Finite Group Databasebeta

← C67×#33127 #30712378

Subgroups — C67×#33127 #30712378

Order 17152.

768 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3095C211yesmaximal, characteristic, complemented
2C67×#3094C211yesmaximal, characteristic
3C2×#468×C67C211yesmaximal, characteristic
4C67×#2873C211yesmaximal, characteristic, complemented
5C67×#1977C212yesmaximal, complemented
6C67×#1952C212yesmaximal, complemented
7C67×#1931C212yesmaximal, complemented
8C67×#1930C212yesmaximal, complemented
9C67×#1901C211yesmaximal, characteristic, complemented
10C67×#1900C211yesmaximal, characteristic
11C67×#1811C211yesmaximal, characteristic, complemented
12C22×SD16×C67C2211yescharacteristic
13C22×M16×C67C2211yescharacteristic
14C22×M16×C67C2211yescharacteristic
15C2×C22⋊Q8×C67C2211yescharacteristic
16C2×#121×C67C2211yescharacteristic
17C2×#117×C67C2211yescharacteristic
18#485×C67—22no—
19#484×C67—21no—
20#483×C67—21no—
21#482×C67—22no—
22#481×C67—21no—
23#480×C67—21no—
24#468×C67—22no—
25#468×C67—22no—
26C2×#108×C67C2212yes—
27#428×C67C2212yes—
28#428×C67C2212yes—
29C2×#107×C67C2211yescharacteristic
30#419×C67C2212yes—
31#418×C67C2212yes—
32C2×#103×C67C2211yescharacteristic
33C2×#102×C67C2211yescharacteristic
34C2×#101×C67C2212yes—
35C2×#100×C67C2212yes—
36#407×C67C2211yescharacteristic
37#407×C67C2211yescharacteristic
38#405×C67—22no—
39#405×C67C2212yes—
40#405×C67C2212yes—
41C4×M16×C67—22no—
42#343×C67C2212yes—
43#343×C67C2212yes—
44#341×C67C2212yes—
45C22×Q8×C67C2311yescharacteristic
46C22×D4×C67—21no—
47C22×D4×C67C2311yescharacteristic
48C23×C4×C67C2311yescharacteristic
49C2×SD16×C67—22no—
50C2×SD16×C67—24no—
51C2×M16×C67—21no—
52C2×M16×C67—24no—
53C2×M16×C67—24no—
54C2×M16×C67D412yes—
55C2×M16×C67D412yes—
56C2×M16×C67C2312yes—
57C2×M16×C67C2312yes—
58C2×M16×C67C2312yes—
59C22×C8×C67—21no—
60C22×C8×C67C2311yescharacteristic
61C22×C8×C67C2311yescharacteristic
62C22⋊Q8×C67—21no—
63C22⋊Q8×C67—21no—
64C22⋊Q8×C67—21no—
65C22⋊Q8×C67—21no—
66#121×C67—21no—
67#121×C67—21no—
68#121×C67—21no—
69#121×C67—21no—
70#117×C67—21no—
71#117×C67—21no—
72#117×C67D412yes—
73#117×C67C2312yes—
74C2×C4⋊C4×C67—21no—
75C2×C4⋊C4×C67D411yescharacteristic
76C2×C4⋊C4×C67D411yescharacteristic
77C2×C4⋊C4×C67C2311yescharacteristic
78C2×C4⋊C4×C67C2311yescharacteristic
79C2×C22⋊C4×C67—21no—
80C2×C22⋊C4×C67—21no—
81C2×C22⋊C4×C67—21no—
82C2×C42×C67—21no—
83#108×C67—24no—
84#107×C67—22no—
85#107×C67—22no—
86#107×C67—22no—
87#106×C67—22no—
88#106×C67—22no—
89#103×C67—22no—
90#103×C67—22no—
91#103×C67—22no—
92#102×C67—22no—
93#102×C67—22no—
94#102×C67—22no—
95#101×C67—24no—
96#100×C67—24no—
97C22⋊C8×C67—21no—
98C22⋊C8×C67—21no—
99C22⋊C8×C67—21no—
100C22⋊C8×C67—21no—
101#97×C67—22no—
102#97×C67—22no—
103C4×C8×C67—22no—
104C4×C8×C67—22no—
105C24×C67—21no—
106C24×C67#4111yescharacteristic
107C2×Q8×C67—21no—
108C2×Q8×C67—21no—
109C2×Q8×C67—22no—
110C2×Q8×C67—22no—
111C2×D4×C67—21no—
112C2×D4×C67—21no—
113C2×D4×C67—22no—
114C2×D4×C67—22no—
115C2×D4×C67—41no—
116C2×D4×C67—41no—
117C2×D4×C67—82no—
118C22×C4×C67—22no—
119C22×C4×C67—22no—
120C22×C4×C67—21no—
121C22×C4×C67—21no—
122C22×C4×C67—21no—
123C22×C4×C67—21no—
124C22×C4×C67—21no—
125C22×C4×C67C2×D411yescharacteristic
126C22×C4×C67C2×D411yescharacteristic
127C22×C4×C67C2×D411yescharacteristic
128C22×C4×C67C2×D411yescharacteristic
129C22×C4×C67#4111yescharacteristic
130C22×C4×C67C2411yescharacteristic
131SD16×C67—44no—
132M16×C67—24no—
133M16×C67—22no—
134M16×C67—24no—
135M16×C67—24no—
136M16×C67—22no—
137M16×C67—22no—
138M16×C67—22no—
139M16×C67—22no—
140M16×C67#4114yes—
141C2×C8×C67—24no—
142C2×C8×C67—22no—
143C2×C8×C67—22no—
144C2×C8×C67—22no—
145C2×C8×C67—22no—
146C2×C8×C67C2×D412yes—
147C2×C8×C67C2×D412yes—
148C2×C8×C67#4114yes—
149C4⋊C4×C67—21no—
150C4⋊C4×C67—21no—
151C4⋊C4×C67—21no—
152C4⋊C4×C67—21no—
153C4⋊C4×C67—21no—
154C4⋊C4×C67—21no—
155C4⋊C4×C67—41no—
156C4⋊C4×C67—41no—
157C4⋊C4×C67C2×D412yes—
158C4⋊C4×C67C2×D412yes—
159C22⋊C4×C67—22no—
160C22⋊C4×C67—22no—
161C22⋊C4×C67—41no—
162C22⋊C4×C67—41no—
163C22⋊C4×C67—41no—
164C22⋊C4×C67—41no—
165C42×C67—22no—
166C42×C67—22no—
167C23×C67—22no—
168C23×C67—41no—
169C23×C67—41no—
170C23×C67—41no—
171C23×C67—42no—
172C23×C67—81no—
173C23×C67—81no—
174C23×C67#12111yescharacteristic
175C23×C67#12111yescharacteristic
176C23×C67C2×#4111yescharacteristic
177Q8×C67—41no—
178Q8×C67—41no—
179Q8×C67—42no—
180D4×C67—41no—
181D4×C67—41no—
182D4×C67—42no—
183D4×C67—82no—
184D4×C67—82no—
185C2×C4×C67—22no—
186C2×C4×C67—21no—
187C2×C4×C67—21no—
188C2×C4×C67—21no—
189C2×C4×C67—21no—
190C2×C4×C67—22no—
191C2×C4×C67—22no—
192C2×C4×C67—22no—
193C2×C4×C67—22no—
194C2×C4×C67—41no—
195C2×C4×C67—41no—
196C2×C4×C67—41no—
197C2×C4×C67—42no—
198C2×C4×C67—81no—
199C2×C4×C67—81no—
200C2×C4×C67—81no—
201C2×C4×C67—81no—
202C2×C4×C67#12111yescharacteristic
203C2×C4×C67#12111yescharacteristic
204C2×C4×C67C22×D411yescharacteristic
205C2×C4×C67C22×D411yescharacteristic
206C2×C4×C67C2×#4112yes—
207C2×C4×C67C2×#4112yes—
208C8×C67—22no—
209C8×C67—24no—
210C8×C67—42no—
211C8×C67—42no—
212C8×C67#12314yes—
213C22×C67—21no—
214C22×C67—21no—
215C22×C67—21no—
216C22×C67—21no—
217C22×C67—21no—
218C22×C67—21no—
219C22×C67—21no—
220C22×C67—21no—
221C22×C67—82no—
222C22×C67—81no—
223C22×C67—81no—
224C22×C67—81no—
225C22×C67—162no—
226C22×C67C2×#12111yescharacteristic
227C22×C67#52912yes—
228C4×C67—21no—
229C4×C67—21no—
230C4×C67—42no—
231C4×C67—42no—
232C4×C67—81no—
233C4×C67—81no—
234C4×C67—81no—
235C4×C67C2×#12311yescharacteristic
236C4×C67#52511yescharacteristic
237C4×C67#53212yes—
238#33127C6711yesmaximal, characteristic, direct product factor, Sylow
239C2×C67—22no—
240C2×C67—41no—
241C2×C67—41no—
242C2×C67—161no—
243C2×C67#285411yescharacteristic
244C2×C67#306811yescharacteristic
245C2×C67#306911yescharacteristic
246#3095C2×C6711yescharacteristic, complemented
247#3094C2×C6711yescharacteristic
248C2×#468C2×C6711yescharacteristic
249#2873C2×C6711yescharacteristic, complemented
250#1977C2×C6712yescomplemented
251#1952C2×C6712yescomplemented
252#1931C2×C6712yescomplemented
253#1930C2×C6712yescomplemented
254#1901C2×C6711yescharacteristic, complemented
255#1900C2×C6711yescharacteristic
256#1811C2×C6711yescharacteristic, complemented
257C67#3312711yescharacteristic, direct product factor, Sylow
258C22×SD16C22×C6711yescharacteristic
259C22×M16C22×C6711yescharacteristic
260C22×M16C22×C6711yescharacteristic
261C2×C22⋊Q8C22×C6711yescharacteristic
262C2×#121C22×C6711yescharacteristic
263C2×#117C22×C6711yescharacteristic
264#485—22no—
265#484—21no—
266#483—21no—
267#482—22no—
268#481—21no—
269#480—21no—
270#468—22no—
271#468—22no—
272C2×#108C22×C6712yes—
273#428C22×C6712yes—
274#428C22×C6712yes—
275C2×#107C22×C6711yescharacteristic
276#419C22×C6712yes—
277#418C22×C6712yes—
278C2×#103C22×C6711yescharacteristic
279C2×#102C22×C6711yescharacteristic
280C2×#101C22×C6712yes—
281C2×#100C22×C6712yes—
282#407C22×C6711yescharacteristic
283#407C22×C6711yescharacteristic
284#405—22no—
285#405C22×C6712yes—
286#405C22×C6712yes—
287C4×M16—22no—
288#343C22×C6712yes—
289#343C22×C6712yes—
290#341C22×C6712yes—
291C22×Q8C23×C6711yescharacteristic
292C22×D4—21no—
293C22×D4C23×C6711yescharacteristic
294C23×C4C23×C6711yescharacteristic
295C2×SD16—22no—
296C2×SD16—24no—
297C2×M16—21no—
298C2×M16—24no—
299C2×M16—24no—
300C2×M16D4×C6712yes—
301C2×M16D4×C6712yes—
302C2×M16C23×C6712yes—
303C2×M16C23×C6712yes—
304C2×M16C23×C6712yes—
305C22×C8—21no—
306C22×C8C23×C6711yescharacteristic
307C22×C8C23×C6711yescharacteristic
308C22⋊Q8—21no—
309C22⋊Q8—21no—
310C22⋊Q8—21no—
311C22⋊Q8—21no—
312#121—21no—
313#121—21no—
314#121—21no—
315#121—21no—
316#117—21no—
317#117—21no—
318#117D4×C6712yes—
319#117C23×C6712yes—
320C2×C4⋊C4—21no—
321C2×C4⋊C4D4×C6711yescharacteristic
322C2×C4⋊C4D4×C6711yescharacteristic
323C2×C4⋊C4C23×C6711yescharacteristic
324C2×C4⋊C4C23×C6711yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C42—21no—
329#108—24no—
330#107—22no—
331#107—22no—
332#107—22no—
333#106—22no—
334#106—22no—
335#103—22no—
336#103—22no—
337#103—22no—
338#102—22no—
339#102—22no—
340#102—22no—
341#101—24no—
342#100—24no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C22⋊C8—21no—
346C22⋊C8—21no—
347#97—22no—
348#97—22no—
349C4×C8—22no—
350C4×C8—22no—
351C24—21no—
352C24#41×C6711yescharacteristic
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—82no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4C2×D4×C6711yescharacteristic
372C22×C4C2×D4×C6711yescharacteristic
373C22×C4C2×D4×C6711yescharacteristic
374C22×C4C2×D4×C6711yescharacteristic
375C22×C4#41×C6711yescharacteristic
376C22×C4C24×C6711yescharacteristic
377SD16—44no—
378M16—24no—
379M16—22no—
380M16—24no—
381M16—24no—
382M16—22no—
383M16—22no—
384M16—22no—
385M16—22no—
386M16#41×C6714yes—
387C2×C8—24no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8C2×D4×C6712yes—
393C2×C8C2×D4×C6712yes—
394C2×C8#41×C6714yes—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4C2×D4×C6712yes—
404C4⋊C4C2×D4×C6712yes—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C42—22no—
412C42—22no—
413C23—22no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—81no—
419C23—81no—
420C23#121×C6711yescharacteristic
421C23#121×C6711yescharacteristic
422C23C2×#41×C6711yescharacteristic
423Q8—41no—
424Q8—41no—
425Q8—42no—
426D4—41no—
427D4—41no—
428D4—42no—
429D4—82no—
430D4—82no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4#121×C6711yescharacteristic
449C2×C4#121×C6711yescharacteristic
450C2×C4C22×D4×C6711yescharacteristic
451C2×C4C22×D4×C6711yescharacteristic
452C2×C4C2×#41×C6712yes—
453C2×C4C2×#41×C6712yes—
454C8—22no—
455C8—24no—
456C8—42no—
457C8—42no—
458C8#123×C6714yes—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—82no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—162no—
472C22C2×#121×C6711yescharacteristic
473C22#529×C6712yes—
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4C2×#123×C6711yescharacteristic
482C4#525×C6711yescharacteristic
483C4#532×C6712yes—
484C2—22no—
485C2—41no—
486C2—41no—
487C2—161no—
488C2C67×#285411yescharacteristic
489C2C67×#306811yescharacteristic
490C2C67×#306911yescharacteristic
subgroupG/H
center, socleC22×C67C2×#121
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC2×#121×C67C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3095C2semidirect
4. C67×#2873C2semidirect
5. C67×#1977C2semidirect
6. C67×#1952C2semidirect
7. C67×#1931C2semidirect
8. C67×#1930C2semidirect
9. C67×#1901C2semidirect
11. C67×#1811C2semidirect
238. #33127C67direct (N × H)
246. #3095C2×C67semidirect
249. #2873C2×C67semidirect
250. #1977C2×C67semidirect
251. #1952C2×C67semidirect
252. #1931C2×C67semidirect
253. #1930C2×C67semidirect
254. #1901C2×C67semidirect
256. #1811C2×C67semidirect
257. C67#33127direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#30952
C67×#30942
C2×#468×C672
C67×#28732
C67×#19772
C67×#19522
C67×#19312
C67×#19302
C67×#19012
C67×#19002
C67×#18112
#3312767