Finite Group Databasebeta

← C67×#61289 #30740540

Subgroups — C67×#61289 #30740540

Order 17152.

1774 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3512C212yesmaximal, complemented
2C67×#3351C218yesmaximal, complemented
3C67×#3339C214yesmaximal, complemented
4C67×#3336C214yesmaximal, complemented
5C2×#499×C67C211yesmaximal, characteristic, complemented
6C67×#3099C214yesmaximal, complemented
7C67×#3080C212yesmaximal, complemented
8C67×#3061C212yesmaximal, complemented
9C4×#135×C67C211yesmaximal, characteristic, complemented
10C67×#2931C211yesmaximal, characteristic, complemented
11C67×#2913C212yesmaximal, complemented
12C2×#135×C67C2211yescharacteristic
13C2×#131×C67C2212yes—
14#553×C67—28no—
15#553×C67C2218yes—
16#552×C67C2218yes—
17#541×C67—22no—
18#540×C67—22no—
19#537×C67C2214yes—
20#532×C67C2211yescharacteristic
21#529×C67—24no—
22C2×#125×C67C2212yes—
23#520×C67C2214yes—
24C4×#41×C67C2212yes—
25#499×C67C2214yes—
26#499×C67C2214yes—
27#485×C67C2218yes—
28#483×C67C2218yes—
29#482×C67C2218yes—
30#481×C67C2218yes—
31#479×C67C2218yes—
32#478×C67C2218yes—
33#468×C67C2218yes—
34#467×C67C2214yes—
35#466×C67C2214yes—
36C4×Q16×C67C2212yes—
37C4×SD16×C67C2214yes—
38C4×D8×C67C2212yes—
39#436×C67C2214yes—
40C8×D4×C67C2214yes—
41#428×C67C2214yes—
42#428×C67C2218yes—
43#427×C67C2214yes—
44#427×C67C2211yescharacteristic
45C2×#107×C67C2212yes—
46C2×#107×C67C2214yes—
47C2×#106×C67C2212yes—
48#423×C67C2211yescharacteristic
49#417×C67C2214yes—
50#416×C67C2212yes—
51#408×C67C2212yes—
52C2×C4×C8×C67C2211yescharacteristic
53C2×#41×C67C2312yes—
54C2×#41×C67C2312yes—
55#135×C67—24no—
56C2×Q16×C67C2311yescharacteristic
57C2×SD16×C67C2312yes—
58C2×D8×C67C2311yescharacteristic
59#131×C67—28no—
60C2×M16×C67C2314yes—
61C2×M16×C67C2312yes—
62C22×C8×C67C2314yes—
63C22×C8×C67C2311yescharacteristic
64C22×C8×C67C2311yescharacteristic
65C4⋊Q8×C67D411yescharacteristic
66#127×C67D411yescharacteristic
67#126×C67—28no—
68#126×C67—28no—
69#125×C67—28no—
70#125×C67—22no—
71#125×C67—22no—
72#125×C67C2318yes—
73#124×C67D412yes—
74#123×C67—28no—
75#123×C67—28no—
76C22⋊Q8×C67—24no—
77C22⋊Q8×C67C2318yes—
78#121×C67—24no—
79#121×C67—28no—
80#121×C67—28no—
81#121×C67D414yes—
82#121×C67C2318yes—
83C4×Q8×C67—24no—
84C4×Q8×C67C2314yes—
85C4×D4×C67—24no—
86C4×D4×C67—28no—
87C4×D4×C67—24no—
88C4×D4×C67—28no—
89C4×D4×C67C2314yes—
90C4×D4×C67C2314yes—
91#117×C67—28no—
92#117×C67—22no—
93#117×C67C2318yes—
94#117×C67C2312yes—
95C2×C4⋊C4×C67—24no—
96C2×C4⋊C4×C67C2318yes—
97C2×C4⋊C4×C67C2314yes—
98C2×C42×C67—22no—
99C2×C42×C67C2311yescharacteristic
100#107×C67C2314yes—
101#107×C67C2314yes—
102#107×C67C2318yes—
103#107×C67C2312yes—
104#106×C67C2318yes—
105#106×C67C2318yes—
106#106×C67C2312yes—
107C4⋊C8×C67C2314yes—
108#103×C67C2318yes—
109#103×C67C2314yes—
110#102×C67C2318yes—
111#102×C67C2314yes—
112C22⋊C8×C67C2314yes—
113C22⋊C8×C67C2314yes—
114C4×C8×C67C2312yes—
115C4×C8×C67C2312yes—
116#41×C67—24no—
117#41×C67—24no—
118#41×C67—24no—
119#41×C67—24no—
120C2×Q8×C67C2×D412yes—
121C2×Q8×C67C2412yes—
122C2×D4×C67—22no—
123C2×D4×C67—28no—
124C2×D4×C67C2×D414yes—
125C2×D4×C67C2×D412yes—
126C2×D4×C67C2×D414yes—
127C2×D4×C67C2412yes—
128C22×C4×C67—24no—
129C22×C4×C67—28no—
130C22×C4×C67—22no—
131C22×C4×C67—24no—
132C22×C4×C67C2×D412yes—
133C22×C4×C67C2414yes—
134C22×C4×C67C2411yescharacteristic
135Q16×C67—22no—
136SD16×C67—24no—
137D8×C67—22no—
138M16×C67—24no—
139M16×C67—28no—
140C2×C8×C67—22no—
141C2×C8×C67—28no—
142C2×C8×C67#4114yes—
143C2×C8×C67C2414yes—
144C2×C8×C67C2412yes—
145C2×C8×C67C2411yescharacteristic
146C2×C8×C67C2411yescharacteristic
147C4⋊C4×C67—28no—
148C4⋊C4×C67—28no—
149C4⋊C4×C67—24no—
150C4⋊C4×C67—28no—
151C4⋊C4×C67—24no—
152C4⋊C4×C67—28no—
153C4⋊C4×C67C2×D414yes—
154C4⋊C4×C67C2414yes—
155C4⋊C4×C67C2418yes—
156C22⋊C4×C67—28no—
157C22⋊C4×C67—22no—
158C22⋊C4×C67—28no—
159C22⋊C4×C67—24no—
160C22⋊C4×C67—28no—
161C22⋊C4×C67C2×D414yes—
162C22⋊C4×C67C2×D414yes—
163C42×C67—24no—
164C42×C67—22no—
165C42×C67—28no—
166C42×C67C2×D411yescharacteristic
167C42×C67C2×D411yescharacteristic
168C42×C67C2412yes—
169C23×C67—22no—
170C23×C67C22×D414yes—
171C23×C67C22×D411yescharacteristic
172Q8×C67—24no—
173Q8×C67—22no—
174Q8×C67—22no—
175D4×C67—28no—
176D4×C67—24no—
177D4×C67—22no—
178D4×C67—22no—
179D4×C67—44no—
180D4×C67—44no—
181D4×C67—88no—
182C2×C4×C67—24no—
183C2×C4×C67—24no—
184C2×C4×C67—24no—
185C2×C4×C67—22no—
186C2×C4×C67—28no—
187C2×C4×C67—42no—
188C2×C4×C67—44no—
189C2×C4×C67—84no—
190C2×C4×C67—88no—
191C2×C4×C67—84no—
192C2×C4×C67C22×D412yes—
193C2×C4×C67C22×D414yes—
194C2×C4×C67C22×D412yes—
195C2×C4×C67C22×D411yescharacteristic
196C2×C4×C67C2×#4112yes—
197C2×C4×C672−1+412yes—
198C2×C4×C67C2511yescharacteristic
199C8×C67—24no—
200C8×C67—22no—
201C8×C67C2×#4112yes—
202C8×C67C2×#4112yes—
203C22×C67—22no—
204C22×C67—24no—
205C22×C67—24no—
206C22×C67—24no—
207C22×C67—44no—
208C22×C67—42no—
209C22×C67#54712yes—
210C22×C67C23×D411yescharacteristic
211C4×C67—22no—
212C4×C67—22no—
213C4×C67—24no—
214C4×C67—44no—
215C4×C67—48no—
216C4×C67—42no—
217C4×C67#54712yes—
218C4×C67C22×#4111yescharacteristic
219C4×C67C2×2−1+411yescharacteristic
220#61289C6711yesmaximal, characteristic, direct product factor, Sylow
221C2×C67—22no—
222C2×C67—44no—
223C2×C67—82no—
224C2×C67C2×#54711yescharacteristic
225C2×C67#358412yes—
226#3512C2×C6712yescomplemented
227#3351C2×C6718yescomplemented
228#3339C2×C6714yescomplemented
229#3336C2×C6714yescomplemented
230C2×#499C2×C6711yescharacteristic, complemented
231#3099C2×C6714yescomplemented
232#3080C2×C6712yescomplemented
233#3061C2×C6712yescomplemented
234C4×#135C2×C6711yescharacteristic, complemented
235#2931C2×C6711yescharacteristic, complemented
236#2913C2×C6712yescomplemented
237C67#6128911yescharacteristic, direct product factor, Sylow
238C2×#135C22×C6711yescharacteristic
239C2×#131C22×C6712yes—
240#553—28no—
241#553C22×C6718yes—
242#552C22×C6718yes—
243#541—22no—
244#540—22no—
245#537C22×C6714yes—
246#532C22×C6711yescharacteristic
247#529—24no—
248C2×#125C22×C6712yes—
249#520C22×C6714yes—
250C4×#41C22×C6712yes—
251#499C22×C6714yes—
252#499C22×C6714yes—
253#485C22×C6718yes—
254#483C22×C6718yes—
255#482C22×C6718yes—
256#481C22×C6718yes—
257#479C22×C6718yes—
258#478C22×C6718yes—
259#468C22×C6718yes—
260#467C22×C6714yes—
261#466C22×C6714yes—
262C4×Q16C22×C6712yes—
263C4×SD16C22×C6714yes—
264C4×D8C22×C6712yes—
265#436C22×C6714yes—
266C8×D4C22×C6714yes—
267#428C22×C6714yes—
268#428C22×C6718yes—
269#427C22×C6714yes—
270#427C22×C6711yescharacteristic
271C2×#107C22×C6714yes—
272C2×#107C22×C6712yes—
273C2×#106C22×C6712yes—
274#423C22×C6711yescharacteristic
275#417C22×C6714yes—
276#416C22×C6712yes—
277#408C22×C6712yes—
278C2×C4×C8C22×C6711yescharacteristic
279C2×#41C23×C6712yes—
280C2×#41C23×C6712yes—
281#135—24no—
282C2×Q16C23×C6711yescharacteristic
283C2×SD16C23×C6712yes—
284C2×D8C23×C6711yescharacteristic
285#131—28no—
286C2×M16C23×C6712yes—
287C2×M16C23×C6714yes—
288C22×C8C23×C6714yes—
289C22×C8C23×C6711yescharacteristic
290C22×C8C23×C6711yescharacteristic
291C4⋊Q8D4×C6711yescharacteristic
292#127D4×C6711yescharacteristic
293#126—28no—
294#126—28no—
295#125—28no—
296#125—22no—
297#125—22no—
298#125C23×C6718yes—
299#124D4×C6712yes—
300#123—28no—
301#123—28no—
302C22⋊Q8—24no—
303C22⋊Q8C23×C6718yes—
304#121—24no—
305#121—28no—
306#121—28no—
307#121D4×C6714yes—
308#121C23×C6718yes—
309C4×Q8—24no—
310C4×Q8C23×C6714yes—
311C4×D4—24no—
312C4×D4—28no—
313C4×D4—24no—
314C4×D4—28no—
315C4×D4C23×C6714yes—
316C4×D4C23×C6714yes—
317#117—28no—
318#117—22no—
319#117C23×C6712yes—
320#117C23×C6718yes—
321C2×C4⋊C4—24no—
322C2×C4⋊C4C23×C6714yes—
323C2×C4⋊C4C23×C6718yes—
324C2×C42—22no—
325C2×C42C23×C6711yescharacteristic
326#107C23×C6712yes—
327#107C23×C6714yes—
328#107C23×C6714yes—
329#107C23×C6718yes—
330#106C23×C6718yes—
331#106C23×C6712yes—
332#106C23×C6718yes—
333C4⋊C8C23×C6714yes—
334#103C23×C6714yes—
335#103C23×C6718yes—
336#102C23×C6714yes—
337#102C23×C6718yes—
338C22⋊C8C23×C6714yes—
339C22⋊C8C23×C6714yes—
340C4×C8C23×C6712yes—
341C4×C8C23×C6712yes—
342#41—24no—
343#41—24no—
344#41—24no—
345#41—24no—
346C2×Q8C2×D4×C6712yes—
347C2×Q8C24×C6712yes—
348C2×D4—22no—
349C2×D4—28no—
350C2×D4C2×D4×C6714yes—
351C2×D4C2×D4×C6712yes—
352C2×D4C2×D4×C6714yes—
353C2×D4C24×C6712yes—
354C22×C4—24no—
355C22×C4—28no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4C2×D4×C6712yes—
359C22×C4C24×C6714yes—
360C22×C4C24×C6711yescharacteristic
361Q16—22no—
362SD16—24no—
363D8—22no—
364M16—24no—
365M16—28no—
366C2×C8—22no—
367C2×C8—28no—
368C2×C8#41×C6714yes—
369C2×C8C24×C6712yes—
370C2×C8C24×C6714yes—
371C2×C8C24×C6711yescharacteristic
372C2×C8C24×C6711yescharacteristic
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4C2×D4×C6714yes—
380C4⋊C4C24×C6714yes—
381C4⋊C4C24×C6718yes—
382C22⋊C4—28no—
383C22⋊C4—22no—
384C22⋊C4—28no—
385C22⋊C4—24no—
386C22⋊C4—28no—
387C22⋊C4C2×D4×C6714yes—
388C22⋊C4C2×D4×C6714yes—
389C42—24no—
390C42—22no—
391C42—28no—
392C42C2×D4×C6711yescharacteristic
393C42C2×D4×C6711yescharacteristic
394C42C24×C6712yes—
395C23—22no—
396C23C22×D4×C6714yes—
397C23C22×D4×C6711yescharacteristic
398Q8—24no—
399Q8—22no—
400Q8—22no—
401D4—28no—
402D4—24no—
403D4—22no—
404D4—22no—
405D4—44no—
406D4—44no—
407D4—88no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—28no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—84no—
416C2×C4—88no—
417C2×C4—84no—
418C2×C4C22×D4×C6712yes—
419C2×C4C22×D4×C6714yes—
420C2×C4C22×D4×C6712yes—
421C2×C4C22×D4×C6711yescharacteristic
422C2×C4C2×#41×C6712yes—
423C2×C42−1+4×C6712yes—
424C2×C4C25×C6711yescharacteristic
425C8—24no—
426C8—22no—
427C8C2×#41×C6712yes—
428C8C2×#41×C6712yes—
429C22—22no—
430C22—24no—
431C22—24no—
432C22—24no—
433C22—44no—
434C22—42no—
435C22#547×C6712yes—
436C22C23×D4×C6711yescharacteristic
437C4—22no—
438C4—22no—
439C4—24no—
440C4—44no—
441C4—48no—
442C4—42no—
443C4#547×C6712yes—
444C4C22×#41×C6711yescharacteristic
445C4C2×2−1+4×C6711yescharacteristic
446C2—22no—
447C2—44no—
448C2—82no—
449C2C2×#547×C6711yescharacteristic
450C2C67×#358412yes—
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×#3512C2semidirect
2. C67×#3351C2semidirect
3. C67×#3339C2semidirect
4. C67×#3336C2semidirect
5. C2×#499×C67C2semidirect
6. C67×#3099C2semidirect
7. C67×#3080C2semidirect
8. C67×#3061C2semidirect
9. C4×#135×C67C2semidirect
10. C67×#2931C2semidirect
11. C67×#2913C2semidirect
220. #61289C67direct (N × H)
226. #3512C2×C67semidirect
227. #3351C2×C67semidirect
228. #3339C2×C67semidirect
229. #3336C2×C67semidirect
230. C2×#499C2×C67semidirect
231. #3099C2×C67semidirect
232. #3080C2×C67semidirect
233. #3061C2×C67semidirect
234. C4×#135C2×C67semidirect
235. #2931C2×C67semidirect
236. #2913C2×C67semidirect
237. C67#61289direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#35122
C67×#33512
C67×#33392
C67×#33362
C2×#499×C672
C67×#30992
C67×#30802
C67×#30612
C4×#135×C672
C67×#29312
C67×#29132
#6128967