Finite Group Databasebeta

← C67×#60176 #30739427

Subgroups — C67×#60176 #30739427

Order 17152.

2818 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#576×C67C211yesmaximal, characteristic
2C4×H(5,𝔽2)×C67C212yesmaximal, complemented
3C67×#2952C213yesmaximal, complemented
4C67×#2948C213yesmaximal, complemented
5C67×#2946C216yesmaximal, complemented
6C4×AGL(1,ℤ/8)×C67C216yesmaximal, complemented
7C67×#2943C213yesmaximal, complemented
8C2×#442×C67C213yesmaximal, complemented
9C67×#2912C211yesmaximal, characteristic, complemented
10C67×#2897C212yesmaximal, complemented
11C67×#2875C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)×C67C2212yes—
13#576×C67C4116yescomplemented
14C2×AGL(1,ℤ/8)×C67C2216yes—
15C2×#135×C67C2213yes—
16C22×D8×C67C2213yes—
17C2×#131×C67C2211yescharacteristic
18C2×#131×C67C2211yescharacteristic
19#520×C67—22no—
20#520×C67C2212yes—
21#520×C67C2216yescomplemented
22#518×C67—26no—
23#518×C67C2216yescomplemented
24C4×#41×C67—22no—
25C4×#41×C67C2211yescharacteristic
26C4×#41×C67C2216yescomplemented
27C2×C4×D4×C67—26no—
28C2×C4×D4×C67C2216yescomplemented
29#442×C67C2213yescomplemented
30#442×C67C2216yescomplemented
31#442×C67C2216yescomplemented
32#442×C67C2216yescomplemented
33#442×C67C2213yescomplemented
34#441×C67C2211yescharacteristic, complemented
35#441×C67C2213yescomplemented
36#440×C67C2216yescomplemented
37#440×C67C2216yescomplemented
38C4×SD16×C67C22112yescomplemented
39C4×D8×C67C22112yescomplemented
40#428×C67C2216yescomplemented
41#427×C67C2213yescomplemented
42C2×#106×C67C2213yescomplemented
43#418×C67C2216yescomplemented
44#417×C67C2216yescomplemented
45#416×C67C2216yescomplemented
46C2×#102×C67C2216yescomplemented
47#405×C67C2213yescomplemented
48#405×C67C2213yescomplemented
49C4×M16×C67C2213yescomplemented
50C2×#97×C67C2213yescomplemented
51H(5,𝔽2)×C67—28no—
52H(5,𝔽2)×C67C2×C4116yescomplemented
53C2×#41×C67—22no—
54C2×#41×C67C2316yes—
55C2×#41×C67C2311yescharacteristic
56C22×D4×C67—26no—
57C22×D4×C67C2316yes—
58C23×C4×C67—26no—
59AGL(1,ℤ/8)×C67C2×C4148yescomplemented
60#135×C67C2×C4124yescomplemented
61C2×Q16×C67C2311yescharacteristic
62C2×SD16×C67C2316yes—
63C2×D8×C67C2×C4124yescomplemented
64C2×D8×C67C2313yes—
65C2×D8×C67C2316yes—
66#131×C67—24no—
67#131×C67C2×C418yescomplemented
68C2×M16×C67C2313yes—
69C2×M16×C67C2313yes—
70C22×C8×C67C2313yes—
71C22×C8×C67C2313yes—
72C4×Q8×C67—22no—
73C4×Q8×C67D412yes—
74C4×Q8×C67C2312yes—
75C4×Q8×C67C2316yes—
76C4×D4×C67—26no—
77C4×D4×C67—26no—
78C4×D4×C67—26no—
79C4×D4×C67—26no—
80C4×D4×C67—26no—
81C4×D4×C67—212no—
82C4×D4×C67—26no—
83C4×D4×C67—26no—
84C4×D4×C67D416yes—
85C4×D4×C67C2316yes—
86C4×D4×C67C23112yes—
87C4×D4×C67C2316yes—
88#117×C67—26no—
89#117×C67—26no—
90#117×C67C2316yes—
91#117×C67C2313yes—
92C2×C4⋊C4×C67—26no—
93C2×C4⋊C4×C67C2316yes—
94C2×C22⋊C4×C67—26no—
95C2×C22⋊C4×C67—212no—
96C2×C42×C67—26no—
97C2×C42×C67C2313yes—
98#107×C67C2316yes—
99#106×C67C2313yes—
100#106×C67C2316yes—
101#106×C67C2311yescharacteristic
102#103×C67C2312yes—
103#103×C67C2316yes—
104#102×C67C2316yes—
105#102×C67C23112yes—
106#102×C67C2316yes—
107#97×C67C2311yescharacteristic
108#97×C67C2313yes—
109#97×C67C2313yes—
110#97×C67C2313yes—
111C4×C8×C67C2316yes—
112C24×C67—26no—
113#41×C67—216no—
114#41×C67—212no—
115#41×C67C22×C414yes—
116#41×C67C22×C4124yescomplemented
117#41×C67#4114yes—
118C2×Q8×C67C2×D411yescharacteristic
119C2×Q8×C67C2412yes—
120C2×D4×C67—26no—
121C2×D4×C67—248no—
122C2×D4×C67—26no—
123C2×D4×C67—212no—
124C2×D4×C67C22×C4124yescomplemented
125C2×D4×C67C2×D413yes—
126C2×D4×C67C2416yes—
127C22×C4×C67—26no—
128C22×C4×C67—26no—
129C22×C4×C67—26no—
130C22×C4×C67—26no—
131C22×C4×C67—212no—
132C22×C4×C67—812no—
133C22×C4×C67C2×D416yes—
134C22×C4×C67C2413yes—
135Q16×C67C22×C414yescomplemented
136SD16×C67C22×C4124yescomplemented
137D8×C67C22×C4112yescomplemented
138D8×C67C22×C4124yescomplemented
139M16×C67—26no—
140M16×C67C22×C4112yescomplemented
141C2×C8×C67—26no—
142C2×C8×C67C22×C4112yescomplemented
143C2×C8×C67C2411yescharacteristic
144C2×C8×C67C2413yes—
145C2×C8×C67C2411yescharacteristic
146C2×C8×C67C2413yes—
147C4⋊C4×C67—26no—
148C4⋊C4×C67—26no—
149C4⋊C4×C67—22no—
150C4⋊C4×C67C2×D416yes—
151C4⋊C4×C67C2416yes—
152C4⋊C4×C67C2412yes—
153C22⋊C4×C67—26no—
154C22⋊C4×C67—26no—
155C22⋊C4×C67—26no—
156C22⋊C4×C67—212no—
157C22⋊C4×C67C2×D416yes—
158C42×C67—22no—
159C42×C67—26no—
160C42×C67—26no—
161C42×C67C2×D416yes—
162C42×C67C2411yescharacteristic
163C42×C67C2413yes—
164C23×C67—26no—
165C23×C67—224no—
166C23×C67—86no—
167C23×C67—86no—
168C23×C67C22×D413yes—
169Q8×C67—22no—
170Q8×C67C4×D414yes—
171Q8×C67C23×C414yes—
172D4×C67—224no—
173D4×C67—224no—
174D4×C67—26no—
175D4×C67C4×D4112yes—
176D4×C67C23×C4112yes—
177C2×C4×C67—22no—
178C2×C4×C67—224no—
179C2×C4×C67—22no—
180C2×C4×C67—26no—
181C2×C4×C67—46no—
182C2×C4×C67—412no—
183C2×C4×C67—86no—
184C2×C4×C67—812no—
185C2×C4×C67—86no—
186C2×C4×C67C23×C416yes—
187C2×C4×C67C22×D412yes—
188C2×C4×C67C22×D416yes—
189C2×C4×C67C22×D413yes—
190C2×C4×C67C2×#4116yes—
191C2×C4×C67C2511yescharacteristic
192C8×C67—21no—
193C8×C67—23no—
194C8×C67C23×C412yes—
195C8×C67C23×C416yes—
196C22×C67—26no—
197C22×C67—212no—
198C22×C67—46no—
199C22×C67—46no—
200C22×C67—824no—
201C22×C67C2×C4×D416yes—
202C22×C67C23×D411yescharacteristic
203C4×C67—22no—
204C4×C67—26no—
205C4×C67—24no—
206C4×C67—46no—
207C4×C67—42no—
208C4×C67C2×C4×D416yes—
209C4×C67C24×C411yescharacteristic
210C4×C67C22×#4111yescharacteristic
211#60176C6711yesmaximal, characteristic, direct product factor, Sylow
212C2×C67—26no—
213C2×C67—412no—
214C2×C67C22×C4×D411yescharacteristic
215C2×C67#358512yes—
216C2×#576C2×C6711yescharacteristic
217C4×H(5,𝔽2)C2×C6712yescomplemented
218#2952C2×C6713yescomplemented
219#2948C2×C6713yescomplemented
220#2946C2×C6716yescomplemented
221C4×AGL(1,ℤ/8)C2×C6716yescomplemented
222#2943C2×C6713yescomplemented
223C2×#442C2×C6713yescomplemented
224#2912C2×C6711yescharacteristic, complemented
225#2897C2×C6712yescomplemented
226#2875C2×C6711yescharacteristic, complemented
227C67#6017611yescharacteristic, direct product factor, Sylow
228C2×H(5,𝔽2)C22×C6712yes—
229#576C4×C67116yescomplemented
230C2×AGL(1,ℤ/8)C22×C6716yes—
231C2×#135C22×C6713yes—
232C22×D8C22×C6713yes—
233C2×#131C22×C6711yescharacteristic
234C2×#131C22×C6711yescharacteristic
235#520—22no—
236#520C22×C6712yes—
237#520C22×C6716yescomplemented
238#518—26no—
239#518C22×C6716yescomplemented
240C4×#41—22no—
241C4×#41C22×C6711yescharacteristic
242C4×#41C22×C6716yescomplemented
243C2×C4×D4—26no—
244C2×C4×D4C22×C6716yescomplemented
245#442C22×C6713yescomplemented
246#442C22×C6716yescomplemented
247#442C22×C6716yescomplemented
248#442C22×C6716yescomplemented
249#442C22×C6713yescomplemented
250#441C22×C6711yescharacteristic, complemented
251#441C22×C6713yescomplemented
252#440C22×C6716yescomplemented
253#440C22×C6716yescomplemented
254C4×SD16C22×C67112yescomplemented
255C4×D8C22×C67112yescomplemented
256#428C22×C6716yescomplemented
257#427C22×C6713yescomplemented
258C2×#106C22×C6713yescomplemented
259#418C22×C6716yescomplemented
260#417C22×C6716yescomplemented
261#416C22×C6716yescomplemented
262C2×#102C22×C6716yescomplemented
263#405C22×C6713yescomplemented
264#405C22×C6713yescomplemented
265C4×M16C22×C6713yescomplemented
266C2×#97C22×C6713yescomplemented
267H(5,𝔽2)—28no—
268H(5,𝔽2)C2×C4×C67116yescomplemented
269C2×#41—22no—
270C2×#41C23×C6716yes—
271C2×#41C23×C6711yescharacteristic
272C22×D4—26no—
273C22×D4C23×C6716yes—
274C23×C4—26no—
275AGL(1,ℤ/8)C2×C4×C67148yescomplemented
276#135C2×C4×C67124yescomplemented
277C2×Q16C23×C6711yescharacteristic
278C2×SD16C23×C6716yes—
279C2×D8C2×C4×C67124yescomplemented
280C2×D8C23×C6713yes—
281C2×D8C23×C6716yes—
282#131—24no—
283#131C2×C4×C6718yescomplemented
284C2×M16C23×C6713yes—
285C2×M16C23×C6713yes—
286C22×C8C23×C6713yes—
287C22×C8C23×C6713yes—
288C4×Q8—22no—
289C4×Q8D4×C6712yes—
290C4×Q8C23×C6712yes—
291C4×Q8C23×C6716yes—
292C4×D4—26no—
293C4×D4—26no—
294C4×D4—26no—
295C4×D4—26no—
296C4×D4—26no—
297C4×D4—212no—
298C4×D4—26no—
299C4×D4—26no—
300C4×D4D4×C6716yes—
301C4×D4C23×C6716yes—
302C4×D4C23×C67112yes—
303C4×D4C23×C6716yes—
304#117—26no—
305#117—26no—
306#117C23×C6716yes—
307#117C23×C6713yes—
308C2×C4⋊C4—26no—
309C2×C4⋊C4C23×C6716yes—
310C2×C22⋊C4—26no—
311C2×C22⋊C4—212no—
312C2×C42—26no—
313C2×C42C23×C6713yes—
314#107C23×C6716yes—
315#106C23×C6713yes—
316#106C23×C6716yes—
317#106C23×C6711yescharacteristic
318#103C23×C6712yes—
319#103C23×C6716yes—
320#102C23×C6716yes—
321#102C23×C67112yes—
322#102C23×C6716yes—
323#97C23×C6711yescharacteristic
324#97C23×C6713yes—
325#97C23×C6713yes—
326#97C23×C6713yes—
327C4×C8C23×C6716yes—
328C24—26no—
329#41—216no—
330#41—212no—
331#41C22×C4×C6714yes—
332#41C22×C4×C67124yescomplemented
333#41#41×C6714yes—
334C2×Q8C2×D4×C6711yescharacteristic
335C2×Q8C24×C6712yes—
336C2×D4—26no—
337C2×D4—248no—
338C2×D4—26no—
339C2×D4—212no—
340C2×D4C22×C4×C67124yescomplemented
341C2×D4C2×D4×C6713yes—
342C2×D4C24×C6716yes—
343C22×C4—26no—
344C22×C4—26no—
345C22×C4—26no—
346C22×C4—26no—
347C22×C4—212no—
348C22×C4—812no—
349C22×C4C2×D4×C6716yes—
350C22×C4C24×C6713yes—
351Q16C22×C4×C6714yescomplemented
352SD16C22×C4×C67124yescomplemented
353D8C22×C4×C67112yescomplemented
354D8C22×C4×C67124yescomplemented
355M16—26no—
356M16C22×C4×C67112yescomplemented
357C2×C8—26no—
358C2×C8C22×C4×C67112yescomplemented
359C2×C8C24×C6713yes—
360C2×C8C24×C6711yescharacteristic
361C2×C8C24×C6713yes—
362C2×C8C24×C6711yescharacteristic
363C4⋊C4—26no—
364C4⋊C4—26no—
365C4⋊C4—22no—
366C4⋊C4C2×D4×C6716yes—
367C4⋊C4C24×C6716yes—
368C4⋊C4C24×C6712yes—
369C22⋊C4—26no—
370C22⋊C4—26no—
371C22⋊C4—26no—
372C22⋊C4—212no—
373C22⋊C4C2×D4×C6716yes—
374C42—22no—
375C42—26no—
376C42—26no—
377C42C2×D4×C6716yes—
378C42C24×C6713yes—
379C42C24×C6711yescharacteristic
380C23—26no—
381C23—224no—
382C23—86no—
383C23—86no—
384C23C22×D4×C6713yes—
385Q8—22no—
386Q8C4×D4×C6714yes—
387Q8C23×C4×C6714yes—
388D4—224no—
389D4—224no—
390D4—26no—
391D4C4×D4×C67112yes—
392D4C23×C4×C67112yes—
393C2×C4—22no—
394C2×C4—224no—
395C2×C4—22no—
396C2×C4—26no—
397C2×C4—46no—
398C2×C4—412no—
399C2×C4—812no—
400C2×C4—86no—
401C2×C4—86no—
402C2×C4C23×C4×C6716yes—
403C2×C4C22×D4×C6712yes—
404C2×C4C22×D4×C6716yes—
405C2×C4C22×D4×C6713yes—
406C2×C4C2×#41×C6716yes—
407C2×C4C25×C6711yescharacteristic
408C8—21no—
409C8—23no—
410C8C23×C4×C6712yes—
411C8C23×C4×C6716yes—
412C22—26no—
413C22—212no—
414C22—46no—
415C22—46no—
416C22—824no—
417C22C2×C4×D4×C6716yes—
418C22C23×D4×C6711yescharacteristic
419C4—22no—
420C4—26no—
421C4—24no—
422C4—46no—
423C4—42no—
424C4C2×C4×D4×C6716yes—
425C4C24×C4×C6711yescharacteristic
426C4C22×#41×C6711yescharacteristic
427C2—26no—
428C2—412no—
429C2C22×C4×D4×C6711yescharacteristic
430C2C67×#358512yes—
subgroupG/H
center, socleC22×C67C23×D4
commutatorC4C24×C4×C67
Frattini, cosocleC2×C4C25×C67
prime radicalC2×#576×C67C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C4×#41×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×H(5,𝔽2)×C67C2semidirect
3. C67×#2952C2semidirect
4. C67×#2948C2semidirect
5. C67×#2946C2semidirect
6. C4×AGL(1,ℤ/8)×C67C2semidirect
7. C67×#2943C2semidirect
8. C2×#442×C67C2semidirect
9. C67×#2912C2semidirect
10. C67×#2897C2semidirect
11. C67×#2875C2semidirect
13. #576×C67C4semidirect
21. #520×C67C22semidirect
23. #518×C67C22semidirect
26. C4×#41×C67C22semidirect
28. C2×C4×D4×C67C22semidirect
29. #442×C67C22semidirect
30. #442×C67C22semidirect
31. #442×C67C22semidirect
32. #442×C67C22semidirect
33. #442×C67C22semidirect
34. #441×C67C22semidirect
35. #441×C67C22semidirect
36. #440×C67C22semidirect
37. #440×C67C22semidirect
38. C4×SD16×C67C22semidirect
39. C4×D8×C67C22semidirect
40. #428×C67C22semidirect
41. #427×C67C22semidirect
42. C2×#106×C67C22semidirect
43. #418×C67C22semidirect
44. #417×C67C22semidirect
45. #416×C67C22semidirect
46. C2×#102×C67C22semidirect
47. #405×C67C22semidirect
48. #405×C67C22semidirect
49. C4×M16×C67C22semidirect
50. C2×#97×C67C22semidirect
52. H(5,𝔽2)×C67C2×C4semidirect
59. AGL(1,ℤ/8)×C67C2×C4semidirect
60. #135×C67C2×C4semidirect
63. C2×D8×C67C2×C4semidirect
67. #131×C67C2×C4semidirect
116. #41×C67C22×C4semidirect
124. C2×D4×C67C22×C4semidirect
135. Q16×C67C22×C4semidirect
136. SD16×C67C22×C4semidirect
137. D8×C67C22×C4semidirect
138. D8×C67C22×C4semidirect
140. M16×C67C22×C4semidirect
142. C2×C8×C67C22×C4semidirect
211. #60176C67direct (N × H)
217. C4×H(5,𝔽2)C2×C67semidirect
218. #2952C2×C67semidirect
219. #2948C2×C67semidirect
220. #2946C2×C67semidirect
221. C4×AGL(1,ℤ/8)C2×C67semidirect
222. #2943C2×C67semidirect
223. C2×#442C2×C67semidirect
224. #2912C2×C67semidirect
225. #2897C2×C67semidirect
226. #2875C2×C67semidirect
227. C67#60176direct (N × H)
229. #576C4×C67semidirect
237. #520C22×C67semidirect
239. #518C22×C67semidirect
242. C4×#41C22×C67semidirect
244. C2×C4×D4C22×C67semidirect
245. #442C22×C67semidirect
246. #442C22×C67semidirect
247. #442C22×C67semidirect
248. #442C22×C67semidirect
249. #442C22×C67semidirect
250. #441C22×C67semidirect
251. #441C22×C67semidirect
252. #440C22×C67semidirect
253. #440C22×C67semidirect
254. C4×SD16C22×C67semidirect
255. C4×D8C22×C67semidirect
256. #428C22×C67semidirect
257. #427C22×C67semidirect
258. C2×#106C22×C67semidirect
259. #418C22×C67semidirect
260. #417C22×C67semidirect
261. #416C22×C67semidirect
262. C2×#102C22×C67semidirect
263. #405C22×C67semidirect
264. #405C22×C67semidirect
265. C4×M16C22×C67semidirect
266. C2×#97C22×C67semidirect
268. H(5,𝔽2)C2×C4×C67semidirect
275. AGL(1,ℤ/8)C2×C4×C67semidirect
276. #135C2×C4×C67semidirect
279. C2×D8C2×C4×C67semidirect
283. #131C2×C4×C67semidirect
332. #41C22×C4×C67semidirect
340. C2×D4C22×C4×C67semidirect
351. Q16C22×C4×C67semidirect
352. SD16C22×C4×C67semidirect
353. D8C22×C4×C67semidirect
354. D8C22×C4×C67semidirect
356. M16C22×C4×C67semidirect
358. C2×C8C22×C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#576×C672
C4×H(5,𝔽2)×C672
C67×#29522
C67×#29482
C67×#29462
C4×AGL(1,ℤ/8)×C672
C67×#29432
C2×#442×C672
C67×#29122
C67×#28972
C67×#28752
#6017667