Finite Group Databasebeta

← C67×#60037 #30739288

Subgroups — C67×#60037 #30739288

Order 17152.

2228 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×#2697C213yesmaximal, complemented
2C67×#2679C211yesmaximal, characteristic
3C67×#2576C216yesmaximal, complemented
4C67×#2566C213yesmaximal, complemented
5C67×#2534C213yesmaximal, complemented
6C67×#2460C216yesmaximal, complemented
7C67×#2374C213yesmaximal, complemented
8C67×#2358C213yesmaximal, complemented
9C4×#120×C67C213yesmaximal, complemented
10C24×C4×C67C2211yescharacteristic, complemented
11C2×#125×C67C2213yescomplemented
12C2×#123×C67C2213yes—
13C2×#120×C67C2213yes—
14C2×#120×C67C2213yes—
15C2×C4×D4×C67C2213yes—
16C2×C4×D4×C67C2216yes—
17C22×C22⋊C4×C67C2213yescomplemented
18#401×C67C2212yescomplemented
19#397×C67C2216yescomplemented
20#396×C67C2216yescomplemented
21#396×C67C2216yescomplemented
22#394×C67C2216yes—
23#394×C67C2216yes—
24#390×C67C2213yes—
25#390×C67C2213yes—
26#388×C67C2216yescomplemented
27#388×C67C2216yescomplemented
28#388×C67C2216yescomplemented
29#387×C67C2216yescomplemented
30#387×C67C2216yescomplemented
31#386×C67C2216yes—
32#386×C67C2213yes—
33#386×C67C2216yes—
34#385×C67C2216yescomplemented
35#385×C67C2213yescomplemented
36#385×C67C2216yescomplemented
37#384×C67C2211yescharacteristic, complemented
38#382×C67C2213yescomplemented
39#381×C67C2213yescomplemented
40#380×C67C2213yescomplemented
41#379×C67C2211yescharacteristic, complemented
42#379×C67C2213yescomplemented
43C4×C4⋊C4×C67C2213yescomplemented
44C4×C22⋊C4×C67C2213yescomplemented
45C4×C22⋊C4×C67C2216yescomplemented
46C4×C22⋊C4×C67C2216yescomplemented
47C25×C67C2311yescharacteristic
48C25×C67C2311yescharacteristic
49C22×D4×C67C2316yes—
50C22×D4×C67C2313yes—
51C23×C4×C67—46no—
52C23×C4×C67C2313yes—
53C23×C4×C67C2313yes—
54C23×C4×C67C2313yes—
55C23×C4×C67C2313yes—
56#125×C67—46no—
57#123×C67—212no—
58#120×C67—212no—
59#120×C67—212no—
60C4×D4×C67—46no—
61C4×D4×C67—412no—
62C2×C4⋊C4×C67C2316yes—
63C2×C4⋊C4×C67C2316yes—
64C2×C4⋊C4×C67C2313yes—
65C2×C4⋊C4×C67C2316yes—
66C2×C4⋊C4×C67C2313yes—
67C2×C4⋊C4×C67C2313yes—
68C2×C22⋊C4×C67—46no—
69C2×C22⋊C4×C67C2316yes—
70C2×C22⋊C4×C67C2316yes—
71C2×C22⋊C4×C67C2316yes—
72C2×C22⋊C4×C67C2313yes—
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×C67C2313yes—
80C2×C22⋊C4×C67C2313yes—
81C2×C42×C67C2313yes—
82C2×C42×C67C2316yes—
83C2×C42×C67C2313yes—
84C2×C42×C67C2311yescharacteristic
85#95×C67C2316yes—
86#95×C67C2316yes—
87#95×C67C2316yes—
88#95×C67C2316yes—
89#95×C67C2316yes—
90#95×C67C2316yes—
91#95×C67C2312yes—
92C24×C67—26no—
93C24×C67—26no—
94C24×C67—43no—
95C24×C67—41no—
96C24×C67—43no—
97C24×C67—41no—
98C24×C67C2413yes—
99C24×C67C2413yes—
100C2×D4×C67—212no—
101C2×D4×C67—212no—
102C2×D4×C67—43no—
103C2×D4×C67—46no—
104C2×D4×C67—43no—
105C2×D4×C67#41112yes—
106C22×C4×C67—26no—
107C22×C4×C67—212no—
108C22×C4×C67—26no—
109C22×C4×C67—26no—
110C22×C4×C67—43no—
111C22×C4×C67—46no—
112C22×C4×C67—43no—
113C22×C4×C67—46no—
114C22×C4×C67—43no—
115C22×C4×C67—416no—
116C22×C4×C67C2416yes—
117C22×C4×C67C2413yes—
118C22×C4×C67C2416yes—
119C22×C4×C67C2413yes—
120C22×C4×C67C2411yescharacteristic
121C22×C4×C67C2413yes—
122C22×C4×C67C2413yes—
123C4⋊C4×C67—212no—
124C4⋊C4×C67—212no—
125C4⋊C4×C67—43no—
126C4⋊C4×C67—43no—
127C4⋊C4×C67—43no—
128C4⋊C4×C67—46no—
129C22⋊C4×C67—26no—
130C22⋊C4×C67—26no—
131C22⋊C4×C67—26no—
132C22⋊C4×C67—212no—
133C22⋊C4×C67—212no—
134C22⋊C4×C67—212no—
135C22⋊C4×C67—212no—
136C22⋊C4×C67—46no—
137C22⋊C4×C67—43no—
138C22⋊C4×C67—43no—
139C22⋊C4×C67—43no—
140C22⋊C4×C67—46no—
141C42×C67—26no—
142C42×C67—41no—
143C42×C67—43no—
144C42×C67—46no—
145C23×C67—26no—
146C23×C67—26no—
147C23×C67—26no—
148C23×C67—26no—
149C23×C67—23no—
150C23×C67—23no—
151C23×C67—23no—
152C23×C67—23no—
153C23×C67—412no—
154C23×C67—412no—
155C23×C67—412no—
156C23×C67—44no—
157C23×C67—86no—
158C23×C67—82no—
159C23×C67C2×#4116yes—
160C23×C67C2×#4116yes—
161C23×C67C2511yescharacteristic
162D4×C67—412no—
163D4×C67—424no—
164C2×C4×C67—23no—
165C2×C4×C67—26no—
166C2×C4×C67—23no—
167C2×C4×C67—26no—
168C2×C4×C67—26no—
169C2×C4×C67—26no—
170C2×C4×C67—26no—
171C2×C4×C67—23no—
172C2×C4×C67—26no—
173C2×C4×C67—26no—
174C2×C4×C67—23no—
175C2×C4×C67—26no—
176C2×C4×C67—23no—
177C2×C4×C67—26no—
178C2×C4×C67—412no—
179C2×C4×C67—412no—
180C2×C4×C67—86no—
181C2×C4×C67—86no—
182C2×C4×C67C2×#4116yes—
183C2×C4×C67H(5,𝔽2)16yes—
184C22×C67—212no—
185C22×C67—212no—
186C22×C67—26no—
187C22×C67—26no—
188C22×C67—416no—
189C22×C67—43no—
190C22×C67—43no—
191C22×C67—46no—
192C22×C67—46no—
193C22×C67—43no—
194C22×C67—43no—
195C22×C67—88no—
196C22×C67C22×#4113yes—
197C22×C67C2×H(5,𝔽2)13yes—
198C22×C67#58511yescharacteristic
199C4×C67—26no—
200C4×C67—26no—
201C4×C67—43no—
202C4×C67—46no—
203C4×C67—46no—
204C4×C67—41no—
205C4×C67—43no—
206#60037C6711yesmaximal, characteristic, direct product factor, Sylow
207C2×C67—46no—
208C2×C67—46no—
209C2×C67#348113yes—
210C2×C67#350513yes—
211C2×C67#353211yescharacteristic
212#2697C2×C6713yescomplemented
213#2679C2×C6711yescharacteristic
214#2576C2×C6716yescomplemented
215#2566C2×C6713yescomplemented
216#2534C2×C6713yescomplemented
217#2460C2×C6716yescomplemented
218#2374C2×C6713yescomplemented
219#2358C2×C6713yescomplemented
220C4×#120C2×C6713yescomplemented
221C67#6003711yescharacteristic, direct product factor, Sylow
222C24×C4C22×C6711yescharacteristic, complemented
223C2×#125C22×C6713yescomplemented
224C2×#123C22×C6713yes—
225C2×#120C22×C6713yes—
226C2×#120C22×C6713yes—
227C2×C4×D4C22×C6713yes—
228C2×C4×D4C22×C6716yes—
229C22×C22⋊C4C22×C6713yescomplemented
230#401C22×C6712yescomplemented
231#397C22×C6716yescomplemented
232#396C22×C6716yescomplemented
233#396C22×C6716yescomplemented
234#394C22×C6716yes—
235#394C22×C6716yes—
236#390C22×C6713yes—
237#390C22×C6713yes—
238#388C22×C6716yescomplemented
239#388C22×C6716yescomplemented
240#388C22×C6716yescomplemented
241#387C22×C6716yescomplemented
242#387C22×C6716yescomplemented
243#386C22×C6716yes—
244#386C22×C6716yes—
245#386C22×C6713yes—
246#385C22×C6716yescomplemented
247#385C22×C6716yescomplemented
248#385C22×C6713yescomplemented
249#384C22×C6711yescharacteristic, complemented
250#382C22×C6713yescomplemented
251#381C22×C6713yescomplemented
252#380C22×C6713yescomplemented
253#379C22×C6713yescomplemented
254#379C22×C6711yescharacteristic, complemented
255C4×C4⋊C4C22×C6713yescomplemented
256C4×C22⋊C4C22×C6713yescomplemented
257C4×C22⋊C4C22×C6716yescomplemented
258C4×C22⋊C4C22×C6716yescomplemented
259C25C23×C6711yescharacteristic
260C25C23×C6711yescharacteristic
261C22×D4C23×C6716yes—
262C22×D4C23×C6713yes—
263C23×C4—46no—
264C23×C4C23×C6713yes—
265C23×C4C23×C6713yes—
266C23×C4C23×C6713yes—
267C23×C4C23×C6713yes—
268#125—46no—
269#123—212no—
270#120—212no—
271#120—212no—
272C4×D4—46no—
273C4×D4—412no—
274C2×C4⋊C4C23×C6716yes—
275C2×C4⋊C4C23×C6716yes—
276C2×C4⋊C4C23×C6713yes—
277C2×C4⋊C4C23×C6716yes—
278C2×C4⋊C4C23×C6713yes—
279C2×C4⋊C4C23×C6713yes—
280C2×C22⋊C4—46no—
281C2×C22⋊C4C23×C6716yes—
282C2×C22⋊C4C23×C6716yes—
283C2×C22⋊C4C23×C6716yes—
284C2×C22⋊C4C23×C6713yes—
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×C6713yes—
292C2×C22⋊C4C23×C6713yes—
293C2×C42C23×C6713yes—
294C2×C42C23×C6716yes—
295C2×C42C23×C6713yes—
296C2×C42C23×C6711yescharacteristic
297#95C23×C6716yes—
298#95C23×C6716yes—
299#95C23×C6716yes—
300#95C23×C6716yes—
301#95C23×C6716yes—
302#95C23×C6716yes—
303#95C23×C6712yes—
304C24—26no—
305C24—26no—
306C24—43no—
307C24—41no—
308C24—43no—
309C24—41no—
310C24C24×C6713yes—
311C24C24×C6713yes—
312C2×D4—212no—
313C2×D4—212no—
314C2×D4—43no—
315C2×D4—46no—
316C2×D4—43no—
317C2×D4#41×C67112yes—
318C22×C4—26no—
319C22×C4—212no—
320C22×C4—26no—
321C22×C4—26no—
322C22×C4—43no—
323C22×C4—46no—
324C22×C4—43no—
325C22×C4—46no—
326C22×C4—43no—
327C22×C4—416no—
328C22×C4C24×C6716yes—
329C22×C4C24×C6716yes—
330C22×C4C24×C6711yescharacteristic
331C22×C4C24×C6713yes—
332C22×C4C24×C6713yes—
333C22×C4C24×C6713yes—
334C22×C4C24×C6713yes—
335C4⋊C4—212no—
336C4⋊C4—212no—
337C4⋊C4—43no—
338C4⋊C4—43no—
339C4⋊C4—43no—
340C4⋊C4—46no—
341C22⋊C4—26no—
342C22⋊C4—26no—
343C22⋊C4—26no—
344C22⋊C4—212no—
345C22⋊C4—212no—
346C22⋊C4—212no—
347C22⋊C4—212no—
348C22⋊C4—46no—
349C22⋊C4—43no—
350C22⋊C4—43no—
351C22⋊C4—46no—
352C22⋊C4—43no—
353C42—26no—
354C42—46no—
355C42—41no—
356C42—43no—
357C23—26no—
358C23—26no—
359C23—26no—
360C23—26no—
361C23—23no—
362C23—23no—
363C23—23no—
364C23—23no—
365C23—412no—
366C23—412no—
367C23—412no—
368C23—44no—
369C23—86no—
370C23—82no—
371C23C2×#41×C6716yes—
372C23C2×#41×C6716yes—
373C23C25×C6711yescharacteristic
374D4—412no—
375D4—424no—
376C2×C4—23no—
377C2×C4—26no—
378C2×C4—23no—
379C2×C4—26no—
380C2×C4—26no—
381C2×C4—26no—
382C2×C4—26no—
383C2×C4—23no—
384C2×C4—26no—
385C2×C4—26no—
386C2×C4—23no—
387C2×C4—26no—
388C2×C4—23no—
389C2×C4—26no—
390C2×C4—412no—
391C2×C4—412no—
392C2×C4—86no—
393C2×C4—86no—
394C2×C4C2×#41×C6716yes—
395C2×C4H(5,𝔽2)×C6716yes—
396C22—212no—
397C22—212no—
398C22—26no—
399C22—26no—
400C22—416no—
401C22—43no—
402C22—43no—
403C22—43no—
404C22—43no—
405C22—46no—
406C22—46no—
407C22—88no—
408C22C22×#41×C6713yes—
409C22C2×H(5,𝔽2)×C6713yes—
410C22#585×C6711yescharacteristic
411C4—26no—
412C4—26no—
413C4—41no—
414C4—43no—
415C4—46no—
416C4—43no—
417C4—46no—
418C2—46no—
419C2—46no—
420C2C67×#348113yes—
421C2C67×#350513yes—
422C2C67×#353211yescharacteristic
subgroupG/H
center, socleC23×C67C25
commutator, Frattini, cosocleC23C25×C67
prime radicalC67×#2679C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2697C2semidirect
3. C67×#2576C2semidirect
4. C67×#2566C2semidirect
5. C67×#2534C2semidirect
6. C67×#2460C2semidirect
7. C67×#2374C2semidirect
8. C67×#2358C2semidirect
9. C4×#120×C67C2semidirect
10. C24×C4×C67C22semidirect
11. C2×#125×C67C22semidirect
17. C22×C22⋊C4×C67C22semidirect
18. #401×C67C22semidirect
19. #397×C67C22semidirect
20. #396×C67C22semidirect
21. #396×C67C22semidirect
26. #388×C67C22semidirect
27. #388×C67C22semidirect
28. #388×C67C22semidirect
29. #387×C67C22semidirect
30. #387×C67C22semidirect
34. #385×C67C22semidirect
35. #385×C67C22semidirect
36. #385×C67C22semidirect
37. #384×C67C22semidirect
38. #382×C67C22semidirect
39. #381×C67C22semidirect
40. #380×C67C22semidirect
41. #379×C67C22semidirect
42. #379×C67C22semidirect
43. C4×C4⋊C4×C67C22semidirect
44. C4×C22⋊C4×C67C22semidirect
45. C4×C22⋊C4×C67C22semidirect
46. C4×C22⋊C4×C67C22semidirect
206. #60037C67direct (N × H)
212. #2697C2×C67semidirect
214. #2576C2×C67semidirect
215. #2566C2×C67semidirect
216. #2534C2×C67semidirect
217. #2460C2×C67semidirect
218. #2374C2×C67semidirect
219. #2358C2×C67semidirect
220. C4×#120C2×C67semidirect
221. C67#60037direct (N × H)
222. C24×C4C22×C67semidirect
223. C2×#125C22×C67semidirect
229. C22×C22⋊C4C22×C67semidirect
230. #401C22×C67semidirect
231. #397C22×C67semidirect
232. #396C22×C67semidirect
233. #396C22×C67semidirect
238. #388C22×C67semidirect
239. #388C22×C67semidirect
240. #388C22×C67semidirect
241. #387C22×C67semidirect
242. #387C22×C67semidirect
246. #385C22×C67semidirect
247. #385C22×C67semidirect
248. #385C22×C67semidirect
249. #384C22×C67semidirect
250. #382C22×C67semidirect
251. #381C22×C67semidirect
252. #380C22×C67semidirect
253. #379C22×C67semidirect
254. #379C22×C67semidirect
255. C4×C4⋊C4C22×C67semidirect
256. C4×C22⋊C4C22×C67semidirect
257. C4×C22⋊C4C22×C67semidirect
258. C4×C22⋊C4C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#26972
C67×#26792
C67×#25762
C67×#25662
C67×#25342
C67×#24602
C67×#23742
C67×#23582
C4×#120×C672
#6003767