Finite Group Databasebeta

← C67×#22478 #30701729

Subgroups — C67×#22478 #30701729

Order 17152.

1132 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#552×C67C211yesmaximal, characteristic
2C67×#2344C212yesmaximal, complemented
3C67×#1996C212yesmaximal, complemented
4C67×#1899C214yesmaximal
5C67×#1820C212yesmaximal, complemented
6C67×#1799C211yesmaximal, characteristic
7C4×#99×C67C211yesmaximal, characteristic
8C67×#1740C212yesmaximal, complemented
9#552×C67—22no—
10#552×C67—24no—
11#552×C67—22no—
12C2×#126×C67C412yescomplemented
13C2×#125×C67C412yescomplemented
14C2×#125×C67C411yescharacteristic, complemented
15C2×#123×C67C2212yes—
16C2×C22⋊Q8×C67C411yescharacteristic, complemented
17C2×C22⋊Q8×C67C2211yescharacteristic
18C2×C4×Q8×C67C411yescharacteristic, complemented
19C2×C4×D4×C67C2211yescharacteristic
20C2×#117×C67C411yescharacteristic, complemented
21C2×#117×C67C2214yes—
22C2×#117×C67C2212yes—
23C22×C4⋊C4×C67C2211yescharacteristic
24C2×#99×C67C2212yes—
25C2×#99×C67C2212yes—
26#389×C67—22no—
27#387×C67—24no—
28#381×C67C2214yes—
29#380×C67C2212yes—
30C4×C4⋊C4×C67—22no—
31C4×C22⋊C4×C67C2212yes—
32#342×C67C2211yescharacteristic
33#342×C67C2214yes—
34#342×C67C2212yes—
35#342×C67C2214yes—
36#342×C67C2211yescharacteristic
37C22×Q8×C67C2×C411yescharacteristic
38C22×D4×C67C2311yescharacteristic
39C23×C4×C67C2×C411yescharacteristic
40C23×C4×C67C2312yes—
41C23×C4×C67C2311yescharacteristic
42#126×C67—24no—
43#126×C67—24no—
44#125×C67—24no—
45#125×C67—24no—
46#125×C67—22no—
47#125×C67—22no—
48#123×C67—24no—
49#123×C67—24no—
50C22⋊Q8×C67—22no—
51C22⋊Q8×C67—22no—
52C22⋊Q8×C67—22no—
53C22⋊Q8×C67—22no—
54C4×Q8×C67—22no—
55C4×Q8×C67—21no—
56C4×Q8×C67—21no—
57C4×D4×C67—22no—
58C4×D4×C67—22no—
59#117×C67—24no—
60#117×C67—24no—
61#117×C67—22no—
62#117×C67—21no—
63#117×C67—21no—
64#117×C67—28no—
65#117×C67—44no—
66C2×C4⋊C4×C67—24no—
67C2×C4⋊C4×C67—22no—
68C2×C4⋊C4×C67C2×C411yescharacteristic
69C2×C4⋊C4×C67C2×C412yes—
70C2×C4⋊C4×C67C2×C412yes—
71C2×C4⋊C4×C67C2×C411yescharacteristic
72C2×C4⋊C4×C67C2×C412yes—
73C2×C4⋊C4×C67C2×C412yes—
74C2×C4⋊C4×C67C2×C412yes—
75C2×C4⋊C4×C67C2×C411yescharacteristic
76C2×C4⋊C4×C67C2×C411yescharacteristic
77C2×C4⋊C4×C67C2×C412yes—
78C2×C4⋊C4×C67D412yes—
79C2×C4⋊C4×C67D414yes—
80C2×C4⋊C4×C67D412yes—
81C2×C22⋊C4×C67C2×C412yes—
82C2×C22⋊C4×C67C2×C411yescharacteristic
83C2×C22⋊C4×C67C2×C412yes—
84C2×C22⋊C4×C67C2314yes—
85C2×C22⋊C4×C67C2312yes—
86C2×C22⋊C4×C67C2311yescharacteristic
87C2×C22⋊C4×C67C2312yes—
88C2×C22⋊C4×C67C2312yes—
89C2×C42×C67—24no—
90C2×C42×C67—22no—
91C2×C42×C67C2×C412yes—
92C2×C42×C67C2×C412yes—
93C2×C42×C67C2×C411yescharacteristic
94#99×C67—24no—
95#99×C67—42no—
96#95×C67—22no—
97#95×C67—24no—
98#95×C67—22no—
99#95×C67—24no—
100C24×C67C22×C411yescharacteristic
101C24×C67C2411yescharacteristic
102C2×Q8×C67—21no—
103C2×Q8×C67—21no—
104C2×Q8×C67—42no—
105C2×D4×C67—22no—
106C2×D4×C67—22no—
107C2×D4×C67—21no—
108C2×D4×C67—21no—
109C22×C4×C67—24no—
110C22×C4×C67—22no—
111C22×C4×C67—22no—
112C22×C4×C67—22no—
113C22×C4×C67—21no—
114C22×C4×C67—21no—
115C22×C4×C67—24no—
116C22×C4×C67—22no—
117C22×C4×C67—42no—
118C22×C4×C67—42no—
119C22×C4×C67C22×C411yescharacteristic
120C22×C4×C67C22×C412yes—
121C22×C4×C67C22×C411yescharacteristic
122C22×C4×C67C22×C412yes—
123C22×C4×C67C22×C411yescharacteristic
124C22×C4×C67C22×C411yescharacteristic
125C22×C4×C67C22×C412yes—
126C22×C4×C67C22×C412yes—
127C22×C4×C67C22×C411yescharacteristic
128C22×C4×C67C2×D412yes—
129C22×C4×C67C2×D414yes—
130C22×C4×C67C2×D414yes—
131C22×C4×C67C2×D412yes—
132C4⋊C4×C67—24no—
133C4⋊C4×C67—21no—
134C4⋊C4×C67—21no—
135C4⋊C4×C67—24no—
136C4⋊C4×C67—22no—
137C4⋊C4×C67—22no—
138C4⋊C4×C67—22no—
139C4⋊C4×C67—24no—
140C4⋊C4×C67—24no—
141C4⋊C4×C67—48no—
142C4⋊C4×C67—44no—
143C4⋊C4×C67C22⋊C412yes—
144C4⋊C4×C67C22⋊C411yescharacteristic
145C4⋊C4×C67C22⋊C411yescharacteristic
146C4⋊C4×C67C22⋊C412yes—
147C4⋊C4×C67C22⋊C412yes—
148C4⋊C4×C67C22⋊C414yes—
149C4⋊C4×C67C22⋊C414yes—
150C22⋊C4×C67—22no—
151C22⋊C4×C67—24no—
152C22⋊C4×C67—22no—
153C22⋊C4×C67—22no—
154C22⋊C4×C67—22no—
155C22⋊C4×C67—22no—
156C22⋊C4×C67—24no—
157C22⋊C4×C67—24no—
158C22⋊C4×C67—22no—
159C22⋊C4×C67—24no—
160C22⋊C4×C67—22no—
161C22⋊C4×C67#4114yes—
162C42×C67—22no—
163C42×C67—22no—
164C42×C67—22no—
165C42×C67—22no—
166C42×C67—21no—
167C42×C67—21no—
168C42×C67—48no—
169C42×C67—44no—
170C23×C67—21no—
171C23×C67—21no—
172C23×C67—21no—
173C23×C67—41no—
174C23×C67—41no—
175C23×C67—41no—
176C23×C67—41no—
177C23×C67C23×C411yescharacteristic
178C23×C67C22×D412yes—
179C23×C67C2×#4112yes—
180C23×C67H(5,𝔽2)11yescharacteristic
181C23×C672−1+411yescharacteristic
182Q8×C67—42no—
183Q8×C67—42no—
184D4×C67—42no—
185D4×C67—42no—
186C2×C4×C67—22no—
187C2×C4×C67—22no—
188C2×C4×C67—22no—
189C2×C4×C67—22no—
190C2×C4×C67—24no—
191C2×C4×C67—24no—
192C2×C4×C67—24no—
193C2×C4×C67—24no—
194C2×C4×C67—24no—
195C2×C4×C67—22no—
196C2×C4×C67—21no—
197C2×C4×C67—21no—
198C2×C4×C67—22no—
199C2×C4×C67—22no—
200C2×C4×C67—21no—
201C2×C4×C67—21no—
202C2×C4×C67—21no—
203C2×C4×C67—24no—
204C2×C4×C67—41no—
205C2×C4×C67—42no—
206C2×C4×C67—42no—
207C2×C4×C67—42no—
208C2×C4×C67—42no—
209C2×C4×C67—42no—
210C2×C4×C67—41no—
211C2×C4×C67—41no—
212C2×C4×C67—42no—
213C2×C4×C67—42no—
214C2×C4×C67—42no—
215C2×C4×C67—44no—
216C2×C4×C67—44no—
217C2×C4×C67—42no—
218C2×C4×C67—44no—
219C2×C4×C67—42no—
220C2×C4×C67C2×C22⋊C412yes—
221C2×C4×C67C2×C22⋊C412yes—
222C2×C4×C67C2×C22⋊C411yescharacteristic
223C2×C4×C67C2×C22⋊C414yes—
224C2×C4×C67C2×C22⋊C412yes—
225C2×C4×C67C2×C22⋊C411yescharacteristic
226C22×C67—22no—
227C22×C67—21no—
228C22×C67—22no—
229C22×C67—21no—
230C22×C67—42no—
231C22×C67—41no—
232C22×C67—41no—
233C22×C67—41no—
234C22×C67—41no—
235C22×C67—41no—
236C22×C67—81no—
237C22×C67—81no—
238C22×C67C22×C22⋊C411yescharacteristic
239C22×C67C4×#4111yescharacteristic
240C22×C67#52011yescharacteristic
241C22×C67#54612yes—
242C22×C67#54712yes—
243C4×C67—22no—
244C4×C67—24no—
245C4×C67—22no—
246C4×C67—21no—
247C4×C67—22no—
248C4×C67—44no—
249C4×C67—41no—
250C4×C67—41no—
251C4×C67—42no—
252C4×C67—41no—
253C4×C67—42no—
254C4×C67—84no—
255C4×C67—82no—
256C4×C67#41012yes—
257#22478C6711yesmaximal, characteristic, direct product factor, Sylow
258C2×C67—22no—
259C2×C67—22no—
260C2×C67—21no—
261C2×C67—21no—
262C2×C67—81no—
263C2×C67#233911yescharacteristic
264C2×C67C2×#41011yescharacteristic
265C2×C67#288411yescharacteristic
266C2×#552C2×C6711yescharacteristic
267#2344C2×C6712yescomplemented
268#1996C2×C6712yescomplemented
269#1899C2×C6714yes—
270#1820C2×C6712yescomplemented
271#1799C2×C6711yescharacteristic
272C4×#99C2×C6711yescharacteristic
273#1740C2×C6712yescomplemented
274C67#2247811yescharacteristic, direct product factor, Sylow
275#552—22no—
276#552—24no—
277#552—22no—
278C2×#126C4×C6712yescomplemented
279C2×#125C4×C6712yescomplemented
280C2×#125C4×C6711yescharacteristic, complemented
281C2×#123C22×C6712yes—
282C2×C22⋊Q8C4×C6711yescharacteristic, complemented
283C2×C22⋊Q8C22×C6711yescharacteristic
284C2×C4×Q8C4×C6711yescharacteristic, complemented
285C2×C4×D4C22×C6711yescharacteristic
286C2×#117C4×C6711yescharacteristic, complemented
287C2×#117C22×C6714yes—
288C2×#117C22×C6712yes—
289C22×C4⋊C4C22×C6711yescharacteristic
290C2×#99C22×C6712yes—
291C2×#99C22×C6712yes—
292#389—22no—
293#387—24no—
294#381C22×C6714yes—
295#380C22×C6712yes—
296C4×C4⋊C4—22no—
297C4×C22⋊C4C22×C6712yes—
298#342C22×C6712yes—
299#342C22×C6711yescharacteristic
300#342C22×C6714yes—
301#342C22×C6714yes—
302#342C22×C6711yescharacteristic
303C22×Q8C2×C4×C6711yescharacteristic
304C22×D4C23×C6711yescharacteristic
305C23×C4C2×C4×C6711yescharacteristic
306C23×C4C23×C6712yes—
307C23×C4C23×C6711yescharacteristic
308#126—24no—
309#126—24no—
310#125—24no—
311#125—24no—
312#125—22no—
313#125—22no—
314#123—24no—
315#123—24no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8—22no—
320C4×Q8—22no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×D4—22no—
324C4×D4—22no—
325#117—24no—
326#117—24no—
327#117—22no—
328#117—21no—
329#117—21no—
330#117—28no—
331#117—44no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4C2×C4×C6711yescharacteristic
335C2×C4⋊C4C2×C4×C6712yes—
336C2×C4⋊C4C2×C4×C6712yes—
337C2×C4⋊C4C2×C4×C6711yescharacteristic
338C2×C4⋊C4C2×C4×C6712yes—
339C2×C4⋊C4C2×C4×C6712yes—
340C2×C4⋊C4C2×C4×C6712yes—
341C2×C4⋊C4C2×C4×C6711yescharacteristic
342C2×C4⋊C4C2×C4×C6711yescharacteristic
343C2×C4⋊C4C2×C4×C6712yes—
344C2×C4⋊C4D4×C6712yes—
345C2×C4⋊C4D4×C6714yes—
346C2×C4⋊C4D4×C6712yes—
347C2×C22⋊C4C2×C4×C6712yes—
348C2×C22⋊C4C2×C4×C6711yescharacteristic
349C2×C22⋊C4C2×C4×C6712yes—
350C2×C22⋊C4C23×C6714yes—
351C2×C22⋊C4C23×C6712yes—
352C2×C22⋊C4C23×C6711yescharacteristic
353C2×C22⋊C4C23×C6712yes—
354C2×C22⋊C4C23×C6712yes—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42C2×C4×C6712yes—
358C2×C42C2×C4×C6712yes—
359C2×C42C2×C4×C6711yescharacteristic
360#99—24no—
361#99—42no—
362#95—22no—
363#95—24no—
364#95—22no—
365#95—24no—
366C24C22×C4×C6711yescharacteristic
367C24C24×C6711yescharacteristic
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—42no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—21no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4C22×C4×C6711yescharacteristic
386C22×C4C22×C4×C6712yes—
387C22×C4C22×C4×C6711yescharacteristic
388C22×C4C22×C4×C6712yes—
389C22×C4C22×C4×C6711yescharacteristic
390C22×C4C22×C4×C6711yescharacteristic
391C22×C4C22×C4×C6712yes—
392C22×C4C22×C4×C6712yes—
393C22×C4C22×C4×C6711yescharacteristic
394C22×C4C2×D4×C6714yes—
395C22×C4C2×D4×C6714yes—
396C22×C4C2×D4×C6712yes—
397C22×C4C2×D4×C6712yes—
398C4⋊C4—24no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—48no—
408C4⋊C4—44no—
409C4⋊C4C22⋊C4×C6712yes—
410C4⋊C4C22⋊C4×C6711yescharacteristic
411C4⋊C4C22⋊C4×C6711yescharacteristic
412C4⋊C4C22⋊C4×C6712yes—
413C4⋊C4C22⋊C4×C6712yes—
414C4⋊C4C22⋊C4×C6714yes—
415C4⋊C4C22⋊C4×C6714yes—
416C22⋊C4—22no—
417C22⋊C4—24no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—24no—
423C22⋊C4—22no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—22no—
427C22⋊C4#41×C6714yes—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—21no—
433C42—21no—
434C42—48no—
435C42—44no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23C23×C4×C6711yescharacteristic
444C23C22×D4×C6712yes—
445C23C2×#41×C6712yes—
446C23H(5,𝔽2)×C6711yescharacteristic
447C232−1+4×C6711yescharacteristic
448Q8—42no—
449Q8—42no—
450D4—42no—
451D4—42no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—24no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4C2×C22⋊C4×C6712yes—
487C2×C4C2×C22⋊C4×C6712yes—
488C2×C4C2×C22⋊C4×C6711yescharacteristic
489C2×C4C2×C22⋊C4×C6714yes—
490C2×C4C2×C22⋊C4×C6712yes—
491C2×C4C2×C22⋊C4×C6711yescharacteristic
492C22—22no—
493C22—21no—
494C22—22no—
495C22—21no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—81no—
503C22—81no—
504C22C22×C22⋊C4×C6711yescharacteristic
505C22C4×#41×C6711yescharacteristic
506C22#520×C6711yescharacteristic
507C22#546×C6712yes—
508C22#547×C6712yes—
509C4—22no—
510C4—24no—
511C4—22no—
512C4—21no—
513C4—22no—
514C4—44no—
515C4—41no—
516C4—41no—
517C4—42no—
518C4—41no—
519C4—42no—
520C4—84no—
521C4—82no—
522C4#410×C6712yes—
523C2—22no—
524C2—22no—
525C2—21no—
526C2—21no—
527C2—81no—
528C2C67×#233911yescharacteristic
529C2C2×#410×C6711yescharacteristic
530C2C67×#288411yescharacteristic
subgroupG/H
center, socleC22×C67C22×C22⋊C4
commutatorC23C23×C4×C67
Frattini, cosocleC24C24×C67
prime radicalC22×D4×C67C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C22×C4⋊C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2344C2semidirect
3. C67×#1996C2semidirect
5. C67×#1820C2semidirect
8. C67×#1740C2semidirect
12. C2×#126×C67C4semidirect
13. C2×#125×C67C4semidirect
14. C2×#125×C67C4semidirect
16. C2×C22⋊Q8×C67C4semidirect
18. C2×C4×Q8×C67C4semidirect
20. C2×#117×C67C4semidirect
257. #22478C67direct (N × H)
267. #2344C2×C67semidirect
268. #1996C2×C67semidirect
270. #1820C2×C67semidirect
273. #1740C2×C67semidirect
274. C67#22478direct (N × H)
278. C2×#126C4×C67semidirect
279. C2×#125C4×C67semidirect
280. C2×#125C4×C67semidirect
282. C2×C22⋊Q8C4×C67semidirect
284. C2×C4×Q8C4×C67semidirect
286. C2×#117C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#552×C672
C67×#23442
C67×#19962
C67×#18992
C67×#18202
C67×#17992
C4×#99×C672
C67×#17402
#2247867