Finite Group Databasebeta

← C67×#32920 #30712171

Subgroups — C67×#32920 #30712171

Order 17152.

1590 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×H(5,𝔽2)×C67C211yesmaximal, characteristic, complemented
2C67×#3038C212yesmaximal, complemented
3C2×#453×C67C212yesmaximal, complemented
4C67×#2541C211yesmaximal, characteristic, complemented
5C67×#1887C212yesmaximal, complemented
6C67×#1883C214yesmaximal, complemented
7C67×#1806C212yesmaximal, complemented
8C67×#1800C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C67C2211yescharacteristic
10C2×AGL(1,ℤ/8)×C67C2214yes—
11C2×#127×C67C2212yes—
12C2×#127×C67C2211yescharacteristic, complemented
13C2×#121×C67—22no—
14#520×C67—22no—
15#520×C67C2212yes—
16#518×C67—22no—
17#518×C67—22no—
18#518×C67C2211yescharacteristic
19C4×#41×C67—22no—
20C4×#41×C67C2212yes—
21C2×C4×D4×C67—22no—
22C2×C4×D4×C67—22no—
23C2×C4×D4×C67C2211yescharacteristic
24#460×C67—22no—
25#460×C67—22no—
26#459×C67—22no—
27#459×C67—22no—
28#453×C67—28no—
29#423×C67C2212yescomplemented
30C2×#104×C67C2214yes—
31#418×C67C2214yescomplemented
32C2×#100×C67C2212yes—
33#392×C67—22no—
34#390×C67C2212yescomplemented
35C4×C4⋊C4×C67C2211yescharacteristic, complemented
36#343×C67C2212yescomplemented
37#339×C67C2214yescomplemented
38H(5,𝔽2)×C67—24no—
39H(5,𝔽2)×C67—22no—
40H(5,𝔽2)×C67D414yescomplemented
41C2×#41×C67—22no—
42C2×#41×C67C2312yes—
43C22×D4×C67—22no—
44C22×D4×C67—22no—
45C22×D4×C67—24no—
46C22×D4×C67—21no—
47C22×D4×C67—21no—
48C22×D4×C67C2312yes—
49C22×D4×C67C2311yescharacteristic
50C23×C4×C67—22no—
51C23×C4×C67D412yes—
52AGL(1,ℤ/8)×C67—28no—
53AGL(1,ℤ/8)×C67—28no—
54C2×SD16×C67—24no—
55C2×D8×C67—24no—
56C2×M16×C67C2314yes—
57#127×C67—24no—
58#127×C67—24no—
59#127×C67—22no—
60#127×C67—22no—
61#121×C67—48no—
62C4×Q8×C67—22no—
63C4×Q8×C67D412yescomplemented
64C4×Q8×C67D412yescomplemented
65C4×D4×C67—24no—
66C4×D4×C67—24no—
67C4×D4×C67—24no—
68C4×D4×C67—24no—
69C4×D4×C67—22no—
70C4×D4×C67—22no—
71C4×D4×C67—24no—
72C4×D4×C67—22no—
73C4×D4×C67—22no—
74C4×D4×C67—22no—
75C4×D4×C67—22no—
76C4×D4×C67—22no—
77C4×D4×C67D412yescomplemented
78C4×D4×C67D412yescomplemented
79#117×C67—24no—
80#117×C67—22no—
81#117×C67—22no—
82#117×C67C2312yes—
83C2×C4⋊C4×C67—22no—
84C2×C4⋊C4×C67—21no—
85C2×C4⋊C4×C67—22no—
86C2×C4⋊C4×C67C2311yescharacteristic
87C2×C22⋊C4×C67—22no—
88C2×C22⋊C4×C67—24no—
89C2×C22⋊C4×C67—22no—
90C2×C22⋊C4×C67—24no—
91C2×C22⋊C4×C67D412yes—
92C2×C42×C67—22no—
93C2×C42×C67—22no—
94C2×C42×C67C2312yes—
95C2×C42×C67C2311yescharacteristic
96C4⋊C8×C67—22no—
97C4⋊C8×C67—22no—
98#104×C67—28no—
99#102×C67—24no—
100#102×C67—24no—
101#100×C67—24no—
102#95×C67—21no—
103C24×C67—22no—
104C24×C67—22no—
105C24×C67C2×D412yes—
106#41×C67—28no—
107#41×C67—28no—
108#41×C67—22no—
109#41×C67—22no—
110#41×C67C2×D414yes—
111#41×C67#4114yes—
112C2×Q8×C67C2×D412yes—
113C2×D4×C67—24no—
114C2×D4×C67—28no—
115C2×D4×C67—28no—
116C2×D4×C67—28no—
117C2×D4×C67—22no—
118C2×D4×C67—28no—
119C2×D4×C67—22no—
120C2×D4×C67—24no—
121C2×D4×C67—22no—
122C2×D4×C67—22no—
123C2×D4×C67—22no—
124C2×D4×C67—24no—
125C2×D4×C67—44no—
126C2×D4×C67—48no—
127C2×D4×C67—48no—
128C2×D4×C67—42no—
129C2×D4×C67—41no—
130C2×D4×C67—41no—
131C2×D4×C67—82no—
132C2×D4×C67—84no—
133C2×D4×C67C2×D412yes—
134C2×D4×C67C2×D412yes—
135C2×D4×C67#4112yes—
136C22×C4×C67—22no—
137C22×C4×C67—22no—
138C22×C4×C67—22no—
139C22×C4×C67—22no—
140C22×C4×C67—24no—
141C22×C4×C67—22no—
142C22×C4×C67—22no—
143C22×C4×C67—22no—
144C22×C4×C67—22no—
145C22×C4×C67—22no—
146C22×C4×C67—44no—
147C22×C4×C67—84no—
148C22×C4×C67C2×D411yescharacteristic
149C22×C4×C67C2×D412yes—
150C22×C4×C67C2×D411yescharacteristic
151C22×C4×C67C2411yescharacteristic
152SD16×C67—48no—
153D8×C67—48no—
154M16×C67—24no—
155M16×C67—24no—
156C2×C8×C67—24no—
157C4⋊C4×C67—24no—
158C4⋊C4×C67—24no—
159C4⋊C4×C67—22no—
160C4⋊C4×C67—22no—
161C4⋊C4×C67—22no—
162C4⋊C4×C67C2×D412yes—
163C4⋊C4×C67C2×D412yes—
164C22⋊C4×C67—24no—
165C22⋊C4×C67—24no—
166C22⋊C4×C67—22no—
167C22⋊C4×C67—22no—
168C22⋊C4×C67—22no—
169C22⋊C4×C67—24no—
170C22⋊C4×C67—48no—
171C42×C67—24no—
172C42×C67—22no—
173C42×C67—22no—
174C42×C67—22no—
175C42×C67—22no—
176C42×C67—22no—
177C42×C67C2×D412yes—
178C42×C67C2×D412yes—
179C23×C67—28no—
180C23×C67—22no—
181C23×C67—22no—
182C23×C67—22no—
183C23×C67—42no—
184C23×C67—42no—
185C23×C67—44no—
186C23×C67—44no—
187C23×C67—42no—
188C23×C67—42no—
189C23×C67—42no—
190C23×C67—82no—
191C23×C67—82no—
192C23×C67#12114yes—
193C23×C67C22×D411yescharacteristic
194Q8×C67—22no—
195Q8×C67#12114yescomplemented
196D4×C67—216no—
197D4×C67—24no—
198D4×C67—24no—
199D4×C67—24no—
200D4×C67—22no—
201D4×C67—24no—
202D4×C67—44no—
203D4×C67—44no—
204D4×C67—82no—
205D4×C67—84no—
206D4×C67—816no—
207D4×C67—82no—
208D4×C67#12114yescomplemented
209C2×C4×C67—28no—
210C2×C4×C67—22no—
211C2×C4×C67—24no—
212C2×C4×C67—24no—
213C2×C4×C67—22no—
214C2×C4×C67—24no—
215C2×C4×C67—22no—
216C2×C4×C67—22no—
217C2×C4×C67—22no—
218C2×C4×C67—41no—
219C2×C4×C67—42no—
220C2×C4×C67—42no—
221C2×C4×C67—44no—
222C2×C4×C67—44no—
223C2×C4×C67—42no—
224C2×C4×C67—82no—
225C2×C4×C67—82no—
226C2×C4×C67—82no—
227C2×C4×C67—84no—
228C2×C4×C67—82no—
229C2×C4×C67#12012yes—
230C2×C4×C67#12011yescharacteristic
231C2×C4×C67#12011yescharacteristic
232C2×C4×C67C22×D411yescharacteristic
233C2×C4×C67C22×D412yes—
234C2×C4×C67C2×#4112yes—
235C2×C4×C67C2×#4111yescharacteristic
236C8×C67—44no—
237C22×C67—21no—
238C22×C67—21no—
239C22×C67—24no—
240C22×C67—24no—
241C22×C67—44no—
242C22×C67—42no—
243C22×C67—42no—
244C22×C67—42no—
245C22×C67—42no—
246C22×C67—82no—
247C22×C67—84no—
248C22×C67—84no—
249C22×C67—88no—
250C22×C67—82no—
251C22×C67—82no—
252C22×C67—82no—
253C22×C67C2×#12011yescharacteristic
254C22×C67C2×#12111yescharacteristic
255C22×C67#52511yescharacteristic
256C4×C67—24no—
257C4×C67—21no—
258C4×C67—22no—
259C4×C67—44no—
260C4×C67—42no—
261C4×C67—42no—
262C4×C67—42no—
263C4×C67#45312yescomplemented
264C4×C67C2×#12112yes—
265C4×C67#52512yes—
266#32920C6711yesmaximal, characteristic, direct product factor, Sylow
267C2×C67—21no—
268C2×C67—21no—
269C2×C67—44no—
270C2×C67—44no—
271C2×C67—162no—
272C2×C67#284711yescharacteristic
273C2×C67C2×#45311yescharacteristic
274C2×C67#302111yescharacteristic
275C4×H(5,𝔽2)C2×C6711yescharacteristic, complemented
276#3038C2×C6712yescomplemented
277C2×#453C2×C6712yescomplemented
278#2541C2×C6711yescharacteristic, complemented
279#1887C2×C6712yescomplemented
280#1883C2×C6714yescomplemented
281#1806C2×C6712yescomplemented
282#1800C2×C6711yescharacteristic, complemented
283C67#3292011yescharacteristic, direct product factor, Sylow
284C2×H(5,𝔽2)C22×C6711yescharacteristic
285C2×AGL(1,ℤ/8)C22×C6714yes—
286C2×#127C22×C6712yes—
287C2×#127C22×C6711yescharacteristic, complemented
288C2×#121—22no—
289#520—22no—
290#520C22×C6712yes—
291#518—22no—
292#518—22no—
293#518C22×C6711yescharacteristic
294C4×#41—22no—
295C4×#41C22×C6712yes—
296C2×C4×D4—22no—
297C2×C4×D4—22no—
298C2×C4×D4C22×C6711yescharacteristic
299#460—22no—
300#460—22no—
301#459—22no—
302#459—22no—
303#453—28no—
304#423C22×C6712yescomplemented
305C2×#104C22×C6714yes—
306#418C22×C6714yescomplemented
307C2×#100C22×C6712yes—
308#392—22no—
309#390C22×C6712yescomplemented
310C4×C4⋊C4C22×C6711yescharacteristic, complemented
311#343C22×C6712yescomplemented
312#339C22×C6714yescomplemented
313H(5,𝔽2)—24no—
314H(5,𝔽2)—22no—
315H(5,𝔽2)D4×C6714yescomplemented
316C2×#41—22no—
317C2×#41C23×C6712yes—
318C22×D4—22no—
319C22×D4—22no—
320C22×D4—24no—
321C22×D4—21no—
322C22×D4—21no—
323C22×D4C23×C6712yes—
324C22×D4C23×C6711yescharacteristic
325C23×C4—22no—
326C23×C4D4×C6712yes—
327AGL(1,ℤ/8)—28no—
328AGL(1,ℤ/8)—28no—
329C2×SD16—24no—
330C2×D8—24no—
331C2×M16C23×C6714yes—
332#127—24no—
333#127—24no—
334#127—22no—
335#127—22no—
336#121—48no—
337C4×Q8—22no—
338C4×Q8D4×C6712yescomplemented
339C4×Q8D4×C6712yescomplemented
340C4×D4—24no—
341C4×D4—24no—
342C4×D4—24no—
343C4×D4—24no—
344C4×D4—22no—
345C4×D4—22no—
346C4×D4—24no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4D4×C6712yescomplemented
353C4×D4D4×C6712yescomplemented
354#117—24no—
355#117—22no—
356#117—22no—
357#117C23×C6712yes—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4C23×C6711yescharacteristic
362C2×C22⋊C4—22no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4D4×C6712yes—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42C23×C6712yes—
370C2×C42C23×C6711yescharacteristic
371C4⋊C8—22no—
372C4⋊C8—22no—
373#104—28no—
374#102—24no—
375#102—24no—
376#100—24no—
377#95—21no—
378C24—22no—
379C24—22no—
380C24C2×D4×C6712yes—
381#41—28no—
382#41—28no—
383#41—22no—
384#41—22no—
385#41C2×D4×C6714yes—
386#41#41×C6714yes—
387C2×Q8C2×D4×C6712yes—
388C2×D4—24no—
389C2×D4—28no—
390C2×D4—28no—
391C2×D4—28no—
392C2×D4—22no—
393C2×D4—28no—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—44no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—42no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—82no—
407C2×D4—84no—
408C2×D4C2×D4×C6712yes—
409C2×D4C2×D4×C6712yes—
410C2×D4#41×C6712yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—44no—
422C22×C4—84no—
423C22×C4C2×D4×C6711yescharacteristic
424C22×C4C2×D4×C6712yes—
425C22×C4C2×D4×C6711yescharacteristic
426C22×C4C24×C6711yescharacteristic
427SD16—48no—
428D8—48no—
429M16—24no—
430M16—24no—
431C2×C8—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4C2×D4×C6712yes—
438C4⋊C4C2×D4×C6712yes—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—24no—
445C22⋊C4—48no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42C2×D4×C6712yes—
453C42C2×D4×C6712yes—
454C23—28no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—82no—
466C23—82no—
467C23#121×C6714yes—
468C23C22×D4×C6711yescharacteristic
469Q8—22no—
470Q8#121×C6714yescomplemented
471D4—216no—
472D4—24no—
473D4—24no—
474D4—24no—
475D4—22no—
476D4—24no—
477D4—44no—
478D4—44no—
479D4—82no—
480D4—84no—
481D4—816no—
482D4—82no—
483D4#121×C6714yescomplemented
484C2×C4—28no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—84no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4#120×C6712yes—
505C2×C4#120×C6711yescharacteristic
506C2×C4#120×C6711yescharacteristic
507C2×C4C22×D4×C6711yescharacteristic
508C2×C4C22×D4×C6712yes—
509C2×C4C2×#41×C6712yes—
510C2×C4C2×#41×C6711yescharacteristic
511C8—44no—
512C22—21no—
513C22—21no—
514C22—24no—
515C22—24no—
516C22—44no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—82no—
522C22—84no—
523C22—84no—
524C22—88no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22C2×#120×C6711yescharacteristic
529C22C2×#121×C6711yescharacteristic
530C22#525×C6711yescharacteristic
531C4—21no—
532C4—22no—
533C4—24no—
534C4—44no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4#453×C6712yescomplemented
539C4C2×#121×C6712yes—
540C4#525×C6712yes—
541C2—21no—
542C2—21no—
543C2—44no—
544C2—44no—
545C2—162no—
546C2C67×#284711yescharacteristic
547C2C2×#453×C6711yescharacteristic
548C2C67×#302111yescharacteristic
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC22×C4C24×C67
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
1. C4×H(5,𝔽2)×C67C2semidirect
2. C67×#3038C2semidirect
3. C2×#453×C67C2semidirect
4. C67×#2541C2semidirect
5. C67×#1887C2semidirect
6. C67×#1883C2semidirect
7. C67×#1806C2semidirect
8. C67×#1800C2semidirect
12. C2×#127×C67C22semidirect
29. #423×C67C22semidirect
31. #418×C67C22semidirect
34. #390×C67C22semidirect
35. C4×C4⋊C4×C67C22semidirect
36. #343×C67C22semidirect
37. #339×C67C22semidirect
40. H(5,𝔽2)×C67D4semidirect
63. C4×Q8×C67D4semidirect
64. C4×Q8×C67D4semidirect
77. C4×D4×C67D4semidirect
78. C4×D4×C67D4semidirect
195. Q8×C67#121semidirect
208. D4×C67#121semidirect
263. C4×C67#453semidirect
266. #32920C67direct (N × H)
275. C4×H(5,𝔽2)C2×C67semidirect
276. #3038C2×C67semidirect
277. C2×#453C2×C67semidirect
278. #2541C2×C67semidirect
279. #1887C2×C67semidirect
280. #1883C2×C67semidirect
281. #1806C2×C67semidirect
282. #1800C2×C67semidirect
283. C67#32920direct (N × H)
287. C2×#127C22×C67semidirect
304. #423C22×C67semidirect
306. #418C22×C67semidirect
309. #390C22×C67semidirect
310. C4×C4⋊C4C22×C67semidirect
311. #343C22×C67semidirect
312. #339C22×C67semidirect
315. H(5,𝔽2)D4×C67semidirect
338. C4×Q8D4×C67semidirect
339. C4×Q8D4×C67semidirect
352. C4×D4D4×C67semidirect
353. C4×D4D4×C67semidirect
470. Q8#121×C67semidirect
483. D4#121×C67semidirect
538. C4#453×C67semidirect

Maximal subgroups

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

subgroupindex
C4×H(5,𝔽2)×C672
C67×#30382
C2×#453×C672
C67×#25412
C67×#18872
C67×#18832
C67×#18062
C67×#18002
#3292067