Finite Group Databasebeta

← C67×#25900 #30705151

Subgroups — C67×#25900 #30705151

Order 17152.

1406 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3496C211yesmaximal, characteristic
2C67×#2504C212yesmaximal, complemented
3C67×#2487C211yesmaximal, characteristic, complemented
4C67×#2044C214yesmaximal, complemented
5C67×#1964C212yesmaximal, complemented
6C67×#1790C214yesmaximal, complemented
7C4×#99×C67C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C67C2212yes—
9#547×C67—22no—
10#546×C67—24no—
11D42×C67—22no—
12#537×C67—21no—
13#534×C67—21no—
14#534×C67C2212yes—
15#532×C67—21no—
16C2×#121×C67—21no—
17C2×#121×C67C2212yes—
18C2×C4×D4×C67C2211yescharacteristic
19C22×C4⋊C4×C67C2211yescharacteristic, complemented
20C2×#99×C67C2212yes—
21C2×#99×C67C2218yescomplemented
22#400×C67—21no—
23#399×C67—21no—
24#397×C67—24no—
25#390×C67—22no—
26#389×C67—21no—
27#386×C67C2212yes—
28#386×C67C2218yescomplemented
29#385×C67C2212yescomplemented
30#382×C67—22no—
31#380×C67—21no—
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×D4×C67—22no—
40C22×D4×C67—24no—
41C22×D4×C67—21no—
42C22×D4×C67D412yes—
43C22×D4×C67C2314yes—
44C22×D4×C67C2311yescharacteristic
45C23×C4×C67D411yescharacteristic
46C23×C4×C67C2312yes—
47C23×C4×C67C2311yescharacteristic
48C4⋊Q8×C67—21no—
49#127×C67—21no—
50#124×C67—22no—
51#123×C67—24no—
52#123×C67—24no—
53C22⋊Q8×C67—24no—
54#121×C67—22no—
55#121×C67—22no—
56#121×C67—24no—
57#121×C67—22no—
58#121×C67—22no—
59#121×C67—22no—
60#121×C67—22no—
61#121×C67—22no—
62#121×C67—22no—
63#120×C67—24no—
64#120×C67—24no—
65C4×D4×C67—22no—
66C4×D4×C67—22no—
67C4×D4×C67—22no—
68C4×D4×C67—22no—
69C2×C4⋊C4×C67—21no—
70C2×C4⋊C4×C67—21no—
71C2×C4⋊C4×C67—22no—
72C2×C4⋊C4×C67—22no—
73C2×C4⋊C4×C67—21no—
74C2×C4⋊C4×C67—21no—
75C2×C4⋊C4×C67—22no—
76C2×C4⋊C4×C67D411yescharacteristic
77C2×C4⋊C4×C67D412yes—
78C2×C22⋊C4×C67—22no—
79C2×C22⋊C4×C67—22no—
80C2×C22⋊C4×C67—28no—
81C2×C22⋊C4×C67D411yescharacteristic
82C2×C22⋊C4×C67D414yes—
83C2×C22⋊C4×C67C2314yes—
84C2×C22⋊C4×C67C2312yes—
85C2×C22⋊C4×C67C2311yescharacteristic
86C2×C42×C67—22no—
87C2×C42×C67D411yescharacteristic
88#99×C67—24no—
89#99×C67—48no—
90#95×C67—22no—
91#95×C67—22no—
92#95×C67—21no—
93#95×C67—28no—
94#95×C67—22no—
95C24×C67—22no—
96C24×C67C2×D414yes—
97C24×C67C2×D411yescharacteristic
98C24×C67C2411yescharacteristic
99#41×C67—416no—
100#41×C67—44no—
101#41×C67—44no—
102C2×Q8×C67—22no—
103C2×D4×C67—22no—
104C2×D4×C67—22no—
105C2×D4×C67—24no—
106C2×D4×C67—22no—
107C2×D4×C67—24no—
108C2×D4×C67—24no—
109C2×D4×C67—28no—
110C2×D4×C67—28no—
111C2×D4×C67—22no—
112C2×D4×C67—22no—
113C2×D4×C67—22no—
114C2×D4×C67—22no—
115C2×D4×C67—44no—
116C2×D4×C67—416no—
117C2×D4×C67—44no—
118C2×D4×C67—44no—
119C2×D4×C67—82no—
120C2×D4×C67#4114yes—
121C22×C4×C67—22no—
122C22×C4×C67—24no—
123C22×C4×C67—21no—
124C22×C4×C67—21no—
125C22×C4×C67—24no—
126C22×C4×C67—24no—
127C22×C4×C67—22no—
128C22×C4×C67—22no—
129C22×C4×C67—42no—
130C22×C4×C67—42no—
131C22×C4×C67—82no—
132C22×C4×C67C2×D414yes—
133C22×C4×C67C2×D412yes—
134C22×C4×C67C2×D411yescharacteristic
135C22×C4×C67C2×D411yescharacteristic
136C22×C4×C67C2×D411yescharacteristic
137C22×C4×C67C2×D411yescharacteristic
138C22×C4×C67C2×D412yes—
139C22×C4×C67C2×D411yescharacteristic
140C4⋊C4×C67—22no—
141C4⋊C4×C67—22no—
142C4⋊C4×C67—22no—
143C4⋊C4×C67—22no—
144C4⋊C4×C67—22no—
145C4⋊C4×C67—42no—
146C4⋊C4×C67—41no—
147C4⋊C4×C67—42no—
148C4⋊C4×C67—41no—
149C4⋊C4×C67—82no—
150C22⋊C4×C67—24no—
151C22⋊C4×C67—24no—
152C22⋊C4×C67—22no—
153C22⋊C4×C67—24no—
154C22⋊C4×C67—22no—
155C22⋊C4×C67—24no—
156C22⋊C4×C67—24no—
157C22⋊C4×C67—24no—
158C22⋊C4×C67—24no—
159C22⋊C4×C67—88no—
160C22⋊C4×C67—82no—
161C42×C67—21no—
162C42×C67—21no—
163C42×C67—44no—
164C23×C67—21no—
165C23×C67—21no—
166C23×C67—24no—
167C23×C67—28no—
168C23×C67—41no—
169C23×C67—41no—
170C23×C67—41no—
171C23×C67—41no—
172C23×C67—44no—
173C23×C67—44no—
174C23×C67—44no—
175C23×C67—82no—
176C23×C67—82no—
177C23×C67#12412yes—
178C23×C67C22×D411yescharacteristic
179C23×C67C22×D412yes—
180C23×C67C2×#4112yes—
181C23×C67H(5,𝔽2)12yes—
182Q8×C67—44no—
183D4×C67—28no—
184D4×C67—44no—
185D4×C67—48no—
186D4×C67—44no—
187D4×C67—44no—
188D4×C67—48no—
189D4×C67—44no—
190D4×C67—82no—
191D4×C67—82no—
192D4×C67—82no—
193C2×C4×C67—21no—
194C2×C4×C67—21no—
195C2×C4×C67—24no—
196C2×C4×C67—22no—
197C2×C4×C67—21no—
198C2×C4×C67—21no—
199C2×C4×C67—24no—
200C2×C4×C67—28no—
201C2×C4×C67—42no—
202C2×C4×C67—41no—
203C2×C4×C67—48no—
204C2×C4×C67—44no—
205C2×C4×C67—42no—
206C2×C4×C67—41no—
207C2×C4×C67—41no—
208C2×C4×C67—44no—
209C2×C4×C67—44no—
210C2×C4×C67—42no—
211C2×C4×C67—42no—
212C2×C4×C67—48no—
213C2×C4×C67—42no—
214C2×C4×C67—42no—
215C2×C4×C67—42no—
216C2×C4×C67—42no—
217C2×C4×C67—44no—
218C2×C4×C67—81no—
219C2×C4×C67#12011yescharacteristic
220C2×C4×C67#12012yes—
221C2×C4×C67#12011yescharacteristic
222C2×C4×C67#12412yes—
223C22×C67—21no—
224C22×C67—22no—
225C22×C67—21no—
226C22×C67—22no—
227C22×C67—22no—
228C22×C67—42no—
229C22×C67—42no—
230C22×C67—42no—
231C22×C67—41no—
232C22×C67—41no—
233C22×C67—44no—
234C22×C67—44no—
235C22×C67—44no—
236C22×C67—81no—
237C22×C67—88no—
238C22×C67—164no—
239C22×C67C2×#12011yescharacteristic
240C22×C67C2×#12411yescharacteristic
241C22×C67#53511yescharacteristic
242C22×C67#54614yes—
243C4×C67—22no—
244C4×C67—21no—
245C4×C67—22no—
246C4×C67—44no—
247C4×C67—41no—
248C4×C67—41no—
249C4×C67—82no—
250C4×C67—84no—
251C4×C67—82no—
252C4×C67#45812yes—
253#25900C6711yesmaximal, characteristic, direct product factor, Sylow
254C2×C67—21no—
255C2×C67—24no—
256C2×C67—21no—
257C2×C67—42no—
258C2×C67—84no—
259C2×C67#243511yescharacteristic
260C2×C67C2×#45811yescharacteristic
261C2×C67#302711yescharacteristic
262#3496C2×C6711yescharacteristic
263#2504C2×C6712yescomplemented
264#2487C2×C6711yescharacteristic, complemented
265#2044C2×C6714yescomplemented
266#1964C2×C6712yescomplemented
267#1790C2×C6714yescomplemented
268C4×#99C2×C6711yescharacteristic, complemented
269C67#2590011yescharacteristic, direct product factor, Sylow
270C2×H(5,𝔽2)C22×C6712yes—
271#547—22no—
272#546—24no—
273D42—22no—
274#537—21no—
275#534—21no—
276#534C22×C6712yes—
277#532—21no—
278C2×#121—21no—
279C2×#121C22×C6712yes—
280C2×C4×D4C22×C6711yescharacteristic
281C22×C4⋊C4C22×C6711yescharacteristic, complemented
282C2×#99C22×C6712yes—
283C2×#99C22×C6718yescomplemented
284#400—21no—
285#399—21no—
286#397—24no—
287#390—22no—
288#389—21no—
289#386C22×C6712yes—
290#386C22×C6718yescomplemented
291#385C22×C6712yescomplemented
292#382—22no—
293#380—21no—
294#380C22×C6711yescharacteristic, complemented
295C4×C22⋊C4C22×C6712yescomplemented
296#342C22×C6712yescomplemented
297H(5,𝔽2)—28no—
298H(5,𝔽2)—44no—
299C2×#41—22no—
300C2×#41—24no—
301C22×D4—22no—
302C22×D4—24no—
303C22×D4—21no—
304C22×D4D4×C6712yes—
305C22×D4C23×C6714yes—
306C22×D4C23×C6711yescharacteristic
307C23×C4D4×C6711yescharacteristic
308C23×C4C23×C6712yes—
309C23×C4C23×C6711yescharacteristic
310C4⋊Q8—21no—
311#127—21no—
312#124—22no—
313#123—24no—
314#123—24no—
315C22⋊Q8—24no—
316#121—22no—
317#121—22no—
318#121—24no—
319#121—22no—
320#121—22no—
321#121—22no—
322#121—22no—
323#121—22no—
324#121—22no—
325#120—24no—
326#120—24no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4D4×C6711yescharacteristic
339C2×C4⋊C4D4×C6712yes—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—28no—
343C2×C22⋊C4D4×C6711yescharacteristic
344C2×C22⋊C4D4×C6714yes—
345C2×C22⋊C4C23×C6714yes—
346C2×C22⋊C4C23×C6712yes—
347C2×C22⋊C4C23×C6711yescharacteristic
348C2×C42—22no—
349C2×C42D4×C6711yescharacteristic
350#99—24no—
351#99—48no—
352#95—22no—
353#95—22no—
354#95—21no—
355#95—28no—
356#95—22no—
357C24—22no—
358C24C2×D4×C6714yes—
359C24C2×D4×C6711yescharacteristic
360C24C24×C6711yescharacteristic
361#41—416no—
362#41—44no—
363#41—44no—
364C2×Q8—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—22no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—44no—
378C2×D4—416no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—82no—
382C2×D4#41×C6714yes—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—82no—
394C22×C4C2×D4×C6714yes—
395C22×C4C2×D4×C6712yes—
396C22×C4C2×D4×C6711yescharacteristic
397C22×C4C2×D4×C6711yescharacteristic
398C22×C4C2×D4×C6711yescharacteristic
399C22×C4C2×D4×C6711yescharacteristic
400C22×C4C2×D4×C6712yes—
401C22×C4C2×D4×C6711yescharacteristic
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—41no—
411C4⋊C4—82no—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—22no—
415C22⋊C4—24no—
416C22⋊C4—22no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—88no—
422C22⋊C4—82no—
423C42—21no—
424C42—21no—
425C42—44no—
426C23—21no—
427C23—21no—
428C23—24no—
429C23—28no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—82no—
438C23—82no—
439C23#124×C6712yes—
440C23C22×D4×C6711yescharacteristic
441C23C22×D4×C6712yes—
442C23C2×#41×C6712yes—
443C23H(5,𝔽2)×C6712yes—
444Q8—44no—
445D4—28no—
446D4—44no—
447D4—48no—
448D4—44no—
449D4—44no—
450D4—48no—
451D4—44no—
452D4—82no—
453D4—82no—
454D4—82no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—48no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—48no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—81no—
481C2×C4#120×C6711yescharacteristic
482C2×C4#120×C6712yes—
483C2×C4#120×C6711yescharacteristic
484C2×C4#124×C6712yes—
485C22—21no—
486C22—22no—
487C22—21no—
488C22—22no—
489C22—22no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—41no—
494C22—41no—
495C22—44no—
496C22—44no—
497C22—44no—
498C22—81no—
499C22—88no—
500C22—164no—
501C22C2×#120×C6711yescharacteristic
502C22C2×#124×C6711yescharacteristic
503C22#535×C6711yescharacteristic
504C22#546×C6714yes—
505C4—21no—
506C4—22no—
507C4—22no—
508C4—41no—
509C4—41no—
510C4—44no—
511C4—82no—
512C4—84no—
513C4—82no—
514C4#458×C6712yes—
515C2—21no—
516C2—24no—
517C2—21no—
518C2—42no—
519C2—84no—
520C2C67×#243511yescharacteristic
521C2C2×#458×C6711yescharacteristic
522C2C67×#302711yescharacteristic
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×#2504C2semidirect
3. C67×#2487C2semidirect
4. C67×#2044C2semidirect
5. C67×#1964C2semidirect
6. C67×#1790C2semidirect
7. C4×#99×C67C2semidirect
19. C22×C4⋊C4×C67C22semidirect
21. C2×#99×C67C22semidirect
28. #386×C67C22semidirect
29. #385×C67C22semidirect
32. #380×C67C22semidirect
33. C4×C22⋊C4×C67C22semidirect
34. #342×C67C22semidirect
253. #25900C67direct (N × H)
263. #2504C2×C67semidirect
264. #2487C2×C67semidirect
265. #2044C2×C67semidirect
266. #1964C2×C67semidirect
267. #1790C2×C67semidirect
268. C4×#99C2×C67semidirect
269. C67#25900direct (N × H)
281. C22×C4⋊C4C22×C67semidirect
283. C2×#99C22×C67semidirect
290. #386C22×C67semidirect
291. #385C22×C67semidirect
294. #380C22×C67semidirect
295. C4×C22⋊C4C22×C67semidirect
296. #342C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34962
C67×#25042
C67×#24872
C67×#20442
C67×#19642
C67×#17902
C4×#99×C672
#2590067