Finite Group Databasebeta

← C67×#18998 #30698249

Subgroups — C67×#18998 #30698249

Order 17152.

926 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#562×C67C211yesmaximal, characteristic
2C67×#3236C211yesmaximal, characteristic, complemented
3C67×#3215C211yesmaximal, characteristic, complemented
4C67×#3140C211yesmaximal, characteristic, complemented
5C67×#1662C214yesmaximal, complemented
6C67×#1661C214yesmaximal, complemented
7C67×#1533C211yesmaximal, characteristic, complemented
8C67×#1528C211yesmaximal, characteristic, complemented
9C67×#1527C211yesmaximal, characteristic, complemented
10#562×C67—22no—
11#562×C67—22no—
12#562×C67—22no—
13#562×C67C2214yes—
14#533×C67C2211yescharacteristic
15C2×#125×C67C2211yescharacteristic
16C2×#124×C67C2211yescharacteristic
17C2×#124×C67C2211yescharacteristic
18C2×#123×C67—21no—
19C2×#123×C67—21no—
20C2×C22⋊Q8×C67—21no—
21C2×C22⋊Q8×C67—21no—
22C2×#121×C67—21no—
23C2×#121×C67—21no—
24#501×C67C2212yescomplemented
25#497×C67C2212yescomplemented
26#491×C67C2212yescomplemented
27#489×C67C2212yescomplemented
28#489×C67C2212yescomplemented
29#489×C67C2212yescomplemented
30#488×C67—22no—
31#484×C67—21no—
32#483×C67—21no—
33#479×C67—22no—
34#464×C67—22no—
35#451×C67—21no—
36#450×C67—21no—
37#432×C67C2211yescharacteristic
38#432×C67C2211yescharacteristic
39#419×C67—21no—
40#418×C67—21no—
41C4×M16×C67C2211yescharacteristic
42#330×C67C2214yescomplemented
43#329×C67C2214yescomplemented
44#329×C67C2214yescomplemented
45C22×Q8×C67D411yescharacteristic
46C22×Q8×C67D411yescharacteristic
47C22×D4×C67—21no—
48C22×D4×C67—21no—
49C22×D4×C67D411yescharacteristic
50C22×D4×C67D411yescharacteristic
51C23×C4×C67—21no—
52C23×C4×C67—21no—
53C2×Q16×C67—22no—
54C2×SD16×C67—22no—
55C2×M16×C67—21no—
56C4⋊Q8×C67C2311yescharacteristic
57C4⋊Q8×C67C2311yescharacteristic
58#125×C67—21no—
59#125×C67—21no—
60#125×C67—21no—
61#125×C67D412yes—
62#125×C67C2312yes—
63#124×C67—21no—
64#124×C67—21no—
65#124×C67—21no—
66#124×C67—21no—
67#124×C67D412yes—
68#124×C67D412yes—
69#124×C67C2312yes—
70#124×C67C2312yes—
71#123×C67—24no—
72#123×C67—24no—
73#123×C67—24no—
74#123×C67—24no—
75C22⋊Q8×C67—24no—
76C22⋊Q8×C67—24no—
77C22⋊Q8×C67—24no—
78C22⋊Q8×C67—24no—
79C22⋊Q8×C67—21no—
80C22⋊Q8×C67—21no—
81#121×C67—24no—
82#121×C67—24no—
83#121×C67—24no—
84#121×C67—24no—
85C4×Q8×C67—21no—
86C4×Q8×C67—21no—
87#117×C67—21no—
88C2×C4⋊C4×C67—21no—
89C2×C4⋊C4×C67—21no—
90C2×C4⋊C4×C67—21no—
91C2×C4⋊C4×C67—21no—
92C2×C4⋊C4×C67D411yescharacteristic
93C2×C4⋊C4×C67D411yescharacteristic
94C2×C22⋊C4×C67—21no—
95C2×C22⋊C4×C67—21no—
96C2×C22⋊C4×C67—21no—
97C2×C22⋊C4×C67—21no—
98C2×C22⋊C4×C67—21no—
99C2×C22⋊C4×C67—21no—
100C2×C42×C67C2311yescharacteristic
101#107×C67—22no—
102#106×C67—22no—
103C4⋊C8×C67—21no—
104C4⋊C8×C67—21no—
105#103×C67—22no—
106#103×C67—22no—
107#103×C67—22no—
108#103×C67—22no—
109#103×C67—22no—
110#102×C67—22no—
111#102×C67—22no—
112#102×C67—22no—
113C22⋊C8×C67—21no—
114C22⋊C8×C67—21no—
115#97×C67C2312yes—
116#97×C67C2312yes—
117#97×C67C2312yes—
118C4×C8×C67—21no—
119C24×C67—21no—
120C24×C67—21no—
121C2×Q8×C67—21no—
122C2×Q8×C67—21no—
123C2×Q8×C67—21no—
124C2×Q8×C67—21no—
125C2×Q8×C67—42no—
126C2×Q8×C67—81no—
127C2×Q8×C67C2×D412yes—
128C2×Q8×C67C2×D412yes—
129C2×D4×C67—24no—
130C2×D4×C67—24no—
131C2×D4×C67—21no—
132C2×D4×C67—21no—
133C2×D4×C67—42no—
134C2×D4×C67—81no—
135C2×D4×C67—82no—
136C2×D4×C67—82no—
137C2×D4×C67C2×D412yes—
138C2×D4×C67C2×D412yes—
139C22×C4×C67—21no—
140C22×C4×C67—21no—
141C22×C4×C67—21no—
142C22×C4×C67—21no—
143C22×C4×C67—24no—
144C22×C4×C67—24no—
145C22×C4×C67—82no—
146C22×C4×C67—82no—
147C22×C4×C67C2×D411yescharacteristic
148C22×C4×C67C2×D411yescharacteristic
149C22×C4×C67C2×D411yescharacteristic
150Q16×C67—82no—
151SD16×C67—82no—
152M16×C67—42no—
153C2×C8×C67—22no—
154C2×C8×C67—22no—
155C2×C8×C67—22no—
156C4⋊C4×C67—24no—
157C4⋊C4×C67—24no—
158C4⋊C4×C67—22no—
159C4⋊C4×C67—22no—
160C4⋊C4×C67—21no—
161C4⋊C4×C67—21no—
162C4⋊C4×C67—22no—
163C4⋊C4×C67—22no—
164C4⋊C4×C67—21no—
165C4⋊C4×C67—21no—
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—24no—
173C22⋊C4×C67—24no—
174C22⋊C4×C67—22no—
175C22⋊C4×C67—22no—
176C22⋊C4×C67—22no—
177C22⋊C4×C67—22no—
178C22⋊C4×C67—22no—
179C22⋊C4×C67—22no—
180C22⋊C4×C67—22no—
181C22⋊C4×C67—22no—
182C22⋊C4×C67—41no—
183C42×C67—41no—
184C42×C67C2×D411yescharacteristic
185C42×C67C2×D411yescharacteristic
186C42×C67C2×D411yescharacteristic
187C42×C67C2411yescharacteristic
188C23×C67—22no—
189C23×C67—22no—
190C23×C67—42no—
191C23×C67—42no—
192C23×C67—42no—
193C23×C67—42no—
194C23×C67—42no—
195C23×C67—42no—
196C23×C67#12011yescharacteristic
197Q8×C67—42no—
198Q8×C67—42no—
199Q8×C67—42no—
200Q8×C67—42no—
201Q8×C67—81no—
202Q8×C67—81no—
203D4×C67—42no—
204D4×C67—42no—
205D4×C67—81no—
206D4×C67—81no—
207D4×C67—84no—
208D4×C67—84no—
209C2×C4×C67—22no—
210C2×C4×C67—22no—
211C2×C4×C67—22no—
212C2×C4×C67—22no—
213C2×C4×C67—22no—
214C2×C4×C67—41no—
215C2×C4×C67—41no—
216C2×C4×C67—42no—
217C2×C4×C67—42no—
218C2×C4×C67—42no—
219C2×C4×C67—42no—
220C2×C4×C67—41no—
221C2×C4×C67—41no—
222C2×C4×C67—84no—
223C2×C4×C67—84no—
224C2×C4×C67#12011yescharacteristic
225C2×C4×C67#12011yescharacteristic
226C2×C4×C67#12011yescharacteristic
227C2×C4×C67C22×D411yescharacteristic
228C2×C4×C67C22×D411yescharacteristic
229C2×C4×C67C22×D411yescharacteristic
230C8×C67—42no—
231C8×C67—42no—
232C8×C67—42no—
233C22×C67—21no—
234C22×C67—21no—
235C22×C67—42no—
236C22×C67—42no—
237C22×C67—42no—
238C22×C67—42no—
239C22×C67—42no—
240C22×C67—42no—
241C22×C67—84no—
242C22×C67—84no—
243C22×C67#45512yes—
244C22×C67C2×#12011yescharacteristic
245C4×C67—21no—
246C4×C67—21no—
247C4×C67—21no—
248C4×C67—21no—
249C4×C67—21no—
250C4×C67—21no—
251C4×C67—42no—
252C4×C67—42no—
253C4×C67—42no—
254C4×C67—42no—
255C4×C67—81no—
256C4×C67—81no—
257#18998C6711yesmaximal, characteristic, direct product factor, Sylow
258C2×C67—22no—
259C2×C67—82no—
260C2×C67—82no—
261C2×C67C2×#45511yescharacteristic
262C2×C67#301911yescharacteristic
263C2×C67#302011yescharacteristic
264C2×#562C2×C6711yescharacteristic
265#3236C2×C6711yescharacteristic, complemented
266#3215C2×C6711yescharacteristic, complemented
267#3140C2×C6711yescharacteristic, complemented
268#1662C2×C6714yescomplemented
269#1661C2×C6714yescomplemented
270#1533C2×C6711yescharacteristic, complemented
271#1528C2×C6711yescharacteristic, complemented
272#1527C2×C6711yescharacteristic, complemented
273C67#1899811yescharacteristic, direct product factor, Sylow
274#562—22no—
275#562—22no—
276#562—22no—
277#562C22×C6714yes—
278#533C22×C6711yescharacteristic
279C2×#125C22×C6711yescharacteristic
280C2×#124C22×C6711yescharacteristic
281C2×#124C22×C6711yescharacteristic
282C2×#123—21no—
283C2×#123—21no—
284C2×C22⋊Q8—21no—
285C2×C22⋊Q8—21no—
286C2×#121—21no—
287C2×#121—21no—
288#501C22×C6712yescomplemented
289#497C22×C6712yescomplemented
290#491C22×C6712yescomplemented
291#489C22×C6712yescomplemented
292#489C22×C6712yescomplemented
293#489C22×C6712yescomplemented
294#488—22no—
295#484—21no—
296#483—21no—
297#479—22no—
298#464—22no—
299#451—21no—
300#450—21no—
301#432C22×C6711yescharacteristic
302#432C22×C6711yescharacteristic
303#419—21no—
304#418—21no—
305C4×M16C22×C6711yescharacteristic
306#330C22×C6714yescomplemented
307#329C22×C6714yescomplemented
308#329C22×C6714yescomplemented
309C22×Q8D4×C6711yescharacteristic
310C22×Q8D4×C6711yescharacteristic
311C22×D4—21no—
312C22×D4—21no—
313C22×D4D4×C6711yescharacteristic
314C22×D4D4×C6711yescharacteristic
315C23×C4—21no—
316C23×C4—21no—
317C2×Q16—22no—
318C2×SD16—22no—
319C2×M16—21no—
320C4⋊Q8C23×C6711yescharacteristic
321C4⋊Q8C23×C6711yescharacteristic
322#125—21no—
323#125—21no—
324#125—21no—
325#125D4×C6712yes—
326#125C23×C6712yes—
327#124—21no—
328#124—21no—
329#124—21no—
330#124—21no—
331#124D4×C6712yes—
332#124D4×C6712yes—
333#124C23×C6712yes—
334#124C23×C6712yes—
335#123—24no—
336#123—24no—
337#123—24no—
338#123—24no—
339C22⋊Q8—24no—
340C22⋊Q8—24no—
341C22⋊Q8—24no—
342C22⋊Q8—24no—
343C22⋊Q8—21no—
344C22⋊Q8—21no—
345#121—24no—
346#121—24no—
347#121—24no—
348#121—24no—
349C4×Q8—21no—
350C4×Q8—21no—
351#117—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4D4×C6711yescharacteristic
357C2×C4⋊C4D4×C6711yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C42C23×C6711yescharacteristic
365#107—22no—
366#106—22no—
367C4⋊C8—21no—
368C4⋊C8—21no—
369#103—22no—
370#103—22no—
371#103—22no—
372#103—22no—
373#103—22no—
374#102—22no—
375#102—22no—
376#102—22no—
377C22⋊C8—21no—
378C22⋊C8—21no—
379#97C23×C6712yes—
380#97C23×C6712yes—
381#97C23×C6712yes—
382C4×C8—21no—
383C24—21no—
384C24—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—42no—
390C2×Q8—81no—
391C2×Q8C2×D4×C6712yes—
392C2×Q8C2×D4×C6712yes—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—42no—
398C2×D4—81no—
399C2×D4—82no—
400C2×D4—82no—
401C2×D4C2×D4×C6712yes—
402C2×D4C2×D4×C6712yes—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4C2×D4×C6711yescharacteristic
412C22×C4C2×D4×C6711yescharacteristic
413C22×C4C2×D4×C6711yescharacteristic
414Q16—82no—
415SD16—82no—
416M16—42no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4C2×D4×C6712yes—
435C4⋊C4C2×D4×C6712yes—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—41no—
447C42—41no—
448C42C2×D4×C6711yescharacteristic
449C42C2×D4×C6711yescharacteristic
450C42C2×D4×C6711yescharacteristic
451C42C24×C6711yescharacteristic
452C23—22no—
453C23—22no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23#120×C6711yescharacteristic
461Q8—42no—
462Q8—42no—
463Q8—42no—
464Q8—42no—
465Q8—81no—
466Q8—81no—
467D4—42no—
468D4—42no—
469D4—81no—
470D4—81no—
471D4—84no—
472D4—84no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4#120×C6711yescharacteristic
489C2×C4#120×C6711yescharacteristic
490C2×C4#120×C6711yescharacteristic
491C2×C4C22×D4×C6711yescharacteristic
492C2×C4C22×D4×C6711yescharacteristic
493C2×C4C22×D4×C6711yescharacteristic
494C8—42no—
495C8—42no—
496C8—42no—
497C22—21no—
498C22—21no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—84no—
506C22—84no—
507C22#455×C6712yes—
508C22C2×#120×C6711yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—81no—
520C4—81no—
521C2—22no—
522C2—82no—
523C2—82no—
524C2C2×#455×C6711yescharacteristic
525C2C67×#301911yescharacteristic
526C2C67×#302011yescharacteristic
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC42C24×C67
prime radicalC2×#562×C67C2
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
2. C67×#3236C2semidirect
3. C67×#3215C2semidirect
4. C67×#3140C2semidirect
5. C67×#1662C2semidirect
6. C67×#1661C2semidirect
7. C67×#1533C2semidirect
8. C67×#1528C2semidirect
9. C67×#1527C2semidirect
24. #501×C67C22semidirect
25. #497×C67C22semidirect
26. #491×C67C22semidirect
27. #489×C67C22semidirect
28. #489×C67C22semidirect
29. #489×C67C22semidirect
42. #330×C67C22semidirect
43. #329×C67C22semidirect
44. #329×C67C22semidirect
257. #18998C67direct (N × H)
265. #3236C2×C67semidirect
266. #3215C2×C67semidirect
267. #3140C2×C67semidirect
268. #1662C2×C67semidirect
269. #1661C2×C67semidirect
270. #1533C2×C67semidirect
271. #1528C2×C67semidirect
272. #1527C2×C67semidirect
273. C67#18998direct (N × H)
288. #501C22×C67semidirect
289. #497C22×C67semidirect
290. #491C22×C67semidirect
291. #489C22×C67semidirect
292. #489C22×C67semidirect
293. #489C22×C67semidirect
306. #330C22×C67semidirect
307. #329C22×C67semidirect
308. #329C22×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#562×C672
C67×#32362
C67×#32152
C67×#31402
C67×#16622
C67×#16612
C67×#15332
C67×#15282
C67×#15272
#1899867