Finite Group Databasebeta

← C67×#33678 #30712929

Subgroups — C67×#33678 #30712929

Order 17152.

568 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3175C211yesmaximal, characteristic, complemented
2C67×#2948C211yesmaximal, characteristic, complemented
3C67×#2237C211yesmaximal, characteristic, complemented
4C67×#2234C211yesmaximal, characteristic, complemented
5C67×#2226C211yesmaximal, characteristic, complemented
6C67×#2224C211yesmaximal, characteristic, complemented
7C67×#2210C211yesmaximal, characteristic, complemented
8C67×#2210C211yesmaximal, characteristic, complemented
9C67×#2187C211yesmaximal, characteristic, complemented
10C67×#2184C211yesmaximal, characteristic
11C67×#2177C211yesmaximal, characteristic, complemented
12C67×#2175C211yesmaximal, characteristic, complemented
13C4×SD32×C67C211yesmaximal, characteristic, complemented
14C4×D16×C67C211yesmaximal, characteristic, complemented
15C67×#2164C211yesmaximal, characteristic, complemented
16C22×D8×C67C2211yescharacteristic
17#542×C67—21no—
18#518×C67—21no—
19C2×SD32×C67C2211yescharacteristic
20C2×SD32×C67C2211yescharacteristic
21C2×D16×C67C2211yescharacteristic
22C2×D16×C67C2211yescharacteristic
23#500×C67C2211yescharacteristic, complemented
24#495×C67C2211yescharacteristic, complemented
25#489×C67—21no—
26#468×C67—21no—
27#467×C67C2211yescharacteristic
28#466×C67C2211yescharacteristic
29#442×C67—21no—
30#442×C67—21no—
31#440×C67—21no—
32C4×Q16×C67C2211yescharacteristic, complemented
33C4×D8×C67C2211yescharacteristic
34C4×D8×C67C2211yescharacteristic
35C4×D8×C67C2211yescharacteristic
36C4×D8×C67C2211yescharacteristic
37C4×D8×C67C2211yescharacteristic, complemented
38#427×C67C2211yescharacteristic
39#418×C67—21no—
40#405×C67C2211yescharacteristic
41#367×C67C2211yescharacteristic, complemented
42#367×C67C2211yescharacteristic, complemented
43#366×C67C2211yescharacteristic, complemented
44#366×C67C2211yescharacteristic, complemented
45C4⋊C16×C67C2211yescharacteristic, complemented
46C4⋊C16×C67C2211yescharacteristic, complemented
47#358×C67C2211yescharacteristic, complemented
48#358×C67C2211yescharacteristic, complemented
49#357×C67C2211yescharacteristic
50#357×C67C2211yescharacteristic
51#357×C67C2211yescharacteristic
52#357×C67C2211yescharacteristic
53#357×C67C2211yescharacteristic, complemented
54#357×C67C2211yescharacteristic, complemented
55C22⋊C16×C67C2211yescharacteristic
56C22⋊C16×C67C2211yescharacteristic
57#346×C67C2211yescharacteristic, complemented
58C4×C16×C67C2211yescharacteristic, complemented
59C22×D4×C67—21no—
60C2×Q16×C67C2311yescharacteristic
61C2×SD16×C67—21no—
62C2×D8×C67—22no—
63C2×D8×C67—22no—
64C2×D8×C67—21no—
65C2×D8×C67C2311yescharacteristic
66C2×D8×C67C2311yescharacteristic
67C2×D8×C67C2311yescharacteristic
68C2×M16×C67D411yescharacteristic
69C22×C8×C67C2311yescharacteristic
70C4⋊Q8×C67—21no—
71#126×C67—41no—
72#124×C67—21no—
73#124×C67—41no—
74#123×C67—41no—
75C22⋊Q8×C67—21no—
76C22⋊Q8×C67—41no—
77#121×C67—21no—
78C4×Q8×C67—21no—
79C4×D4×C67—21no—
80C4×D4×C67—21no—
81C4×D4×C67—21no—
82C4×D4×C67—21no—
83C4×D4×C67—21no—
84C4×D4×C67—41no—
85C4×D4×C67—41no—
86#117×C67—21no—
87#117×C67D411yescharacteristic
88C2×C22⋊C4×C67—41no—
89C2×C22⋊C4×C67—41no—
90SD32×C67—22no—
91SD32×C67—22no—
92D16×C67—22no—
93D16×C67—22no—
94C2×C16×C67C2311yescharacteristic
95C2×C16×C67C2311yescharacteristic
96C2×C16×C67C2311yescharacteristic
97C2×C16×C67C2311yescharacteristic
98#107×C67—21no—
99#107×C67C2311yescharacteristic
100#107×C67C2311yescharacteristic
101#107×C67C2311yescharacteristic
102#107×C67C2311yescharacteristic
103#106×C67—21no—
104#103×C67—21no—
105#103×C67—21no—
106#102×C67—21no—
107#102×C67—21no—
108#102×C67—21no—
109#102×C67—21no—
110#102×C67—21no—
111#102×C67—21no—
112#97×C67D411yescharacteristic
113#97×C67D411yescharacteristic
114C4×C8×C67C2311yescharacteristic
115C4×C8×C67C2311yescharacteristic
116C24×C67—41no—
117C2×Q8×C67—21no—
118C2×Q8×C67—41no—
119C2×D4×C67—24no—
120C2×D4×C67—21no—
121C2×D4×C67—21no—
122C2×D4×C67—21no—
123C2×D4×C67—42no—
124C2×D4×C67—41no—
125C2×D4×C67—41no—
126C22×C4×C67—41no—
127C22×C4×C67—41no—
128C22×C4×C67—41no—
129C22×C4×C67—41no—
130C22×C4×C67—41no—
131C22×C4×C67C2×D411yescharacteristic
132Q16×C67—21no—
133Q16×C67—21no—
134SD16×C67—42no—
135D8×C67—24no—
136D8×C67—21no—
137D8×C67—21no—
138D8×C67—21no—
139D8×C67—21no—
140D8×C67#4112yes—
141D8×C67#4112yes—
142M16×C67—41no—
143C2×C8×C67—21no—
144C2×C8×C67—21no—
145C2×C8×C67C2×D411yescharacteristic
146C2×C8×C67C2×D411yescharacteristic
147C2×C8×C67C2×D411yescharacteristic
148C2×C8×C67C2411yescharacteristic
149C4⋊C4×C67—21no—
150C4⋊C4×C67—21no—
151C4⋊C4×C67—21no—
152C4⋊C4×C67—21no—
153C4⋊C4×C67—41no—
154C4⋊C4×C67—41no—
155C4⋊C4×C67—41no—
156C4⋊C4×C67D811yescharacteristic
157C4⋊C4×C67D811yescharacteristic
158C22⋊C4×C67—41no—
159C22⋊C4×C67—41no—
160C22⋊C4×C67—41no—
161C22⋊C4×C67—41no—
162C22⋊C4×C67—41no—
163C22⋊C4×C67—41no—
164C22⋊C4×C67—41no—
165C22⋊C4×C67—41no—
166C22⋊C4×C67—41no—
167C22⋊C4×C67—41no—
168C22⋊C4×C67D811yescharacteristic
169C22⋊C4×C67D811yescharacteristic
170C42×C67—41no—
171C42×C67—41no—
172C42×C67C2×D411yescharacteristic
173C42×C67C2×D411yescharacteristic
174C16×C67—21no—
175C16×C67—21no—
176C16×C67—21no—
177C16×C67—21no—
178C23×C67—44no—
179C23×C67—41no—
180C23×C67—41no—
181C23×C67—41no—
182C23×C67—161no—
183C23×C67—161no—
184C23×C67C2×D811yescharacteristic
185Q8×C67—41no—
186Q8×C67—41no—
187Q8×C67—161no—
188D4×C67—22no—
189D4×C67—22no—
190D4×C67—44no—
191D4×C67—41no—
192D4×C67—41no—
193D4×C67—41no—
194D4×C67—41no—
195D4×C67—161no—
196C2×C4×C67—21no—
197C2×C4×C67—21no—
198C2×C4×C67—41no—
199C2×C4×C67—41no—
200C2×C4×C67—41no—
201C2×C4×C67—41no—
202C2×C4×C67—41no—
203C2×C4×C67—161no—
204C2×C4×C67—161no—
205C2×C4×C67—161no—
206C2×C4×C67—161no—
207C2×C4×C67—161no—
208C2×C4×C67C2×D811yescharacteristic
209C2×C4×C67C2×D811yescharacteristic
210C2×C4×C67C2×D811yescharacteristic
211C2×C4×C67C2×D811yescharacteristic
212C2×C4×C67C2×D811yescharacteristic
213C2×C4×C67C22×D411yescharacteristic
214C8×C67—21no—
215C8×C67—21no—
216C8×C67—21no—
217C8×C67C2×#4111yescharacteristic
218C8×C67C2×#4111yescharacteristic
219C22×C67—21no—
220C22×C67—21no—
221C22×C67—21no—
222C22×C67—42no—
223C22×C67—42no—
224C22×C67—81no—
225C22×C67—81no—
226C22×C67—81no—
227C22×C67—81no—
228C22×C67—81no—
229C22×C67—81no—
230C22×C67—81no—
231C22×C67—164no—
232C22×C67C22×D811yescharacteristic
233C4×C67—21no—
234C4×C67—21no—
235C4×C67—21no—
236C4×C67—21no—
237C4×C67—21no—
238C4×C67—81no—
239C4×C67—81no—
240C4×C67—81no—
241C4×C67—81no—
242C4×C67—81no—
243C4×C67#52511yescharacteristic
244C4×C67#57511yescharacteristic
245#33678C6711yesmaximal, characteristic, direct product factor, Sylow
246C2×C67—41no—
247C2×C67—82no—
248C2×C67—82no—
249C2×C67—161no—
250C2×C67#310011yescharacteristic
251C2×C67#341511yescharacteristic
252C2×C67#341611yescharacteristic
253#3175C2×C6711yescharacteristic, complemented
254#2948C2×C6711yescharacteristic, complemented
255#2237C2×C6711yescharacteristic, complemented
256#2234C2×C6711yescharacteristic, complemented
257#2226C2×C6711yescharacteristic, complemented
258#2224C2×C6711yescharacteristic, complemented
259#2210C2×C6711yescharacteristic, complemented
260#2210C2×C6711yescharacteristic, complemented
261#2187C2×C6711yescharacteristic, complemented
262#2184C2×C6711yescharacteristic
263#2177C2×C6711yescharacteristic, complemented
264#2175C2×C6711yescharacteristic, complemented
265C4×SD32C2×C6711yescharacteristic, complemented
266C4×D16C2×C6711yescharacteristic, complemented
267#2164C2×C6711yescharacteristic, complemented
268C67#3367811yescharacteristic, direct product factor, Sylow
269C22×D8C22×C6711yescharacteristic
270#542—21no—
271#518—21no—
272C2×SD32C22×C6711yescharacteristic
273C2×SD32C22×C6711yescharacteristic
274C2×D16C22×C6711yescharacteristic
275C2×D16C22×C6711yescharacteristic
276#500C22×C6711yescharacteristic, complemented
277#495C22×C6711yescharacteristic, complemented
278#489—21no—
279#468—21no—
280#467C22×C6711yescharacteristic
281#466C22×C6711yescharacteristic
282#442—21no—
283#442—21no—
284#440—21no—
285C4×Q16C22×C6711yescharacteristic, complemented
286C4×D8C22×C6711yescharacteristic
287C4×D8C22×C6711yescharacteristic
288C4×D8C22×C6711yescharacteristic
289C4×D8C22×C6711yescharacteristic
290C4×D8C22×C6711yescharacteristic, complemented
291#427C22×C6711yescharacteristic
292#418—21no—
293#405C22×C6711yescharacteristic
294#367C22×C6711yescharacteristic, complemented
295#367C22×C6711yescharacteristic, complemented
296#366C22×C6711yescharacteristic, complemented
297#366C22×C6711yescharacteristic, complemented
298C4⋊C16C22×C6711yescharacteristic, complemented
299C4⋊C16C22×C6711yescharacteristic, complemented
300#358C22×C6711yescharacteristic, complemented
301#358C22×C6711yescharacteristic, complemented
302#357C22×C6711yescharacteristic
303#357C22×C6711yescharacteristic
304#357C22×C6711yescharacteristic
305#357C22×C6711yescharacteristic
306#357C22×C6711yescharacteristic, complemented
307#357C22×C6711yescharacteristic, complemented
308C22⋊C16C22×C6711yescharacteristic
309C22⋊C16C22×C6711yescharacteristic
310#346C22×C6711yescharacteristic, complemented
311C4×C16C22×C6711yescharacteristic, complemented
312C22×D4—21no—
313C2×Q16C23×C6711yescharacteristic
314C2×SD16—21no—
315C2×D8—22no—
316C2×D8—22no—
317C2×D8—21no—
318C2×D8C23×C6711yescharacteristic
319C2×D8C23×C6711yescharacteristic
320C2×D8C23×C6711yescharacteristic
321C2×M16D4×C6711yescharacteristic
322C22×C8C23×C6711yescharacteristic
323C4⋊Q8—21no—
324#126—41no—
325#124—21no—
326#124—41no—
327#123—41no—
328C22⋊Q8—21no—
329C22⋊Q8—41no—
330#121—21no—
331C4×Q8—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—41no—
338C4×D4—41no—
339#117—21no—
340#117D4×C6711yescharacteristic
341C2×C22⋊C4—41no—
342C2×C22⋊C4—41no—
343SD32—22no—
344SD32—22no—
345D16—22no—
346D16—22no—
347C2×C16C23×C6711yescharacteristic
348C2×C16C23×C6711yescharacteristic
349C2×C16C23×C6711yescharacteristic
350C2×C16C23×C6711yescharacteristic
351#107—21no—
352#107C23×C6711yescharacteristic
353#107C23×C6711yescharacteristic
354#107C23×C6711yescharacteristic
355#107C23×C6711yescharacteristic
356#106—21no—
357#103—21no—
358#103—21no—
359#102—21no—
360#102—21no—
361#102—21no—
362#102—21no—
363#102—21no—
364#102—21no—
365#97D4×C6711yescharacteristic
366#97D4×C6711yescharacteristic
367C4×C8C23×C6711yescharacteristic
368C4×C8C23×C6711yescharacteristic
369C24—41no—
370C2×Q8—21no—
371C2×Q8—41no—
372C2×D4—24no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—42no—
377C2×D4—41no—
378C2×D4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4C2×D4×C6711yescharacteristic
385Q16—21no—
386Q16—21no—
387SD16—42no—
388D8—24no—
389D8—21no—
390D8—21no—
391D8—21no—
392D8—21no—
393D8#41×C6712yes—
394D8#41×C6712yes—
395M16—41no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8C2×D4×C6711yescharacteristic
399C2×C8C2×D4×C6711yescharacteristic
400C2×C8C2×D4×C6711yescharacteristic
401C2×C8C24×C6711yescharacteristic
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4D8×C6711yescharacteristic
410C4⋊C4D8×C6711yescharacteristic
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4D8×C6711yescharacteristic
422C22⋊C4D8×C6711yescharacteristic
423C42—41no—
424C42—41no—
425C42C2×D4×C6711yescharacteristic
426C42C2×D4×C6711yescharacteristic
427C16—21no—
428C16—21no—
429C16—21no—
430C16—21no—
431C23—44no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—161no—
436C23—161no—
437C23C2×D8×C6711yescharacteristic
438Q8—41no—
439Q8—41no—
440Q8—161no—
441D4—22no—
442D4—22no—
443D4—44no—
444D4—41no—
445D4—41no—
446D4—41no—
447D4—41no—
448D4—161no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—161no—
457C2×C4—161no—
458C2×C4—161no—
459C2×C4—161no—
460C2×C4—161no—
461C2×C4C2×D8×C6711yescharacteristic
462C2×C4C2×D8×C6711yescharacteristic
463C2×C4C2×D8×C6711yescharacteristic
464C2×C4C2×D8×C6711yescharacteristic
465C2×C4C2×D8×C6711yescharacteristic
466C2×C4C22×D4×C6711yescharacteristic
467C8—21no—
468C8—21no—
469C8—21no—
470C8C2×#41×C6711yescharacteristic
471C8C2×#41×C6711yescharacteristic
472C22—21no—
473C22—21no—
474C22—21no—
475C22—42no—
476C22—42no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—164no—
485C22C22×D8×C6711yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4#525×C6711yescharacteristic
497C4#575×C6711yescharacteristic
498C2—41no—
499C2—82no—
500C2—82no—
501C2—161no—
502C2C67×#310011yescharacteristic
503C2C67×#341511yescharacteristic
504C2C67×#341611yescharacteristic
subgroupG/H
center, socleC22×C67C22×D8
commutator, Frattini, cosocleC2×C8C24×C67
prime radicalC67×#2184C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ #117×C67 ◁ #405×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3175C2semidirect
2. C67×#2948C2semidirect
3. C67×#2237C2semidirect
4. C67×#2234C2semidirect
5. C67×#2226C2semidirect
6. C67×#2224C2semidirect
7. C67×#2210C2semidirect
8. C67×#2210C2semidirect
9. C67×#2187C2semidirect
11. C67×#2177C2semidirect
12. C67×#2175C2semidirect
13. C4×SD32×C67C2semidirect
14. C4×D16×C67C2semidirect
15. C67×#2164C2semidirect
23. #500×C67C22semidirect
24. #495×C67C22semidirect
32. C4×Q16×C67C22semidirect
37. C4×D8×C67C22semidirect
41. #367×C67C22semidirect
42. #367×C67C22semidirect
43. #366×C67C22semidirect
44. #366×C67C22semidirect
45. C4⋊C16×C67C22semidirect
46. C4⋊C16×C67C22semidirect
47. #358×C67C22semidirect
48. #358×C67C22semidirect
53. #357×C67C22semidirect
54. #357×C67C22semidirect
57. #346×C67C22semidirect
58. C4×C16×C67C22semidirect
245. #33678C67direct (N × H)
253. #3175C2×C67semidirect
254. #2948C2×C67semidirect
255. #2237C2×C67semidirect
256. #2234C2×C67semidirect
257. #2226C2×C67semidirect
258. #2224C2×C67semidirect
259. #2210C2×C67semidirect
260. #2210C2×C67semidirect
261. #2187C2×C67semidirect
263. #2177C2×C67semidirect
264. #2175C2×C67semidirect
265. C4×SD32C2×C67semidirect
266. C4×D16C2×C67semidirect
267. #2164C2×C67semidirect
268. C67#33678direct (N × H)
276. #500C22×C67semidirect
277. #495C22×C67semidirect
285. C4×Q16C22×C67semidirect
290. C4×D8C22×C67semidirect
294. #367C22×C67semidirect
295. #367C22×C67semidirect
296. #366C22×C67semidirect
297. #366C22×C67semidirect
298. C4⋊C16C22×C67semidirect
299. C4⋊C16C22×C67semidirect
300. #358C22×C67semidirect
301. #358C22×C67semidirect
306. #357C22×C67semidirect
307. #357C22×C67semidirect
310. #346C22×C67semidirect
311. C4×C16C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#31752
C67×#29482
C67×#22372
C67×#22342
C67×#22262
C67×#22242
C67×#22102
C67×#21872
C67×#21842
C67×#21772
C67×#21752
C4×SD32×C672
C4×D16×C672
C67×#21642
#3367867