Finite Group Databasebeta

← C67×#34512 #30713763

Subgroups — C67×#34512 #30713763

Order 17152.

2576 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×H(3,𝔽4)×C67C211yesmaximal, characteristic
2C67×#2375C213yesmaximal, complemented
3C67×#2372C213yesmaximal, complemented
4C67×#2367C216yesmaximal, complemented
5C67×#2358C213yesmaximal, complemented
6C67×#2337C216yesmaximal, complemented
7C67×#2326C213yesmaximal, complemented
8C67×#2321C213yesmaximal, complemented
9C4×#120×C67C213yesmaximal, complemented
10C24×C4×C67C2211yescharacteristic, complemented
11H(3,𝔽4)×C67C4116yescomplemented
12C2×#124×C67C2213yes—
13C2×#124×C67C2216yes—
14C2×#120×C67C2213yes—
15C2×#120×C67C2213yes—
16C2×C4×Q8×C67C2213yescomplemented
17C2×C4×D4×C67C2213yes—
18C2×C4×D4×C67C2216yes—
19C22×C22⋊C4×C67C2213yescomplemented
20#391×C67C2213yescomplemented
21#391×C67C2216yescomplemented
22#390×C67C2216yes—
23#390×C67C2213yes—
24#388×C67C2216yescomplemented
25#388×C67C2216yescomplemented
26#388×C67C2216yescomplemented
27#388×C67C2216yescomplemented
28#388×C67C2216yescomplemented
29#388×C67C2216yescomplemented
30#386×C67C2216yes—
31#386×C67C2213yes—
32#386×C67C2213yes—
33#386×C67C2216yes—
34#385×C67C2216yescomplemented
35#385×C67C2213yescomplemented
36#385×C67C2213yescomplemented
37#385×C67C2216yescomplemented
38#384×C67C2211yescharacteristic, complemented
39#383×C67C2212yescomplemented
40#379×C67C2213yescomplemented
41#379×C67C2211yescharacteristic, complemented
42C4×C4⋊C4×C67C2216yescomplemented
43C4×C22⋊C4×C67C2213yescomplemented
44C4×C22⋊C4×C67C2216yescomplemented
45C4×C22⋊C4×C67C2213yescomplemented
46C4×C22⋊C4×C67C2216yescomplemented
47#376×C67C2213yescomplemented
48C25×C67C2311yescharacteristic
49C25×C67C2311yescharacteristic
50C22×Q8×C67C2313yes—
51C22×D4×C67C2313yes—
52C22×D4×C67C2316yes—
53C23×C4×C67—46no—
54C23×C4×C67C2313yes—
55C23×C4×C67C2313yes—
56C23×C4×C67C2313yes—
57C23×C4×C67C2313yes—
58#124×C67C2×C4124yescomplemented
59#124×C67C2×C4148yescomplemented
60#120×C67C2×C4124yescomplemented
61#120×C67C2×C4124yescomplemented
62C4×Q8×C67—46no—
63C4×D4×C67—46no—
64C4×D4×C67—412no—
65C2×C4⋊C4×C67C2316yes—
66C2×C4⋊C4×C67C2313yes—
67C2×C4⋊C4×C67C2313yes—
68C2×C4⋊C4×C67C2316yes—
69C2×C22⋊C4×C67—46no—
70C2×C22⋊C4×C67C2316yes—
71C2×C22⋊C4×C67C2316yes—
72C2×C22⋊C4×C67C2313yes—
73C2×C22⋊C4×C67C2316yes—
74C2×C22⋊C4×C67C2313yes—
75C2×C22⋊C4×C67C2313yes—
76C2×C22⋊C4×C67C2316yes—
77C2×C22⋊C4×C67C2313yes—
78C2×C22⋊C4×C67C2316yes—
79C2×C22⋊C4×C67C2313yes—
80C2×C22⋊C4×C67C2316yes—
81C2×C22⋊C4×C67C2313yes—
82C2×C42×C67C2313yes—
83C2×C42×C67C2316yes—
84C2×C42×C67C2316yes—
85C2×C42×C67C2313yes—
86C2×C42×C67C2311yescharacteristic
87C2×C42×C67C2312yes—
88#95×C67C2316yes—
89#95×C67C2316yes—
90#95×C67C2316yes—
91#95×C67C2316yes—
92#95×C67C2316yes—
93#95×C67C2316yes—
94C24×C67—43no—
95C24×C67—43no—
96C24×C67—43no—
97C24×C67—43no—
98C24×C67C22×C414yes—
99C24×C67C22×C414yescomplemented
100C24×C67C2413yes—
101C24×C67C2413yes—
102C2×Q8×C67—43no—
103C2×Q8×C67—43no—
104C2×Q8×C67C22×C4112yescomplemented
105C2×D4×C67—43no—
106C2×D4×C67—46no—
107C2×D4×C67—43no—
108C2×D4×C67—46no—
109C2×D4×C67C22×C4112yes—
110C2×D4×C67C22×C4124yes—
111C22×C4×C67—212no—
112C22×C4×C67—26no—
113C22×C4×C67—26no—
114C22×C4×C67—43no—
115C22×C4×C67—46no—
116C22×C4×C67—43no—
117C22×C4×C67—46no—
118C22×C4×C67—46no—
119C22×C4×C67—416no—
120C22×C4×C67C2413yes—
121C22×C4×C67C2413yes—
122C22×C4×C67C2416yes—
123C22×C4×C67C2411yescharacteristic
124C22×C4×C67C2413yes—
125C22×C4×C67C2416yes—
126C22×C4×C67C2413yes—
127C4⋊C4×C67—212no—
128C4⋊C4×C67—46no—
129C4⋊C4×C67—43no—
130C4⋊C4×C67—43no—
131C22⋊C4×C67—212no—
132C22⋊C4×C67—46no—
133C22⋊C4×C67—46no—
134C22⋊C4×C67—43no—
135C22⋊C4×C67—43no—
136C22⋊C4×C67—43no—
137C22⋊C4×C67—43no—
138C22⋊C4×C67—46no—
139C22⋊C4×C67C22×C4124yes—
140C22⋊C4×C67C22×C4112yes—
141C22⋊C4×C67C22×C4124yescomplemented
142C22⋊C4×C67C22×C4112yescomplemented
143C42×C67—26no—
144C42×C67—43no—
145C42×C67—46no—
146C42×C67—46no—
147C42×C67C22×C414yescomplemented
148C42×C67C22×C418yescomplemented
149C23×C67—26no—
150C23×C67—23no—
151C23×C67—26no—
152C23×C67—23no—
153C23×C67—26no—
154C23×C67—23no—
155C23×C67—23no—
156C23×C67—26no—
157C23×C67—412no—
158C23×C67—412no—
159C23×C67—44no—
160C23×C67—412no—
161C23×C67—82no—
162C23×C67—86no—
163C23×C67C23×C416yes—
164C23×C67C23×C416yes—
165C23×C67C2511yescharacteristic
166Q8×C67—412no—
167D4×C67—412no—
168D4×C67—424no—
169C2×C4×C67—23no—
170C2×C4×C67—26no—
171C2×C4×C67—26no—
172C2×C4×C67—23no—
173C2×C4×C67—26no—
174C2×C4×C67—23no—
175C2×C4×C67—26no—
176C2×C4×C67—26no—
177C2×C4×C67—23no—
178C2×C4×C67—23no—
179C2×C4×C67—26no—
180C2×C4×C67—26no—
181C2×C4×C67—26no—
182C2×C4×C67—412no—
183C2×C4×C67—412no—
184C2×C4×C67—86no—
185C2×C4×C67—86no—
186C2×C4×C67C23×C416yes—
187C2×C4×C67C23×C4112yes—
188C2×C4×C67H(5,𝔽2)16yes—
189C22×C67—26no—
190C22×C67—212no—
191C22×C67—212no—
192C22×C67—26no—
193C22×C67—416no—
194C22×C67—43no—
195C22×C67—43no—
196C22×C67—43no—
197C22×C67—43no—
198C22×C67—46no—
199C22×C67—46no—
200C22×C67—88no—
201C22×C67C24×C411yescharacteristic
202C22×C67C2×H(5,𝔽2)13yes—
203C22×C67#58513yes—
204C4×C67—26no—
205C4×C67—212no—
206C4×C67—26no—
207C4×C67—41no—
208C4×C67—43no—
209C4×C67—43no—
210C4×C67—46no—
211#34512C6711yesmaximal, characteristic, direct product factor, Sylow
212C2×C67—46no—
213C2×C67—46no—
214C2×C67C4×H(5,𝔽2)13yes—
215C2×C67#353211yescharacteristic
216C2×C67#354013yes—
217C2×H(3,𝔽4)C2×C6711yescharacteristic
218#2375C2×C6713yescomplemented
219#2372C2×C6713yescomplemented
220#2367C2×C6716yescomplemented
221#2358C2×C6713yescomplemented
222#2337C2×C6716yescomplemented
223#2326C2×C6713yescomplemented
224#2321C2×C6713yescomplemented
225C4×#120C2×C6713yescomplemented
226C67#3451211yescharacteristic, direct product factor, Sylow
227C24×C4C22×C6711yescharacteristic, complemented
228H(3,𝔽4)C4×C67116yescomplemented
229C2×#124C22×C6713yes—
230C2×#124C22×C6716yes—
231C2×#120C22×C6713yes—
232C2×#120C22×C6713yes—
233C2×C4×Q8C22×C6713yescomplemented
234C2×C4×D4C22×C6713yes—
235C2×C4×D4C22×C6716yes—
236C22×C22⋊C4C22×C6713yescomplemented
237#391C22×C6713yescomplemented
238#391C22×C6716yescomplemented
239#390C22×C6716yes—
240#390C22×C6713yes—
241#388C22×C6716yescomplemented
242#388C22×C6716yescomplemented
243#388C22×C6716yescomplemented
244#388C22×C6716yescomplemented
245#388C22×C6716yescomplemented
246#388C22×C6716yescomplemented
247#386C22×C6716yes—
248#386C22×C6713yes—
249#386C22×C6716yes—
250#386C22×C6713yes—
251#385C22×C6713yescomplemented
252#385C22×C6716yescomplemented
253#385C22×C6716yescomplemented
254#385C22×C6713yescomplemented
255#384C22×C6711yescharacteristic, complemented
256#383C22×C6712yescomplemented
257#379C22×C6711yescharacteristic, complemented
258#379C22×C6713yescomplemented
259C4×C4⋊C4C22×C6716yescomplemented
260C4×C22⋊C4C22×C6713yescomplemented
261C4×C22⋊C4C22×C6713yescomplemented
262C4×C22⋊C4C22×C6716yescomplemented
263C4×C22⋊C4C22×C6716yescomplemented
264#376C22×C6713yescomplemented
265C25C23×C6711yescharacteristic
266C25C23×C6711yescharacteristic
267C22×Q8C23×C6713yes—
268C22×D4C23×C6713yes—
269C22×D4C23×C6716yes—
270C23×C4—46no—
271C23×C4C23×C6713yes—
272C23×C4C23×C6713yes—
273C23×C4C23×C6713yes—
274C23×C4C23×C6713yes—
275#124C2×C4×C67124yescomplemented
276#124C2×C4×C67148yescomplemented
277#120C2×C4×C67124yescomplemented
278#120C2×C4×C67124yescomplemented
279C4×Q8—46no—
280C4×D4—46no—
281C4×D4—412no—
282C2×C4⋊C4C23×C6716yes—
283C2×C4⋊C4C23×C6713yes—
284C2×C4⋊C4C23×C6713yes—
285C2×C4⋊C4C23×C6716yes—
286C2×C22⋊C4—46no—
287C2×C22⋊C4C23×C6716yes—
288C2×C22⋊C4C23×C6716yes—
289C2×C22⋊C4C23×C6713yes—
290C2×C22⋊C4C23×C6716yes—
291C2×C22⋊C4C23×C6713yes—
292C2×C22⋊C4C23×C6713yes—
293C2×C22⋊C4C23×C6716yes—
294C2×C22⋊C4C23×C6713yes—
295C2×C22⋊C4C23×C6716yes—
296C2×C22⋊C4C23×C6713yes—
297C2×C22⋊C4C23×C6716yes—
298C2×C22⋊C4C23×C6713yes—
299C2×C42C23×C6713yes—
300C2×C42C23×C6716yes—
301C2×C42C23×C6716yes—
302C2×C42C23×C6713yes—
303C2×C42C23×C6711yescharacteristic
304C2×C42C23×C6712yes—
305#95C23×C6716yes—
306#95C23×C6716yes—
307#95C23×C6716yes—
308#95C23×C6716yes—
309#95C23×C6716yes—
310#95C23×C6716yes—
311C24—43no—
312C24—43no—
313C24—43no—
314C24—43no—
315C24C22×C4×C6714yes—
316C24C22×C4×C6714yescomplemented
317C24C24×C6713yes—
318C24C24×C6713yes—
319C2×Q8—43no—
320C2×Q8—43no—
321C2×Q8C22×C4×C67112yescomplemented
322C2×D4—43no—
323C2×D4—46no—
324C2×D4—43no—
325C2×D4—46no—
326C2×D4C22×C4×C67112yes—
327C2×D4C22×C4×C67124yes—
328C22×C4—212no—
329C22×C4—26no—
330C22×C4—26no—
331C22×C4—43no—
332C22×C4—46no—
333C22×C4—43no—
334C22×C4—46no—
335C22×C4—46no—
336C22×C4—416no—
337C22×C4C24×C6716yes—
338C22×C4C24×C6711yescharacteristic
339C22×C4C24×C6713yes—
340C22×C4C24×C6713yes—
341C22×C4C24×C6713yes—
342C22×C4C24×C6716yes—
343C22×C4C24×C6713yes—
344C4⋊C4—212no—
345C4⋊C4—43no—
346C4⋊C4—43no—
347C4⋊C4—46no—
348C22⋊C4—212no—
349C22⋊C4—46no—
350C22⋊C4—43no—
351C22⋊C4—43no—
352C22⋊C4—46no—
353C22⋊C4—46no—
354C22⋊C4—43no—
355C22⋊C4—43no—
356C22⋊C4C22×C4×C67124yes—
357C22⋊C4C22×C4×C67112yes—
358C22⋊C4C22×C4×C67124yescomplemented
359C22⋊C4C22×C4×C67112yescomplemented
360C42—26no—
361C42—46no—
362C42—46no—
363C42—43no—
364C42C22×C4×C6714yescomplemented
365C42C22×C4×C6718yescomplemented
366C23—26no—
367C23—23no—
368C23—26no—
369C23—23no—
370C23—26no—
371C23—23no—
372C23—23no—
373C23—26no—
374C23—412no—
375C23—412no—
376C23—44no—
377C23—412no—
378C23—82no—
379C23—86no—
380C23C23×C4×C6716yes—
381C23C23×C4×C6716yes—
382C23C25×C6711yescharacteristic
383Q8—412no—
384D4—412no—
385D4—424no—
386C2×C4—23no—
387C2×C4—26no—
388C2×C4—26no—
389C2×C4—23no—
390C2×C4—26no—
391C2×C4—23no—
392C2×C4—26no—
393C2×C4—26no—
394C2×C4—23no—
395C2×C4—23no—
396C2×C4—26no—
397C2×C4—26no—
398C2×C4—26no—
399C2×C4—412no—
400C2×C4—412no—
401C2×C4—86no—
402C2×C4—86no—
403C2×C4C23×C4×C6716yes—
404C2×C4C23×C4×C67112yes—
405C2×C4H(5,𝔽2)×C6716yes—
406C22—26no—
407C22—212no—
408C22—212no—
409C22—26no—
410C22—416no—
411C22—43no—
412C22—43no—
413C22—43no—
414C22—43no—
415C22—46no—
416C22—46no—
417C22—88no—
418C22C24×C4×C6711yescharacteristic
419C22C2×H(5,𝔽2)×C6713yes—
420C22#585×C6713yes—
421C4—26no—
422C4—26no—
423C4—212no—
424C4—41no—
425C4—43no—
426C4—46no—
427C4—43no—
428C2—46no—
429C2—46no—
430C2C4×H(5,𝔽2)×C6713yes—
431C2C67×#353211yescharacteristic
432C2C67×#354013yes—
subgroupG/H
center, socleC23×C67C25
commutatorC22C24×C4×C67
Frattini, cosocleC23C25×C67
prime radicalC2×H(3,𝔽4)×C67C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2375C2semidirect
3. C67×#2372C2semidirect
4. C67×#2367C2semidirect
5. C67×#2358C2semidirect
6. C67×#2337C2semidirect
7. C67×#2326C2semidirect
8. C67×#2321C2semidirect
9. C4×#120×C67C2semidirect
10. C24×C4×C67C22semidirect
11. H(3,𝔽4)×C67C4semidirect
16. C2×C4×Q8×C67C22semidirect
19. C22×C22⋊C4×C67C22semidirect
20. #391×C67C22semidirect
21. #391×C67C22semidirect
24. #388×C67C22semidirect
25. #388×C67C22semidirect
26. #388×C67C22semidirect
27. #388×C67C22semidirect
28. #388×C67C22semidirect
29. #388×C67C22semidirect
34. #385×C67C22semidirect
35. #385×C67C22semidirect
36. #385×C67C22semidirect
37. #385×C67C22semidirect
38. #384×C67C22semidirect
39. #383×C67C22semidirect
40. #379×C67C22semidirect
41. #379×C67C22semidirect
42. C4×C4⋊C4×C67C22semidirect
43. C4×C22⋊C4×C67C22semidirect
44. C4×C22⋊C4×C67C22semidirect
45. C4×C22⋊C4×C67C22semidirect
46. C4×C22⋊C4×C67C22semidirect
47. #376×C67C22semidirect
58. #124×C67C2×C4semidirect
59. #124×C67C2×C4semidirect
60. #120×C67C2×C4semidirect
61. #120×C67C2×C4semidirect
99. C24×C67C22×C4semidirect
104. C2×Q8×C67C22×C4semidirect
141. C22⋊C4×C67C22×C4semidirect
142. C22⋊C4×C67C22×C4semidirect
147. C42×C67C22×C4semidirect
148. C42×C67C22×C4semidirect
211. #34512C67direct (N × H)
218. #2375C2×C67semidirect
219. #2372C2×C67semidirect
220. #2367C2×C67semidirect
221. #2358C2×C67semidirect
222. #2337C2×C67semidirect
223. #2326C2×C67semidirect
224. #2321C2×C67semidirect
225. C4×#120C2×C67semidirect
226. C67#34512direct (N × H)
227. C24×C4C22×C67semidirect
228. H(3,𝔽4)C4×C67semidirect
233. C2×C4×Q8C22×C67semidirect
236. C22×C22⋊C4C22×C67semidirect
237. #391C22×C67semidirect
238. #391C22×C67semidirect
241. #388C22×C67semidirect
242. #388C22×C67semidirect
243. #388C22×C67semidirect
244. #388C22×C67semidirect
245. #388C22×C67semidirect
246. #388C22×C67semidirect
251. #385C22×C67semidirect
252. #385C22×C67semidirect
253. #385C22×C67semidirect
254. #385C22×C67semidirect
255. #384C22×C67semidirect
256. #383C22×C67semidirect
257. #379C22×C67semidirect
258. #379C22×C67semidirect
259. C4×C4⋊C4C22×C67semidirect
260. C4×C22⋊C4C22×C67semidirect
261. C4×C22⋊C4C22×C67semidirect
262. C4×C22⋊C4C22×C67semidirect
263. C4×C22⋊C4C22×C67semidirect
264. #376C22×C67semidirect
275. #124C2×C4×C67semidirect
276. #124C2×C4×C67semidirect
277. #120C2×C4×C67semidirect
278. #120C2×C4×C67semidirect
316. C24C22×C4×C67semidirect
321. C2×Q8C22×C4×C67semidirect
358. C22⋊C4C22×C4×C67semidirect
359. C22⋊C4C22×C4×C67semidirect
364. C42C22×C4×C67semidirect
365. C42C22×C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×H(3,𝔽4)×C672
C67×#23752
C67×#23722
C67×#23672
C67×#23582
C67×#23372
C67×#23262
C67×#23212
C4×#120×C672
#3451267