Finite Group Databasebeta

← C67×#61293 #30740544

Subgroups — C67×#61293 #30740544

Order 17152.

1874 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3506C212yesmaximal, complemented
2C67×#3347C218yesmaximal, complemented
3C67×#3345C218yesmaximal, complemented
4C67×#3162C211yesmaximal, characteristic, complemented
5C67×#3096C214yesmaximal, complemented
6C67×#3077C212yesmaximal, complemented
7C67×#3058C212yesmaximal, complemented
8C2×#442×C67C211yesmaximal, characteristic, complemented
9C67×#2932C211yesmaximal, characteristic, complemented
10C67×#2912C212yesmaximal, complemented
11C22×D8×C67C2211yescharacteristic
12C2×#131×C67C2212yes—
13#559×C67—24no—
14#556×C67—22no—
15#555×C67C2214yescomplemented
16#553×C67C2218yescomplemented
17#550×C67—22no—
18#547×C67C2214yescomplemented
19#538×C67—24no—
20#535×C67C2214yes—
21#533×C67C2212yes—
22#532×C67C2211yescharacteristic
23#525×C67—24no—
24#520×C67C2214yes—
25C2×C4×D4×C67C2212yes—
26#501×C67C2214yescomplemented
27#501×C67C2214yescomplemented
28#482×C67C2218yescomplemented
29#482×C67C2218yescomplemented
30#481×C67C2218yescomplemented
31#480×C67C2218yescomplemented
32#478×C67C2218yescomplemented
33#476×C67C2214yescomplemented
34#474×C67C2214yescomplemented
35#469×C67C2214yes—
36#469×C67C2214yes—
37#466×C67C2218yes—
38#442×C67C2218yes—
39#435×C67C2218yescomplemented
40#428×C67C2214yes—
41#428×C67C2218yescomplemented
42#427×C67C2212yes—
43#427×C67C2212yes—
44#427×C67C2214yescomplemented
45C2×#106×C67C2211yescharacteristic
46C2×#106×C67C2214yescomplemented
47#423×C67C2211yescharacteristic
48#418×C67C2214yes—
49C2×#102×C67C2212yes—
50#408×C67C2212yes—
51C2×#97×C67C2211yescharacteristic
52C2×#41×C67—24no—
53C2×#41×C67C2312yes—
54C22×D4×C67C2312yes—
55C23×C4×C67—22no—
56C2×D8×C67—24no—
57C2×D8×C67C2314yes—
58#131×C67—28no—
59C2×M16×C67C2314yes—
60C2×M16×C67C2312yes—
61C22×C8×C67C2314yes—
62C22×C8×C67C2311yescharacteristic
63C22×C8×C67C2311yescharacteristic
64C4⋊Q8×C67D411yescharacteristic
65C4⋊Q8×C67C2312yes—
66C4⋊Q8×C67C2312yes—
67#127×C67—24no—
68#127×C67D411yescharacteristic
69#126×C67—28no—
70#125×C67C2314yes—
71#124×C67—24no—
72#124×C67—28no—
73#124×C67D412yes—
74#123×C67—28no—
75#123×C67—28no—
76C22⋊Q8×C67—22no—
77C22⋊Q8×C67—22no—
78C22⋊Q8×C67—24no—
79#121×C67—24no—
80#121×C67—28no—
81#121×C67D414yes—
82#121×C67C2318yes—
83#121×C67C2318yes—
84#120×C67—28no—
85C4×Q8×C67—24no—
86C4×Q8×C67—22no—
87C4×Q8×C67—22no—
88C4×D4×C67—28no—
89C4×D4×C67—24no—
90C4×D4×C67—28no—
91C4×D4×C67—22no—
92C4×D4×C67—22no—
93C4×D4×C67C2314yes—
94C4×D4×C67C2314yes—
95C4×D4×C67C2314yes—
96#117×C67—24no—
97#117×C67C2318yes—
98#117×C67C2314yes—
99C2×C4⋊C4×C67C2318yes—
100C2×C4⋊C4×C67C2312yes—
101C2×C22⋊C4×C67—22no—
102C2×C42×C67C2311yescharacteristic
103#107×C67C2314yes—
104#107×C67C2318yes—
105#107×C67C2314yes—
106#106×C67C2312yes—
107#106×C67C2312yes—
108#106×C67C2318yes—
109#106×C67C2312yes—
110#106×C67C2312yes—
111#106×C67C2314yes—
112C4⋊C8×C67C2314yes—
113#102×C67C2318yes—
114#102×C67C2318yes—
115#102×C67C2318yes—
116C22⋊C8×C67C2318yes—
117#97×C67C2314yes—
118C24×C67—22no—
119#41×C67—24no—
120#41×C67—24no—
121#41×C67—88no—
122C2×Q8×C67—22no—
123C2×Q8×C67—22no—
124C2×Q8×C67C2×D412yes—
125C2×D4×C67—28no—
126C2×D4×C67—28no—
127C2×D4×C67—22no—
128C2×D4×C67—24no—
129C2×D4×C67—24no—
130C2×D4×C67C2×D414yes—
131C2×D4×C67C2×D412yes—
132C2×D4×C67C2×D414yes—
133C2×D4×C67C2414yes—
134C22×C4×C67—22no—
135C22×C4×C67—28no—
136C22×C4×C67—24no—
137C22×C4×C67—24no—
138C22×C4×C67—84no—
139C22×C4×C67C2×D412yes—
140C22×C4×C67C2414yes—
141C22×C4×C67C2411yescharacteristic
142D8×C67—28no—
143M16×C67—24no—
144M16×C67—28no—
145C2×C8×C67—22no—
146C2×C8×C67—28no—
147C2×C8×C67#4114yescomplemented
148C2×C8×C67C2414yes—
149C2×C8×C67C2412yes—
150C2×C8×C67C2412yes—
151C4⋊C4×C67—22no—
152C4⋊C4×C67—28no—
153C4⋊C4×C67—22no—
154C4⋊C4×C67—24no—
155C4⋊C4×C67—24no—
156C4⋊C4×C67—22no—
157C4⋊C4×C67C2×D414yes—
158C4⋊C4×C67C2414yes—
159C4⋊C4×C67C2414yes—
160C4⋊C4×C67C2414yes—
161C22⋊C4×C67—24no—
162C22⋊C4×C67—28no—
163C22⋊C4×C67—24no—
164C22⋊C4×C67—24no—
165C22⋊C4×C67—28no—
166C22⋊C4×C67—28no—
167C22⋊C4×C67C2×D418yes—
168C42×C67—28no—
169C42×C67—24no—
170C42×C67C2×D411yescharacteristic
171C42×C67C2×D411yescharacteristic
172C42×C67C2412yes—
173C23×C67—24no—
174C23×C67—44no—
175C23×C67—82no—
176C23×C67—82no—
177C23×C67C22×D414yes—
178C23×C67C22×D411yescharacteristic
179Q8×C67—24no—
180Q8×C67—44no—
181Q8×C67—82no—
182D4×C67—28no—
183D4×C67—24no—
184D4×C67—24no—
185D4×C67—24no—
186D4×C67—44no—
187D4×C67—44no—
188D4×C67—88no—
189D4×C67—88no—
190C2×C4×C67—24no—
191C2×C4×C67—24no—
192C2×C4×C67—24no—
193C2×C4×C67—24no—
194C2×C4×C67—24no—
195C2×C4×C67—42no—
196C2×C4×C67—48no—
197C2×C4×C67—84no—
198C2×C4×C67—82no—
199C2×C4×C67—88no—
200C2×C4×C67C22×D414yes—
201C2×C4×C67C22×D412yes—
202C2×C4×C67C22×D412yes—
203C2×C4×C67C22×D411yescharacteristic
204C2×C4×C67C2×#4112yes—
205C2×C4×C672−1+412yes—
206C2×C4×C67C2511yescharacteristic
207C8×C67—24no—
208C8×C67—22no—
209C8×C67C2×#4114yes—
210C22×C67—22no—
211C22×C67—28no—
212C22×C67—24no—
213C22×C67—44no—
214C22×C67—44no—
215C22×C67—44no—
216C22×C67—88no—
217C22×C67#54712yes—
218C22×C67C23×D411yescharacteristic
219C4×C67—22no—
220C4×C67—22no—
221C4×C67—24no—
222C4×C67—44no—
223C4×C67—44no—
224C4×C67—44no—
225C4×C67#54712yes—
226C4×C67C22×#4111yescharacteristic
227C4×C67C2×2−1+411yescharacteristic
228#61293C6711yesmaximal, characteristic, direct product factor, Sylow
229C2×C67—22no—
230C2×C67—44no—
231C2×C67—84no—
232C2×C67C2×#54711yescharacteristic
233C2×C67#358512yes—
234#3506C2×C6712yescomplemented
235#3347C2×C6718yescomplemented
236#3345C2×C6718yescomplemented
237#3162C2×C6711yescharacteristic, complemented
238#3096C2×C6714yescomplemented
239#3077C2×C6712yescomplemented
240#3058C2×C6712yescomplemented
241C2×#442C2×C6711yescharacteristic, complemented
242#2932C2×C6711yescharacteristic, complemented
243#2912C2×C6712yescomplemented
244C67#6129311yescharacteristic, direct product factor, Sylow
245C22×D8C22×C6711yescharacteristic
246C2×#131C22×C6712yes—
247#559—24no—
248#556—22no—
249#555C22×C6714yescomplemented
250#553C22×C6718yescomplemented
251#550—22no—
252#547C22×C6714yescomplemented
253#538—24no—
254#535C22×C6714yes—
255#533C22×C6712yes—
256#532C22×C6711yescharacteristic
257#525—24no—
258#520C22×C6714yes—
259C2×C4×D4C22×C6712yes—
260#501C22×C6714yescomplemented
261#501C22×C6714yescomplemented
262#482C22×C6718yescomplemented
263#482C22×C6718yescomplemented
264#481C22×C6718yescomplemented
265#480C22×C6718yescomplemented
266#478C22×C6718yescomplemented
267#476C22×C6714yescomplemented
268#474C22×C6714yescomplemented
269#469C22×C6714yes—
270#469C22×C6714yes—
271#466C22×C6718yes—
272#442C22×C6718yes—
273#435C22×C6718yescomplemented
274#428C22×C6714yes—
275#428C22×C6718yescomplemented
276#427C22×C6712yes—
277#427C22×C6712yes—
278#427C22×C6714yescomplemented
279C2×#106C22×C6711yescharacteristic
280C2×#106C22×C6714yescomplemented
281#423C22×C6711yescharacteristic
282#418C22×C6714yes—
283C2×#102C22×C6712yes—
284#408C22×C6712yes—
285C2×#97C22×C6711yescharacteristic
286C2×#41—24no—
287C2×#41C23×C6712yes—
288C22×D4C23×C6712yes—
289C23×C4—22no—
290C2×D8—24no—
291C2×D8C23×C6714yes—
292#131—28no—
293C2×M16C23×C6712yes—
294C2×M16C23×C6714yes—
295C22×C8C23×C6714yes—
296C22×C8C23×C6711yescharacteristic
297C22×C8C23×C6711yescharacteristic
298C4⋊Q8D4×C6711yescharacteristic
299C4⋊Q8C23×C6712yes—
300C4⋊Q8C23×C6712yes—
301#127—24no—
302#127D4×C6711yescharacteristic
303#126—28no—
304#125C23×C6714yes—
305#124—24no—
306#124—28no—
307#124D4×C6712yes—
308#123—28no—
309#123—28no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—24no—
313#121—24no—
314#121—28no—
315#121D4×C6714yes—
316#121C23×C6718yes—
317#121C23×C6718yes—
318#120—28no—
319C4×Q8—24no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×D4—28no—
323C4×D4—24no—
324C4×D4—28no—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4C23×C6714yes—
328C4×D4C23×C6714yes—
329C4×D4C23×C6714yes—
330#117—24no—
331#117C23×C6714yes—
332#117C23×C6718yes—
333C2×C4⋊C4C23×C6712yes—
334C2×C4⋊C4C23×C6718yes—
335C2×C22⋊C4—22no—
336C2×C42C23×C6711yescharacteristic
337#107C23×C6714yes—
338#107C23×C6714yes—
339#107C23×C6718yes—
340#106C23×C6714yes—
341#106C23×C6712yes—
342#106C23×C6718yes—
343#106C23×C6712yes—
344#106C23×C6712yes—
345#106C23×C6712yes—
346C4⋊C8C23×C6714yes—
347#102C23×C6718yes—
348#102C23×C6718yes—
349#102C23×C6718yes—
350C22⋊C8C23×C6718yes—
351#97C23×C6714yes—
352C24—22no—
353#41—24no—
354#41—24no—
355#41—88no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8C2×D4×C6712yes—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4C2×D4×C6714yes—
365C2×D4C2×D4×C6712yes—
366C2×D4C2×D4×C6714yes—
367C2×D4C24×C6714yes—
368C22×C4—22no—
369C22×C4—28no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—84no—
373C22×C4C2×D4×C6712yes—
374C22×C4C24×C6714yes—
375C22×C4C24×C6711yescharacteristic
376D8—28no—
377M16—24no—
378M16—28no—
379C2×C8—22no—
380C2×C8—28no—
381C2×C8#41×C6714yescomplemented
382C2×C8C24×C6712yes—
383C2×C8C24×C6714yes—
384C2×C8C24×C6712yes—
385C4⋊C4—22no—
386C4⋊C4—28no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4C2×D4×C6714yes—
392C4⋊C4C24×C6714yes—
393C4⋊C4C24×C6714yes—
394C4⋊C4C24×C6714yes—
395C22⋊C4—24no—
396C22⋊C4—28no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C22⋊C4C2×D4×C6718yes—
402C42—28no—
403C42—24no—
404C42C2×D4×C6711yescharacteristic
405C42C2×D4×C6711yescharacteristic
406C42C24×C6712yes—
407C23—24no—
408C23—44no—
409C23—82no—
410C23—82no—
411C23C22×D4×C6714yes—
412C23C22×D4×C6711yescharacteristic
413Q8—24no—
414Q8—44no—
415Q8—82no—
416D4—28no—
417D4—24no—
418D4—24no—
419D4—24no—
420D4—44no—
421D4—44no—
422D4—88no—
423D4—88no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—42no—
430C2×C4—48no—
431C2×C4—84no—
432C2×C4—82no—
433C2×C4—88no—
434C2×C4C22×D4×C6714yes—
435C2×C4C22×D4×C6712yes—
436C2×C4C22×D4×C6712yes—
437C2×C4C22×D4×C6711yescharacteristic
438C2×C4C2×#41×C6712yes—
439C2×C42−1+4×C6712yes—
440C2×C4C25×C6711yescharacteristic
441C8—24no—
442C8—22no—
443C8C2×#41×C6714yes—
444C22—22no—
445C22—28no—
446C22—24no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—88no—
451C22#547×C6712yes—
452C22C23×D4×C6711yescharacteristic
453C4—22no—
454C4—22no—
455C4—24no—
456C4—44no—
457C4—44no—
458C4—44no—
459C4#547×C6712yes—
460C4C22×#41×C6711yescharacteristic
461C4C2×2−1+4×C6711yescharacteristic
462C2—22no—
463C2—44no—
464C2—84no—
465C2C2×#547×C6711yescharacteristic
466C2C67×#358512yes—
subgroupG/H
center, socleC22×C67C23×D4
commutator, Frattini, cosocleC2×C4C25×C67
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ #532×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3506C2semidirect
2. C67×#3347C2semidirect
3. C67×#3345C2semidirect
4. C67×#3162C2semidirect
5. C67×#3096C2semidirect
6. C67×#3077C2semidirect
7. C67×#3058C2semidirect
8. C2×#442×C67C2semidirect
9. C67×#2932C2semidirect
10. C67×#2912C2semidirect
15. #555×C67C22semidirect
16. #553×C67C22semidirect
18. #547×C67C22semidirect
26. #501×C67C22semidirect
27. #501×C67C22semidirect
28. #482×C67C22semidirect
29. #482×C67C22semidirect
30. #481×C67C22semidirect
31. #480×C67C22semidirect
32. #478×C67C22semidirect
33. #476×C67C22semidirect
34. #474×C67C22semidirect
39. #435×C67C22semidirect
41. #428×C67C22semidirect
44. #427×C67C22semidirect
46. C2×#106×C67C22semidirect
147. C2×C8×C67#41semidirect
228. #61293C67direct (N × H)
234. #3506C2×C67semidirect
235. #3347C2×C67semidirect
236. #3345C2×C67semidirect
237. #3162C2×C67semidirect
238. #3096C2×C67semidirect
239. #3077C2×C67semidirect
240. #3058C2×C67semidirect
241. C2×#442C2×C67semidirect
242. #2932C2×C67semidirect
243. #2912C2×C67semidirect
244. C67#61293direct (N × H)
249. #555C22×C67semidirect
250. #553C22×C67semidirect
252. #547C22×C67semidirect
260. #501C22×C67semidirect
261. #501C22×C67semidirect
262. #482C22×C67semidirect
263. #482C22×C67semidirect
264. #481C22×C67semidirect
265. #480C22×C67semidirect
266. #478C22×C67semidirect
267. #476C22×C67semidirect
268. #474C22×C67semidirect
273. #435C22×C67semidirect
275. #428C22×C67semidirect
278. #427C22×C67semidirect
280. C2×#106C22×C67semidirect
381. C2×C8#41×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#35062
C67×#33472
C67×#33452
C67×#31622
C67×#30962
C67×#30772
C67×#30582
C2×#442×C672
C67×#29322
C67×#29122
#6129367