Finite Group Databasebeta

← C67×#17392 #30696643

Subgroups — C67×#17392 #30696643

Order 17152.

1450 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
1C22×#123×C67C211yesmaximal, characteristic
2C2×#407×C67C212yesmaximal, complemented
3C67×#1468C218yesmaximal, complemented
4C67×#1463C212yesmaximal, complemented
5C67×#1462C212yesmaximal, complemented
6C23×D4×C67C411yescharacteristic
7C24×C4×C67C2211yescharacteristic
8C22×M16×C67—22no—
9C2×#123×C67—22no—
10C2×#123×C67—22no—
11C2×#123×C67—22no—
12C2×#123×C67—22no—
13C2×#123×C67C414yes—
14C2×#123×C67C2214yes—
15C22×C4⋊C4×C67C411yescharacteristic
16C22×C4⋊C4×C67C2211yescharacteristic
17C22×C22⋊C4×C67C411yescharacteristic
18C22×C22⋊C4×C67C411yescharacteristic
19C22×C22⋊C4×C67C2211yescharacteristic
20#407×C67—22no—
21#407×C67—22no—
22#407×C67C2214yes—
23#407×C67C2214yescomplemented
24C2×C22⋊C8×C67—22no—
25C2×C22⋊C8×C67C2214yescomplemented
26#324×C67C2218yescomplemented
27#323×C67C2218yescomplemented
28C25×C67C2×C411yescharacteristic
29C25×C67C2×C411yescharacteristic
30C22×D4×C67—22no—
31C22×D4×C67—21no—
32C22×D4×C67—21no—
33C22×D4×C67—44no—
34C22×D4×C67C2×C414yes—
35C23×C4×C67—44no—
36C23×C4×C67C2×C411yescharacteristic
37C23×C4×C67C2×C411yescharacteristic
38C23×C4×C67C2×C411yescharacteristic
39C23×C4×C67C2×C411yescharacteristic
40C23×C4×C67C2×C412yes—
41C23×C4×C67D412yes—
42C23×C4×C67D412yes—
43C23×C4×C67D412yes—
44C23×C4×C67D412yes—
45C23×C4×C67C2311yescharacteristic
46C23×C4×C67C2312yes—
47C2×M16×C67—24no—
48C2×M16×C67—24no—
49C2×M16×C67—48no—
50C22×C8×C67—24no—
51#123×C67—24no—
52#123×C67—24no—
53#123×C67—24no—
54#123×C67—24no—
55#123×C67—24no—
56#123×C67—24no—
57#123×C67—24no—
58#123×C67—24no—
59C2×C4⋊C4×C67—22no—
60C2×C4⋊C4×C67—21no—
61C2×C4⋊C4×C67—21no—
62C2×C4⋊C4×C67—22no—
63C2×C4⋊C4×C67—22no—
64C2×C4⋊C4×C67C2×C414yes—
65C2×C4⋊C4×C67C2×C412yes—
66C2×C4⋊C4×C67C2312yes—
67C2×C22⋊C4×C67—22no—
68C2×C22⋊C4×C67—21no—
69C2×C22⋊C4×C67—21no—
70C2×C22⋊C4×C67—22no—
71C2×C22⋊C4×C67—21no—
72C2×C22⋊C4×C67—21no—
73C2×C22⋊C4×C67—22no—
74C2×C22⋊C4×C67—22no—
75C2×C22⋊C4×C67C2×C414yes—
76C2×C22⋊C4×C67C2×C414yes—
77C2×C22⋊C4×C67C2×C412yes—
78C2×C22⋊C4×C67C2312yes—
79C22⋊C8×C67—24no—
80C22⋊C8×C67—24no—
81C22⋊C8×C67—24no—
82C22⋊C8×C67C2318yes—
83C24×C67—24no—
84C24×C67—21no—
85C24×C67—21no—
86C24×C67—42no—
87C24×C67—42no—
88C24×C67—42no—
89C24×C67—42no—
90C24×C67—42no—
91C24×C67C22⋊C412yes—
92C24×C67C22⋊C411yescharacteristic
93C24×C67C22⋊C411yescharacteristic
94C24×C67C22×C411yescharacteristic
95C24×C67C22×C412yes—
96C24×C67C22×C412yes—
97C2×D4×C67—24no—
98C2×D4×C67—24no—
99C2×D4×C67—48no—
100C2×D4×C67—44no—
101C2×D4×C67—44no—
102C2×D4×C67—48no—
103C22×C4×C67—28no—
104C22×C4×C67—21no—
105C22×C4×C67—21no—
106C22×C4×C67—21no—
107C22×C4×C67—21no—
108C22×C4×C67—21no—
109C22×C4×C67—21no—
110C22×C4×C67—42no—
111C22×C4×C67—44no—
112C22×C4×C67—44no—
113C22×C4×C67—44no—
114C22×C4×C67—44no—
115C22×C4×C67—44no—
116C22×C4×C67—42no—
117C22×C4×C67—42no—
118C22×C4×C67—42no—
119C22×C4×C67—42no—
120C22×C4×C67C22⋊C412yes—
121C22×C4×C67C22⋊C412yes—
122C22×C4×C67C22⋊C412yes—
123C22×C4×C67C22⋊C411yescharacteristic
124C22×C4×C67C22⋊C411yescharacteristic
125C22×C4×C67C22⋊C412yes—
126C22×C4×C67C22⋊C412yes—
127C22×C4×C67C22×C412yes—
128C22×C4×C67C22×C412yes—
129C22×C4×C67C22×C412yes—
130C22×C4×C67C22×C412yes—
131C22×C4×C67C22×C411yescharacteristic
132C22×C4×C67C2×D412yes—
133C22×C4×C67C2×D412yes—
134C22×C4×C67C2×D412yes—
135C22×C4×C67C2×D412yes—
136C22×C4×C67C2×D412yes—
137C22×C4×C67C2×D412yes—
138C22×C4×C67C2411yescharacteristic
139M16×C67—48no—
140M16×C67—48no—
141C2×C8×C67—28no—
142C2×C8×C67—48no—
143C4⋊C4×C67—24no—
144C4⋊C4×C67—24no—
145C4⋊C4×C67—24no—
146C4⋊C4×C67—24no—
147C22⋊C4×C67—24no—
148C22⋊C4×C67—24no—
149C22⋊C4×C67—24no—
150C22⋊C4×C67—24no—
151C22⋊C4×C67—24no—
152C22⋊C4×C67—24no—
153C23×C67—22no—
154C23×C67—22no—
155C23×C67—22no—
156C23×C67—24no—
157C23×C67—24no—
158C23×C67—22no—
159C23×C67—24no—
160C23×C67—24no—
161C23×C67—22no—
162C23×C67—24no—
163C23×C67—41no—
164C23×C67—42no—
165C23×C67—41no—
166C23×C67—41no—
167C23×C67—41no—
168C23×C67—42no—
169C23×C67—416no—
170C23×C67—42no—
171C23×C67—42no—
172C23×C67—42no—
173C23×C67—42no—
174C23×C67—42no—
175C23×C67—42no—
176C23×C67—42no—
177C23×C67—42no—
178C23×C67—88no—
179C23×C67#9914yes—
180C23×C67C2×C22⋊C412yes—
181C23×C67C2×C22⋊C412yes—
182C23×C67C2×C22⋊C411yescharacteristic
183C23×C67C2×C22⋊C411yescharacteristic
184C23×C67C23×C411yescharacteristic
185D4×C67—48no—
186D4×C67—48no—
187C2×C4×C67—24no—
188C2×C4×C67—24no—
189C2×C4×C67—24no—
190C2×C4×C67—24no—
191C2×C4×C67—24no—
192C2×C4×C67—24no—
193C2×C4×C67—22no—
194C2×C4×C67—22no—
195C2×C4×C67—22no—
196C2×C4×C67—24no—
197C2×C4×C67—41no—
198C2×C4×C67—41no—
199C2×C4×C67—42no—
200C2×C4×C67—42no—
201C2×C4×C67—42no—
202C2×C4×C67—41no—
203C2×C4×C67—42no—
204C2×C4×C67—42no—
205C2×C4×C67—42no—
206C2×C4×C67—41no—
207C2×C4×C67—42no—
208C2×C4×C67—42no—
209C2×C4×C67—42no—
210C2×C4×C67—42no—
211C2×C4×C67—42no—
212C2×C4×C67—42no—
213C2×C4×C67—42no—
214C2×C4×C67—42no—
215C2×C4×C67C2×C22⋊C412yes—
216C2×C4×C67C2×C22⋊C412yes—
217C2×C4×C67C2×C22⋊C412yes—
218C2×C4×C67C22×D412yes—
219C8×C67—48no—
220C22×C67—28no—
221C22×C67—21no—
222C22×C67—21no—
223C22×C67—22no—
224C22×C67—21no—
225C22×C67—21no—
226C22×C67—21no—
227C22×C67—21no—
228C22×C67—22no—
229C22×C67—22no—
230C22×C67—22no—
231C22×C67—22no—
232C22×C67—44no—
233C22×C67—48no—
234C22×C67—48no—
235C22×C67—44no—
236C22×C67—42no—
237C22×C67—42no—
238C22×C67—42no—
239C22×C67—84no—
240C22×C67—84no—
241C22×C67—84no—
242C22×C67C2×#9912yes—
243C22×C67C2×#9914yes—
244C22×C67#41312yes—
245C22×C67C22×C22⋊C411yescharacteristic
246C22×C67#56812yes—
247C4×C67—22no—
248C4×C67—22no—
249C4×C67—22no—
250C4×C67—22no—
251C4×C67—42no—
252C4×C67—42no—
253C4×C67—42no—
254#17392C6711yesmaximal, characteristic, direct product factor, Sylow
255C2×C67—22no—
256C2×C67—22no—
257C2×C67—22no—
258C2×C67—24no—
259C2×C67—42no—
260C2×C67—82no—
261C2×C67#287911yescharacteristic
262C2×C67C22×#9911yescharacteristic
263C2×C67C2×#41311yescharacteristic
264C22×#123C2×C6711yescharacteristic
265C2×#407C2×C6712yescomplemented
266#1468C2×C6718yescomplemented
267#1463C2×C6712yescomplemented
268#1462C2×C6712yescomplemented
269C67#1739211yescharacteristic, direct product factor, Sylow
270C23×D4C4×C6711yescharacteristic
271C24×C4C22×C6711yescharacteristic
272C22×M16—22no—
273C2×#123—22no—
274C2×#123—22no—
275C2×#123—22no—
276C2×#123—22no—
277C2×#123C4×C6714yes—
278C2×#123C22×C6714yes—
279C22×C4⋊C4C4×C6711yescharacteristic
280C22×C4⋊C4C22×C6711yescharacteristic
281C22×C22⋊C4C4×C6711yescharacteristic
282C22×C22⋊C4C4×C6711yescharacteristic
283C22×C22⋊C4C22×C6711yescharacteristic
284#407—22no—
285#407—22no—
286#407C22×C6714yes—
287#407C22×C6714yescomplemented
288C2×C22⋊C8—22no—
289C2×C22⋊C8C22×C6714yescomplemented
290#324C22×C6718yescomplemented
291#323C22×C6718yescomplemented
292C25C2×C4×C6711yescharacteristic
293C25C2×C4×C6711yescharacteristic
294C22×D4—22no—
295C22×D4—21no—
296C22×D4—21no—
297C22×D4—44no—
298C22×D4C2×C4×C6714yes—
299C23×C4—44no—
300C23×C4C2×C4×C6711yescharacteristic
301C23×C4C2×C4×C6711yescharacteristic
302C23×C4C2×C4×C6711yescharacteristic
303C23×C4C2×C4×C6711yescharacteristic
304C23×C4C2×C4×C6712yes—
305C23×C4D4×C6712yes—
306C23×C4D4×C6712yes—
307C23×C4D4×C6712yes—
308C23×C4D4×C6712yes—
309C23×C4C23×C6711yescharacteristic
310C23×C4C23×C6712yes—
311C2×M16—24no—
312C2×M16—24no—
313C2×M16—48no—
314C22×C8—24no—
315#123—24no—
316#123—24no—
317#123—24no—
318#123—24no—
319#123—24no—
320#123—24no—
321#123—24no—
322#123—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4C2×C4×C6714yes—
329C2×C4⋊C4C2×C4×C6712yes—
330C2×C4⋊C4C23×C6712yes—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4C2×C4×C6714yes—
340C2×C22⋊C4C2×C4×C6714yes—
341C2×C22⋊C4C2×C4×C6712yes—
342C2×C22⋊C4C23×C6712yes—
343C22⋊C8—24no—
344C22⋊C8—24no—
345C22⋊C8—24no—
346C22⋊C8C23×C6718yes—
347C24—24no—
348C24—21no—
349C24—21no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24—42no—
355C24C22⋊C4×C6712yes—
356C24C22⋊C4×C6711yescharacteristic
357C24C22⋊C4×C6711yescharacteristic
358C24C22×C4×C6711yescharacteristic
359C24C22×C4×C6712yes—
360C24C22×C4×C6712yes—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—48no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—48no—
367C22×C4—28no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4C22⋊C4×C6712yes—
385C22×C4C22⋊C4×C6712yes—
386C22×C4C22⋊C4×C6712yes—
387C22×C4C22⋊C4×C6711yescharacteristic
388C22×C4C22⋊C4×C6711yescharacteristic
389C22×C4C22⋊C4×C6712yes—
390C22×C4C22⋊C4×C6712yes—
391C22×C4C22×C4×C6712yes—
392C22×C4C22×C4×C6712yes—
393C22×C4C22×C4×C6712yes—
394C22×C4C22×C4×C6712yes—
395C22×C4C22×C4×C6711yescharacteristic
396C22×C4C2×D4×C6712yes—
397C22×C4C2×D4×C6712yes—
398C22×C4C2×D4×C6712yes—
399C22×C4C2×D4×C6712yes—
400C22×C4C2×D4×C6712yes—
401C22×C4C2×D4×C6712yes—
402C22×C4C24×C6711yescharacteristic
403M16—48no—
404M16—48no—
405C2×C8—28no—
406C2×C8—48no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—22no—
423C23—24no—
424C23—24no—
425C23—22no—
426C23—24no—
427C23—41no—
428C23—42no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—416no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—88no—
443C23#99×C6714yes—
444C23C2×C22⋊C4×C6712yes—
445C23C2×C22⋊C4×C6712yes—
446C23C2×C22⋊C4×C6711yescharacteristic
447C23C2×C22⋊C4×C6711yescharacteristic
448C23C23×C4×C6711yescharacteristic
449D4—48no—
450D4—48no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4C2×C22⋊C4×C6712yes—
480C2×C4C2×C22⋊C4×C6712yes—
481C2×C4C2×C22⋊C4×C6712yes—
482C2×C4C22×D4×C6712yes—
483C8—48no—
484C22—28no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—44no—
497C22—48no—
498C22—48no—
499C22—44no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—84no—
504C22—84no—
505C22—84no—
506C22C2×#99×C6712yes—
507C22C2×#99×C6714yes—
508C22#413×C6712yes—
509C22C22×C22⋊C4×C6711yescharacteristic
510C22#568×C6712yes—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—22no—
515C4—42no—
516C4—42no—
517C4—42no—
518C2—22no—
519C2—22no—
520C2—22no—
521C2—24no—
522C2—42no—
523C2—82no—
524C2C67×#287911yescharacteristic
525C2C22×#99×C6711yescharacteristic
526C2C2×#413×C6711yescharacteristic
subgroupG/H
center, socleC22×C67C22×C22⋊C4
commutatorC23C23×C4×C67
Frattini, cosocleC22×C4C24×C67
prime radicalC23×D4×C67C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C24×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#407×C67C2semidirect
3. C67×#1468C2semidirect
4. C67×#1463C2semidirect
5. C67×#1462C2semidirect
23. #407×C67C22semidirect
25. C2×C22⋊C8×C67C22semidirect
26. #324×C67C22semidirect
27. #323×C67C22semidirect
254. #17392C67direct (N × H)
265. C2×#407C2×C67semidirect
266. #1468C2×C67semidirect
267. #1463C2×C67semidirect
268. #1462C2×C67semidirect
269. C67#17392direct (N × H)
287. #407C22×C67semidirect
289. C2×C22⋊C8C22×C67semidirect
290. #324C22×C67semidirect
291. #323C22×C67semidirect

Maximal subgroups

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

subgroupindex
C22×#123×C672
C2×#407×C672
C67×#14682
C67×#14632
C67×#14622
#1739267