Finite Group Databasebeta

← C67×#34262 #30713513

Subgroups — C67×#34262 #30713513

Order 17152.

2198 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#519×C67C211yesmaximal, characteristic
2C67×#2356C213yesmaximal, complemented
3C67×#2355C213yesmaximal, complemented
4C67×#2351C211yesmaximal, characteristic, complemented
5Q8×C4⋊C4×C67C211yesmaximal, characteristic, complemented
6C67×#2347C213yesmaximal
7C67×#2346C213yesmaximal
8Q8×C22⋊C4×C67C211yesmaximal, characteristic
9Q8×C22⋊C4×C67C211yesmaximal, characteristic
10C67×#2320C213yesmaximal, complemented
11C67×#2314C213yesmaximal
12C67×#2298C213yesmaximal
13C67×#2295C213yesmaximal, complemented
14C67×#2276C211yesmaximal, characteristic, complemented
15C42×Q8×C67C211yesmaximal, characteristic, complemented
16C23×Q8×C67C2211yescharacteristic
17#519×C67C4116yescomplemented
18C2×C4×Q8×C67C2211yescharacteristic
19C2×C4×Q8×C67C2211yescharacteristic
20C2×C4×Q8×C67C2211yescharacteristic
21C2×C4×Q8×C67C2211yescharacteristic
22C2×C4×Q8×C67C2211yescharacteristic
23C2×C4×Q8×C67C2213yes—
24C2×C4×Q8×C67C2211yescharacteristic
25C2×C4×Q8×C67C2213yes—
26C2×#117×C67C2213yes—
27C2×#117×C67C2213yes—
28#391×C67C2213yes—
29#391×C67C2213yes—
30#391×C67C2213yes—
31#391×C67C2213yes—
32#389×C67C2213yes—
33#389×C67C2213yes—
34#389×C67C2213yes—
35#389×C67C2213yes—
36#388×C67C2211yescharacteristic
37#388×C67C2213yes—
38#388×C67C2213yes—
39#388×C67C2213yes—
40#388×C67C2216yes—
41#387×C67C2213yes—
42#387×C67C2216yes—
43#387×C67C2213yes—
44#387×C67C2213yes—
45#387×C67C2213yes—
46#387×C67C2216yes—
47#384×C67C2213yes—
48#383×C67C2216yes—
49#382×C67C2213yes—
50#382×C67C2216yes—
51#380×C67C2213yes—
52#380×C67C2213yes—
53C4×C4⋊C4×C67C2213yes—
54C4×C4⋊C4×C67C2213yes—
55C4×C4⋊C4×C67C2213yes—
56C4×C4⋊C4×C67C2213yes—
57C4×C4⋊C4×C67C2213yes—
58C4×C4⋊C4×C67C2213yes—
59C4×C4⋊C4×C67C2216yes—
60C4×C4⋊C4×C67C2213yes—
61C4×C4⋊C4×C67C2213yes—
62C4×C22⋊C4×C67C2213yes—
63C4×C22⋊C4×C67C2213yes—
64#376×C67C2213yes—
65#376×C67C2216yes—
66C43×C67C2213yes—
67C22×Q8×C67—24no—
68C22×Q8×C67—24no—
69C22×Q8×C67C2×C418yes—
70C22×Q8×C67C2313yes—
71C22×Q8×C67C2311yescharacteristic
72C23×C4×C67C2313yes—
73C4×Q8×C67—24no—
74C4×Q8×C67—24no—
75C4×Q8×C67—24no—
76C4×Q8×C67—24no—
77C4×Q8×C67C2×C4124yes—
78C4×Q8×C67C2×C4124yes—
79C4×Q8×C67C2×C418yes—
80C4×Q8×C67C2×C418yes—
81#117×C67C2×C4124yes—
82#117×C67C2×C4124yes—
83C2×C4⋊C4×C67C2313yes—
84C2×C4⋊C4×C67C2313yes—
85C2×C4⋊C4×C67C2313yes—
86C2×C4⋊C4×C67C2313yes—
87C2×C4⋊C4×C67C2311yescharacteristic
88C2×C4⋊C4×C67C2313yes—
89C2×C4⋊C4×C67C2313yes—
90C2×C4⋊C4×C67C2316yes—
91C2×C4⋊C4×C67C2313yes—
92C2×C4⋊C4×C67C2313yes—
93C2×C4⋊C4×C67C2313yes—
94C2×C4⋊C4×C67C2313yes—
95C2×C4⋊C4×C67C2313yes—
96C2×C22⋊C4×C67C2311yescharacteristic
97C2×C22⋊C4×C67C2311yescharacteristic
98C2×C22⋊C4×C67C2313yes—
99C2×C22⋊C4×C67C2313yes—
100C2×C22⋊C4×C67C2311yescharacteristic
101C2×C22⋊C4×C67C2313yes—
102C2×C42×C67C2313yes—
103C2×C42×C67C2313yes—
104C2×C42×C67C2313yes—
105C2×C42×C67C2313yes—
106C2×C42×C67C2311yescharacteristic
107C2×C42×C67C2313yes—
108C2×C42×C67C2313yes—
109C2×C42×C67C2313yes—
110C2×C42×C67C2316yes—
111C2×C42×C67C2313yes—
112C2×C42×C67C2316yes—
113C2×C42×C67C2313yes—
114#95×C67C2311yescharacteristic
115#95×C67C2313yes—
116#95×C67C2313yes—
117#95×C67C2313yes—
118#95×C67C2313yes—
119#95×C67C2313yes—
120#95×C67C2316yes—
121#95×C67C2316yes—
122#95×C67C2313yes—
123#95×C67C2313yes—
124#95×C67C2313yes—
125#95×C67C2316yes—
126#95×C67C2311yescharacteristic
127#95×C67C2316yes—
128#95×C67C2313yes—
129#95×C67C2313yes—
130C24×C67C2411yescharacteristic
131C2×Q8×C67—232no—
132C2×Q8×C67—26no—
133C2×Q8×C67—26no—
134C2×Q8×C67C22×C4112yes—
135C2×Q8×C67C22×C414yes—
136C2×Q8×C67#4114yes—
137C2×Q8×C67#4114yes—
138C22×C4×C67—26no—
139C22×C4×C67—26no—
140C22×C4×C67C22×C4112yes—
141C22×C4×C67C2413yes—
142C22×C4×C67C2413yes—
143C22×C4×C67C2411yescharacteristic
144C22×C4×C67C2413yes—
145C22×C4×C67C2413yes—
146C22×C4×C67C2411yescharacteristic
147C22×C4×C67C2411yescharacteristic
148C22×C4×C67C2413yes—
149C22×C4×C67C2413yes—
150C22×C4×C67C2411yescharacteristic
151C22×C4×C67C2413yes—
152C22×C4×C67C2411yescharacteristic
153C22×C4×C67C2411yescharacteristic
154C22×C4×C67C2413yes—
155C4⋊C4×C67—26no—
156C4⋊C4×C67—26no—
157C4⋊C4×C67—26no—
158C4⋊C4×C67—26no—
159C4⋊C4×C67—22no—
160C4⋊C4×C67—26no—
161C4⋊C4×C67—43no—
162C4⋊C4×C67—43no—
163C4⋊C4×C67—46no—
164C4⋊C4×C67C22×C4112yes—
165C4⋊C4×C67C22×C4112yes—
166C4⋊C4×C67C22×C4112yes—
167C4⋊C4×C67C22×C4112yes—
168C22⋊C4×C67—22no—
169C22⋊C4×C67—22no—
170C22⋊C4×C67—43no—
171C22⋊C4×C67—43no—
172C22⋊C4×C67C22×C414yes—
173C22⋊C4×C67C22×C4112yes—
174C42×C67—26no—
175C42×C67—26no—
176C42×C67—26no—
177C42×C67—26no—
178C42×C67—22no—
179C42×C67—26no—
180C42×C67—43no—
181C42×C67—46no—
182C42×C67—43no—
183C42×C67C22×C4124yes—
184C42×C67C22×C4112yes—
185C42×C67C22×C4112yes—
186C23×C67—21no—
187C23×C67—21no—
188C23×C67—21no—
189C23×C67—21no—
190C23×C67—21no—
191C23×C67—21no—
192C23×C67C23×C412yes—
193C23×C67C2511yescharacteristic
194Q8×C67—224no—
195Q8×C67#117116yes—
196C2×C4×C67—21no—
197C2×C4×C67—21no—
198C2×C4×C67—23no—
199C2×C4×C67—23no—
200C2×C4×C67—21no—
201C2×C4×C67—23no—
202C2×C4×C67—21no—
203C2×C4×C67—23no—
204C2×C4×C67—21no—
205C2×C4×C67—23no—
206C2×C4×C67—21no—
207C2×C4×C67—21no—
208C2×C4×C67—21no—
209C2×C4×C67—23no—
210C2×C4×C67—224no—
211C2×C4×C67—21no—
212C2×C4×C67—21no—
213C2×C4×C67—23no—
214C2×C4×C67—23no—
215C2×C4×C67—23no—
216C2×C4×C67—23no—
217C2×C4×C67—23no—
218C2×C4×C67—21no—
219C2×C4×C67—21no—
220C2×C4×C67—23no—
221C2×C4×C67—21no—
222C2×C4×C67—23no—
223C2×C4×C67—23no—
224C2×C4×C67—23no—
225C2×C4×C67—23no—
226C2×C4×C67—23no—
227C2×C4×C67—23no—
228C2×C4×C67—21no—
229C2×C4×C67C23×C416yes—
230C2×C4×C67C23×C412yes—
231C2×C4×C67C23×C416yes—
232C2×C4×C67C23×C412yes—
233C2×C4×C67C23×C416yes—
234C2×C4×C67C23×C416yes—
235C2×C4×C67C2×#4116yes—
236C2×C4×C67C2×#4116yes—
237C2×C4×C672−1+412yes—
238C2×C4×C672−1+412yes—
239C22×C67—22no—
240C22×C67—22no—
241C22×C67—22no—
242C22×C67—41no—
243C22×C67—41no—
244C22×C67—41no—
245C22×C67—41no—
246C22×C67C24×C411yescharacteristic
247C22×C67C22×#4111yescharacteristic
248C22×C67C22×#4111yescharacteristic
249C22×C67C2×2−1+411yescharacteristic
250C22×C67C2×2−1+411yescharacteristic
251C22×C67#58511yescharacteristic
252C22×C67#58511yescharacteristic
253C4×C67—26no—
254C4×C67—22no—
255C4×C67—26no—
256C4×C67—26no—
257C4×C67—22no—
258C4×C67—22no—
259C4×C67—22no—
260C4×C67—43no—
261C4×C67—41no—
262C4×C67—43no—
263C4×C67—41no—
264C4×C67—43no—
265C4×C67—43no—
266C4×C67C2×#117112yes—
267#34262C6711yesmaximal, characteristic, direct product factor, Sylow
268C2×C67—42no—
269C2×C67C22×#11711yescharacteristic
270C2×C67C4×2−1+411yescharacteristic
271C2×C67C4×2−1+411yescharacteristic
272C2×C67#351611yescharacteristic
273C2×C67#351811yescharacteristic
274C2×C67#352311yescharacteristic
275C2×C67#352411yescharacteristic
276C2×#519C2×C6711yescharacteristic
277#2356C2×C6713yescomplemented
278#2355C2×C6713yescomplemented
279#2351C2×C6711yescharacteristic, complemented
280Q8×C4⋊C4C2×C6711yescharacteristic, complemented
281#2347C2×C6713yes—
282#2346C2×C6713yes—
283Q8×C22⋊C4C2×C6711yescharacteristic
284Q8×C22⋊C4C2×C6711yescharacteristic
285#2320C2×C6713yescomplemented
286#2314C2×C6713yes—
287#2298C2×C6713yes—
288#2295C2×C6713yescomplemented
289#2276C2×C6711yescharacteristic, complemented
290C42×Q8C2×C6711yescharacteristic, complemented
291C67#3426211yescharacteristic, direct product factor, Sylow
292C23×Q8C22×C6711yescharacteristic
293#519C4×C67116yescomplemented
294C2×C4×Q8C22×C6711yescharacteristic
295C2×C4×Q8C22×C6711yescharacteristic
296C2×C4×Q8C22×C6711yescharacteristic
297C2×C4×Q8C22×C6711yescharacteristic
298C2×C4×Q8C22×C6713yes—
299C2×C4×Q8C22×C6711yescharacteristic
300C2×C4×Q8C22×C6713yes—
301C2×C4×Q8C22×C6711yescharacteristic
302C2×#117C22×C6713yes—
303C2×#117C22×C6713yes—
304#391C22×C6713yes—
305#391C22×C6713yes—
306#391C22×C6713yes—
307#391C22×C6713yes—
308#389C22×C6713yes—
309#389C22×C6713yes—
310#389C22×C6713yes—
311#389C22×C6713yes—
312#388C22×C6713yes—
313#388C22×C6716yes—
314#388C22×C6713yes—
315#388C22×C6713yes—
316#388C22×C6711yescharacteristic
317#387C22×C6716yes—
318#387C22×C6716yes—
319#387C22×C6713yes—
320#387C22×C6713yes—
321#387C22×C6713yes—
322#387C22×C6713yes—
323#384C22×C6713yes—
324#383C22×C6716yes—
325#382C22×C6713yes—
326#382C22×C6716yes—
327#380C22×C6713yes—
328#380C22×C6713yes—
329C4×C4⋊C4C22×C6713yes—
330C4×C4⋊C4C22×C6713yes—
331C4×C4⋊C4C22×C6713yes—
332C4×C4⋊C4C22×C6713yes—
333C4×C4⋊C4C22×C6713yes—
334C4×C4⋊C4C22×C6716yes—
335C4×C4⋊C4C22×C6713yes—
336C4×C4⋊C4C22×C6713yes—
337C4×C4⋊C4C22×C6713yes—
338C4×C22⋊C4C22×C6713yes—
339C4×C22⋊C4C22×C6713yes—
340#376C22×C6713yes—
341#376C22×C6716yes—
342C43C22×C6713yes—
343C22×Q8—24no—
344C22×Q8—24no—
345C22×Q8C2×C4×C6718yes—
346C22×Q8C23×C6713yes—
347C22×Q8C23×C6711yescharacteristic
348C23×C4C23×C6713yes—
349C4×Q8—24no—
350C4×Q8—24no—
351C4×Q8—24no—
352C4×Q8—24no—
353C4×Q8C2×C4×C67124yes—
354C4×Q8C2×C4×C67124yes—
355C4×Q8C2×C4×C6718yes—
356C4×Q8C2×C4×C6718yes—
357#117C2×C4×C67124yes—
358#117C2×C4×C67124yes—
359C2×C4⋊C4C23×C6713yes—
360C2×C4⋊C4C23×C6713yes—
361C2×C4⋊C4C23×C6713yes—
362C2×C4⋊C4C23×C6713yes—
363C2×C4⋊C4C23×C6711yescharacteristic
364C2×C4⋊C4C23×C6713yes—
365C2×C4⋊C4C23×C6713yes—
366C2×C4⋊C4C23×C6716yes—
367C2×C4⋊C4C23×C6713yes—
368C2×C4⋊C4C23×C6713yes—
369C2×C4⋊C4C23×C6713yes—
370C2×C4⋊C4C23×C6713yes—
371C2×C4⋊C4C23×C6713yes—
372C2×C22⋊C4C23×C6711yescharacteristic
373C2×C22⋊C4C23×C6711yescharacteristic
374C2×C22⋊C4C23×C6713yes—
375C2×C22⋊C4C23×C6713yes—
376C2×C22⋊C4C23×C6711yescharacteristic
377C2×C22⋊C4C23×C6713yes—
378C2×C42C23×C6713yes—
379C2×C42C23×C6713yes—
380C2×C42C23×C6713yes—
381C2×C42C23×C6713yes—
382C2×C42C23×C6711yescharacteristic
383C2×C42C23×C6713yes—
384C2×C42C23×C6713yes—
385C2×C42C23×C6713yes—
386C2×C42C23×C6716yes—
387C2×C42C23×C6713yes—
388C2×C42C23×C6716yes—
389C2×C42C23×C6713yes—
390#95C23×C6711yescharacteristic
391#95C23×C6713yes—
392#95C23×C6713yes—
393#95C23×C6713yes—
394#95C23×C6713yes—
395#95C23×C6713yes—
396#95C23×C6716yes—
397#95C23×C6716yes—
398#95C23×C6713yes—
399#95C23×C6713yes—
400#95C23×C6713yes—
401#95C23×C6716yes—
402#95C23×C6711yescharacteristic
403#95C23×C6716yes—
404#95C23×C6713yes—
405#95C23×C6713yes—
406C24C24×C6711yescharacteristic
407C2×Q8—232no—
408C2×Q8—26no—
409C2×Q8—26no—
410C2×Q8C22×C4×C67112yes—
411C2×Q8C22×C4×C6714yes—
412C2×Q8#41×C6714yes—
413C2×Q8#41×C6714yes—
414C22×C4—26no—
415C22×C4—26no—
416C22×C4C22×C4×C67112yes—
417C22×C4C24×C6711yescharacteristic
418C22×C4C24×C6711yescharacteristic
419C22×C4C24×C6713yes—
420C22×C4C24×C6713yes—
421C22×C4C24×C6713yes—
422C22×C4C24×C6713yes—
423C22×C4C24×C6713yes—
424C22×C4C24×C6711yescharacteristic
425C22×C4C24×C6713yes—
426C22×C4C24×C6713yes—
427C22×C4C24×C6711yescharacteristic
428C22×C4C24×C6711yescharacteristic
429C22×C4C24×C6711yescharacteristic
430C22×C4C24×C6713yes—
431C4⋊C4—26no—
432C4⋊C4—26no—
433C4⋊C4—26no—
434C4⋊C4—26no—
435C4⋊C4—22no—
436C4⋊C4—26no—
437C4⋊C4—43no—
438C4⋊C4—43no—
439C4⋊C4—46no—
440C4⋊C4C22×C4×C67112yes—
441C4⋊C4C22×C4×C67112yes—
442C4⋊C4C22×C4×C67112yes—
443C4⋊C4C22×C4×C67112yes—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—43no—
447C22⋊C4—43no—
448C22⋊C4C22×C4×C6714yes—
449C22⋊C4C22×C4×C67112yes—
450C42—26no—
451C42—26no—
452C42—26no—
453C42—26no—
454C42—22no—
455C42—26no—
456C42—43no—
457C42—46no—
458C42—43no—
459C42C22×C4×C67124yes—
460C42C22×C4×C67112yes—
461C42C22×C4×C67112yes—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23C23×C4×C6712yes—
469C23C25×C6711yescharacteristic
470Q8—224no—
471Q8#117×C67116yes—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—23no—
475C2×C4—23no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—23no—
479C2×C4—23no—
480C2×C4—21no—
481C2×C4—23no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—23no—
486C2×C4—224no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—23no—
490C2×C4—23no—
491C2×C4—23no—
492C2×C4—23no—
493C2×C4—23no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—23no—
498C2×C4—23no—
499C2×C4—23no—
500C2×C4—23no—
501C2×C4—23no—
502C2×C4—23no—
503C2×C4—23no—
504C2×C4—21no—
505C2×C4C23×C4×C6716yes—
506C2×C4C23×C4×C6712yes—
507C2×C4C23×C4×C6716yes—
508C2×C4C23×C4×C6712yes—
509C2×C4C23×C4×C6716yes—
510C2×C4C23×C4×C6716yes—
511C2×C4C2×#41×C6716yes—
512C2×C4C2×#41×C6716yes—
513C2×C42−1+4×C6712yes—
514C2×C42−1+4×C6712yes—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22C24×C4×C6711yescharacteristic
523C22C22×#41×C6711yescharacteristic
524C22C22×#41×C6711yescharacteristic
525C22C2×2−1+4×C6711yescharacteristic
526C22C2×2−1+4×C6711yescharacteristic
527C22#585×C6711yescharacteristic
528C22#585×C6711yescharacteristic
529C4—22no—
530C4—26no—
531C4—22no—
532C4—26no—
533C4—26no—
534C4—22no—
535C4—22no—
536C4—41no—
537C4—43no—
538C4—43no—
539C4—43no—
540C4—41no—
541C4—43no—
542C4C2×#117×C67112yes—
543C2—42no—
544C2C22×#117×C6711yescharacteristic
545C2C4×2−1+4×C6711yescharacteristic
546C2C4×2−1+4×C6711yescharacteristic
547C2C67×#351611yescharacteristic
548C2C67×#351811yescharacteristic
549C2C67×#352311yescharacteristic
550C2C67×#352411yescharacteristic
subgroupG/H
center, socleC23×C67C25
commutatorC22C24×C4×C67
Frattini, cosocleC23C25×C67
prime radicalC24×C67C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2356C2semidirect
3. C67×#2355C2semidirect
4. C67×#2351C2semidirect
5. Q8×C4⋊C4×C67C2semidirect
10. C67×#2320C2semidirect
13. C67×#2295C2semidirect
14. C67×#2276C2semidirect
15. C42×Q8×C67C2semidirect
17. #519×C67C4semidirect
267. #34262C67direct (N × H)
277. #2356C2×C67semidirect
278. #2355C2×C67semidirect
279. #2351C2×C67semidirect
280. Q8×C4⋊C4C2×C67semidirect
285. #2320C2×C67semidirect
288. #2295C2×C67semidirect
289. #2276C2×C67semidirect
290. C42×Q8C2×C67semidirect
291. C67#34262direct (N × H)
293. #519C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#519×C672
C67×#23562
C67×#23552
C67×#23512
Q8×C4⋊C4×C672
C67×#23472
C67×#23462
Q8×C22⋊C4×C672
C67×#23202
C67×#23142
C67×#22982
C67×#22952
C67×#22762
C42×Q8×C672
#3426267