Finite Group Databasebeta

← C67×#29527 #30708778

Subgroups — C67×#29527 #30708778

Order 17152.

634 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2644C211yesmaximal, characteristic
2C67×#2331C211yesmaximal, characteristic, complemented
3C67×#2101C211yesmaximal, characteristic, complemented
4C67×#2092C211yesmaximal, characteristic
5C67×#2057C211yesmaximal, characteristic, complemented
6C67×#2023C211yesmaximal, characteristic
7C67×#1943C211yesmaximal, characteristic, complemented
8C67×#1942C211yesmaximal, characteristic, complemented
9C67×#1936C211yesmaximal, characteristic
10C67×#1934C211yesmaximal, characteristic
11C67×#1925C211yesmaximal, characteristic
12C67×#1925C211yesmaximal, characteristic
13C67×#1921C211yesmaximal, characteristic, complemented
14C67×#1919C211yesmaximal, characteristic, complemented
15C67×#1746C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8×C67C2211yescharacteristic
17C2×#107×C67C2211yescharacteristic
18C2×#107×C67C2211yescharacteristic
19C2×#107×C67C2211yescharacteristic
20C2×#106×C67C2211yescharacteristic
21C2×#102×C67C2211yescharacteristic
22C2×#102×C67C2211yescharacteristic
23C2×#102×C67C2211yescharacteristic
24C2×#102×C67C2211yescharacteristic
25C2×#97×C67C2211yescharacteristic
26C2×C4×C8×C67C2211yescharacteristic
27#397×C67—21no—
28#397×C67—21no—
29#397×C67—21no—
30#396×C67—21no—
31#391×C67—21no—
32#391×C67—21no—
33#390×C67C2211yescharacteristic
34#390×C67C2211yescharacteristic
35#390×C67C2211yescharacteristic
36#389×C67C2211yescharacteristic
37#389×C67C2211yescharacteristic
38#389×C67C2211yescharacteristic
39#389×C67C2211yescharacteristic
40#389×C67C2211yescharacteristic
41#387×C67—21no—
42#387×C67—21no—
43#382×C67C2211yescharacteristic
44C4×C4⋊C4×C67C2211yescharacteristic
45C4×C4⋊C4×C67C2211yescharacteristic
46#376×C67C2211yescharacteristic
47#340×C67C2211yescharacteristic
48#340×C67C2211yescharacteristic
49#340×C67C2211yescharacteristic
50#340×C67C2211yescharacteristic
51#340×C67C2211yescharacteristic
52#340×C67C2211yescharacteristic
53#340×C67C2211yescharacteristic
54#340×C67C2211yescharacteristic
55#336×C67C2211yescharacteristic
56#336×C67C2211yescharacteristic
57#336×C67C2211yescharacteristic
58#336×C67C2211yescharacteristic
59C22×Q8×C67—21no—
60C22×D4×C67C2311yescharacteristic
61C22×C8×C67C2311yescharacteristic
62C22×C8×C67C2311yescharacteristic
63C22×C8×C67C2311yescharacteristic
64C22×C8×C67C2311yescharacteristic
65C4⋊Q8×C67—22no—
66C4⋊Q8×C67—22no—
67C2×C4⋊C4×C67—21no—
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×C67C2311yescharacteristic
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×C67—21no—
87C2×C42×C67—21no—
88C2×C42×C67C2311yescharacteristic
89C2×C42×C67C2311yescharacteristic
90C2×C42×C67C2311yescharacteristic
91#107×C67—22no—
92#107×C67—22no—
93#107×C67—22no—
94#106×C67—22no—
95#102×C67—22no—
96#102×C67—22no—
97#102×C67—22no—
98#102×C67—22no—
99#97×C67—22no—
100C4×C8×C67—22no—
101#95×C67—21no—
102#95×C67—21no—
103#95×C67—21no—
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×C67D411yescharacteristic
111#95×C67D411yescharacteristic
112#95×C67D411yescharacteristic
113#95×C67D411yescharacteristic
114C24×C67—21no—
115C2×Q8×C67—24no—
116C2×Q8×C67—81no—
117C2×Q8×C67—81no—
118C2×D4×C67—21no—
119C2×D4×C67—21no—
120C2×D4×C67—21no—
121C2×D4×C67—21no—
122C2×D4×C67—21no—
123C2×D4×C67—21no—
124C22×C4×C67—21no—
125C22×C4×C67—21no—
126C22×C4×C67—21no—
127C22×C4×C67—21no—
128C22×C4×C67—21no—
129C22×C4×C67—21no—
130C22×C4×C67—21no—
131C22×C4×C67C2×D411yescharacteristic
132C22×C4×C67C2×D411yescharacteristic
133C22×C4×C67C2×D411yescharacteristic
134C22×C4×C67C2×D411yescharacteristic
135C22×C4×C67C2×D411yescharacteristic
136C22×C4×C67C2×D411yescharacteristic
137C22×C4×C67C2411yescharacteristic
138C2×C8×C67—21no—
139C2×C8×C67—21no—
140C2×C8×C67—21no—
141C2×C8×C67—21no—
142C2×C8×C67—21no—
143C2×C8×C67—21no—
144C2×C8×C67—21no—
145C2×C8×C67—21no—
146C2×C8×C67—21no—
147C2×C8×C67—21no—
148C2×C8×C67#4112yes—
149C2×C8×C67#4112yes—
150C4⋊C4×C67—24no—
151C4⋊C4×C67—21no—
152C4⋊C4×C67—21no—
153C4⋊C4×C67—21no—
154C4⋊C4×C67—21no—
155C4⋊C4×C67—21no—
156C4⋊C4×C67—21no—
157C4⋊C4×C67—21no—
158C4⋊C4×C67—21no—
159C4⋊C4×C67—21no—
160C4⋊C4×C67—21no—
161C4⋊C4×C67—21no—
162C4⋊C4×C67—21no—
163C4⋊C4×C67—81no—
164C4⋊C4×C67—81no—
165C4⋊C4×C67—81no—
166C4⋊C4×C67—81no—
167C4⋊C4×C67—81no—
168C4⋊C4×C67#4112yes—
169C4⋊C4×C67#4112yes—
170C22⋊C4×C67—81no—
171C22⋊C4×C67—81no—
172C22⋊C4×C67—81no—
173C22⋊C4×C67—81no—
174C22⋊C4×C67—81no—
175C22⋊C4×C67—81no—
176C42×C67—21no—
177C42×C67—21no—
178C42×C67—21no—
179C42×C67—21no—
180C42×C67—21no—
181C42×C67—21no—
182C42×C67—81no—
183C42×C67—81no—
184C23×C67—41no—
185C23×C67—41no—
186C23×C67—41no—
187C23×C67—41no—
188C23×C67—41no—
189C23×C67—41no—
190C23×C67—41no—
191C23×C67C22×D411yescharacteristic
192Q8×C67—84no—
193D4×C67—41no—
194D4×C67—41no—
195D4×C67—41no—
196D4×C67—41no—
197C2×C4×C67—21no—
198C2×C4×C67—21no—
199C2×C4×C67—22no—
200C2×C4×C67—22no—
201C2×C4×C67—21no—
202C2×C4×C67—21no—
203C2×C4×C67—21no—
204C2×C4×C67—21no—
205C2×C4×C67—21no—
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—41no—
218C2×C4×C67—41no—
219C2×C4×C67—41no—
220C2×C4×C67—41no—
221C2×C4×C67—41no—
222C2×C4×C67—41no—
223C2×C4×C67—41no—
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×C67C2×#4111yescharacteristic
237C2×C4×C67C2×#4111yescharacteristic
238C2×C4×C67C2×#4111yescharacteristic
239C2×C4×C67C2×#4111yescharacteristic
240C2×C4×C672−1+411yescharacteristic
241C2×C4×C672−1+411yescharacteristic
242C8×C67—22no—
243C8×C67—22no—
244C8×C67—41no—
245C8×C67—41no—
246C22×C67—81no—
247C22×C67—81no—
248C22×C67—81no—
249C22×C67—81no—
250C22×C67—81no—
251C22×C67—81no—
252C22×C67—81no—
253C22×C67#52511yescharacteristic
254C22×C67#53211yescharacteristic
255C22×C67#53611yescharacteristic
256C22×C67#57611yescharacteristic
257C22×C67#57611yescharacteristic
258C22×C67#57711yescharacteristic
259C22×C67#57811yescharacteristic
260C4×C67—41no—
261C4×C67—41no—
262C4×C67—42no—
263C4×C67—42no—
264C4×C67—41no—
265C4×C67—41no—
266C4×C67—41no—
267C4×C67—41no—
268C4×C67—81no—
269C4×C67—81no—
270C4×C67—81no—
271C4×C67—81no—
272C4×C67—81no—
273C4×C67#55211yescharacteristic
274C4×C67#55211yescharacteristic
275C4×C67#55611yescharacteristic
276C4×C67#55611yescharacteristic
277#29527C6711yesmaximal, characteristic, direct product factor, Sylow
278C2×C67—161no—
279C2×C67#260411yescharacteristic
280C2×C67#312111yescharacteristic
281C2×C67#312211yescharacteristic
282C2×C67#317511yescharacteristic
283C2×C67#317911yescharacteristic
284C2×C67#324111yescharacteristic
285C2×C67#324311yescharacteristic
286#2644C2×C6711yescharacteristic
287#2331C2×C6711yescharacteristic, complemented
288#2101C2×C6711yescharacteristic, complemented
289#2092C2×C6711yescharacteristic
290#2057C2×C6711yescharacteristic, complemented
291#2023C2×C6711yescharacteristic
292#1943C2×C6711yescharacteristic, complemented
293#1942C2×C6711yescharacteristic, complemented
294#1936C2×C6711yescharacteristic
295#1934C2×C6711yescharacteristic
296#1925C2×C6711yescharacteristic
297#1925C2×C6711yescharacteristic
298#1921C2×C6711yescharacteristic, complemented
299#1919C2×C6711yescharacteristic, complemented
300#1746C2×C6711yescharacteristic, complemented
301C67#2952711yescharacteristic, direct product factor, Sylow
302C2×C4⋊Q8C22×C6711yescharacteristic
303C2×#107C22×C6711yescharacteristic
304C2×#107C22×C6711yescharacteristic
305C2×#107C22×C6711yescharacteristic
306C2×#106C22×C6711yescharacteristic
307C2×#102C22×C6711yescharacteristic
308C2×#102C22×C6711yescharacteristic
309C2×#102C22×C6711yescharacteristic
310C2×#102C22×C6711yescharacteristic
311C2×#97C22×C6711yescharacteristic
312C2×C4×C8C22×C6711yescharacteristic
313#397—21no—
314#397—21no—
315#397—21no—
316#396—21no—
317#391—21no—
318#391—21no—
319#390C22×C6711yescharacteristic
320#390C22×C6711yescharacteristic
321#390C22×C6711yescharacteristic
322#389C22×C6711yescharacteristic
323#389C22×C6711yescharacteristic
324#389C22×C6711yescharacteristic
325#389C22×C6711yescharacteristic
326#389C22×C6711yescharacteristic
327#387—21no—
328#387—21no—
329#382C22×C6711yescharacteristic
330C4×C4⋊C4C22×C6711yescharacteristic
331C4×C4⋊C4C22×C6711yescharacteristic
332#376C22×C6711yescharacteristic
333#340C22×C6711yescharacteristic
334#340C22×C6711yescharacteristic
335#340C22×C6711yescharacteristic
336#340C22×C6711yescharacteristic
337#340C22×C6711yescharacteristic
338#340C22×C6711yescharacteristic
339#340C22×C6711yescharacteristic
340#340C22×C6711yescharacteristic
341#336C22×C6711yescharacteristic
342#336C22×C6711yescharacteristic
343#336C22×C6711yescharacteristic
344#336C22×C6711yescharacteristic
345C22×Q8—21no—
346C22×D4C23×C6711yescharacteristic
347C22×C8C23×C6711yescharacteristic
348C22×C8C23×C6711yescharacteristic
349C22×C8C23×C6711yescharacteristic
350C22×C8C23×C6711yescharacteristic
351C4⋊Q8—22no—
352C4⋊Q8—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4C23×C6711yescharacteristic
360C2×C4⋊C4C23×C6711yescharacteristic
361C2×C4⋊C4C23×C6711yescharacteristic
362C2×C4⋊C4C23×C6711yescharacteristic
363C2×C4⋊C4C23×C6711yescharacteristic
364C2×C4⋊C4C23×C6711yescharacteristic
365C2×C4⋊C4C23×C6711yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42C23×C6711yescharacteristic
375C2×C42C23×C6711yescharacteristic
376C2×C42C23×C6711yescharacteristic
377#107—22no—
378#107—22no—
379#107—22no—
380#106—22no—
381#102—22no—
382#102—22no—
383#102—22no—
384#102—22no—
385#97—22no—
386C4×C8—22no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95D4×C6711yescharacteristic
397#95D4×C6711yescharacteristic
398#95D4×C6711yescharacteristic
399#95D4×C6711yescharacteristic
400C24—21no—
401C2×Q8—24no—
402C2×Q8—81no—
403C2×Q8—81no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4C2×D4×C6711yescharacteristic
418C22×C4C2×D4×C6711yescharacteristic
419C22×C4C2×D4×C6711yescharacteristic
420C22×C4C2×D4×C6711yescharacteristic
421C22×C4C2×D4×C6711yescharacteristic
422C22×C4C2×D4×C6711yescharacteristic
423C22×C4C24×C6711yescharacteristic
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8#41×C6712yes—
435C2×C8#41×C6712yes—
436C4⋊C4—24no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4#41×C6712yes—
455C4⋊C4#41×C6712yes—
456C22⋊C4—81no—
457C22⋊C4—81no—
458C22⋊C4—81no—
459C22⋊C4—81no—
460C22⋊C4—81no—
461C22⋊C4—81no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—81no—
469C42—81no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23C22×D4×C6711yescharacteristic
478Q8—84no—
479D4—41no—
480D4—41no—
481D4—41no—
482D4—41no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4C2×#41×C6711yescharacteristic
523C2×C4C2×#41×C6711yescharacteristic
524C2×C4C2×#41×C6711yescharacteristic
525C2×C4C2×#41×C6711yescharacteristic
526C2×C42−1+4×C6711yescharacteristic
527C2×C42−1+4×C6711yescharacteristic
528C8—22no—
529C8—22no—
530C8—41no—
531C8—41no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22#525×C6711yescharacteristic
540C22#532×C6711yescharacteristic
541C22#536×C6711yescharacteristic
542C22#576×C6711yescharacteristic
543C22#576×C6711yescharacteristic
544C22#577×C6711yescharacteristic
545C22#578×C6711yescharacteristic
546C4—41no—
547C4—41no—
548C4—42no—
549C4—42no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—81no—
555C4—81no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4#552×C6711yescharacteristic
560C4#552×C6711yescharacteristic
561C4#556×C6711yescharacteristic
562C4#556×C6711yescharacteristic
563C2—161no—
564C2C67×#260411yescharacteristic
565C2C67×#312111yescharacteristic
566C2C67×#312211yescharacteristic
567C2C67×#317511yescharacteristic
568C2C67×#317911yescharacteristic
569C2C67×#324111yescharacteristic
570C2C67×#324311yescharacteristic
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
2. C67×#2331C2semidirect
3. C67×#2101C2semidirect
5. C67×#2057C2semidirect
7. C67×#1943C2semidirect
8. C67×#1942C2semidirect
13. C67×#1921C2semidirect
14. C67×#1919C2semidirect
15. C67×#1746C2semidirect
277. #29527C67direct (N × H)
287. #2331C2×C67semidirect
288. #2101C2×C67semidirect
290. #2057C2×C67semidirect
292. #1943C2×C67semidirect
293. #1942C2×C67semidirect
298. #1921C2×C67semidirect
299. #1919C2×C67semidirect
300. #1746C2×C67semidirect
301. C67#29527direct (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×#23312
C67×#21012
C67×#20922
C67×#20572
C67×#20232
C67×#19432
C67×#19422
C67×#19362
C67×#19342
C67×#19252
C67×#19212
C67×#19192
C67×#17462
#2952767