Finite Group Databasebeta

← C67×#18123 #30697374

Subgroups — C67×#18123 #30697374

Order 17152.

1208 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3449C211yesmaximal, characteristic
2C67×#3103C212yesmaximal, complemented
3C67×#2396C211yesmaximal, characteristic, complemented
4C67×#1616C212yesmaximal, complemented
5C67×#1613C212yesmaximal, complemented
6C67×#1604C212yesmaximal, complemented
7C67×#1601C212yesmaximal, complemented
8C67×#1502C212yesmaximal, complemented
9C67×#1458C211yesmaximal, characteristic, complemented
10C22×#41×C67—21no—
11#537×C67—21no—
12#537×C67C2212yes—
13#536×C67—22no—
14#535×C67—22no—
15#534×C67—21no—
16#534×C67C2212yes—
17#532×C67—21no—
18#532×C67C2212yes—
19#525×C67—24no—
20C2×C4×Q8×C67C2211yescharacteristic, complemented
21C2×#117×C67C2211yescharacteristic
22#481×C67C2214yescomplemented
23#475×C67—22no—
24#460×C67—22no—
25#448×C67C2214yescomplemented
26C4×SD16×C67—24no—
27#431×C67C2212yescomplemented
28#400×C67C2211yescharacteristic, complemented
29#399×C67C2211yescharacteristic
30#398×C67—21no—
31#396×C67—21no—
32#395×C67C2211yescharacteristic, complemented
33#393×C67C2211yescharacteristic
34C4×C4⋊C4×C67—21no—
35C4×C4⋊C4×C67C2211yescharacteristic, complemented
36C4×C22⋊C4×C67—21no—
37#327×C67C2214yescomplemented
38#327×C67C2214yescomplemented
39#324×C67C2212yescomplemented
40#323×C67C2212yes—
41C2×#41×C67—22no—
42C2×#41×C67—22no—
43C2×#41×C67—22no—
44C2×#41×C67—22no—
45C2×#41×C67—44no—
46C2×#41×C67—44no—
47C22×Q8×C67—21no—
48C22×Q8×C67C2311yescharacteristic
49C22×D4×C67—21no—
50C22×D4×C67—22no—
51C23×C4×C67—22no—
52C23×C4×C67D411yescharacteristic
53C2×SD16×C67—24no—
54C4⋊Q8×C67—21no—
55C4⋊Q8×C67D412yes—
56#127×C67—21no—
57#127×C67D412yes—
58#124×C67—22no—
59#124×C67—22no—
60#123×C67—24no—
61#123×C67—24no—
62C22⋊Q8×C67—24no—
63C22⋊Q8×C67—24no—
64#121×C67—24no—
65#121×C67—24no—
66#121×C67—22no—
67#121×C67—22no—
68#121×C67—22no—
69#121×C67C2314yes—
70#120×C67—24no—
71#120×C67—24no—
72C4×Q8×C67—22no—
73C4×Q8×C67—22no—
74C4×D4×C67—22no—
75C4×D4×C67—22no—
76C4×D4×C67—22no—
77C4×D4×C67D414yes—
78#117×C67—22no—
79#117×C67—22no—
80C2×C4⋊C4×C67—21no—
81C2×C4⋊C4×C67—21no—
82C2×C4⋊C4×C67—21no—
83C2×C4⋊C4×C67—21no—
84C2×C4⋊C4×C67D411yescharacteristic
85C2×C4⋊C4×C67C2311yescharacteristic
86C2×C4⋊C4×C67C2311yescharacteristic
87C2×C22⋊C4×C67—21no—
88C2×C22⋊C4×C67—21no—
89C2×C22⋊C4×C67D411yescharacteristic
90C2×C22⋊C4×C67C2311yescharacteristic
91C2×C42×C67—21no—
92C2×C42×C67—21no—
93C2×C42×C67D411yescharacteristic
94C2×C42×C67C2311yescharacteristic
95#106×C67—24no—
96C4⋊C8×C67—22no—
97#103×C67—24no—
98#102×C67—24no—
99C22⋊C8×C67C2314yes—
100C4×C8×C67—22no—
101#95×C67—21no—
102#95×C67—21no—
103#95×C67—21no—
104#95×C67—21no—
105#95×C67C2311yescharacteristic
106#95×C67C2311yescharacteristic
107C24×C67—22no—
108C24×C67C2×D411yescharacteristic
109#41×C67—44no—
110#41×C67—48no—
111#41×C67—48no—
112#41×C67—44no—
113#41×C67—44no—
114#41×C67—44no—
115C2×Q8×C67—22no—
116C2×Q8×C67—22no—
117C2×Q8×C67—22no—
118C2×Q8×C67—42no—
119C2×Q8×C67—44no—
120C2×D4×C67—24no—
121C2×D4×C67—22no—
122C2×D4×C67—24no—
123C2×D4×C67—22no—
124C2×D4×C67—22no—
125C2×D4×C67—22no—
126C2×D4×C67—22no—
127C2×D4×C67—44no—
128C2×D4×C67—44no—
129C2×D4×C67—44no—
130C2×D4×C67C2×D414yes—
131C22×C4×C67—21no—
132C22×C4×C67—21no—
133C22×C4×C67—22no—
134C22×C4×C67—24no—
135C22×C4×C67—22no—
136C22×C4×C67—21no—
137C22×C4×C67—21no—
138C22×C4×C67—24no—
139C22×C4×C67—21no—
140C22×C4×C67—21no—
141C22×C4×C67—48no—
142C22×C4×C67SD1614yescomplemented
143C22×C4×C67C2×D411yescharacteristic
144C22×C4×C67C2×D411yescharacteristic
145C22×C4×C67C2×D411yescharacteristic
146C22×C4×C67C2×D411yescharacteristic
147C22×C4×C67C2×D411yescharacteristic
148C22×C4×C67C2411yescharacteristic
149SD16×C67—48no—
150C2×C8×C67—24no—
151C4⋊C4×C67—22no—
152C4⋊C4×C67—22no—
153C4⋊C4×C67—22no—
154C4⋊C4×C67—22no—
155C4⋊C4×C67—22no—
156C4⋊C4×C67—22no—
157C4⋊C4×C67—42no—
158C4⋊C4×C67—42no—
159C4⋊C4×C67—42no—
160C4⋊C4×C67—42no—
161C4⋊C4×C67C2×D414yes—
162C22⋊C4×C67—24no—
163C22⋊C4×C67—24no—
164C22⋊C4×C67—22no—
165C22⋊C4×C67—24no—
166C22⋊C4×C67—22no—
167C22⋊C4×C67—24no—
168C22⋊C4×C67—42no—
169C22⋊C4×C67—42no—
170C42×C67—21no—
171C42×C67—21no—
172C42×C67—41no—
173C42×C67—41no—
174C42×C67—42no—
175C42×C67C2×D412yes—
176C42×C67C2×D412yes—
177C23×C67—21no—
178C23×C67—21no—
179C23×C67—24no—
180C23×C67—41no—
181C23×C67—41no—
182C23×C67—44no—
183C23×C67—82no—
184C23×C67—82no—
185C23×C67C2×SD1612yes—
186C23×C67C22×D411yescharacteristic
187Q8×C67—24no—
188Q8×C67—42no—
189Q8×C67—44no—
190Q8×C67—44no—
191D4×C67—44no—
192D4×C67—44no—
193D4×C67—48no—
194D4×C67—44no—
195D4×C67—44no—
196D4×C67—44no—
197D4×C67—82no—
198D4×C67—82no—
199D4×C67—82no—
200C2×C4×C67—21no—
201C2×C4×C67—21no—
202C2×C4×C67—24no—
203C2×C4×C67—24no—
204C2×C4×C67—22no—
205C2×C4×C67—22no—
206C2×C4×C67—21no—
207C2×C4×C67—21no—
208C2×C4×C67—24no—
209C2×C4×C67—41no—
210C2×C4×C67—41no—
211C2×C4×C67—48no—
212C2×C4×C67—41no—
213C2×C4×C67—44no—
214C2×C4×C67—41no—
215C2×C4×C67—41no—
216C2×C4×C67—41no—
217C2×C4×C67—48no—
218C2×C4×C67—41no—
219C2×C4×C67—41no—
220C2×C4×C67—41no—
221C2×C4×C67—41no—
222C2×C4×C67—41no—
223C2×C4×C67—41no—
224C2×C4×C67—41no—
225C2×C4×C67—41no—
226C2×C4×C67#12012yes—
227C2×C4×C67#12011yescharacteristic
228C2×C4×C67#12011yescharacteristic
229C2×C4×C67C2×SD1612yes—
230C2×C4×C67C2×SD1612yes—
231C2×C4×C67AGL(1,ℤ/8)12yes—
232C2×C4×C67C22×D412yes—
233C8×C67—44no—
234C22×C67—21no—
235C22×C67—21no—
236C22×C67—22no—
237C22×C67—42no—
238C22×C67—41no—
239C22×C67—41no—
240C22×C67—44no—
241C22×C67—44no—
242C22×C67—44no—
243C22×C67—81no—
244C22×C67—88no—
245C22×C67C2×#12011yescharacteristic
246C22×C67C22×SD1611yescharacteristic
247C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
248C4×C67—22no—
249C4×C67—22no—
250C4×C67—22no—
251C4×C67—42no—
252C4×C67—44no—
253C4×C67—41no—
254C4×C67—42no—
255C4×C67—41no—
256C4×C67—81no—
257C4×C67—81no—
258C4×C67#45012yes—
259C4×C67#45012yes—
260#18123C6711yesmaximal, characteristic, direct product factor, Sylow
261C2×C67—21no—
262C2×C67—21no—
263C2×C67—42no—
264C2×C67—84no—
265C2×C67C2×#45011yescharacteristic
266C2×C67#301811yescharacteristic
267C2×C67#302511yescharacteristic
268#3449C2×C6711yescharacteristic
269#3103C2×C6712yescomplemented
270#2396C2×C6711yescharacteristic, complemented
271#1616C2×C6712yescomplemented
272#1613C2×C6712yescomplemented
273#1604C2×C6712yescomplemented
274#1601C2×C6712yescomplemented
275#1502C2×C6712yescomplemented
276#1458C2×C6711yescharacteristic, complemented
277C67#1812311yescharacteristic, direct product factor, Sylow
278C22×#41—21no—
279#537—21no—
280#537C22×C6712yes—
281#536—22no—
282#535—22no—
283#534—21no—
284#534C22×C6712yes—
285#532—21no—
286#532C22×C6712yes—
287#525—24no—
288C2×C4×Q8C22×C6711yescharacteristic, complemented
289C2×#117C22×C6711yescharacteristic
290#481C22×C6714yescomplemented
291#475—22no—
292#460—22no—
293#448C22×C6714yescomplemented
294C4×SD16—24no—
295#431C22×C6712yescomplemented
296#400C22×C6711yescharacteristic, complemented
297#399C22×C6711yescharacteristic
298#398—21no—
299#396—21no—
300#395C22×C6711yescharacteristic, complemented
301#393C22×C6711yescharacteristic
302C4×C4⋊C4—21no—
303C4×C4⋊C4C22×C6711yescharacteristic, complemented
304C4×C22⋊C4—21no—
305#327C22×C6714yescomplemented
306#327C22×C6714yescomplemented
307#324C22×C6712yescomplemented
308#323C22×C6712yes—
309C2×#41—22no—
310C2×#41—22no—
311C2×#41—22no—
312C2×#41—22no—
313C2×#41—44no—
314C2×#41—44no—
315C22×Q8—21no—
316C22×Q8C23×C6711yescharacteristic
317C22×D4—21no—
318C22×D4—22no—
319C23×C4—22no—
320C23×C4D4×C6711yescharacteristic
321C2×SD16—24no—
322C4⋊Q8—21no—
323C4⋊Q8D4×C6712yes—
324#127—21no—
325#127D4×C6712yes—
326#124—22no—
327#124—22no—
328#123—24no—
329#123—24no—
330C22⋊Q8—24no—
331C22⋊Q8—24no—
332#121—24no—
333#121—24no—
334#121—22no—
335#121—22no—
336#121—22no—
337#121C23×C6714yes—
338#120—24no—
339#120—24no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4D4×C6714yes—
346#117—22no—
347#117—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4D4×C6711yescharacteristic
353C2×C4⋊C4C23×C6711yescharacteristic
354C2×C4⋊C4C23×C6711yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4D4×C6711yescharacteristic
358C2×C22⋊C4C23×C6711yescharacteristic
359C2×C42—21no—
360C2×C42—21no—
361C2×C42D4×C6711yescharacteristic
362C2×C42C23×C6711yescharacteristic
363#106—24no—
364C4⋊C8—22no—
365#103—24no—
366#102—24no—
367C22⋊C8C23×C6714yes—
368C4×C8—22no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95C23×C6711yescharacteristic
374#95C23×C6711yescharacteristic
375C24—22no—
376C24C2×D4×C6711yescharacteristic
377#41—44no—
378#41—48no—
379#41—48no—
380#41—44no—
381#41—44no—
382#41—44no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4C2×D4×C6714yes—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—48no—
410C22×C4SD16×C6714yescomplemented
411C22×C4C2×D4×C6711yescharacteristic
412C22×C4C2×D4×C6711yescharacteristic
413C22×C4C2×D4×C6711yescharacteristic
414C22×C4C2×D4×C6711yescharacteristic
415C22×C4C2×D4×C6711yescharacteristic
416C22×C4C24×C6711yescharacteristic
417SD16—48no—
418C2×C8—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4C2×D4×C6714yes—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—22no—
433C22⋊C4—24no—
434C22⋊C4—22no—
435C22⋊C4—24no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C42—21no—
439C42—21no—
440C42—41no—
441C42—41no—
442C42—42no—
443C42C2×D4×C6712yes—
444C42C2×D4×C6712yes—
445C23—21no—
446C23—21no—
447C23—24no—
448C23—41no—
449C23—41no—
450C23—44no—
451C23—82no—
452C23—82no—
453C23C2×SD16×C6712yes—
454C23C22×D4×C6711yescharacteristic
455Q8—24no—
456Q8—44no—
457Q8—42no—
458Q8—44no—
459D4—44no—
460D4—44no—
461D4—48no—
462D4—44no—
463D4—44no—
464D4—44no—
465D4—82no—
466D4—82no—
467D4—82no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—48no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—48no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4#120×C6712yes—
495C2×C4#120×C6711yescharacteristic
496C2×C4#120×C6711yescharacteristic
497C2×C4C2×SD16×C6712yes—
498C2×C4C2×SD16×C6712yes—
499C2×C4AGL(1,ℤ/8)×C6712yes—
500C2×C4C22×D4×C6712yes—
501C8—44no—
502C22—21no—
503C22—21no—
504C22—22no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—81no—
512C22—88no—
513C22C2×#120×C6711yescharacteristic
514C22C22×SD16×C6711yescharacteristic
515C22C2×AGL(1,ℤ/8)×C6711yescharacteristic
516C4—22no—
517C4—22no—
518C4—22no—
519C4—42no—
520C4—41no—
521C4—42no—
522C4—41no—
523C4—44no—
524C4—81no—
525C4—81no—
526C4#450×C6712yes—
527C4#450×C6712yes—
528C2—21no—
529C2—21no—
530C2—42no—
531C2—84no—
532C2C2×#450×C6711yescharacteristic
533C2C67×#301811yescharacteristic
534C2C67×#302511yescharacteristic
subgroupG/H
centerC2×C4×C67#120
commutator, Frattini, cosocleC22×C4C24×C67
socleC22×C67C2×#120
prime radicalC67×#3449C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#3103C2semidirect
3. C67×#2396C2semidirect
4. C67×#1616C2semidirect
5. C67×#1613C2semidirect
6. C67×#1604C2semidirect
7. C67×#1601C2semidirect
8. C67×#1502C2semidirect
9. C67×#1458C2semidirect
20. C2×C4×Q8×C67C22semidirect
22. #481×C67C22semidirect
25. #448×C67C22semidirect
27. #431×C67C22semidirect
28. #400×C67C22semidirect
32. #395×C67C22semidirect
35. C4×C4⋊C4×C67C22semidirect
37. #327×C67C22semidirect
38. #327×C67C22semidirect
39. #324×C67C22semidirect
142. C22×C4×C67SD16semidirect
260. #18123C67direct (N × H)
269. #3103C2×C67semidirect
270. #2396C2×C67semidirect
271. #1616C2×C67semidirect
272. #1613C2×C67semidirect
273. #1604C2×C67semidirect
274. #1601C2×C67semidirect
275. #1502C2×C67semidirect
276. #1458C2×C67semidirect
277. C67#18123direct (N × H)
288. C2×C4×Q8C22×C67semidirect
290. #481C22×C67semidirect
293. #448C22×C67semidirect
295. #431C22×C67semidirect
296. #400C22×C67semidirect
300. #395C22×C67semidirect
303. C4×C4⋊C4C22×C67semidirect
305. #327C22×C67semidirect
306. #327C22×C67semidirect
307. #324C22×C67semidirect
410. C22×C4SD16×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34492
C67×#31032
C67×#23962
C67×#16162
C67×#16132
C67×#16042
C67×#16012
C67×#15022
C67×#14582
#1812367