Finite Group Databasebeta

← C67×#32865 #30712116

Subgroups — C67×#32865 #30712116

Order 17152.

596 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2645C211yesmaximal, characteristic, complemented
2C67×#2644C211yesmaximal, characteristic
3C67×#2088C211yesmaximal, characteristic, complemented
4C67×#2087C211yesmaximal, characteristic, complemented
5C67×#2086C211yesmaximal, characteristic, complemented
6C67×#2068C211yesmaximal, characteristic, complemented
7C67×#2068C211yesmaximal, characteristic, complemented
8C67×#2067C211yesmaximal, characteristic
9C67×#2067C211yesmaximal, characteristic
10C67×#2066C211yesmaximal, characteristic, complemented
11C67×#2065C211yesmaximal, characteristic
12C67×#2055C211yesmaximal, characteristic
13C67×#2055C211yesmaximal, characteristic
14C67×#2055C211yesmaximal, characteristic
15C67×#1843C211yesmaximal, characteristic, complemented
16C22×SD16×C67C2211yescharacteristic
17C2×#107×C67C2211yescharacteristic
18C2×#107×C67C2211yescharacteristic
19C2×#106×C67C2211yescharacteristic
20C2×C4⋊C8×C67C2211yescharacteristic
21C2×C4⋊C8×C67C2211yescharacteristic
22C2×C4⋊C8×C67C2211yescharacteristic
23C2×C4⋊C8×C67C2211yescharacteristic
24C2×C4⋊C8×C67C2211yescharacteristic
25C2×C4⋊C8×C67C2211yescharacteristic
26C2×#103×C67C2211yescharacteristic
27C2×#103×C67C2211yescharacteristic
28C2×#103×C67C2211yescharacteristic
29C2×#102×C67C2211yescharacteristic
30C2×#102×C67C2211yescharacteristic
31C2×#102×C67C2211yescharacteristic
32#401×C67—21no—
33#400×C67—21no—
34#400×C67—21no—
35#400×C67—21no—
36#397×C67—21no—
37#397×C67—21no—
38#397×C67—21no—
39#396×C67—21no—
40#391×C67C2211yescharacteristic
41#391×C67C2211yescharacteristic
42#391×C67C2211yescharacteristic
43#390×C67C2211yescharacteristic
44#390×C67C2211yescharacteristic
45#390×C67C2211yescharacteristic
46#389×C67C2211yescharacteristic
47#389×C67C2211yescharacteristic
48#389×C67C2211yescharacteristic
49#389×C67C2211yescharacteristic
50#389×C67C2211yescharacteristic
51#389×C67C2211yescharacteristic
52#376×C67C2211yescharacteristic
53#340×C67C2211yescharacteristic
54#340×C67C2211yescharacteristic
55#340×C67C2211yescharacteristic
56#340×C67C2211yescharacteristic
57#340×C67C2211yescharacteristic
58#340×C67C2211yescharacteristic
59C22×Q8×C67C2311yescharacteristic
60C22×D4×C67C2311yescharacteristic
61C2×SD16×C67—41no—
62C2×SD16×C67—41no—
63C2×SD16×C67—41no—
64C22×C8×C67C2311yescharacteristic
65C22×C8×C67C2311yescharacteristic
66C22×C8×C67C2311yescharacteristic
67C22×C8×C67C2311yescharacteristic
68C2×C4⋊C4×C67—21no—
69C2×C4⋊C4×C67—21no—
70C2×C4⋊C4×C67—21no—
71C2×C4⋊C4×C67—21no—
72C2×C4⋊C4×C67—21no—
73C2×C4⋊C4×C67—21no—
74C2×C4⋊C4×C67C2311yescharacteristic
75C2×C4⋊C4×C67C2311yescharacteristic
76C2×C4⋊C4×C67C2311yescharacteristic
77C2×C4⋊C4×C67C2311yescharacteristic
78C2×C4⋊C4×C67C2311yescharacteristic
79C2×C4⋊C4×C67C2311yescharacteristic
80C2×C22⋊C4×C67—21no—
81C2×C22⋊C4×C67—21no—
82C2×C22⋊C4×C67—21no—
83C2×C22⋊C4×C67—21no—
84C2×C22⋊C4×C67—21no—
85C2×C22⋊C4×C67—21no—
86C2×C42×C67C2311yescharacteristic
87C2×C42×C67C2311yescharacteristic
88C2×C42×C67C2311yescharacteristic
89#107×C67—41no—
90#107×C67—41no—
91#106×C67—41no—
92C4⋊C8×C67—41no—
93C4⋊C8×C67—41no—
94C4⋊C8×C67—41no—
95C4⋊C8×C67—41no—
96C4⋊C8×C67—41no—
97C4⋊C8×C67—41no—
98#103×C67—41no—
99#103×C67—41no—
100#103×C67—41no—
101#102×C67—41no—
102#102×C67—41no—
103#102×C67—41no—
104#95×C67—21no—
105#95×C67—21no—
106#95×C67—21no—
107#95×C67—21no—
108#95×C67—21no—
109#95×C67—21no—
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—
121C2×Q8×C67—21no—
122C2×Q8×C67—21no—
123C2×Q8×C67—21no—
124C2×Q8×C67—21no—
125C2×Q8×C67—21no—
126C2×Q8×C67—21no—
127C2×D4×C67—21no—
128C2×D4×C67—21no—
129C2×D4×C67—21no—
130C2×D4×C67—21no—
131C2×D4×C67—21no—
132C2×D4×C67—21no—
133C22×C4×C67—21no—
134C22×C4×C67—21no—
135C22×C4×C67—21no—
136C22×C4×C67—21no—
137C22×C4×C67—21no—
138C22×C4×C67—21no—
139C22×C4×C67—21no—
140C22×C4×C67C2×D411yescharacteristic
141C22×C4×C67C2×D411yescharacteristic
142C22×C4×C67C2×D411yescharacteristic
143C22×C4×C67C2×D411yescharacteristic
144C22×C4×C67C2×D411yescharacteristic
145C22×C4×C67C2×D411yescharacteristic
146C22×C4×C67C2411yescharacteristic
147SD16×C67—44no—
148C2×C8×C67—21no—
149C2×C8×C67—21no—
150C2×C8×C67—21no—
151C2×C8×C67—21no—
152C2×C8×C67—21no—
153C2×C8×C67—21no—
154C2×C8×C67—21no—
155C2×C8×C67—21no—
156C2×C8×C67—21no—
157C2×C8×C67—21no—
158C2×C8×C67—21no—
159C2×C8×C67—21no—
160C4⋊C4×C67—21no—
161C4⋊C4×C67—21no—
162C4⋊C4×C67—21no—
163C4⋊C4×C67—21no—
164C4⋊C4×C67—21no—
165C4⋊C4×C67—21no—
166C4⋊C4×C67—21no—
167C4⋊C4×C67—21no—
168C4⋊C4×C67—21no—
169C4⋊C4×C67—21no—
170C4⋊C4×C67—21no—
171C4⋊C4×C67—21no—
172C4⋊C4×C67—81no—
173C4⋊C4×C67—81no—
174C4⋊C4×C67—81no—
175C4⋊C4×C67—81no—
176C4⋊C4×C67—81no—
177C4⋊C4×C67—81no—
178C22⋊C4×C67—81no—
179C22⋊C4×C67—81no—
180C22⋊C4×C67—81no—
181C22⋊C4×C67—81no—
182C22⋊C4×C67—81no—
183C22⋊C4×C67—81no—
184C42×C67—21no—
185C42×C67—21no—
186C42×C67—21no—
187C42×C67—21no—
188C42×C67—21no—
189C42×C67—21no—
190C23×C67—41no—
191C23×C67—41no—
192C23×C67—41no—
193C23×C67—41no—
194C23×C67—41no—
195C23×C67—41no—
196C23×C67—41no—
197C23×C67C22×D411yescharacteristic
198Q8×C67—41no—
199Q8×C67—41no—
200Q8×C67—41no—
201Q8×C67—41no—
202D4×C67—41no—
203D4×C67—41no—
204D4×C67—41no—
205D4×C67—41no—
206C2×C4×C67—21no—
207C2×C4×C67—21no—
208C2×C4×C67—21no—
209C2×C4×C67—21no—
210C2×C4×C67—21no—
211C2×C4×C67—21no—
212C2×C4×C67—21no—
213C2×C4×C67—21no—
214C2×C4×C67—21no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
218C2×C4×C67—21no—
219C2×C4×C67—21no—
220C2×C4×C67—21no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
223C2×C4×C67—21no—
224C2×C4×C67—41no—
225C2×C4×C67—41no—
226C2×C4×C67—41no—
227C2×C4×C67—41no—
228C2×C4×C67—41no—
229C2×C4×C67—41no—
230C2×C4×C67—41no—
231C2×C4×C67—41no—
232C2×C4×C67—41no—
233C2×C4×C67—41no—
234C2×C4×C67—41no—
235C2×C4×C67—41no—
236C2×C4×C67—41no—
237C2×C4×C67—41no—
238C2×C4×C67—41no—
239C2×C4×C67—41no—
240C2×C4×C67—41no—
241C2×C4×C67—41no—
242C2×C4×C67—41no—
243C2×C4×C67—41no—
244C2×C4×C67—41no—
245C2×C4×C67H(5,𝔽2)11yescharacteristic
246C2×C4×C67H(5,𝔽2)11yescharacteristic
247C2×C4×C67H(5,𝔽2)11yescharacteristic
248C2×C4×C672−1+411yescharacteristic
249C2×C4×C672−1+411yescharacteristic
250C2×C4×C672−1+411yescharacteristic
251C8×C67—41no—
252C8×C67—41no—
253C8×C67—41no—
254C8×C67—41no—
255C22×C67—81no—
256C22×C67—81no—
257C22×C67—81no—
258C22×C67—81no—
259C22×C67—81no—
260C22×C67—81no—
261C22×C67—81no—
262C22×C67#53711yescharacteristic
263C22×C67#53711yescharacteristic
264C22×C67#53711yescharacteristic
265C22×C67#57611yescharacteristic
266C22×C67#57711yescharacteristic
267C22×C67#57711yescharacteristic
268C22×C67#57811yescharacteristic
269C4×C67—41no—
270C4×C67—41no—
271C4×C67—41no—
272C4×C67—41no—
273C4×C67—41no—
274C4×C67—41no—
275C4×C67—81no—
276C4×C67—81no—
277C4×C67—81no—
278C4×C67—81no—
279C4×C67—81no—
280C4×C67—81no—
281C4×C67—81no—
282C4×C67#56011yescharacteristic
283C4×C67#56311yescharacteristic
284C4×C67#56311yescharacteristic
285C4×C67#56311yescharacteristic
286#32865C6711yesmaximal, characteristic, direct product factor, Sylow
287C2×C67—161no—
288C2×C67#282311yescharacteristic
289C2×C67#325711yescharacteristic
290C2×C67#325711yescharacteristic
291C2×C67#325811yescharacteristic
292C2×C67#325811yescharacteristic
293C2×C67#326011yescharacteristic
294C2×C67#326211yescharacteristic
295#2645C2×C6711yescharacteristic, complemented
296#2644C2×C6711yescharacteristic
297#2088C2×C6711yescharacteristic, complemented
298#2087C2×C6711yescharacteristic, complemented
299#2086C2×C6711yescharacteristic, complemented
300#2068C2×C6711yescharacteristic, complemented
301#2068C2×C6711yescharacteristic, complemented
302#2067C2×C6711yescharacteristic
303#2067C2×C6711yescharacteristic
304#2066C2×C6711yescharacteristic, complemented
305#2065C2×C6711yescharacteristic
306#2055C2×C6711yescharacteristic
307#2055C2×C6711yescharacteristic
308#2055C2×C6711yescharacteristic
309#1843C2×C6711yescharacteristic, complemented
310C67#3286511yescharacteristic, direct product factor, Sylow
311C22×SD16C22×C6711yescharacteristic
312C2×#107C22×C6711yescharacteristic
313C2×#107C22×C6711yescharacteristic
314C2×#106C22×C6711yescharacteristic
315C2×C4⋊C8C22×C6711yescharacteristic
316C2×C4⋊C8C22×C6711yescharacteristic
317C2×C4⋊C8C22×C6711yescharacteristic
318C2×C4⋊C8C22×C6711yescharacteristic
319C2×C4⋊C8C22×C6711yescharacteristic
320C2×C4⋊C8C22×C6711yescharacteristic
321C2×#103C22×C6711yescharacteristic
322C2×#103C22×C6711yescharacteristic
323C2×#103C22×C6711yescharacteristic
324C2×#102C22×C6711yescharacteristic
325C2×#102C22×C6711yescharacteristic
326C2×#102C22×C6711yescharacteristic
327#401—21no—
328#400—21no—
329#400—21no—
330#400—21no—
331#397—21no—
332#397—21no—
333#397—21no—
334#396—21no—
335#391C22×C6711yescharacteristic
336#391C22×C6711yescharacteristic
337#391C22×C6711yescharacteristic
338#390C22×C6711yescharacteristic
339#390C22×C6711yescharacteristic
340#390C22×C6711yescharacteristic
341#389C22×C6711yescharacteristic
342#389C22×C6711yescharacteristic
343#389C22×C6711yescharacteristic
344#389C22×C6711yescharacteristic
345#389C22×C6711yescharacteristic
346#389C22×C6711yescharacteristic
347#376C22×C6711yescharacteristic
348#340C22×C6711yescharacteristic
349#340C22×C6711yescharacteristic
350#340C22×C6711yescharacteristic
351#340C22×C6711yescharacteristic
352#340C22×C6711yescharacteristic
353#340C22×C6711yescharacteristic
354C22×Q8C23×C6711yescharacteristic
355C22×D4C23×C6711yescharacteristic
356C2×SD16—41no—
357C2×SD16—41no—
358C2×SD16—41no—
359C22×C8C23×C6711yescharacteristic
360C22×C8C23×C6711yescharacteristic
361C22×C8C23×C6711yescharacteristic
362C22×C8C23×C6711yescharacteristic
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4C23×C6711yescharacteristic
370C2×C4⋊C4C23×C6711yescharacteristic
371C2×C4⋊C4C23×C6711yescharacteristic
372C2×C4⋊C4C23×C6711yescharacteristic
373C2×C4⋊C4C23×C6711yescharacteristic
374C2×C4⋊C4C23×C6711yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C42C23×C6711yescharacteristic
382C2×C42C23×C6711yescharacteristic
383C2×C42C23×C6711yescharacteristic
384#107—41no—
385#107—41no—
386#106—41no—
387C4⋊C8—41no—
388C4⋊C8—41no—
389C4⋊C8—41no—
390C4⋊C8—41no—
391C4⋊C8—41no—
392C4⋊C8—41no—
393#103—41no—
394#103—41no—
395#103—41no—
396#102—41no—
397#102—41no—
398#102—41no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95D4×C6711yescharacteristic
412#95D4×C6711yescharacteristic
413#95D4×C6711yescharacteristic
414#95D4×C6711yescharacteristic
415C24—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4C2×D4×C6711yescharacteristic
436C22×C4C2×D4×C6711yescharacteristic
437C22×C4C2×D4×C6711yescharacteristic
438C22×C4C2×D4×C6711yescharacteristic
439C22×C4C2×D4×C6711yescharacteristic
440C22×C4C2×D4×C6711yescharacteristic
441C22×C4C24×C6711yescharacteristic
442SD16—44no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C22⋊C4—81no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C22×D4×C6711yescharacteristic
493Q8—41no—
494Q8—41no—
495Q8—41no—
496Q8—41no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—41no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4H(5,𝔽2)×C6711yescharacteristic
541C2×C4H(5,𝔽2)×C6711yescharacteristic
542C2×C4H(5,𝔽2)×C6711yescharacteristic
543C2×C42−1+4×C6711yescharacteristic
544C2×C42−1+4×C6711yescharacteristic
545C2×C42−1+4×C6711yescharacteristic
546C8—41no—
547C8—41no—
548C8—41no—
549C8—41no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22#537×C6711yescharacteristic
558C22#537×C6711yescharacteristic
559C22#537×C6711yescharacteristic
560C22#576×C6711yescharacteristic
561C22#577×C6711yescharacteristic
562C22#577×C6711yescharacteristic
563C22#578×C6711yescharacteristic
564C4—41no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4#560×C6711yescharacteristic
578C4#563×C6711yescharacteristic
579C4#563×C6711yescharacteristic
580C4#563×C6711yescharacteristic
581C2—161no—
582C2C67×#282311yescharacteristic
583C2C67×#325711yescharacteristic
584C2C67×#325711yescharacteristic
585C2C67×#325811yescharacteristic
586C2C67×#325811yescharacteristic
587C2C67×#326011yescharacteristic
588C2C67×#326211yescharacteristic
subgroupG/H
center, socleC23×C67C22×D4
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC22×D4×C67C23
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×#2645C2semidirect
3. C67×#2088C2semidirect
4. C67×#2087C2semidirect
5. C67×#2086C2semidirect
6. C67×#2068C2semidirect
7. C67×#2068C2semidirect
10. C67×#2066C2semidirect
15. C67×#1843C2semidirect
286. #32865C67direct (N × H)
295. #2645C2×C67semidirect
297. #2088C2×C67semidirect
298. #2087C2×C67semidirect
299. #2086C2×C67semidirect
300. #2068C2×C67semidirect
301. #2068C2×C67semidirect
304. #2066C2×C67semidirect
309. #1843C2×C67semidirect
310. C67#32865direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#26452
C67×#26442
C67×#20882
C67×#20872
C67×#20862
C67×#20682
C67×#20672
C67×#20662
C67×#20652
C67×#20552
C67×#18432
#3286567