Finite Group Databasebeta

← C67×#25899 #30705150

Subgroups — C67×#25899 #30705150

Order 17152.

1574 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3496C211yesmaximal, characteristic
2C67×#2436C212yesmaximal, complemented
3C67×#2409C211yesmaximal, characteristic, complemented
4C67×#2044C214yesmaximal, complemented
5C67×#1964C212yesmaximal, complemented
6C67×#1790C214yesmaximal, complemented
7C4×#99×C67C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C67C2212yes—
9C23×Q8×C67C2211yescharacteristic, complemented
10#547×C67—22no—
11#546×C67—24no—
12D42×C67—22no—
13#537×C67—21no—
14#534×C67—21no—
15#534×C67C2212yes—
16#532×C67—21no—
17C2×#124×C67—22no—
18C2×C22⋊Q8×C67—21no—
19C2×#121×C67—21no—
20C2×#121×C67C2212yes—
21C2×C4×D4×C67C2211yescharacteristic
22C2×#99×C67C2212yes—
23C2×#99×C67C2218yescomplemented
24#395×C67—21no—
25#393×C67—21no—
26#393×C67—24no—
27#391×C67—22no—
28#391×C67—21no—
29#386×C67C2212yes—
30#386×C67C2218yescomplemented
31#385×C67C2212yescomplemented
32#380×C67C2211yescharacteristic, complemented
33C4×C22⋊C4×C67C2212yescomplemented
34#342×C67C2212yescomplemented
35H(5,𝔽2)×C67—28no—
36H(5,𝔽2)×C67—44no—
37C2×#41×C67—22no—
38C2×#41×C67—24no—
39C22×Q8×C67—21no—
40C22×Q8×C67—22no—
41C22×Q8×C67—22no—
42C22×Q8×C67—21no—
43C22×Q8×C67—44no—
44C22×D4×C67—22no—
45C22×D4×C67—24no—
46C22×D4×C67—21no—
47C22×D4×C67D412yes—
48C22×D4×C67C2314yes—
49C22×D4×C67C2311yescharacteristic
50C23×C4×C67D411yescharacteristic
51C23×C4×C67C2312yes—
52C23×C4×C67C2311yescharacteristic
53C4⋊Q8×C67—21no—
54#127×C67—21no—
55#124×C67—22no—
56#124×C67—48no—
57#123×C67—24no—
58#123×C67—24no—
59C22⋊Q8×C67—24no—
60C22⋊Q8×C67—44no—
61#121×C67—22no—
62#121×C67—22no—
63#121×C67—24no—
64#121×C67—22no—
65#121×C67—22no—
66#121×C67—22no—
67#121×C67—22no—
68#121×C67—22no—
69#121×C67—22no—
70#120×C67—24no—
71#120×C67—24no—
72C4×D4×C67—22no—
73C4×D4×C67—22no—
74C4×D4×C67—22no—
75C4×D4×C67—22no—
76C2×C4⋊C4×C67—21no—
77C2×C4⋊C4×C67—21no—
78C2×C4⋊C4×C67—22no—
79C2×C4⋊C4×C67D411yescharacteristic
80C2×C4⋊C4×C67D412yes—
81C2×C22⋊C4×C67—22no—
82C2×C22⋊C4×C67—22no—
83C2×C22⋊C4×C67—28no—
84C2×C22⋊C4×C67D411yescharacteristic
85C2×C22⋊C4×C67D414yes—
86C2×C22⋊C4×C67C2314yes—
87C2×C22⋊C4×C67C2312yes—
88C2×C22⋊C4×C67C2311yescharacteristic
89C2×C42×C67—22no—
90C2×C42×C67D411yescharacteristic
91#99×C67—24no—
92#99×C67—48no—
93#95×C67—22no—
94#95×C67—21no—
95#95×C67—22no—
96#95×C67—22no—
97#95×C67—28no—
98C24×C67—22no—
99C24×C67C2×D414yes—
100C24×C67C2×D411yescharacteristic
101C24×C67C2411yescharacteristic
102#41×C67—416no—
103#41×C67—44no—
104#41×C67—44no—
105C2×Q8×C67—22no—
106C2×Q8×C67—24no—
107C2×Q8×C67—28no—
108C2×Q8×C67—44no—
109C2×Q8×C67—42no—
110C2×Q8×C67—48no—
111C2×Q8×C67—44no—
112C2×Q8×C67—41no—
113C2×Q8×C67—42no—
114C2×Q8×C67—41no—
115C2×D4×C67—22no—
116C2×D4×C67—22no—
117C2×D4×C67—24no—
118C2×D4×C67—22no—
119C2×D4×C67—24no—
120C2×D4×C67—24no—
121C2×D4×C67—28no—
122C2×D4×C67—28no—
123C2×D4×C67—22no—
124C2×D4×C67—22no—
125C2×D4×C67—22no—
126C2×D4×C67—22no—
127C2×D4×C67—44no—
128C2×D4×C67—416no—
129C2×D4×C67—44no—
130C2×D4×C67—44no—
131C2×D4×C67—82no—
132C2×D4×C67#4114yes—
133C22×C4×C67—22no—
134C22×C4×C67—24no—
135C22×C4×C67—24no—
136C22×C4×C67—22no—
137C22×C4×C67—21no—
138C22×C4×C67—21no—
139C22×C4×C67—24no—
140C22×C4×C67—22no—
141C22×C4×C67—44no—
142C22×C4×C67—42no—
143C22×C4×C67—42no—
144C22×C4×C67C2×D414yes—
145C22×C4×C67C2×D412yes—
146C22×C4×C67C2×D411yescharacteristic
147C22×C4×C67C2×D411yescharacteristic
148C22×C4×C67C2×D411yescharacteristic
149C22×C4×C67C2×D411yescharacteristic
150C22×C4×C67C2×D412yes—
151C22×C4×C67C2×D411yescharacteristic
152C4⋊C4×C67—22no—
153C4⋊C4×C67—22no—
154C4⋊C4×C67—22no—
155C4⋊C4×C67—22no—
156C4⋊C4×C67—22no—
157C4⋊C4×C67—42no—
158C4⋊C4×C67—44no—
159C22⋊C4×C67—24no—
160C22⋊C4×C67—24no—
161C22⋊C4×C67—22no—
162C22⋊C4×C67—24no—
163C22⋊C4×C67—22no—
164C22⋊C4×C67—24no—
165C22⋊C4×C67—24no—
166C22⋊C4×C67—24no—
167C22⋊C4×C67—24no—
168C22⋊C4×C67—44no—
169C22⋊C4×C67—416no—
170C42×C67—21no—
171C42×C67—21no—
172C42×C67—44no—
173C23×C67—21no—
174C23×C67—21no—
175C23×C67—24no—
176C23×C67—28no—
177C23×C67—41no—
178C23×C67—41no—
179C23×C67—41no—
180C23×C67—41no—
181C23×C67—44no—
182C23×C67—44no—
183C23×C67—44no—
184C23×C67—82no—
185C23×C67—82no—
186C23×C67#12412yescomplemented
187C23×C67C22×D411yescharacteristic
188C23×C67C22×D412yes—
189C23×C67C2×#4112yes—
190C23×C67H(5,𝔽2)12yes—
191Q8×C67—44no—
192Q8×C67—48no—
193Q8×C67—44no—
194Q8×C67—44no—
195D4×C67—28no—
196D4×C67—44no—
197D4×C67—48no—
198D4×C67—44no—
199D4×C67—44no—
200D4×C67—48no—
201D4×C67—44no—
202D4×C67—82no—
203D4×C67—82no—
204D4×C67—82no—
205C2×C4×C67—21no—
206C2×C4×C67—21no—
207C2×C4×C67—24no—
208C2×C4×C67—22no—
209C2×C4×C67—28no—
210C2×C4×C67—24no—
211C2×C4×C67—21no—
212C2×C4×C67—21no—
213C2×C4×C67—24no—
214C2×C4×C67—28no—
215C2×C4×C67—42no—
216C2×C4×C67—41no—
217C2×C4×C67—48no—
218C2×C4×C67—42no—
219C2×C4×C67—44no—
220C2×C4×C67—44no—
221C2×C4×C67—42no—
222C2×C4×C67—42no—
223C2×C4×C67—41no—
224C2×C4×C67—41no—
225C2×C4×C67—44no—
226C2×C4×C67—42no—
227C2×C4×C67—42no—
228C2×C4×C67—44no—
229C2×C4×C67—42no—
230C2×C4×C67—44no—
231C2×C4×C67—81no—
232C2×C4×C67#12011yescharacteristic
233C2×C4×C67#12012yes—
234C2×C4×C67#12011yescharacteristic
235C2×C4×C67#12412yescomplemented
236C22×C67—21no—
237C22×C67—22no—
238C22×C67—21no—
239C22×C67—22no—
240C22×C67—22no—
241C22×C67—42no—
242C22×C67—42no—
243C22×C67—42no—
244C22×C67—41no—
245C22×C67—41no—
246C22×C67—44no—
247C22×C67—44no—
248C22×C67—44no—
249C22×C67—81no—
250C22×C67—88no—
251C22×C67—164no—
252C22×C67C2×#12011yescharacteristic
253C22×C67C2×#12411yescharacteristic
254C22×C67#53511yescharacteristic
255C22×C67#54614yes—
256C4×C67—22no—
257C4×C67—21no—
258C4×C67—22no—
259C4×C67—44no—
260C4×C67—44no—
261C4×C67—41no—
262C4×C67—41no—
263C4×C67—82no—
264C4×C67—84no—
265C4×C67B(4,𝔽2)12yes—
266#25899C6711yesmaximal, characteristic, direct product factor, Sylow
267C2×C67—21no—
268C2×C67—24no—
269C2×C67—21no—
270C2×C67—42no—
271C2×C67—84no—
272C2×C67#243511yescharacteristic
273C2×C67C2×B(4,𝔽2)11yescharacteristic
274C2×C67#302811yescharacteristic
275#3496C2×C6711yescharacteristic
276#2436C2×C6712yescomplemented
277#2409C2×C6711yescharacteristic, complemented
278#2044C2×C6714yescomplemented
279#1964C2×C6712yescomplemented
280#1790C2×C6714yescomplemented
281C4×#99C2×C6711yescharacteristic, complemented
282C67#2589911yescharacteristic, direct product factor, Sylow
283C2×H(5,𝔽2)C22×C6712yes—
284C23×Q8C22×C6711yescharacteristic, complemented
285#547—22no—
286#546—24no—
287D42—22no—
288#537—21no—
289#534—21no—
290#534C22×C6712yes—
291#532—21no—
292C2×#124—22no—
293C2×C22⋊Q8—21no—
294C2×#121—21no—
295C2×#121C22×C6712yes—
296C2×C4×D4C22×C6711yescharacteristic
297C2×#99C22×C6712yes—
298C2×#99C22×C6718yescomplemented
299#395—21no—
300#393—21no—
301#393—24no—
302#391—22no—
303#391—21no—
304#386C22×C6712yes—
305#386C22×C6718yescomplemented
306#385C22×C6712yescomplemented
307#380C22×C6711yescharacteristic, complemented
308C4×C22⋊C4C22×C6712yescomplemented
309#342C22×C6712yescomplemented
310H(5,𝔽2)—28no—
311H(5,𝔽2)—44no—
312C2×#41—22no—
313C2×#41—24no—
314C22×Q8—21no—
315C22×Q8—22no—
316C22×Q8—22no—
317C22×Q8—21no—
318C22×Q8—44no—
319C22×D4—22no—
320C22×D4—24no—
321C22×D4—21no—
322C22×D4D4×C6712yes—
323C22×D4C23×C6714yes—
324C22×D4C23×C6711yescharacteristic
325C23×C4D4×C6711yescharacteristic
326C23×C4C23×C6712yes—
327C23×C4C23×C6711yescharacteristic
328C4⋊Q8—21no—
329#127—21no—
330#124—22no—
331#124—48no—
332#123—24no—
333#123—24no—
334C22⋊Q8—24no—
335C22⋊Q8—44no—
336#121—22no—
337#121—22no—
338#121—24no—
339#121—22no—
340#121—22no—
341#121—22no—
342#121—22no—
343#121—22no—
344#121—22no—
345#120—24no—
346#120—24no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—22no—
350C4×D4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4D4×C6711yescharacteristic
355C2×C4⋊C4D4×C6712yes—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—28no—
359C2×C22⋊C4D4×C6711yescharacteristic
360C2×C22⋊C4D4×C6714yes—
361C2×C22⋊C4C23×C6714yes—
362C2×C22⋊C4C23×C6712yes—
363C2×C22⋊C4C23×C6711yescharacteristic
364C2×C42—22no—
365C2×C42D4×C6711yescharacteristic
366#99—24no—
367#99—48no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—28no—
373C24—22no—
374C24C2×D4×C6714yes—
375C24C2×D4×C6711yescharacteristic
376C24C24×C6711yescharacteristic
377#41—416no—
378#41—44no—
379#41—44no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—28no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—48no—
386C2×Q8—44no—
387C2×Q8—41no—
388C2×Q8—42no—
389C2×Q8—41no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—28no—
397C2×D4—28no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—44no—
403C2×D4—416no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—82no—
407C2×D4#41×C6714yes—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4C2×D4×C6714yes—
420C22×C4C2×D4×C6712yes—
421C22×C4C2×D4×C6711yescharacteristic
422C22×C4C2×D4×C6711yescharacteristic
423C22×C4C2×D4×C6711yescharacteristic
424C22×C4C2×D4×C6711yescharacteristic
425C22×C4C2×D4×C6712yes—
426C22×C4C2×D4×C6711yescharacteristic
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—44no—
444C22⋊C4—416no—
445C42—21no—
446C42—21no—
447C42—44no—
448C23—21no—
449C23—21no—
450C23—24no—
451C23—28no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—82no—
460C23—82no—
461C23#124×C6712yescomplemented
462C23C22×D4×C6711yescharacteristic
463C23C22×D4×C6712yes—
464C23C2×#41×C6712yes—
465C23H(5,𝔽2)×C6712yes—
466Q8—44no—
467Q8—48no—
468Q8—44no—
469Q8—44no—
470D4—28no—
471D4—44no—
472D4—48no—
473D4—44no—
474D4—44no—
475D4—48no—
476D4—44no—
477D4—82no—
478D4—82no—
479D4—82no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—81no—
507C2×C4#120×C6711yescharacteristic
508C2×C4#120×C6712yes—
509C2×C4#120×C6711yescharacteristic
510C2×C4#124×C6712yescomplemented
511C22—21no—
512C22—22no—
513C22—21no—
514C22—22no—
515C22—22no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—41no—
520C22—41no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—81no—
525C22—88no—
526C22—164no—
527C22C2×#120×C6711yescharacteristic
528C22C2×#124×C6711yescharacteristic
529C22#535×C6711yescharacteristic
530C22#546×C6714yes—
531C4—21no—
532C4—22no—
533C4—22no—
534C4—44no—
535C4—41no—
536C4—41no—
537C4—44no—
538C4—82no—
539C4—84no—
540C4B(4,𝔽2)×C6712yes—
541C2—21no—
542C2—24no—
543C2—21no—
544C2—42no—
545C2—84no—
546C2C67×#243511yescharacteristic
547C2C2×B(4,𝔽2)×C6711yescharacteristic
548C2C67×#302811yescharacteristic
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC24C24×C67
prime radicalC67×#3496C2
LCSG ▷ C24 ▷ 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×#2436C2semidirect
3. C67×#2409C2semidirect
4. C67×#2044C2semidirect
5. C67×#1964C2semidirect
6. C67×#1790C2semidirect
7. C4×#99×C67C2semidirect
9. C23×Q8×C67C22semidirect
23. C2×#99×C67C22semidirect
30. #386×C67C22semidirect
31. #385×C67C22semidirect
32. #380×C67C22semidirect
33. C4×C22⋊C4×C67C22semidirect
34. #342×C67C22semidirect
186. C23×C67#124semidirect
235. C2×C4×C67#124semidirect
266. #25899C67direct (N × H)
276. #2436C2×C67semidirect
277. #2409C2×C67semidirect
278. #2044C2×C67semidirect
279. #1964C2×C67semidirect
280. #1790C2×C67semidirect
281. C4×#99C2×C67semidirect
282. C67#25899direct (N × H)
284. C23×Q8C22×C67semidirect
298. C2×#99C22×C67semidirect
305. #386C22×C67semidirect
306. #385C22×C67semidirect
307. #380C22×C67semidirect
308. C4×C22⋊C4C22×C67semidirect
309. #342C22×C67semidirect
461. C23#124×C67semidirect
510. C2×C4#124×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34962
C67×#24362
C67×#24092
C67×#20442
C67×#19642
C67×#17902
C4×#99×C672
#2589967