Finite Group Databasebeta

← C2×C67×#1609 #30697356

Subgroups — C2×C67×#1609 #30697356

Order 17152.

1060 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#539×C67C211yesmaximal, characteristic
2C2×#483×C67C211yesmaximal, characteristic, complemented
3C2×#480×C67C211yesmaximal, characteristic
4C2×#398×C67C211yesmaximal, characteristic, complemented
5C67×#1609C218yesmaximal, direct product factor
6C2×#328×C67C211yesmaximal, characteristic, complemented
7C2×#327×C67C211yesmaximal, characteristic
8C2×#324×C67C211yesmaximal, characteristic, complemented
9#539×C67—22no—
10#539×C67—22no—
11#539×C67—22no—
12#539×C67C2214yes—
13C2×#126×C67—21no—
14C2×#125×C67—21no—
15C2×#123×C67—21no—
16C2×#123×C67—21no—
17C2×C22⋊Q8×C67—21no—
18C2×C22⋊Q8×C67C2211yescharacteristic
19C2×#121×C67C2211yescharacteristic
20C2×C4×D4×C67—21no—
21C22×C4⋊C4×C67—21no—
22C22×C4⋊C4×C67C2211yescharacteristic
23C22×C4⋊C4×C67C2211yescharacteristic
24#483×C67—22no—
25#483×C67C2214yescomplemented
26#480×C67—22no—
27#480×C67C2214yes—
28C2×#107×C67—21no—
29C2×#106×C67—21no—
30C2×#103×C67—21no—
31C2×#102×C67—21no—
32C2×C22⋊C8×C67C2211yescharacteristic
33C2×C22⋊C8×C67C2211yescharacteristic
34#398×C67—22no—
35#398×C67C2214yescomplemented
36C2×#95×C67—21no—
37C2×#95×C67C2211yescharacteristic
38#328×C67C2214yescomplemented
39#327×C67C2214yes—
40#324×C67C2214yescomplemented
41C22×Q8×C67D411yescharacteristic
42C22×D4×C67—21no—
43C22×D4×C67D411yescharacteristic
44C23×C4×C67—21no—
45C23×C4×C67—21no—
46C23×C4×C67—21no—
47C23×C4×C67D411yescharacteristic
48C23×C4×C67D411yescharacteristic
49C23×C4×C67C2311yescharacteristic
50C22×C8×C67—21no—
51C22×C8×C67—21no—
52#126×C67—24no—
53#126×C67—24no—
54#125×C67—24no—
55#125×C67—24no—
56#123×C67—24no—
57#123×C67—24no—
58#123×C67—24no—
59#123×C67—24no—
60C22⋊Q8×C67—24no—
61C22⋊Q8×C67—24no—
62C22⋊Q8×C67—21no—
63C22⋊Q8×C67—21no—
64C22⋊Q8×C67D412yes—
65C22⋊Q8×C67C2312yes—
66#121×C67—21no—
67#121×C67—21no—
68#121×C67D412yes—
69#121×C67C2312yes—
70C4×D4×C67—24no—
71C4×D4×C67—24no—
72C2×C4⋊C4×C67—21no—
73C2×C4⋊C4×C67—21no—
74C2×C4⋊C4×C67—21no—
75C2×C4⋊C4×C67—21no—
76C2×C4⋊C4×C67—21no—
77C2×C4⋊C4×C67—21no—
78C2×C4⋊C4×C67—21no—
79C2×C4⋊C4×C67—21no—
80C2×C4⋊C4×C67—21no—
81C2×C4⋊C4×C67—21no—
82C2×C4⋊C4×C67—21no—
83C2×C4⋊C4×C67—21no—
84C2×C4⋊C4×C67—24no—
85C2×C4⋊C4×C67—21no—
86C2×C4⋊C4×C67—44no—
87C2×C4⋊C4×C67D412yes—
88C2×C4⋊C4×C67D411yescharacteristic
89C2×C4⋊C4×C67D411yescharacteristic
90C2×C4⋊C4×C67C2312yes—
91C2×C4⋊C4×C67C2312yes—
92C2×C22⋊C4×C67—21no—
93C2×C22⋊C4×C67—21no—
94C2×C22⋊C4×C67—21no—
95C2×C22⋊C4×C67—21no—
96C2×C22⋊C4×C67—21no—
97C2×C42×C67—21no—
98#107×C67—24no—
99#106×C67—24no—
100#103×C67—24no—
101#102×C67—24no—
102C22⋊C8×C67—21no—
103C22⋊C8×C67—21no—
104C22⋊C8×C67C2312yes—
105C22⋊C8×C67C2312yes—
106#95×C67—24no—
107#95×C67—21no—
108#95×C67C2312yes—
109C24×C67—21no—
110C24×C67C2×D411yescharacteristic
111C2×Q8×C67—21no—
112C2×Q8×C67—42no—
113C2×Q8×C67C2×D412yes—
114C2×D4×C67—22no—
115C2×D4×C67—22no—
116C2×D4×C67—21no—
117C2×D4×C67—42no—
118C2×D4×C67—42no—
119C2×D4×C67—42no—
120C2×D4×C67C2×D412yes—
121C22×C4×C67—24no—
122C22×C4×C67—21no—
123C22×C4×C67—21no—
124C22×C4×C67—21no—
125C22×C4×C67—21no—
126C22×C4×C67—21no—
127C22×C4×C67—21no—
128C22×C4×C67—22no—
129C22×C4×C67—22no—
130C22×C4×C67—22no—
131C22×C4×C67—24no—
132C22×C4×C67—21no—
133C22×C4×C67—22no—
134C22×C4×C67—42no—
135C22×C4×C67—42no—
136C22×C4×C67—42no—
137C22×C4×C67—42no—
138C22×C4×C67—41no—
139C22×C4×C67—41no—
140C22×C4×C67—42no—
141C22×C4×C67—42no—
142C22×C4×C67—82no—
143C22×C4×C67C2×D412yes—
144C22×C4×C67C2×D411yescharacteristic
145C22×C4×C67C2×D411yescharacteristic
146C22×C4×C67C2×D412yes—
147C22×C4×C67C2×D411yescharacteristic
148C22×C4×C67C2×D411yescharacteristic
149C22×C4×C67C2×D411yescharacteristic
150C22×C4×C67C2411yescharacteristic
151C2×C8×C67—22no—
152C2×C8×C67—22no—
153C2×C8×C67—42no—
154C2×C8×C67—42no—
155C4⋊C4×C67—24no—
156C4⋊C4×C67—24no—
157C4⋊C4×C67—24no—
158C4⋊C4×C67—22no—
159C4⋊C4×C67—22no—
160C4⋊C4×C67—22no—
161C4⋊C4×C67—21no—
162C4⋊C4×C67—21no—
163C4⋊C4×C67—22no—
164C4⋊C4×C67—22no—
165C4⋊C4×C67—22no—
166C4⋊C4×C67—22no—
167C4⋊C4×C67—22no—
168C4⋊C4×C67—48no—
169C4⋊C4×C67—41no—
170C4⋊C4×C67—41no—
171C4⋊C4×C67C2×D412yes—
172C4⋊C4×C67C2×D412yes—
173C22⋊C4×C67—24no—
174C22⋊C4×C67—24no—
175C22⋊C4×C67—24no—
176C22⋊C4×C67—22no—
177C22⋊C4×C67—22no—
178C22⋊C4×C67—22no—
179C22⋊C4×C67—22no—
180C42×C67—24no—
181C23×C67—22no—
182C23×C67—22no—
183C23×C67—22no—
184C23×C67—42no—
185C23×C67—42no—
186C23×C67—42no—
187C23×C67#12011yescharacteristic
188C23×C67#12011yescharacteristic
189C23×C67#13512yes—
190C23×C67AGL(1,ℤ/8)12yes—
191C23×C67C22×D411yescharacteristic
192Q8×C67—42no—
193Q8×C67—81no—
194D4×C67—42no—
195D4×C67—42no—
196D4×C67—42no—
197D4×C67—44no—
198D4×C67—81no—
199C2×C4×C67—22no—
200C2×C4×C67—22no—
201C2×C4×C67—24no—
202C2×C4×C67—22no—
203C2×C4×C67—22no—
204C2×C4×C67—22no—
205C2×C4×C67—22no—
206C2×C4×C67—22no—
207C2×C4×C67—22no—
208C2×C4×C67—22no—
209C2×C4×C67—41no—
210C2×C4×C67—41no—
211C2×C4×C67—41no—
212C2×C4×C67—41no—
213C2×C4×C67—48no—
214C2×C4×C67—42no—
215C2×C4×C67—42no—
216C2×C4×C67—42no—
217C2×C4×C67—42no—
218C2×C4×C67—42no—
219C2×C4×C67—42no—
220C2×C4×C67—42no—
221C2×C4×C67—42no—
222C2×C4×C67—42no—
223C2×C4×C67—42no—
224C2×C4×C67—42no—
225C2×C4×C67—42no—
226C2×C4×C67—81no—
227C2×C4×C67—84no—
228C2×C4×C67—82no—
229C2×C4×C67—82no—
230C2×C4×C67#12011yescharacteristic
231C2×C4×C67#12011yescharacteristic
232C2×C4×C67C22×D411yescharacteristic
233C2×C4×C67C22×D411yescharacteristic
234C8×C67—42no—
235C8×C67—42no—
236C22×C67—24no—
237C22×C67—24no—
238C22×C67—21no—
239C22×C67—21no—
240C22×C67—21no—
241C22×C67—21no—
242C22×C67—42no—
243C22×C67—42no—
244C22×C67—42no—
245C22×C67—84no—
246C22×C67#44912yes—
247C22×C67#44912yes—
248C22×C67#45612yes—
249C22×C67#45812yes—
250C22×C67C2×#12011yescharacteristic
251C22×C67C2×#13511yescharacteristic
252C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
253C4×C67—21no—
254C4×C67—21no—
255C4×C67—21no—
256C4×C67—21no—
257C4×C67—44no—
258C4×C67—42no—
259C4×C67—42no—
260C4×C67—42no—
261C4×C67—42no—
262C4×C67—42no—
263C4×C67—82no—
264C2×#1609C6711yesmaximal, characteristic, direct product factor, Sylow
265C2×C67—21no—
266C2×C67—21no—
267C2×C67—22no—
268C2×C67—82no—
269C2×C67#160914yesdirect product factor
270C2×C67C2×#44911yescharacteristic
271C2×C67C2×#45611yescharacteristic
272C2×C67C2×#45811yescharacteristic
273C2×#539C2×C6711yescharacteristic
274C2×#483C2×C6711yescharacteristic, complemented
275C2×#480C2×C6711yescharacteristic
276C2×#398C2×C6711yescharacteristic, complemented
277#1609C2×C6718yesdirect product factor
278C2×#328C2×C6711yescharacteristic, complemented
279C2×#327C2×C6711yescharacteristic
280C2×#324C2×C6711yescharacteristic, complemented
281C67C2×#160911yescharacteristic, direct product factor, Sylow
282#539—22no—
283#539—22no—
284#539—22no—
285#539C22×C6714yes—
286C2×#126—21no—
287C2×#125—21no—
288C2×#123—21no—
289C2×#123—21no—
290C2×C22⋊Q8—21no—
291C2×C22⋊Q8C22×C6711yescharacteristic
292C2×#121C22×C6711yescharacteristic
293C2×C4×D4—21no—
294C22×C4⋊C4—21no—
295C22×C4⋊C4C22×C6711yescharacteristic
296C22×C4⋊C4C22×C6711yescharacteristic
297#483—22no—
298#483C22×C6714yescomplemented
299#480—22no—
300#480C22×C6714yes—
301C2×#107—21no—
302C2×#106—21no—
303C2×#103—21no—
304C2×#102—21no—
305C2×C22⋊C8C22×C6711yescharacteristic
306C2×C22⋊C8C22×C6711yescharacteristic
307#398—22no—
308#398C22×C6714yescomplemented
309C2×#95—21no—
310C2×#95C22×C6711yescharacteristic
311#328C22×C6714yescomplemented
312#327C22×C6714yes—
313#324C22×C6714yescomplemented
314C22×Q8D4×C6711yescharacteristic
315C22×D4—21no—
316C22×D4D4×C6711yescharacteristic
317C23×C4—21no—
318C23×C4—21no—
319C23×C4—21no—
320C23×C4D4×C6711yescharacteristic
321C23×C4D4×C6711yescharacteristic
322C23×C4C23×C6711yescharacteristic
323C22×C8—21no—
324C22×C8—21no—
325#126—24no—
326#126—24no—
327#125—24no—
328#125—24no—
329#123—24no—
330#123—24no—
331#123—24no—
332#123—24no—
333C22⋊Q8—24no—
334C22⋊Q8—24no—
335C22⋊Q8—21no—
336C22⋊Q8—21no—
337C22⋊Q8D4×C6712yes—
338C22⋊Q8C23×C6712yes—
339#121—21no—
340#121—21no—
341#121D4×C6712yes—
342#121C23×C6712yes—
343C4×D4—24no—
344C4×D4—24no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4D4×C6712yes—
361C2×C4⋊C4D4×C6711yescharacteristic
362C2×C4⋊C4D4×C6711yescharacteristic
363C2×C4⋊C4C23×C6712yes—
364C2×C4⋊C4C23×C6712yes—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C42—21no—
371#107—24no—
372#106—24no—
373#103—24no—
374#102—24no—
375C22⋊C8—21no—
376C22⋊C8—21no—
377C22⋊C8C23×C6712yes—
378C22⋊C8C23×C6712yes—
379#95—24no—
380#95—21no—
381#95C23×C6712yes—
382C24—21no—
383C24C2×D4×C6711yescharacteristic
384C2×Q8—21no—
385C2×Q8—42no—
386C2×Q8C2×D4×C6712yes—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4C2×D4×C6712yes—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—82no—
416C22×C4C2×D4×C6712yes—
417C22×C4C2×D4×C6711yescharacteristic
418C22×C4C2×D4×C6711yescharacteristic
419C22×C4C2×D4×C6712yes—
420C22×C4C2×D4×C6711yescharacteristic
421C22×C4C2×D4×C6711yescharacteristic
422C22×C4C2×D4×C6711yescharacteristic
423C22×C4C24×C6711yescharacteristic
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—42no—
427C2×C8—42no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—48no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4C2×D4×C6712yes—
445C4⋊C4C2×D4×C6712yes—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C42—24no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23#120×C6711yescharacteristic
461C23#120×C6711yescharacteristic
462C23#135×C6712yes—
463C23AGL(1,ℤ/8)×C6712yes—
464C23C22×D4×C6711yescharacteristic
465Q8—42no—
466Q8—81no—
467D4—42no—
468D4—44no—
469D4—42no—
470D4—42no—
471D4—81no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—81no—
500C2×C4—84no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4#120×C6711yescharacteristic
504C2×C4#120×C6711yescharacteristic
505C2×C4C22×D4×C6711yescharacteristic
506C2×C4C22×D4×C6711yescharacteristic
507C8—42no—
508C8—42no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—24no—
514C22—24no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—84no—
519C22#449×C6712yes—
520C22#449×C6712yes—
521C22#456×C6712yes—
522C22#458×C6712yes—
523C22C2×#120×C6711yescharacteristic
524C22C2×#135×C6711yescharacteristic
525C22C2×AGL(1,ℤ/8)×C6711yescharacteristic
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—44no—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—82no—
537C2—21no—
538C2—22no—
539C2—21no—
540C2—82no—
541C2C67×#160914yesdirect product factor
542C2C2×#449×C6711yescharacteristic
543C2C2×#456×C6711yescharacteristic
544C2C2×#458×C6711yescharacteristic
subgroupG/H
center, socleC23×C67#120
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC2×#121×C67C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#483×C67C2semidirect
4. C2×#398×C67C2semidirect
5. C67×#1609C2direct (N × H)
6. C2×#328×C67C2semidirect
8. C2×#324×C67C2semidirect
25. #483×C67C22semidirect
35. #398×C67C22semidirect
38. #328×C67C22semidirect
40. #324×C67C22semidirect
264. C2×#1609C67direct (N × H)
269. C2×C67#1609direct (N × H)
274. C2×#483C2×C67semidirect
276. C2×#398C2×C67semidirect
277. #1609C2×C67direct (N × H)
278. C2×#328C2×C67semidirect
280. C2×#324C2×C67semidirect
281. C67C2×#1609direct (N × H)
298. #483C22×C67semidirect
308. #398C22×C67semidirect
311. #328C22×C67semidirect
313. #324C22×C67semidirect
541. C2C67×#1609direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#539×C672
C2×#483×C672
C2×#480×C672
C2×#398×C672
C67×#16092
C2×#328×C672
C2×#327×C672
C2×#324×C672
C2×#160967