Finite Group Databasebeta

← C67×#60317 #30739568

Subgroups — C67×#60317 #30739568

Order 17152.

1892 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3481C211yesmaximal, characteristic
2C67×#2993C214yesmaximal, complemented
3C67×#2991C212yesmaximal, complemented
4C67×#2979C211yesmaximal, characteristic, complemented
5C67×#2978C214yesmaximal, complemented
6C67×#2975C214yesmaximal, complemented
7C67×#2969C214yesmaximal, complemented
8C67×#2965C212yesmaximal, complemented
9D4×M16×C67C214yesmaximal, complemented
10C67×#2931C211yesmaximal, characteristic, complemented
11C67×#2930C214yesmaximal, complemented
12C2×#131×C67C2212yescomplemented
13C22×M16×C67C2214yescomplemented
14#559×C67C411yescharacteristic
15#554×C67C412yes—
16#553×C67C414yes—
17#551×C67C414yes—
18#546×C67C414yes—
19D42×C67C411yescharacteristic
20#532×C67C2211yescharacteristic
21#529×C67C2214yes—
22#525×C67C2214yes—
23C4×#41×C67C2212yes—
24C2×C4×D4×C67C2214yes—
25#446×C67C2212yescomplemented
26#446×C67C2212yescomplemented
27C8×Q8×C67C2214yescomplemented
28#436×C67C2212yes—
29#436×C67C2214yes—
30#436×C67C2212yes—
31#436×C67C2218yescomplemented
32#436×C67C2218yescomplemented
33#436×C67C2214yescomplemented
34#436×C67C2218yescomplemented
35#436×C67C2214yescomplemented
36#435×C67C2214yescomplemented
37#435×C67C2218yescomplemented
38#435×C67C2218yescomplemented
39#435×C67C2214yescomplemented
40C8×D4×C67C2214yescomplemented
41C8×D4×C67C2214yescomplemented
42#433×C67C2218yescomplemented
43#431×C67C2214yescomplemented
44#431×C67C2212yescomplemented
45#431×C67C2214yescomplemented
46#424×C67C2214yescomplemented
47#423×C67C2211yescharacteristic, complemented
48#423×C67C2214yescomplemented
49#408×C67C2214yescomplemented
50#408×C67C2212yescomplemented
51#407×C67C2214yescomplemented
52#407×C67C2218yescomplemented
53#405×C67C2214yescomplemented
54C4×M16×C67C2214yescomplemented
55C2×C4×C8×C67C2211yescharacteristic, complemented
56C2×#41×C67C2×C412yes—
57C2×#41×C67C2312yes—
58C22×D4×C67C2×C414yes—
59C23×C4×C67C2314yes—
60#131×C67—44no—
61C2×M16×C67—48no—
62C2×M16×C67C2314yes—
63C2×M16×C67C2318yes—
64C2×M16×C67C2314yes—
65C2×M16×C67C2312yes—
66C2×M16×C67C2314yes—
67C2×M16×C67C2314yes—
68C22×C8×C67C2314yes—
69C22×C8×C67C2311yescharacteristic
70C22×C8×C67C2314yes—
71C22×C8×C67C2311yescharacteristic
72C4⋊Q8×C67C2×C411yescharacteristic
73#127×C67C2×C411yescharacteristic
74#126×C67C2×C418yes—
75#125×C67C2×C414yes—
76#124×C67C2×C414yes—
77#124×C67C2×C412yes—
78#123×C67C2×C414yes—
79#123×C67C2×C414yes—
80#123×C67C2×C418yes—
81C22⋊Q8×C67C2×C418yes—
82C22⋊Q8×C67C2×C414yes—
83#121×C67C2×C418yes—
84#121×C67C2×C414yes—
85#121×C67C2×C414yes—
86#120×C67C2×C414yes—
87#120×C67C2×C414yes—
88C4×Q8×C67C2×C412yes—
89C4×Q8×C67C2312yes—
90C4×D4×C67C2×C414yes—
91C4×D4×C67C2×C418yes—
92C4×D4×C67C2×C414yes—
93C4×D4×C67C2×C412yes—
94C4×D4×C67C2×C414yes—
95C4×D4×C67C2312yes—
96C4×D4×C67C2318yes—
97C4×D4×C67C2314yes—
98C4×D4×C67C2314yes—
99#117×C67C2×C414yes—
100#117×C67C2×C412yes—
101#117×C67C2314yes—
102C2×C4⋊C4×C67C2×C414yes—
103C2×C22⋊C4×C67C2×C418yes—
104C2×C42×C67C2314yes—
105C2×C42×C67C2311yescharacteristic
106C4⋊C8×C67C2312yes—
107C4⋊C8×C67C2314yes—
108C4⋊C8×C67C2318yes—
109C4⋊C8×C67C2314yes—
110C4⋊C8×C67C2312yes—
111C4⋊C8×C67C2314yes—
112C22⋊C8×C67C2314yes—
113C22⋊C8×C67C2314yes—
114C22⋊C8×C67C2318yes—
115C22⋊C8×C67C2318yes—
116C22⋊C8×C67C2318yes—
117C22⋊C8×C67C2314yes—
118#97×C67C2314yes—
119#97×C67C2314yes—
120C4×C8×C67C2312yes—
121C4×C8×C67C2312yes—
122C4×C8×C67C2314yes—
123C4×C8×C67C2318yes—
124C24×C67C22×C414yes—
125#41×C67—42no—
126#41×C67—42no—
127#41×C67—44no—
128C2×Q8×C67C22×C412yes—
129C2×Q8×C67C22×C412yes—
130C2×D4×C67—48no—
131C2×D4×C67C22×C412yes—
132C2×D4×C67C22×C414yes—
133C2×D4×C67C22×C414yes—
134C2×D4×C67C22×C418yes—
135C2×D4×C67C22×C412yes—
136C2×D4×C67C22×C414yes—
137C22×C4×C67—44no—
138C22×C4×C67—44no—
139C22×C4×C67C22×C414yes—
140C22×C4×C67C22×C412yes—
141C22×C4×C67C22×C414yes—
142C22×C4×C67C22×C418yes—
143C22×C4×C67C22×C414yes—
144C22×C4×C67C2411yescharacteristic
145C22×C4×C67C2414yes—
146C22×C4×C67C2414yes—
147M16×C67—28no—
148M16×C67—48no—
149M16×C67—44no—
150M16×C67—44no—
151M16×C67—44no—
152M16×C67—42no—
153C2×C8×C67—28no—
154C2×C8×C67—22no—
155C2×C8×C67—41no—
156C2×C8×C67—44no—
157C2×C8×C67C2414yes—
158C2×C8×C67C2414yes—
159C2×C8×C67C2411yescharacteristic
160C2×C8×C67C2412yes—
161C2×C8×C67C2414yes—
162C2×C8×C67C2411yescharacteristic
163C4⋊C4×C67C22×C412yes—
164C4⋊C4×C67C22×C414yes—
165C4⋊C4×C67C22×C418yes—
166C4⋊C4×C67C22×C412yes—
167C4⋊C4×C67C22×C414yes—
168C4⋊C4×C67C22×C414yes—
169C4⋊C4×C67C22×C418yes—
170C22⋊C4×C67C22×C414yes—
171C22⋊C4×C67C22×C418yes—
172C22⋊C4×C67C22×C418yes—
173C22⋊C4×C67C22×C418yes—
174C22⋊C4×C67C22×C414yes—
175C22⋊C4×C67C22×C414yes—
176C22⋊C4×C67C22×C414yes—
177C22⋊C4×C67C22×C414yes—
178C42×C67C22×C411yescharacteristic
179C42×C67C22×C414yes—
180C42×C67C22×C414yes—
181C42×C67C22×C411yescharacteristic
182C42×C67C2412yes—
183C42×C67C2414yes—
184C23×C67—44no—
185C23×C67—48no—
186C23×C67C23×C414yes—
187C23×C67C23×C414yes—
188C23×C67C23×C411yescharacteristic
189Q8×C67—24no—
190Q8×C67—42no—
191D4×C67—24no—
192D4×C67—28no—
193D4×C67—48no—
194D4×C67—44no—
195D4×C67—42no—
196D4×C67—44no—
197C2×C4×C67—28no—
198C2×C4×C67—24no—
199C2×C4×C67—24no—
200C2×C4×C67—21no—
201C2×C4×C67—24no—
202C2×C4×C67—21no—
203C2×C4×C67—24no—
204C2×C4×C67—24no—
205C2×C4×C67—44no—
206C2×C4×C67—48no—
207C2×C4×C67—44no—
208C2×C4×C67C23×C412yes—
209C2×C4×C67C23×C414yes—
210C2×C4×C67C23×C411yescharacteristic
211C2×C4×C67C23×C414yes—
212C2×C4×C67C23×C414yes—
213C2×C4×C67C23×C414yes—
214C2×C4×C67C23×C412yes—
215C2×C4×C67C2511yescharacteristic
216C8×C67—24no—
217C8×C67—21no—
218C8×C67—24no—
219C8×C67—22no—
220C8×C67—24no—
221C8×C67H(5,𝔽2)12yes—
222C22×C67—24no—
223C22×C67—24no—
224C22×C67—21no—
225C22×C67—22no—
226C22×C67—24no—
227C22×C67—24no—
228C22×C67—28no—
229C22×C67—88no—
230C22×C67C24×C411yescharacteristic
231C4×C67—24no—
232C4×C67—24no—
233C4×C67—21no—
234C4×C67—22no—
235C4×C67—24no—
236C4×C67—24no—
237C4×C67#56814yes—
238C4×C67C2×H(5,𝔽2)11yescharacteristic
239C4×C67#58511yescharacteristic
240#60317C6711yesmaximal, characteristic, direct product factor, Sylow
241C2×C67—41no—
242C2×C67—44no—
243C2×C67—44no—
244C2×C67C4×H(5,𝔽2)11yescharacteristic
245C2×C67C2×#56812yes—
246#3481C2×C6711yescharacteristic
247#2993C2×C6714yescomplemented
248#2991C2×C6712yescomplemented
249#2979C2×C6711yescharacteristic, complemented
250#2978C2×C6714yescomplemented
251#2975C2×C6714yescomplemented
252#2969C2×C6714yescomplemented
253#2965C2×C6712yescomplemented
254D4×M16C2×C6714yescomplemented
255#2931C2×C6711yescharacteristic, complemented
256#2930C2×C6714yescomplemented
257C67#6031711yescharacteristic, direct product factor, Sylow
258C2×#131C22×C6712yescomplemented
259C22×M16C22×C6714yescomplemented
260#559C4×C6711yescharacteristic
261#554C4×C6712yes—
262#553C4×C6714yes—
263#551C4×C6714yes—
264#546C4×C6714yes—
265D42C4×C6711yescharacteristic
266#532C22×C6711yescharacteristic
267#529C22×C6714yes—
268#525C22×C6714yes—
269C4×#41C22×C6712yes—
270C2×C4×D4C22×C6714yes—
271#446C22×C6712yescomplemented
272#446C22×C6712yescomplemented
273C8×Q8C22×C6714yescomplemented
274#436C22×C6712yes—
275#436C22×C6714yes—
276#436C22×C6712yes—
277#436C22×C6718yescomplemented
278#436C22×C6718yescomplemented
279#436C22×C6714yescomplemented
280#436C22×C6718yescomplemented
281#436C22×C6714yescomplemented
282#435C22×C6714yescomplemented
283#435C22×C6718yescomplemented
284#435C22×C6718yescomplemented
285#435C22×C6714yescomplemented
286C8×D4C22×C6714yescomplemented
287C8×D4C22×C6714yescomplemented
288#433C22×C6718yescomplemented
289#431C22×C6714yescomplemented
290#431C22×C6712yescomplemented
291#431C22×C6714yescomplemented
292#424C22×C6714yescomplemented
293#423C22×C6711yescharacteristic, complemented
294#423C22×C6714yescomplemented
295#408C22×C6714yescomplemented
296#408C22×C6712yescomplemented
297#407C22×C6714yescomplemented
298#407C22×C6718yescomplemented
299#405C22×C6714yescomplemented
300C4×M16C22×C6714yescomplemented
301C2×C4×C8C22×C6711yescharacteristic, complemented
302C2×#41C2×C4×C6712yes—
303C2×#41C23×C6712yes—
304C22×D4C2×C4×C6714yes—
305C23×C4C23×C6714yes—
306#131—44no—
307C2×M16—48no—
308C2×M16C23×C6714yes—
309C2×M16C23×C6718yes—
310C2×M16C23×C6714yes—
311C2×M16C23×C6712yes—
312C2×M16C23×C6714yes—
313C2×M16C23×C6714yes—
314C22×C8C23×C6714yes—
315C22×C8C23×C6711yescharacteristic
316C22×C8C23×C6714yes—
317C22×C8C23×C6711yescharacteristic
318C4⋊Q8C2×C4×C6711yescharacteristic
319#127C2×C4×C6711yescharacteristic
320#126C2×C4×C6718yes—
321#125C2×C4×C6714yes—
322#124C2×C4×C6714yes—
323#124C2×C4×C6712yes—
324#123C2×C4×C6714yes—
325#123C2×C4×C6714yes—
326#123C2×C4×C6718yes—
327C22⋊Q8C2×C4×C6718yes—
328C22⋊Q8C2×C4×C6714yes—
329#121C2×C4×C6718yes—
330#121C2×C4×C6714yes—
331#121C2×C4×C6714yes—
332#120C2×C4×C6714yes—
333#120C2×C4×C6714yes—
334C4×Q8C2×C4×C6712yes—
335C4×Q8C23×C6712yes—
336C4×D4C2×C4×C6714yes—
337C4×D4C2×C4×C6718yes—
338C4×D4C2×C4×C6714yes—
339C4×D4C2×C4×C6712yes—
340C4×D4C2×C4×C6714yes—
341C4×D4C23×C6712yes—
342C4×D4C23×C6718yes—
343C4×D4C23×C6714yes—
344C4×D4C23×C6714yes—
345#117C2×C4×C6714yes—
346#117C2×C4×C6712yes—
347#117C23×C6714yes—
348C2×C4⋊C4C2×C4×C6714yes—
349C2×C22⋊C4C2×C4×C6718yes—
350C2×C42C23×C6714yes—
351C2×C42C23×C6711yescharacteristic
352C4⋊C8C23×C6712yes—
353C4⋊C8C23×C6714yes—
354C4⋊C8C23×C6718yes—
355C4⋊C8C23×C6714yes—
356C4⋊C8C23×C6712yes—
357C4⋊C8C23×C6714yes—
358C22⋊C8C23×C6714yes—
359C22⋊C8C23×C6714yes—
360C22⋊C8C23×C6718yes—
361C22⋊C8C23×C6718yes—
362C22⋊C8C23×C6718yes—
363C22⋊C8C23×C6714yes—
364#97C23×C6714yes—
365#97C23×C6714yes—
366C4×C8C23×C6712yes—
367C4×C8C23×C6712yes—
368C4×C8C23×C6714yes—
369C4×C8C23×C6718yes—
370C24C22×C4×C6714yes—
371#41—42no—
372#41—42no—
373#41—44no—
374C2×Q8C22×C4×C6712yes—
375C2×Q8C22×C4×C6712yes—
376C2×D4—48no—
377C2×D4C22×C4×C6712yes—
378C2×D4C22×C4×C6714yes—
379C2×D4C22×C4×C6714yes—
380C2×D4C22×C4×C6718yes—
381C2×D4C22×C4×C6712yes—
382C2×D4C22×C4×C6714yes—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4C22×C4×C6714yes—
386C22×C4C22×C4×C6712yes—
387C22×C4C22×C4×C6714yes—
388C22×C4C22×C4×C6718yes—
389C22×C4C22×C4×C6714yes—
390C22×C4C24×C6711yescharacteristic
391C22×C4C24×C6714yes—
392C22×C4C24×C6714yes—
393M16—28no—
394M16—48no—
395M16—44no—
396M16—42no—
397M16—44no—
398M16—44no—
399C2×C8—28no—
400C2×C8—22no—
401C2×C8—44no—
402C2×C8—41no—
403C2×C8C24×C6714yes—
404C2×C8C24×C6712yes—
405C2×C8C24×C6711yescharacteristic
406C2×C8C24×C6714yes—
407C2×C8C24×C6714yes—
408C2×C8C24×C6711yescharacteristic
409C4⋊C4C22×C4×C6712yes—
410C4⋊C4C22×C4×C6714yes—
411C4⋊C4C22×C4×C6718yes—
412C4⋊C4C22×C4×C6712yes—
413C4⋊C4C22×C4×C6714yes—
414C4⋊C4C22×C4×C6714yes—
415C4⋊C4C22×C4×C6718yes—
416C22⋊C4C22×C4×C6714yes—
417C22⋊C4C22×C4×C6718yes—
418C22⋊C4C22×C4×C6718yes—
419C22⋊C4C22×C4×C6718yes—
420C22⋊C4C22×C4×C6714yes—
421C22⋊C4C22×C4×C6714yes—
422C22⋊C4C22×C4×C6714yes—
423C22⋊C4C22×C4×C6714yes—
424C42C22×C4×C6711yescharacteristic
425C42C22×C4×C6714yes—
426C42C22×C4×C6714yes—
427C42C22×C4×C6711yescharacteristic
428C42C24×C6712yes—
429C42C24×C6714yes—
430C23—44no—
431C23—48no—
432C23C23×C4×C6714yes—
433C23C23×C4×C6714yes—
434C23C23×C4×C6711yescharacteristic
435Q8—24no—
436Q8—42no—
437D4—24no—
438D4—28no—
439D4—48no—
440D4—44no—
441D4—44no—
442D4—42no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—44no—
452C2×C4—48no—
453C2×C4—44no—
454C2×C4C23×C4×C6712yes—
455C2×C4C23×C4×C6714yes—
456C2×C4C23×C4×C6711yescharacteristic
457C2×C4C23×C4×C6714yes—
458C2×C4C23×C4×C6714yes—
459C2×C4C23×C4×C6714yes—
460C2×C4C23×C4×C6712yes—
461C2×C4C25×C6711yescharacteristic
462C8—24no—
463C8—21no—
464C8—24no—
465C8—22no—
466C8—24no—
467C8H(5,𝔽2)×C6712yes—
468C22—22no—
469C22—24no—
470C22—24no—
471C22—28no—
472C22—24no—
473C22—24no—
474C22—21no—
475C22—88no—
476C22C24×C4×C6711yescharacteristic
477C4—22no—
478C4—24no—
479C4—24no—
480C4—24no—
481C4—24no—
482C4—21no—
483C4#568×C6714yes—
484C4C2×H(5,𝔽2)×C6711yescharacteristic
485C4#585×C6711yescharacteristic
486C2—44no—
487C2—44no—
488C2—41no—
489C2C4×H(5,𝔽2)×C6711yescharacteristic
490C2C2×#568×C6712yes—
subgroupG/H
centerC2×C4×C67C25
commutatorC22C24×C4×C67
Frattini, cosocleC2×C4C25×C67
socleC22×C67C24×C4
prime radicalC67×#3481C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2993C2semidirect
3. C67×#2991C2semidirect
4. C67×#2979C2semidirect
5. C67×#2978C2semidirect
6. C67×#2975C2semidirect
7. C67×#2969C2semidirect
8. C67×#2965C2semidirect
9. D4×M16×C67C2semidirect
10. C67×#2931C2semidirect
11. C67×#2930C2semidirect
12. C2×#131×C67C22semidirect
13. C22×M16×C67C22semidirect
25. #446×C67C22semidirect
26. #446×C67C22semidirect
27. C8×Q8×C67C22semidirect
31. #436×C67C22semidirect
32. #436×C67C22semidirect
33. #436×C67C22semidirect
34. #436×C67C22semidirect
35. #436×C67C22semidirect
36. #435×C67C22semidirect
37. #435×C67C22semidirect
38. #435×C67C22semidirect
39. #435×C67C22semidirect
40. C8×D4×C67C22semidirect
41. C8×D4×C67C22semidirect
42. #433×C67C22semidirect
43. #431×C67C22semidirect
44. #431×C67C22semidirect
45. #431×C67C22semidirect
46. #424×C67C22semidirect
47. #423×C67C22semidirect
48. #423×C67C22semidirect
49. #408×C67C22semidirect
50. #408×C67C22semidirect
51. #407×C67C22semidirect
52. #407×C67C22semidirect
53. #405×C67C22semidirect
54. C4×M16×C67C22semidirect
55. C2×C4×C8×C67C22semidirect
240. #60317C67direct (N × H)
247. #2993C2×C67semidirect
248. #2991C2×C67semidirect
249. #2979C2×C67semidirect
250. #2978C2×C67semidirect
251. #2975C2×C67semidirect
252. #2969C2×C67semidirect
253. #2965C2×C67semidirect
254. D4×M16C2×C67semidirect
255. #2931C2×C67semidirect
256. #2930C2×C67semidirect
257. C67#60317direct (N × H)
258. C2×#131C22×C67semidirect
259. C22×M16C22×C67semidirect
271. #446C22×C67semidirect
272. #446C22×C67semidirect
273. C8×Q8C22×C67semidirect
277. #436C22×C67semidirect
278. #436C22×C67semidirect
279. #436C22×C67semidirect
280. #436C22×C67semidirect
281. #436C22×C67semidirect
282. #435C22×C67semidirect
283. #435C22×C67semidirect
284. #435C22×C67semidirect
285. #435C22×C67semidirect
286. C8×D4C22×C67semidirect
287. C8×D4C22×C67semidirect
288. #433C22×C67semidirect
289. #431C22×C67semidirect
290. #431C22×C67semidirect
291. #431C22×C67semidirect
292. #424C22×C67semidirect
293. #423C22×C67semidirect
294. #423C22×C67semidirect
295. #408C22×C67semidirect
296. #408C22×C67semidirect
297. #407C22×C67semidirect
298. #407C22×C67semidirect
299. #405C22×C67semidirect
300. C4×M16C22×C67semidirect
301. C2×C4×C8C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34812
C67×#29932
C67×#29912
C67×#29792
C67×#29782
C67×#29752
C67×#29692
C67×#29652
D4×M16×C672
C67×#29312
C67×#29302
#6031767