Finite Group Databasebeta

← C67×#61586 #30740837

Subgroups — C67×#61586 #30740837

Order 17152.

2098 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3501C211yesmaximal, characteristic, complemented
2C2×#547×C67C211yesmaximal, characteristic
3C67×#3331C218yesmaximal, complemented
4C67×#3276C218yesmaximal, complemented
5C67×#3209C212yesmaximal, complemented
6C67×#3195C214yesmaximal, complemented
7C67×#3101C212yesmaximal, complemented
8C67×#3007C214yesmaximal, complemented
9C67×#2975C211yesmaximal, characteristic, complemented
10C22×#41×C67C2212yes—
11C2×#137×C67C2218yescomplemented
12#554×C67—24no—
13#554×C67C2214yescomplemented
14D4×Q8×C67C2214yescomplemented
15#547×C67—24no—
16#547×C67C2218yes—
17#544×C67—21no—
18#543×C67—22no—
19#533×C67C2212yes—
20C2×C4⋊Q8×C67C2211yescharacteristic
21C2×#123×C67—22no—
22C2×C22⋊Q8×C67C2212yes—
23C2×C22⋊Q8×C67C2214yes—
24C2×#121×C67—21no—
25#518×C67—21no—
26C2×C4×D4×C67C2211yescharacteristic
27C2×C4×D4×C67C2211yescharacteristic
28C22×C4⋊C4×C67—21no—
29#494×C67C2214yescomplemented
30#487×C67C2214yescomplemented
31#483×C67C2214yes—
32#483×C67C2218yescomplemented
33#476×C67C2214yescomplemented
34#470×C67C22116yescomplemented
35#462×C67C2218yescomplemented
36#461×C67C2214yescomplemented
37#452×C67C22116yescomplemented
38#451×C67C2218yes—
39#451×C67C2218yescomplemented
40#450×C67C2214yes—
41C4×SD16×C67C2218yescomplemented
42#436×C67C2218yescomplemented
43#431×C67C2212yes—
44#417×C67C2218yescomplemented
45#407×C67C2214yes—
46C2×#41×C67—24no—
47C2×#41×C67—48no—
48C2×#41×C67C2318yes—
49C22×Q8×C67C2312yes—
50C22×Q8×C67C2314yes—
51C22×D4×C67—22no—
52C22×D4×C67D411yescharacteristic
53C22×D4×C67C2311yescharacteristic
54C23×C4×C67—21no—
55C23×C4×C67—22no—
56C23×C4×C67C2312yes—
57#137×C67—416no—
58C2×Q16×C67C23116yes—
59C2×SD16×C67C2318yes—
60C2×M16×C67C2318yes—
61C4⋊Q8×C67—24no—
62C4⋊Q8×C67—22no—
63C4⋊Q8×C67C2312yes—
64C4⋊Q8×C67C2314yes—
65C4⋊Q8×C67C2312yes—
66#125×C67—24no—
67#125×C67—22no—
68#123×C67—216no—
69C22⋊Q8×C67—24no—
70C22⋊Q8×C67—24no—
71C22⋊Q8×C67—24no—
72C22⋊Q8×C67—22no—
73C22⋊Q8×C67—28no—
74C22⋊Q8×C67—28no—
75C22⋊Q8×C67C2314yes—
76C22⋊Q8×C67C2318yes—
77C22⋊Q8×C67C2318yes—
78#121×C67—28no—
79C4×Q8×C67—22no—
80C4×Q8×C67C2314yes—
81C4×D4×C67—24no—
82C4×D4×C67—24no—
83C4×D4×C67—22no—
84C4×D4×C67D414yes—
85C4×D4×C67C2314yes—
86C4×D4×C67C2314yes—
87#117×C67—22no—
88C2×C4⋊C4×C67—21no—
89C2×C4⋊C4×C67—22no—
90C2×C4⋊C4×C67—24no—
91C2×C4⋊C4×C67—21no—
92C2×C4⋊C4×C67—28no—
93C2×C4⋊C4×C67D411yescharacteristic
94C2×C4⋊C4×C67C2311yescharacteristic
95C2×C4⋊C4×C67C2314yes—
96C2×C22⋊C4×C67—24no—
97C2×C22⋊C4×C67—22no—
98C2×C22⋊C4×C67—21no—
99C2×C22⋊C4×C67D412yes—
100C2×C42×C67C2311yescharacteristic
101#106×C67C2318yes—
102C4⋊C8×C67C2314yes—
103#103×C67C23116yes—
104#103×C67C2318yes—
105#102×C67C2318yes—
106C22⋊C8×C67C2314yes—
107C22⋊C8×C67C2318yes—
108C4×C8×C67C2314yes—
109C24×C67—21no—
110C24×C67C2×D412yes—
111#41×C67—216no—
112#41×C67—48no—
113#41×C67—416no—
114C2×Q8×C67—22no—
115C2×Q8×C67—22no—
116C2×Q8×C67—24no—
117C2×Q8×C67—44no—
118C2×Q8×C67—48no—
119C2×Q8×C67C2414yes—
120C2×Q8×C67C2414yes—
121C2×D4×C67—24no—
122C2×D4×C67—28no—
123C2×D4×C67—24no—
124C2×D4×C67—21no—
125C2×D4×C67—48no—
126C2×D4×C67C2×D414yes—
127C2×D4×C67C2412yes—
128C22×C4×C67—21no—
129C22×C4×C67—22no—
130C22×C4×C67—24no—
131C22×C4×C67—24no—
132C22×C4×C67—24no—
133C22×C4×C67—28no—
134C22×C4×C67—44no—
135C22×C4×C67—48no—
136C22×C4×C67—44no—
137C22×C4×C67C2×D411yescharacteristic
138C22×C4×C67C2×D412yes—
139C22×C4×C67C2×D414yes—
140C22×C4×C67C2×D411yescharacteristic
141C22×C4×C67C2411yescharacteristic
142C22×C4×C67C2414yes—
143Q16×C67—416no—
144SD16×C67—216no—
145M16×C67—48no—
146C2×C8×C67C2418yes—
147C4⋊C4×C67—24no—
148C4⋊C4×C67—28no—
149C4⋊C4×C67—24no—
150C4⋊C4×C67—28no—
151C4⋊C4×C67—24no—
152C4⋊C4×C67—21no—
153C4⋊C4×C67—24no—
154C4⋊C4×C67—22no—
155C4⋊C4×C67—22no—
156C4⋊C4×C67—22no—
157C4⋊C4×C67—24no—
158C4⋊C4×C67C2×D414yes—
159C4⋊C4×C67C2414yes—
160C4⋊C4×C67C2412yes—
161C22⋊C4×C67—24no—
162C22⋊C4×C67—28no—
163C22⋊C4×C67—28no—
164C22⋊C4×C67—24no—
165C22⋊C4×C67C2×D418yes—
166C42×C67—24no—
167C42×C67C2×D412yes—
168C42×C67C2412yes—
169C23×C67—24no—
170C23×C67—22no—
171C23×C67—24no—
172C23×C67—42no—
173C23×C67—42no—
174C23×C67—42no—
175C23×C67—42no—
176C23×C67C22×D414yes—
177C23×C67C22×D411yescharacteristic
178Q8×C67—28no—
179Q8×C67—28no—
180Q8×C67—44no—
181Q8×C67—82no—
182Q8×C67—84no—
183D4×C67—24no—
184D4×C67—28no—
185D4×C67—22no—
186D4×C67—416no—
187D4×C67H(5,𝔽2)14yes—
188C2×C4×C67—28no—
189C2×C4×C67—22no—
190C2×C4×C67—24no—
191C2×C4×C67—24no—
192C2×C4×C67—22no—
193C2×C4×C67—24no—
194C2×C4×C67—42no—
195C2×C4×C67—44no—
196C2×C4×C67—41no—
197C2×C4×C67—416no—
198C2×C4×C67—44no—
199C2×C4×C67—42no—
200C2×C4×C67—48no—
201C2×C4×C67—84no—
202C2×C4×C67—84no—
203C2×C4×C67#13714yescomplemented
204C2×C4×C67C22×D411yescharacteristic
205C2×C4×C67C22×D412yes—
206C2×C4×C67C22×D414yes—
207C2×C4×C67C22×D412yes—
208C2×C4×C67C2511yescharacteristic
209C8×C67—28no—
210C22×C67—21no—
211C22×C67—21no—
212C22×C67—24no—
213C22×C67—24no—
214C22×C67—24no—
215C22×C67—24no—
216C22×C67—48no—
217C22×C67—42no—
218C22×C67—42no—
219C22×C67—84no—
220C22×C67C2×#13712yes—
221C22×C67C23×D411yescharacteristic
222C4×C67—22no—
223C4×C67—24no—
224C4×C67—21no—
225C4×C67—44no—
226C4×C67—44no—
227C4×C67—44no—
228C4×C67—42no—
229C4×C67C2×#13714yes—
230C4×C67C2×H(5,𝔽2)12yes—
231#61586C6711yesmaximal, characteristic, direct product factor, Sylow
232C2×C67—22no—
233C2×C67—44no—
234C2×C67—44no—
235C2×C67#348411yescharacteristic
236C2×C67C22×#13711yescharacteristic
237C2×C67#358611yescharacteristic
238#3501C2×C6711yescharacteristic, complemented
239C2×#547C2×C6711yescharacteristic
240#3331C2×C6718yescomplemented
241#3276C2×C6718yescomplemented
242#3209C2×C6712yescomplemented
243#3195C2×C6714yescomplemented
244#3101C2×C6712yescomplemented
245#3007C2×C6714yescomplemented
246#2975C2×C6711yescharacteristic, complemented
247C67#6158611yescharacteristic, direct product factor, Sylow
248C22×#41C22×C6712yes—
249C2×#137C22×C6718yescomplemented
250#554—24no—
251#554C22×C6714yescomplemented
252D4×Q8C22×C6714yescomplemented
253#547—24no—
254#547C22×C6718yes—
255#544—21no—
256#543—22no—
257#533C22×C6712yes—
258C2×C4⋊Q8C22×C6711yescharacteristic
259C2×#123—22no—
260C2×C22⋊Q8C22×C6712yes—
261C2×C22⋊Q8C22×C6714yes—
262C2×#121—21no—
263#518—21no—
264C2×C4×D4C22×C6711yescharacteristic
265C2×C4×D4C22×C6711yescharacteristic
266C22×C4⋊C4—21no—
267#494C22×C6714yescomplemented
268#487C22×C6714yescomplemented
269#483C22×C6714yes—
270#483C22×C6718yescomplemented
271#476C22×C6714yescomplemented
272#470C22×C67116yescomplemented
273#462C22×C6718yescomplemented
274#461C22×C6714yescomplemented
275#452C22×C67116yescomplemented
276#451C22×C6718yes—
277#451C22×C6718yescomplemented
278#450C22×C6714yes—
279C4×SD16C22×C6718yescomplemented
280#436C22×C6718yescomplemented
281#431C22×C6712yes—
282#417C22×C6718yescomplemented
283#407C22×C6714yes—
284C2×#41—24no—
285C2×#41—48no—
286C2×#41C23×C6718yes—
287C22×Q8C23×C6712yes—
288C22×Q8C23×C6714yes—
289C22×D4—22no—
290C22×D4D4×C6711yescharacteristic
291C22×D4C23×C6711yescharacteristic
292C23×C4—21no—
293C23×C4—22no—
294C23×C4C23×C6712yes—
295#137—416no—
296C2×Q16C23×C67116yes—
297C2×SD16C23×C6718yes—
298C2×M16C23×C6718yes—
299C4⋊Q8—24no—
300C4⋊Q8—22no—
301C4⋊Q8C23×C6714yes—
302C4⋊Q8C23×C6712yes—
303C4⋊Q8C23×C6712yes—
304#125—24no—
305#125—22no—
306#123—216no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—24no—
310C22⋊Q8—22no—
311C22⋊Q8—28no—
312C22⋊Q8—28no—
313C22⋊Q8C23×C6714yes—
314C22⋊Q8C23×C6718yes—
315C22⋊Q8C23×C6718yes—
316#121—28no—
317C4×Q8—22no—
318C4×Q8C23×C6714yes—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—22no—
322C4×D4D4×C6714yes—
323C4×D4C23×C6714yes—
324C4×D4C23×C6714yes—
325#117—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—28no—
331C2×C4⋊C4D4×C6711yescharacteristic
332C2×C4⋊C4C23×C6711yescharacteristic
333C2×C4⋊C4C23×C6714yes—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4D4×C6712yes—
338C2×C42C23×C6711yescharacteristic
339#106C23×C6718yes—
340C4⋊C8C23×C6714yes—
341#103C23×C67116yes—
342#103C23×C6718yes—
343#102C23×C6718yes—
344C22⋊C8C23×C6714yes—
345C22⋊C8C23×C6718yes—
346C4×C8C23×C6714yes—
347C24—21no—
348C24C2×D4×C6712yes—
349#41—216no—
350#41—48no—
351#41—416no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—24no—
355C2×Q8—44no—
356C2×Q8—48no—
357C2×Q8C24×C6714yes—
358C2×Q8C24×C6714yes—
359C2×D4—24no—
360C2×D4—28no—
361C2×D4—24no—
362C2×D4—21no—
363C2×D4—48no—
364C2×D4C2×D4×C6714yes—
365C2×D4C24×C6712yes—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—44no—
373C22×C4—48no—
374C22×C4—44no—
375C22×C4C2×D4×C6711yescharacteristic
376C22×C4C2×D4×C6712yes—
377C22×C4C2×D4×C6714yes—
378C22×C4C2×D4×C6711yescharacteristic
379C22×C4C24×C6711yescharacteristic
380C22×C4C24×C6714yes—
381Q16—416no—
382SD16—216no—
383M16—48no—
384C2×C8C24×C6718yes—
385C4⋊C4—24no—
386C4⋊C4—28no—
387C4⋊C4—24no—
388C4⋊C4—28no—
389C4⋊C4—24no—
390C4⋊C4—21no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4C2×D4×C6714yes—
397C4⋊C4C24×C6714yes—
398C4⋊C4C24×C6712yes—
399C22⋊C4—24no—
400C22⋊C4—28no—
401C22⋊C4—28no—
402C22⋊C4—24no—
403C22⋊C4C2×D4×C6718yes—
404C42—24no—
405C42C2×D4×C6712yes—
406C42C24×C6712yes—
407C23—24no—
408C23—22no—
409C23—24no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23C22×D4×C6714yes—
415C23C22×D4×C6711yescharacteristic
416Q8—28no—
417Q8—28no—
418Q8—44no—
419Q8—84no—
420Q8—82no—
421D4—24no—
422D4—28no—
423D4—22no—
424D4—416no—
425D4H(5,𝔽2)×C6714yes—
426C2×C4—28no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—416no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—48no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4#137×C6714yescomplemented
442C2×C4C22×D4×C6711yescharacteristic
443C2×C4C22×D4×C6712yes—
444C2×C4C22×D4×C6714yes—
445C2×C4C22×D4×C6712yes—
446C2×C4C25×C6711yescharacteristic
447C8—28no—
448C22—21no—
449C22—21no—
450C22—24no—
451C22—24no—
452C22—24no—
453C22—24no—
454C22—48no—
455C22—42no—
456C22—42no—
457C22—84no—
458C22C2×#137×C6712yes—
459C22C23×D4×C6711yescharacteristic
460C4—22no—
461C4—24no—
462C4—21no—
463C4—42no—
464C4—44no—
465C4—44no—
466C4—44no—
467C4C2×#137×C6714yes—
468C4C2×H(5,𝔽2)×C6712yes—
469C2—22no—
470C2—44no—
471C2—44no—
472C2C67×#348411yescharacteristic
473C2C22×#137×C6711yescharacteristic
474C2C67×#358611yescharacteristic
subgroupG/H
center, socleC22×C67C23×D4
commutator, Frattini, cosocleC2×C4C25×C67
prime radicalC2×#547×C67C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C4×D4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3501C2semidirect
3. C67×#3331C2semidirect
4. C67×#3276C2semidirect
5. C67×#3209C2semidirect
6. C67×#3195C2semidirect
7. C67×#3101C2semidirect
8. C67×#3007C2semidirect
9. C67×#2975C2semidirect
11. C2×#137×C67C22semidirect
13. #554×C67C22semidirect
14. D4×Q8×C67C22semidirect
29. #494×C67C22semidirect
30. #487×C67C22semidirect
32. #483×C67C22semidirect
33. #476×C67C22semidirect
34. #470×C67C22semidirect
35. #462×C67C22semidirect
36. #461×C67C22semidirect
37. #452×C67C22semidirect
39. #451×C67C22semidirect
41. C4×SD16×C67C22semidirect
42. #436×C67C22semidirect
44. #417×C67C22semidirect
203. C2×C4×C67#137semidirect
231. #61586C67direct (N × H)
238. #3501C2×C67semidirect
240. #3331C2×C67semidirect
241. #3276C2×C67semidirect
242. #3209C2×C67semidirect
243. #3195C2×C67semidirect
244. #3101C2×C67semidirect
245. #3007C2×C67semidirect
246. #2975C2×C67semidirect
247. C67#61586direct (N × H)
249. C2×#137C22×C67semidirect
251. #554C22×C67semidirect
252. D4×Q8C22×C67semidirect
267. #494C22×C67semidirect
268. #487C22×C67semidirect
270. #483C22×C67semidirect
271. #476C22×C67semidirect
272. #470C22×C67semidirect
273. #462C22×C67semidirect
274. #461C22×C67semidirect
275. #452C22×C67semidirect
277. #451C22×C67semidirect
279. C4×SD16C22×C67semidirect
280. #436C22×C67semidirect
282. #417C22×C67semidirect
441. C2×C4#137×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#35012
C2×#547×C672
C67×#33312
C67×#32762
C67×#32092
C67×#31952
C67×#31012
C67×#30072
C67×#29752
#6158667