Finite Group Databasebeta

← C67×#61588 #30740839

Subgroups — C67×#61588 #30740839

Order 17152.

2150 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3501C211yesmaximal, characteristic, complemented
2C2×#550×C67C211yesmaximal, characteristic
3C67×#3333C218yesmaximal, complemented
4C67×#3278C218yesmaximal, complemented
5C67×#3208C212yesmaximal, complemented
6C67×#3193C214yesmaximal, complemented
7C67×#3104C212yesmaximal, complemented
8C67×#3007C214yesmaximal, complemented
9C67×#2975C211yesmaximal, characteristic, complemented
10C22×#41×C67C2212yes—
11C2×#137×C67C2218yescomplemented
12#554×C67—24no—
13#554×C67C2214yescomplemented
14#550×C67—24no—
15#550×C67C2218yes—
16D4×Q8×C67C2214yescomplemented
17#544×C67—21no—
18#543×C67—22no—
19#533×C67C2212yes—
20C2×#127×C67—21no—
21C2×#127×C67C2211yescharacteristic
22C2×C22⋊Q8×C67C2214yes—
23C2×#121×C67—22no—
24C2×#121×C67C2212yes—
25#518×C67—21no—
26C2×C4×Q8×C67C2211yescharacteristic
27C2×C4×D4×C67—21no—
28C2×C4×D4×C67C2211yescharacteristic
29#492×C67C2214yescomplemented
30#486×C67C2214yescomplemented
31#484×C67C2214yes—
32#480×C67C2218yescomplemented
33#477×C67C2214yescomplemented
34#468×C67C22116yescomplemented
35#462×C67C2214yescomplemented
36#460×C67C2218yescomplemented
37#452×C67C22116yescomplemented
38#451×C67C2214yes—
39#450×C67C2218yes—
40#448×C67C2218yescomplemented
41C4×Q16×C67C2218yescomplemented
42#436×C67C2218yescomplemented
43#431×C67C2212yes—
44#417×C67C2218yescomplemented
45#407×C67C2214yes—
46C2×#41×C67—24no—
47C2×#41×C67—48no—
48C2×#41×C67C2318yes—
49C22×Q8×C67C2311yescharacteristic
50C22×Q8×C67C2314yes—
51C22×D4×C67—22no—
52C22×D4×C67—22no—
53C22×D4×C67—21no—
54C22×D4×C67—21no—
55C22×D4×C67D411yescharacteristic
56C22×D4×C67C2312yes—
57C23×C4×C67—22no—
58C23×C4×C67C2312yes—
59#137×C67—416no—
60C2×Q16×C67C2318yes—
61C2×SD16×C67C23116yes—
62C2×M16×C67C2318yes—
63C4⋊Q8×C67—24no—
64C4⋊Q8×C67C2312yes—
65C4⋊Q8×C67C2312yes—
66#127×C67—28no—
67#127×C67—22no—
68#127×C67C2314yes—
69#125×C67—24no—
70#125×C67—22no—
71C22⋊Q8×C67—24no—
72C22⋊Q8×C67—24no—
73C22⋊Q8×C67—22no—
74C22⋊Q8×C67—28no—
75C22⋊Q8×C67—28no—
76C22⋊Q8×C67C2314yes—
77C22⋊Q8×C67C2318yes—
78#121×C67—24no—
79#121×C67—216no—
80#121×C67C2318yes—
81C4×Q8×C67—22no—
82C4×Q8×C67—22no—
83C4×Q8×C67C2314yes—
84C4×Q8×C67C2314yes—
85C4×D4×C67—24no—
86C4×D4×C67—24no—
87C4×D4×C67—28no—
88C4×D4×C67D414yes—
89C4×D4×C67C2314yes—
90#117×C67—22no—
91C2×C4⋊C4×C67—21no—
92C2×C4⋊C4×C67—24no—
93C2×C4⋊C4×C67D411yescharacteristic
94C2×C4⋊C4×C67C2314yes—
95C2×C4⋊C4×C67C2311yescharacteristic
96C2×C22⋊C4×C67—24no—
97C2×C22⋊C4×C67—22no—
98C2×C22⋊C4×C67D412yes—
99C2×C42×C67—21no—
100C2×C42×C67C2311yescharacteristic
101#107×C67C2318yes—
102C4⋊C8×C67C2314yes—
103#103×C67C2318yes—
104#103×C67C2318yes—
105#102×C67C23116yes—
106C22⋊C8×C67C2318yes—
107C22⋊C8×C67C2314yes—
108C4×C8×C67C2314yes—
109C24×C67—22no—
110C24×C67C2×D412yes—
111#41×C67—216no—
112#41×C67—48no—
113#41×C67—416no—
114C2×Q8×C67—24no—
115C2×Q8×C67—21no—
116C2×Q8×C67—22no—
117C2×Q8×C67—24no—
118C2×Q8×C67—48no—
119C2×Q8×C67C2414yes—
120C2×Q8×C67C2412yes—
121C2×D4×C67—28no—
122C2×D4×C67—28no—
123C2×D4×C67—24no—
124C2×D4×C67—24no—
125C2×D4×C67—22no—
126C2×D4×C67—24no—
127C2×D4×C67—44no—
128C2×D4×C67—48no—
129C2×D4×C67—44no—
130C2×D4×C67—82no—
131C2×D4×C67—84no—
132C2×D4×C67C2×D414yes—
133C2×D4×C67C2414yes—
134C22×C4×C67—21no—
135C22×C4×C67—21no—
136C22×C4×C67—24no—
137C22×C4×C67—24no—
138C22×C4×C67—28no—
139C22×C4×C67—44no—
140C22×C4×C67—48no—
141C22×C4×C67C2×D411yescharacteristic
142C22×C4×C67C2×D412yes—
143C22×C4×C67C2×D414yes—
144C22×C4×C67C2×D411yescharacteristic
145C22×C4×C67C2411yescharacteristic
146C22×C4×C67C2414yes—
147Q16×C67—216no—
148SD16×C67—416no—
149M16×C67—48no—
150C2×C8×C67C2418yes—
151C4⋊C4×C67—24no—
152C4⋊C4×C67—24no—
153C4⋊C4×C67—28no—
154C4⋊C4×C67—21no—
155C4⋊C4×C67—24no—
156C4⋊C4×C67—22no—
157C4⋊C4×C67—22no—
158C4⋊C4×C67—22no—
159C4⋊C4×C67—24no—
160C4⋊C4×C67C2×D414yes—
161C4⋊C4×C67C2414yes—
162C4⋊C4×C67C2412yes—
163C22⋊C4×C67—24no—
164C22⋊C4×C67—28no—
165C22⋊C4×C67—28no—
166C22⋊C4×C67C2×D418yes—
167C42×C67—24no—
168C42×C67—24no—
169C42×C67C2×D412yes—
170C42×C67C2412yes—
171C23×C67—24no—
172C23×C67—24no—
173C23×C67—42no—
174C23×C67—42no—
175C23×C67—44no—
176C23×C67—44no—
177C23×C67—44no—
178C23×C67C22×D414yes—
179C23×C67C22×D411yescharacteristic
180Q8×C67—24no—
181Q8×C67—22no—
182Q8×C67—28no—
183Q8×C67—82no—
184Q8×C67—84no—
185Q8×C67H(5,𝔽2)14yes—
186D4×C67—28no—
187D4×C67—28no—
188D4×C67—416no—
189D4×C67—44no—
190D4×C67—48no—
191D4×C67—84no—
192D4×C67—88no—
193C2×C4×C67—24no—
194C2×C4×C67—22no—
195C2×C4×C67—28no—
196C2×C4×C67—22no—
197C2×C4×C67—22no—
198C2×C4×C67—24no—
199C2×C4×C67—24no—
200C2×C4×C67—42no—
201C2×C4×C67—44no—
202C2×C4×C67—41no—
203C2×C4×C67—416no—
204C2×C4×C67—42no—
205C2×C4×C67—84no—
206C2×C4×C67—84no—
207C2×C4×C67#13714yescomplemented
208C2×C4×C67C22×D411yescharacteristic
209C2×C4×C67C22×D412yes—
210C2×C4×C67C22×D414yes—
211C2×C4×C67C22×D412yes—
212C2×C4×C67C2511yescharacteristic
213C8×C67—28no—
214C22×C67—21no—
215C22×C67—21no—
216C22×C67—24no—
217C22×C67—24no—
218C22×C67—24no—
219C22×C67—44no—
220C22×C67—44no—
221C22×C67—44no—
222C22×C67—84no—
223C22×C67—88no—
224C22×C67C2×#13712yes—
225C22×C67C23×D411yescharacteristic
226C4×C67—22no—
227C4×C67—24no—
228C4×C67—24no—
229C4×C67—21no—
230C4×C67—42no—
231C4×C67—44no—
232C4×C67—44no—
233C4×C67C2×#13714yes—
234C4×C67C2×H(5,𝔽2)12yes—
235#61588C6711yesmaximal, characteristic, direct product factor, Sylow
236C2×C67—22no—
237C2×C67—44no—
238C2×C67—84no—
239C2×C67#348411yescharacteristic
240C2×C67C22×#13711yescharacteristic
241C2×C67#358511yescharacteristic
242#3501C2×C6711yescharacteristic, complemented
243C2×#550C2×C6711yescharacteristic
244#3333C2×C6718yescomplemented
245#3278C2×C6718yescomplemented
246#3208C2×C6712yescomplemented
247#3193C2×C6714yescomplemented
248#3104C2×C6712yescomplemented
249#3007C2×C6714yescomplemented
250#2975C2×C6711yescharacteristic, complemented
251C67#6158811yescharacteristic, direct product factor, Sylow
252C22×#41C22×C6712yes—
253C2×#137C22×C6718yescomplemented
254#554—24no—
255#554C22×C6714yescomplemented
256#550—24no—
257#550C22×C6718yes—
258D4×Q8C22×C6714yescomplemented
259#544—21no—
260#543—22no—
261#533C22×C6712yes—
262C2×#127—21no—
263C2×#127C22×C6711yescharacteristic
264C2×C22⋊Q8C22×C6714yes—
265C2×#121—22no—
266C2×#121C22×C6712yes—
267#518—21no—
268C2×C4×Q8C22×C6711yescharacteristic
269C2×C4×D4—21no—
270C2×C4×D4C22×C6711yescharacteristic
271#492C22×C6714yescomplemented
272#486C22×C6714yescomplemented
273#484C22×C6714yes—
274#480C22×C6718yescomplemented
275#477C22×C6714yescomplemented
276#468C22×C67116yescomplemented
277#462C22×C6714yescomplemented
278#460C22×C6718yescomplemented
279#452C22×C67116yescomplemented
280#451C22×C6714yes—
281#450C22×C6718yes—
282#448C22×C6718yescomplemented
283C4×Q16C22×C6718yescomplemented
284#436C22×C6718yescomplemented
285#431C22×C6712yes—
286#417C22×C6718yescomplemented
287#407C22×C6714yes—
288C2×#41—24no—
289C2×#41—48no—
290C2×#41C23×C6718yes—
291C22×Q8C23×C6711yescharacteristic
292C22×Q8C23×C6714yes—
293C22×D4—22no—
294C22×D4—22no—
295C22×D4—21no—
296C22×D4—21no—
297C22×D4D4×C6711yescharacteristic
298C22×D4C23×C6712yes—
299C23×C4—22no—
300C23×C4C23×C6712yes—
301#137—416no—
302C2×Q16C23×C6718yes—
303C2×SD16C23×C67116yes—
304C2×M16C23×C6718yes—
305C4⋊Q8—24no—
306C4⋊Q8C23×C6712yes—
307C4⋊Q8C23×C6712yes—
308#127—28no—
309#127—22no—
310#127C23×C6714yes—
311#125—24no—
312#125—22no—
313C22⋊Q8—24no—
314C22⋊Q8—24no—
315C22⋊Q8—22no—
316C22⋊Q8—28no—
317C22⋊Q8—28no—
318C22⋊Q8C23×C6714yes—
319C22⋊Q8C23×C6718yes—
320#121—24no—
321#121—216no—
322#121C23×C6718yes—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×Q8C23×C6714yes—
326C4×Q8C23×C6714yes—
327C4×D4—24no—
328C4×D4—24no—
329C4×D4—28no—
330C4×D4D4×C6714yes—
331C4×D4C23×C6714yes—
332#117—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4D4×C6711yescharacteristic
336C2×C4⋊C4C23×C6711yescharacteristic
337C2×C4⋊C4C23×C6714yes—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4D4×C6712yes—
341C2×C42—21no—
342C2×C42C23×C6711yescharacteristic
343#107C23×C6718yes—
344C4⋊C8C23×C6714yes—
345#103C23×C6718yes—
346#103C23×C6718yes—
347#102C23×C67116yes—
348C22⋊C8C23×C6714yes—
349C22⋊C8C23×C6718yes—
350C4×C8C23×C6714yes—
351C24—22no—
352C24C2×D4×C6712yes—
353#41—216no—
354#41—48no—
355#41—416no—
356C2×Q8—24no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—48no—
361C2×Q8C24×C6714yes—
362C2×Q8C24×C6712yes—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—22no—
368C2×D4—24no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—44no—
372C2×D4—82no—
373C2×D4—84no—
374C2×D4C2×D4×C6714yes—
375C2×D4C24×C6714yes—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—44no—
382C22×C4—48no—
383C22×C4C2×D4×C6711yescharacteristic
384C22×C4C2×D4×C6712yes—
385C22×C4C2×D4×C6714yes—
386C22×C4C2×D4×C6711yescharacteristic
387C22×C4C24×C6711yescharacteristic
388C22×C4C24×C6714yes—
389Q16—216no—
390SD16—416no—
391M16—48no—
392C2×C8C24×C6718yes—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—28no—
396C4⋊C4—21no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4C2×D4×C6714yes—
403C4⋊C4C24×C6714yes—
404C4⋊C4C24×C6712yes—
405C22⋊C4—24no—
406C22⋊C4—28no—
407C22⋊C4—28no—
408C22⋊C4C2×D4×C6718yes—
409C42—24no—
410C42—24no—
411C42C2×D4×C6712yes—
412C42C24×C6712yes—
413C23—24no—
414C23—24no—
415C23—42no—
416C23—42no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23C22×D4×C6714yes—
421C23C22×D4×C6711yescharacteristic
422Q8—24no—
423Q8—22no—
424Q8—28no—
425Q8—84no—
426Q8—82no—
427Q8H(5,𝔽2)×C6714yes—
428D4—28no—
429D4—28no—
430D4—416no—
431D4—44no—
432D4—48no—
433D4—84no—
434D4—88no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—28no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—41no—
445C2×C4—416no—
446C2×C4—42no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4#137×C6714yescomplemented
450C2×C4C22×D4×C6711yescharacteristic
451C2×C4C22×D4×C6712yes—
452C2×C4C22×D4×C6714yes—
453C2×C4C22×D4×C6712yes—
454C2×C4C25×C6711yescharacteristic
455C8—28no—
456C22—21no—
457C22—21no—
458C22—24no—
459C22—24no—
460C22—24no—
461C22—44no—
462C22—44no—
463C22—44no—
464C22—84no—
465C22—88no—
466C22C2×#137×C6712yes—
467C22C23×D4×C6711yescharacteristic
468C4—22no—
469C4—24no—
470C4—24no—
471C4—21no—
472C4—42no—
473C4—44no—
474C4—44no—
475C4C2×#137×C6714yes—
476C4C2×H(5,𝔽2)×C6712yes—
477C2—22no—
478C2—44no—
479C2—84no—
480C2C67×#348411yescharacteristic
481C2C22×#137×C6711yescharacteristic
482C2C67×#358511yescharacteristic
subgroupG/H
center, socleC22×C67C23×D4
commutator, Frattini, cosocleC2×C4C25×C67
prime radicalC2×#550×C67C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C4×D4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3501C2semidirect
3. C67×#3333C2semidirect
4. C67×#3278C2semidirect
5. C67×#3208C2semidirect
6. C67×#3193C2semidirect
7. C67×#3104C2semidirect
8. C67×#3007C2semidirect
9. C67×#2975C2semidirect
11. C2×#137×C67C22semidirect
13. #554×C67C22semidirect
16. D4×Q8×C67C22semidirect
29. #492×C67C22semidirect
30. #486×C67C22semidirect
32. #480×C67C22semidirect
33. #477×C67C22semidirect
34. #468×C67C22semidirect
35. #462×C67C22semidirect
36. #460×C67C22semidirect
37. #452×C67C22semidirect
40. #448×C67C22semidirect
41. C4×Q16×C67C22semidirect
42. #436×C67C22semidirect
44. #417×C67C22semidirect
207. C2×C4×C67#137semidirect
235. #61588C67direct (N × H)
242. #3501C2×C67semidirect
244. #3333C2×C67semidirect
245. #3278C2×C67semidirect
246. #3208C2×C67semidirect
247. #3193C2×C67semidirect
248. #3104C2×C67semidirect
249. #3007C2×C67semidirect
250. #2975C2×C67semidirect
251. C67#61588direct (N × H)
253. C2×#137C22×C67semidirect
255. #554C22×C67semidirect
258. D4×Q8C22×C67semidirect
271. #492C22×C67semidirect
272. #486C22×C67semidirect
274. #480C22×C67semidirect
275. #477C22×C67semidirect
276. #468C22×C67semidirect
277. #462C22×C67semidirect
278. #460C22×C67semidirect
279. #452C22×C67semidirect
282. #448C22×C67semidirect
283. C4×Q16C22×C67semidirect
284. #436C22×C67semidirect
286. #417C22×C67semidirect
449. C2×C4#137×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#35012
C2×#550×C672
C67×#33332
C67×#32782
C67×#32082
C67×#31932
C67×#31042
C67×#30072
C67×#29752
#6158867