Finite Group Databasebeta

← C67×#19200 #30698451

Subgroups — C67×#19200 #30698451

Order 17152.

660 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3246C211yesmaximal, characteristic, complemented
2C67×#3245C211yesmaximal, characteristic
3C67×#3131C211yesmaximal, characteristic, complemented
4C67×#2931C211yesmaximal, characteristic, complemented
5C67×#1703C212yesmaximal, complemented
6C67×#1701C212yesmaximal, complemented
7C67×#1679C212yesmaximal, complemented
8C67×#1677C212yesmaximal, complemented
9C2×#334×C67C211yesmaximal, characteristic, complemented
10C67×#1550C211yesmaximal, characteristic, complemented
11C67×#1547C211yesmaximal, characteristic, complemented
12C2×#131×C67—21no—
13#533×C67C2211yescharacteristic
14#532×C67C2211yescharacteristic
15#532×C67C2211yescharacteristic
16#488×C67—21no—
17#488×C67—21no—
18#487×C67C2211yescharacteristic
19#487×C67C2211yescharacteristic
20#486×C67C2211yescharacteristic
21#486×C67C2211yescharacteristic
22#476×C67C2212yes—
23#475×C67—21no—
24#474×C67—21no—
25#474×C67C2212yes—
26#466×C67—22no—
27#465×C67—22no—
28#461×C67C2212yes—
29#460×C67—21no—
30#459×C67—21no—
31#459×C67C2212yes—
32#436×C67—21no—
33#436×C67C2212yes—
34C8×D4×C67—21no—
35C8×D4×C67C2212yes—
36#423×C67C2211yescharacteristic
37C2×C4⋊C8×C67C2211yescharacteristic
38C2×C4⋊C8×C67C2211yescharacteristic
39#416×C67—21no—
40#416×C67—21no—
41#408×C67—21no—
42C2×C4×C8×C67C2211yescharacteristic
43#334×C67C2212yes—
44#334×C67C2212yes—
45#333×C67C2212yes—
46#332×C67C2212yes—
47#332×C67C2212yes—
48#331×C67C2212yes—
49C2×#41×C67—21no—
50C2×#41×C67—21no—
51C2×SD16×C67—22no—
52C2×D8×C67—22no—
53#131×C67—44no—
54C2×M16×C67—21no—
55C2×M16×C67—22no—
56C22×C8×C67—21no—
57C22×C8×C67—21no—
58C22×C8×C67—22no—
59C22×C8×C67D411yescharacteristic
60C22×C8×C67D411yescharacteristic
61C4⋊Q8×C67D411yescharacteristic
62C4⋊Q8×C67C2311yescharacteristic
63C4⋊Q8×C67C2311yescharacteristic
64C4⋊Q8×C67C2311yescharacteristic
65#127×C67D411yescharacteristic
66#127×C67C2311yescharacteristic
67#125×C67—21no—
68#124×C67—21no—
69#124×C67—21no—
70C22⋊Q8×C67—21no—
71C22⋊Q8×C67—21no—
72#121×C67—21no—
73#121×C67—21no—
74#121×C67—21no—
75#121×C67D412yes—
76C4×Q8×C67—21no—
77C4×Q8×C67—21no—
78C4×D4×C67—21no—
79C4×D4×C67—21no—
80C4×D4×C67—21no—
81C4×D4×C67C2312yes—
82#117×C67—21no—
83C2×C42×C67C2311yescharacteristic
84#107×C67—22no—
85#106×C67—22no—
86C4⋊C8×C67—21no—
87C4⋊C8×C67—21no—
88C4⋊C8×C67—21no—
89C4⋊C8×C67C2312yes—
90C4⋊C8×C67C2312yes—
91C4⋊C8×C67C2312yes—
92#103×C67—21no—
93#103×C67—21no—
94#103×C67—21no—
95#103×C67—21no—
96#103×C67—22no—
97#102×C67—22no—
98#102×C67—22no—
99#102×C67—21no—
100#102×C67—21no—
101#102×C67—21no—
102#102×C67—21no—
103#102×C67—22no—
104C22⋊C8×C67—22no—
105C22⋊C8×C67—22no—
106C4×C8×C67D411yescharacteristic
107C4×C8×C67D411yescharacteristic
108C4×C8×C67C2311yescharacteristic
109C4×C8×C67C2311yescharacteristic
110#41×C67—42no—
111#41×C67—42no—
112#41×C67—42no—
113#41×C67—42no—
114C2×Q8×C67—21no—
115C2×Q8×C67—21no—
116C2×Q8×C67—21no—
117C2×Q8×C67—21no—
118C2×D4×C67—22no—
119C2×D4×C67—21no—
120C2×D4×C67—21no—
121C2×D4×C67—21no—
122C2×D4×C67—21no—
123C2×D4×C67—21no—
124C2×D4×C67—41no—
125C2×D4×C67C2×D412yes—
126C22×C4×C67—22no—
127C22×C4×C67—41no—
128C22×C4×C67C2×D411yescharacteristic
129C22×C4×C67C2×D411yescharacteristic
130C22×C4×C67#4111yescharacteristic
131SD16×C67—82no—
132D8×C67—82no—
133M16×C67—44no—
134M16×C67—42no—
135C2×C8×C67—22no—
136C2×C8×C67—22no—
137C2×C8×C67—22no—
138C2×C8×C67—22no—
139C2×C8×C67—22no—
140C2×C8×C67—44no—
141C2×C8×C67—81no—
142C2×C8×C67—81no—
143C2×C8×C67C2×D411yescharacteristic
144C2×C8×C67C2×D411yescharacteristic
145C2×C8×C67C2×D411yescharacteristic
146C2×C8×C67C2×D411yescharacteristic
147C4⋊C4×C67—21no—
148C4⋊C4×C67—21no—
149C4⋊C4×C67—21no—
150C4⋊C4×C67—21no—
151C4⋊C4×C67—21no—
152C4⋊C4×C67—21no—
153C4⋊C4×C67—21no—
154C4⋊C4×C67—41no—
155C4⋊C4×C67—41no—
156C4⋊C4×C67C2×D412yes—
157C22⋊C4×C67—22no—
158C22⋊C4×C67—22no—
159C22⋊C4×C67—41no—
160C22⋊C4×C67—41no—
161C22⋊C4×C67—41no—
162C42×C67—41no—
163C42×C67C2×D411yescharacteristic
164C42×C67C2×D411yescharacteristic
165C42×C67#4111yescharacteristic
166C42×C67C2411yescharacteristic
167C23×C67—22no—
168C23×C67—41no—
169C23×C67#12111yescharacteristic
170Q8×C67—42no—
171Q8×C67—41no—
172Q8×C67—41no—
173Q8×C67—41no—
174Q8×C67—41no—
175Q8×C67—81no—
176D4×C67—22no—
177D4×C67—22no—
178D4×C67—42no—
179D4×C67—44no—
180D4×C67—41no—
181D4×C67—41no—
182D4×C67—41no—
183D4×C67—41no—
184D4×C67—42no—
185D4×C67—81no—
186D4×C67—82no—
187C2×C4×C67—21no—
188C2×C4×C67—21no—
189C2×C4×C67—22no—
190C2×C4×C67—41no—
191C2×C4×C67—42no—
192C2×C4×C67—42no—
193C2×C4×C67—41no—
194C2×C4×C67—41no—
195C2×C4×C67—41no—
196C2×C4×C67—82no—
197C2×C4×C67#12111yescharacteristic
198C2×C4×C67#12111yescharacteristic
199C2×C4×C67#12111yescharacteristic
200C2×C4×C67#13512yes—
201C2×C4×C67AGL(1,ℤ/8)12yes—
202C2×C4×C67C22×D411yescharacteristic
203C2×C4×C67C22×D411yescharacteristic
204C2×C4×C67C2×#4111yescharacteristic
205C8×C67—21no—
206C8×C67—21no—
207C8×C67—21no—
208C8×C67—21no—
209C8×C67—42no—
210C8×C67—42no—
211C8×C67—42no—
212C22×C67—21no—
213C22×C67—21no—
214C22×C67—42no—
215C22×C67—42no—
216C22×C67—42no—
217C22×C67—81no—
218C22×C67—81no—
219C22×C67—81no—
220C22×C67#46312yes—
221C22×C67C2×#12111yescharacteristic
222C4×C67—21no—
223C4×C67—21no—
224C4×C67—21no—
225C4×C67—42no—
226C4×C67—81no—
227C4×C67—81no—
228C4×C67—81no—
229C4×C67#46312yes—
230C4×C67C2×#13511yescharacteristic
231C4×C67C2×AGL(1,ℤ/8)11yescharacteristic
232C4×C67#57711yescharacteristic
233C4×C67#57711yescharacteristic
234#19200C6711yesmaximal, characteristic, direct product factor, Sylow
235C2×C67—22no—
236C2×C67—82no—
237C2×C67—161no—
238C2×C67C2×#46311yescharacteristic
239C2×C67#306011yescharacteristic
240C2×C67#306711yescharacteristic
241#3246C2×C6711yescharacteristic, complemented
242#3245C2×C6711yescharacteristic
243#3131C2×C6711yescharacteristic, complemented
244#2931C2×C6711yescharacteristic, complemented
245#1703C2×C6712yescomplemented
246#1701C2×C6712yescomplemented
247#1679C2×C6712yescomplemented
248#1677C2×C6712yescomplemented
249C2×#334C2×C6711yescharacteristic, complemented
250#1550C2×C6711yescharacteristic, complemented
251#1547C2×C6711yescharacteristic, complemented
252C67#1920011yescharacteristic, direct product factor, Sylow
253C2×#131—21no—
254#533C22×C6711yescharacteristic
255#532C22×C6711yescharacteristic
256#532C22×C6711yescharacteristic
257#488—21no—
258#488—21no—
259#487C22×C6711yescharacteristic
260#487C22×C6711yescharacteristic
261#486C22×C6711yescharacteristic
262#486C22×C6711yescharacteristic
263#476C22×C6712yes—
264#475—21no—
265#474—21no—
266#474C22×C6712yes—
267#466—22no—
268#465—22no—
269#461C22×C6712yes—
270#460—21no—
271#459—21no—
272#459C22×C6712yes—
273#436—21no—
274#436C22×C6712yes—
275C8×D4—21no—
276C8×D4C22×C6712yes—
277#423C22×C6711yescharacteristic
278C2×C4⋊C8C22×C6711yescharacteristic
279C2×C4⋊C8C22×C6711yescharacteristic
280#416—21no—
281#416—21no—
282#408—21no—
283C2×C4×C8C22×C6711yescharacteristic
284#334C22×C6712yes—
285#334C22×C6712yes—
286#333C22×C6712yes—
287#332C22×C6712yes—
288#332C22×C6712yes—
289#331C22×C6712yes—
290C2×#41—21no—
291C2×#41—21no—
292C2×SD16—22no—
293C2×D8—22no—
294#131—44no—
295C2×M16—21no—
296C2×M16—22no—
297C22×C8—21no—
298C22×C8—21no—
299C22×C8—22no—
300C22×C8D4×C6711yescharacteristic
301C22×C8D4×C6711yescharacteristic
302C4⋊Q8D4×C6711yescharacteristic
303C4⋊Q8C23×C6711yescharacteristic
304C4⋊Q8C23×C6711yescharacteristic
305C4⋊Q8C23×C6711yescharacteristic
306#127D4×C6711yescharacteristic
307#127C23×C6711yescharacteristic
308#125—21no—
309#124—21no—
310#124—21no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313#121—21no—
314#121—21no—
315#121—21no—
316#121D4×C6712yes—
317C4×Q8—21no—
318C4×Q8—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4C23×C6712yes—
323#117—21no—
324C2×C42C23×C6711yescharacteristic
325#107—22no—
326#106—22no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8C23×C6712yes—
331C4⋊C8C23×C6712yes—
332C4⋊C8C23×C6712yes—
333#103—21no—
334#103—21no—
335#103—21no—
336#103—21no—
337#103—22no—
338#102—22no—
339#102—22no—
340#102—21no—
341#102—21no—
342#102—21no—
343#102—21no—
344#102—22no—
345C22⋊C8—22no—
346C22⋊C8—22no—
347C4×C8D4×C6711yescharacteristic
348C4×C8D4×C6711yescharacteristic
349C4×C8C23×C6711yescharacteristic
350C4×C8C23×C6711yescharacteristic
351#41—42no—
352#41—42no—
353#41—42no—
354#41—42no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—41no—
366C2×D4C2×D4×C6712yes—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4C2×D4×C6711yescharacteristic
370C22×C4C2×D4×C6711yescharacteristic
371C22×C4#41×C6711yescharacteristic
372SD16—82no—
373D8—82no—
374M16—44no—
375M16—42no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—44no—
382C2×C8—81no—
383C2×C8—81no—
384C2×C8C2×D4×C6711yescharacteristic
385C2×C8C2×D4×C6711yescharacteristic
386C2×C8C2×D4×C6711yescharacteristic
387C2×C8C2×D4×C6711yescharacteristic
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4C2×D4×C6712yes—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C42—41no—
404C42C2×D4×C6711yescharacteristic
405C42C2×D4×C6711yescharacteristic
406C42#41×C6711yescharacteristic
407C42C24×C6711yescharacteristic
408C23—22no—
409C23—41no—
410C23#121×C6711yescharacteristic
411Q8—42no—
412Q8—41no—
413Q8—41no—
414Q8—41no—
415Q8—41no—
416Q8—81no—
417D4—22no—
418D4—22no—
419D4—42no—
420D4—44no—
421D4—42no—
422D4—41no—
423D4—41no—
424D4—41no—
425D4—41no—
426D4—81no—
427D4—82no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—82no—
438C2×C4#121×C6711yescharacteristic
439C2×C4#121×C6711yescharacteristic
440C2×C4#121×C6711yescharacteristic
441C2×C4#135×C6712yes—
442C2×C4AGL(1,ℤ/8)×C6712yes—
443C2×C4C22×D4×C6711yescharacteristic
444C2×C4C22×D4×C6711yescharacteristic
445C2×C4C2×#41×C6711yescharacteristic
446C8—21no—
447C8—21no—
448C8—21no—
449C8—21no—
450C8—42no—
451C8—42no—
452C8—42no—
453C22—21no—
454C22—21no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22#463×C6712yes—
462C22C2×#121×C6711yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4#463×C6712yes—
471C4C2×#135×C6711yescharacteristic
472C4C2×AGL(1,ℤ/8)×C6711yescharacteristic
473C4#577×C6711yescharacteristic
474C4#577×C6711yescharacteristic
475C2—22no—
476C2—82no—
477C2—161no—
478C2C2×#463×C6711yescharacteristic
479C2C67×#306011yescharacteristic
480C2C67×#306711yescharacteristic
subgroupG/H
center, socleC22×C67C2×#121
commutator, Frattini, cosocleC42C24×C67
prime radicalC67×#3245C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3246C2semidirect
3. C67×#3131C2semidirect
4. C67×#2931C2semidirect
5. C67×#1703C2semidirect
6. C67×#1701C2semidirect
7. C67×#1679C2semidirect
8. C67×#1677C2semidirect
9. C2×#334×C67C2semidirect
10. C67×#1550C2semidirect
11. C67×#1547C2semidirect
234. #19200C67direct (N × H)
241. #3246C2×C67semidirect
243. #3131C2×C67semidirect
244. #2931C2×C67semidirect
245. #1703C2×C67semidirect
246. #1701C2×C67semidirect
247. #1679C2×C67semidirect
248. #1677C2×C67semidirect
249. C2×#334C2×C67semidirect
250. #1550C2×C67semidirect
251. #1547C2×C67semidirect
252. C67#19200direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#32462
C67×#32452
C67×#31312
C67×#29312
C67×#17032
C67×#17012
C67×#16792
C67×#16772
C2×#334×C672
C67×#15502
C67×#15472
#1920067