Finite Group Databasebeta

← C67×#24014 #30703265

Subgroups — C67×#24014 #30703265

Order 17152.

1276 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#536×C67C211yesmaximal, characteristic
2C2×#484×C67C211yesmaximal, characteristic, complemented
3C2×#483×C67C211yesmaximal, characteristic, complemented
4C67×#2311C211yesmaximal, characteristic
5C67×#2036C212yesmaximal, complemented
6C67×#2035C212yesmaximal, complemented
7C67×#1873C211yesmaximal, characteristic, complemented
8C67×#1873C211yesmaximal, characteristic, complemented
9C67×#1854C212yesmaximal, complemented
10C67×#1787C212yesmaximal, complemented
11C67×#1780C211yesmaximal, characteristic
12C22×#41×C67C2211yescharacteristic
13C23×Q8×C67C2211yescharacteristic
14#536×C67—24no—
15#536×C67—24no—
16C2×C4⋊Q8×C67—21no—
17C2×#124×C67—21no—
18C2×#123×C67—21no—
19C2×#123×C67—21no—
20C2×C22⋊Q8×C67C2211yescharacteristic
21C2×C22⋊Q8×C67C2211yescharacteristic
22C2×C22⋊Q8×C67C2211yescharacteristic
23C2×C22⋊Q8×C67C2211yescharacteristic
24C2×#117×C67C2211yescharacteristic
25C2×#117×C67C2211yescharacteristic
26C22×C4⋊C4×C67C2211yescharacteristic
27#484×C67—24no—
28#483×C67—24no—
29C2×#107×C67C2212yes—
30C2×#106×C67C2212yes—
31C2×#103×C67C2212yes—
32C2×#103×C67C2212yes—
33C2×#103×C67C2212yes—
34C2×#103×C67C2212yes—
35C2×C22⋊C8×C67C2212yes—
36C2×C22⋊C8×C67C2211yescharacteristic
37C2×C22⋊C8×C67C2211yescharacteristic
38C2×C22⋊C8×C67C2212yes—
39#391×C67—21no—
40#390×C67—21no—
41#386×C67—21no—
42#386×C67—21no—
43#380×C67C2212yes—
44#380×C67C2212yes—
45#340×C67C2212yes—
46#340×C67C2212yes—
47C2×#41×C67—22no—
48C2×#41×C67—22no—
49C2×#41×C67—24no—
50C2×#41×C67—42no—
51C22×Q8×C67—22no—
52C22×Q8×C67—22no—
53C22×Q8×C67—21no—
54C22×Q8×C67—24no—
55C22×Q8×C67—42no—
56C22×Q8×C67D411yescharacteristic
57C22×Q8×C67C2311yescharacteristic
58C22×Q8×C67C2311yescharacteristic
59C22×D4×C67D411yescharacteristic
60C22×D4×C67D411yescharacteristic
61C22×D4×C67D411yescharacteristic
62C23×C4×C67—21no—
63C23×C4×C67—21no—
64C23×C4×C67C2311yescharacteristic
65C23×C4×C67C2311yescharacteristic
66C23×C4×C67C2311yescharacteristic
67C22×C8×C67C2312yes—
68C22×C8×C67C2312yes—
69C4⋊Q8×C67—28no—
70#124×C67—28no—
71#123×C67—28no—
72#123×C67—28no—
73C22⋊Q8×C67—22no—
74C22⋊Q8×C67—22no—
75C22⋊Q8×C67—22no—
76C22⋊Q8×C67—22no—
77C22⋊Q8×C67—22no—
78C22⋊Q8×C67—22no—
79C22⋊Q8×C67D414yes—
80C22⋊Q8×C67D414yes—
81#117×C67—22no—
82#117×C67—22no—
83#117×C67—42no—
84C2×C4⋊C4×C67—21no—
85C2×C4⋊C4×C67—21no—
86C2×C4⋊C4×C67—21no—
87C2×C4⋊C4×C67—21no—
88C2×C4⋊C4×C67—22no—
89C2×C4⋊C4×C67—22no—
90C2×C4⋊C4×C67—21no—
91C2×C4⋊C4×C67C2312yes—
92C2×C4⋊C4×C67C2312yes—
93C2×C4⋊C4×C67C2311yescharacteristic
94C2×C4⋊C4×C67C2311yescharacteristic
95C2×C22⋊C4×C67—21no—
96C2×C22⋊C4×C67—21no—
97C2×C22⋊C4×C67—21no—
98C2×C22⋊C4×C67—21no—
99C2×C22⋊C4×C67—21no—
100C2×C22⋊C4×C67—21no—
101C2×C22⋊C4×C67—22no—
102C2×C22⋊C4×C67—22no—
103C2×C42×C67—21no—
104C2×C42×C67—21no—
105#107×C67—24no—
106#106×C67—24no—
107#103×C67—24no—
108#103×C67—24no—
109#103×C67—42no—
110#103×C67—42no—
111C22⋊C8×C67—22no—
112C22⋊C8×C67—22no—
113C22⋊C8×C67—24no—
114C22⋊C8×C67—42no—
115#95×C67—22no—
116#95×C67—22no—
117C24×C67C2×D411yescharacteristic
118C24×C67C2×D411yescharacteristic
119C24×C67C2×D411yescharacteristic
120#41×C67—48no—
121#41×C67—48no—
122C2×Q8×C67—24no—
123C2×Q8×C67—22no—
124C2×Q8×C67—22no—
125C2×Q8×C67—21no—
126C2×Q8×C67—21no—
127C2×Q8×C67—22no—
128C2×Q8×C67—22no—
129C2×Q8×C67—22no—
130C2×Q8×C67—22no—
131C2×Q8×C67—41no—
132C2×Q8×C67—48no—
133C2×Q8×C67—48no—
134C2×Q8×C67—42no—
135C2×Q8×C67—42no—
136C2×Q8×C67C2×D412yes—
137C2×Q8×C67C2×D412yes—
138C2×D4×C67—22no—
139C2×D4×C67—22no—
140C2×D4×C67—22no—
141C2×D4×C67—22no—
142C2×D4×C67—22no—
143C2×D4×C67—22no—
144C2×D4×C67—41no—
145C2×D4×C67—41no—
146C2×D4×C67—41no—
147C22×C4×C67—21no—
148C22×C4×C67—21no—
149C22×C4×C67—22no—
150C22×C4×C67—22no—
151C22×C4×C67—22no—
152C22×C4×C67—24no—
153C22×C4×C67—21no—
154C22×C4×C67—21no—
155C22×C4×C67—21no—
156C22×C4×C67—21no—
157C22×C4×C67—21no—
158C22×C4×C67—21no—
159C22×C4×C67—22no—
160C22×C4×C67—22no—
161C22×C4×C67—22no—
162C22×C4×C67—22no—
163C22×C4×C67—42no—
164C22×C4×C67—42no—
165C22×C4×C67—42no—
166C22×C4×C67—82no—
167C22×C4×C67C2×D411yescharacteristic
168C22×C4×C67C2×D411yescharacteristic
169C22×C4×C67C2×D412yes—
170C22×C4×C67C2×D412yes—
171C22×C4×C67C2×D411yescharacteristic
172C22×C4×C67#4114yes—
173C22×C4×C67C2411yescharacteristic
174C2×C8×C67—22no—
175C2×C8×C67—22no—
176C2×C8×C67—22no—
177C2×C8×C67—22no—
178C2×C8×C67—22no—
179C2×C8×C67—22no—
180C4⋊C4×C67—24no—
181C4⋊C4×C67—24no—
182C4⋊C4×C67—24no—
183C4⋊C4×C67—24no—
184C4⋊C4×C67—21no—
185C4⋊C4×C67—21no—
186C4⋊C4×C67—22no—
187C4⋊C4×C67—22no—
188C4⋊C4×C67—22no—
189C4⋊C4×C67—22no—
190C4⋊C4×C67—42no—
191C4⋊C4×C67C2×D412yes—
192C4⋊C4×C67C2×D412yes—
193C22⋊C4×C67—24no—
194C22⋊C4×C67—24no—
195C22⋊C4×C67—24no—
196C22⋊C4×C67—24no—
197C22⋊C4×C67—24no—
198C22⋊C4×C67—42no—
199C22⋊C4×C67—82no—
200C22⋊C4×C67—82no—
201C42×C67—24no—
202C42×C67—42no—
203C23×C67—21no—
204C23×C67—21no—
205C23×C67—21no—
206C23×C67—21no—
207C23×C67—22no—
208C23×C67—22no—
209C23×C67—22no—
210C23×C67—22no—
211C23×C67—21no—
212C23×C67—21no—
213C23×C67—21no—
214C23×C67—22no—
215C23×C67—22no—
216C23×C67#12012yes—
217C23×C67#12312yes—
218C23×C67C22×D411yescharacteristic
219Q8×C67—24no—
220Q8×C67—24no—
221Q8×C67—44no—
222Q8×C67—48no—
223Q8×C67—42no—
224Q8×C67—42no—
225D4×C67—44no—
226D4×C67—44no—
227D4×C67—44no—
228C2×C4×C67—21no—
229C2×C4×C67—21no—
230C2×C4×C67—21no—
231C2×C4×C67—21no—
232C2×C4×C67—24no—
233C2×C4×C67—24no—
234C2×C4×C67—24no—
235C2×C4×C67—24no—
236C2×C4×C67—22no—
237C2×C4×C67—22no—
238C2×C4×C67—22no—
239C2×C4×C67—22no—
240C2×C4×C67—21no—
241C2×C4×C67—21no—
242C2×C4×C67—21no—
243C2×C4×C67—48no—
244C2×C4×C67—41no—
245C2×C4×C67—41no—
246C2×C4×C67—41no—
247C2×C4×C67—41no—
248C2×C4×C67—42no—
249C2×C4×C67—42no—
250C2×C4×C67—42no—
251C2×C4×C67—42no—
252C2×C4×C67—42no—
253C2×C4×C67—42no—
254C2×C4×C67—42no—
255C2×C4×C67—42no—
256C2×C4×C67—42no—
257C2×C4×C67—42no—
258C2×C4×C67—84no—
259C2×C4×C67#12012yes—
260C2×C4×C67#12312yes—
261C2×C4×C67C22×D411yescharacteristic
262C2×C4×C67C22×D411yescharacteristic
263C2×C4×C67C2×#4112yes—
264C2×C4×C67H(5,𝔽2)12yes—
265C8×C67—42no—
266C8×C67—42no—
267C22×C67—24no—
268C22×C67—22no—
269C22×C67—22no—
270C22×C67—22no—
271C22×C67—22no—
272C22×C67—42no—
273C22×C67—42no—
274C22×C67—42no—
275C22×C67—41no—
276C22×C67—41no—
277C22×C67—42no—
278C22×C67—41no—
279C22×C67—42no—
280C22×C67—41no—
281C22×C67—41no—
282C22×C67C2×#12011yescharacteristic
283C22×C67C2×#12311yescharacteristic
284C22×C67#53411yescharacteristic
285C22×C67#57712yes—
286C22×C67#57812yes—
287C4×C67—22no—
288C4×C67—22no—
289C4×C67—41no—
290C4×C67—42no—
291C4×C67—42no—
292C4×C67—42no—
293C4×C67—42no—
294C4×C67—82no—
295C4×C67—82no—
296C4×C67#54612yes—
297C4×C67#54612yes—
298#24014C6711yesmaximal, characteristic, direct product factor, Sylow
299C2×C67—42no—
300C2×C67—42no—
301C2×C67—81no—
302C2×C67#240511yescharacteristic
303C2×C67#301011yescharacteristic
304C2×C67#301111yescharacteristic
305C2×C67#309712yes—
306C2×C67#320112yes—
307C2×#536C2×C6711yescharacteristic
308C2×#484C2×C6711yescharacteristic, complemented
309C2×#483C2×C6711yescharacteristic, complemented
310#2311C2×C6711yescharacteristic
311#2036C2×C6712yescomplemented
312#2035C2×C6712yescomplemented
313#1873C2×C6711yescharacteristic, complemented
314#1873C2×C6711yescharacteristic, complemented
315#1854C2×C6712yescomplemented
316#1787C2×C6712yescomplemented
317#1780C2×C6711yescharacteristic
318C67#2401411yescharacteristic, direct product factor, Sylow
319C22×#41C22×C6711yescharacteristic
320C23×Q8C22×C6711yescharacteristic
321#536—24no—
322#536—24no—
323C2×C4⋊Q8—21no—
324C2×#124—21no—
325C2×#123—21no—
326C2×#123—21no—
327C2×C22⋊Q8C22×C6711yescharacteristic
328C2×C22⋊Q8C22×C6711yescharacteristic
329C2×C22⋊Q8C22×C6711yescharacteristic
330C2×C22⋊Q8C22×C6711yescharacteristic
331C2×#117C22×C6711yescharacteristic
332C2×#117C22×C6711yescharacteristic
333C22×C4⋊C4C22×C6711yescharacteristic
334#484—24no—
335#483—24no—
336C2×#107C22×C6712yes—
337C2×#106C22×C6712yes—
338C2×#103C22×C6712yes—
339C2×#103C22×C6712yes—
340C2×#103C22×C6712yes—
341C2×#103C22×C6712yes—
342C2×C22⋊C8C22×C6711yescharacteristic
343C2×C22⋊C8C22×C6711yescharacteristic
344C2×C22⋊C8C22×C6712yes—
345C2×C22⋊C8C22×C6712yes—
346#391—21no—
347#390—21no—
348#386—21no—
349#386—21no—
350#380C22×C6712yes—
351#380C22×C6712yes—
352#340C22×C6712yes—
353#340C22×C6712yes—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—24no—
357C2×#41—42no—
358C22×Q8—22no—
359C22×Q8—22no—
360C22×Q8—21no—
361C22×Q8—24no—
362C22×Q8—42no—
363C22×Q8D4×C6711yescharacteristic
364C22×Q8C23×C6711yescharacteristic
365C22×Q8C23×C6711yescharacteristic
366C22×D4D4×C6711yescharacteristic
367C22×D4D4×C6711yescharacteristic
368C22×D4D4×C6711yescharacteristic
369C23×C4—21no—
370C23×C4—21no—
371C23×C4C23×C6711yescharacteristic
372C23×C4C23×C6711yescharacteristic
373C23×C4C23×C6711yescharacteristic
374C22×C8C23×C6712yes—
375C22×C8C23×C6712yes—
376C4⋊Q8—28no—
377#124—28no—
378#123—28no—
379#123—28no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8D4×C6714yes—
387C22⋊Q8D4×C6714yes—
388#117—22no—
389#117—22no—
390#117—42no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4C23×C6712yes—
399C2×C4⋊C4C23×C6712yes—
400C2×C4⋊C4C23×C6711yescharacteristic
401C2×C4⋊C4C23×C6711yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C42—21no—
411C2×C42—21no—
412#107—24no—
413#106—24no—
414#103—24no—
415#103—24no—
416#103—42no—
417#103—42no—
418C22⋊C8—22no—
419C22⋊C8—22no—
420C22⋊C8—24no—
421C22⋊C8—42no—
422#95—22no—
423#95—22no—
424C24C2×D4×C6711yescharacteristic
425C24C2×D4×C6711yescharacteristic
426C24C2×D4×C6711yescharacteristic
427#41—48no—
428#41—48no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—41no—
439C2×Q8—48no—
440C2×Q8—48no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8C2×D4×C6712yes—
444C2×Q8C2×D4×C6712yes—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4—41no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—82no—
474C22×C4C2×D4×C6711yescharacteristic
475C22×C4C2×D4×C6711yescharacteristic
476C22×C4C2×D4×C6712yes—
477C22×C4C2×D4×C6712yes—
478C22×C4C2×D4×C6711yescharacteristic
479C22×C4#41×C6714yes—
480C22×C4C24×C6711yescharacteristic
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—42no—
498C4⋊C4C2×D4×C6712yes—
499C4⋊C4C2×D4×C6712yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—42no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C42—24no—
509C42—42no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—22no—
522C23—22no—
523C23#120×C6712yes—
524C23#123×C6712yes—
525C23C22×D4×C6711yescharacteristic
526Q8—24no—
527Q8—24no—
528Q8—44no—
529Q8—48no—
530Q8—42no—
531Q8—42no—
532D4—44no—
533D4—44no—
534D4—44no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—48no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—84no—
566C2×C4#120×C6712yes—
567C2×C4#123×C6712yes—
568C2×C4C22×D4×C6711yescharacteristic
569C2×C4C22×D4×C6711yescharacteristic
570C2×C4C2×#41×C6712yes—
571C2×C4H(5,𝔽2)×C6712yes—
572C8—42no—
573C8—42no—
574C22—22no—
575C22—22no—
576C22—24no—
577C22—22no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—41no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22—42no—
589C22C2×#120×C6711yescharacteristic
590C22C2×#123×C6711yescharacteristic
591C22#534×C6711yescharacteristic
592C22#577×C6712yes—
593C22#578×C6712yes—
594C4—22no—
595C4—22no—
596C4—42no—
597C4—42no—
598C4—41no—
599C4—42no—
600C4—42no—
601C4—82no—
602C4—82no—
603C4#546×C6712yes—
604C4#546×C6712yes—
605C2—42no—
606C2—42no—
607C2—81no—
608C2C67×#240511yescharacteristic
609C2C67×#301011yescharacteristic
610C2C67×#301111yescharacteristic
611C2C67×#309712yes—
612C2C67×#320112yes—
subgroupG/H
center, socleC23×C67C22×D4
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC22×#41×C67C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C67 ◁ C22×#41×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#484×C67C2semidirect
3. C2×#483×C67C2semidirect
5. C67×#2036C2semidirect
6. C67×#2035C2semidirect
7. C67×#1873C2semidirect
8. C67×#1873C2semidirect
9. C67×#1854C2semidirect
10. C67×#1787C2semidirect
298. #24014C67direct (N × H)
308. C2×#484C2×C67semidirect
309. C2×#483C2×C67semidirect
311. #2036C2×C67semidirect
312. #2035C2×C67semidirect
313. #1873C2×C67semidirect
314. #1873C2×C67semidirect
315. #1854C2×C67semidirect
316. #1787C2×C67semidirect
318. C67#24014direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#536×C672
C2×#484×C672
C2×#483×C672
C67×#23112
C67×#20362
C67×#20352
C67×#18732
C67×#18542
C67×#17872
C67×#17802
#2401467