Finite Group Databasebeta

← C67×#33000 #30712251

Subgroups — C67×#33000 #30712251

Order 17152.

1300 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#520×C67C211yesmaximal, characteristic
2C67×#3095C211yesmaximal, characteristic, complemented
3C67×#3007C211yesmaximal, characteristic
4C67×#2502C211yesmaximal, characteristic, complemented
5C67×#1882C214yesmaximal, complemented
6C67×#1859C214yesmaximal, complemented
7C67×#1826C211yesmaximal, characteristic, complemented
8C67×#1821C211yesmaximal, characteristic, complemented
9C67×#1789C211yesmaximal, characteristic
10C22×#41×C67C2211yescharacteristic
11C2×#137×C67C2212yes—
12C2×C22⋊Q8×C67C2211yescharacteristic
13#520×C67—22no—
14#520×C67—22no—
15#520×C67—22no—
16#520×C67C2214yes—
17C2×C4×Q8×C67—21no—
18C2×C4×D4×C67—21no—
19C2×C4×D4×C67—22no—
20C2×#117×C67—21no—
21C2×#117×C67C2211yescharacteristic
22C22×C4⋊C4×C67—21no—
23C22×C4⋊C4×C67C2211yescharacteristic
24#485×C67—22no—
25#484×C67—21no—
26#483×C67—21no—
27#452×C67—22no—
28#451×C67—21no—
29#450×C67—21no—
30#428×C67C2212yescomplemented
31C2×#104×C67C2214yes—
32#419×C67C2212yescomplemented
33#417×C67C2212yes—
34#407×C67C2211yescharacteristic
35#407×C67C2211yescharacteristic
36#398×C67—22no—
37#391×C67C2212yescomplemented
38#385×C67—22no—
39#384×C67C2212yescomplemented
40C4×C22⋊C4×C67C2211yescharacteristic
41#343×C67C2214yescomplemented
42#339×C67C2214yescomplemented
43C2×#41×C67—28no—
44C2×#41×C67—21no—
45C2×#41×C67—21no—
46C2×#41×C67D412yes—
47C2×#41×C67C2312yes—
48C22×Q8×C67D411yescharacteristic
49C22×Q8×C67C2311yescharacteristic
50C22×D4×C67—21no—
51C22×D4×C67D411yescharacteristic
52C23×C4×C67—21no—
53C23×C4×C67—22no—
54C23×C4×C67D412yes—
55C23×C4×C67C2311yescharacteristic
56#137×C67—28no—
57C2×Q16×C67—22no—
58C2×SD16×C67—22no—
59C2×M16×C67C2312yes—
60C2×M16×C67C2312yes—
61C22⋊Q8×C67—21no—
62C22⋊Q8×C67—21no—
63C22⋊Q8×C67—21no—
64C22⋊Q8×C67—21no—
65C4×Q8×C67—24no—
66C4×Q8×C67—24no—
67C4×D4×C67—28no—
68C4×D4×C67—24no—
69C4×D4×C67—24no—
70C4×D4×C67—28no—
71#117×C67—24no—
72#117×C67—21no—
73#117×C67—21no—
74#117×C67—24no—
75#117×C67D412yes—
76#117×C67C2312yes—
77C2×C4⋊C4×C67—22no—
78C2×C4⋊C4×C67—24no—
79C2×C4⋊C4×C67—21no—
80C2×C4⋊C4×C67—21no—
81C2×C4⋊C4×C67—21no—
82C2×C4⋊C4×C67—21no—
83C2×C4⋊C4×C67—24no—
84C2×C4⋊C4×C67—21no—
85C2×C4⋊C4×C67—24no—
86C2×C4⋊C4×C67D412yes—
87C2×C4⋊C4×C67D411yescharacteristic
88C2×C4⋊C4×C67D411yescharacteristic
89C2×C4⋊C4×C67C2311yescharacteristic
90C2×C4⋊C4×C67C2312yes—
91C2×C22⋊C4×C67—22no—
92C2×C22⋊C4×C67—21no—
93C2×C22⋊C4×C67—21no—
94C2×C22⋊C4×C67—21no—
95C2×C42×C67—22no—
96C2×C42×C67—21no—
97C2×C42×C67C2312yes—
98#107×C67—22no—
99#106×C67—22no—
100#104×C67—28no—
101#103×C67—22no—
102#103×C67—22no—
103#103×C67—22no—
104#102×C67—22no—
105C22⋊C8×C67—21no—
106C22⋊C8×C67—21no—
107C22⋊C8×C67—21no—
108C22⋊C8×C67—21no—
109#95×C67—21no—
110#95×C67—24no—
111C24×C67—21no—
112C24×C67C2×D411yescharacteristic
113#41×C67—28no—
114#41×C67—28no—
115#41×C67—24no—
116#41×C67—24no—
117#41×C67—24no—
118#41×C67#4118yes—
119C2×Q8×C67—24no—
120C2×Q8×C67—21no—
121C2×Q8×C67—21no—
122C2×Q8×C67—21no—
123C2×Q8×C67—24no—
124C2×Q8×C67C2×D412yes—
125C2×D4×C67—28no—
126C2×D4×C67—22no—
127C2×D4×C67—22no—
128C2×D4×C67—24no—
129C2×D4×C67—21no—
130C2×D4×C67C2×D412yes—
131C22×C4×C67—28no—
132C22×C4×C67—21no—
133C22×C4×C67—22no—
134C22×C4×C67—22no—
135C22×C4×C67—24no—
136C22×C4×C67—21no—
137C22×C4×C67—22no—
138C22×C4×C67—21no—
139C22×C4×C67—22no—
140C22×C4×C67—28no—
141C22×C4×C67—22no—
142C22×C4×C67—21no—
143C22×C4×C67—21no—
144C22×C4×C67—82no—
145C22×C4×C67—84no—
146C22×C4×C67C2×D411yescharacteristic
147C22×C4×C67C2×D411yescharacteristic
148C22×C4×C67C2×D411yescharacteristic
149C22×C4×C67C2×D411yescharacteristic
150C22×C4×C67C2×D411yescharacteristic
151C22×C4×C67C2×D414yes—
152C22×C4×C67C2411yescharacteristic
153Q16×C67—44no—
154SD16×C67—44no—
155M16×C67—24no—
156M16×C67—24no—
157C2×C8×C67—22no—
158C2×C8×C67—22no—
159C4⋊C4×C67—28no—
160C4⋊C4×C67—22no—
161C4⋊C4×C67—21no—
162C4⋊C4×C67—22no—
163C4⋊C4×C67—21no—
164C4⋊C4×C67—22no—
165C4⋊C4×C67—22no—
166C4⋊C4×C67—21no—
167C4⋊C4×C67—21no—
168C4⋊C4×C67—41no—
169C4⋊C4×C67—41no—
170C4⋊C4×C67—84no—
171C4⋊C4×C67C2×D412yes—
172C4⋊C4×C67C2×D412yes—
173C22⋊C4×C67—24no—
174C22⋊C4×C67—28no—
175C22⋊C4×C67—22no—
176C22⋊C4×C67—22no—
177C22⋊C4×C67—41no—
178C22⋊C4×C67—41no—
179C42×C67—28no—
180C42×C67—22no—
181C42×C67—22no—
182C42×C67—24no—
183C23×C67—24no—
184C23×C67—22no—
185C23×C67—22no—
186C23×C67—22no—
187C23×C67—41no—
188C23×C67—42no—
189C23×C67—42no—
190C23×C67#12011yescharacteristic
191C23×C67#12011yescharacteristic
192C23×C67C22×D411yescharacteristic
193Q8×C67—24no—
194Q8×C67—22no—
195Q8×C67—42no—
196Q8×C67—42no—
197Q8×C67#12314yes—
198D4×C67—28no—
199D4×C67—24no—
200D4×C67—28no—
201D4×C67—22no—
202D4×C67#12314yes—
203C2×C4×C67—24no—
204C2×C4×C67—28no—
205C2×C4×C67—22no—
206C2×C4×C67—22no—
207C2×C4×C67—22no—
208C2×C4×C67—22no—
209C2×C4×C67—22no—
210C2×C4×C67—28no—
211C2×C4×C67—22no—
212C2×C4×C67—24no—
213C2×C4×C67—22no—
214C2×C4×C67—24no—
215C2×C4×C67—22no—
216C2×C4×C67—42no—
217C2×C4×C67—44no—
218C2×C4×C67—42no—
219C2×C4×C67—42no—
220C2×C4×C67—42no—
221C2×C4×C67—44no—
222C2×C4×C67—42no—
223C2×C4×C67—41no—
224C2×C4×C67—41no—
225C2×C4×C67—42no—
226C2×C4×C67—82no—
227C2×C4×C67—88no—
228C2×C4×C67#12011yescharacteristic
229C2×C4×C67#12011yescharacteristic
230C2×C4×C67C22×D411yescharacteristic
231C2×C4×C67C22×D411yescharacteristic
232C2×C4×C67C2×#4114yes—
233C8×C67—42no—
234C8×C67—42no—
235C22×C67—21no—
236C22×C67—21no—
237C22×C67—21no—
238C22×C67—21no—
239C22×C67—21no—
240C22×C67—24no—
241C22×C67—21no—
242C22×C67—21no—
243C22×C67—44no—
244C22×C67—48no—
245C22×C67—42no—
246C22×C67—42no—
247C22×C67#45612yes—
248C22×C67C2×#12011yescharacteristic
249C22×C67#52512yes—
250C4×C67—21no—
251C4×C67—24no—
252C4×C67—21no—
253C4×C67—44no—
254C4×C67—42no—
255C4×C67—44no—
256C4×C67—42no—
257C4×C67—81no—
258C4×C67—81no—
259C4×C67C2×#12312yes—
260C4×C67#52512yes—
261#33000C6711yesmaximal, characteristic, direct product factor, Sylow
262C2×C67—22no—
263C2×C67—22no—
264C2×C67—41no—
265C2×C67—44no—
266C2×C67#284911yescharacteristic
267C2×C67C2×#45611yescharacteristic
268C2×C67#301911yescharacteristic
269C2×#520C2×C6711yescharacteristic
270#3095C2×C6711yescharacteristic, complemented
271#3007C2×C6711yescharacteristic
272#2502C2×C6711yescharacteristic, complemented
273#1882C2×C6714yescomplemented
274#1859C2×C6714yescomplemented
275#1826C2×C6711yescharacteristic, complemented
276#1821C2×C6711yescharacteristic, complemented
277#1789C2×C6711yescharacteristic
278C67#3300011yescharacteristic, direct product factor, Sylow
279C22×#41C22×C6711yescharacteristic
280C2×#137C22×C6712yes—
281C2×C22⋊Q8C22×C6711yescharacteristic
282#520—22no—
283#520—22no—
284#520—22no—
285#520C22×C6714yes—
286C2×C4×Q8—21no—
287C2×C4×D4—21no—
288C2×C4×D4—22no—
289C2×#117—21no—
290C2×#117C22×C6711yescharacteristic
291C22×C4⋊C4—21no—
292C22×C4⋊C4C22×C6711yescharacteristic
293#485—22no—
294#484—21no—
295#483—21no—
296#452—22no—
297#451—21no—
298#450—21no—
299#428C22×C6712yescomplemented
300C2×#104C22×C6714yes—
301#419C22×C6712yescomplemented
302#417C22×C6712yes—
303#407C22×C6711yescharacteristic
304#407C22×C6711yescharacteristic
305#398—22no—
306#391C22×C6712yescomplemented
307#385—22no—
308#384C22×C6712yescomplemented
309C4×C22⋊C4C22×C6711yescharacteristic
310#343C22×C6714yescomplemented
311#339C22×C6714yescomplemented
312C2×#41—28no—
313C2×#41—21no—
314C2×#41—21no—
315C2×#41D4×C6712yes—
316C2×#41C23×C6712yes—
317C22×Q8D4×C6711yescharacteristic
318C22×Q8C23×C6711yescharacteristic
319C22×D4—21no—
320C22×D4D4×C6711yescharacteristic
321C23×C4—21no—
322C23×C4—22no—
323C23×C4D4×C6712yes—
324C23×C4C23×C6711yescharacteristic
325#137—28no—
326C2×Q16—22no—
327C2×SD16—22no—
328C2×M16C23×C6712yes—
329C2×M16C23×C6712yes—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C4×Q8—24no—
335C4×Q8—24no—
336C4×D4—28no—
337C4×D4—24no—
338C4×D4—24no—
339C4×D4—28no—
340#117—24no—
341#117—21no—
342#117—21no—
343#117—24no—
344#117D4×C6712yes—
345#117C23×C6712yes—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4D4×C6712yes—
356C2×C4⋊C4D4×C6711yescharacteristic
357C2×C4⋊C4D4×C6711yescharacteristic
358C2×C4⋊C4C23×C6711yescharacteristic
359C2×C4⋊C4C23×C6712yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C42—22no—
365C2×C42—21no—
366C2×C42C23×C6712yes—
367#107—22no—
368#106—22no—
369#104—28no—
370#103—22no—
371#103—22no—
372#103—22no—
373#102—22no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8—21no—
377C22⋊C8—21no—
378#95—21no—
379#95—24no—
380C24—21no—
381C24C2×D4×C6711yescharacteristic
382#41—28no—
383#41—28no—
384#41—24no—
385#41—24no—
386#41—24no—
387#41#41×C6718yes—
388C2×Q8—24no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—24no—
393C2×Q8C2×D4×C6712yes—
394C2×D4—28no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—21no—
399C2×D4C2×D4×C6712yes—
400C22×C4—28no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—82no—
414C22×C4—84no—
415C22×C4C2×D4×C6711yescharacteristic
416C22×C4C2×D4×C6711yescharacteristic
417C22×C4C2×D4×C6711yescharacteristic
418C22×C4C2×D4×C6711yescharacteristic
419C22×C4C2×D4×C6711yescharacteristic
420C22×C4C2×D4×C6714yes—
421C22×C4C24×C6711yescharacteristic
422Q16—44no—
423SD16—44no—
424M16—24no—
425M16—24no—
426C2×C8—22no—
427C2×C8—22no—
428C4⋊C4—28no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—84no—
440C4⋊C4C2×D4×C6712yes—
441C4⋊C4C2×D4×C6712yes—
442C22⋊C4—28no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C42—28no—
449C42—22no—
450C42—22no—
451C42—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23#120×C6711yescharacteristic
460C23#120×C6711yescharacteristic
461C23C22×D4×C6711yescharacteristic
462Q8—24no—
463Q8—22no—
464Q8—42no—
465Q8—42no—
466Q8#123×C6714yes—
467D4—28no—
468D4—24no—
469D4—28no—
470D4—22no—
471D4#123×C6714yes—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—28no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—82no—
496C2×C4—88no—
497C2×C4#120×C6711yescharacteristic
498C2×C4#120×C6711yescharacteristic
499C2×C4C22×D4×C6711yescharacteristic
500C2×C4C22×D4×C6711yescharacteristic
501C2×C4C2×#41×C6714yes—
502C8—42no—
503C8—42no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—24no—
512C22—44no—
513C22—48no—
514C22—42no—
515C22—42no—
516C22#456×C6712yes—
517C22C2×#120×C6711yescharacteristic
518C22#525×C6712yes—
519C4—21no—
520C4—24no—
521C4—21no—
522C4—44no—
523C4—42no—
524C4—44no—
525C4—42no—
526C4—81no—
527C4—81no—
528C4C2×#123×C6712yes—
529C4#525×C6712yes—
530C2—22no—
531C2—22no—
532C2—41no—
533C2—44no—
534C2C67×#284911yescharacteristic
535C2C2×#456×C6711yescharacteristic
536C2C67×#301911yescharacteristic
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC22×#41×C67C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#3095C2semidirect
4. C67×#2502C2semidirect
5. C67×#1882C2semidirect
6. C67×#1859C2semidirect
7. C67×#1826C2semidirect
8. C67×#1821C2semidirect
30. #428×C67C22semidirect
32. #419×C67C22semidirect
37. #391×C67C22semidirect
39. #384×C67C22semidirect
41. #343×C67C22semidirect
42. #339×C67C22semidirect
261. #33000C67direct (N × H)
270. #3095C2×C67semidirect
272. #2502C2×C67semidirect
273. #1882C2×C67semidirect
274. #1859C2×C67semidirect
275. #1826C2×C67semidirect
276. #1821C2×C67semidirect
278. C67#33000direct (N × H)
299. #428C22×C67semidirect
301. #419C22×C67semidirect
306. #391C22×C67semidirect
308. #384C22×C67semidirect
310. #343C22×C67semidirect
311. #339C22×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#520×C672
C67×#30952
C67×#30072
C67×#25022
C67×#18822
C67×#18592
C67×#18262
C67×#18212
C67×#17892
#3300067