Finite Group Databasebeta

← C67×#60204 #30739455

Subgroups — C67×#60204 #30739455

Order 17152.

2050 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3444C211yesmaximal, characteristic
2C67×#2966C218yesmaximal, complemented
3C67×#2965C214yesmaximal, complemented
4C8×#41×C67C214yesmaximal, complemented
5C67×#2960C212yesmaximal, complemented
6C67×#2931C212yesmaximal, complemented
7C67×#2929C214yesmaximal, complemented
8C67×#2922C212yesmaximal, complemented
9C67×#2920C211yesmaximal, characteristic, complemented
10C4×#131×C67C212yesmaximal, complemented
11C22×C4×C8×C67C211yesmaximal, characteristic, complemented
12C2×#131×C67C2214yes—
13C23×C8×C67C2212yes—
14#533×C67C412yes—
15#533×C67C2211yescharacteristic
16#532×C67C412yes—
17#532×C67C2211yescharacteristic
18#529×C67C414yes—
19#529×C67C414yes—
20#529×C67C2214yes—
21#525×C67C414yes—
22#525×C67C2212yes—
23C4×#41×C67C2214yes—
24C4×#41×C67C2212yes—
25C22×C42×C67C2211yescharacteristic
26#446×C67C2218yescomplemented
27C8×Q8×C67C2218yescomplemented
28#436×C67C2218yescomplemented
29#435×C67C22116yes—
30C8×D4×C67C22116yes—
31C8×D4×C67C2218yescomplemented
32#433×C67C2218yescomplemented
33#432×C67C2214yes—
34#431×C67C2214yes—
35#431×C67C2218yescomplemented
36#424×C67C2214yes—
37#423×C67C2212yes—
38#408×C67C2214yes—
39#407×C67C2212yes—
40#405×C67C2214yes—
41#405×C67C2218yescomplemented
42C4×M16×C67C2212yes—
43C4×M16×C67C2214yescomplemented
44C2×C4×C8×C67C2211yescharacteristic
45C2×C4×C8×C67C2212yes—
46C2×C4×C8×C67C2212yes—
47C2×C4×C8×C67C2214yes—
48C2×C4×C8×C67C2211yescharacteristic
49C2×C4×C8×C67C2212yes—
50C2×C4×C8×C67C2214yescomplemented
51C2×#41×C67C2×C414yes—
52C2×#41×C67C2312yes—
53C23×C4×C67C2×C412yes—
54C23×C4×C67C2311yescharacteristic
55#131×C67—216no—
56C2×M16×C67C2314yes—
57C2×M16×C67C2318yes—
58C22×C8×C67—28no—
59C22×C8×C67C2312yes—
60C22×C8×C67C2314yes—
61C22×C8×C67C2314yes—
62C22×C8×C67C2318yes—
63C22×C8×C67C2312yes—
64C4⋊Q8×C67C2×C412yes—
65C4⋊Q8×C67C2×C411yescharacteristic
66#127×C67C2×C411yescharacteristic
67#126×C67C2×C418yes—
68#125×C67C2×C412yes—
69#125×C67C2×C414yes—
70#124×C67C2×C412yes—
71#124×C67C2×C414yes—
72#123×C67C2×C418yes—
73#123×C67C2×C414yes—
74C22⋊Q8×C67C2×C414yes—
75C22⋊Q8×C67C2×C414yes—
76C22⋊Q8×C67C2×C414yes—
77#121×C67C2×C414yes—
78#121×C67C2×C414yes—
79#121×C67C2×C414yes—
80#120×C67C2×C414yes—
81C4×Q8×C67C2×C414yes—
82C4×Q8×C67C2×C414yes—
83C4×Q8×C67C2314yes—
84C4×D4×C67C2×C418yes—
85C4×D4×C67C2×C418yes—
86C4×D4×C67C2×C414yes—
87C4×D4×C67C2×C414yes—
88C4×D4×C67C2318yes—
89C4×D4×C67C2314yes—
90#117×C67C2×C414yes—
91#117×C67C2×C418yes—
92#117×C67C2312yes—
93#117×C67C2314yes—
94C2×C42×C67C2×C412yes—
95C2×C42×C67C2×C412yes—
96C2×C42×C67C2×C412yes—
97C2×C42×C67C2311yescharacteristic
98C2×C42×C67C2312yes—
99C2×C42×C67C2312yes—
100C2×C42×C67C2311yescharacteristic
101C2×C42×C67C2314yes—
102C4⋊C8×C67C2318yes—
103C4⋊C8×C67C23116yes—
104C22⋊C8×C67C2318yes—
105C22⋊C8×C67C23116yes—
106#97×C67C2314yes—
107#97×C67C2318yes—
108C4×C8×C67C2311yescharacteristic
109C4×C8×C67C2312yes—
110C4×C8×C67C2312yes—
111C4×C8×C67C2312yes—
112C4×C8×C67C2314yes—
113C4×C8×C67C2314yes—
114C4×C8×C67C2314yes—
115C4×C8×C67C2318yes—
116C4×C8×C67C2311yescharacteristic
117C24×C67C22×C411yescharacteristic
118#41×C67—216no—
119#41×C67—28no—
120C2×Q8×C67C22×C412yes—
121C2×Q8×C67C22×C414yes—
122C2×D4×C67C22×C414yes—
123C2×D4×C67C22×C412yes—
124C2×D4×C67C22×C418yes—
125C2×D4×C67C22×C414yes—
126C22×C4×C67—28no—
127C22×C4×C67—24no—
128C22×C4×C67C22×C412yes—
129C22×C4×C67C22×C411yescharacteristic
130C22×C4×C67C22×C412yes—
131C22×C4×C67C22×C412yes—
132C22×C4×C67C22×C412yes—
133C22×C4×C67C22×C412yes—
134C22×C4×C67C22×C412yes—
135C22×C4×C67C22×C418yes—
136C22×C4×C67C22×C412yes—
137C22×C4×C67C2412yes—
138C22×C4×C67C2414yes—
139C22×C4×C67C2411yescharacteristic
140M16×C67—28no—
141M16×C67—216no—
142C2×C8×C67—24no—
143C2×C8×C67—28no—
144C2×C8×C67—28no—
145C2×C8×C67—216no—
146C2×C8×C67#4118yescomplemented
147C2×C8×C67C2414yes—
148C2×C8×C67C2418yes—
149C2×C8×C67C2412yes—
150C2×C8×C67C2412yes—
151C4⋊C4×C67C22×C414yes—
152C4⋊C4×C67C22×C418yes—
153C4⋊C4×C67C22×C414yes—
154C4⋊C4×C67C22×C418yes—
155C4⋊C4×C67C22×C414yes—
156C4⋊C4×C67C22×C418yes—
157C22⋊C4×C67C22×C418yes—
158C22⋊C4×C67C22×C418yes—
159C22⋊C4×C67C22×C418yes—
160C22⋊C4×C67C22×C414yes—
161C22⋊C4×C67C22×C418yes—
162C42×C67C22×C412yes—
163C42×C67C22×C412yes—
164C42×C67C22×C412yes—
165C42×C67C22×C412yes—
166C42×C67C22×C411yescharacteristic
167C42×C67C22×C412yes—
168C42×C67C22×C411yescharacteristic
169C42×C67C22×C418yes—
170C42×C67C2412yes—
171C42×C67C2414yes—
172C42×C67C2411yescharacteristic
173C42×C67C2411yescharacteristic
174C23×C67—22no—
175C23×C67—24no—
176C23×C67C23×C414yes—
177C23×C67C23×C411yescharacteristic
178C23×C67C23×C412yes—
179Q8×C67—24no—
180Q8×C67—28no—
181D4×C67—28no—
182D4×C67—216no—
183D4×C67—24no—
184D4×C67—28no—
185C2×C4×C67—22no—
186C2×C4×C67—24no—
187C2×C4×C67—24no—
188C2×C4×C67—24no—
189C2×C4×C67—24no—
190C2×C4×C67—24no—
191C2×C4×C67—24no—
192C2×C4×C67—216no—
193C2×C4×C67—24no—
194C2×C4×C67—28no—
195C2×C4×C67#13118yescomplemented
196C2×C4×C67C23×C412yes—
197C2×C4×C67C23×C412yes—
198C2×C4×C67C23×C412yes—
199C2×C4×C67C23×C414yes—
200C2×C4×C67C23×C418yes—
201C2×C4×C67C23×C411yescharacteristic
202C2×C4×C67C23×C412yes—
203C2×C4×C67C23×C412yes—
204C2×C4×C67C2×#4114yes—
205C2×C4×C67C2511yescharacteristic
206C8×C67—24no—
207C8×C67—28no—
208C8×C67C2×#4114yes—
209C8×C67C2×#4114yes—
210C22×C67—28no—
211C22×C67—22no—
212C22×C67—22no—
213C22×C67—22no—
214C22×C67—24no—
215C22×C67—28no—
216C22×C67C4×#4112yes—
217C22×C67C2×#13114yes—
218C22×C67C24×C411yescharacteristic
219C4×C67—22no—
220C4×C67—24no—
221C4×C67—22no—
222C4×C67—22no—
223C4×C67—28no—
224C4×C67C4×#4112yes—
225C4×C67C2×#13114yes—
226C4×C67C2×#13114yes—
227C4×C67C22×#4112yes—
228#60204C6711yesmaximal, characteristic, direct product factor, Sylow
229C2×C67—22no—
230C2×C67—24no—
231C2×C67—44no—
232C2×C67C2×C4×#4111yescharacteristic
233C2×C67C22×#13112yes—
234#3444C2×C6711yescharacteristic
235#2966C2×C6718yescomplemented
236#2965C2×C6714yescomplemented
237C8×#41C2×C6714yescomplemented
238#2960C2×C6712yescomplemented
239#2931C2×C6712yescomplemented
240#2929C2×C6714yescomplemented
241#2922C2×C6712yescomplemented
242#2920C2×C6711yescharacteristic, complemented
243C4×#131C2×C6712yescomplemented
244C22×C4×C8C2×C6711yescharacteristic, complemented
245C67#6020411yescharacteristic, direct product factor, Sylow
246C2×#131C22×C6714yes—
247C23×C8C22×C6712yes—
248#533C4×C6712yes—
249#533C22×C6711yescharacteristic
250#532C4×C6712yes—
251#532C22×C6711yescharacteristic
252#529C4×C6714yes—
253#529C4×C6714yes—
254#529C22×C6714yes—
255#525C4×C6714yes—
256#525C22×C6712yes—
257C4×#41C22×C6714yes—
258C4×#41C22×C6712yes—
259C22×C42C22×C6711yescharacteristic
260#446C22×C6718yescomplemented
261C8×Q8C22×C6718yescomplemented
262#436C22×C6718yescomplemented
263#435C22×C67116yes—
264C8×D4C22×C67116yes—
265C8×D4C22×C6718yescomplemented
266#433C22×C6718yescomplemented
267#432C22×C6714yes—
268#431C22×C6714yes—
269#431C22×C6718yescomplemented
270#424C22×C6714yes—
271#423C22×C6712yes—
272#408C22×C6714yes—
273#407C22×C6712yes—
274#405C22×C6714yes—
275#405C22×C6718yescomplemented
276C4×M16C22×C6712yes—
277C4×M16C22×C6714yescomplemented
278C2×C4×C8C22×C6711yescharacteristic
279C2×C4×C8C22×C6711yescharacteristic
280C2×C4×C8C22×C6712yes—
281C2×C4×C8C22×C6712yes—
282C2×C4×C8C22×C6712yes—
283C2×C4×C8C22×C6714yes—
284C2×C4×C8C22×C6714yescomplemented
285C2×#41C2×C4×C6714yes—
286C2×#41C23×C6712yes—
287C23×C4C2×C4×C6712yes—
288C23×C4C23×C6711yescharacteristic
289#131—216no—
290C2×M16C23×C6714yes—
291C2×M16C23×C6718yes—
292C22×C8—28no—
293C22×C8C23×C6712yes—
294C22×C8C23×C6714yes—
295C22×C8C23×C6714yes—
296C22×C8C23×C6718yes—
297C22×C8C23×C6712yes—
298C4⋊Q8C2×C4×C6712yes—
299C4⋊Q8C2×C4×C6711yescharacteristic
300#127C2×C4×C6711yescharacteristic
301#126C2×C4×C6718yes—
302#125C2×C4×C6712yes—
303#125C2×C4×C6714yes—
304#124C2×C4×C6712yes—
305#124C2×C4×C6714yes—
306#123C2×C4×C6718yes—
307#123C2×C4×C6714yes—
308C22⋊Q8C2×C4×C6714yes—
309C22⋊Q8C2×C4×C6714yes—
310C22⋊Q8C2×C4×C6714yes—
311#121C2×C4×C6714yes—
312#121C2×C4×C6714yes—
313#121C2×C4×C6714yes—
314#120C2×C4×C6714yes—
315C4×Q8C2×C4×C6714yes—
316C4×Q8C2×C4×C6714yes—
317C4×Q8C23×C6714yes—
318C4×D4C2×C4×C6718yes—
319C4×D4C2×C4×C6718yes—
320C4×D4C2×C4×C6714yes—
321C4×D4C2×C4×C6714yes—
322C4×D4C23×C6718yes—
323C4×D4C23×C6714yes—
324#117C2×C4×C6714yes—
325#117C2×C4×C6718yes—
326#117C23×C6712yes—
327#117C23×C6714yes—
328C2×C42C2×C4×C6712yes—
329C2×C42C2×C4×C6712yes—
330C2×C42C2×C4×C6712yes—
331C2×C42C23×C6714yes—
332C2×C42C23×C6711yescharacteristic
333C2×C42C23×C6712yes—
334C2×C42C23×C6712yes—
335C2×C42C23×C6711yescharacteristic
336C4⋊C8C23×C6718yes—
337C4⋊C8C23×C67116yes—
338C22⋊C8C23×C6718yes—
339C22⋊C8C23×C67116yes—
340#97C23×C6714yes—
341#97C23×C6718yes—
342C4×C8C23×C6711yescharacteristic
343C4×C8C23×C6711yescharacteristic
344C4×C8C23×C6712yes—
345C4×C8C23×C6712yes—
346C4×C8C23×C6712yes—
347C4×C8C23×C6714yes—
348C4×C8C23×C6714yes—
349C4×C8C23×C6714yes—
350C4×C8C23×C6718yes—
351C24C22×C4×C6711yescharacteristic
352#41—216no—
353#41—28no—
354C2×Q8C22×C4×C6712yes—
355C2×Q8C22×C4×C6714yes—
356C2×D4C22×C4×C6714yes—
357C2×D4C22×C4×C6712yes—
358C2×D4C22×C4×C6718yes—
359C2×D4C22×C4×C6714yes—
360C22×C4—28no—
361C22×C4—24no—
362C22×C4C22×C4×C6712yes—
363C22×C4C22×C4×C6711yescharacteristic
364C22×C4C22×C4×C6712yes—
365C22×C4C22×C4×C6712yes—
366C22×C4C22×C4×C6712yes—
367C22×C4C22×C4×C6712yes—
368C22×C4C22×C4×C6712yes—
369C22×C4C22×C4×C6718yes—
370C22×C4C22×C4×C6712yes—
371C22×C4C24×C6712yes—
372C22×C4C24×C6714yes—
373C22×C4C24×C6711yescharacteristic
374M16—28no—
375M16—216no—
376C2×C8—24no—
377C2×C8—28no—
378C2×C8—28no—
379C2×C8—216no—
380C2×C8#41×C6718yescomplemented
381C2×C8C24×C6714yes—
382C2×C8C24×C6718yes—
383C2×C8C24×C6712yes—
384C2×C8C24×C6712yes—
385C4⋊C4C22×C4×C6714yes—
386C4⋊C4C22×C4×C6718yes—
387C4⋊C4C22×C4×C6714yes—
388C4⋊C4C22×C4×C6718yes—
389C4⋊C4C22×C4×C6714yes—
390C4⋊C4C22×C4×C6718yes—
391C22⋊C4C22×C4×C6718yes—
392C22⋊C4C22×C4×C6718yes—
393C22⋊C4C22×C4×C6718yes—
394C22⋊C4C22×C4×C6714yes—
395C22⋊C4C22×C4×C6718yes—
396C42C22×C4×C6712yes—
397C42C22×C4×C6712yes—
398C42C22×C4×C6712yes—
399C42C22×C4×C6712yes—
400C42C22×C4×C6711yescharacteristic
401C42C22×C4×C6712yes—
402C42C22×C4×C6711yescharacteristic
403C42C22×C4×C6718yes—
404C42C24×C6711yescharacteristic
405C42C24×C6711yescharacteristic
406C42C24×C6712yes—
407C42C24×C6714yes—
408C23—22no—
409C23—24no—
410C23C23×C4×C6714yes—
411C23C23×C4×C6711yescharacteristic
412C23C23×C4×C6712yes—
413Q8—24no—
414Q8—28no—
415D4—28no—
416D4—216no—
417D4—24no—
418D4—28no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—216no—
427C2×C4—24no—
428C2×C4—28no—
429C2×C4#131×C6718yescomplemented
430C2×C4C23×C4×C6712yes—
431C2×C4C23×C4×C6712yes—
432C2×C4C23×C4×C6712yes—
433C2×C4C23×C4×C6714yes—
434C2×C4C23×C4×C6718yes—
435C2×C4C23×C4×C6711yescharacteristic
436C2×C4C23×C4×C6712yes—
437C2×C4C23×C4×C6712yes—
438C2×C4C2×#41×C6714yes—
439C2×C4C25×C6711yescharacteristic
440C8—24no—
441C8—28no—
442C8C2×#41×C6714yes—
443C8C2×#41×C6714yes—
444C22—28no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—28no—
449C22—24no—
450C22C4×#41×C6712yes—
451C22C2×#131×C6714yes—
452C22C24×C4×C6711yescharacteristic
453C4—22no—
454C4—22no—
455C4—28no—
456C4—22no—
457C4—24no—
458C4C4×#41×C6712yes—
459C4C2×#131×C6714yes—
460C4C2×#131×C6714yes—
461C4C22×#41×C6712yes—
462C2—22no—
463C2—24no—
464C2—44no—
465C2C2×C4×#41×C6711yescharacteristic
466C2C22×#131×C6712yes—
subgroupG/H
centerC4×C8×C67C23
commutatorC22C24×C4×C67
Frattini, cosocleC2×C4C25×C67
socleC22×C67C24×C4
prime radicalC67×#3444C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C4×C8×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2966C2semidirect
3. C67×#2965C2semidirect
4. C8×#41×C67C2semidirect
5. C67×#2960C2semidirect
6. C67×#2931C2semidirect
7. C67×#2929C2semidirect
8. C67×#2922C2semidirect
9. C67×#2920C2semidirect
10. C4×#131×C67C2semidirect
11. C22×C4×C8×C67C2semidirect
26. #446×C67C22semidirect
27. C8×Q8×C67C22semidirect
28. #436×C67C22semidirect
31. C8×D4×C67C22semidirect
32. #433×C67C22semidirect
35. #431×C67C22semidirect
41. #405×C67C22semidirect
43. C4×M16×C67C22semidirect
50. C2×C4×C8×C67C22semidirect
146. C2×C8×C67#41semidirect
195. C2×C4×C67#131semidirect
228. #60204C67direct (N × H)
235. #2966C2×C67semidirect
236. #2965C2×C67semidirect
237. C8×#41C2×C67semidirect
238. #2960C2×C67semidirect
239. #2931C2×C67semidirect
240. #2929C2×C67semidirect
241. #2922C2×C67semidirect
242. #2920C2×C67semidirect
243. C4×#131C2×C67semidirect
244. C22×C4×C8C2×C67semidirect
245. C67#60204direct (N × H)
260. #446C22×C67semidirect
261. C8×Q8C22×C67semidirect
262. #436C22×C67semidirect
265. C8×D4C22×C67semidirect
266. #433C22×C67semidirect
269. #431C22×C67semidirect
275. #405C22×C67semidirect
277. C4×M16C22×C67semidirect
284. C2×C4×C8C22×C67semidirect
380. C2×C8#41×C67semidirect
429. C2×C4#131×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34442
C67×#29662
C67×#29652
C8×#41×C672
C67×#29602
C67×#29312
C67×#29292
C67×#29222
C67×#29202
C4×#131×C672
C22×C4×C8×C672
#6020467