Finite Group Databasebeta

← C67×#60207 #30739458

Subgroups — C67×#60207 #30739458

Order 17152.

1962 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3444C211yesmaximal, characteristic
2C67×#2970C218yesmaximal, complemented
3C67×#2968C218yesmaximal, complemented
4C67×#2961C212yesmaximal, complemented
5C67×#2932C212yesmaximal, complemented
6C67×#2929C214yesmaximal, complemented
7C67×#2923C212yesmaximal, complemented
8C67×#2920C211yesmaximal, characteristic, complemented
9C67×#2875C212yesmaximal, complemented
10C22×#97×C67C211yesmaximal, characteristic, complemented
11C2×#131×C67C2214yes—
12C23×C8×C67C2212yes—
13#533×C67C412yes—
14#533×C67C2211yescharacteristic
15#532×C67C412yes—
16#532×C67C2211yescharacteristic
17#529×C67C414yes—
18#529×C67C414yes—
19#529×C67C2214yes—
20#525×C67C414yes—
21#525×C67C2212yes—
22C4×#41×C67C2214yes—
23C4×#41×C67C2212yes—
24C22×C42×C67C2211yescharacteristic
25#446×C67C22116yescomplemented
26#435×C67C22116yes—
27#435×C67C22116yescomplemented
28C8×D4×C67C22116yes—
29#433×C67C2218yescomplemented
30#432×C67C2214yes—
31#432×C67C2218yescomplemented
32#431×C67C2214yes—
33#424×C67C2214yes—
34#423×C67C2212yes—
35#408×C67C2214yes—
36#407×C67C2212yes—
37#405×C67C2214yes—
38#405×C67C2218yescomplemented
39C4×M16×C67C2212yes—
40C4×M16×C67C2214yescomplemented
41C2×#97×C67C2211yescharacteristic
42C2×#97×C67C2211yescharacteristic
43C2×#97×C67C2212yes—
44C2×#97×C67C2214yes—
45C2×#97×C67C2214yes—
46C2×#97×C67C2214yescomplemented
47C2×#41×C67C2×C414yes—
48C2×#41×C67C2312yes—
49C23×C4×C67C2×C412yes—
50C23×C4×C67C2311yescharacteristic
51#131×C67—216no—
52C2×M16×C67C2314yes—
53C2×M16×C67C2318yes—
54C22×C8×C67—28no—
55C22×C8×C67C2312yes—
56C22×C8×C67C2318yes—
57C22×C8×C67C2318yes—
58C22×C8×C67C2312yes—
59C4⋊Q8×C67C2×C412yes—
60C4⋊Q8×C67C2×C411yescharacteristic
61#127×C67C2×C411yescharacteristic
62#126×C67C2×C418yes—
63#125×C67C2×C412yes—
64#125×C67C2×C414yes—
65#124×C67C2×C412yes—
66#124×C67C2×C414yes—
67#123×C67C2×C418yes—
68#123×C67C2×C414yes—
69C22⋊Q8×C67C2×C414yes—
70C22⋊Q8×C67C2×C414yes—
71C22⋊Q8×C67C2×C414yes—
72#121×C67C2×C414yes—
73#121×C67C2×C414yes—
74#121×C67C2×C414yes—
75#120×C67C2×C414yes—
76C4×Q8×C67C2×C414yes—
77C4×Q8×C67C2×C414yes—
78C4×Q8×C67C2314yes—
79C4×D4×C67C2×C418yes—
80C4×D4×C67C2×C418yes—
81C4×D4×C67C2×C414yes—
82C4×D4×C67C2×C414yes—
83C4×D4×C67C2318yes—
84C4×D4×C67C2314yes—
85#117×C67C2×C414yes—
86#117×C67C2×C418yes—
87#117×C67C2312yes—
88#117×C67C2314yes—
89C2×C42×C67C2×C412yes—
90C2×C42×C67C2×C412yes—
91C2×C42×C67C2×C412yes—
92C2×C42×C67C2311yescharacteristic
93C2×C42×C67C2312yes—
94C2×C42×C67C2312yes—
95C2×C42×C67C2311yescharacteristic
96C2×C42×C67C2314yes—
97C4⋊C8×C67C2318yes—
98C4⋊C8×C67C23116yes—
99C22⋊C8×C67C2318yes—
100C22⋊C8×C67C23116yes—
101#97×C67C2312yes—
102#97×C67C2312yes—
103#97×C67C2312yes—
104#97×C67C2312yes—
105#97×C67C2318yes—
106#97×C67C2314yes—
107#97×C67C2318yes—
108C4×C8×C67C2314yes—
109C4×C8×C67C2318yes—
110C24×C67C22×C411yescharacteristic
111#41×C67—28no—
112#41×C67—48no—
113C2×Q8×C67C22×C412yes—
114C2×Q8×C67C22×C414yes—
115C2×D4×C67C22×C414yes—
116C2×D4×C67C22×C412yes—
117C2×D4×C67C22×C418yes—
118C2×D4×C67C22×C414yes—
119C22×C4×C67—24no—
120C22×C4×C67—44no—
121C22×C4×C67C22×C412yes—
122C22×C4×C67C22×C411yescharacteristic
123C22×C4×C67C22×C412yes—
124C22×C4×C67C22×C412yes—
125C22×C4×C67C22×C412yes—
126C22×C4×C67C22×C412yes—
127C22×C4×C67C22×C412yes—
128C22×C4×C67C22×C412yes—
129C22×C4×C67C22×C418yes—
130C22×C4×C67C2412yes—
131C22×C4×C67C2414yes—
132C22×C4×C67C2411yescharacteristic
133M16×C67—28no—
134M16×C67—216no—
135C2×C8×C67—24no—
136C2×C8×C67—216no—
137C2×C8×C67—216no—
138C2×C8×C67#4118yescomplemented
139C2×C8×C67C2414yes—
140C2×C8×C67C2418yes—
141C2×C8×C67C2414yes—
142C4⋊C4×C67C22×C414yes—
143C4⋊C4×C67C22×C418yes—
144C4⋊C4×C67C22×C414yes—
145C4⋊C4×C67C22×C418yes—
146C4⋊C4×C67C22×C414yes—
147C4⋊C4×C67C22×C418yes—
148C22⋊C4×C67C22×C418yes—
149C22⋊C4×C67C22×C418yes—
150C22⋊C4×C67C22×C418yes—
151C22⋊C4×C67C22×C414yes—
152C22⋊C4×C67C22×C418yes—
153C42×C67C22×C412yes—
154C42×C67C22×C412yes—
155C42×C67C22×C412yes—
156C42×C67C22×C412yes—
157C42×C67C22×C411yescharacteristic
158C42×C67C22×C412yes—
159C42×C67C22×C411yescharacteristic
160C42×C67C22×C418yes—
161C42×C67C2412yes—
162C42×C67C2414yes—
163C42×C67C2411yescharacteristic
164C42×C67C2411yescharacteristic
165C23×C67—22no—
166C23×C67—24no—
167C23×C67C23×C414yes—
168C23×C67C23×C411yescharacteristic
169C23×C67C23×C412yes—
170Q8×C67—24no—
171Q8×C67—44no—
172D4×C67—28no—
173D4×C67—216no—
174D4×C67—24no—
175D4×C67—44no—
176C2×C4×C67—22no—
177C2×C4×C67—24no—
178C2×C4×C67—24no—
179C2×C4×C67—24no—
180C2×C4×C67—24no—
181C2×C4×C67—28no—
182C2×C4×C67—42no—
183C2×C4×C67—42no—
184C2×C4×C67—42no—
185C2×C4×C67—42no—
186C2×C4×C67—48no—
187C2×C4×C67C23×C412yes—
188C2×C4×C67C23×C412yes—
189C2×C4×C67C23×C412yes—
190C2×C4×C67C23×C412yes—
191C2×C4×C67C23×C412yes—
192C2×C4×C67C23×C414yes—
193C2×C4×C67C23×C418yes—
194C2×C4×C67C23×C411yescharacteristic
195C2×C4×C67C2×#4114yes—
196C2×C4×C67C2511yescharacteristic
197C8×C67—24no—
198C8×C67—28no—
199C8×C67C2×#4118yes—
200C22×C67—28no—
201C22×C67—22no—
202C22×C67—22no—
203C22×C67—22no—
204C22×C67—24no—
205C22×C67—28no—
206C22×C67C4×#4112yes—
207C22×C67#56814yes—
208C22×C67C24×C411yescharacteristic
209C4×C67—22no—
210C4×C67—24no—
211C4×C67—22no—
212C4×C67—22no—
213C4×C67—22no—
214C4×C67—22no—
215C4×C67—28no—
216C4×C67C4×#4112yes—
217C4×C67C22×#4112yes—
218#60207C6711yesmaximal, characteristic, direct product factor, Sylow
219C2×C67—22no—
220C2×C67—24no—
221C2×C67—44no—
222C2×C67C2×C4×#4111yescharacteristic
223C2×C67C2×#56812yes—
224#3444C2×C6711yescharacteristic
225#2970C2×C6718yescomplemented
226#2968C2×C6718yescomplemented
227#2961C2×C6712yescomplemented
228#2932C2×C6712yescomplemented
229#2929C2×C6714yescomplemented
230#2923C2×C6712yescomplemented
231#2920C2×C6711yescharacteristic, complemented
232#2875C2×C6712yescomplemented
233C22×#97C2×C6711yescharacteristic, complemented
234C67#6020711yescharacteristic, direct product factor, Sylow
235C2×#131C22×C6714yes—
236C23×C8C22×C6712yes—
237#533C4×C6712yes—
238#533C22×C6711yescharacteristic
239#532C4×C6712yes—
240#532C22×C6711yescharacteristic
241#529C4×C6714yes—
242#529C4×C6714yes—
243#529C22×C6714yes—
244#525C4×C6714yes—
245#525C22×C6712yes—
246C4×#41C22×C6714yes—
247C4×#41C22×C6712yes—
248C22×C42C22×C6711yescharacteristic
249#446C22×C67116yescomplemented
250#435C22×C67116yes—
251#435C22×C67116yescomplemented
252C8×D4C22×C67116yes—
253#433C22×C6718yescomplemented
254#432C22×C6714yes—
255#432C22×C6718yescomplemented
256#431C22×C6714yes—
257#424C22×C6714yes—
258#423C22×C6712yes—
259#408C22×C6714yes—
260#407C22×C6712yes—
261#405C22×C6714yes—
262#405C22×C6718yescomplemented
263C4×M16C22×C6712yes—
264C4×M16C22×C6714yescomplemented
265C2×#97C22×C6711yescharacteristic
266C2×#97C22×C6711yescharacteristic
267C2×#97C22×C6712yes—
268C2×#97C22×C6714yes—
269C2×#97C22×C6714yes—
270C2×#97C22×C6714yescomplemented
271C2×#41C2×C4×C6714yes—
272C2×#41C23×C6712yes—
273C23×C4C2×C4×C6712yes—
274C23×C4C23×C6711yescharacteristic
275#131—216no—
276C2×M16C23×C6714yes—
277C2×M16C23×C6718yes—
278C22×C8—28no—
279C22×C8C23×C6712yes—
280C22×C8C23×C6718yes—
281C22×C8C23×C6718yes—
282C22×C8C23×C6712yes—
283C4⋊Q8C2×C4×C6712yes—
284C4⋊Q8C2×C4×C6711yescharacteristic
285#127C2×C4×C6711yescharacteristic
286#126C2×C4×C6718yes—
287#125C2×C4×C6712yes—
288#125C2×C4×C6714yes—
289#124C2×C4×C6712yes—
290#124C2×C4×C6714yes—
291#123C2×C4×C6718yes—
292#123C2×C4×C6714yes—
293C22⋊Q8C2×C4×C6714yes—
294C22⋊Q8C2×C4×C6714yes—
295C22⋊Q8C2×C4×C6714yes—
296#121C2×C4×C6714yes—
297#121C2×C4×C6714yes—
298#121C2×C4×C6714yes—
299#120C2×C4×C6714yes—
300C4×Q8C2×C4×C6714yes—
301C4×Q8C2×C4×C6714yes—
302C4×Q8C23×C6714yes—
303C4×D4C2×C4×C6718yes—
304C4×D4C2×C4×C6718yes—
305C4×D4C2×C4×C6714yes—
306C4×D4C2×C4×C6714yes—
307C4×D4C23×C6718yes—
308C4×D4C23×C6714yes—
309#117C2×C4×C6714yes—
310#117C2×C4×C6718yes—
311#117C23×C6712yes—
312#117C23×C6714yes—
313C2×C42C2×C4×C6712yes—
314C2×C42C2×C4×C6712yes—
315C2×C42C2×C4×C6712yes—
316C2×C42C23×C6714yes—
317C2×C42C23×C6711yescharacteristic
318C2×C42C23×C6712yes—
319C2×C42C23×C6712yes—
320C2×C42C23×C6711yescharacteristic
321C4⋊C8C23×C6718yes—
322C4⋊C8C23×C67116yes—
323C22⋊C8C23×C6718yes—
324C22⋊C8C23×C67116yes—
325#97C23×C6712yes—
326#97C23×C6712yes—
327#97C23×C6712yes—
328#97C23×C6712yes—
329#97C23×C6718yes—
330#97C23×C6714yes—
331#97C23×C6718yes—
332C4×C8C23×C6714yes—
333C4×C8C23×C6718yes—
334C24C22×C4×C6711yescharacteristic
335#41—28no—
336#41—48no—
337C2×Q8C22×C4×C6712yes—
338C2×Q8C22×C4×C6714yes—
339C2×D4C22×C4×C6714yes—
340C2×D4C22×C4×C6712yes—
341C2×D4C22×C4×C6718yes—
342C2×D4C22×C4×C6714yes—
343C22×C4—24no—
344C22×C4—44no—
345C22×C4C22×C4×C6712yes—
346C22×C4C22×C4×C6711yescharacteristic
347C22×C4C22×C4×C6712yes—
348C22×C4C22×C4×C6712yes—
349C22×C4C22×C4×C6712yes—
350C22×C4C22×C4×C6712yes—
351C22×C4C22×C4×C6712yes—
352C22×C4C22×C4×C6712yes—
353C22×C4C22×C4×C6718yes—
354C22×C4C24×C6712yes—
355C22×C4C24×C6714yes—
356C22×C4C24×C6711yescharacteristic
357M16—28no—
358M16—216no—
359C2×C8—24no—
360C2×C8—216no—
361C2×C8—216no—
362C2×C8#41×C6718yescomplemented
363C2×C8C24×C6718yes—
364C2×C8C24×C6714yes—
365C2×C8C24×C6714yes—
366C4⋊C4C22×C4×C6714yes—
367C4⋊C4C22×C4×C6718yes—
368C4⋊C4C22×C4×C6714yes—
369C4⋊C4C22×C4×C6718yes—
370C4⋊C4C22×C4×C6714yes—
371C4⋊C4C22×C4×C6718yes—
372C22⋊C4C22×C4×C6718yes—
373C22⋊C4C22×C4×C6718yes—
374C22⋊C4C22×C4×C6718yes—
375C22⋊C4C22×C4×C6714yes—
376C22⋊C4C22×C4×C6718yes—
377C42C22×C4×C6712yes—
378C42C22×C4×C6712yes—
379C42C22×C4×C6712yes—
380C42C22×C4×C6712yes—
381C42C22×C4×C6711yescharacteristic
382C42C22×C4×C6712yes—
383C42C22×C4×C6711yescharacteristic
384C42C22×C4×C6718yes—
385C42C24×C6711yescharacteristic
386C42C24×C6714yes—
387C42C24×C6711yescharacteristic
388C42C24×C6712yes—
389C23—22no—
390C23—24no—
391C23C23×C4×C6714yes—
392C23C23×C4×C6711yescharacteristic
393C23C23×C4×C6712yes—
394Q8—24no—
395Q8—44no—
396D4—28no—
397D4—216no—
398D4—24no—
399D4—44no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—28no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—48no—
411C2×C4C23×C4×C6712yes—
412C2×C4C23×C4×C6712yes—
413C2×C4C23×C4×C6712yes—
414C2×C4C23×C4×C6712yes—
415C2×C4C23×C4×C6712yes—
416C2×C4C23×C4×C6714yes—
417C2×C4C23×C4×C6718yes—
418C2×C4C23×C4×C6711yescharacteristic
419C2×C4C2×#41×C6714yes—
420C2×C4C25×C6711yescharacteristic
421C8—24no—
422C8—28no—
423C8C2×#41×C6718yes—
424C22—28no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—28no—
429C22—24no—
430C22C4×#41×C6712yes—
431C22#568×C6714yes—
432C22C24×C4×C6711yescharacteristic
433C4—22no—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—28no—
438C4—22no—
439C4—24no—
440C4C4×#41×C6712yes—
441C4C22×#41×C6712yes—
442C2—22no—
443C2—24no—
444C2—44no—
445C2C2×C4×#41×C6711yescharacteristic
446C2C2×#568×C6712yes—
subgroupG/H
centerC2×C4×C67C25
commutatorC22C24×C4×C67
Frattini, cosocleC2×C4C25×C67
socleC22×C67C24×C4
prime radicalC67×#3444C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2970C2semidirect
3. C67×#2968C2semidirect
4. C67×#2961C2semidirect
5. C67×#2932C2semidirect
6. C67×#2929C2semidirect
7. C67×#2923C2semidirect
8. C67×#2920C2semidirect
9. C67×#2875C2semidirect
10. C22×#97×C67C2semidirect
25. #446×C67C22semidirect
27. #435×C67C22semidirect
29. #433×C67C22semidirect
31. #432×C67C22semidirect
38. #405×C67C22semidirect
40. C4×M16×C67C22semidirect
46. C2×#97×C67C22semidirect
138. C2×C8×C67#41semidirect
218. #60207C67direct (N × H)
225. #2970C2×C67semidirect
226. #2968C2×C67semidirect
227. #2961C2×C67semidirect
228. #2932C2×C67semidirect
229. #2929C2×C67semidirect
230. #2923C2×C67semidirect
231. #2920C2×C67semidirect
232. #2875C2×C67semidirect
233. C22×#97C2×C67semidirect
234. C67#60207direct (N × H)
249. #446C22×C67semidirect
251. #435C22×C67semidirect
253. #433C22×C67semidirect
255. #432C22×C67semidirect
262. #405C22×C67semidirect
264. C4×M16C22×C67semidirect
270. C2×#97C22×C67semidirect
362. 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×#34442
C67×#29702
C67×#29682
C67×#29612
C67×#29322
C67×#29292
C67×#29232
C67×#29202
C67×#28752
C22×#97×C672
#6020767