Finite Group Databasebeta

← C67×#19364 #30698615

Subgroups — C67×#19364 #30698615

Order 17152.

660 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3218C211yesmaximal, characteristic, complemented
2C67×#3217C211yesmaximal, characteristic
3C2×#486×C67C211yesmaximal, characteristic, complemented
4C67×#2966C211yesmaximal, characteristic, complemented
5C67×#1702C212yesmaximal, complemented
6C67×#1701C212yesmaximal, complemented
7C67×#1682C212yesmaximal, complemented
8C67×#1681C212yesmaximal, complemented
9C67×#1568C211yesmaximal, characteristic, complemented
10C67×#1556C211yesmaximal, characteristic, complemented
11C67×#1555C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8×C67C2211yescharacteristic
13C2×#127×C67C2211yescharacteristic
14#529×C67C2211yescharacteristic
15#497×C67—21no—
16#496×C67—21no—
17#491×C67—21no—
18#490×C67—21no—
19#486×C67—21no—
20#486×C67—21no—
21#486×C67C2212yes—
22#486×C67C2212yes—
23#462×C67C2212yescomplemented
24#461×C67C2212yescomplemented
25#460×C67C2212yes—
26#459×C67C2212yes—
27#451×C67—21no—
28#450×C67—21no—
29#448×C67—21no—
30#447×C67—21no—
31#446×C67C2211yescharacteristic, complemented
32C8×Q8×C67C2211yescharacteristic, complemented
33#436×C67C2211yescharacteristic, complemented
34#435×C67—21no—
35C8×D4×C67—21no—
36C8×D4×C67C2211yescharacteristic, complemented
37#433×C67—21no—
38#432×C67C2211yescharacteristic
39#432×C67C2211yescharacteristic
40#432×C67C2211yescharacteristic
41C2×#102×C67—21no—
42C2×#102×C67—21no—
43#405×C67—21no—
44C2×C4×C8×C67C2211yescharacteristic
45#335×C67C2212yescomplemented
46#334×C67C2212yescomplemented
47#332×C67C2212yescomplemented
48#332×C67C2212yescomplemented
49#331×C67C2212yes—
50#331×C67C2212yes—
51C22×Q8×C67—21no—
52C22×D4×C67—21no—
53C22×D4×C67—21no—
54C22×D4×C67—21no—
55C2×Q16×C67—22no—
56C2×SD16×C67—22no—
57C2×SD16×C67—22no—
58C2×D8×C67—22no—
59C2×M16×C67—21no—
60C22×C8×C67—21no—
61C22×C8×C67D411yescharacteristic
62C22×C8×C67D411yescharacteristic
63C4⋊Q8×C67—21no—
64C4⋊Q8×C67—21no—
65C4⋊Q8×C67—21no—
66C4⋊Q8×C67C2312yes—
67#127×C67—21no—
68#127×C67—21no—
69#127×C67—21no—
70#127×C67C2312yes—
71#126×C67—21no—
72#125×C67D411yescharacteristic
73#124×C67D411yescharacteristic
74#123×C67—21no—
75C22⋊Q8×C67D411yescharacteristic
76#121×C67D411yescharacteristic
77C4×Q8×C67C2311yescharacteristic
78C4×D4×C67—21no—
79C4×D4×C67C2311yescharacteristic
80#117×C67—21no—
81C2×C4⋊C4×C67—21no—
82C2×C4⋊C4×C67—21no—
83C2×C42×C67C2311yescharacteristic
84C4⋊C8×C67—21no—
85C4⋊C8×C67—21no—
86C4⋊C8×C67C2312yes—
87C4⋊C8×C67C2312yes—
88C4⋊C8×C67C2311yescharacteristic
89C4⋊C8×C67C2311yescharacteristic
90#103×C67—22no—
91#103×C67—22no—
92#102×C67—22no—
93#102×C67—22no—
94#102×C67—22no—
95#102×C67—22no—
96#102×C67—22no—
97#102×C67—22no—
98C22⋊C8×C67—21no—
99C22⋊C8×C67—21no—
100C22⋊C8×C67—21no—
101C22⋊C8×C67—21no—
102C22⋊C8×C67—21no—
103#97×C67—21no—
104#97×C67—21no—
105#97×C67—21no—
106#97×C67—21no—
107C4×C8×C67—21no—
108C4×C8×C67D411yescharacteristic
109C4×C8×C67D411yescharacteristic
110C4×C8×C67C2311yescharacteristic
111C4×C8×C67C2311yescharacteristic
112C24×C67—41no—
113C2×Q8×C67—22no—
114C2×Q8×C67—41no—
115C2×Q8×C67—82no—
116C2×Q8×C67C2×D411yescharacteristic
117C2×D4×C67—21no—
118C2×D4×C67—22no—
119C2×D4×C67—22no—
120C2×D4×C67—22no—
121C2×D4×C67—42no—
122C2×D4×C67—42no—
123C2×D4×C67—41no—
124C2×D4×C67—41no—
125C2×D4×C67—41no—
126C2×D4×C67—82no—
127C2×D4×C67—82no—
128C2×D4×C67C2×D411yescharacteristic
129C22×C4×C67—21no—
130C22×C4×C67—41no—
131C22×C4×C67C2×D411yescharacteristic
132C22×C4×C67C2×D411yescharacteristic
133C22×C4×C67#4111yescharacteristic
134Q16×C67—82no—
135SD16×C67—82no—
136SD16×C67—82no—
137D8×C67—82no—
138M16×C67—42no—
139C2×C8×C67—22no—
140C2×C8×C67—22no—
141C2×C8×C67—21no—
142C2×C8×C67—21no—
143C2×C8×C67—41no—
144C2×C8×C67—41no—
145C2×C8×C67—42no—
146C2×C8×C67C2×D411yescharacteristic
147C2×C8×C67C2×D411yescharacteristic
148C2×C8×C67C2×D411yescharacteristic
149C2×C8×C67C2×D411yescharacteristic
150C4⋊C4×C67—21no—
151C4⋊C4×C67—21no—
152C4⋊C4×C67—21no—
153C4⋊C4×C67—21no—
154C4⋊C4×C67—22no—
155C4⋊C4×C67—22no—
156C4⋊C4×C67—41no—
157C4⋊C4×C67—41no—
158C4⋊C4×C67C2×D411yescharacteristic
159C4⋊C4×C67C2×D411yescharacteristic
160C22⋊C4×C67—21no—
161C22⋊C4×C67—21no—
162C22⋊C4×C67—21no—
163C22⋊C4×C67—21no—
164C22⋊C4×C67—21no—
165C42×C67—21no—
166C42×C67C2×D411yescharacteristic
167C42×C67C2×D411yescharacteristic
168C42×C67#4111yescharacteristic
169C42×C67C2411yescharacteristic
170C23×C67—21no—
171C23×C67—42no—
172C23×C67—82no—
173C23×C67—82no—
174C23×C67—82no—
175C23×C67#12111yescharacteristic
176Q8×C67—22no—
177Q8×C67—42no—
178Q8×C67—42no—
179Q8×C67—82no—
180D4×C67—22no—
181D4×C67—42no—
182D4×C67—42no—
183D4×C67—42no—
184D4×C67—42no—
185D4×C67—42no—
186D4×C67—82no—
187D4×C67—82no—
188D4×C67—82no—
189D4×C67—82no—
190C2×C4×C67—22no—
191C2×C4×C67—21no—
192C2×C4×C67—21no—
193C2×C4×C67—22no—
194C2×C4×C67—21no—
195C2×C4×C67—21no—
196C2×C4×C67—21no—
197C2×C4×C67—41no—
198C2×C4×C67—41no—
199C2×C4×C67—42no—
200C2×C4×C67—82no—
201C2×C4×C67#12111yescharacteristic
202C2×C4×C67#12111yescharacteristic
203C2×C4×C67#12111yescharacteristic
204C2×C4×C67C22×D411yescharacteristic
205C2×C4×C67C22×D411yescharacteristic
206C2×C4×C67C2×#4111yescharacteristic
207C8×C67—21no—
208C8×C67—21no—
209C8×C67—21no—
210C8×C67—21no—
211C8×C67—41no—
212C8×C67—41no—
213C8×C67—42no—
214C8×C67—42no—
215C22×C67—21no—
216C22×C67—21no—
217C22×C67—41no—
218C22×C67—41no—
219C22×C67—41no—
220C22×C67—82no—
221C22×C67—82no—
222C22×C67—82no—
223C22×C67—164no—
224C22×C67#47112yes—
225C22×C67C2×#12111yescharacteristic
226C4×C67—21no—
227C4×C67—21no—
228C4×C67—21no—
229C4×C67—21no—
230C4×C67—41no—
231C4×C67—41no—
232C4×C67—41no—
233C4×C67—82no—
234C4×C67#57511yescharacteristic
235C4×C67#57511yescharacteristic
236C4×C67#57612yes—
237#19364C6711yesmaximal, characteristic, direct product factor, Sylow
238C2×C67—22no—
239C2×C67—81no—
240C2×C67—162no—
241C2×C67#304011yescharacteristic
242C2×C67#305811yescharacteristic
243C2×C67C2×#47111yescharacteristic
244#3218C2×C6711yescharacteristic, complemented
245#3217C2×C6711yescharacteristic
246C2×#486C2×C6711yescharacteristic, complemented
247#2966C2×C6711yescharacteristic, complemented
248#1702C2×C6712yescomplemented
249#1701C2×C6712yescomplemented
250#1682C2×C6712yescomplemented
251#1681C2×C6712yescomplemented
252#1568C2×C6711yescharacteristic, complemented
253#1556C2×C6711yescharacteristic, complemented
254#1555C2×C6711yescharacteristic, complemented
255C67#1936411yescharacteristic, direct product factor, Sylow
256C2×C4⋊Q8C22×C6711yescharacteristic
257C2×#127C22×C6711yescharacteristic
258#529C22×C6711yescharacteristic
259#497—21no—
260#496—21no—
261#491—21no—
262#490—21no—
263#486—21no—
264#486—21no—
265#486C22×C6712yes—
266#486C22×C6712yes—
267#462C22×C6712yescomplemented
268#461C22×C6712yescomplemented
269#460C22×C6712yes—
270#459C22×C6712yes—
271#451—21no—
272#450—21no—
273#448—21no—
274#447—21no—
275#446C22×C6711yescharacteristic, complemented
276C8×Q8C22×C6711yescharacteristic, complemented
277#436C22×C6711yescharacteristic, complemented
278#435—21no—
279C8×D4—21no—
280C8×D4C22×C6711yescharacteristic, complemented
281#433—21no—
282#432C22×C6711yescharacteristic
283#432C22×C6711yescharacteristic
284#432C22×C6711yescharacteristic
285C2×#102—21no—
286C2×#102—21no—
287#405—21no—
288C2×C4×C8C22×C6711yescharacteristic
289#335C22×C6712yescomplemented
290#334C22×C6712yescomplemented
291#332C22×C6712yescomplemented
292#332C22×C6712yescomplemented
293#331C22×C6712yes—
294#331C22×C6712yes—
295C22×Q8—21no—
296C22×D4—21no—
297C22×D4—21no—
298C22×D4—21no—
299C2×Q16—22no—
300C2×SD16—22no—
301C2×SD16—22no—
302C2×D8—22no—
303C2×M16—21no—
304C22×C8—21no—
305C22×C8D4×C6711yescharacteristic
306C22×C8D4×C6711yescharacteristic
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8C23×C6712yes—
311#127—21no—
312#127—21no—
313#127—21no—
314#127C23×C6712yes—
315#126—21no—
316#125D4×C6711yescharacteristic
317#124D4×C6711yescharacteristic
318#123—21no—
319C22⋊Q8D4×C6711yescharacteristic
320#121D4×C6711yescharacteristic
321C4×Q8C23×C6711yescharacteristic
322C4×D4—21no—
323C4×D4C23×C6711yescharacteristic
324#117—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C42C23×C6711yescharacteristic
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8C23×C6711yescharacteristic
331C4⋊C8C23×C6711yescharacteristic
332C4⋊C8C23×C6712yes—
333C4⋊C8C23×C6712yes—
334#103—22no—
335#103—22no—
336#102—22no—
337#102—22no—
338#102—22no—
339#102—22no—
340#102—22no—
341#102—22no—
342C22⋊C8—21no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C22⋊C8—21no—
346C22⋊C8—21no—
347#97—21no—
348#97—21no—
349#97—21no—
350#97—21no—
351C4×C8—21no—
352C4×C8D4×C6711yescharacteristic
353C4×C8D4×C6711yescharacteristic
354C4×C8C23×C6711yescharacteristic
355C4×C8C23×C6711yescharacteristic
356C24—41no—
357C2×Q8—22no—
358C2×Q8—41no—
359C2×Q8—82no—
360C2×Q8C2×D4×C6711yescharacteristic
361C2×D4—21no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4C2×D4×C6711yescharacteristic
373C22×C4—21no—
374C22×C4—41no—
375C22×C4C2×D4×C6711yescharacteristic
376C22×C4C2×D4×C6711yescharacteristic
377C22×C4#41×C6711yescharacteristic
378Q16—82no—
379SD16—82no—
380SD16—82no—
381D8—82no—
382M16—42no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8C2×D4×C6711yescharacteristic
391C2×C8C2×D4×C6711yescharacteristic
392C2×C8C2×D4×C6711yescharacteristic
393C2×C8C2×D4×C6711yescharacteristic
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4C2×D4×C6711yescharacteristic
403C4⋊C4C2×D4×C6711yescharacteristic
404C22⋊C4—21no—
405C22⋊C4—21no—
406C22⋊C4—21no—
407C22⋊C4—21no—
408C22⋊C4—21no—
409C42—21no—
410C42C2×D4×C6711yescharacteristic
411C42C2×D4×C6711yescharacteristic
412C42#41×C6711yescharacteristic
413C42C24×C6711yescharacteristic
414C23—21no—
415C23—42no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23#121×C6711yescharacteristic
420Q8—22no—
421Q8—42no—
422Q8—42no—
423Q8—82no—
424D4—22no—
425D4—42no—
426D4—42no—
427D4—42no—
428D4—42no—
429D4—42no—
430D4—82no—
431D4—82no—
432D4—82no—
433D4—82no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—82no—
445C2×C4#121×C6711yescharacteristic
446C2×C4#121×C6711yescharacteristic
447C2×C4#121×C6711yescharacteristic
448C2×C4C22×D4×C6711yescharacteristic
449C2×C4C22×D4×C6711yescharacteristic
450C2×C4C2×#41×C6711yescharacteristic
451C8—21no—
452C8—21no—
453C8—21no—
454C8—21no—
455C8—41no—
456C8—41no—
457C8—42no—
458C8—42no—
459C22—21no—
460C22—21no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—164no—
468C22#471×C6712yes—
469C22C2×#121×C6711yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—82no—
478C4#575×C6711yescharacteristic
479C4#575×C6711yescharacteristic
480C4#576×C6712yes—
481C2—22no—
482C2—81no—
483C2—162no—
484C2C67×#304011yescharacteristic
485C2C67×#305811yescharacteristic
486C2C2×#471×C6711yescharacteristic
subgroupG/H
center, socleC22×C67C2×#121
commutator, Frattini, cosocleC42C24×C67
prime radicalC67×#3217C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3218C2semidirect
3. C2×#486×C67C2semidirect
4. C67×#2966C2semidirect
5. C67×#1702C2semidirect
6. C67×#1701C2semidirect
7. C67×#1682C2semidirect
8. C67×#1681C2semidirect
9. C67×#1568C2semidirect
10. C67×#1556C2semidirect
11. C67×#1555C2semidirect
23. #462×C67C22semidirect
24. #461×C67C22semidirect
31. #446×C67C22semidirect
32. C8×Q8×C67C22semidirect
33. #436×C67C22semidirect
36. C8×D4×C67C22semidirect
45. #335×C67C22semidirect
46. #334×C67C22semidirect
47. #332×C67C22semidirect
48. #332×C67C22semidirect
237. #19364C67direct (N × H)
244. #3218C2×C67semidirect
246. C2×#486C2×C67semidirect
247. #2966C2×C67semidirect
248. #1702C2×C67semidirect
249. #1701C2×C67semidirect
250. #1682C2×C67semidirect
251. #1681C2×C67semidirect
252. #1568C2×C67semidirect
253. #1556C2×C67semidirect
254. #1555C2×C67semidirect
255. C67#19364direct (N × H)
267. #462C22×C67semidirect
268. #461C22×C67semidirect
275. #446C22×C67semidirect
276. C8×Q8C22×C67semidirect
277. #436C22×C67semidirect
280. C8×D4C22×C67semidirect
289. #335C22×C67semidirect
290. #334C22×C67semidirect
291. #332C22×C67semidirect
292. #332C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#32182
C67×#32172
C2×#486×C672
C67×#29662
C67×#17022
C67×#17012
C67×#16822
C67×#16812
C67×#15682
C67×#15562
C67×#15552
#1936467