Finite Group Databasebeta

← C67×#19414 #30698665

Subgroups — C67×#19414 #30698665

Order 17152.

662 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×#3227C211yesmaximal, characteristic, complemented
2C67×#3079C211yesmaximal, characteristic, complemented
3C67×#3035C211yesmaximal, characteristic
4C67×#2988C211yesmaximal, characteristic, complemented
5C67×#1678C212yesmaximal, complemented
6C67×#1677C212yesmaximal, complemented
7C67×#1672C212yesmaximal, complemented
8C67×#1671C212yesmaximal, complemented
9C67×#1570C211yesmaximal, characteristic, complemented
10C67×#1547C211yesmaximal, characteristic, complemented
11C2×#333×C67C211yesmaximal, characteristic, complemented
12C2×#135×C67—21no—
13#532×C67C2211yescharacteristic
14C2×C4⋊Q8×C67C2211yescharacteristic
15C4×#41×C67C2211yescharacteristic
16#498×C67C2212yes—
17#486×C67C2212yes—
18#481×C67—22no—
19#477×C67C2212yes—
20#476×C67—21no—
21#476×C67—21no—
22#476×C67C2212yes—
23#475×C67—21no—
24#474×C67—21no—
25#464×C67—21no—
26#463×C67—21no—
27#462×C67C2211yescharacteristic
28#461×C67C2211yescharacteristic
29#460×C67C2211yescharacteristic
30#459×C67C2211yescharacteristic
31C8×Q8×C67C2212yes—
32C8×D4×C67—22no—
33C8×D4×C67C2212yes—
34#431×C67—21no—
35#431×C67C2211yescharacteristic
36#428×C67—21no—
37#423×C67C2211yescharacteristic
38C2×C4⋊C8×C67—21no—
39C2×C4⋊C8×C67C2211yescharacteristic
40C2×C4⋊C8×C67C2211yescharacteristic
41#417×C67—21no—
42#416×C67—21no—
43#334×C67C2212yes—
44#334×C67C2212yes—
45#333×C67C2212yes—
46#333×C67C2212yes—
47#332×C67C2212yes—
48#331×C67C2212yes—
49C2×#41×C67—21no—
50C2×#41×C67D411yescharacteristic
51C22×Q8×C67—21no—
52#135×C67—44no—
53C2×Q16×C67—21no—
54C2×SD16×C67—21no—
55C2×SD16×C67—21no—
56C2×D8×C67—21no—
57C2×M16×C67—21no—
58C22×C8×C67—21no—
59C22×C8×C67—21no—
60C22×C8×C67—22no—
61C4⋊Q8×C67—21no—
62C4⋊Q8×C67—21no—
63C4⋊Q8×C67—21no—
64C4⋊Q8×C67C2311yescharacteristic
65C4⋊Q8×C67C2312yes—
66#127×C67C2311yescharacteristic
67#124×C67—21no—
68#121×C67—21no—
69#121×C67—21no—
70C4×Q8×C67D411yescharacteristic
71C4×Q8×C67C2311yescharacteristic
72C4×D4×C67—21no—
73C4×D4×C67—21no—
74C4×D4×C67—21no—
75C4×D4×C67—21no—
76C4×D4×C67D411yescharacteristic
77C4×D4×C67C2311yescharacteristic
78#117×C67—21no—
79#117×C67D411yescharacteristic
80C2×C4⋊C4×C67—21no—
81C2×C4⋊C4×C67—21no—
82C2×C42×C67—21no—
83C2×C42×C67C2311yescharacteristic
84#107×C67—22no—
85#106×C67—22no—
86#106×C67—22no—
87#106×C67—22no—
88C4⋊C8×C67—22no—
89C4⋊C8×C67—21no—
90C4⋊C8×C67—21no—
91C4⋊C8×C67—21no—
92C4⋊C8×C67D411yescharacteristic
93C4⋊C8×C67D411yescharacteristic
94C4⋊C8×C67C2311yescharacteristic
95C4⋊C8×C67C2311yescharacteristic
96C4⋊C8×C67C2312yes—
97C4⋊C8×C67C2312yes—
98#103×C67—22no—
99#103×C67—21no—
100#103×C67—21no—
101#102×C67—22no—
102#102×C67—22no—
103#102×C67—22no—
104#102×C67—21no—
105#102×C67—21no—
106C22⋊C8×C67—22no—
107C22⋊C8×C67D412yes—
108C4×C8×C67—22no—
109C4×C8×C67C2312yes—
110#41×C67—22no—
111#41×C67—42no—
112#41×C67—42no—
113#41×C67SD1614yes—
114C2×Q8×C67—21no—
115C2×Q8×C67—22no—
116C2×Q8×C67—41no—
117C2×Q8×C67—82no—
118C2×Q8×C67C2×D411yescharacteristic
119C2×D4×C67—21no—
120C2×D4×C67—21no—
121C2×D4×C67—21no—
122C2×D4×C67—21no—
123C2×D4×C67—41no—
124C2×D4×C67C2×D411yescharacteristic
125C22×C4×C67—21no—
126C22×C4×C67—21no—
127C22×C4×C67—21no—
128C22×C4×C67—41no—
129C22×C4×C67—41no—
130C22×C4×C67C2×D411yescharacteristic
131C22×C4×C67C2×D411yescharacteristic
132C22×C4×C67#4111yescharacteristic
133Q16×C67—42no—
134SD16×C67—42no—
135SD16×C67—42no—
136D8×C67—42no—
137M16×C67—42no—
138C2×C8×C67—22no—
139C2×C8×C67—21no—
140C2×C8×C67—21no—
141C2×C8×C67—22no—
142C2×C8×C67—42no—
143C2×C8×C67—82no—
144C2×C8×C67—81no—
145C2×C8×C67C2×D412yes—
146C2×C8×C67C2×D412yes—
147C4⋊C4×C67—21no—
148C4⋊C4×C67—21no—
149C4⋊C4×C67—22no—
150C4⋊C4×C67—22no—
151C4⋊C4×C67—21no—
152C4⋊C4×C67—21no—
153C4⋊C4×C67—41no—
154C4⋊C4×C67—41no—
155C4⋊C4×C67C2×D411yescharacteristic
156C4⋊C4×C67C2×D411yescharacteristic
157C22⋊C4×C67—21no—
158C22⋊C4×C67—21no—
159C22⋊C4×C67—21no—
160C22⋊C4×C67—41no—
161C22⋊C4×C67—41no—
162C42×C67—21no—
163C42×C67—21no—
164C42×C67—21no—
165C42×C67C2×D411yescharacteristic
166C42×C67C2×D411yescharacteristic
167C42×C67#4111yescharacteristic
168C42×C67C2411yescharacteristic
169C23×C67—21no—
170C23×C67—41no—
171C23×C67#12111yescharacteristic
172Q8×C67—21no—
173Q8×C67—41no—
174Q8×C67—41no—
175Q8×C67—82no—
176Q8×C67—82no—
177Q8×C67C2×SD1612yes—
178D4×C67—24no—
179D4×C67—21no—
180D4×C67—41no—
181D4×C67—41no—
182D4×C67—41no—
183D4×C67—41no—
184D4×C67—81no—
185D4×C67—82no—
186D4×C67C2×SD1612yes—
187C2×C4×C67—21no—
188C2×C4×C67—21no—
189C2×C4×C67—21no—
190C2×C4×C67—24no—
191C2×C4×C67—21no—
192C2×C4×C67—21no—
193C2×C4×C67—41no—
194C2×C4×C67—41no—
195C2×C4×C67—41no—
196C2×C4×C67—41no—
197C2×C4×C67—42no—
198C2×C4×C67—81no—
199C2×C4×C67—82no—
200C2×C4×C67—82no—
201C2×C4×C67#12111yescharacteristic
202C2×C4×C67#12111yescharacteristic
203C2×C4×C67#12111yescharacteristic
204C2×C4×C67C2×SD1612yes—
205C2×C4×C67#13512yes—
206C2×C4×C67C22×D411yescharacteristic
207C2×C4×C67C22×D411yescharacteristic
208C2×C4×C67C2×#4111yescharacteristic
209C8×C67—22no—
210C8×C67—22no—
211C8×C67—41no—
212C8×C67—41no—
213C8×C67—42no—
214C8×C67—42no—
215C22×C67—21no—
216C22×C67—21no—
217C22×C67—22no—
218C22×C67—41no—
219C22×C67—41no—
220C22×C67—81no—
221C22×C67—81no—
222C22×C67—81no—
223C22×C67#46512yes—
224C22×C67C2×#12111yescharacteristic
225C4×C67—21no—
226C4×C67—22no—
227C4×C67—21no—
228C4×C67—21no—
229C4×C67—41no—
230C4×C67—41no—
231C4×C67—81no—
232C4×C67—82no—
233C4×C67#46512yes—
234C4×C67C22×SD1611yescharacteristic
235C4×C67C2×#13511yescharacteristic
236C4×C67#57711yescharacteristic
237C4×C67#57811yescharacteristic
238#19414C6711yesmaximal, characteristic, direct product factor, Sylow
239C2×C67—22no—
240C2×C67—42no—
241C2×C67—161no—
242C2×C67#304611yescharacteristic
243C2×C67C2×#46511yescharacteristic
244C2×C67#306811yescharacteristic
245#3227C2×C6711yescharacteristic, complemented
246#3079C2×C6711yescharacteristic, complemented
247#3035C2×C6711yescharacteristic
248#2988C2×C6711yescharacteristic, complemented
249#1678C2×C6712yescomplemented
250#1677C2×C6712yescomplemented
251#1672C2×C6712yescomplemented
252#1671C2×C6712yescomplemented
253#1570C2×C6711yescharacteristic, complemented
254#1547C2×C6711yescharacteristic, complemented
255C2×#333C2×C6711yescharacteristic, complemented
256C67#1941411yescharacteristic, direct product factor, Sylow
257C2×#135—21no—
258#532C22×C6711yescharacteristic
259C2×C4⋊Q8C22×C6711yescharacteristic
260C4×#41C22×C6711yescharacteristic
261#498C22×C6712yes—
262#486C22×C6712yes—
263#481—22no—
264#477C22×C6712yes—
265#476—21no—
266#476—21no—
267#476C22×C6712yes—
268#475—21no—
269#474—21no—
270#464—21no—
271#463—21no—
272#462C22×C6711yescharacteristic
273#461C22×C6711yescharacteristic
274#460C22×C6711yescharacteristic
275#459C22×C6711yescharacteristic
276C8×Q8C22×C6712yes—
277C8×D4—22no—
278C8×D4C22×C6712yes—
279#431—21no—
280#431C22×C6711yescharacteristic
281#428—21no—
282#423C22×C6711yescharacteristic
283C2×C4⋊C8—21no—
284C2×C4⋊C8C22×C6711yescharacteristic
285C2×C4⋊C8C22×C6711yescharacteristic
286#417—21no—
287#416—21no—
288#334C22×C6712yes—
289#334C22×C6712yes—
290#333C22×C6712yes—
291#333C22×C6712yes—
292#332C22×C6712yes—
293#331C22×C6712yes—
294C2×#41—21no—
295C2×#41D4×C6711yescharacteristic
296C22×Q8—21no—
297#135—44no—
298C2×Q16—21no—
299C2×SD16—21no—
300C2×SD16—21no—
301C2×D8—21no—
302C2×M16—21no—
303C22×C8—21no—
304C22×C8—21no—
305C22×C8—22no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8C23×C6712yes—
310C4⋊Q8C23×C6711yescharacteristic
311#127C23×C6711yescharacteristic
312#124—21no—
313#121—21no—
314#121—21no—
315C4×Q8D4×C6711yescharacteristic
316C4×Q8C23×C6711yescharacteristic
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4D4×C6711yescharacteristic
322C4×D4C23×C6711yescharacteristic
323#117—21no—
324#117D4×C6711yescharacteristic
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C42—21no—
328C2×C42C23×C6711yescharacteristic
329#107—22no—
330#106—22no—
331#106—22no—
332#106—22no—
333C4⋊C8—22no—
334C4⋊C8—21no—
335C4⋊C8—21no—
336C4⋊C8—21no—
337C4⋊C8D4×C6711yescharacteristic
338C4⋊C8D4×C6711yescharacteristic
339C4⋊C8C23×C6712yes—
340C4⋊C8C23×C6712yes—
341C4⋊C8C23×C6711yescharacteristic
342C4⋊C8C23×C6711yescharacteristic
343#103—22no—
344#103—21no—
345#103—21no—
346#102—22no—
347#102—22no—
348#102—22no—
349#102—21no—
350#102—21no—
351C22⋊C8—22no—
352C22⋊C8D4×C6712yes—
353C4×C8—22no—
354C4×C8C23×C6712yes—
355#41—22no—
356#41—42no—
357#41—42no—
358#41SD16×C6714yes—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—41no—
362C2×Q8—82no—
363C2×Q8C2×D4×C6711yescharacteristic
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4C2×D4×C6711yescharacteristic
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4C2×D4×C6711yescharacteristic
376C22×C4C2×D4×C6711yescharacteristic
377C22×C4#41×C6711yescharacteristic
378Q16—42no—
379SD16—42no—
380SD16—42no—
381D8—42no—
382M16—42no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—42no—
388C2×C8—82no—
389C2×C8—81no—
390C2×C8C2×D4×C6712yes—
391C2×C8C2×D4×C6712yes—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4C2×D4×C6711yescharacteristic
401C4⋊C4C2×D4×C6711yescharacteristic
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—21no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42C2×D4×C6711yescharacteristic
411C42C2×D4×C6711yescharacteristic
412C42#41×C6711yescharacteristic
413C42C24×C6711yescharacteristic
414C23—21no—
415C23—41no—
416C23#121×C6711yescharacteristic
417Q8—21no—
418Q8—41no—
419Q8—41no—
420Q8—82no—
421Q8—82no—
422Q8C2×SD16×C6712yes—
423D4—24no—
424D4—21no—
425D4—41no—
426D4—41no—
427D4—41no—
428D4—41no—
429D4—81no—
430D4—82no—
431D4C2×SD16×C6712yes—
432C2×C4—21no—
433C2×C4—24no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4#121×C6711yescharacteristic
447C2×C4#121×C6711yescharacteristic
448C2×C4#121×C6711yescharacteristic
449C2×C4C2×SD16×C6712yes—
450C2×C4#135×C6712yes—
451C2×C4C22×D4×C6711yescharacteristic
452C2×C4C22×D4×C6711yescharacteristic
453C2×C4C2×#41×C6711yescharacteristic
454C8—22no—
455C8—22no—
456C8—41no—
457C8—41no—
458C8—42no—
459C8—42no—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22#465×C6712yes—
469C22C2×#121×C6711yescharacteristic
470C4—21no—
471C4—22no—
472C4—21no—
473C4—21no—
474C4—41no—
475C4—41no—
476C4—81no—
477C4—82no—
478C4#465×C6712yes—
479C4C22×SD16×C6711yescharacteristic
480C4C2×#135×C6711yescharacteristic
481C4#577×C6711yescharacteristic
482C4#578×C6711yescharacteristic
483C2—22no—
484C2—42no—
485C2—161no—
486C2C67×#304611yescharacteristic
487C2C2×#465×C6711yescharacteristic
488C2C67×#306811yescharacteristic
subgroupG/H
center, socleC22×C67C2×#121
commutator, Frattini, cosocleC42C24×C67
prime radicalC67×#3035C2
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×#3227C2semidirect
2. C67×#3079C2semidirect
4. C67×#2988C2semidirect
5. C67×#1678C2semidirect
6. C67×#1677C2semidirect
7. C67×#1672C2semidirect
8. C67×#1671C2semidirect
9. C67×#1570C2semidirect
10. C67×#1547C2semidirect
11. C2×#333×C67C2semidirect
238. #19414C67direct (N × H)
245. #3227C2×C67semidirect
246. #3079C2×C67semidirect
248. #2988C2×C67semidirect
249. #1678C2×C67semidirect
250. #1677C2×C67semidirect
251. #1672C2×C67semidirect
252. #1671C2×C67semidirect
253. #1570C2×C67semidirect
254. #1547C2×C67semidirect
255. C2×#333C2×C67semidirect
256. C67#19414direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#32272
C67×#30792
C67×#30352
C67×#29882
C67×#16782
C67×#16772
C67×#16722
C67×#16712
C67×#15702
C67×#15472
C2×#333×C672
#1941467