Finite Group Databasebeta

← C67×#36676 #30715927

Subgroups — C67×#36676 #30715927

Order 17152.

2006 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2842C211yesmaximal, characteristic, complemented
2C67×#2834C211yesmaximal, characteristic, complemented
3C67×#2827C216yesmaximal, complemented
4C67×#2826C213yesmaximal, complemented
5C67×#2825C212yesmaximal, complemented
6C67×#2812C213yesmaximal, complemented
7C67×#2401C212yesmaximal, complemented
8C67×#2397C211yesmaximal, characteristic, complemented
9C67×#2395C213yesmaximal, complemented
10C67×#2392C216yesmaximal, complemented
11C4×#120×C67C213yesmaximal, complemented
12C24×C4×C67C2211yescharacteristic
13C2×#120×C67C2213yes—
14C2×C4×D4×C67C2216yes—
15C2×C4×D4×C67C2213yes—
16C2×C4×D4×C67C2213yescomplemented
17C2×#117×C67C2213yescomplemented
18#401×C67C2212yescomplemented
19#400×C67C2216yescomplemented
20#400×C67C2213yescomplemented
21#399×C67C2213yes—
22#399×C67C2211yescharacteristic
23#399×C67C2213yescomplemented
24#398×C67C2213yescomplemented
25#398×C67C2216yescomplemented
26#397×C67C2216yes—
27#397×C67C2213yes—
28#397×C67C2216yescomplemented
29#397×C67C2216yescomplemented
30#396×C67C2212yes—
31#396×C67C2216yes—
32#396×C67C2216yescomplemented
33#394×C67C2216yes—
34#394×C67C2216yes—
35#394×C67C2216yes—
36#394×C67C2216yescomplemented
37#392×C67C2216yes—
38#392×C67C2213yes—
39#392×C67C2211yescharacteristic, complemented
40#386×C67C2216yes—
41#386×C67C2213yes—
42#385×C67C2213yes—
43#385×C67C2216yes—
44#379×C67C2213yes—
45C4×C4⋊C4×C67C2216yescomplemented
46C4×C22⋊C4×C67C2216yes—
47C4×C22⋊C4×C67C2213yes—
48C4×C22⋊C4×C67C2213yescomplemented
49#376×C67C2211yescharacteristic, complemented
50C25×C67C2311yescharacteristic
51C22×D4×C67C2316yes—
52C22×D4×C67C2313yes—
53C22×D4×C67C2313yes—
54C23×C4×C67—46no—
55C23×C4×C67C2313yes—
56C23×C4×C67C2312yes—
57C23×C4×C67C2311yescharacteristic
58C23×C4×C67C2311yescharacteristic
59C23×C4×C67C2312yes—
60#120×C67—212no—
61C4×D4×C67—412no—
62C4×D4×C67—46no—
63C4×D4×C67—46no—
64#117×C67—46no—
65C2×C4⋊C4×C67C2313yes—
66C2×C4⋊C4×C67C2313yes—
67C2×C4⋊C4×C67C2316yes—
68C2×C4⋊C4×C67C2313yes—
69C2×C4⋊C4×C67C2313yes—
70C2×C4⋊C4×C67C2316yes—
71C2×C4⋊C4×C67C2316yes—
72C2×C22⋊C4×C67C2316yes—
73C2×C22⋊C4×C67C2313yes—
74C2×C22⋊C4×C67C2316yes—
75C2×C22⋊C4×C67C2313yes—
76C2×C22⋊C4×C67C2316yes—
77C2×C22⋊C4×C67C2313yes—
78C2×C22⋊C4×C67C2316yes—
79C2×C22⋊C4×C67C2316yes—
80C2×C22⋊C4×C67C2316yes—
81C2×C22⋊C4×C67C2313yes—
82C2×C22⋊C4×C67C2316yes—
83C2×C42×C67C2316yes—
84C2×C42×C67C2313yes—
85#95×C67C2313yes—
86#95×C67C2311yescharacteristic
87#95×C67C2316yes—
88#95×C67C2316yes—
89#95×C67C2316yes—
90#95×C67C2313yes—
91#95×C67C2313yes—
92#95×C67C2316yes—
93#95×C67C2316yes—
94C24×C67—43no—
95C24×C67—43no—
96C24×C67#4114yes—
97C24×C67C2413yes—
98C24×C67C2411yescharacteristic
99C24×C67C2412yes—
100C2×D4×C67—212no—
101C2×D4×C67—26no—
102C2×D4×C67—43no—
103C2×D4×C67—43no—
104C2×D4×C67—46no—
105C2×D4×C67—46no—
106C2×D4×C67—46no—
107C2×D4×C67—43no—
108C22×C4×C67—26no—
109C22×C4×C67—212no—
110C22×C4×C67—212no—
111C22×C4×C67—26no—
112C22×C4×C67—43no—
113C22×C4×C67—46no—
114C22×C4×C67—416no—
115C22×C4×C67C2416yes—
116C22×C4×C67C2413yes—
117C22×C4×C67C2416yes—
118C22×C4×C67C2413yes—
119C22×C4×C67C2413yes—
120C22×C4×C67C2411yescharacteristic
121C22×C4×C67C2412yes—
122C22×C4×C67C2411yescharacteristic
123C4⋊C4×C67—46no—
124C4⋊C4×C67—46no—
125C4⋊C4×C67—43no—
126C4⋊C4×C67—43no—
127C4⋊C4×C67—43no—
128C4⋊C4×C67—43no—
129C4⋊C4×C67—46no—
130C22⋊C4×C67—212no—
131C22⋊C4×C67—26no—
132C22⋊C4×C67—43no—
133C22⋊C4×C67—46no—
134C22⋊C4×C67—46no—
135C22⋊C4×C67—46no—
136C22⋊C4×C67—46no—
137C22⋊C4×C67—46no—
138C22⋊C4×C67—46no—
139C22⋊C4×C67—43no—
140C22⋊C4×C67—43no—
141C42×C67—212no—
142C42×C67—26no—
143C23×C67—26no—
144C23×C67—23no—
145C23×C67—23no—
146C23×C67—26no—
147C23×C67—23no—
148C23×C67—23no—
149C23×C67—26no—
150C23×C67—26no—
151C23×C67—23no—
152C23×C67—412no—
153C23×C67—44no—
154C23×C67—412no—
155C23×C67C2×#4112yes—
156C23×C67C2×#4114yes—
157C23×C67C2511yescharacteristic
158D4×C67—412no—
159D4×C67—424no—
160D4×C67—412no—
161C2×C4×C67—23no—
162C2×C4×C67—26no—
163C2×C4×C67—26no—
164C2×C4×C67—23no—
165C2×C4×C67—23no—
166C2×C4×C67—26no—
167C2×C4×C67—26no—
168C2×C4×C67—23no—
169C2×C4×C67—26no—
170C2×C4×C67—23no—
171C2×C4×C67—26no—
172C2×C4×C67—26no—
173C2×C4×C67—26no—
174C2×C4×C67—26no—
175C2×C4×C67—23no—
176C2×C4×C67—44no—
177C2×C4×C67—48no—
178C2×C4×C67—412no—
179C2×C4×C67—48no—
180C2×C4×C67—44no—
181C2×C4×C67H(5,𝔽2)16yes—
182C22×C67—26no—
183C22×C67—212no—
184C22×C67—416no—
185C22×C67—42no—
186C22×C67—41no—
187C22×C67—46no—
188C22×C67—43no—
189C22×C67—43no—
190C22×C67—46no—
191C22×C67—43no—
192C22×C67—46no—
193C22×C67—43no—
194C22×C67C22×#4111yescharacteristic
195C22×C67C2×H(5,𝔽2)13yes—
196C22×C67#58513yes—
197C4×C67—43no—
198C4×C67—46no—
199C4×C67—43no—
200C4×C67—42no—
201C4×C67—41no—
202C4×C67—46no—
203C4×C67—43no—
204C4×C67#56014yes—
205#36676C6711yesmaximal, characteristic, direct product factor, Sylow
206C2×C67—42no—
207C2×C67—44no—
208C2×C67—83no—
209C2×C67#345913yes—
210C2×C67C2×#56011yescharacteristic
211C2×C67#352913yes—
212#2842C2×C6711yescharacteristic, complemented
213#2834C2×C6711yescharacteristic, complemented
214#2827C2×C6716yescomplemented
215#2826C2×C6713yescomplemented
216#2825C2×C6712yescomplemented
217#2812C2×C6713yescomplemented
218#2401C2×C6712yescomplemented
219#2397C2×C6711yescharacteristic, complemented
220#2395C2×C6713yescomplemented
221#2392C2×C6716yescomplemented
222C4×#120C2×C6713yescomplemented
223C67#3667611yescharacteristic, direct product factor, Sylow
224C24×C4C22×C6711yescharacteristic
225C2×#120C22×C6713yes—
226C2×C4×D4C22×C6716yes—
227C2×C4×D4C22×C6713yes—
228C2×C4×D4C22×C6713yescomplemented
229C2×#117C22×C6713yescomplemented
230#401C22×C6712yescomplemented
231#400C22×C6716yescomplemented
232#400C22×C6713yescomplemented
233#399C22×C6713yes—
234#399C22×C6711yescharacteristic
235#399C22×C6713yescomplemented
236#398C22×C6713yescomplemented
237#398C22×C6716yescomplemented
238#397C22×C6716yes—
239#397C22×C6713yes—
240#397C22×C6716yescomplemented
241#397C22×C6716yescomplemented
242#396C22×C6712yes—
243#396C22×C6716yes—
244#396C22×C6716yescomplemented
245#394C22×C6716yes—
246#394C22×C6716yes—
247#394C22×C6716yes—
248#394C22×C6716yescomplemented
249#392C22×C6716yes—
250#392C22×C6713yes—
251#392C22×C6711yescharacteristic, complemented
252#386C22×C6716yes—
253#386C22×C6713yes—
254#385C22×C6713yes—
255#385C22×C6716yes—
256#379C22×C6713yes—
257C4×C4⋊C4C22×C6716yescomplemented
258C4×C22⋊C4C22×C6716yes—
259C4×C22⋊C4C22×C6713yes—
260C4×C22⋊C4C22×C6713yescomplemented
261#376C22×C6711yescharacteristic, complemented
262C25C23×C6711yescharacteristic
263C22×D4C23×C6716yes—
264C22×D4C23×C6713yes—
265C22×D4C23×C6713yes—
266C23×C4—46no—
267C23×C4C23×C6713yes—
268C23×C4C23×C6712yes—
269C23×C4C23×C6711yescharacteristic
270C23×C4C23×C6711yescharacteristic
271C23×C4C23×C6712yes—
272#120—212no—
273C4×D4—412no—
274C4×D4—46no—
275C4×D4—46no—
276#117—46no—
277C2×C4⋊C4C23×C6713yes—
278C2×C4⋊C4C23×C6713yes—
279C2×C4⋊C4C23×C6716yes—
280C2×C4⋊C4C23×C6713yes—
281C2×C4⋊C4C23×C6713yes—
282C2×C4⋊C4C23×C6716yes—
283C2×C4⋊C4C23×C6716yes—
284C2×C22⋊C4C23×C6716yes—
285C2×C22⋊C4C23×C6713yes—
286C2×C22⋊C4C23×C6716yes—
287C2×C22⋊C4C23×C6713yes—
288C2×C22⋊C4C23×C6716yes—
289C2×C22⋊C4C23×C6713yes—
290C2×C22⋊C4C23×C6716yes—
291C2×C22⋊C4C23×C6716yes—
292C2×C22⋊C4C23×C6716yes—
293C2×C22⋊C4C23×C6713yes—
294C2×C22⋊C4C23×C6716yes—
295C2×C42C23×C6716yes—
296C2×C42C23×C6713yes—
297#95C23×C6713yes—
298#95C23×C6711yescharacteristic
299#95C23×C6716yes—
300#95C23×C6716yes—
301#95C23×C6716yes—
302#95C23×C6713yes—
303#95C23×C6713yes—
304#95C23×C6716yes—
305#95C23×C6716yes—
306C24—43no—
307C24—43no—
308C24#41×C6714yes—
309C24C24×C6713yes—
310C24C24×C6711yescharacteristic
311C24C24×C6712yes—
312C2×D4—212no—
313C2×D4—26no—
314C2×D4—43no—
315C2×D4—43no—
316C2×D4—46no—
317C2×D4—46no—
318C2×D4—46no—
319C2×D4—43no—
320C22×C4—26no—
321C22×C4—212no—
322C22×C4—212no—
323C22×C4—26no—
324C22×C4—43no—
325C22×C4—46no—
326C22×C4—416no—
327C22×C4C24×C6713yes—
328C22×C4C24×C6716yes—
329C22×C4C24×C6713yes—
330C22×C4C24×C6712yes—
331C22×C4C24×C6711yescharacteristic
332C22×C4C24×C6716yes—
333C22×C4C24×C6713yes—
334C22×C4C24×C6711yescharacteristic
335C4⋊C4—43no—
336C4⋊C4—43no—
337C4⋊C4—43no—
338C4⋊C4—43no—
339C4⋊C4—46no—
340C4⋊C4—46no—
341C4⋊C4—46no—
342C22⋊C4—212no—
343C22⋊C4—26no—
344C22⋊C4—46no—
345C22⋊C4—46no—
346C22⋊C4—46no—
347C22⋊C4—46no—
348C22⋊C4—43no—
349C22⋊C4—43no—
350C22⋊C4—43no—
351C22⋊C4—46no—
352C22⋊C4—46no—
353C42—212no—
354C42—26no—
355C23—26no—
356C23—23no—
357C23—23no—
358C23—26no—
359C23—23no—
360C23—23no—
361C23—26no—
362C23—26no—
363C23—23no—
364C23—412no—
365C23—44no—
366C23—412no—
367C23C2×#41×C6712yes—
368C23C2×#41×C6714yes—
369C23C25×C6711yescharacteristic
370D4—412no—
371D4—424no—
372D4—412no—
373C2×C4—23no—
374C2×C4—26no—
375C2×C4—26no—
376C2×C4—23no—
377C2×C4—23no—
378C2×C4—26no—
379C2×C4—26no—
380C2×C4—23no—
381C2×C4—26no—
382C2×C4—23no—
383C2×C4—26no—
384C2×C4—26no—
385C2×C4—26no—
386C2×C4—26no—
387C2×C4—23no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—44no—
392C2×C4—412no—
393C2×C4H(5,𝔽2)×C6716yes—
394C22—26no—
395C22—212no—
396C22—416no—
397C22—42no—
398C22—41no—
399C22—46no—
400C22—43no—
401C22—43no—
402C22—46no—
403C22—43no—
404C22—43no—
405C22—46no—
406C22C22×#41×C6711yescharacteristic
407C22C2×H(5,𝔽2)×C6713yes—
408C22#585×C6713yes—
409C4—42no—
410C4—41no—
411C4—43no—
412C4—43no—
413C4—46no—
414C4—43no—
415C4—46no—
416C4#560×C6714yes—
417C2—42no—
418C2—44no—
419C2—83no—
420C2C67×#345913yes—
421C2C2×#560×C6711yescharacteristic
422C2C67×#352913yes—
subgroupG/H
centerC22×C4×C67C24
commutator, Frattini, cosocleC23C25×C67
socleC23×C67C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2842C2semidirect
2. C67×#2834C2semidirect
3. C67×#2827C2semidirect
4. C67×#2826C2semidirect
5. C67×#2825C2semidirect
6. C67×#2812C2semidirect
7. C67×#2401C2semidirect
8. C67×#2397C2semidirect
9. C67×#2395C2semidirect
10. C67×#2392C2semidirect
11. C4×#120×C67C2semidirect
16. C2×C4×D4×C67C22semidirect
17. C2×#117×C67C22semidirect
18. #401×C67C22semidirect
19. #400×C67C22semidirect
20. #400×C67C22semidirect
23. #399×C67C22semidirect
24. #398×C67C22semidirect
25. #398×C67C22semidirect
28. #397×C67C22semidirect
29. #397×C67C22semidirect
32. #396×C67C22semidirect
36. #394×C67C22semidirect
39. #392×C67C22semidirect
45. C4×C4⋊C4×C67C22semidirect
48. C4×C22⋊C4×C67C22semidirect
49. #376×C67C22semidirect
205. #36676C67direct (N × H)
212. #2842C2×C67semidirect
213. #2834C2×C67semidirect
214. #2827C2×C67semidirect
215. #2826C2×C67semidirect
216. #2825C2×C67semidirect
217. #2812C2×C67semidirect
218. #2401C2×C67semidirect
219. #2397C2×C67semidirect
220. #2395C2×C67semidirect
221. #2392C2×C67semidirect
222. C4×#120C2×C67semidirect
223. C67#36676direct (N × H)
228. C2×C4×D4C22×C67semidirect
229. C2×#117C22×C67semidirect
230. #401C22×C67semidirect
231. #400C22×C67semidirect
232. #400C22×C67semidirect
235. #399C22×C67semidirect
236. #398C22×C67semidirect
237. #398C22×C67semidirect
240. #397C22×C67semidirect
241. #397C22×C67semidirect
244. #396C22×C67semidirect
248. #394C22×C67semidirect
251. #392C22×C67semidirect
257. C4×C4⋊C4C22×C67semidirect
260. C4×C22⋊C4C22×C67semidirect
261. #376C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#28422
C67×#28342
C67×#28272
C67×#28262
C67×#28252
C67×#28122
C67×#24012
C67×#23972
C67×#23952
C67×#23922
C4×#120×C672
#3667667