Finite Group Databasebeta

← C67×#60320 #30739571

Subgroups — C67×#60320 #30739571

Order 17152.

1876 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3483C211yesmaximal, characteristic
2C67×#2990C212yesmaximal, complemented
3C67×#2989C214yesmaximal, complemented
4C67×#2981C214yesmaximal, complemented
5C67×#2980C214yesmaximal, complemented
6C67×#2979C211yesmaximal, characteristic, complemented
7C67×#2970C214yesmaximal, complemented
8C67×#2965C212yesmaximal, complemented
9C67×#2935C214yesmaximal, complemented
10C67×#2932C214yesmaximal, complemented
11C67×#2931C211yesmaximal, characteristic, complemented
12C2×#131×C67C2212yes—
13C2×#131×C67C2214yes—
14Q82×C67C411yescharacteristic
15#555×C67C414yes—
16#555×C67C411yescharacteristic
17#555×C67C414yes—
18#550×C67C412yes—
19#550×C67C414yes—
20#532×C67C2211yescharacteristic
21#532×C67C2214yes—
22#532×C67C2214yes—
23C4×#41×C67C2212yes—
24C4×#41×C67C2214yes—
25#446×C67C2212yes—
26#446×C67C2212yes—
27#446×C67C2214yes—
28#436×C67C2214yes—
29#436×C67C2212yes—
30#436×C67C2214yes—
31#436×C67C2212yes—
32#435×C67C2214yes—
33#435×C67C2218yes—
34#435×C67C2218yes—
35#435×C67C2218yes—
36#435×C67C2218yes—
37#435×C67C2214yes—
38#435×C67C2218yes—
39C8×D4×C67C2214yes—
40C8×D4×C67C2214yes—
41C8×D4×C67C2214yes—
42#432×C67C2218yes—
43#432×C67C2214yes—
44#431×C67C2212yes—
45#431×C67C2214yes—
46#423×C67C2211yescharacteristic
47#423×C67C2214yes—
48C2×C4⋊C8×C67C2214yes—
49#408×C67C2212yes—
50#408×C67C2214yes—
51#408×C67C2214yes—
52#408×C67C2218yes—
53C4×M16×C67C2214yes—
54C2×#97×C67C2214yes—
55C2×C4×C8×C67C2211yescharacteristic
56C2×#41×C67C2×C412yes—
57C2×#41×C67C2×C414yes—
58C2×#41×C67C2312yes—
59C2×#41×C67C2314yes—
60#131×C67—44no—
61#131×C67—48no—
62C2×M16×C67C2314yes—
63C2×M16×C67C2312yes—
64C2×M16×C67C2314yes—
65C2×M16×C67C2314yes—
66C2×M16×C67C2314yes—
67C22×C8×C67C2311yescharacteristic
68C22×C8×C67C2314yes—
69C22×C8×C67C2318yes—
70C22×C8×C67C2311yescharacteristic
71C22×C8×C67C2314yes—
72C4⋊Q8×C67C2×C411yescharacteristic
73C4⋊Q8×C67C2×C414yes—
74C4⋊Q8×C67C2×C414yes—
75#127×C67C2×C411yescharacteristic
76#127×C67C2×C414yes—
77#127×C67C2×C414yes—
78#124×C67C2×C412yes—
79#124×C67C2×C414yes—
80#124×C67C2×C414yes—
81#124×C67C2×C418yes—
82#121×C67C2×C414yes—
83#121×C67C2×C414yes—
84#121×C67C2×C414yes—
85#121×C67C2×C418yes—
86#121×C67C2×C418yes—
87#121×C67C2×C418yes—
88C4×Q8×C67C2×C412yes—
89C4×Q8×C67C2×C414yes—
90C4×Q8×C67C2312yes—
91C4×D4×C67C2×C412yes—
92C4×D4×C67C2×C414yes—
93C4×D4×C67C2×C414yes—
94C4×D4×C67C2×C418yes—
95C4×D4×C67C2312yes—
96C4×D4×C67C2314yes—
97C4×D4×C67C2314yes—
98C4×D4×C67C2318yes—
99#117×C67C2×C412yes—
100#117×C67C2×C414yes—
101#117×C67C2×C414yes—
102#117×C67C2×C418yes—
103C2×C42×C67C2311yescharacteristic
104C2×C42×C67C2314yes—
105C2×C42×C67C2314yes—
106C4⋊C8×C67C2312yes—
107C4⋊C8×C67C2314yes—
108C4⋊C8×C67C2318yes—
109C4⋊C8×C67C2314yes—
110C4⋊C8×C67C2312yes—
111C4⋊C8×C67C2314yes—
112C22⋊C8×C67C2314yes—
113C22⋊C8×C67C2314yes—
114C22⋊C8×C67C2314yes—
115C22⋊C8×C67C2318yes—
116C22⋊C8×C67C2318yes—
117C22⋊C8×C67C2318yes—
118#97×C67C2314yes—
119#97×C67C2318yes—
120#97×C67C2314yes—
121C4×C8×C67C2312yes—
122C4×C8×C67C2312yes—
123C4×C8×C67C2314yes—
124#41×C67—42no—
125#41×C67—44no—
126#41×C67—42no—
127#41×C67—44no—
128#41×C67—44no—
129#41×C67—48no—
130C2×Q8×C67C22×C412yes—
131C2×Q8×C67C22×C412yes—
132C2×Q8×C67C22×C414yes—
133C2×D4×C67C22×C412yes—
134C2×D4×C67C22×C414yes—
135C2×D4×C67C22×C414yes—
136C2×D4×C67C22×C418yes—
137C2×D4×C67C22×C414yes—
138C2×D4×C67C22×C414yes—
139C2×D4×C67C22×C418yes—
140C2×D4×C67C22×C412yes—
141C22×C4×C67C22×C412yes—
142C22×C4×C67C22×C414yes—
143C22×C4×C67C22×C418yes—
144C22×C4×C67C22×C414yes—
145C22×C4×C67C2411yescharacteristic
146C22×C4×C67C2414yes—
147C22×C4×C67C2414yes—
148M16×C67—28no—
149M16×C67—44no—
150M16×C67—42no—
151M16×C67—44no—
152M16×C67—44no—
153C2×C8×C67—28no—
154C2×C8×C67—22no—
155C2×C8×C67—44no—
156C2×C8×C67—41no—
157C2×C8×C67—48no—
158C2×C8×C67C2414yes—
159C2×C8×C67C2414yes—
160C2×C8×C67C2411yescharacteristic
161C2×C8×C67C2414yes—
162C2×C8×C67C2412yes—
163C2×C8×C67C2411yescharacteristic
164C4⋊C4×C67C22×C414yes—
165C4⋊C4×C67C22×C412yes—
166C4⋊C4×C67C22×C414yes—
167C4⋊C4×C67C22×C418yes—
168C4⋊C4×C67C22×C412yes—
169C4⋊C4×C67C22×C414yes—
170C22⋊C4×C67C22×C414yes—
171C22⋊C4×C67C22×C418yes—
172C22⋊C4×C67C22×C414yes—
173C22⋊C4×C67C22×C418yes—
174C22⋊C4×C67C22×C414yes—
175C22⋊C4×C67C22×C418yes—
176C42×C67C22×C411yescharacteristic
177C42×C67C22×C414yes—
178C42×C67C22×C414yes—
179C42×C67C22×C414yes—
180C42×C67C22×C414yes—
181C42×C67C22×C411yescharacteristic
182C42×C67C2412yes—
183C42×C67C2414yes—
184C23×C67C23×C411yescharacteristic
185C23×C67C23×C414yes—
186C23×C67C23×C414yes—
187Q8×C67—24no—
188Q8×C67—42no—
189Q8×C67—44no—
190D4×C67—24no—
191D4×C67—28no—
192D4×C67—44no—
193D4×C67—42no—
194D4×C67—44no—
195D4×C67—48no—
196D4×C67—44no—
197D4×C67—48no—
198C2×C4×C67—24no—
199C2×C4×C67—28no—
200C2×C4×C67—24no—
201C2×C4×C67—21no—
202C2×C4×C67—24no—
203C2×C4×C67—24no—
204C2×C4×C67—21no—
205C2×C4×C67—24no—
206C2×C4×C67—48no—
207C2×C4×C67—44no—
208C2×C4×C67C23×C412yes—
209C2×C4×C67C23×C414yes—
210C2×C4×C67C23×C414yes—
211C2×C4×C67C23×C411yescharacteristic
212C2×C4×C67C23×C414yes—
213C2×C4×C67C23×C414yes—
214C2×C4×C67C23×C412yes—
215C2×C4×C67C2511yescharacteristic
216C8×C67—24no—
217C8×C67—21no—
218C8×C67—24no—
219C8×C67—24no—
220C8×C67—22no—
221C8×C67H(5,𝔽2)12yes—
222C22×C67—21no—
223C22×C67—24no—
224C22×C67—24no—
225C22×C67—22no—
226C22×C67—24no—
227C22×C67—24no—
228C22×C67—28no—
229C22×C67C24×C411yescharacteristic
230C4×C67—24no—
231C4×C67—21no—
232C4×C67—24no—
233C4×C67—22no—
234C4×C67—24no—
235C4×C67—24no—
236C4×C67#56814yes—
237C4×C67C2×H(5,𝔽2)11yescharacteristic
238C4×C67#58511yescharacteristic
239#60320C6711yesmaximal, characteristic, direct product factor, Sylow
240C2×C67—41no—
241C2×C67—44no—
242C2×C67—44no—
243C2×C67C4×H(5,𝔽2)11yescharacteristic
244C2×C67C2×#56812yes—
245#3483C2×C6711yescharacteristic
246#2990C2×C6712yescomplemented
247#2989C2×C6714yescomplemented
248#2981C2×C6714yescomplemented
249#2980C2×C6714yescomplemented
250#2979C2×C6711yescharacteristic, complemented
251#2970C2×C6714yescomplemented
252#2965C2×C6712yescomplemented
253#2935C2×C6714yescomplemented
254#2932C2×C6714yescomplemented
255#2931C2×C6711yescharacteristic, complemented
256C67#6032011yescharacteristic, direct product factor, Sylow
257C2×#131C22×C6712yes—
258C2×#131C22×C6714yes—
259Q82C4×C6711yescharacteristic
260#555C4×C6714yes—
261#555C4×C6711yescharacteristic
262#555C4×C6714yes—
263#550C4×C6712yes—
264#550C4×C6714yes—
265#532C22×C6711yescharacteristic
266#532C22×C6714yes—
267#532C22×C6714yes—
268C4×#41C22×C6712yes—
269C4×#41C22×C6714yes—
270#446C22×C6712yes—
271#446C22×C6712yes—
272#446C22×C6714yes—
273#436C22×C6714yes—
274#436C22×C6712yes—
275#436C22×C6714yes—
276#436C22×C6712yes—
277#435C22×C6714yes—
278#435C22×C6718yes—
279#435C22×C6718yes—
280#435C22×C6718yes—
281#435C22×C6718yes—
282#435C22×C6714yes—
283#435C22×C6718yes—
284C8×D4C22×C6714yes—
285C8×D4C22×C6714yes—
286C8×D4C22×C6714yes—
287#432C22×C6718yes—
288#432C22×C6714yes—
289#431C22×C6712yes—
290#431C22×C6714yes—
291#423C22×C6711yescharacteristic
292#423C22×C6714yes—
293C2×C4⋊C8C22×C6714yes—
294#408C22×C6712yes—
295#408C22×C6714yes—
296#408C22×C6714yes—
297#408C22×C6718yes—
298C4×M16C22×C6714yes—
299C2×#97C22×C6714yes—
300C2×C4×C8C22×C6711yescharacteristic
301C2×#41C2×C4×C6712yes—
302C2×#41C2×C4×C6714yes—
303C2×#41C23×C6712yes—
304C2×#41C23×C6714yes—
305#131—44no—
306#131—48no—
307C2×M16C23×C6714yes—
308C2×M16C23×C6712yes—
309C2×M16C23×C6714yes—
310C2×M16C23×C6714yes—
311C2×M16C23×C6714yes—
312C22×C8C23×C6711yescharacteristic
313C22×C8C23×C6714yes—
314C22×C8C23×C6718yes—
315C22×C8C23×C6711yescharacteristic
316C22×C8C23×C6714yes—
317C4⋊Q8C2×C4×C6711yescharacteristic
318C4⋊Q8C2×C4×C6714yes—
319C4⋊Q8C2×C4×C6714yes—
320#127C2×C4×C6711yescharacteristic
321#127C2×C4×C6714yes—
322#127C2×C4×C6714yes—
323#124C2×C4×C6712yes—
324#124C2×C4×C6714yes—
325#124C2×C4×C6714yes—
326#124C2×C4×C6718yes—
327#121C2×C4×C6714yes—
328#121C2×C4×C6714yes—
329#121C2×C4×C6714yes—
330#121C2×C4×C6718yes—
331#121C2×C4×C6718yes—
332#121C2×C4×C6718yes—
333C4×Q8C2×C4×C6712yes—
334C4×Q8C2×C4×C6714yes—
335C4×Q8C23×C6712yes—
336C4×D4C2×C4×C6712yes—
337C4×D4C2×C4×C6714yes—
338C4×D4C2×C4×C6714yes—
339C4×D4C2×C4×C6718yes—
340C4×D4C23×C6712yes—
341C4×D4C23×C6714yes—
342C4×D4C23×C6714yes—
343C4×D4C23×C6718yes—
344#117C2×C4×C6712yes—
345#117C2×C4×C6714yes—
346#117C2×C4×C6714yes—
347#117C2×C4×C6718yes—
348C2×C42C23×C6711yescharacteristic
349C2×C42C23×C6714yes—
350C2×C42C23×C6714yes—
351C4⋊C8C23×C6712yes—
352C4⋊C8C23×C6714yes—
353C4⋊C8C23×C6718yes—
354C4⋊C8C23×C6714yes—
355C4⋊C8C23×C6712yes—
356C4⋊C8C23×C6714yes—
357C22⋊C8C23×C6714yes—
358C22⋊C8C23×C6714yes—
359C22⋊C8C23×C6714yes—
360C22⋊C8C23×C6718yes—
361C22⋊C8C23×C6718yes—
362C22⋊C8C23×C6718yes—
363#97C23×C6714yes—
364#97C23×C6718yes—
365#97C23×C6714yes—
366C4×C8C23×C6712yes—
367C4×C8C23×C6712yes—
368C4×C8C23×C6714yes—
369#41—42no—
370#41—44no—
371#41—42no—
372#41—44no—
373#41—44no—
374#41—48no—
375C2×Q8C22×C4×C6712yes—
376C2×Q8C22×C4×C6712yes—
377C2×Q8C22×C4×C6714yes—
378C2×D4C22×C4×C6712yes—
379C2×D4C22×C4×C6714yes—
380C2×D4C22×C4×C6714yes—
381C2×D4C22×C4×C6718yes—
382C2×D4C22×C4×C6714yes—
383C2×D4C22×C4×C6714yes—
384C2×D4C22×C4×C6718yes—
385C2×D4C22×C4×C6712yes—
386C22×C4C22×C4×C6712yes—
387C22×C4C22×C4×C6714yes—
388C22×C4C22×C4×C6718yes—
389C22×C4C22×C4×C6714yes—
390C22×C4C24×C6711yescharacteristic
391C22×C4C24×C6714yes—
392C22×C4C24×C6714yes—
393M16—28no—
394M16—42no—
395M16—44no—
396M16—44no—
397M16—44no—
398C2×C8—28no—
399C2×C8—22no—
400C2×C8—48no—
401C2×C8—44no—
402C2×C8—41no—
403C2×C8C24×C6712yes—
404C2×C8C24×C6711yescharacteristic
405C2×C8C24×C6714yes—
406C2×C8C24×C6714yes—
407C2×C8C24×C6714yes—
408C2×C8C24×C6711yescharacteristic
409C4⋊C4C22×C4×C6714yes—
410C4⋊C4C22×C4×C6712yes—
411C4⋊C4C22×C4×C6714yes—
412C4⋊C4C22×C4×C6718yes—
413C4⋊C4C22×C4×C6712yes—
414C4⋊C4C22×C4×C6714yes—
415C22⋊C4C22×C4×C6714yes—
416C22⋊C4C22×C4×C6718yes—
417C22⋊C4C22×C4×C6714yes—
418C22⋊C4C22×C4×C6718yes—
419C22⋊C4C22×C4×C6714yes—
420C22⋊C4C22×C4×C6718yes—
421C42C22×C4×C6711yescharacteristic
422C42C22×C4×C6714yes—
423C42C22×C4×C6714yes—
424C42C22×C4×C6714yes—
425C42C22×C4×C6714yes—
426C42C22×C4×C6711yescharacteristic
427C42C24×C6712yes—
428C42C24×C6714yes—
429C23C23×C4×C6711yescharacteristic
430C23C23×C4×C6714yes—
431C23C23×C4×C6714yes—
432Q8—24no—
433Q8—42no—
434Q8—44no—
435D4—24no—
436D4—28no—
437D4—44no—
438D4—48no—
439D4—44no—
440D4—42no—
441D4—44no—
442D4—48no—
443C2×C4—24no—
444C2×C4—28no—
445C2×C4—24no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—48no—
452C2×C4—44no—
453C2×C4C23×C4×C6712yes—
454C2×C4C23×C4×C6714yes—
455C2×C4C23×C4×C6714yes—
456C2×C4C23×C4×C6711yescharacteristic
457C2×C4C23×C4×C6714yes—
458C2×C4C23×C4×C6714yes—
459C2×C4C23×C4×C6712yes—
460C2×C4C25×C6711yescharacteristic
461C8—24no—
462C8—21no—
463C8—24no—
464C8—24no—
465C8—22no—
466C8H(5,𝔽2)×C6712yes—
467C22—22no—
468C22—24no—
469C22—24no—
470C22—28no—
471C22—21no—
472C22—24no—
473C22—24no—
474C22C24×C4×C6711yescharacteristic
475C4—22no—
476C4—24no—
477C4—24no—
478C4—24no—
479C4—21no—
480C4—24no—
481C4#568×C6714yes—
482C4C2×H(5,𝔽2)×C6711yescharacteristic
483C4#585×C6711yescharacteristic
484C2—41no—
485C2—44no—
486C2—44no—
487C2C4×H(5,𝔽2)×C6711yescharacteristic
488C2C2×#568×C6712yes—
subgroupG/H
centerC2×C4×C67C25
commutatorC22C24×C4×C67
Frattini, cosocleC2×C4C25×C67
socleC22×C67C24×C4
prime radicalC67×#3483C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2990C2semidirect
3. C67×#2989C2semidirect
4. C67×#2981C2semidirect
5. C67×#2980C2semidirect
6. C67×#2979C2semidirect
7. C67×#2970C2semidirect
8. C67×#2965C2semidirect
9. C67×#2935C2semidirect
10. C67×#2932C2semidirect
11. C67×#2931C2semidirect
239. #60320C67direct (N × H)
246. #2990C2×C67semidirect
247. #2989C2×C67semidirect
248. #2981C2×C67semidirect
249. #2980C2×C67semidirect
250. #2979C2×C67semidirect
251. #2970C2×C67semidirect
252. #2965C2×C67semidirect
253. #2935C2×C67semidirect
254. #2932C2×C67semidirect
255. #2931C2×C67semidirect
256. C67#60320direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#34832
C67×#29902
C67×#29892
C67×#29812
C67×#29802
C67×#29792
C67×#29702
C67×#29652
C67×#29352
C67×#29322
C67×#29312
#6032067