Finite Group Databasebeta

← C67×#61288 #30740539

Subgroups — C67×#61288 #30740539

Order 17152.

1846 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3511C212yesmaximal, complemented
2C67×#3356C214yesmaximal, complemented
3C67×#3354C214yesmaximal, complemented
4C67×#3340C214yesmaximal, complemented
5C67×#3335C214yesmaximal, complemented
6C2×#498×C67C211yesmaximal, characteristic, complemented
7C67×#3099C214yesmaximal, complemented
8C67×#3079C212yesmaximal, complemented
9C67×#3060C212yesmaximal, complemented
10C4×#135×C67C211yesmaximal, characteristic, complemented
11C67×#2931C211yesmaximal, characteristic, complemented
12C67×#2912C212yesmaximal, complemented
13C2×#135×C67C2211yescharacteristic
14C2×#131×C67C2212yes—
15#556×C67C2214yes—
16#555×C67C2214yes—
17#554×C67C2214yes—
18#553×C67—24no—
19#548×C67—24no—
20#547×C67C2214yes—
21#542×C67—24no—
22#537×C67C2214yes—
23#533×C67—22no—
24#532×C67—22no—
25#532×C67C2211yescharacteristic
26C2×C4⋊Q8×C67C2212yes—
27#520×C67C2214yes—
28C4×#41×C67C2212yes—
29#498×C67C2214yes—
30#498×C67C2214yes—
31#485×C67C2218yes—
32#483×C67C2218yes—
33#482×C67C2218yes—
34#481×C67C2218yes—
35#477×C67C2214yes—
36#476×C67C2214yes—
37#475×C67C2214yes—
38#474×C67C2214yes—
39#470×C67C2214yes—
40#469×C67C2214yes—
41#465×C67C2218yes—
42C4×Q16×C67C2212yes—
43C4×SD16×C67C2212yes—
44C4×SD16×C67C2212yes—
45C4×D8×C67C2212yes—
46#436×C67C2214yes—
47C8×D4×C67C2214yes—
48#428×C67C2214yes—
49#428×C67C2218yes—
50#427×C67C2214yes—
51#427×C67C2211yescharacteristic
52C2×#106×C67C2212yes—
53C2×#106×C67C2214yes—
54C2×#106×C67C2212yes—
55#423×C67C2211yescharacteristic
56#417×C67C2214yes—
57#416×C67C2212yes—
58#408×C67C2212yes—
59C2×C4×C8×C67C2211yescharacteristic
60C2×#41×C67—24no—
61C2×#41×C67C2312yes—
62C2×#41×C67C2312yes—
63C22×Q8×C67—22no—
64#135×C67—24no—
65C2×Q16×C67C2311yescharacteristic
66C2×SD16×C67C2312yes—
67C2×D8×C67C2311yescharacteristic
68#131×C67—28no—
69C2×M16×C67C2314yes—
70C2×M16×C67C2312yes—
71C22×C8×C67C2314yes—
72C22×C8×C67C2311yescharacteristic
73C22×C8×C67C2311yescharacteristic
74C4⋊Q8×C67—22no—
75C4⋊Q8×C67—22no—
76C4⋊Q8×C67D411yescharacteristic
77C4⋊Q8×C67C2314yes—
78C4⋊Q8×C67C2314yes—
79#127×C67D411yescharacteristic
80#126×C67—28no—
81#125×C67—24no—
82#124×C67—24no—
83#124×C67—28no—
84#124×C67D412yes—
85#123×C67—28no—
86C22⋊Q8×C67—24no—
87C22⋊Q8×C67—24no—
88C22⋊Q8×C67—28no—
89C22⋊Q8×C67C2318yes—
90#121×C67—24no—
91#121×C67—24no—
92#121×C67—28no—
93#121×C67D414yes—
94#121×C67C2318yes—
95C4×Q8×C67—24no—
96C4×Q8×C67—24no—
97C4×Q8×C67C2312yes—
98C4×Q8×C67C2312yes—
99C4×D4×C67—24no—
100C4×D4×C67—28no—
101C4×D4×C67—24no—
102C4×D4×C67—22no—
103C4×D4×C67—22no—
104C4×D4×C67C2312yes—
105C4×D4×C67C2312yes—
106C4×D4×C67C2314yes—
107#117×C67—24no—
108#117×C67—22no—
109#117×C67C2318yes—
110#117×C67C2312yes—
111C2×C4⋊C4×C67—22no—
112C2×C4⋊C4×C67C2318yes—
113C2×C4⋊C4×C67C2314yes—
114C2×C42×C67—22no—
115C2×C42×C67C2311yescharacteristic
116#107×C67C2318yes—
117#107×C67C2312yes—
118#106×C67C2314yes—
119#106×C67C2314yes—
120#106×C67C2318yes—
121#106×C67C2318yes—
122#106×C67C2311yescharacteristic
123#106×C67C2311yescharacteristic
124C4⋊C8×C67C2314yes—
125#103×C67C2318yes—
126#103×C67C2312yes—
127#103×C67C2312yes—
128#102×C67C2318yes—
129#102×C67C2312yes—
130#102×C67C2312yes—
131C22⋊C8×C67C2318yes—
132C4×C8×C67C2312yes—
133C4×C8×C67C2312yes—
134#41×C67—24no—
135#41×C67—24no—
136#41×C67—24no—
137#41×C67—24no—
138#41×C67—88no—
139C2×Q8×C67—24no—
140C2×Q8×C67—24no—
141C2×Q8×C67—84no—
142C2×Q8×C67C2×D412yes—
143C2×Q8×C67C2412yes—
144C2×D4×C67—22no—
145C2×D4×C67—28no—
146C2×D4×C67C2×D414yes—
147C2×D4×C67C2×D412yes—
148C2×D4×C67C2×D414yes—
149C2×D4×C67C2412yes—
150C22×C4×C67—24no—
151C22×C4×C67—28no—
152C22×C4×C67—22no—
153C22×C4×C67—22no—
154C22×C4×C67—22no—
155C22×C4×C67C2×D412yes—
156C22×C4×C67C2414yes—
157C22×C4×C67C2411yescharacteristic
158Q16×C67—22no—
159SD16×C67—24no—
160D8×C67—22no—
161M16×C67—24no—
162M16×C67—28no—
163C2×C8×C67—22no—
164C2×C8×C67—28no—
165C2×C8×C67#4114yescomplemented
166C2×C8×C67C2414yes—
167C2×C8×C67C2412yes—
168C2×C8×C67C2411yescharacteristic
169C2×C8×C67C2411yescharacteristic
170C4⋊C4×C67—24no—
171C4⋊C4×C67—24no—
172C4⋊C4×C67—24no—
173C4⋊C4×C67—28no—
174C4⋊C4×C67—22no—
175C4⋊C4×C67—22no—
176C4⋊C4×C67—28no—
177C4⋊C4×C67C2×D414yes—
178C4⋊C4×C67C2412yes—
179C4⋊C4×C67C2418yes—
180C4⋊C4×C67C2412yes—
181C22⋊C4×C67—22no—
182C22⋊C4×C67—28no—
183C22⋊C4×C67—28no—
184C22⋊C4×C67—22no—
185C22⋊C4×C67—22no—
186C22⋊C4×C67—28no—
187C22⋊C4×C67C2×D418yes—
188C42×C67—22no—
189C42×C67—22no—
190C42×C67—22no—
191C42×C67—28no—
192C42×C67C2×D411yescharacteristic
193C42×C67C2×D411yescharacteristic
194C42×C67C2412yes—
195C23×C67—22no—
196C23×C67C22×D414yes—
197C23×C67C22×D411yescharacteristic
198Q8×C67—24no—
199Q8×C67—22no—
200Q8×C67—22no—
201Q8×C67—48no—
202Q8×C67—84no—
203D4×C67—28no—
204D4×C67—24no—
205D4×C67—28no—
206D4×C67—22no—
207D4×C67—22no—
208D4×C67—44no—
209D4×C67—88no—
210C2×C4×C67—24no—
211C2×C4×C67—24no—
212C2×C4×C67—24no—
213C2×C4×C67—22no—
214C2×C4×C67—22no—
215C2×C4×C67—22no—
216C2×C4×C67—28no—
217C2×C4×C67—44no—
218C2×C4×C67—44no—
219C2×C4×C67—44no—
220C2×C4×C67—84no—
221C2×C4×C67—88no—
222C2×C4×C67C22×D414yes—
223C2×C4×C67C22×D412yes—
224C2×C4×C67C22×D411yescharacteristic
225C2×C4×C67C22×D412yes—
226C2×C4×C67C2×#4112yes—
227C2×C4×C672−1+412yes—
228C2×C4×C67C2511yescharacteristic
229C8×C67—24no—
230C8×C67—22no—
231C8×C67C2×#4112yes—
232C8×C67C2×#4112yes—
233C22×C67—22no—
234C22×C67—24no—
235C22×C67—28no—
236C22×C67—42no—
237C22×C67—42no—
238C22×C67—42no—
239C22×C67#54712yes—
240C22×C67C23×D411yescharacteristic
241C4×C67—24no—
242C4×C67—22no—
243C4×C67—42no—
244C4×C67—42no—
245C4×C67—42no—
246C4×C67—48no—
247C4×C67#54712yes—
248C4×C67#57714yes—
249C4×C67C22×#4111yescharacteristic
250C4×C67C2×2−1+411yescharacteristic
251#61288C6711yesmaximal, characteristic, direct product factor, Sylow
252C2×C67—22no—
253C2×C67—44no—
254C2×C67—82no—
255C2×C67C2×#54711yescharacteristic
256C2×C67C2×#57712yes—
257#3511C2×C6712yescomplemented
258#3356C2×C6714yescomplemented
259#3354C2×C6714yescomplemented
260#3340C2×C6714yescomplemented
261#3335C2×C6714yescomplemented
262C2×#498C2×C6711yescharacteristic, complemented
263#3099C2×C6714yescomplemented
264#3079C2×C6712yescomplemented
265#3060C2×C6712yescomplemented
266C4×#135C2×C6711yescharacteristic, complemented
267#2931C2×C6711yescharacteristic, complemented
268#2912C2×C6712yescomplemented
269C67#6128811yescharacteristic, direct product factor, Sylow
270C2×#135C22×C6711yescharacteristic
271C2×#131C22×C6712yes—
272#556C22×C6714yes—
273#555C22×C6714yes—
274#554C22×C6714yes—
275#553—24no—
276#548—24no—
277#547C22×C6714yes—
278#542—24no—
279#537C22×C6714yes—
280#533—22no—
281#532—22no—
282#532C22×C6711yescharacteristic
283C2×C4⋊Q8C22×C6712yes—
284#520C22×C6714yes—
285C4×#41C22×C6712yes—
286#498C22×C6714yes—
287#498C22×C6714yes—
288#485C22×C6718yes—
289#483C22×C6718yes—
290#482C22×C6718yes—
291#481C22×C6718yes—
292#477C22×C6714yes—
293#476C22×C6714yes—
294#475C22×C6714yes—
295#474C22×C6714yes—
296#470C22×C6714yes—
297#469C22×C6714yes—
298#465C22×C6718yes—
299C4×Q16C22×C6712yes—
300C4×SD16C22×C6712yes—
301C4×SD16C22×C6712yes—
302C4×D8C22×C6712yes—
303#436C22×C6714yes—
304C8×D4C22×C6714yes—
305#428C22×C6714yes—
306#428C22×C6718yes—
307#427C22×C6714yes—
308#427C22×C6711yescharacteristic
309C2×#106C22×C6714yes—
310C2×#106C22×C6712yes—
311C2×#106C22×C6712yes—
312#423C22×C6711yescharacteristic
313#417C22×C6714yes—
314#416C22×C6712yes—
315#408C22×C6712yes—
316C2×C4×C8C22×C6711yescharacteristic
317C2×#41—24no—
318C2×#41C23×C6712yes—
319C2×#41C23×C6712yes—
320C22×Q8—22no—
321#135—24no—
322C2×Q16C23×C6711yescharacteristic
323C2×SD16C23×C6712yes—
324C2×D8C23×C6711yescharacteristic
325#131—28no—
326C2×M16C23×C6712yes—
327C2×M16C23×C6714yes—
328C22×C8C23×C6714yes—
329C22×C8C23×C6711yescharacteristic
330C22×C8C23×C6711yescharacteristic
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8D4×C6711yescharacteristic
334C4⋊Q8C23×C6714yes—
335C4⋊Q8C23×C6714yes—
336#127D4×C6711yescharacteristic
337#126—28no—
338#125—24no—
339#124—24no—
340#124—28no—
341#124D4×C6712yes—
342#123—28no—
343C22⋊Q8—24no—
344C22⋊Q8—24no—
345C22⋊Q8—28no—
346C22⋊Q8C23×C6718yes—
347#121—24no—
348#121—24no—
349#121—28no—
350#121D4×C6714yes—
351#121C23×C6718yes—
352C4×Q8—24no—
353C4×Q8—24no—
354C4×Q8C23×C6712yes—
355C4×Q8C23×C6712yes—
356C4×D4—24no—
357C4×D4—28no—
358C4×D4—24no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4C23×C6712yes—
362C4×D4C23×C6712yes—
363C4×D4C23×C6714yes—
364#117—24no—
365#117—22no—
366#117C23×C6712yes—
367#117C23×C6718yes—
368C2×C4⋊C4—22no—
369C2×C4⋊C4C23×C6714yes—
370C2×C4⋊C4C23×C6718yes—
371C2×C42—22no—
372C2×C42C23×C6711yescharacteristic
373#107C23×C6712yes—
374#107C23×C6718yes—
375#106C23×C6711yescharacteristic
376#106C23×C6714yes—
377#106C23×C6714yes—
378#106C23×C6718yes—
379#106C23×C6718yes—
380#106C23×C6711yescharacteristic
381C4⋊C8C23×C6714yes—
382#103C23×C6712yes—
383#103C23×C6712yes—
384#103C23×C6718yes—
385#102C23×C6712yes—
386#102C23×C6712yes—
387#102C23×C6718yes—
388C22⋊C8C23×C6718yes—
389C4×C8C23×C6712yes—
390C4×C8C23×C6712yes—
391#41—24no—
392#41—24no—
393#41—24no—
394#41—24no—
395#41—88no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—84no—
399C2×Q8C2×D4×C6712yes—
400C2×Q8C24×C6712yes—
401C2×D4—22no—
402C2×D4—28no—
403C2×D4C2×D4×C6714yes—
404C2×D4C2×D4×C6712yes—
405C2×D4C2×D4×C6714yes—
406C2×D4C24×C6712yes—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4C2×D4×C6712yes—
413C22×C4C24×C6714yes—
414C22×C4C24×C6711yescharacteristic
415Q16—22no—
416SD16—24no—
417D8—22no—
418M16—24no—
419M16—28no—
420C2×C8—22no—
421C2×C8—28no—
422C2×C8#41×C6714yescomplemented
423C2×C8C24×C6712yes—
424C2×C8C24×C6714yes—
425C2×C8C24×C6711yescharacteristic
426C2×C8C24×C6711yescharacteristic
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—28no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—28no—
434C4⋊C4C2×D4×C6714yes—
435C4⋊C4C24×C6712yes—
436C4⋊C4C24×C6718yes—
437C4⋊C4C24×C6712yes—
438C22⋊C4—22no—
439C22⋊C4—28no—
440C22⋊C4—28no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—28no—
444C22⋊C4C2×D4×C6718yes—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—28no—
449C42C2×D4×C6711yescharacteristic
450C42C2×D4×C6711yescharacteristic
451C42C24×C6712yes—
452C23—22no—
453C23C22×D4×C6714yes—
454C23C22×D4×C6711yescharacteristic
455Q8—24no—
456Q8—22no—
457Q8—22no—
458Q8—48no—
459Q8—84no—
460D4—28no—
461D4—24no—
462D4—28no—
463D4—22no—
464D4—22no—
465D4—44no—
466D4—88no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—28no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—84no—
478C2×C4—88no—
479C2×C4C22×D4×C6714yes—
480C2×C4C22×D4×C6712yes—
481C2×C4C22×D4×C6711yescharacteristic
482C2×C4C22×D4×C6712yes—
483C2×C4C2×#41×C6712yes—
484C2×C42−1+4×C6712yes—
485C2×C4C25×C6711yescharacteristic
486C8—24no—
487C8—22no—
488C8C2×#41×C6712yes—
489C8C2×#41×C6712yes—
490C22—22no—
491C22—28no—
492C22—24no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22#547×C6712yes—
497C22C23×D4×C6711yescharacteristic
498C4—22no—
499C4—24no—
500C4—42no—
501C4—42no—
502C4—48no—
503C4—42no—
504C4#547×C6712yes—
505C4#577×C6714yes—
506C4C22×#41×C6711yescharacteristic
507C4C2×2−1+4×C6711yescharacteristic
508C2—22no—
509C2—44no—
510C2—82no—
511C2C2×#547×C6711yescharacteristic
512C2C2×#577×C6712yes—
subgroupG/H
center, socleC22×C67C23×D4
commutator, Frattini, cosocleC2×C4C25×C67
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ #532×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3511C2semidirect
2. C67×#3356C2semidirect
3. C67×#3354C2semidirect
4. C67×#3340C2semidirect
5. C67×#3335C2semidirect
6. C2×#498×C67C2semidirect
7. C67×#3099C2semidirect
8. C67×#3079C2semidirect
9. C67×#3060C2semidirect
10. C4×#135×C67C2semidirect
11. C67×#2931C2semidirect
12. C67×#2912C2semidirect
165. C2×C8×C67#41semidirect
251. #61288C67direct (N × H)
257. #3511C2×C67semidirect
258. #3356C2×C67semidirect
259. #3354C2×C67semidirect
260. #3340C2×C67semidirect
261. #3335C2×C67semidirect
262. C2×#498C2×C67semidirect
263. #3099C2×C67semidirect
264. #3079C2×C67semidirect
265. #3060C2×C67semidirect
266. C4×#135C2×C67semidirect
267. #2931C2×C67semidirect
268. #2912C2×C67semidirect
269. C67#61288direct (N × H)
422. C2×C8#41×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#35112
C67×#33562
C67×#33542
C67×#33402
C67×#33352
C2×#498×C672
C67×#30992
C67×#30792
C67×#30602
C4×#135×C672
C67×#29312
C67×#29122
#6128867