×C67C2211yescharacteristic
55#336×C67C2211yescharacteristic
56#336×C67C2211yescharacteristic
57#336×C67C2211yescharacteristic
58#336×C67C2211yescharacteristic
59C22×D4×C67C2311yescharacteristic
60C22×D4×C67C2311yescharacteristic
61C23×C4×C67—21no—
62C23×C4×C67—21no—
63C23×C4×C67—21no—
64C2×D8×C67—22no—
65C2×D8×C67—22no—
66C2×D8×C67—22no—
67C22×C8×C67C2311yescharacteristic
68C22×C8×C67C2311yescharacteristic
69C22×C8×C67C2311yescharacteristic
70C22×C8×C67C2311yescharacteristic
71C4×D4×C67—22no—
72C4×D4×C67—22no—
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—21no—
80C2×C4⋊C4×C67C2311yescharacteristic
81C2×C4⋊C4×C67C2311yescharacteristic
82C2×C4⋊C4×C67C2311yescharacteristic
83C2×C4⋊C4×C67C2311yescharacteristic
84C2×C4⋊C4×C67C2311yescharacteristic
85C2×C4⋊C4×C67C2311yescharacteristic
86C2×C22⋊C4×C67—21no—
87C2×C22⋊C4×C67—21no—
88C2×C22⋊C4×C67—21no—
89C2×C22⋊C4×C67—21no—
90C2×C22⋊C4×C67—21no—
91C2×C22⋊C4×C67—21no—
92C2×C22⋊C4×C67—21no—
93C2×C22⋊C4×C67—21no—
94C2×C22⋊C4×C67—21no—
95C2×C42×C67—21no—
96C2×C42×C67—21no—
97C2×C42×C67—21no—
98C2×C42×C67C2311yescharacteristic
99C2×C42×C67C2311yescharacteristic
100C2×C42×C67C2311yescharacteristic
101#107×C67—22no—
102#106×C67—22no—
103#106×C67—22no—
104#102×C67—22no—
105#102×C67—22no—
106#102×C67—22no—
107#102×C67—22no—
108#102×C67—22no—
109#102×C67—22no—
110#95×C67—21no—
111#95×C67—21no—
112#95×C67—21no—
113#95×C67—21no—
114#95×C67—21no—
115#95×C67—21no—
116#95×C67D411yescharacteristic
117#95×C67D411yescharacteristic
118#95×C67D411yescharacteristic
119#95×C67D411yescharacteristic
120C24×C67—21no—
121C24×C67—21no—
122C2×D4×C67—21no—
123C2×D4×C67—21no—
124C2×D4×C67—21no—
125C2×D4×C67—21no—
126C2×D4×C67—21no—
127C2×D4×C67—21no—
128C2×D4×C67—21no—
129C2×D4×C67—21no—
130C2×D4×C67—21no—
131C2×D4×C67#4112yes—
132C2×D4×C67#4112yes—
133C2×D4×C67#4112yes—
134C22×C4×C67—21no—
135C22×C4×C67—21no—
136C22×C4×C67—21no—
137C22×C4×C67—24no—
138C22×C4×C67—24no—
139C22×C4×C67—24no—
140C22×C4×C67—21no—
141C22×C4×C67—21no—
142C22×C4×C67—21no—
143C22×C4×C67—81no—
144C22×C4×C67—81no—
145C22×C4×C67—81no—
146C22×C4×C67—81no—
147C22×C4×C67—81no—
148C22×C4×C67—81no—
149C22×C4×C67C2×D411yescharacteristic
150C22×C4×C67C2×D411yescharacteristic
151C22×C4×C67C2×D411yescharacteristic
152C22×C4×C67C2×D411yescharacteristic
153C22×C4×C67C2×D411yescharacteristic
154C22×C4×C67C2×D411yescharacteristic
155C22×C4×C67C2411yescharacteristic
156D8×C67—44no—
157C2×C8×C67—21no—
158C2×C8×C67—21no—
159C2×C8×C67—21no—
160C2×C8×C67—21no—
161C2×C8×C67—21no—
162C2×C8×C67—21no—
163C2×C8×C67—21no—
164C2×C8×C67—21no—
165C2×C8×C67—21no—
166C2×C8×C67—21no—
167C2×C8×C67—21no—
168C2×C8×C67—21no—
169C4⋊C4×C67—21no—
170C4⋊C4×C67—21no—
171C4⋊C4×C67—21no—
172C4⋊C4×C67—21no—
173C4⋊C4×C67—21no—
174C4⋊C4×C67—21no—
175C4⋊C4×C67—21no—
176C4⋊C4×C67—21no—
177C4⋊C4×C67—21no—
178C4⋊C4×C67—81no—
179C4⋊C4×C67—81no—
180C4⋊C4×C67—81no—
181C4⋊C4×C67#4112yes—
182C4⋊C4×C67#4112yes—
183C4⋊C4×C67#4112yes—
184C22⋊C4×C67—24no—
185C22⋊C4×C67—24no—
186C22⋊C4×C67—24no—
187C22⋊C4×C67—81no—
188C22⋊C4×C67—81no—
189C22⋊C4×C67—81no—
190C22⋊C4×C67—81no—
191C22⋊C4×C67—81no—
192C22⋊C4×C67—81no—
193C42×C67—21no—
194C42×C67—21no—
195C42×C67—21no—
196C42×C67—21no—
197C42×C67—21no—
198C42×C67—21no—
199C42×C67—81no—
200C42×C67—81no—
201C42×C67—81no—
202C23×C67—22no—
203C23×C67—22no—
204C23×C67—22no—
205C23×C67—41no—
206C23×C67—41no—
207C23×C67—41no—
208C23×C67—41no—
209C23×C67—41no—
210C23×C67—41no—
211C23×C67—41no—
212C23×C67—41no—
213C23×C67—41no—
214C23×C67—41no—
215C23×C67—41no—
216C23×C67C22×D411yescharacteristic
217D4×C67—22no—
218D4×C67—22no—
219D4×C67—22no—
220D4×C67—41no—
221D4×C67—41no—
222D4×C67—41no—
223D4×C67—41no—
224D4×C67#12614yes—
225C2×C4×C67—21no—
226C2×C4×C67—21no—
227C2×C4×C67—21no—
228C2×C4×C67—21no—
229C2×C4×C67—21no—
230C2×C4×C67—21no—
231C2×C4×C67—22no—
232C2×C4×C67—22no—
233C2×C4×C67—22no—
234C2×C4×C67—21no—
235C2×C4×C67—21no—
236C2×C4×C67—21no—
237C2×C4×C67—21no—
238C2×C4×C67—21no—
239C2×C4×C67—21no—
240C2×C4×C67—21no—
241C2×C4×C67—21no—
242C2×C4×C67—21no—
243C2×C4×C67—21no—
244C2×C4×C67—21no—
245C2×C4×C67—21no—
246C2×C4×C67—41no—
247C2×C4×C67—41no—
248C2×C4×C67—41no—
249C2×C4×C67—41no—
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—41no—
257C2×C4×C67—41no—
258C2×C4×C67—41no—
259C2×C4×C67—41no—
260C2×C4×C67—41no—
261C2×C4×C67—84no—
262C2×C4×C67—84no—
263C2×C4×C67—84no—
264C2×C4×C67C2×#4111yescharacteristic
265C2×C4×C67C2×#4111yescharacteristic
266C2×C4×C67C2×#4111yescharacteristic
267C2×C4×C67C2×#4111yescharacteristic
268C2×C4×C67C2×#4111yescharacteristic
269C2×C4×C67C2×#4111yescharacteristic
270C8×C67—41no—
271C8×C67—41no—
272C8×C67—41no—
273C8×C67—41no—
274C22×C67—24no—
275C22×C67—42no—
276C22×C67—42no—
277C22×C67—42no—
278C22×C67—42no—
279C22×C67—42no—
280C22×C67—42no—
281C22×C67—81no—
282C22×C67—81no—
283C22×C67—81no—
284C22×C67—81no—
285C22×C67—81no—
286C22×C67—81no—
287C22×C67—81no—
288C22×C67#52511yescharacteristic
289C22×C67#52511yescharacteristic
290C22×C67#52511yescharacteristic
291C22×C67#57611yescharacteristic
292C22×C67#57611yescharacteristic
293C22×C67#57711yescharacteristic
294C22×C67#57711yescharacteristic
295C4×C67—41no—
296C4×C67—41no—
297C4×C67—42no—
298C4×C67—42no—
299C4×C67—41no—
300C4×C67—41no—
301C4×C67—41no—
302C4×C67—42no—
303C4×C67—41no—
304C4×C67—81no—
305C4×C67—81no—
306C4×C67—81no—
307C4×C67C2×#12611yescharacteristic
308C4×C67#52911yescharacteristic
309C4×C67#52911yescharacteristic
310C4×C67#52911yescharacteristic
311#33086C6711yesmaximal, characteristic, direct product factor, Sylow
312C2×C67—44no—
313C2×C67—161no—
314C2×C67#285211yescharacteristic
315C2×C67#311811yescharacteristic
316C2×C67#311811yescharacteristic
317C2×C67#311911yescharacteristic
318C2×C67#311911yescharacteristic
319C2×C67#312211yescharacteristic
320C2×C67#312511yescharacteristic
321#2644C2×C6711yescharacteristic, complemented
322#2275C2×C6711yescharacteristic, complemented
323#1926C2×C6711yescharacteristic, complemented
324#1926C2×C6711yescharacteristic, complemented
325#1925C2×C6711yescharacteristic, complemented
326#1920C2×C6711yescharacteristic, complemented
327#1919C2×C6711yescharacteristic, complemented
328#1918C2×C6711yescharacteristic, complemented
329#1879C2×C6711yescharacteristic
330#1879C2×C6711yescharacteristic
331#1879C2×C6711yescharacteristic
332#1864C2×C6711yescharacteristic, complemented
333#1864C2×C6711yescharacteristic, complemented
334#1862C2×C6711yescharacteristic, complemented
335#1840C2×C6711yescharacteristic, complemented
336C67#3308611yescharacteristic, direct product factor, Sylow
337C22×D8C22×C6711yescharacteristic
338C2×C4×D4C22×C6711yescharacteristic
339C2×C4×D4C22×C6711yescharacteristic
340C2×C4×D4C22×C6711yescharacteristic
341C2×#107C22×C6711yescharacteristic
342C2×#106C22×C6711yescharacteristic
343C2×#106C22×C6711yescharacteristic
344C2×#102C22×C6711yescharacteristic
345C2×#102C22×C6711yescharacteristic
346C2×#102C22×C6711yescharacteristic
347C2×#102C22×C6711yescharacteristic
348C2×#102C22×C6711yescharacteristic
349C2×#102C22×C6711yescharacteristic
350#397—21no—
351#397—21no—
352#397—21no—
353#396—21no—
354#390C22×C6711yescharacteristic
355#390C22×C6711yescharacteristic
356#390C22×C6711yescharacteristic
357#389C22×C6711yescharacteristic
358#389C22×C6711yescharacteristic
359#389C22×C6711yescharacteristic
360C4×C4⋊C4C22×C6711yescharacteristic
361C4×C4⋊C4C22×C6711yescharacteristic
362C4×C4⋊C4C22×C6711yescharacteristic
363C4×C22⋊C4—21no—
364C4×C22⋊C4—21no—
365C4×C22⋊C4—21no—
366#376C22×C6711yescharacteristic
367C2×#95—21no—
368#340C22×C6711yescharacteristic
369#340C22×C6711yescharacteristic
370#340C22×C6711yescharacteristic
371#340C22×C6711yescharacteristic
372#340C22×C6711yescharacteristic
373#340C22×C6711yescharacteristic
374#336C22×C6711yescharacteristic
375#336C22×C6711yescharacteristic
376#336C22×C6711yescharacteristic
377#336C22×C6711yescharacteristic
378#336C22×C6711yescharacteristic
379#336C22×C6711yescharacteristic
380C22×D4C23×C6711yescharacteristic
381C22×D4C23×C6711yescharacteristic
382C23×C4—21no—
383C23×C4—21no—
384C23×C4—21no—
385C2×D8—22no—
386C2×D8—22no—
387C2×D8—22no—
388C22×C8C23×C6711yescharacteristic
389C22×C8C23×C6711yescharacteristic
390C22×C8C23×C6711yescharacteristic
391C22×C8C23×C6711yescharacteristic
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4C23×C6711yescharacteristic
402C2×C4⋊C4C23×C6711yescharacteristic
403C2×C4⋊C4C23×C6711yescharacteristic
404C2×C4⋊C4C23×C6711yescharacteristic
405C2×C4⋊C4C23×C6711yescharacteristic
406C2×C4⋊C4C23×C6711yescharacteristic
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C42—21no—
417C2×C42—21no—
418C2×C42—21no—
419C2×C42C23×C6711yescharacteristic
420C2×C42C23×C6711yescharacteristic
421C2×C42C23×C6711yescharacteristic
422#107—22no—
423#106—22no—
424#106—22no—
425#102—22no—
426#102—22no—
427#102—22no—
428#102—22no—
429#102—22no—
430#102—22no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95D4×C6711yescharacteristic
438#95D4×C6711yescharacteristic
439#95D4×C6711yescharacteristic
440#95D4×C6711yescharacteristic
441C24—21no—
442C24—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4#41×C6712yes—
453C2×D4#41×C6712yes—
454C2×D4#41×C6712yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—81no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4C2×D4×C6711yescharacteristic
471C22×C4C2×D4×C6711yescharacteristic
472C22×C4C2×D4×C6711yescharacteristic
473C22×C4C2×D4×C6711yescharacteristic
474C22×C4C2×D4×C6711yescharacteristic
475C22×C4C2×D4×C6711yescharacteristic
476C22×C4C24×C6711yescharacteristic
477D8—44no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4#41×C6712yes—
503C4⋊C4#41×C6712yes—
504C4⋊C4#41×C6712yes—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—81no—
509C22⋊C4—81no—
510C22⋊C4—81no—
511C22⋊C4—81no—
512C22⋊C4—81no—
513C22⋊C4—81no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—81no—
521C42—81no—
522C42—81no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C22×D4×C6711yescharacteristic
538D4—22no—
539D4—22no—
540D4—22no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4#126×C6714yes—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
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—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4C2×#41×C6711yescharacteristic
586C2×C4C2×#41×C6711yescharacteristic
587C2×C4C2×#41×C6711yescharacteristic
588C2×C4C2×#41×C6711yescharacteristic
589C2×C4C2×#41×C6711yescharacteristic
590C2×C4C2×#41×C6711yescharacteristic
591C8—41no—
592C8—41no—
593C8—41no—
594C8—41no—
595C22—24no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22#525×C6711yescharacteristic
610C22#525×C6711yescharacteristic
611C22#525×C6711yescharacteristic
612C22#576×C6711yescharacteristic
613C22#576×C6711yescharacteristic
614C22#577×C6711yescharacteristic
615C22#577×C6711yescharacteristic
616C4—41no—
617C4—41no—
618C4—42no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4C2×#126×C6711yescharacteristic
629C4#529×C6711yescharacteristic
630C4#529×C6711yescharacteristic
631C4#529×C6711yescharacteristic
632C2—44no—
633C2—161no—
634C2C67×#285211yescharacteristic
635C2C67×#311811yescharacteristic
636C2C67×#311811yescharacteristic
637C2C67×#311911yescharacteristic
638C2C67×#311911yescharacteristic
639C2C67×#312211yescharacteristic
640C2C67×#312511yescharacteristic
subgroupG/H
center, socleC23×C67C22×D4
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC22×D8×C67C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C67 ◁ #376×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2644C2semidirect
2. C67×#2275C2semidirect
3. C67×#1926C2semidirect
4. C67×#1926C2semidirect
5. C67×#1925C2semidirect
6. C67×#1920C2semidirect
7. C67×#1919C2semidirect
8. C67×#1918C2semidirect
12. C67×#1864C2semidirect
13. C67×#1864C2semidirect
14. C67×#1862C2semidirect
15. C67×#1840C2semidirect
311. #33086C67direct (N × H)
321. #2644C2×C67semidirect
322. #2275C2×C67semidirect
323. #1926C2×C67semidirect
324. #1926C2×C67semidirect
325. #1925C2×C67semidirect
326. #1920C2×C67semidirect
327. #1919C2×C67semidirect
328. #1918C2×C67semidirect
332. #1864C2×C67semidirect
333. #1864C2×C67semidirect
334. #1862C2×C67semidirect
335. #1840C2×C67semidirect
336. C67#33086direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#26442
C67×#22752
C67×#19262
C67×#19252
C67×#19202
C67×#19192
C67×#19182
C67×#18792
C67×#18642
C67×#18622
C67×#18402
#3308667