Finite Group Databasebeta

← C67×#22406 #30701657

Subgroups — C67×#22406 #30701657

Order 17152.

1474 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547×C67C211yesmaximal, characteristic
2C2×#410×C67C212yesmaximal, complemented
3C67×#2344C212yesmaximal, complemented
4C67×#1964C212yesmaximal, complemented
5C67×#1899C214yesmaximal, complemented
6C67×#1799C211yesmaximal, characteristic, complemented
7C67×#1782C212yesmaximal, complemented
8C4×#99×C67C211yesmaximal, characteristic, complemented
9C22×#41×C67C2212yes—
10#547×C67—22no—
11#547×C67—24no—
12#547×C67—22no—
13C2×C4⋊Q8×C67C411yescharacteristic, complemented
14C2×#123×C67C412yescomplemented
15C2×#123×C67C2212yes—
16C2×C22⋊Q8×C67C412yescomplemented
17C2×#121×C67C411yescharacteristic, complemented
18C2×#121×C67C2211yescharacteristic
19C2×C4×D4×C67C411yescharacteristic, complemented
20C2×C4×D4×C67C2211yescharacteristic
21C2×#117×C67C2214yes—
22C22×C4⋊C4×C67C411yescharacteristic, complemented
23C22×C4⋊C4×C67C2211yescharacteristic
24#410×C67—28no—
25C2×#99×C67C2211yescharacteristic
26C2×#99×C67C2212yes—
27C2×#99×C67C2214yes—
28C2×#99×C67C2214yes—
29C2×#99×C67C2211yescharacteristic
30#389×C67—22no—
31#387×C67—24no—
32#381×C67C2214yes—
33#380×C67C2212yes—
34C4×C4⋊C4×C67—22no—
35C4×C22⋊C4×C67C2212yes—
36#342×C67C2212yes—
37#342×C67C2212yes—
38C2×#41×C67—24no—
39C2×#41×C67—24no—
40C2×#41×C67—24no—
41C2×#41×C67—24no—
42C2×#41×C67—44no—
43C22×Q8×C67C2×C412yes—
44C22×D4×C67C2×C412yes—
45C22×D4×C67C2×C411yescharacteristic
46C22×D4×C67C2312yes—
47C22×D4×C67C2311yescharacteristic
48C23×C4×C67C2×C412yes—
49C23×C4×C67C2×C412yes—
50C23×C4×C67C2×C411yescharacteristic
51C23×C4×C67C2312yes—
52C23×C4×C67C2311yescharacteristic
53C4⋊Q8×C67—22no—
54C4⋊Q8×C67—22no—
55#123×C67—24no—
56#123×C67—24no—
57#123×C67—24no—
58#123×C67—24no—
59C22⋊Q8×C67—24no—
60C22⋊Q8×C67—24no—
61#121×C67—22no—
62#121×C67—22no—
63#121×C67—22no—
64#121×C67—22no—
65C4×D4×C67—22no—
66C4×D4×C67—22no—
67C4×D4×C67—22no—
68C4×D4×C67—21no—
69C4×D4×C67—21no—
70#117×C67—28no—
71#117×C67—44no—
72C2×C4⋊C4×C67—22no—
73C2×C4⋊C4×C67—21no—
74C2×C4⋊C4×C67—21no—
75C2×C4⋊C4×C67—24no—
76C2×C4⋊C4×C67—22no—
77C2×C4⋊C4×C67C2×C411yescharacteristic
78C2×C4⋊C4×C67C2×C412yes—
79C2×C4⋊C4×C67C2×C411yescharacteristic
80C2×C4⋊C4×C67C2×C412yes—
81C2×C4⋊C4×C67C2×C411yescharacteristic
82C2×C4⋊C4×C67C2×C411yescharacteristic
83C2×C4⋊C4×C67C2×C412yes—
84C2×C4⋊C4×C67D412yes—
85C2×C4⋊C4×C67D414yes—
86C2×C4⋊C4×C67D412yes—
87C2×C22⋊C4×C67C2×C412yes—
88C2×C22⋊C4×C67C2×C411yescharacteristic
89C2×C22⋊C4×C67C2×C412yes—
90C2×C22⋊C4×C67C2×C412yes—
91C2×C22⋊C4×C67C2314yes—
92C2×C22⋊C4×C67C2312yes—
93C2×C22⋊C4×C67C2311yescharacteristic
94C2×C22⋊C4×C67C2312yes—
95C2×C42×C67—24no—
96C2×C42×C67—22no—
97C2×C42×C67C2×C411yescharacteristic
98#99×C67—24no—
99#99×C67—22no—
100#99×C67—28no—
101#99×C67—44no—
102#99×C67D414yescomplemented
103#95×C67—22no—
104#95×C67—24no—
105#95×C67—22no—
106#95×C67—24no—
107C24×C67C22×C412yes—
108C24×C67C22×C411yescharacteristic
109C24×C67C2411yescharacteristic
110#41×C67—48no—
111#41×C67—48no—
112#41×C67—48no—
113#41×C67—48no—
114C2×Q8×C67—24no—
115C2×Q8×C67—24no—
116C2×Q8×C67—42no—
117C2×D4×C67—22no—
118C2×D4×C67—22no—
119C2×D4×C67—24no—
120C2×D4×C67—24no—
121C2×D4×C67—22no—
122C2×D4×C67—22no—
123C2×D4×C67—24no—
124C2×D4×C67—21no—
125C2×D4×C67—21no—
126C2×D4×C67—24no—
127C2×D4×C67—42no—
128C2×D4×C67—42no—
129C2×D4×C67C2×D412yes—
130C2×D4×C67#4112yes—
131C22×C4×C67—22no—
132C22×C4×C67—22no—
133C22×C4×C67—24no—
134C22×C4×C67—24no—
135C22×C4×C67—22no—
136C22×C4×C67—21no—
137C22×C4×C67—21no—
138C22×C4×C67—22no—
139C22×C4×C67—22no—
140C22×C4×C67—22no—
141C22×C4×C67—22no—
142C22×C4×C67—24no—
143C22×C4×C67—24no—
144C22×C4×C67—22no—
145C22×C4×C67—42no—
146C22×C4×C67—42no—
147C22×C4×C67—42no—
148C22×C4×C67—42no—
149C22×C4×C67C22×C412yes—
150C22×C4×C67C22×C411yescharacteristic
151C22×C4×C67C22×C412yes—
152C22×C4×C67C22×C412yes—
153C22×C4×C67C22×C411yescharacteristic
154C22×C4×C67C22×C411yescharacteristic
155C22×C4×C67C22×C411yescharacteristic
156C22×C4×C67C22×C411yescharacteristic
157C22×C4×C67C2×D412yes—
158C22×C4×C67C2×D414yes—
159C22×C4×C67C2×D414yes—
160C22×C4×C67C2×D412yes—
161C4⋊C4×C67—21no—
162C4⋊C4×C67—21no—
163C4⋊C4×C67—22no—
164C4⋊C4×C67—22no—
165C4⋊C4×C67—22no—
166C4⋊C4×C67—24no—
167C4⋊C4×C67—48no—
168C4⋊C4×C67—44no—
169C4⋊C4×C67C22⋊C412yes—
170C4⋊C4×C67C22⋊C411yescharacteristic
171C4⋊C4×C67C22⋊C411yescharacteristic
172C4⋊C4×C67C22⋊C412yes—
173C4⋊C4×C67C22⋊C412yes—
174C4⋊C4×C67C22⋊C414yes—
175C4⋊C4×C67C22⋊C414yes—
176C22⋊C4×C67—22no—
177C22⋊C4×C67—22no—
178C22⋊C4×C67—22no—
179C22⋊C4×C67—22no—
180C22⋊C4×C67—22no—
181C22⋊C4×C67—24no—
182C22⋊C4×C67—24no—
183C22⋊C4×C67—24no—
184C22⋊C4×C67—22no—
185C22⋊C4×C67—24no—
186C22⋊C4×C67—22no—
187C22⋊C4×C67C2×D414yes—
188C42×C67—21no—
189C42×C67—21no—
190C42×C67—48no—
191C42×C67—44no—
192C23×C67—21no—
193C23×C67—21no—
194C23×C67—22no—
195C23×C67—22no—
196C23×C67—22no—
197C23×C67—41no—
198C23×C67—41no—
199C23×C67—41no—
200C23×C67—41no—
201C23×C67—42no—
202C23×C67—42no—
203C23×C67C4×D412yes—
204C23×C67C23×C411yescharacteristic
205C23×C67C22×D412yes—
206C23×C67C22×D411yescharacteristic
207C23×C67C2×#4111yescharacteristic
208C23×C67H(5,𝔽2)12yes—
209Q8×C67—48no—
210D4×C67—24no—
211D4×C67—24no—
212D4×C67—48no—
213D4×C67—42no—
214D4×C67—44no—
215D4×C67—44no—
216D4×C67—42no—
217C2×C4×C67—21no—
218C2×C4×C67—22no—
219C2×C4×C67—22no—
220C2×C4×C67—22no—
221C2×C4×C67—22no—
222C2×C4×C67—21no—
223C2×C4×C67—24no—
224C2×C4×C67—24no—
225C2×C4×C67—24no—
226C2×C4×C67—24no—
227C2×C4×C67—24no—
228C2×C4×C67—22no—
229C2×C4×C67—21no—
230C2×C4×C67—21no—
231C2×C4×C67—22no—
232C2×C4×C67—22no—
233C2×C4×C67—22no—
234C2×C4×C67—24no—
235C2×C4×C67—41no—
236C2×C4×C67—42no—
237C2×C4×C67—44no—
238C2×C4×C67—44no—
239C2×C4×C67—44no—
240C2×C4×C67—44no—
241C2×C4×C67—42no—
242C2×C4×C67—42no—
243C2×C4×C67—42no—
244C2×C4×C67—42no—
245C2×C4×C67—41no—
246C2×C4×C67—41no—
247C2×C4×C67—42no—
248C2×C4×C67—44no—
249C2×C4×C67—44no—
250C2×C4×C67—42no—
251C2×C4×C67—44no—
252C2×C4×C67—42no—
253C2×C4×C67C2×C22⋊C411yescharacteristic
254C2×C4×C67C2×C22⋊C412yes—
255C2×C4×C67C2×C22⋊C412yes—
256C2×C4×C67C2×C22⋊C412yes—
257C2×C4×C67C2×C22⋊C414yes—
258C2×C4×C67C2×C22⋊C411yescharacteristic
259C2×C4×C67C4×D412yes—
260C22×C67—22no—
261C22×C67—21no—
262C22×C67—22no—
263C22×C67—21no—
264C22×C67—22no—
265C22×C67—41no—
266C22×C67—41no—
267C22×C67—42no—
268C22×C67—42no—
269C22×C67—42no—
270C22×C67—41no—
271C22×C67—41no—
272C22×C67—42no—
273C22×C67—42no—
274C22×C67—42no—
275C22×C67—82no—
276C22×C67—82no—
277C22×C67C22×C22⋊C411yescharacteristic
278C22×C67C2×C4×D411yescharacteristic
279C22×C67#51811yescharacteristic
280C22×C67D4212yes—
281C22×C67#54612yes—
282C4×C67—22no—
283C4×C67—22no—
284C4×C67—24no—
285C4×C67—22no—
286C4×C67—21no—
287C4×C67—22no—
288C4×C67—44no—
289C4×C67—42no—
290C4×C67—41no—
291C4×C67—41no—
292C4×C67—84no—
293C4×C67—82no—
294C4×C67#41012yescomplemented
295#22406C6711yesmaximal, characteristic, direct product factor, Sylow
296C2×C67—22no—
297C2×C67—22no—
298C2×C67—21no—
299C2×C67—21no—
300C2×C67—42no—
301C2×C67—82no—
302C2×C67D4×C22⋊C411yescharacteristic
303C2×C67C2×#41011yescharacteristic
304C2×C67#288411yescharacteristic
305C2×#547C2×C6711yescharacteristic
306C2×#410C2×C6712yescomplemented
307#2344C2×C6712yescomplemented
308#1964C2×C6712yescomplemented
309#1899C2×C6714yescomplemented
310#1799C2×C6711yescharacteristic, complemented
311#1782C2×C6712yescomplemented
312C4×#99C2×C6711yescharacteristic, complemented
313C67#2240611yescharacteristic, direct product factor, Sylow
314C22×#41C22×C6712yes—
315#547—22no—
316#547—24no—
317#547—22no—
318C2×C4⋊Q8C4×C6711yescharacteristic, complemented
319C2×#123C4×C6712yescomplemented
320C2×#123C22×C6712yes—
321C2×C22⋊Q8C4×C6712yescomplemented
322C2×#121C4×C6711yescharacteristic, complemented
323C2×#121C22×C6711yescharacteristic
324C2×C4×D4C4×C6711yescharacteristic, complemented
325C2×C4×D4C22×C6711yescharacteristic
326C2×#117C22×C6714yes—
327C22×C4⋊C4C4×C6711yescharacteristic, complemented
328C22×C4⋊C4C22×C6711yescharacteristic
329#410—28no—
330C2×#99C22×C6712yes—
331C2×#99C22×C6714yes—
332C2×#99C22×C6711yescharacteristic
333C2×#99C22×C6714yes—
334C2×#99C22×C6711yescharacteristic
335#389—22no—
336#387—24no—
337#381C22×C6714yes—
338#380C22×C6712yes—
339C4×C4⋊C4—22no—
340C4×C22⋊C4C22×C6712yes—
341#342C22×C6712yes—
342#342C22×C6712yes—
343C2×#41—24no—
344C2×#41—24no—
345C2×#41—24no—
346C2×#41—24no—
347C2×#41—44no—
348C22×Q8C2×C4×C6712yes—
349C22×D4C2×C4×C6712yes—
350C22×D4C2×C4×C6711yescharacteristic
351C22×D4C23×C6712yes—
352C22×D4C23×C6711yescharacteristic
353C23×C4C2×C4×C6712yes—
354C23×C4C2×C4×C6712yes—
355C23×C4C2×C4×C6711yescharacteristic
356C23×C4C23×C6712yes—
357C23×C4C23×C6711yescharacteristic
358C4⋊Q8—22no—
359C4⋊Q8—22no—
360#123—24no—
361#123—24no—
362#123—24no—
363#123—24no—
364C22⋊Q8—24no—
365C22⋊Q8—24no—
366#121—22no—
367#121—22no—
368#121—22no—
369#121—22no—
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—21no—
374C4×D4—21no—
375#117—28no—
376#117—44no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4C2×C4×C6711yescharacteristic
383C2×C4⋊C4C2×C4×C6712yes—
384C2×C4⋊C4C2×C4×C6711yescharacteristic
385C2×C4⋊C4C2×C4×C6712yes—
386C2×C4⋊C4C2×C4×C6711yescharacteristic
387C2×C4⋊C4C2×C4×C6711yescharacteristic
388C2×C4⋊C4C2×C4×C6712yes—
389C2×C4⋊C4D4×C6712yes—
390C2×C4⋊C4D4×C6714yes—
391C2×C4⋊C4D4×C6712yes—
392C2×C22⋊C4C2×C4×C6712yes—
393C2×C22⋊C4C2×C4×C6711yescharacteristic
394C2×C22⋊C4C2×C4×C6712yes—
395C2×C22⋊C4C2×C4×C6712yes—
396C2×C22⋊C4C23×C6714yes—
397C2×C22⋊C4C23×C6712yes—
398C2×C22⋊C4C23×C6711yescharacteristic
399C2×C22⋊C4C23×C6712yes—
400C2×C42—24no—
401C2×C42—22no—
402C2×C42C2×C4×C6711yescharacteristic
403#99—24no—
404#99—22no—
405#99—28no—
406#99—44no—
407#99D4×C6714yescomplemented
408#95—22no—
409#95—24no—
410#95—22no—
411#95—24no—
412C24C22×C4×C6712yes—
413C24C22×C4×C6711yescharacteristic
414C24C24×C6711yescharacteristic
415#41—48no—
416#41—48no—
417#41—48no—
418#41—48no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—42no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—24no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4C2×D4×C6712yes—
435C2×D4#41×C6712yes—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4C22×C4×C6712yes—
455C22×C4C22×C4×C6711yescharacteristic
456C22×C4C22×C4×C6712yes—
457C22×C4C22×C4×C6712yes—
458C22×C4C22×C4×C6711yescharacteristic
459C22×C4C22×C4×C6711yescharacteristic
460C22×C4C22×C4×C6711yescharacteristic
461C22×C4C22×C4×C6711yescharacteristic
462C22×C4C2×D4×C6714yes—
463C22×C4C2×D4×C6714yes—
464C22×C4C2×D4×C6712yes—
465C22×C4C2×D4×C6712yes—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4C22⋊C4×C6712yes—
475C4⋊C4C22⋊C4×C6711yescharacteristic
476C4⋊C4C22⋊C4×C6711yescharacteristic
477C4⋊C4C22⋊C4×C6712yes—
478C4⋊C4C22⋊C4×C6712yes—
479C4⋊C4C22⋊C4×C6714yes—
480C4⋊C4C22⋊C4×C6714yes—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—24no—
492C22⋊C4C2×D4×C6714yes—
493C42—21no—
494C42—21no—
495C42—48no—
496C42—44no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23C4×D4×C6712yes—
509C23C23×C4×C6711yescharacteristic
510C23C22×D4×C6712yes—
511C23C22×D4×C6711yescharacteristic
512C23C2×#41×C6711yescharacteristic
513C23H(5,𝔽2)×C6712yes—
514Q8—48no—
515D4—24no—
516D4—24no—
517D4—48no—
518D4—42no—
519D4—44no—
520D4—44no—
521D4—42no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4C2×C22⋊C4×C6711yescharacteristic
559C2×C4C2×C22⋊C4×C6712yes—
560C2×C4C2×C22⋊C4×C6712yes—
561C2×C4C2×C22⋊C4×C6712yes—
562C2×C4C2×C22⋊C4×C6714yes—
563C2×C4C2×C22⋊C4×C6711yescharacteristic
564C2×C4C4×D4×C6712yes—
565C22—22no—
566C22—21no—
567C22—22no—
568C22—21no—
569C22—22no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—41no—
576C22—41no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—82no—
581C22—82no—
582C22C22×C22⋊C4×C6711yescharacteristic
583C22C2×C4×D4×C6711yescharacteristic
584C22#518×C6711yescharacteristic
585C22D42×C6712yes—
586C22#546×C6712yes—
587C4—22no—
588C4—22no—
589C4—24no—
590C4—22no—
591C4—21no—
592C4—22no—
593C4—44no—
594C4—42no—
595C4—41no—
596C4—41no—
597C4—84no—
598C4—82no—
599C4#410×C6712yescomplemented
600C2—22no—
601C2—22no—
602C2—21no—
603C2—21no—
604C2—42no—
605C2—82no—
606C2D4×C22⋊C4×C6711yescharacteristic
607C2C2×#410×C6711yescharacteristic
608C2C67×#288411yescharacteristic
subgroupG/H
center, socleC22×C67C22×C22⋊C4
commutatorC23C23×C4×C67
Frattini, cosocleC24C24×C67
prime radicalC2×#547×C67C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C22×C4⋊C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#410×C67C2semidirect
3. C67×#2344C2semidirect
4. C67×#1964C2semidirect
5. C67×#1899C2semidirect
6. C67×#1799C2semidirect
7. C67×#1782C2semidirect
8. C4×#99×C67C2semidirect
13. C2×C4⋊Q8×C67C4semidirect
14. C2×#123×C67C4semidirect
16. C2×C22⋊Q8×C67C4semidirect
17. C2×#121×C67C4semidirect
19. C2×C4×D4×C67C4semidirect
22. C22×C4⋊C4×C67C4semidirect
102. #99×C67D4semidirect
294. C4×C67#410semidirect
295. #22406C67direct (N × H)
306. C2×#410C2×C67semidirect
307. #2344C2×C67semidirect
308. #1964C2×C67semidirect
309. #1899C2×C67semidirect
310. #1799C2×C67semidirect
311. #1782C2×C67semidirect
312. C4×#99C2×C67semidirect
313. C67#22406direct (N × H)
318. C2×C4⋊Q8C4×C67semidirect
319. C2×#123C4×C67semidirect
321. C2×C22⋊Q8C4×C67semidirect
322. C2×#121C4×C67semidirect
324. C2×C4×D4C4×C67semidirect
327. C22×C4⋊C4C4×C67semidirect
407. #99D4×C67semidirect
599. C4#410×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#547×C672
C2×#410×C672
C67×#23442
C67×#19642
C67×#18992
C67×#17992
C67×#17822
C4×#99×C672
#2240667