Finite Group Databasebeta

← C67×#33679 #30712930

Subgroups — C67×#33679 #30712930

Order 17152.

558 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3176C211yesmaximal, characteristic, complemented
2C67×#2951C211yesmaximal, characteristic, complemented
3C67×#2233C211yesmaximal, characteristic, complemented
4C67×#2232C211yesmaximal, characteristic, complemented
5C67×#2229C211yesmaximal, characteristic, complemented
6C67×#2228C211yesmaximal, characteristic, complemented
7C67×#2207C211yesmaximal, characteristic, complemented
8C67×#2206C211yesmaximal, characteristic, complemented
9C67×#2186C211yesmaximal, characteristic, complemented
10C67×#2186C211yesmaximal, characteristic, complemented
11C67×#2177C211yesmaximal, characteristic, complemented
12C67×#2175C211yesmaximal, characteristic, complemented
13C4×SD32×C67C211yesmaximal, characteristic, complemented
14C4×D16×C67C211yesmaximal, characteristic, complemented
15C67×#2164C211yesmaximal, characteristic, complemented
16C2×#135×C67C2211yescharacteristic
17#540×C67—21no—
18#520×C67—21no—
19C2×SD32×C67C2211yescharacteristic
20C2×SD32×C67C2211yescharacteristic
21C2×D16×C67C2211yescharacteristic
22C2×D16×C67C2211yescharacteristic
23#499×C67C2211yescharacteristic
24#493×C67C2211yescharacteristic
25#490×C67—21no—
26#469×C67—21no—
27#466×C67C2211yescharacteristic
28#466×C67C2211yescharacteristic
29#442×C67—21no—
30#440×C67—21no—
31#440×C67—21no—
32C4×Q16×C67C2211yescharacteristic
33C4×Q16×C67C2211yescharacteristic
34C4×D8×C67C2211yescharacteristic
35C4×D8×C67C2211yescharacteristic
36C4×D8×C67C2211yescharacteristic
37C4×D8×C67C2211yescharacteristic
38C2×#107×C67C2211yescharacteristic
39#417×C67—21no—
40#405×C67C2211yescharacteristic
41#367×C67C2211yescharacteristic
42#367×C67C2211yescharacteristic
43#366×C67C2211yescharacteristic
44#366×C67C2211yescharacteristic
45C4⋊C16×C67C2211yescharacteristic
46C4⋊C16×C67C2211yescharacteristic
47#358×C67C2211yescharacteristic
48#358×C67C2211yescharacteristic
49#357×C67C2211yescharacteristic
50#357×C67C2211yescharacteristic
51#357×C67C2211yescharacteristic
52#357×C67C2211yescharacteristic
53#357×C67C2211yescharacteristic
54#357×C67C2211yescharacteristic
55C22⋊C16×C67C2211yescharacteristic
56C22⋊C16×C67C2211yescharacteristic
57#346×C67C2211yescharacteristic
58C4×C16×C67C2211yescharacteristic
59C2×#41×C67—21no—
60#135×C67—22no—
61#135×C67—22no—
62C2×Q16×C67C2311yescharacteristic
63C2×SD16×C67—21no—
64C2×D8×C67—21no—
65C2×D8×C67C2311yescharacteristic
66C2×D8×C67C2311yescharacteristic
67C2×D8×C67C2311yescharacteristic
68C2×M16×C67D411yescharacteristic
69C22×C8×C67C2311yescharacteristic
70#127×C67—21no—
71#125×C67—21no—
72#123×C67—41no—
73#123×C67—41no—
74#121×C67—21no—
75#121×C67—21no—
76#121×C67—41no—
77#121×C67—41no—
78C4×Q8×C67—21no—
79C4×Q8×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—41no—
87#117×C67D411yescharacteristic
88C2×C4⋊C4×C67—21no—
89C2×C4⋊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
116#41×C67—24no—
117#41×C67—42no—
118C2×Q8×C67—21no—
119C2×D4×C67—21no—
120C2×D4×C67—21no—
121C2×D4×C67—21no—
122C2×D4×C67—41no—
123C2×D4×C67—41no—
124C2×D4×C67—41no—
125C2×D4×C67—41no—
126C2×D4×C67—41no—
127C22×C4×C67—41no—
128C22×C4×C67—41no—
129C22×C4×C67—41no—
130C22×C4×C67—41no—
131C22×C4×C67—41no—
132C22×C4×C67—41no—
133C22×C4×C67C2×D411yescharacteristic
134Q16×C67—21no—
135Q16×C67#4112yes—
136SD16×C67—24no—
137D8×C67—21no—
138D8×C67—21no—
139D8×C67—21no—
140D8×C67—21no—
141D8×C67—21no—
142D8×C67—42no—
143D8×C67#4112yes—
144M16×C67—41no—
145C2×C8×C67—21no—
146C2×C8×C67—21no—
147C2×C8×C67C2×D411yescharacteristic
148C2×C8×C67C2×D411yescharacteristic
149C2×C8×C67C2×D411yescharacteristic
150C2×C8×C67C2411yescharacteristic
151C4⋊C4×C67—21no—
152C4⋊C4×C67—21no—
153C4⋊C4×C67—21no—
154C4⋊C4×C67—21no—
155C4⋊C4×C67—41no—
156C4⋊C4×C67—41no—
157C4⋊C4×C67—41no—
158C4⋊C4×C67—41no—
159C4⋊C4×C67—41no—
160C4⋊C4×C67D811yescharacteristic
161C4⋊C4×C67D811yescharacteristic
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—41no—
179C23×C67—41no—
180C23×C67—41no—
181C23×C67C2×D811yescharacteristic
182Q8×C67—22no—
183Q8×C67—41no—
184D4×C67—22no—
185D4×C67—44no—
186D4×C67—41no—
187D4×C67—41no—
188D4×C67—41no—
189D4×C67—41no—
190D4×C67—41no—
191D4×C67—161no—
192D4×C67—161no—
193D4×C67—161no—
194D4×C67—161no—
195C2×C4×C67—21no—
196C2×C4×C67—21no—
197C2×C4×C67—44no—
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—81no—
224C22×C67—81no—
225C22×C67—81no—
226C22×C67—81no—
227C22×C67—81no—
228C22×C67—81no—
229C22×C67—81no—
230C22×C67—81no—
231C22×C67C22×D811yescharacteristic
232C4×C67—21no—
233C4×C67—21no—
234C4×C67—21no—
235C4×C67—21no—
236C4×C67—21no—
237C4×C67—42no—
238C4×C67—81no—
239C4×C67—81no—
240C4×C67—81no—
241C4×C67—81no—
242C4×C67#52511yescharacteristic
243C4×C67#57511yescharacteristic
244#33679C6711yesmaximal, characteristic, direct product factor, Sylow
245C2×C67—41no—
246C2×C67—82no—
247C2×C67—161no—
248C2×C67—161no—
249C2×C67#310011yescharacteristic
250C2×C67#341511yescharacteristic
251C2×C67#341611yescharacteristic
252#3176C2×C6711yescharacteristic, complemented
253#2951C2×C6711yescharacteristic, complemented
254#2233C2×C6711yescharacteristic, complemented
255#2232C2×C6711yescharacteristic, complemented
256#2229C2×C6711yescharacteristic, complemented
257#2228C2×C6711yescharacteristic, complemented
258#2207C2×C6711yescharacteristic, complemented
259#2206C2×C6711yescharacteristic, complemented
260#2186C2×C6711yescharacteristic, complemented
261#2186C2×C6711yescharacteristic, complemented
262#2177C2×C6711yescharacteristic, complemented
263#2175C2×C6711yescharacteristic, complemented
264C4×SD32C2×C6711yescharacteristic, complemented
265C4×D16C2×C6711yescharacteristic, complemented
266#2164C2×C6711yescharacteristic, complemented
267C67#3367911yescharacteristic, direct product factor, Sylow
268C2×#135C22×C6711yescharacteristic
269#540—21no—
270#520—21no—
271C2×SD32C22×C6711yescharacteristic
272C2×SD32C22×C6711yescharacteristic
273C2×D16C22×C6711yescharacteristic
274C2×D16C22×C6711yescharacteristic
275#499C22×C6711yescharacteristic
276#493C22×C6711yescharacteristic
277#490—21no—
278#469—21no—
279#466C22×C6711yescharacteristic
280#466C22×C6711yescharacteristic
281#442—21no—
282#440—21no—
283#440—21no—
284C4×Q16C22×C6711yescharacteristic
285C4×Q16C22×C6711yescharacteristic
286C4×D8C22×C6711yescharacteristic
287C4×D8C22×C6711yescharacteristic
288C4×D8C22×C6711yescharacteristic
289C4×D8C22×C6711yescharacteristic
290C2×#107C22×C6711yescharacteristic
291#417—21no—
292#405C22×C6711yescharacteristic
293#367C22×C6711yescharacteristic
294#367C22×C6711yescharacteristic
295#366C22×C6711yescharacteristic
296#366C22×C6711yescharacteristic
297C4⋊C16C22×C6711yescharacteristic
298C4⋊C16C22×C6711yescharacteristic
299#358C22×C6711yescharacteristic
300#358C22×C6711yescharacteristic
301#357C22×C6711yescharacteristic
302#357C22×C6711yescharacteristic
303#357C22×C6711yescharacteristic
304#357C22×C6711yescharacteristic
305#357C22×C6711yescharacteristic
306#357C22×C6711yescharacteristic
307C22⋊C16C22×C6711yescharacteristic
308C22⋊C16C22×C6711yescharacteristic
309#346C22×C6711yescharacteristic
310C4×C16C22×C6711yescharacteristic
311C2×#41—21no—
312#135—22no—
313#135—22no—
314C2×Q16C23×C6711yescharacteristic
315C2×SD16—21no—
316C2×D8—21no—
317C2×D8C23×C6711yescharacteristic
318C2×D8C23×C6711yescharacteristic
319C2×D8C23×C6711yescharacteristic
320C2×M16D4×C6711yescharacteristic
321C22×C8C23×C6711yescharacteristic
322#127—21no—
323#125—21no—
324#123—41no—
325#123—41no—
326#121—21no—
327#121—21no—
328#121—41no—
329#121—41no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—41no—
337C4×D4—41no—
338#117—41no—
339#117D4×C6711yescharacteristic
340C2×C4⋊C4—21no—
341C2×C4⋊C4—41no—
342SD32—22no—
343SD32—22no—
344D16—22no—
345D16—22no—
346C2×C16C23×C6711yescharacteristic
347C2×C16C23×C6711yescharacteristic
348C2×C16C23×C6711yescharacteristic
349C2×C16C23×C6711yescharacteristic
350#107—21no—
351#107C23×C6711yescharacteristic
352#107C23×C6711yescharacteristic
353#107C23×C6711yescharacteristic
354#107C23×C6711yescharacteristic
355#106—21no—
356#103—21no—
357#103—21no—
358#102—21no—
359#102—21no—
360#102—21no—
361#102—21no—
362#102—21no—
363#102—21no—
364#97D4×C6711yescharacteristic
365#97D4×C6711yescharacteristic
366C4×C8C23×C6711yescharacteristic
367C4×C8C23×C6711yescharacteristic
368#41—24no—
369#41—42no—
370C2×Q8—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4C2×D4×C6711yescharacteristic
386Q16—21no—
387Q16#41×C6712yes—
388SD16—24no—
389D8—21no—
390D8—21no—
391D8—21no—
392D8—21no—
393D8—21no—
394D8—42no—
395D8#41×C6712yes—
396M16—41no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8C2×D4×C6711yescharacteristic
400C2×C8C2×D4×C6711yescharacteristic
401C2×C8C2×D4×C6711yescharacteristic
402C2×C8C24×C6711yescharacteristic
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4D8×C6711yescharacteristic
413C4⋊C4D8×C6711yescharacteristic
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4D8×C6711yescharacteristic
421C22⋊C4D8×C6711yescharacteristic
422C42—41no—
423C42—41no—
424C42C2×D4×C6711yescharacteristic
425C42C2×D4×C6711yescharacteristic
426C16—21no—
427C16—21no—
428C16—21no—
429C16—21no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23C2×D8×C6711yescharacteristic
434Q8—22no—
435Q8—41no—
436D4—22no—
437D4—44no—
438D4—41no—
439D4—41no—
440D4—41no—
441D4—41no—
442D4—41no—
443D4—161no—
444D4—161no—
445D4—161no—
446D4—161no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—44no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—161no—
456C2×C4—161no—
457C2×C4—161no—
458C2×C4—161no—
459C2×C4—161no—
460C2×C4C2×D8×C6711yescharacteristic
461C2×C4C2×D8×C6711yescharacteristic
462C2×C4C2×D8×C6711yescharacteristic
463C2×C4C2×D8×C6711yescharacteristic
464C2×C4C2×D8×C6711yescharacteristic
465C2×C4C22×D4×C6711yescharacteristic
466C8—21no—
467C8—21no—
468C8—21no—
469C8C2×#41×C6711yescharacteristic
470C8C2×#41×C6711yescharacteristic
471C22—21no—
472C22—21no—
473C22—21no—
474C22—42no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22C22×D8×C6711yescharacteristic
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—42no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4#525×C6711yescharacteristic
495C4#575×C6711yescharacteristic
496C2—41no—
497C2—82no—
498C2—161no—
499C2—161no—
500C2C67×#310011yescharacteristic
501C2C67×#341511yescharacteristic
502C2C67×#341611yescharacteristic
subgroupG/H
center, socleC22×C67C22×D8
commutator, Frattini, cosocleC2×C8C24×C67
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×#3176C2semidirect
2. C67×#2951C2semidirect
3. C67×#2233C2semidirect
4. C67×#2232C2semidirect
5. C67×#2229C2semidirect
6. C67×#2228C2semidirect
7. C67×#2207C2semidirect
8. C67×#2206C2semidirect
9. C67×#2186C2semidirect
10. C67×#2186C2semidirect
11. C67×#2177C2semidirect
12. C67×#2175C2semidirect
13. C4×SD32×C67C2semidirect
14. C4×D16×C67C2semidirect
15. C67×#2164C2semidirect
244. #33679C67direct (N × H)
252. #3176C2×C67semidirect
253. #2951C2×C67semidirect
254. #2233C2×C67semidirect
255. #2232C2×C67semidirect
256. #2229C2×C67semidirect
257. #2228C2×C67semidirect
258. #2207C2×C67semidirect
259. #2206C2×C67semidirect
260. #2186C2×C67semidirect
261. #2186C2×C67semidirect
262. #2177C2×C67semidirect
263. #2175C2×C67semidirect
264. C4×SD32C2×C67semidirect
265. C4×D16C2×C67semidirect
266. #2164C2×C67semidirect
267. C67#33679direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#31762
C67×#29512
C67×#22332
C67×#22322
C67×#22292
C67×#22282
C67×#22072
C67×#22062
C67×#21862
C67×#21772
C67×#21752
C4×SD32×C672
C4×D16×C672
C67×#21642
#3367967