Finite Group Databasebeta

← C67×#23992 #30703243

Subgroups — C67×#23992 #30703243

Order 17152.

1128 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3509C211yesmaximal, characteristic, complemented
2C67×#3028C211yesmaximal, characteristic
3C67×#3022C211yesmaximal, characteristic
4C67×#2901C211yesmaximal, characteristic
5C67×#2031C212yesmaximal, complemented
6C67×#2005C212yesmaximal, complemented
7C67×#1915C211yesmaximal, characteristic, complemented
8C67×#1912C212yesmaximal, complemented
9C67×#1909C211yesmaximal, characteristic, complemented
10C67×#1905C212yesmaximal, complemented
11C67×#1827C211yesmaximal, characteristic, complemented
12#585×C67C2211yescharacteristic
13C2×#137×C67C2212yes—
14#568×C67C2211yescharacteristic
15#557×C67—22no—
16#556×C67—22no—
17#552×C67—24no—
18#541×C67—22no—
19#536×C67C2211yescharacteristic
20#536×C67C2211yescharacteristic
21#536×C67C2212yes—
22#533×C67—22no—
23#533×C67C2211yescharacteristic, complemented
24C2×C4⋊Q8×C67C2211yescharacteristic, complemented
25#519×C67C2211yescharacteristic, complemented
26#519×C67C2211yescharacteristic, complemented
27C4×#41×C67C2211yescharacteristic
28#458×C67—24no—
29#456×C67—22no—
30#455×C67—22no—
31#421×C67—22no—
32#421×C67C2212yes—
33#421×C67C2212yes—
34C2×#104×C67—22no—
35C2×#104×C67C2212yes—
36#419×C67C2212yescomplemented
37#413×C67C2211yescharacteristic
38C2×#101×C67C2212yescomplemented
39#410×C67C2212yes—
40#410×C67C2211yescharacteristic
41#344×C67C2212yescomplemented
42#339×C67C2212yescomplemented
43H(3,ℤ/4)×C67C2212yescomplemented
44H(3,ℤ/4)×C67C2212yescomplemented
452−1+4×C67—22no—
462−1+4×C67—21no—
47H(5,𝔽2)×C67—22no—
48H(5,𝔽2)×C67—22no—
49H(5,𝔽2)×C67—21no—
50C2×#41×C67—22no—
51C2×#41×C67—42no—
52C2×#41×C67C2312yes—
53C2×#41×C67C2311yescharacteristic
54C22×Q8×C67C2311yescharacteristic
55C22×Q8×C67C2311yescharacteristic
56#137×C67—24no—
57#137×C67—42no—
58C2×Q16×C67—22no—
59C2×SD16×C67—22no—
60#131×C67—42no—
61C2×M16×C67—22no—
62C2×M16×C67C2312yes—
63C2×M16×C67C2312yes—
64C4⋊Q8×C67—22no—
65C4⋊Q8×C67—21no—
66C4⋊Q8×C67—22no—
67C4⋊Q8×C67—21no—
68C4⋊Q8×C67—21no—
69C4⋊Q8×C67—21no—
70C4⋊Q8×C67D412yes—
71#126×C67—24no—
72#126×C67—24no—
73#125×C67—24no—
74#125×C67—24no—
75#125×C67—24no—
76#125×C67—21no—
77#124×C67—21no—
78#124×C67—22no—
79#124×C67D412yes—
80#123×C67—22no—
81#123×C67—24no—
82#123×C67D414yes—
83C22⋊Q8×C67—24no—
84C22⋊Q8×C67—22no—
85C22⋊Q8×C67—22no—
86C22⋊Q8×C67—22no—
87C22⋊Q8×C67—42no—
88C4×Q8×C67—24no—
89C4×Q8×C67—21no—
90C4×Q8×C67—22no—
91C4×Q8×C67—22no—
92C4×Q8×C67—22no—
93C4×Q8×C67—22no—
94C4×Q8×C67—22no—
95C4×D4×C67—22no—
96C4×D4×C67—21no—
97#117×C67—24no—
98#117×C67—22no—
99#117×C67—22no—
100#117×C67D412yes—
101#117×C67C2312yes—
102#117×C67C2311yescharacteristic
103#117×C67C2311yescharacteristic
104#117×C67C2311yescharacteristic
105C2×C4⋊C4×C67—22no—
106C2×C42×C67D412yes—
107C2×C42×C67C2311yescharacteristic
108#104×C67—24no—
109#104×C67—22no—
110#104×C67—24no—
111#104×C67—22no—
112#103×C67—42no—
113#101×C67—21no—
114#101×C67—42no—
115#100×C67—21no—
116#99×C67—22no—
117#99×C67—24no—
118#41×C67—22no—
119#41×C67—22no—
120#41×C67—24no—
121#41×C67—22no—
122#41×C67—22no—
123#41×C67—22no—
124#41×C67—22no—
125#41×C67—44no—
126#41×C67—42no—
127#41×C67—42no—
128#41×C67—42no—
129#41×C67—42no—
130C2×Q8×C67—22no—
131C2×Q8×C67—21no—
132C2×Q8×C67—21no—
133C2×Q8×C67—21no—
134C2×Q8×C67—22no—
135C2×Q8×C67—22no—
136C2×Q8×C67—22no—
137C2×Q8×C67—22no—
138C2×Q8×C67—22no—
139C2×Q8×C67—42no—
140C2×Q8×C67C2×D412yes—
141C2×Q8×C67C2×D411yescharacteristic
142C2×Q8×C67#4112yes—
143C2×D4×C67—22no—
144C2×D4×C67—22no—
145C2×D4×C67—42no—
146C2×D4×C67—44no—
147C2×D4×C67—42no—
148C2×D4×C67C2×D412yes—
149C2×D4×C67C2×D411yescharacteristic
150C2×D4×C67#4112yes—
151C22×C4×C67—22no—
152C22×C4×C67—21no—
153C22×C4×C67—21no—
154C22×C4×C67—22no—
155C22×C4×C67—42no—
156C22×C4×C67C2×D412yes—
157C22×C4×C67C2×D412yes—
158C22×C4×C67C2411yescharacteristic
159Q16×C67—44no—
160SD16×C67—44no—
161M16×C67—22no—
162M16×C67—22no—
163M16×C67—22no—
164M16×C67—22no—
165M16×C67—42no—
166C2×C8×C67—22no—
167C2×C8×C67—22no—
168C2×C8×C67—42no—
169C4⋊C4×C67—24no—
170C4⋊C4×C67—24no—
171C4⋊C4×C67—22no—
172C4⋊C4×C67—24no—
173C4⋊C4×C67—24no—
174C4⋊C4×C67—22no—
175C4⋊C4×C67—22no—
176C4⋊C4×C67—22no—
177C4⋊C4×C67—24no—
178C4⋊C4×C67—24no—
179C4⋊C4×C67—21no—
180C4⋊C4×C67—21no—
181C4⋊C4×C67—21no—
182C4⋊C4×C67—22no—
183C4⋊C4×C67—44no—
184C22⋊C4×C67—24no—
185C22⋊C4×C67—24no—
186C22⋊C4×C67—22no—
187C22⋊C4×C67—21no—
188C22⋊C4×C67—21no—
189C22⋊C4×C67—22no—
190C22⋊C4×C67—44no—
191C22⋊C4×C67C2×D412yes—
192C42×C67—24no—
193C42×C67—24no—
194C42×C67—22no—
195C42×C67—21no—
196C42×C67—21no—
197C42×C67—22no—
198C42×C67C2×D411yescharacteristic
199C42×C67C2×D412yes—
200C42×C67C2×D411yescharacteristic
201C42×C67C2×D412yes—
202C23×C67—22no—
203C23×C67—42no—
204C23×C67#12012yes—
205C23×C67C22×D411yescharacteristic
206Q8×C67—22no—
207Q8×C67—22no—
208Q8×C67—24no—
209Q8×C67—22no—
210Q8×C67—44no—
211Q8×C67—42no—
212Q8×C67—41no—
213Q8×C67—41no—
214Q8×C67—41no—
215Q8×C67—41no—
216Q8×C67—42no—
217D4×C67—24no—
218D4×C67—22no—
219D4×C67—22no—
220D4×C67—22no—
221D4×C67—42no—
222D4×C67—44no—
223D4×C67—42no—
224D4×C67—42no—
225C2×C4×C67—21no—
226C2×C4×C67—22no—
227C2×C4×C67—22no—
228C2×C4×C67—22no—
229C2×C4×C67—22no—
230C2×C4×C67—22no—
231C2×C4×C67—22no—
232C2×C4×C67—22no—
233C2×C4×C67—21no—
234C2×C4×C67—24no—
235C2×C4×C67—22no—
236C2×C4×C67—42no—
237C2×C4×C67—44no—
238C2×C4×C67—44no—
239C2×C4×C67—42no—
240C2×C4×C67—42no—
241C2×C4×C67—42no—
242C2×C4×C67—42no—
243C2×C4×C67—42no—
244C2×C4×C67#12012yes—
245C2×C4×C67C22×D411yescharacteristic
246C2×C4×C67C22×D411yescharacteristic
247C2×C4×C67C2×#4112yes—
248C2×C4×C67H(5,𝔽2)12yes—
249C8×C67—42no—
250C8×C67—42no—
251C22×C67—22no—
252C22×C67—41no—
253C22×C67—42no—
254C22×C67—42no—
255C22×C67—82no—
256C22×C67—162no—
257C22×C67C2×#12011yescharacteristic
258C22×C67#54612yes—
259C4×C67—22no—
260C4×C67—22no—
261C4×C67—42no—
262C4×C67—41no—
263C4×C67—42no—
264C4×C67—42no—
265C4×C67—42no—
266C4×C67—42no—
267C4×C67—42no—
268C4×C67—82no—
269C4×C67#52511yescharacteristic
270C4×C67#53511yescharacteristic
271C4×C67#54612yes—
272#23992C6711yesmaximal, characteristic, direct product factor, Sylow
273C2×C67—21no—
274C2×C67—22no—
275C2×C67—42no—
276C2×C67—82no—
277C2×C67#240411yescharacteristic
278#3509C2×C6711yescharacteristic, complemented
279#3028C2×C6711yescharacteristic
280#3022C2×C6711yescharacteristic
281#2901C2×C6711yescharacteristic
282#2031C2×C6712yescomplemented
283#2005C2×C6712yescomplemented
284#1915C2×C6711yescharacteristic, complemented
285#1912C2×C6712yescomplemented
286#1909C2×C6711yescharacteristic, complemented
287#1905C2×C6712yescomplemented
288#1827C2×C6711yescharacteristic, complemented
289C67#2399211yescharacteristic, direct product factor, Sylow
290#585C22×C6711yescharacteristic
291C2×#137C22×C6712yes—
292#568C22×C6711yescharacteristic
293#557—22no—
294#556—22no—
295#552—24no—
296#541—22no—
297#536C22×C6711yescharacteristic
298#536C22×C6711yescharacteristic
299#536C22×C6712yes—
300#533—22no—
301#533C22×C6711yescharacteristic, complemented
302C2×C4⋊Q8C22×C6711yescharacteristic, complemented
303#519C22×C6711yescharacteristic, complemented
304#519C22×C6711yescharacteristic, complemented
305C4×#41C22×C6711yescharacteristic
306#458—24no—
307#456—22no—
308#455—22no—
309#421—22no—
310#421C22×C6712yes—
311#421C22×C6712yes—
312C2×#104—22no—
313C2×#104C22×C6712yes—
314#419C22×C6712yescomplemented
315#413C22×C6711yescharacteristic
316C2×#101C22×C6712yescomplemented
317#410C22×C6711yescharacteristic
318#410C22×C6712yes—
319#344C22×C6712yescomplemented
320#339C22×C6712yescomplemented
321H(3,ℤ/4)C22×C6712yescomplemented
322H(3,ℤ/4)C22×C6712yescomplemented
3232−1+4—22no—
3242−1+4—21no—
325H(5,𝔽2)—22no—
326H(5,𝔽2)—22no—
327H(5,𝔽2)—21no—
328C2×#41—22no—
329C2×#41—42no—
330C2×#41C23×C6712yes—
331C2×#41C23×C6711yescharacteristic
332C22×Q8C23×C6711yescharacteristic
333C22×Q8C23×C6711yescharacteristic
334#137—24no—
335#137—42no—
336C2×Q16—22no—
337C2×SD16—22no—
338#131—42no—
339C2×M16—22no—
340C2×M16C23×C6712yes—
341C2×M16C23×C6712yes—
342C4⋊Q8—22no—
343C4⋊Q8—21no—
344C4⋊Q8—22no—
345C4⋊Q8—21no—
346C4⋊Q8—21no—
347C4⋊Q8—21no—
348C4⋊Q8D4×C6712yes—
349#126—24no—
350#126—24no—
351#125—24no—
352#125—24no—
353#125—24no—
354#125—21no—
355#124—21no—
356#124—22no—
357#124D4×C6712yes—
358#123—22no—
359#123—24no—
360#123D4×C6714yes—
361C22⋊Q8—24no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—42no—
366C4×Q8—24no—
367C4×Q8—21no—
368C4×Q8—22no—
369C4×Q8—22no—
370C4×Q8—22no—
371C4×Q8—22no—
372C4×Q8—22no—
373C4×D4—22no—
374C4×D4—21no—
375#117—24no—
376#117—22no—
377#117—22no—
378#117D4×C6712yes—
379#117C23×C6712yes—
380#117C23×C6711yescharacteristic
381#117C23×C6711yescharacteristic
382#117C23×C6711yescharacteristic
383C2×C4⋊C4—22no—
384C2×C42D4×C6712yes—
385C2×C42C23×C6711yescharacteristic
386#104—22no—
387#104—24no—
388#104—22no—
389#104—24no—
390#103—42no—
391#101—21no—
392#101—42no—
393#100—21no—
394#99—22no—
395#99—24no—
396#41—22no—
397#41—22no—
398#41—24no—
399#41—22no—
400#41—22no—
401#41—22no—
402#41—22no—
403#41—44no—
404#41—42no—
405#41—42no—
406#41—42no—
407#41—42no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—42no—
418C2×Q8C2×D4×C6712yes—
419C2×Q8C2×D4×C6711yescharacteristic
420C2×Q8#41×C6712yes—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4C2×D4×C6712yes—
427C2×D4C2×D4×C6711yescharacteristic
428C2×D4#41×C6712yes—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—42no—
434C22×C4C2×D4×C6712yes—
435C22×C4C2×D4×C6712yes—
436C22×C4C24×C6711yescharacteristic
437Q16—44no—
438SD16—44no—
439M16—22no—
440M16—22no—
441M16—22no—
442M16—22no—
443M16—42no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—42no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—44no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—22no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—22no—
468C22⋊C4—44no—
469C22⋊C4C2×D4×C6712yes—
470C42—24no—
471C42—24no—
472C42—22no—
473C42—21no—
474C42—21no—
475C42—22no—
476C42C2×D4×C6711yescharacteristic
477C42C2×D4×C6712yes—
478C42C2×D4×C6711yescharacteristic
479C42C2×D4×C6712yes—
480C23—22no—
481C23—42no—
482C23#120×C6712yes—
483C23C22×D4×C6711yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—24no—
487Q8—22no—
488Q8—42no—
489Q8—44no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493Q8—41no—
494Q8—42no—
495D4—24no—
496D4—22no—
497D4—22no—
498D4—22no—
499D4—42no—
500D4—44no—
501D4—42no—
502D4—42no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4#120×C6712yes—
523C2×C4C22×D4×C6711yescharacteristic
524C2×C4C22×D4×C6711yescharacteristic
525C2×C4C2×#41×C6712yes—
526C2×C4H(5,𝔽2)×C6712yes—
527C8—42no—
528C8—42no—
529C22—22no—
530C22—41no—
531C22—42no—
532C22—42no—
533C22—82no—
534C22—162no—
535C22C2×#120×C6711yescharacteristic
536C22#546×C6712yes—
537C4—22no—
538C4—22no—
539C4—42no—
540C4—41no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—82no—
547C4#525×C6711yescharacteristic
548C4#535×C6711yescharacteristic
549C4#546×C6712yes—
550C2—21no—
551C2—22no—
552C2—42no—
553C2—82no—
554C2C67×#240411yescharacteristic
subgroupG/H
center, socleC2×C67#2404
commutator, Frattini, cosocleC22×C4C24×C67
prime radical#585×C67C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C67 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3509C2semidirect
5. C67×#2031C2semidirect
6. C67×#2005C2semidirect
7. C67×#1915C2semidirect
8. C67×#1912C2semidirect
9. C67×#1909C2semidirect
10. C67×#1905C2semidirect
11. C67×#1827C2semidirect
23. #533×C67C22semidirect
24. C2×C4⋊Q8×C67C22semidirect
25. #519×C67C22semidirect
26. #519×C67C22semidirect
36. #419×C67C22semidirect
38. C2×#101×C67C22semidirect
41. #344×C67C22semidirect
42. #339×C67C22semidirect
43. H(3,ℤ/4)×C67C22semidirect
44. H(3,ℤ/4)×C67C22semidirect
272. #23992C67direct (N × H)
278. #3509C2×C67semidirect
282. #2031C2×C67semidirect
283. #2005C2×C67semidirect
284. #1915C2×C67semidirect
285. #1912C2×C67semidirect
286. #1909C2×C67semidirect
287. #1905C2×C67semidirect
288. #1827C2×C67semidirect
289. C67#23992direct (N × H)
301. #533C22×C67semidirect
302. C2×C4⋊Q8C22×C67semidirect
303. #519C22×C67semidirect
304. #519C22×C67semidirect
314. #419C22×C67semidirect
316. C2×#101C22×C67semidirect
319. #344C22×C67semidirect
320. #339C22×C67semidirect
321. H(3,ℤ/4)C22×C67semidirect
322. H(3,ℤ/4)C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#35092
C67×#30282
C67×#30222
C67×#29012
C67×#20312
C67×#20052
C67×#19152
C67×#19122
C67×#19092
C67×#19052
C67×#18272
#2399267