Finite Group Databasebeta

← C67×#60624 #30739875

Subgroups — C67×#60624 #30739875

Order 17152.

2038 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3504C211yesmaximal, characteristic, complemented
2C67×#3500C211yesmaximal, characteristic
3C67×#3142C213yesmaximal, complemented
4C67×#3140C213yesmaximal, complemented
5C67×#3137C216yesmaximal, complemented
6C67×#3136C216yesmaximal, complemented
7C67×#3131C213yesmaximal, complemented
8C2×#488×C67C213yesmaximal, complemented
9C67×#2898C212yesmaximal, complemented
10C67×#2897C212yesmaximal, complemented
11C4×#131×C67C211yesmaximal, characteristic, complemented
12C2×2−1+4×C67C2211yescharacteristic
13C2×H(5,𝔽2)×C67C2211yescharacteristic
14C2×#131×C67C2212yes—
15#557×C67—22no—
16#554×C67—26no—
17#548×C67—22no—
18#546×C67—26no—
19#544×C67C2216yescomplemented
20#536×C67C2213yes—
21#535×C67C2213yes—
22#533×C67C2213yescomplemented
23#532×C67C2213yes—
24C2×#125×C67C2213yescomplemented
25C2×#124×C67C2213yes—
26#520×C67C2212yes—
27C4×#41×C67C2211yescharacteristic
28#491×C67C2216yescomplemented
29#490×C67C2216yescomplemented
30#489×C67C22112yescomplemented
31#488×C67C2213yescomplemented
32#488×C67C22112yescomplemented
33#488×C67C2216yescomplemented
34#488×C67C2211yescharacteristic, complemented
35#488×C67C2213yescomplemented
36#488×C67C2213yescomplemented
37#487×C67C2216yescomplemented
38#486×C67C2216yescomplemented
39#419×C67C2216yescomplemented
40#418×C67C2216yes—
41#417×C67C2216yes—
42#417×C67C2216yescomplemented
43#416×C67C2216yes—
44#416×C67C2216yescomplemented
45C2×#103×C67C2216yescomplemented
46C2×#102×C67C2216yes—
47#405×C67C2216yescomplemented
48C4×M16×C67C2213yescomplemented
49C2×C4×C8×C67C2213yescomplemented
502−1+4×C67—28no—
51H(5,𝔽2)×C67—28no—
52C2×#41×C67—21no—
53C2×#41×C67—23no—
54C2×#41×C67C2313yes—
55C2×#41×C67C2313yes—
56C2×#41×C67C2311yescharacteristic
57C22×Q8×C67—21no—
58C22×Q8×C67C2313yes—
59C22×D4×C67—23no—
60C22×D4×C67C2313yes—
61#131×C67—28no—
62C2×M16×C67C2316yes—
63C22×C8×C67C2316yes—
64C4⋊Q8×C67—26no—
65C4⋊Q8×C67C2313yes—
66C4⋊Q8×C67C2316yes—
67#127×C67C2313yes—
68#125×C67—26no—
69#125×C67—26no—
70#125×C67C2313yes—
71#125×C67C2311yescharacteristic
72#125×C67C2316yes—
73#124×C67—26no—
74#124×C67C2313yes—
75#124×C67C2311yescharacteristic
76#124×C67C2316yes—
77#123×C67—26no—
78C22⋊Q8×C67—26no—
79C22⋊Q8×C67—26no—
80C22⋊Q8×C67—212no—
81#121×C67—26no—
82#121×C67—26no—
83#120×C67—26no—
84C4×Q8×C67—22no—
85C4×Q8×C67—26no—
86C4×Q8×C67D412yes—
87C4×D4×C67—26no—
88C4×D4×C67—26no—
89C4×D4×C67D416yes—
90#117×C67C2316yes—
91#117×C67C2313yes—
92C2×C4⋊C4×C67—26no—
93C2×C4⋊C4×C67C2316yes—
94C2×C22⋊C4×C67—26no—
95C2×C42×C67C2313yes—
96#103×C67C2312yes—
97#103×C67C2316yes—
98#103×C67C2316yes—
99#103×C67C2316yes—
100#103×C67C23112yes—
101#102×C67C2312yes—
102#102×C67C2316yes—
103#102×C67C2316yes—
104#102×C67C2316yes—
105#102×C67C23112yes—
106#97×C67C2316yes—
107C4×C8×C67C2316yes—
108C4×C8×C67C2311yescharacteristic
109C4×C8×C67C2313yes—
110C24×C67—23no—
111#41×C67—212no—
112#41×C67—212no—
113#41×C67—44no—
114#41×C67—412no—
115#41×C67#4118yes—
116C2×Q8×C67—23no—
117C2×Q8×C67—26no—
118C2×Q8×C67—212no—
119C2×Q8×C67—44no—
120C2×Q8×C67C2×D411yescharacteristic
121C2×Q8×C67C2413yes—
122C2×Q8×C67C2411yescharacteristic
123C2×D4×C67—23no—
124C2×D4×C67—23no—
125C2×D4×C67—23no—
126C2×D4×C67—26no—
127C2×D4×C67—212no—
128C2×D4×C67—412no—
129C2×D4×C67C2×D413yes—
130C2×D4×C67C2413yes—
131C2×D4×C67C2411yescharacteristic
132C22×C4×C67—23no—
133C22×C4×C67—26no—
134C22×C4×C67—23no—
135C22×C4×C67—23no—
136C22×C4×C67C2×D416yes—
137C22×C4×C67C2413yes—
138M16×C67—212no—
139C2×C8×C67—212no—
140C2×C8×C67C2412yes—
141C2×C8×C67C2416yes—
142C4⋊C4×C67—26no—
143C4⋊C4×C67—212no—
144C4⋊C4×C67—26no—
145C4⋊C4×C67—26no—
146C4⋊C4×C67—22no—
147C4⋊C4×C67—26no—
148C4⋊C4×C67C2×D416yes—
149C4⋊C4×C67C2416yes—
150C4⋊C4×C67C2412yes—
151C22⋊C4×C67—26no—
152C22⋊C4×C67—26no—
153C22⋊C4×C67—212no—
154C22⋊C4×C67—22no—
155C22⋊C4×C67C2×D416yes—
156C42×C67—26no—
157C42×C67C2×D416yes—
158C42×C67C2411yescharacteristic
159C42×C67C2413yes—
160C23×C67—23no—
161C23×C67—21no—
162C23×C67—46no—
163C23×C67—86no—
164C23×C67C22×D413yes—
165Q8×C67—26no—
166Q8×C67—22no—
167Q8×C67—46no—
168Q8×C67—86no—
169Q8×C67#12414yes—
170D4×C67—26no—
171D4×C67—22no—
172D4×C67—46no—
173D4×C67—46no—
174D4×C67—46no—
175D4×C67—86no—
176D4×C67#124112yes—
177C2×C4×C67—22no—
178C2×C4×C67—26no—
179C2×C4×C67—23no—
180C2×C4×C67—21no—
181C2×C4×C67—46no—
182C2×C4×C67—46no—
183C2×C4×C67—46no—
184C2×C4×C67—46no—
185C2×C4×C67—86no—
186C2×C4×C67C22×D412yes—
187C2×C4×C67C22×D416yes—
188C2×C4×C67C22×D413yes—
189C2×C4×C67C2×#41112yes—
190C2×C4×C67C2511yescharacteristic
191C8×C67—22no—
192C8×C67—26no—
193C22×C67—26no—
194C22×C67—42no—
195C22×C67—46no—
196C22×C67—41no—
197C22×C67—43no—
198C22×C67—166no—
199C22×C67C2×#12416yes—
200C22×C67C23×D411yescharacteristic
201C4×C67—22no—
202C4×C67—26no—
203C4×C67—46no—
204C4×C67—42no—
205C4×C67—41no—
206C4×C67—43no—
207C4×C67C2×#12416yes—
208C4×C67C22×#4112yes—
209#60624C6711yesmaximal, characteristic, direct product factor, Sylow
210C2×C67—26no—
211C2×C67—81no—
212C2×C67—83no—
213C2×C67C22×#12411yescharacteristic
214C2×C67#358412yes—
215#3504C2×C6711yescharacteristic, complemented
216#3500C2×C6711yescharacteristic
217#3142C2×C6713yescomplemented
218#3140C2×C6713yescomplemented
219#3137C2×C6716yescomplemented
220#3136C2×C6716yescomplemented
221#3131C2×C6713yescomplemented
222C2×#488C2×C6713yescomplemented
223#2898C2×C6712yescomplemented
224#2897C2×C6712yescomplemented
225C4×#131C2×C6711yescharacteristic, complemented
226C67#6062411yescharacteristic, direct product factor, Sylow
227C2×2−1+4C22×C6711yescharacteristic
228C2×H(5,𝔽2)C22×C6711yescharacteristic
229C2×#131C22×C6712yes—
230#557—22no—
231#554—26no—
232#548—22no—
233#546—26no—
234#544C22×C6716yescomplemented
235#536C22×C6713yes—
236#535C22×C6713yes—
237#533C22×C6713yescomplemented
238#532C22×C6713yes—
239C2×#125C22×C6713yescomplemented
240C2×#124C22×C6713yes—
241#520C22×C6712yes—
242C4×#41C22×C6711yescharacteristic
243#491C22×C6716yescomplemented
244#490C22×C6716yescomplemented
245#489C22×C67112yescomplemented
246#488C22×C6713yescomplemented
247#488C22×C67112yescomplemented
248#488C22×C6716yescomplemented
249#488C22×C6711yescharacteristic, complemented
250#488C22×C6713yescomplemented
251#488C22×C6713yescomplemented
252#487C22×C6716yescomplemented
253#486C22×C6716yescomplemented
254#419C22×C6716yescomplemented
255#418C22×C6716yes—
256#417C22×C6716yes—
257#417C22×C6716yescomplemented
258#416C22×C6716yes—
259#416C22×C6716yescomplemented
260C2×#103C22×C6716yescomplemented
261C2×#102C22×C6716yes—
262#405C22×C6716yescomplemented
263C4×M16C22×C6713yescomplemented
264C2×C4×C8C22×C6713yescomplemented
2652−1+4—28no—
266H(5,𝔽2)—28no—
267C2×#41—21no—
268C2×#41—23no—
269C2×#41C23×C6713yes—
270C2×#41C23×C6713yes—
271C2×#41C23×C6711yescharacteristic
272C22×Q8—21no—
273C22×Q8C23×C6713yes—
274C22×D4—23no—
275C22×D4C23×C6713yes—
276#131—28no—
277C2×M16C23×C6716yes—
278C22×C8C23×C6716yes—
279C4⋊Q8—26no—
280C4⋊Q8C23×C6713yes—
281C4⋊Q8C23×C6716yes—
282#127C23×C6713yes—
283#125—26no—
284#125—26no—
285#125C23×C6713yes—
286#125C23×C6716yes—
287#125C23×C6711yescharacteristic
288#124—26no—
289#124C23×C6713yes—
290#124C23×C6716yes—
291#124C23×C6711yescharacteristic
292#123—26no—
293C22⋊Q8—26no—
294C22⋊Q8—26no—
295C22⋊Q8—212no—
296#121—26no—
297#121—26no—
298#120—26no—
299C4×Q8—22no—
300C4×Q8—26no—
301C4×Q8D4×C6712yes—
302C4×D4—26no—
303C4×D4—26no—
304C4×D4D4×C6716yes—
305#117C23×C6716yes—
306#117C23×C6713yes—
307C2×C4⋊C4—26no—
308C2×C4⋊C4C23×C6716yes—
309C2×C22⋊C4—26no—
310C2×C42C23×C6713yes—
311#103C23×C6716yes—
312#103C23×C67112yes—
313#103C23×C6712yes—
314#103C23×C6716yes—
315#103C23×C6716yes—
316#102C23×C6716yes—
317#102C23×C67112yes—
318#102C23×C6716yes—
319#102C23×C6712yes—
320#102C23×C6716yes—
321#97C23×C6716yes—
322C4×C8C23×C6711yescharacteristic
323C4×C8C23×C6713yes—
324C4×C8C23×C6716yes—
325C24—23no—
326#41—212no—
327#41—212no—
328#41—44no—
329#41—412no—
330#41#41×C6718yes—
331C2×Q8—23no—
332C2×Q8—26no—
333C2×Q8—212no—
334C2×Q8—44no—
335C2×Q8C2×D4×C6711yescharacteristic
336C2×Q8C24×C6713yes—
337C2×Q8C24×C6711yescharacteristic
338C2×D4—23no—
339C2×D4—23no—
340C2×D4—23no—
341C2×D4—26no—
342C2×D4—212no—
343C2×D4—412no—
344C2×D4C2×D4×C6713yes—
345C2×D4C24×C6713yes—
346C2×D4C24×C6711yescharacteristic
347C22×C4—23no—
348C22×C4—26no—
349C22×C4—23no—
350C22×C4—23no—
351C22×C4C2×D4×C6716yes—
352C22×C4C24×C6713yes—
353M16—212no—
354C2×C8—212no—
355C2×C8C24×C6712yes—
356C2×C8C24×C6716yes—
357C4⋊C4—26no—
358C4⋊C4—212no—
359C4⋊C4—26no—
360C4⋊C4—26no—
361C4⋊C4—22no—
362C4⋊C4—26no—
363C4⋊C4C2×D4×C6716yes—
364C4⋊C4C24×C6716yes—
365C4⋊C4C24×C6712yes—
366C22⋊C4—26no—
367C22⋊C4—26no—
368C22⋊C4—212no—
369C22⋊C4—22no—
370C22⋊C4C2×D4×C6716yes—
371C42—26no—
372C42C2×D4×C6716yes—
373C42C24×C6711yescharacteristic
374C42C24×C6713yes—
375C23—23no—
376C23—21no—
377C23—46no—
378C23—86no—
379C23C22×D4×C6713yes—
380Q8—26no—
381Q8—22no—
382Q8—46no—
383Q8—86no—
384Q8#124×C6714yes—
385D4—26no—
386D4—22no—
387D4—46no—
388D4—46no—
389D4—46no—
390D4—86no—
391D4#124×C67112yes—
392C2×C4—22no—
393C2×C4—26no—
394C2×C4—23no—
395C2×C4—21no—
396C2×C4—46no—
397C2×C4—46no—
398C2×C4—46no—
399C2×C4—46no—
400C2×C4—86no—
401C2×C4C22×D4×C6712yes—
402C2×C4C22×D4×C6716yes—
403C2×C4C22×D4×C6713yes—
404C2×C4C2×#41×C67112yes—
405C2×C4C25×C6711yescharacteristic
406C8—22no—
407C8—26no—
408C22—26no—
409C22—42no—
410C22—46no—
411C22—41no—
412C22—43no—
413C22—166no—
414C22C2×#124×C6716yes—
415C22C23×D4×C6711yescharacteristic
416C4—22no—
417C4—26no—
418C4—46no—
419C4—42no—
420C4—41no—
421C4—43no—
422C4C2×#124×C6716yes—
423C4C22×#41×C6712yes—
424C2—26no—
425C2—83no—
426C2—81no—
427C2C22×#124×C6711yescharacteristic
428C2C67×#358412yes—
subgroupG/H
center, socleC22×C67C23×D4
commutator, Frattini, cosocleC2×C4C25×C67
prime radicalC67×#3500C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C4×#41×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3504C2semidirect
3. C67×#3142C2semidirect
4. C67×#3140C2semidirect
5. C67×#3137C2semidirect
6. C67×#3136C2semidirect
7. C67×#3131C2semidirect
8. C2×#488×C67C2semidirect
9. C67×#2898C2semidirect
10. C67×#2897C2semidirect
11. C4×#131×C67C2semidirect
19. #544×C67C22semidirect
22. #533×C67C22semidirect
24. C2×#125×C67C22semidirect
28. #491×C67C22semidirect
29. #490×C67C22semidirect
30. #489×C67C22semidirect
31. #488×C67C22semidirect
32. #488×C67C22semidirect
33. #488×C67C22semidirect
34. #488×C67C22semidirect
35. #488×C67C22semidirect
36. #488×C67C22semidirect
37. #487×C67C22semidirect
38. #486×C67C22semidirect
39. #419×C67C22semidirect
42. #417×C67C22semidirect
44. #416×C67C22semidirect
45. C2×#103×C67C22semidirect
47. #405×C67C22semidirect
48. C4×M16×C67C22semidirect
49. C2×C4×C8×C67C22semidirect
209. #60624C67direct (N × H)
215. #3504C2×C67semidirect
217. #3142C2×C67semidirect
218. #3140C2×C67semidirect
219. #3137C2×C67semidirect
220. #3136C2×C67semidirect
221. #3131C2×C67semidirect
222. C2×#488C2×C67semidirect
223. #2898C2×C67semidirect
224. #2897C2×C67semidirect
225. C4×#131C2×C67semidirect
226. C67#60624direct (N × H)
234. #544C22×C67semidirect
237. #533C22×C67semidirect
239. C2×#125C22×C67semidirect
243. #491C22×C67semidirect
244. #490C22×C67semidirect
245. #489C22×C67semidirect
246. #488C22×C67semidirect
247. #488C22×C67semidirect
248. #488C22×C67semidirect
249. #488C22×C67semidirect
250. #488C22×C67semidirect
251. #488C22×C67semidirect
252. #487C22×C67semidirect
253. #486C22×C67semidirect
254. #419C22×C67semidirect
257. #417C22×C67semidirect
259. #416C22×C67semidirect
260. C2×#103C22×C67semidirect
262. #405C22×C67semidirect
263. C4×M16C22×C67semidirect
264. 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×#35042
C67×#35002
C67×#31422
C67×#31402
C67×#31372
C67×#31362
C67×#31312
C2×#488×C672
C67×#28982
C67×#28972
C4×#131×C672
#6062467