Finite Group Databasebeta

← C67×#23986 #30703237

Subgroups — C67×#23986 #30703237

Order 17152.

1426 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3491C211yesmaximal, characteristic, complemented
2C67×#2849C212yesmaximal, complemented
3C67×#2280C211yesmaximal, characteristic, complemented
4C67×#2030C211yesmaximal, characteristic, complemented
5C67×#2029C211yesmaximal, characteristic, complemented
6C67×#2022C211yesmaximal, characteristic, complemented
7C67×#2021C211yesmaximal, characteristic
8C67×#1899C212yesmaximal, complemented
9C67×#1890C212yesmaximal, complemented
10C67×#1889C212yesmaximal, complemented
11C67×#1820C211yesmaximal, characteristic, complemented
12C24×C4×C67C2211yescharacteristic
13#552×C67—24no—
14#551×C67—24no—
15#543×C67C2211yescharacteristic
16#539×C67—22no—
17#534×C67C2211yescharacteristic
18#529×C67—22no—
19C2×#123×C67C2212yes—
20C2×#123×C67C2212yes—
21C2×#120×C67C2212yes—
22#518×C67C2212yes—
23C2×#117×C67C2211yescharacteristic
24C2×#117×C67C2211yescharacteristic
25C2×#99×C67C2212yes—
26C2×#99×C67C2212yes—
27#386×C67—24no—
28#385×C67—24no—
29#381×C67C2211yescharacteristic
30#381×C67C2211yescharacteristic
31#381×C67C2212yes—
32#381×C67C2212yes—
33#380×C67—22no—
34#379×C67C2211yescharacteristic
35#379×C67C2211yescharacteristic
36#379×C67C2212yes—
37C4×C22⋊C4×C67—22no—
38#342×C67C2211yescharacteristic
39#342×C67C2211yescharacteristic
40#342×C67C2212yes—
41#342×C67C2212yes—
42#342×C67C2212yes—
43C25×C67C2311yescharacteristic
44C22×D4×C67—22no—
45C22×D4×C67C2312yes—
46C23×C4×C67—22no—
47C23×C4×C67—21no—
48C23×C4×C67—21no—
49C23×C4×C67—22no—
50C23×C4×C67—24no—
51C23×C4×C67—42no—
52C23×C4×C67C2311yescharacteristic
53C23×C4×C67C2311yescharacteristic
54#126×C67—24no—
55#126×C67—24no—
56#125×C67—22no—
57#125×C67—24no—
58#124×C67—22no—
59#123×C67—24no—
60#123×C67—22no—
61#123×C67—22no—
62#123×C67—24no—
63#123×C67—22no—
64#123×C67—24no—
65#123×C67—42no—
66C22⋊Q8×C67—24no—
67C22⋊Q8×C67—22no—
68C22⋊Q8×C67—24no—
69#121×C67—22no—
70#120×C67—22no—
71#120×C67—24no—
72#120×C67—42no—
73C4×Q8×C67—22no—
74C4×D4×C67—22no—
75C4×D4×C67—24no—
76C4×D4×C67—24no—
77#117×C67—24no—
78#117×C67—24no—
79#117×C67—22no—
80#117×C67—42no—
81C2×C4⋊C4×C67—21no—
82C2×C4⋊C4×C67—22no—
83C2×C4⋊C4×C67D412yescomplemented
84C2×C4⋊C4×C67D414yescomplemented
85C2×C22⋊C4×C67—22no—
86C2×C22⋊C4×C67—22no—
87C2×C22⋊C4×C67—22no—
88C2×C22⋊C4×C67—22no—
89C2×C22⋊C4×C67—22no—
90C2×C22⋊C4×C67—24no—
91C2×C22⋊C4×C67D414yescomplemented
92C2×C22⋊C4×C67C2311yescharacteristic
93C2×C22⋊C4×C67C2311yescharacteristic
94C2×C22⋊C4×C67C2312yes—
95C2×C22⋊C4×C67C2312yes—
96C2×C22⋊C4×C67C2311yescharacteristic
97C2×C22⋊C4×C67C2311yescharacteristic
98C2×C22⋊C4×C67C2312yes—
99C2×C42×C67—21no—
100C2×C42×C67D412yescomplemented
101#99×C67—42no—
102#99×C67—42no—
103#95×C67—22no—
104#95×C67—22no—
105#95×C67—24no—
106#95×C67—24no—
107C24×C67—21no—
108C24×C67—22no—
109C24×C67—21no—
110C24×C67—22no—
111C24×C67—44no—
112C24×C67C2×D412yes—
113C24×C67#4112yes—
114C24×C67C2411yescharacteristic
115C2×Q8×C67—22no—
116C2×D4×C67—22no—
117C2×D4×C67—22no—
118C2×D4×C67—22no—
119C2×D4×C67—22no—
120C2×D4×C67—22no—
121C2×D4×C67—22no—
122C2×D4×C67—22no—
123C2×D4×C67—44no—
124C2×D4×C67—48no—
125C22×C4×C67—21no—
126C22×C4×C67—21no—
127C22×C4×C67—22no—
128C22×C4×C67—24no—
129C22×C4×C67—24no—
130C22×C4×C67—22no—
131C22×C4×C67—21no—
132C22×C4×C67—21no—
133C22×C4×C67—22no—
134C22×C4×C67—24no—
135C22×C4×C67—21no—
136C22×C4×C67—21no—
137C22×C4×C67—22no—
138C22×C4×C67—22no—
139C22×C4×C67—48no—
140C22×C4×C67—48no—
141C22×C4×C67—42no—
142C22×C4×C67—42no—
143C22×C4×C67—42no—
144C22×C4×C67—42no—
145C22×C4×C67C2×D412yes—
146C22×C4×C67C2×D411yescharacteristic
147C22×C4×C67C2×D414yes—
148C22×C4×C67C2×D411yescharacteristic
149C22×C4×C67C2×D412yes—
150C22×C4×C67C2×D412yes—
151C22×C4×C67C2×D414yes—
152C22×C4×C67#4112yes—
153C4⋊C4×C67—24no—
154C4⋊C4×C67—24no—
155C4⋊C4×C67—22no—
156C4⋊C4×C67—22no—
157C4⋊C4×C67—24no—
158C4⋊C4×C67—24no—
159C4⋊C4×C67—24no—
160C4⋊C4×C67—44no—
161C4⋊C4×C67—42no—
162C22⋊C4×C67—24no—
163C22⋊C4×C67—24no—
164C22⋊C4×C67—22no—
165C22⋊C4×C67—22no—
166C22⋊C4×C67—22no—
167C22⋊C4×C67—22no—
168C22⋊C4×C67—24no—
169C22⋊C4×C67—24no—
170C22⋊C4×C67—22no—
171C22⋊C4×C67—22no—
172C22⋊C4×C67—22no—
173C22⋊C4×C67—22no—
174C22⋊C4×C67—22no—
175C22⋊C4×C67—22no—
176C22⋊C4×C67—44no—
177C22⋊C4×C67—44no—
178C22⋊C4×C67—44no—
179C22⋊C4×C67—44no—
180C22⋊C4×C67—48no—
181C42×C67—24no—
182C42×C67—22no—
183C42×C67—22no—
184C42×C67—42no—
185C23×C67—24no—
186C23×C67—28no—
187C23×C67—22no—
188C23×C67—24no—
189C23×C67—22no—
190C23×C67—22no—
191C23×C67—22no—
192C23×C67—44no—
193C23×C67—42no—
194C23×C67—42no—
195C23×C67—44no—
196C23×C67—48no—
197C23×C67—44no—
198C23×C67—44no—
199C23×C67—41no—
200C23×C67—41no—
201C23×C67—42no—
202C23×C67C22×D411yescharacteristic
203C23×C67C22×D412yes—
204C23×C67C2×#4111yescharacteristic
205C23×C67C2×#4111yescharacteristic
206C23×C67H(5,𝔽2)12yes—
207Q8×C67—82no—
208D4×C67—42no—
209D4×C67—42no—
210D4×C67—42no—
211D4×C67—42no—
212D4×C67—82no—
213D4×C67—88no—
214C2×C4×C67—21no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
218C2×C4×C67—24no—
219C2×C4×C67—24no—
220C2×C4×C67—22no—
221C2×C4×C67—22no—
222C2×C4×C67—22no—
223C2×C4×C67—24no—
224C2×C4×C67—22no—
225C2×C4×C67—22no—
226C2×C4×C67—24no—
227C2×C4×C67—22no—
228C2×C4×C67—48no—
229C2×C4×C67—48no—
230C2×C4×C67—41no—
231C2×C4×C67—41no—
232C2×C4×C67—42no—
233C2×C4×C67—42no—
234C2×C4×C67—44no—
235C2×C4×C67—42no—
236C2×C4×C67—42no—
237C2×C4×C67—42no—
238C2×C4×C67—44no—
239C2×C4×C67—44no—
240C2×C4×C67—41no—
241C2×C4×C67—41no—
242C2×C4×C67—42no—
243C2×C4×C67—42no—
244C2×C4×C67—84no—
245C2×C4×C67#12012yes—
246C2×C4×C67#12012yes—
247C22×C67—22no—
248C22×C67—24no—
249C22×C67—28no—
250C22×C67—24no—
251C22×C67—41no—
252C22×C67—41no—
253C22×C67—42no—
254C22×C67—44no—
255C22×C67—44no—
256C22×C67—48no—
257C22×C67—42no—
258C22×C67—42no—
259C22×C67—44no—
260C22×C67—42no—
261C22×C67—44no—
262C22×C67—82no—
263C22×C67—82no—
264C22×C67C2×#12011yescharacteristic
265C22×C67#52511yescharacteristic
266C22×C67#53511yescharacteristic
267C22×C67#54612yes—
268C22×C67#54612yes—
269C4×C67—22no—
270C4×C67—22no—
271C4×C67—42no—
272C4×C67—42no—
273C4×C67—42no—
274C4×C67—44no—
275C4×C67—42no—
276C4×C67—81no—
277C4×C67—81no—
278C4×C67—82no—
279C4×C67—82no—
280#23986C6711yesmaximal, characteristic, direct product factor, Sylow
281C2×C67—21no—
282C2×C67—22no—
283C2×C67—22no—
284C2×C67—21no—
285C2×C67—44no—
286C2×C67—82no—
287C2×C67#240411yescharacteristic
288C2×C67#302512yes—
289#3491C2×C6711yescharacteristic, complemented
290#2849C2×C6712yescomplemented
291#2280C2×C6711yescharacteristic, complemented
292#2030C2×C6711yescharacteristic, complemented
293#2029C2×C6711yescharacteristic, complemented
294#2022C2×C6711yescharacteristic, complemented
295#2021C2×C6711yescharacteristic
296#1899C2×C6712yescomplemented
297#1890C2×C6712yescomplemented
298#1889C2×C6712yescomplemented
299#1820C2×C6711yescharacteristic, complemented
300C67#2398611yescharacteristic, direct product factor, Sylow
301C24×C4C22×C6711yescharacteristic
302#552—24no—
303#551—24no—
304#543C22×C6711yescharacteristic
305#539—22no—
306#534C22×C6711yescharacteristic
307#529—22no—
308C2×#123C22×C6712yes—
309C2×#123C22×C6712yes—
310C2×#120C22×C6712yes—
311#518C22×C6712yes—
312C2×#117C22×C6711yescharacteristic
313C2×#117C22×C6711yescharacteristic
314C2×#99C22×C6712yes—
315C2×#99C22×C6712yes—
316#386—24no—
317#385—24no—
318#381C22×C6712yes—
319#381C22×C6712yes—
320#381C22×C6711yescharacteristic
321#381C22×C6711yescharacteristic
322#380—22no—
323#379C22×C6712yes—
324#379C22×C6711yescharacteristic
325#379C22×C6711yescharacteristic
326C4×C22⋊C4—22no—
327#342C22×C6712yes—
328#342C22×C6711yescharacteristic
329#342C22×C6711yescharacteristic
330#342C22×C6712yes—
331#342C22×C6712yes—
332C25C23×C6711yescharacteristic
333C22×D4—22no—
334C22×D4C23×C6712yes—
335C23×C4—22no—
336C23×C4—21no—
337C23×C4—21no—
338C23×C4—22no—
339C23×C4—24no—
340C23×C4—42no—
341C23×C4C23×C6711yescharacteristic
342C23×C4C23×C6711yescharacteristic
343#126—24no—
344#126—24no—
345#125—22no—
346#125—24no—
347#124—22no—
348#123—24no—
349#123—22no—
350#123—22no—
351#123—24no—
352#123—22no—
353#123—24no—
354#123—42no—
355C22⋊Q8—24no—
356C22⋊Q8—22no—
357C22⋊Q8—24no—
358#121—22no—
359#120—22no—
360#120—24no—
361#120—42no—
362C4×Q8—22no—
363C4×D4—22no—
364C4×D4—24no—
365C4×D4—24no—
366#117—24no—
367#117—24no—
368#117—22no—
369#117—42no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4D4×C6712yescomplemented
373C2×C4⋊C4D4×C6714yescomplemented
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4D4×C6714yescomplemented
381C2×C22⋊C4C23×C6711yescharacteristic
382C2×C22⋊C4C23×C6711yescharacteristic
383C2×C22⋊C4C23×C6712yes—
384C2×C22⋊C4C23×C6712yes—
385C2×C22⋊C4C23×C6711yescharacteristic
386C2×C22⋊C4C23×C6711yescharacteristic
387C2×C22⋊C4C23×C6712yes—
388C2×C42—21no—
389C2×C42D4×C6712yescomplemented
390#99—42no—
391#99—42no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—24no—
396C24—21no—
397C24—22no—
398C24—21no—
399C24—22no—
400C24—44no—
401C24C2×D4×C6712yes—
402C24#41×C6712yes—
403C24C24×C6711yescharacteristic
404C2×Q8—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—44no—
413C2×D4—48no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4C2×D4×C6712yes—
435C22×C4C2×D4×C6714yes—
436C22×C4C2×D4×C6711yescharacteristic
437C22×C4C2×D4×C6711yescharacteristic
438C22×C4C2×D4×C6712yes—
439C22×C4C2×D4×C6712yes—
440C22×C4C2×D4×C6714yes—
441C22×C4#41×C6712yes—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C42—24no—
471C42—22no—
472C42—22no—
473C42—42no—
474C23—24no—
475C23—28no—
476C23—22no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—48no—
486C23—44no—
487C23—44no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23C22×D4×C6711yescharacteristic
492C23C22×D4×C6712yes—
493C23C2×#41×C6711yescharacteristic
494C23C2×#41×C6711yescharacteristic
495C23H(5,𝔽2)×C6712yes—
496Q8—82no—
497D4—42no—
498D4—42no—
499D4—42no—
500D4—42no—
501D4—82no—
502D4—88no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—84no—
534C2×C4#120×C6712yes—
535C2×C4#120×C6712yes—
536C22—22no—
537C22—24no—
538C22—28no—
539C22—24no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—44no—
544C22—44no—
545C22—48no—
546C22—42no—
547C22—44no—
548C22—42no—
549C22—44no—
550C22—42no—
551C22—82no—
552C22—82no—
553C22C2×#120×C6711yescharacteristic
554C22#525×C6711yescharacteristic
555C22#535×C6711yescharacteristic
556C22#546×C6712yes—
557C22#546×C6712yes—
558C4—22no—
559C4—22no—
560C4—42no—
561C4—44no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—82no—
566C4—81no—
567C4—81no—
568C4—82no—
569C2—21no—
570C2—22no—
571C2—22no—
572C2—21no—
573C2—44no—
574C2—82no—
575C2C67×#240411yescharacteristic
576C2C67×#302512yes—
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC24C24×C67
prime radicalC67×#2021C2
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
1. C67×#3491C2semidirect
2. C67×#2849C2semidirect
3. C67×#2280C2semidirect
4. C67×#2030C2semidirect
5. C67×#2029C2semidirect
6. C67×#2022C2semidirect
8. C67×#1899C2semidirect
9. C67×#1890C2semidirect
10. C67×#1889C2semidirect
11. C67×#1820C2semidirect
83. C2×C4⋊C4×C67D4semidirect
84. C2×C4⋊C4×C67D4semidirect
91. C2×C22⋊C4×C67D4semidirect
100. C2×C42×C67D4semidirect
280. #23986C67direct (N × H)
289. #3491C2×C67semidirect
290. #2849C2×C67semidirect
291. #2280C2×C67semidirect
292. #2030C2×C67semidirect
293. #2029C2×C67semidirect
294. #2022C2×C67semidirect
296. #1899C2×C67semidirect
297. #1890C2×C67semidirect
298. #1889C2×C67semidirect
299. #1820C2×C67semidirect
300. C67#23986direct (N × H)
372. C2×C4⋊C4D4×C67semidirect
373. C2×C4⋊C4D4×C67semidirect
380. C2×C22⋊C4D4×C67semidirect
389. C2×C42D4×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34912
C67×#28492
C67×#22802
C67×#20302
C67×#20292
C67×#20222
C67×#20212
C67×#18992
C67×#18902
C67×#18892
C67×#18202
#2398667