Finite Group Databasebeta

← C67×#25876 #30705127

Subgroups — C67×#25876 #30705127

Order 17152.

1222 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3499C211yesmaximal, characteristic
2C67×#3141C211yesmaximal, characteristic, complemented
3C67×#3079C211yesmaximal, characteristic, complemented
4C67×#2862C211yesmaximal, characteristic, complemented
5C67×#2038C214yesmaximal, complemented
6C67×#1966C211yesmaximal, characteristic, complemented
7C67×#1962C211yesmaximal, characteristic, complemented
8C67×#1793C214yesmaximal, complemented
9C4×#104×C67C211yesmaximal, characteristic, complemented
10C2×2−1+4×C67C2212yes—
11D4×Q8×C67—21no—
12D4×Q8×C67—21no—
13#547×C67—21no—
14#547×C67—21no—
15#547×C67—22no—
16#547×C67—22no—
17#536×C67—22no—
18#536×C67C2212yes—
19#532×C67—21no—
20#532×C67C2211yescharacteristic
21C2×C4⋊Q8×C67—21no—
22C2×C4⋊Q8×C67C2211yescharacteristic
23C2×C4⋊Q8×C67C2211yescharacteristic
24C4×#41×C67C2211yescharacteristic
25#489×C67—22no—
26#487×C67—21no—
27#486×C67—21no—
28#477×C67—21no—
29#476×C67—21no—
30#475×C67—21no—
31#474×C67—21no—
32#428×C67C2212yes—
33#423×C67C2211yescharacteristic
34C2×#104×C67C2212yes—
35C2×#104×C67C2214yes—
36#417×C67C2214yes—
37#417×C67C2212yes—
38C2×#101×C67C2214yes—
39C4×M16×C67C2211yescharacteristic
40#391×C67—22no—
41#391×C67C2214yes—
42#382×C67—22no—
43C43×C67C2211yescharacteristic
44#339×C67C2212yes—
452−1+4×C67—28no—
462−1+4×C67—44no—
47C2×#41×C67—22no—
48C2×#41×C67—22no—
49C2×#41×C67—24no—
50C2×#41×C67C2312yes—
51C2×#41×C67C2311yescharacteristic
52C22×Q8×C67—22no—
53C22×Q8×C67—22no—
54C22×Q8×C67—21no—
55C22×Q8×C67C2312yes—
56C2×M16×C67C2312yes—
57C2×M16×C67C2312yes—
58C4⋊Q8×C67—22no—
59C4⋊Q8×C67—22no—
60C4⋊Q8×C67—22no—
61C4⋊Q8×C67—21no—
62C4⋊Q8×C67—21no—
63C4⋊Q8×C67—21no—
64C4⋊Q8×C67—21no—
65C4⋊Q8×C67—21no—
66C4⋊Q8×C67—21no—
67C4⋊Q8×C67—21no—
68C4⋊Q8×C67—21no—
69C4⋊Q8×C67D411yescharacteristic
70#127×C67D411yescharacteristic
71#124×C67—22no—
72#124×C67D412yes—
73#123×C67—24no—
74#123×C67—24no—
75#123×C67—24no—
76C22⋊Q8×C67—24no—
77C22⋊Q8×C67—24no—
78C22⋊Q8×C67—22no—
79C22⋊Q8×C67—22no—
80#121×C67—22no—
81#121×C67—22no—
82#121×C67—21no—
83#121×C67—21no—
84C4×Q8×C67D411yescharacteristic
85C4×Q8×C67D411yescharacteristic
86C4×D4×C67—22no—
87C4×D4×C67—22no—
88C4×D4×C67—21no—
89C4×D4×C67—21no—
90C4×D4×C67—21no—
91C4×D4×C67—21no—
92C4×D4×C67D411yescharacteristic
93C4×D4×C67D411yescharacteristic
94#117×C67—21no—
95#117×C67C2311yescharacteristic
96C2×C4⋊C4×C67—22no—
97C2×C4⋊C4×C67—22no—
98C2×C4⋊C4×C67—21no—
99C2×C4⋊C4×C67D412yes—
100C2×C4⋊C4×C67D412yes—
101C2×C4⋊C4×C67C2312yes—
102C2×C42×C67—21no—
103C2×C42×C67—21no—
104C2×C42×C67—21no—
105C2×C42×C67C2312yes—
106C2×C42×C67C2311yescharacteristic
107#107×C67—22no—
108#106×C67—22no—
109C4⋊C8×C67—21no—
110C4⋊C8×C67—21no—
111#104×C67—24no—
112#104×C67—44no—
113#103×C67—22no—
114#103×C67—24no—
115#102×C67—22no—
116#102×C67—24no—
117#101×C67—44no—
118#97×C67—21no—
119C4×C8×C67—21no—
120#95×C67—28no—
121#41×C67—22no—
122#41×C67—24no—
123#41×C67—24no—
124#41×C67—48no—
125#41×C67—416no—
126#41×C67—44no—
127#41×C67—44no—
128#41×C67#4114yes—
129C2×Q8×C67—24no—
130C2×Q8×C67—22no—
131C2×Q8×C67—22no—
132C2×Q8×C67—24no—
133C2×Q8×C67—24no—
134C2×Q8×C67—48no—
135C2×Q8×C67—44no—
136C2×Q8×C67—44no—
137C2×Q8×C67—44no—
138C2×Q8×C67—41no—
139C2×Q8×C67—41no—
140C2×Q8×C67C2×D412yes—
141C2×Q8×C67C2×D411yescharacteristic
142C2×D4×C67—21no—
143C2×D4×C67—24no—
144C2×D4×C67—22no—
145C2×D4×C67—21no—
146C2×D4×C67—21no—
147C2×D4×C67C2×D412yes—
148C2×D4×C67C2×D411yescharacteristic
149C22×C4×C67—21no—
150C22×C4×C67—24no—
151C22×C4×C67—24no—
152C22×C4×C67—22no—
153C22×C4×C67—21no—
154C22×C4×C67—21no—
155C22×C4×C67—22no—
156C22×C4×C67—22no—
157C22×C4×C67C2×D414yes—
158C22×C4×C67C2×D411yescharacteristic
159C22×C4×C67C2×D411yescharacteristic
160C22×C4×C67C2411yescharacteristic
161M16×C67—22no—
162M16×C67—22no—
163M16×C67—24no—
164C2×C8×C67—22no—
165C2×C8×C67—22no—
166C4⋊C4×C67—24no—
167C4⋊C4×C67—24no—
168C4⋊C4×C67—21no—
169C4⋊C4×C67—21no—
170C4⋊C4×C67—22no—
171C4⋊C4×C67—22no—
172C4⋊C4×C67—22no—
173C4⋊C4×C67—22no—
174C4⋊C4×C67—21no—
175C4⋊C4×C67—21no—
176C4⋊C4×C67—22no—
177C4⋊C4×C67—22no—
178C4⋊C4×C67—41no—
179C4⋊C4×C67—41no—
180C4⋊C4×C67C2×D411yescharacteristic
181C4⋊C4×C67C2×D411yescharacteristic
182C22⋊C4×C67—24no—
183C22⋊C4×C67—24no—
184C22⋊C4×C67—21no—
185C22⋊C4×C67—21no—
186C22⋊C4×C67—21no—
187C22⋊C4×C67—22no—
188C22⋊C4×C67—22no—
189C42×C67—24no—
190C42×C67—21no—
191C42×C67—21no—
192C42×C67—21no—
193C42×C67—22no—
194C42×C67—22no—
195C42×C67—42no—
196C42×C67C2×D411yescharacteristic
197C42×C67C2×D411yescharacteristic
198C42×C67C2×D411yescharacteristic
199C42×C67C2×D411yescharacteristic
200C23×C67—21no—
201C23×C67—22no—
202C23×C67C22×D411yescharacteristic
203Q8×C67—21no—
204Q8×C67—24no—
205Q8×C67—48no—
206Q8×C67—42no—
207Q8×C67—42no—
208Q8×C67—44no—
209Q8×C67—82no—
210Q8×C67—82no—
211Q8×C67#12412yes—
212D4×C67—24no—
213D4×C67—21no—
214D4×C67—24no—
215D4×C67—48no—
216D4×C67—44no—
217D4×C67—42no—
218D4×C67—42no—
219D4×C67#12412yes—
220C2×C4×C67—21no—
221C2×C4×C67—24no—
222C2×C4×C67—22no—
223C2×C4×C67—22no—
224C2×C4×C67—24no—
225C2×C4×C67—21no—
226C2×C4×C67—21no—
227C2×C4×C67—22no—
228C2×C4×C67—24no—
229C2×C4×C67—28no—
230C2×C4×C67—48no—
231C2×C4×C67—42no—
232C2×C4×C67—48no—
233C2×C4×C67—42no—
234C2×C4×C67—42no—
235C2×C4×C67—42no—
236C2×C4×C67—42no—
237C2×C4×C67—42no—
238C2×C4×C67—44no—
239C2×C4×C67—42no—
240C2×C4×C67#12011yescharacteristic
241C2×C4×C67#12011yescharacteristic
242C2×C4×C67#12011yescharacteristic
243C2×C4×C67#12011yescharacteristic
244C2×C4×C67C22×D411yescharacteristic
245C2×C4×C67C22×D411yescharacteristic
246C2×C4×C67C2×#4112yes—
247C2×C4×C67H(5,𝔽2)12yes—
248C8×C67—42no—
249C8×C67—42no—
250C22×C67—21no—
251C22×C67—21no—
252C22×C67—22no—
253C22×C67—41no—
254C22×C67—41no—
255C22×C67—42no—
256C22×C67—42no—
257C22×C67—42no—
258C22×C67C2×#12011yescharacteristic
259C22×C67#54612yes—
260C4×C67—22no—
261C4×C67—21no—
262C4×C67—21no—
263C4×C67—44no—
264C4×C67—44no—
265C4×C67—42no—
266C4×C67—41no—
267C4×C67—41no—
268C4×C67—82no—
269C4×C67#45412yes—
270C4×C67#45612yes—
271C4×C67C2×#12411yescharacteristic
272C4×C67#53511yescharacteristic
273C4×C67#54612yes—
274#25876C6711yesmaximal, characteristic, direct product factor, Sylow
275C2×C67—22no—
276C2×C67—42no—
277C2×C67—82no—
278C2×C67#243511yescharacteristic
279C2×C67C2×#45411yescharacteristic
280C2×C67C2×#45611yescharacteristic
281#3499C2×C6711yescharacteristic
282#3141C2×C6711yescharacteristic, complemented
283#3079C2×C6711yescharacteristic, complemented
284#2862C2×C6711yescharacteristic, complemented
285#2038C2×C6714yescomplemented
286#1966C2×C6711yescharacteristic, complemented
287#1962C2×C6711yescharacteristic, complemented
288#1793C2×C6714yescomplemented
289C4×#104C2×C6711yescharacteristic, complemented
290C67#2587611yescharacteristic, direct product factor, Sylow
291C2×2−1+4C22×C6712yes—
292D4×Q8—21no—
293D4×Q8—21no—
294#547—21no—
295#547—21no—
296#547—22no—
297#547—22no—
298#536—22no—
299#536C22×C6712yes—
300#532—21no—
301#532C22×C6711yescharacteristic
302C2×C4⋊Q8—21no—
303C2×C4⋊Q8C22×C6711yescharacteristic
304C2×C4⋊Q8C22×C6711yescharacteristic
305C4×#41C22×C6711yescharacteristic
306#489—22no—
307#487—21no—
308#486—21no—
309#477—21no—
310#476—21no—
311#475—21no—
312#474—21no—
313#428C22×C6712yes—
314#423C22×C6711yescharacteristic
315C2×#104C22×C6712yes—
316C2×#104C22×C6714yes—
317#417C22×C6712yes—
318#417C22×C6714yes—
319C2×#101C22×C6714yes—
320C4×M16C22×C6711yescharacteristic
321#391—22no—
322#391C22×C6714yes—
323#382—22no—
324C43C22×C6711yescharacteristic
325#339C22×C6712yes—
3262−1+4—28no—
3272−1+4—44no—
328C2×#41—22no—
329C2×#41—22no—
330C2×#41—24no—
331C2×#41C23×C6712yes—
332C2×#41C23×C6711yescharacteristic
333C22×Q8—22no—
334C22×Q8—22no—
335C22×Q8—21no—
336C22×Q8C23×C6712yes—
337C2×M16C23×C6712yes—
338C2×M16C23×C6712yes—
339C4⋊Q8—22no—
340C4⋊Q8—22no—
341C4⋊Q8—22no—
342C4⋊Q8—21no—
343C4⋊Q8—21no—
344C4⋊Q8—21no—
345C4⋊Q8—21no—
346C4⋊Q8—21no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8—21no—
350C4⋊Q8D4×C6711yescharacteristic
351#127D4×C6711yescharacteristic
352#124—22no—
353#124D4×C6712yes—
354#123—24no—
355#123—24no—
356#123—24no—
357C22⋊Q8—24no—
358C22⋊Q8—24no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361#121—22no—
362#121—22no—
363#121—21no—
364#121—21no—
365C4×Q8D4×C6711yescharacteristic
366C4×Q8D4×C6711yescharacteristic
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—21no—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4—21no—
373C4×D4D4×C6711yescharacteristic
374C4×D4D4×C6711yescharacteristic
375#117—21no—
376#117C23×C6711yescharacteristic
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4D4×C6712yes—
381C2×C4⋊C4D4×C6712yes—
382C2×C4⋊C4C23×C6712yes—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42C23×C6712yes—
387C2×C42C23×C6711yescharacteristic
388#107—22no—
389#106—22no—
390C4⋊C8—21no—
391C4⋊C8—21no—
392#104—24no—
393#104—44no—
394#103—22no—
395#103—24no—
396#102—22no—
397#102—24no—
398#101—44no—
399#97—21no—
400C4×C8—21no—
401#95—28no—
402#41—22no—
403#41—24no—
404#41—24no—
405#41—48no—
406#41—416no—
407#41—44no—
408#41—44no—
409#41#41×C6714yes—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—48no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—41no—
420C2×Q8—41no—
421C2×Q8C2×D4×C6712yes—
422C2×Q8C2×D4×C6711yescharacteristic
423C2×D4—21no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4C2×D4×C6712yes—
429C2×D4C2×D4×C6711yescharacteristic
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4C2×D4×C6714yes—
439C22×C4C2×D4×C6711yescharacteristic
440C22×C4C2×D4×C6711yescharacteristic
441C22×C4C24×C6711yescharacteristic
442M16—22no—
443M16—22no—
444M16—24no—
445C2×C8—22no—
446C2×C8—22no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4C2×D4×C6711yescharacteristic
462C4⋊C4C2×D4×C6711yescharacteristic
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C42—24no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—22no—
475C42—22no—
476C42—42no—
477C42C2×D4×C6711yescharacteristic
478C42C2×D4×C6711yescharacteristic
479C42C2×D4×C6711yescharacteristic
480C42C2×D4×C6711yescharacteristic
481C23—21no—
482C23—22no—
483C23C22×D4×C6711yescharacteristic
484Q8—21no—
485Q8—24no—
486Q8—48no—
487Q8—42no—
488Q8—42no—
489Q8—44no—
490Q8—82no—
491Q8—82no—
492Q8#124×C6712yes—
493D4—24no—
494D4—21no—
495D4—24no—
496D4—48no—
497D4—44no—
498D4—42no—
499D4—42no—
500D4#124×C6712yes—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4#120×C6711yescharacteristic
522C2×C4#120×C6711yescharacteristic
523C2×C4#120×C6711yescharacteristic
524C2×C4#120×C6711yescharacteristic
525C2×C4C22×D4×C6711yescharacteristic
526C2×C4C22×D4×C6711yescharacteristic
527C2×C4C2×#41×C6712yes—
528C2×C4H(5,𝔽2)×C6712yes—
529C8—42no—
530C8—42no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—41no—
535C22—41no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22C2×#120×C6711yescharacteristic
540C22#546×C6712yes—
541C4—21no—
542C4—21no—
543C4—22no—
544C4—44no—
545C4—41no—
546C4—41no—
547C4—44no—
548C4—42no—
549C4—82no—
550C4#454×C6712yes—
551C4#456×C6712yes—
552C4C2×#124×C6711yescharacteristic
553C4#535×C6711yescharacteristic
554C4#546×C6712yes—
555C2—22no—
556C2—42no—
557C2—82no—
558C2C67×#243511yescharacteristic
559C2C2×#454×C6711yescharacteristic
560C2C2×#456×C6711yescharacteristic
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC67×#3499C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#3141C2semidirect
3. C67×#3079C2semidirect
4. C67×#2862C2semidirect
5. C67×#2038C2semidirect
6. C67×#1966C2semidirect
7. C67×#1962C2semidirect
8. C67×#1793C2semidirect
9. C4×#104×C67C2semidirect
274. #25876C67direct (N × H)
282. #3141C2×C67semidirect
283. #3079C2×C67semidirect
284. #2862C2×C67semidirect
285. #2038C2×C67semidirect
286. #1966C2×C67semidirect
287. #1962C2×C67semidirect
288. #1793C2×C67semidirect
289. C4×#104C2×C67semidirect
290. C67#25876direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#34992
C67×#31412
C67×#30792
C67×#28622
C67×#20382
C67×#19662
C67×#19622
C67×#17932
C4×#104×C672
#2587667