Finite Group Databasebeta

← C67×#60140 #30739391

Subgroups — C67×#60140 #30739391

Order 17152.

2722 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#131×C67C212yesmaximal, complemented
2C67×#3449C211yesmaximal, characteristic
3C67×#2932C214yesmaximal, complemented
4C67×#2931C214yesmaximal, complemented
5C67×#2930C218yesmaximal, complemented
6C67×#2929C218yesmaximal, complemented
7C67×#2905C211yesmaximal, characteristic, complemented
8C67×#2879C212yesmaximal, complemented
9C2×#405×C67C211yesmaximal, characteristic, complemented
10C22×#41×C67C2212yes—
11C2×#131×C67—216no—
12C2×#131×C67C2214yes—
13C2×#131×C67C2214yes—
14C2×#131×C67C2214yescomplemented
15C2×#131×C67C2214yescomplemented
16C22×M16×C67C2212yes—
17C22×M16×C67C2211yescharacteristic
18C22×M16×C67C2214yescomplemented
19C23×C8×C67C2211yescharacteristic
20C23×C8×C67C2214yescomplemented
21#537×C67C414yes—
22#536×C67C414yes—
23#535×C67C414yes—
24#534×C67C414yes—
25#532×C67C2214yes—
26#525×C67C2218yes—
27C2×#117×C67C2211yescharacteristic
28#436×C67C22116yescomplemented
29#435×C67C22132yescomplemented
30C8×D4×C67C22116yescomplemented
31#424×C67C2218yescomplemented
32#423×C67C2214yescomplemented
33#408×C67C2218yes—
34#408×C67C2218yescomplemented
35#407×C67C2218yescomplemented
36#405×C67C2214yescomplemented
37#405×C67C2214yescomplemented
38C2×#97×C67C2212yescomplemented
39C2×C4×C8×C67C2212yescomplemented
40C2×#41×C67—28no—
41C2×#41×C67—28no—
42C2×#41×C67C2×C414yes—
43C2×#41×C67C2×C414yes—
44C2×#41×C67C2314yes—
45C2×#41×C67C2314yes—
46C22×Q8×C67C2×C412yes—
47C22×D4×C67C2×C414yes—
48C22×D4×C67C2×C412yes—
49C23×C4×C67C2314yes—
50C23×C4×C67C2311yescharacteristic
51#131×C67—216no—
52#131×C67—216no—
53#131×C67—216no—
54#131×C67—216no—
55C2×M16×C67—28no—
56C2×M16×C67—24no—
57C2×M16×C67—216no—
58C2×M16×C67C2314yes—
59C2×M16×C67C2314yes—
60C2×M16×C67C2314yes—
61C2×M16×C67C2318yes—
62C2×M16×C67C2318yes—
63C22×C8×C67—216no—
64C22×C8×C67D418yescomplemented
65C22×C8×C67C2314yes—
66C22×C8×C67C2318yes—
67C22×C8×C67C2312yes—
68C22×C8×C67C2318yes—
69C22×C8×C67C2312yes—
70C22×C8×C67C2311yescharacteristic
71C22×C8×C67C2312yes—
72C22×C8×C67C2311yescharacteristic
73C4⋊Q8×C67C2×C414yes—
74#127×C67C2×C414yes—
75#124×C67C2×C418yes—
76#123×C67C2×C4116yes—
77C22⋊Q8×C67C2×C4116yes—
78#121×C67C2×C4116yes—
79#121×C67C2×C4116yes—
80#120×C67C2×C4116yes—
81C4×D4×C67C23116yes—
82#117×C67C2×C414yes—
83#117×C67C2314yes—
84C2×C4⋊C4×C67C2×C412yes—
85C2×C22⋊C4×C67C2×C412yes—
86C2×C42×C67C2312yes—
87C4⋊C8×C67C23116yes—
88C22⋊C8×C67C23132yes—
89#97×C67C2318yes—
90C4×C8×C67C2314yes—
91C4×C8×C67C2314yes—
92C24×C67C22×C414yes—
93C24×C67C22×C411yescharacteristic
94#41×C67—216no—
95#41×C67—216no—
96#41×C67—28no—
97#41×C67—28no—
98#41×C67—28no—
99#41×C67—28no—
100C2×Q8×C67—28no—
101C2×Q8×C67C22×C414yes—
102C2×Q8×C67C22×C414yes—
103C2×D4×C67—216no—
104C2×D4×C67—28no—
105C2×D4×C67C22×C418yes—
106C2×D4×C67C22×C418yes—
107C2×D4×C67C22×C414yes—
108C2×D4×C67C22×C414yes—
109C2×D4×C67C22×C4116yes—
110C22×C4×C67—28no—
111C22×C4×C67—28no—
112C22×C4×C67C22×C418yes—
113C22×C4×C67C22×C414yes—
114C22×C4×C67C22×C414yes—
115C22×C4×C67C22×C411yescharacteristic
116C22×C4×C67C22×C412yes—
117C22×C4×C67C2×D414yes—
118C22×C4×C67#4114yes—
119C22×C4×C67C2418yes—
120C22×C4×C67C2411yescharacteristic
121C22×C4×C67C2412yes—
122M16×C67—28no—
123M16×C67—28no—
124M16×C67—28no—
125M16×C67—216no—
126M16×C67—216no—
127C2×C8×C67—28no—
128C2×C8×C67—24no—
129C2×C8×C67—216no—
130C2×C8×C67—216no—
131C2×C8×C67C2×D418yes—
132C2×C8×C67C2×D418yes—
133C2×C8×C67C2×D414yes—
134C2×C8×C67C2×D414yes—
135C2×C8×C67C2418yes—
136C2×C8×C67C2414yes—
137C2×C8×C67C2412yes—
138C2×C8×C67C2411yescharacteristic
139C2×C8×C67C2411yescharacteristic
140C4⋊C4×C67C22×C418yes—
141C4⋊C4×C67C22×C4116yes—
142C22⋊C4×C67C22×C418yes—
143C22⋊C4×C67C22×C4132yes—
144C42×C67C22×C414yes—
145C42×C67C2414yes—
146C23×C67—28no—
147C23×C67—42no—
148C23×C67—48no—
149C23×C67C4×D414yes—
150C23×C67C23×C418yes—
151C23×C67C23×C412yes—
152C23×C67C23×C411yescharacteristic
153Q8×C67—28no—
154Q8×C67—28no—
155D4×C67—216no—
156D4×C67—216no—
157D4×C67—28no—
158D4×C67—28no—
159D4×C67—416no—
160C2×C4×C67—28no—
161C2×C4×C67—216no—
162C2×C4×C67—28no—
163C2×C4×C67—28no—
164C2×C4×C67—44no—
165C2×C4×C67C4×D418yes—
166C2×C4×C67C4×D414yes—
167C2×C4×C67C23×C418yes—
168C2×C4×C67C23×C418yes—
169C2×C4×C67C23×C411yescharacteristic
170C2×C4×C67C23×C412yes—
171C2×C4×C67C22×D414yes—
172C2×C4×C67C22×D412yes—
173C2×C4×C67C2×#4112yes—
174C2×C4×C67C2×#4114yes—
175C2×C4×C67C2511yescharacteristic
176C8×C67—28no—
177C8×C67—24no—
178C8×C67C22×D412yes—
179C8×C67C22×D414yes—
180C8×C67C22×D412yes—
181C22×C67—22no—
182C22×C67—24no—
183C22×C67—28no—
184C22×C67—216no—
185C22×C67—44no—
186C22×C67—416no—
187C22×C67C2×C4×D414yes—
188C22×C67C2×C4×D412yes—
189C22×C67C24×C411yescharacteristic
190C4×C67—28no—
191C4×C67—28no—
192C4×C67C2×C4×D414yes—
193C4×C67C2×C4×D412yes—
194C4×C67C23×D411yescharacteristic
195C4×C67C22×#4111yescharacteristic
196#60140C6711yesmaximal, characteristic, direct product factor, Sylow
197C2×C67—24no—
198C2×C67—22no—
199C2×C67—48no—
200C2×C67C22×C4×D411yescharacteristic
201C2×C67#357312yes—
202C22×#131C2×C6712yescomplemented
203#3449C2×C6711yescharacteristic
204#2932C2×C6714yescomplemented
205#2931C2×C6714yescomplemented
206#2930C2×C6718yescomplemented
207#2929C2×C6718yescomplemented
208#2905C2×C6711yescharacteristic, complemented
209#2879C2×C6712yescomplemented
210C2×#405C2×C6711yescharacteristic, complemented
211C67#6014011yescharacteristic, direct product factor, Sylow
212C22×#41C22×C6712yes—
213C2×#131—216no—
214C2×#131C22×C6714yes—
215C2×#131C22×C6714yes—
216C2×#131C22×C6714yescomplemented
217C2×#131C22×C6714yescomplemented
218C22×M16C22×C6711yescharacteristic
219C22×M16C22×C6712yes—
220C22×M16C22×C6714yescomplemented
221C23×C8C22×C6711yescharacteristic
222C23×C8C22×C6714yescomplemented
223#537C4×C6714yes—
224#536C4×C6714yes—
225#535C4×C6714yes—
226#534C4×C6714yes—
227#532C22×C6714yes—
228#525C22×C6718yes—
229C2×#117C22×C6711yescharacteristic
230#436C22×C67116yescomplemented
231#435C22×C67132yescomplemented
232C8×D4C22×C67116yescomplemented
233#424C22×C6718yescomplemented
234#423C22×C6714yescomplemented
235#408C22×C6718yes—
236#408C22×C6718yescomplemented
237#407C22×C6718yescomplemented
238#405C22×C6714yescomplemented
239#405C22×C6714yescomplemented
240C2×#97C22×C6712yescomplemented
241C2×C4×C8C22×C6712yescomplemented
242C2×#41—28no—
243C2×#41—28no—
244C2×#41C2×C4×C6714yes—
245C2×#41C2×C4×C6714yes—
246C2×#41C23×C6714yes—
247C2×#41C23×C6714yes—
248C22×Q8C2×C4×C6712yes—
249C22×D4C2×C4×C6714yes—
250C22×D4C2×C4×C6712yes—
251C23×C4C23×C6714yes—
252C23×C4C23×C6711yescharacteristic
253#131—216no—
254#131—216no—
255#131—216no—
256#131—216no—
257C2×M16—28no—
258C2×M16—24no—
259C2×M16—216no—
260C2×M16C23×C6714yes—
261C2×M16C23×C6714yes—
262C2×M16C23×C6714yes—
263C2×M16C23×C6718yes—
264C2×M16C23×C6718yes—
265C22×C8—216no—
266C22×C8D4×C6718yescomplemented
267C22×C8C23×C6714yes—
268C22×C8C23×C6718yes—
269C22×C8C23×C6718yes—
270C22×C8C23×C6712yes—
271C22×C8C23×C6712yes—
272C22×C8C23×C6712yes—
273C22×C8C23×C6711yescharacteristic
274C22×C8C23×C6711yescharacteristic
275C4⋊Q8C2×C4×C6714yes—
276#127C2×C4×C6714yes—
277#124C2×C4×C6718yes—
278#123C2×C4×C67116yes—
279C22⋊Q8C2×C4×C67116yes—
280#121C2×C4×C67116yes—
281#121C2×C4×C67116yes—
282#120C2×C4×C67116yes—
283C4×D4C23×C67116yes—
284#117C2×C4×C6714yes—
285#117C23×C6714yes—
286C2×C4⋊C4C2×C4×C6712yes—
287C2×C22⋊C4C2×C4×C6712yes—
288C2×C42C23×C6712yes—
289C4⋊C8C23×C67116yes—
290C22⋊C8C23×C67132yes—
291#97C23×C6718yes—
292C4×C8C23×C6714yes—
293C4×C8C23×C6714yes—
294C24C22×C4×C6714yes—
295C24C22×C4×C6711yescharacteristic
296#41—216no—
297#41—216no—
298#41—28no—
299#41—28no—
300#41—28no—
301#41—28no—
302C2×Q8—28no—
303C2×Q8C22×C4×C6714yes—
304C2×Q8C22×C4×C6714yes—
305C2×D4—216no—
306C2×D4—28no—
307C2×D4C22×C4×C6718yes—
308C2×D4C22×C4×C6718yes—
309C2×D4C22×C4×C6714yes—
310C2×D4C22×C4×C6714yes—
311C2×D4C22×C4×C67116yes—
312C22×C4—28no—
313C22×C4—28no—
314C22×C4C22×C4×C6718yes—
315C22×C4C22×C4×C6714yes—
316C22×C4C22×C4×C6714yes—
317C22×C4C22×C4×C6711yescharacteristic
318C22×C4C22×C4×C6712yes—
319C22×C4C2×D4×C6714yes—
320C22×C4#41×C6714yes—
321C22×C4C24×C6718yes—
322C22×C4C24×C6711yescharacteristic
323C22×C4C24×C6712yes—
324M16—28no—
325M16—28no—
326M16—28no—
327M16—216no—
328M16—216no—
329C2×C8—28no—
330C2×C8—24no—
331C2×C8—216no—
332C2×C8—216no—
333C2×C8C2×D4×C6718yes—
334C2×C8C2×D4×C6718yes—
335C2×C8C2×D4×C6714yes—
336C2×C8C2×D4×C6714yes—
337C2×C8C24×C6714yes—
338C2×C8C24×C6718yes—
339C2×C8C24×C6711yescharacteristic
340C2×C8C24×C6711yescharacteristic
341C2×C8C24×C6712yes—
342C4⋊C4C22×C4×C6718yes—
343C4⋊C4C22×C4×C67116yes—
344C22⋊C4C22×C4×C6718yes—
345C22⋊C4C22×C4×C67132yes—
346C42C22×C4×C6714yes—
347C42C24×C6714yes—
348C23—28no—
349C23—42no—
350C23—48no—
351C23C4×D4×C6714yes—
352C23C23×C4×C6718yes—
353C23C23×C4×C6712yes—
354C23C23×C4×C6711yescharacteristic
355Q8—28no—
356Q8—28no—
357D4—216no—
358D4—216no—
359D4—28no—
360D4—28no—
361D4—416no—
362C2×C4—28no—
363C2×C4—216no—
364C2×C4—28no—
365C2×C4—28no—
366C2×C4—44no—
367C2×C4C4×D4×C6718yes—
368C2×C4C4×D4×C6714yes—
369C2×C4C23×C4×C6718yes—
370C2×C4C23×C4×C6718yes—
371C2×C4C23×C4×C6711yescharacteristic
372C2×C4C23×C4×C6712yes—
373C2×C4C22×D4×C6714yes—
374C2×C4C22×D4×C6712yes—
375C2×C4C2×#41×C6712yes—
376C2×C4C2×#41×C6714yes—
377C2×C4C25×C6711yescharacteristic
378C8—28no—
379C8—24no—
380C8C22×D4×C6712yes—
381C8C22×D4×C6714yes—
382C8C22×D4×C6712yes—
383C22—22no—
384C22—24no—
385C22—216no—
386C22—28no—
387C22—44no—
388C22—416no—
389C22C2×C4×D4×C6714yes—
390C22C2×C4×D4×C6712yes—
391C22C24×C4×C6711yescharacteristic
392C4—28no—
393C4—28no—
394C4C2×C4×D4×C6714yes—
395C4C2×C4×D4×C6712yes—
396C4C23×D4×C6711yescharacteristic
397C4C22×#41×C6711yescharacteristic
398C2—24no—
399C2—22no—
400C2—48no—
401C2C22×C4×D4×C6711yescharacteristic
402C2C67×#357312yes—
subgroupG/H
centerC2×C8×C67C24
commutatorC22C24×C4×C67
Frattini, cosocleC2×C4C25×C67
socleC22×C67C24×C4
prime radicalC67×#3449C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C8×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×#131×C67C2semidirect
3. C67×#2932C2semidirect
4. C67×#2931C2semidirect
5. C67×#2930C2semidirect
6. C67×#2929C2semidirect
7. C67×#2905C2semidirect
8. C67×#2879C2semidirect
9. C2×#405×C67C2semidirect
14. C2×#131×C67C22semidirect
15. C2×#131×C67C22semidirect
18. C22×M16×C67C22semidirect
20. C23×C8×C67C22semidirect
28. #436×C67C22semidirect
29. #435×C67C22semidirect
30. C8×D4×C67C22semidirect
31. #424×C67C22semidirect
32. #423×C67C22semidirect
34. #408×C67C22semidirect
35. #407×C67C22semidirect
36. #405×C67C22semidirect
37. #405×C67C22semidirect
38. C2×#97×C67C22semidirect
39. C2×C4×C8×C67C22semidirect
64. C22×C8×C67D4semidirect
196. #60140C67direct (N × H)
202. C22×#131C2×C67semidirect
204. #2932C2×C67semidirect
205. #2931C2×C67semidirect
206. #2930C2×C67semidirect
207. #2929C2×C67semidirect
208. #2905C2×C67semidirect
209. #2879C2×C67semidirect
210. C2×#405C2×C67semidirect
211. C67#60140direct (N × H)
216. C2×#131C22×C67semidirect
217. C2×#131C22×C67semidirect
220. C22×M16C22×C67semidirect
222. C23×C8C22×C67semidirect
230. #436C22×C67semidirect
231. #435C22×C67semidirect
232. C8×D4C22×C67semidirect
233. #424C22×C67semidirect
234. #423C22×C67semidirect
236. #408C22×C67semidirect
237. #407C22×C67semidirect
238. #405C22×C67semidirect
239. #405C22×C67semidirect
240. C2×#97C22×C67semidirect
241. C2×C4×C8C22×C67semidirect
266. C22×C8D4×C67semidirect

Maximal subgroups

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

subgroupindex
C22×#131×C672
C67×#34492
C67×#29322
C67×#29312
C67×#29302
C67×#29292
C67×#29052
C67×#28792
C2×#405×C672
#6014067