Finite Group Databasebeta

← C67×#10960 #30690211

Subgroups — C67×#10960 #30690211

Order 17152.

1526 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#99×C67C212yesmaximal
2C2×#386×C67C211yesmaximal, characteristic
3C2×C4×C22⋊C4×C67C211yesmaximal, characteristic, complemented
4C2×#342×C67C212yesmaximal, complemented
5C67×C22⋊#95C211yesmaximal, characteristic, complemented
6C23×D4×C67C2211yescharacteristic
7C24×C4×C67C2211yescharacteristic
8C22×C22⋊C4×C67C412yescomplemented
9C22×C22⋊C4×C67C2211yescharacteristic
10C22×C22⋊C4×C67C2212yes—
11C22×C22⋊C4×C67C2212yes—
12C22×C42×C67—21no—
13C2×#99×C67—24no—
14C2×#99×C67—24no—
15C2×#99×C67C418yescomplemented
16#386×C67—21no—
17#386×C67—22no—
18#386×C67—21no—
19C4×C22⋊C4×C67—24no—
20C2×#95×C67—21no—
21C2×#95×C67—22no—
22C2×#95×C67C412yescomplemented
23#342×C67—24no—
24C25×C67C2×C411yescharacteristic
25C25×C67C2311yescharacteristic
26C22×D4×C67—21no—
27C22×D4×C67—21no—
28C22×D4×C67—22no—
29C22×D4×C67—22no—
30C22×D4×C67—22no—
31C22×D4×C67—22no—
32C22×D4×C67—22no—
33C22×D4×C67C2×C414yes—
34C23×C4×C67—22no—
35C23×C4×C67—22no—
36C23×C4×C67—22no—
37C23×C4×C67—22no—
38C23×C4×C67—44no—
39C23×C4×C67C2×C411yescharacteristic
40C23×C4×C67C2×C412yes—
41C23×C4×C67C2×C412yes—
42C23×C4×C67D411yescharacteristic
43C23×C4×C67D412yes—
44C23×C4×C67D411yescharacteristic
45C2×C22⋊C4×C67—22no—
46C2×C22⋊C4×C67—22no—
47C2×C22⋊C4×C67—22no—
48C2×C22⋊C4×C67—22no—
49C2×C22⋊C4×C67—22no—
50C2×C22⋊C4×C67—22no—
51C2×C22⋊C4×C67—22no—
52C2×C22⋊C4×C67—22no—
53C2×C22⋊C4×C67—21no—
54C2×C22⋊C4×C67—21no—
55C2×C22⋊C4×C67—22no—
56C2×C22⋊C4×C67—22no—
57C2×C22⋊C4×C67—22no—
58C2×C22⋊C4×C67—22no—
59C2×C22⋊C4×C67—44no—
60C2×C22⋊C4×C67C2×C414yes—
61C2×C22⋊C4×C67C2×C414yes—
62C2×C22⋊C4×C67D414yes—
63C2×C22⋊C4×C67D414yes—
64C2×C42×C67—24no—
65C2×C42×C67—44no—
66#99×C67—216no—
67#95×C67—24no—
68#95×C67—22no—
69#95×C67—22no—
70#95×C67—44no—
71C24×C67—21no—
72C24×C67—21no—
73C24×C67—22no—
74C24×C67—22no—
75C24×C67—42no—
76C24×C67—42no—
77C24×C67—42no—
78C24×C67—42no—
79C24×C67C4212yescomplemented
80C24×C67C22×C411yescharacteristic
81C24×C67C22×C412yes—
82C24×C67C2×D411yescharacteristic
83C24×C67C2×D411yescharacteristic
84C24×C67C2×D412yes—
85C24×C67C2×D412yes—
86C24×C67#4112yes—
87C24×C67#4112yes—
88C24×C67#4112yes—
89C2×D4×C67—24no—
90C2×D4×C67—24no—
91C2×D4×C67—24no—
92C2×D4×C67—24no—
93C2×D4×C67—24no—
94C2×D4×C67—24no—
95C2×D4×C67—41no—
96C2×D4×C67—42no—
97C2×D4×C67—42no—
98C2×D4×C67—42no—
99C2×D4×C67—41no—
100C2×D4×C67—42no—
101C2×D4×C67—42no—
102C2×D4×C67—41no—
103C2×D4×C67—42no—
104C2×D4×C67—41no—
105C22×C4×C67—21no—
106C22×C4×C67—21no—
107C22×C4×C67—24no—
108C22×C4×C67—21no—
109C22×C4×C67—21no—
110C22×C4×C67—24no—
111C22×C4×C67—24no—
112C22×C4×C67—22no—
113C22×C4×C67—22no—
114C22×C4×C67—24no—
115C22×C4×C67—21no—
116C22×C4×C67—24no—
117C22×C4×C67—24no—
118C22×C4×C67—21no—
119C22×C4×C67—24no—
120C22×C4×C67—24no—
121C22×C4×C67—22no—
122C22×C4×C67—24no—
123C22×C4×C67—24no—
124C22×C4×C67—42no—
125C22×C4×C67—42no—
126C22×C4×C67—42no—
127C22×C4×C67—48no—
128C22×C4×C67—48no—
129C22×C4×C67—44no—
130C22×C4×C67—44no—
131C22×C4×C67—44no—
132C22×C4×C67—44no—
133C22×C4×C67—42no—
134C22×C4×C67—42no—
135C22×C4×C67—42no—
136C22×C4×C67—42no—
137C22×C4×C67C4212yescomplemented
138C22×C4×C67C22⋊C412yes—
139C22×C4×C67C22⋊C412yes—
140C22⋊C4×C67—24no—
141C22⋊C4×C67—24no—
142C22⋊C4×C67—24no—
143C22⋊C4×C67—24no—
144C22⋊C4×C67—24no—
145C22⋊C4×C67—44no—
146C22⋊C4×C67—42no—
147C22⋊C4×C67—42no—
148C22⋊C4×C67—42no—
149C22⋊C4×C67—41no—
150C22⋊C4×C67—41no—
151C22⋊C4×C67—44no—
152C22⋊C4×C67C22⋊C418yescomplemented
153C42×C67—48no—
154C23×C67—22no—
155C23×C67—22no—
156C23×C67—22no—
157C23×C67—22no—
158C23×C67—22no—
159C23×C67—22no—
160C23×C67—22no—
161C23×C67—24no—
162C23×C67—42no—
163C23×C67—42no—
164C23×C67—44no—
165C23×C67—42no—
166C23×C67—44no—
167C23×C67—42no—
168C23×C67—44no—
169C23×C67—44no—
170C23×C67—44no—
171C23×C67—44no—
172C23×C67—41no—
173C23×C67—42no—
174C23×C67—41no—
175C23×C67—42no—
176C23×C67—42no—
177C23×C67—82no—
178C23×C67—82no—
179C23×C67—84no—
180C23×C67C2×C4211yescharacteristic
181C23×C67C2×C22⋊C412yes—
182C23×C67C2×C22⋊C411yescharacteristic
183C23×C67#11711yescharacteristic
184C23×C67#11712yes—
185C23×C67C4×D412yes—
186C23×C67C4×D412yes—
187C23×C67C4×D412yes—
188C23×C67C4×D412yes—
189C23×C67C4×D412yes—
190C23×C67C4×D412yes—
191C23×C67#12011yescharacteristic
192C23×C67#12112yes—
193C23×C67#12112yes—
194C23×C67#12112yes—
195C23×C67#12311yescharacteristic
196C23×C67#12312yes—
197C23×C67#12411yescharacteristic
198C23×C67#12412yes—
199C23×C67#12612yes—
200C23×C67#12711yescharacteristic
201D4×C67—44no—
202D4×C67—44no—
203D4×C67—44no—
204D4×C67—44no—
205C2×C4×C67—24no—
206C2×C4×C67—24no—
207C2×C4×C67—22no—
208C2×C4×C67—22no—
209C2×C4×C67—24no—
210C2×C4×C67—22no—
211C2×C4×C67—28no—
212C2×C4×C67—22no—
213C2×C4×C67—22no—
214C2×C4×C67—22no—
215C2×C4×C67—48no—
216C2×C4×C67—48no—
217C2×C4×C67—44no—
218C2×C4×C67—44no—
219C2×C4×C67—48no—
220C2×C4×C67—48no—
221C2×C4×C67—42no—
222C2×C4×C67—42no—
223C2×C4×C67—41no—
224C2×C4×C67—42no—
225C2×C4×C67—41no—
226C2×C4×C67—48no—
227C2×C4×C67—44no—
228C2×C4×C67—42no—
229C2×C4×C67—42no—
230C2×C4×C67—42no—
231C2×C4×C67—82no—
232C2×C4×C67—82no—
233C2×C4×C67—84no—
234C2×C4×C67—84no—
235C2×C4×C67#9914yescomplemented
236C22×C67—21no—
237C22×C67—21no—
238C22×C67—22no—
239C22×C67—22no—
240C22×C67—21no—
241C22×C67—22no—
242C22×C67—22no—
243C22×C67—21no—
244C22×C67—24no—
245C22×C67—22no—
246C22×C67—24no—
247C22×C67—22no—
248C22×C67—24no—
249C22×C67—24no—
250C22×C67—22no—
251C22×C67—22no—
252C22×C67—42no—
253C22×C67—42no—
254C22×C67—42no—
255C22×C67—42no—
256C22×C67—42no—
257C22×C67—42no—
258C22×C67—42no—
259C22×C67—42no—
260C22×C67—42no—
261C22×C67—44no—
262C22×C67—84no—
263C22×C67—84no—
264C22×C67—81no—
265C22×C67—81no—
266C22×C67—82no—
267C22×C67C4×C22⋊C412yes—
268C22×C67C4×C22⋊C411yescharacteristic
269C22×C67#38612yes—
270C22×C67#38611yescharacteristic
271C22×C67#38812yes—
272C22×C67#38812yes—
273C22×C67#38812yes—
274C22×C67#39012yes—
275C22×C67#39011yescharacteristic
276C22×C67C2×#9912yes—
277C22×C67#41012yes—
278C4×C67—24no—
279C4×C67—48no—
280C4×C67—42no—
281C4×C67—42no—
282C4×C67—84no—
283C4×C67—82no—
284#10960C6711yesmaximal, characteristic, direct product factor, Sylow
285C2×C67—22no—
286C2×C67—22no—
287C2×C67—22no—
288C2×C67—22no—
289C2×C67—22no—
290C2×C67—22no—
291C2×C67—82no—
292C2×C67#143711yescharacteristic
293C2×C67C4×#9912yes—
294C2×C67#189811yescharacteristic
295C2×C67#189911yescharacteristic
296C2×C67#196412yes—
297C22×#99C2×C6712yes—
298C2×#386C2×C6711yescharacteristic
299C2×C4×C22⋊C4C2×C6711yescharacteristic, complemented
300C2×#342C2×C6712yescomplemented
301C22⋊#95C2×C6711yescharacteristic, complemented
302C67#1096011yescharacteristic, direct product factor, Sylow
303C23×D4C22×C6711yescharacteristic
304C24×C4C22×C6711yescharacteristic
305C22×C22⋊C4C4×C6712yescomplemented
306C22×C22⋊C4C22×C6711yescharacteristic
307C22×C22⋊C4C22×C6712yes—
308C22×C22⋊C4C22×C6712yes—
309C22×C42—21no—
310C2×#99—24no—
311C2×#99—24no—
312C2×#99C4×C6718yescomplemented
313#386—21no—
314#386—22no—
315#386—21no—
316C4×C22⋊C4—24no—
317C2×#95—21no—
318C2×#95—22no—
319C2×#95C4×C6712yescomplemented
320#342—24no—
321C25C2×C4×C6711yescharacteristic
322C25C23×C6711yescharacteristic
323C22×D4—21no—
324C22×D4—21no—
325C22×D4—22no—
326C22×D4—22no—
327C22×D4—22no—
328C22×D4—22no—
329C22×D4—22no—
330C22×D4C2×C4×C6714yes—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4—22no—
335C23×C4—44no—
336C23×C4C2×C4×C6711yescharacteristic
337C23×C4C2×C4×C6712yes—
338C23×C4C2×C4×C6712yes—
339C23×C4D4×C6711yescharacteristic
340C23×C4D4×C6712yes—
341C23×C4D4×C6711yescharacteristic
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—44no—
357C2×C22⋊C4C2×C4×C6714yes—
358C2×C22⋊C4C2×C4×C6714yes—
359C2×C22⋊C4D4×C6714yes—
360C2×C22⋊C4D4×C6714yes—
361C2×C42—24no—
362C2×C42—44no—
363#99—216no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—44no—
368C24—21no—
369C24—21no—
370C24—22no—
371C24—22no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24C42×C6712yescomplemented
377C24C22×C4×C6711yescharacteristic
378C24C22×C4×C6712yes—
379C24C2×D4×C6711yescharacteristic
380C24C2×D4×C6711yescharacteristic
381C24C2×D4×C6712yes—
382C24C2×D4×C6712yes—
383C24#41×C6712yes—
384C24#41×C6712yes—
385C24#41×C6712yes—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—41no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—41no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—41no—
400C2×D4—42no—
401C2×D4—41no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4C42×C6712yescomplemented
435C22×C4C22⋊C4×C6712yes—
436C22×C4C22⋊C4×C6712yes—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—44no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—44no—
449C22⋊C4C22⋊C4×C6718yescomplemented
450C42—48no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—42no—
463C23—44no—
464C23—44no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—41no—
470C23—42no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—82no—
475C23—82no—
476C23—84no—
477C23C2×C42×C6711yescharacteristic
478C23C2×C22⋊C4×C6712yes—
479C23C2×C22⋊C4×C6711yescharacteristic
480C23#117×C6711yescharacteristic
481C23#117×C6712yes—
482C23C4×D4×C6712yes—
483C23C4×D4×C6712yes—
484C23C4×D4×C6712yes—
485C23C4×D4×C6712yes—
486C23C4×D4×C6712yes—
487C23C4×D4×C6712yes—
488C23#120×C6711yescharacteristic
489C23#121×C6712yes—
490C23#121×C6712yes—
491C23#121×C6712yes—
492C23#123×C6711yescharacteristic
493C23#123×C6712yes—
494C23#124×C6711yescharacteristic
495C23#124×C6712yes—
496C23#126×C6712yes—
497C23#127×C6711yescharacteristic
498D4—44no—
499D4—44no—
500D4—44no—
501D4—44no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—48no—
518C2×C4—44no—
519C2×C4—48no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—48no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—84no—
531C2×C4—84no—
532C2×C4#99×C6714yescomplemented
533C22—24no—
534C22—24no—
535C22—22no—
536C22—22no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—22no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—21no—
545C22—24no—
546C22—22no—
547C22—24no—
548C22—22no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—42no—
559C22—84no—
560C22—84no—
561C22—81no—
562C22—81no—
563C22—82no—
564C22C4×C22⋊C4×C6712yes—
565C22C4×C22⋊C4×C6711yescharacteristic
566C22#386×C6712yes—
567C22#386×C6711yescharacteristic
568C22#388×C6712yes—
569C22#388×C6712yes—
570C22#388×C6712yes—
571C22#390×C6712yes—
572C22#390×C6711yescharacteristic
573C22C2×#99×C6712yes—
574C22#410×C6712yes—
575C4—24no—
576C4—48no—
577C4—42no—
578C4—42no—
579C4—84no—
580C4—82no—
581C2—22no—
582C2—22no—
583C2—22no—
584C2—22no—
585C2—22no—
586C2—22no—
587C2—82no—
588C2C67×#143711yescharacteristic
589C2C4×#99×C6712yes—
590C2C67×#189811yescharacteristic
591C2C67×#189911yescharacteristic
592C2C67×#196412yes—
subgroupG/H
center, socleC23×C67C2×C22⋊C4
commutatorC23C2×C42×C67
Frattini, cosocleC25C23×C67
prime radicalC23×D4×C67C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C67 ◁ C24×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C4×C22⋊C4×C67C2semidirect
4. C2×#342×C67C2semidirect
5. C67×C22⋊#95C2semidirect
8. C22×C22⋊C4×C67C4semidirect
15. C2×#99×C67C4semidirect
22. C2×#95×C67C4semidirect
79. C24×C67C42semidirect
137. C22×C4×C67C42semidirect
152. C22⋊C4×C67C22⋊C4semidirect
235. C2×C4×C67#99semidirect
284. #10960C67direct (N × H)
299. C2×C4×C22⋊C4C2×C67semidirect
300. C2×#342C2×C67semidirect
301. C22⋊#95C2×C67semidirect
302. C67#10960direct (N × H)
305. C22×C22⋊C4C4×C67semidirect
312. C2×#99C4×C67semidirect
319. C2×#95C4×C67semidirect
376. C24C42×C67semidirect
434. C22×C4C42×C67semidirect
449. C22⋊C4C22⋊C4×C67semidirect
532. C2×C4#99×C67semidirect

Maximal subgroups

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

subgroupindex
C22×#99×C672
C2×#386×C672
C2×C4×C22⋊C4×C672
C2×#342×C672
C67×C22⋊#952
#1096067