Finite Group Databasebeta

← C67×#32999 #30712250

Subgroups — C67×#32999 #30712250

Order 17152.

1284 conjugacy classes of subgroups.

The subgroup lattice has 536 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×#2503C211yesmaximal, characteristic, complemented
5C67×#1882C214yesmaximal, complemented
6C67×#1860C214yesmaximal, 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#400×C67—22no—
37#391×C67C2212yescomplemented
38#382×C67C2212yescomplemented
39#381×C67—22no—
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×C67D412yes—
86C2×C4⋊C4×C67D411yescharacteristic
87C2×C4⋊C4×C67D411yescharacteristic
88C2×C4⋊C4×C67C2311yescharacteristic
89C2×C4⋊C4×C67C2312yes—
90C2×C22⋊C4×C67—22no—
91C2×C22⋊C4×C67—21no—
92C2×C22⋊C4×C67—21no—
93C2×C22⋊C4×C67—21no—
94C2×C42×C67—22no—
95C2×C42×C67—21no—
96C2×C42×C67C2312yes—
97#107×C67—22no—
98#106×C67—22no—
99#104×C67—28no—
100#103×C67—22no—
101#103×C67—22no—
102#103×C67—22no—
103#102×C67—22no—
104C22⋊C8×C67—21no—
105C22⋊C8×C67—21no—
106C22⋊C8×C67—21no—
107C22⋊C8×C67—21no—
108#95×C67—21no—
109#95×C67—24no—
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—28no—
140C22×C4×C67—22no—
141C22×C4×C67—22no—
142C22×C4×C67—21no—
143C22×C4×C67—21no—
144C22×C4×C67—44no—
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×C67C2×D412yes—
171C4⋊C4×C67C2×D412yes—
172C22⋊C4×C67—28no—
173C22⋊C4×C67—22no—
174C22⋊C4×C67—22no—
175C22⋊C4×C67—42no—
176C22⋊C4×C67—41no—
177C22⋊C4×C67—41no—
178C42×C67—28no—
179C42×C67—22no—
180C42×C67—22no—
181C42×C67—24no—
182C23×C67—24no—
183C23×C67—22no—
184C23×C67—22no—
185C23×C67—22no—
186C23×C67—41no—
187C23×C67—42no—
188C23×C67—42no—
189C23×C67#12011yescharacteristic
190C23×C67#12011yescharacteristic
191C23×C67C22×D411yescharacteristic
192Q8×C67—24no—
193Q8×C67—22no—
194Q8×C67—42no—
195Q8×C67—42no—
196Q8×C67#12314yes—
197D4×C67—28no—
198D4×C67—24no—
199D4×C67—28no—
200D4×C67—22no—
201D4×C67#12314yes—
202C2×C4×C67—24no—
203C2×C4×C67—28no—
204C2×C4×C67—22no—
205C2×C4×C67—22no—
206C2×C4×C67—22no—
207C2×C4×C67—22no—
208C2×C4×C67—22no—
209C2×C4×C67—28no—
210C2×C4×C67—22no—
211C2×C4×C67—22no—
212C2×C4×C67—24no—
213C2×C4×C67—22no—
214C2×C4×C67—42no—
215C2×C4×C67—44no—
216C2×C4×C67—42no—
217C2×C4×C67—44no—
218C2×C4×C67—42no—
219C2×C4×C67—42no—
220C2×C4×C67—42no—
221C2×C4×C67—41no—
222C2×C4×C67—41no—
223C2×C4×C67—42no—
224C2×C4×C67—42no—
225C2×C4×C67—82no—
226C2×C4×C67—88no—
227C2×C4×C67#12011yescharacteristic
228C2×C4×C67#12011yescharacteristic
229C2×C4×C67C22×D411yescharacteristic
230C2×C4×C67C22×D411yescharacteristic
231C2×C4×C67C2×#4114yes—
232C8×C67—42no—
233C8×C67—42no—
234C22×C67—21no—
235C22×C67—21no—
236C22×C67—21no—
237C22×C67—21no—
238C22×C67—21no—
239C22×C67—24no—
240C22×C67—21no—
241C22×C67—21no—
242C22×C67—44no—
243C22×C67—48no—
244C22×C67—42no—
245C22×C67—42no—
246C22×C67#45512yes—
247C22×C67C2×#12011yescharacteristic
248C22×C67#52512yes—
249C4×C67—21no—
250C4×C67—24no—
251C4×C67—21no—
252C4×C67—44no—
253C4×C67—42no—
254C4×C67—42no—
255C4×C67—82no—
256C4×C67—81no—
257C4×C67—81no—
258C4×C67C2×#12312yes—
259C4×C67#52512yes—
260#32999C6711yesmaximal, characteristic, direct product factor, Sylow
261C2×C67—22no—
262C2×C67—22no—
263C2×C67—41no—
264C2×C67—44no—
265C2×C67#284911yescharacteristic
266C2×C67C2×#45511yescharacteristic
267C2×C67#302011yescharacteristic
268C2×#520C2×C6711yescharacteristic
269#3095C2×C6711yescharacteristic, complemented
270#3007C2×C6711yescharacteristic
271#2503C2×C6711yescharacteristic, complemented
272#1882C2×C6714yescomplemented
273#1860C2×C6714yescomplemented
274#1826C2×C6711yescharacteristic, complemented
275#1821C2×C6711yescharacteristic, complemented
276#1789C2×C6711yescharacteristic
277C67#3299911yescharacteristic, direct product factor, Sylow
278C22×#41C22×C6711yescharacteristic
279C2×#137C22×C6712yes—
280C2×C22⋊Q8C22×C6711yescharacteristic
281#520—22no—
282#520—22no—
283#520—22no—
284#520C22×C6714yes—
285C2×C4×Q8—21no—
286C2×C4×D4—21no—
287C2×C4×D4—22no—
288C2×#117—21no—
289C2×#117C22×C6711yescharacteristic
290C22×C4⋊C4—21no—
291C22×C4⋊C4C22×C6711yescharacteristic
292#485—22no—
293#484—21no—
294#483—21no—
295#452—22no—
296#451—21no—
297#450—21no—
298#428C22×C6712yescomplemented
299C2×#104C22×C6714yes—
300#419C22×C6712yescomplemented
301#417C22×C6712yes—
302#407C22×C6711yescharacteristic
303#407C22×C6711yescharacteristic
304#400—22no—
305#391C22×C6712yescomplemented
306#382C22×C6712yescomplemented
307#381—22no—
308C4×C22⋊C4C22×C6711yescharacteristic
309#343C22×C6714yescomplemented
310#339C22×C6714yescomplemented
311C2×#41—28no—
312C2×#41—21no—
313C2×#41—21no—
314C2×#41D4×C6712yes—
315C2×#41C23×C6712yes—
316C22×Q8D4×C6711yescharacteristic
317C22×Q8C23×C6711yescharacteristic
318C22×D4—21no—
319C22×D4D4×C6711yescharacteristic
320C23×C4—21no—
321C23×C4—22no—
322C23×C4D4×C6712yes—
323C23×C4C23×C6711yescharacteristic
324#137—28no—
325C2×Q16—22no—
326C2×SD16—22no—
327C2×M16C23×C6712yes—
328C2×M16C23×C6712yes—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C4×Q8—24no—
334C4×Q8—24no—
335C4×D4—28no—
336C4×D4—24no—
337C4×D4—24no—
338C4×D4—28no—
339#117—24no—
340#117—21no—
341#117—21no—
342#117—24no—
343#117D4×C6712yes—
344#117C23×C6712yes—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4D4×C6712yes—
354C2×C4⋊C4D4×C6711yescharacteristic
355C2×C4⋊C4D4×C6711yescharacteristic
356C2×C4⋊C4C23×C6711yescharacteristic
357C2×C4⋊C4C23×C6712yes—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C42—22no—
363C2×C42—21no—
364C2×C42C23×C6712yes—
365#107—22no—
366#106—22no—
367#104—28no—
368#103—22no—
369#103—22no—
370#103—22no—
371#102—22no—
372C22⋊C8—21no—
373C22⋊C8—21no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376#95—21no—
377#95—24no—
378#95—24no—
379C24—21no—
380C24C2×D4×C6711yescharacteristic
381#41—28no—
382#41—28no—
383#41—24no—
384#41—24no—
385#41—24no—
386#41#41×C6718yes—
387C2×Q8—24no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—24no—
392C2×Q8C2×D4×C6712yes—
393C2×D4—28no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—21no—
398C2×D4C2×D4×C6712yes—
399C22×C4—28no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—44no—
413C22×C4—84no—
414C22×C4C2×D4×C6711yescharacteristic
415C22×C4C2×D4×C6711yescharacteristic
416C22×C4C2×D4×C6711yescharacteristic
417C22×C4C2×D4×C6711yescharacteristic
418C22×C4C2×D4×C6711yescharacteristic
419C22×C4C2×D4×C6714yes—
420C22×C4C24×C6711yescharacteristic
421Q16—44no—
422SD16—44no—
423M16—24no—
424M16—24no—
425C2×C8—22no—
426C2×C8—22no—
427C4⋊C4—28no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4C2×D4×C6712yes—
439C4⋊C4C2×D4×C6712yes—
440C22⋊C4—28no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—42no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C42—28no—
447C42—22no—
448C42—22no—
449C42—24no—
450C23—24no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—41no—
455C23—42no—
456C23—42no—
457C23#120×C6711yescharacteristic
458C23#120×C6711yescharacteristic
459C23C22×D4×C6711yescharacteristic
460Q8—24no—
461Q8—22no—
462Q8—42no—
463Q8—42no—
464Q8#123×C6714yes—
465D4—28no—
466D4—24no—
467D4—28no—
468D4—22no—
469D4#123×C6714yes—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—28no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—82no—
494C2×C4—88no—
495C2×C4#120×C6711yescharacteristic
496C2×C4#120×C6711yescharacteristic
497C2×C4C22×D4×C6711yescharacteristic
498C2×C4C22×D4×C6711yescharacteristic
499C2×C4C2×#41×C6714yes—
500C8—42no—
501C8—42no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—24no—
510C22—44no—
511C22—48no—
512C22—42no—
513C22—42no—
514C22#455×C6712yes—
515C22C2×#120×C6711yescharacteristic
516C22#525×C6712yes—
517C4—21no—
518C4—24no—
519C4—21no—
520C4—44no—
521C4—42no—
522C4—42no—
523C4—82no—
524C4—81no—
525C4—81no—
526C4C2×#123×C6712yes—
527C4#525×C6712yes—
528C2—22no—
529C2—22no—
530C2—41no—
531C2—44no—
532C2C67×#284911yescharacteristic
533C2C2×#455×C6711yescharacteristic
534C2C67×#302011yescharacteristic
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×#2503C2semidirect
5. C67×#1882C2semidirect
6. C67×#1860C2semidirect
7. C67×#1826C2semidirect
8. C67×#1821C2semidirect
30. #428×C67C22semidirect
32. #419×C67C22semidirect
37. #391×C67C22semidirect
38. #382×C67C22semidirect
41. #343×C67C22semidirect
42. #339×C67C22semidirect
260. #32999C67direct (N × H)
269. #3095C2×C67semidirect
271. #2503C2×C67semidirect
272. #1882C2×C67semidirect
273. #1860C2×C67semidirect
274. #1826C2×C67semidirect
275. #1821C2×C67semidirect
277. C67#32999direct (N × H)
298. #428C22×C67semidirect
300. #419C22×C67semidirect
305. #391C22×C67semidirect
306. #382C22×C67semidirect
309. #343C22×C67semidirect
310. #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×#25032
C67×#18822
C67×#18602
C67×#18262
C67×#18212
C67×#17892
#3299967