Finite Group Databasebeta

← C67×#33063 #30712314

Subgroups — C67×#33063 #30712314

Order 17152.

772 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#489×C67C211yesmaximal, characteristic, complemented
2C67×#3079C211yesmaximal, characteristic, complemented
3C67×#3040C211yesmaximal, characteristic
4C67×#2875C211yesmaximal, characteristic, complemented
5C67×#1997C211yesmaximal, characteristic, complemented
6C67×#1967C211yesmaximal, characteristic, complemented
7C67×#1931C212yesmaximal, complemented
8C67×#1930C212yesmaximal, complemented
9C67×#1908C212yesmaximal, complemented
10C67×#1907C212yesmaximal
11C67×#1838C211yesmaximal, characteristic, complemented
12C2×#137×C67C2211yescharacteristic
13C2×AGL(1,ℤ/8)×C67C2211yescharacteristic
14C2×#131×C67C2212yes—
15C2×C4⋊Q8×C67C2211yescharacteristic
16C2×#124×C67C2211yescharacteristic
17C4×#41×C67C2211yescharacteristic
18#489×C67—21no—
19#489×C67—21no—
20#489×C67—22no—
21#477×C67—21no—
22#476×C67—21no—
23#475×C67—21no—
24#474×C67—21no—
25#464×C67—21no—
26#464×C67—21no—
27#463×C67—21no—
28#463×C67—21no—
29#428×C67C2212yes—
30#423×C67C2211yescharacteristic
31#423×C67C2211yescharacteristic
32#419×C67C2211yescharacteristic
33#418×C67C2211yescharacteristic
34#417×C67C2212yes—
35C2×#103×C67C2212yes—
36C2×#102×C67C2212yes—
37C2×#101×C67C2212yes—
38C2×#100×C67C2212yes—
39#405×C67—22no—
40#405×C67C2212yes—
41C4×M16×C67—21no—
42C4×M16×C67C2211yescharacteristic
43C2×#97×C67—21no—
44C2×#97×C67C2211yescharacteristic
45#343×C67C2212yes—
46#343×C67C2212yes—
47#341×C67C2212yes—
48#341×C67C2212yes—
49C2×#41×C67C2311yescharacteristic
50C22×Q8×C67—21no—
51C22×Q8×C67C2311yescharacteristic
52C22×D4×C67C2311yescharacteristic
53#137×C67—24no—
54AGL(1,ℤ/8)×C67—24no—
55C2×Q16×C67—21no—
56C2×SD16×C67—21no—
57C2×SD16×C67—21no—
58C2×D8×C67—21no—
59#131×C67—24no—
60#131×C67—24no—
61C2×M16×C67—22no—
62C2×M16×C67C2311yescharacteristic
63C2×M16×C67C2311yescharacteristic
64C2×M16×C67C2312yes—
65C2×M16×C67C2312yes—
66C22×C8×C67—22no—
67C22×C8×C67C2312yes—
68C4⋊Q8×C67—21no—
69C4⋊Q8×C67—21no—
70C4⋊Q8×C67—21no—
71C4⋊Q8×C67—21no—
72#124×C67—21no—
73#124×C67—21no—
74#124×C67—21no—
75#124×C67—21no—
76C4×Q8×C67D411yescharacteristic
77C4×Q8×C67D411yescharacteristic
78C4×D4×C67—21no—
79C4×D4×C67—21no—
80C4×D4×C67D411yescharacteristic
81C4×D4×C67D411yescharacteristic
82#117×C67—21no—
83#117×C67C2311yescharacteristic
84C2×C4⋊C4×C67—21no—
85C2×C4⋊C4×C67C2312yes—
86C2×C22⋊C4×C67—21no—
87C2×C22⋊C4×C67—21no—
88C2×C42×C67—21no—
89C2×C42×C67C2311yescharacteristic
90#107×C67—22no—
91#106×C67—22no—
92C4⋊C8×C67—21no—
93C4⋊C8×C67—21no—
94C4⋊C8×C67—21no—
95C4⋊C8×C67—21no—
96#103×C67—22no—
97#103×C67—22no—
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—
106#101×C67—24no—
107#100×C67—24no—
108#97×C67—22no—
109#97×C67—21no—
110#97×C67D411yescharacteristic
111#97×C67D411yescharacteristic
112#97×C67D412yes—
113C4×C8×C67—22no—
114C4×C8×C67—21no—
115C24×C67—21no—
116#41×C67—22no—
117#41×C67#4114yes—
118C2×Q8×C67—21no—
119C2×Q8×C67—21no—
120C2×Q8×C67—22no—
121C2×Q8×C67—22no—
122C2×Q8×C67—41no—
123C2×Q8×C67—41no—
124C2×Q8×C67—82no—
125C2×Q8×C67C2×D411yescharacteristic
126C2×D4×C67—21no—
127C2×D4×C67—21no—
128C2×D4×C67—21no—
129C2×D4×C67—22no—
130C2×D4×C67—22no—
131C2×D4×C67C2×D411yescharacteristic
132C22×C4×C67—21no—
133C22×C4×C67—21no—
134C22×C4×C67—21no—
135C22×C4×C67—21no—
136C22×C4×C67—22no—
137C22×C4×C67#4111yescharacteristic
138C22×C4×C67#4111yescharacteristic
139C22×C4×C67C2411yescharacteristic
140Q16×C67—42no—
141SD16×C67—42no—
142SD16×C67—42no—
143D8×C67—42no—
144M16×C67—28no—
145M16×C67—22no—
146M16×C67—22no—
147M16×C67—22no—
148M16×C67—22no—
149M16×C67—22no—
150M16×C67#4114yes—
151C2×C8×C67—28no—
152C2×C8×C67—22no—
153C2×C8×C67—22no—
154C2×C8×C67—22no—
155C2×C8×C67—21no—
156C2×C8×C67—22no—
157C2×C8×C67—21no—
158C2×C8×C67C2×D412yes—
159C2×C8×C67C2×D412yes—
160C4⋊C4×C67—21no—
161C4⋊C4×C67—21no—
162C4⋊C4×C67—22no—
163C4⋊C4×C67—22no—
164C4⋊C4×C67—41no—
165C4⋊C4×C67—41no—
166C4⋊C4×C67C2×D411yescharacteristic
167C4⋊C4×C67C2×D411yescharacteristic
168C22⋊C4×C67—21no—
169C22⋊C4×C67—21no—
170C22⋊C4×C67—21no—
171C22⋊C4×C67—41no—
172C22⋊C4×C67—41no—
173C22⋊C4×C67—41no—
174C22⋊C4×C67—41no—
175C42×C67—21no—
176C42×C67—21no—
177C42×C67—21no—
178C42×C67C2×D411yescharacteristic
179C42×C67C2×D411yescharacteristic
180C42×C67C2×D411yescharacteristic
181C42×C67C2×D411yescharacteristic
182C23×C67—21no—
183C23×C67—41no—
184C23×C67—42no—
185C23×C67—81no—
186C23×C67—81no—
187C23×C67C2×#4111yescharacteristic
188Q8×C67—21no—
189Q8×C67—42no—
190Q8×C67—41no—
191Q8×C67—41no—
192Q8×C67—82no—
193Q8×C67—82no—
194Q8×C67#12412yes—
195D4×C67—24no—
196D4×C67—21no—
197D4×C67—42no—
198D4×C67—41no—
199D4×C67—41no—
200D4×C67#12412yes—
201C2×C4×C67—21no—
202C2×C4×C67—24no—
203C2×C4×C67—21no—
204C2×C4×C67—21no—
205C2×C4×C67—41no—
206C2×C4×C67—41no—
207C2×C4×C67—41no—
208C2×C4×C67—42no—
209C2×C4×C67—42no—
210C2×C4×C67—81no—
211C2×C4×C67—81no—
212C2×C4×C67—82no—
213C2×C4×C67#12111yescharacteristic
214C2×C4×C67#12111yescharacteristic
215C2×C4×C67#12111yescharacteristic
216C2×C4×C67#12111yescharacteristic
217C2×C4×C67C22×D411yescharacteristic
218C2×C4×C67C22×D411yescharacteristic
219C2×C4×C67C2×#4112yes—
220C2×C4×C67C2×#4112yes—
221C8×C67—22no—
222C8×C67—24no—
223C8×C67—22no—
224C8×C67—41no—
225C8×C67—41no—
226C8×C67—42no—
227C22×C67—21no—
228C22×C67—21no—
229C22×C67—22no—
230C22×C67—41no—
231C22×C67—41no—
232C22×C67—81no—
233C22×C67—81no—
234C22×C67—81no—
235C22×C67—162no—
236C22×C67C2×#12111yescharacteristic
237C22×C67#52912yes—
238C4×C67—22no—
239C4×C67—21no—
240C4×C67—21no—
241C4×C67—21no—
242C4×C67—21no—
243C4×C67—41no—
244C4×C67—41no—
245C4×C67—81no—
246C4×C67—82no—
247C4×C67#52512yes—
248C4×C67C2×#12411yescharacteristic
249C4×C67#53211yescharacteristic
250#33063C6711yesmaximal, characteristic, direct product factor, Sylow
251C2×C67—22no—
252C2×C67—42no—
253C2×C67—161no—
254C2×C67#285011yescharacteristic
255C2×C67#306811yescharacteristic
256C2×C67#306911yescharacteristic
257C2×#489C2×C6711yescharacteristic, complemented
258#3079C2×C6711yescharacteristic, complemented
259#3040C2×C6711yescharacteristic
260#2875C2×C6711yescharacteristic, complemented
261#1997C2×C6711yescharacteristic, complemented
262#1967C2×C6711yescharacteristic, complemented
263#1931C2×C6712yescomplemented
264#1930C2×C6712yescomplemented
265#1908C2×C6712yescomplemented
266#1907C2×C6712yes—
267#1838C2×C6711yescharacteristic, complemented
268C67#3306311yescharacteristic, direct product factor, Sylow
269C2×#137C22×C6711yescharacteristic
270C2×AGL(1,ℤ/8)C22×C6711yescharacteristic
271C2×#131C22×C6712yes—
272C2×C4⋊Q8C22×C6711yescharacteristic
273C2×#124C22×C6711yescharacteristic
274C4×#41C22×C6711yescharacteristic
275#489—21no—
276#489—21no—
277#489—22no—
278#477—21no—
279#476—21no—
280#475—21no—
281#474—21no—
282#464—21no—
283#464—21no—
284#463—21no—
285#463—21no—
286#428C22×C6712yes—
287#423C22×C6711yescharacteristic
288#423C22×C6711yescharacteristic
289#419C22×C6711yescharacteristic
290#418C22×C6711yescharacteristic
291#417C22×C6712yes—
292C2×#103C22×C6712yes—
293C2×#102C22×C6712yes—
294C2×#101C22×C6712yes—
295C2×#100C22×C6712yes—
296#405—22no—
297#405C22×C6712yes—
298C4×M16—21no—
299C4×M16C22×C6711yescharacteristic
300C2×#97—21no—
301C2×#97C22×C6711yescharacteristic
302#343C22×C6712yes—
303#343C22×C6712yes—
304#341C22×C6712yes—
305#341C22×C6712yes—
306C2×#41C23×C6711yescharacteristic
307C22×Q8—21no—
308C22×Q8C23×C6711yescharacteristic
309C22×D4C23×C6711yescharacteristic
310#137—24no—
311AGL(1,ℤ/8)—24no—
312C2×Q16—21no—
313C2×SD16—21no—
314C2×SD16—21no—
315C2×D8—21no—
316#131—24no—
317#131—24no—
318C2×M16—22no—
319C2×M16C23×C6711yescharacteristic
320C2×M16C23×C6711yescharacteristic
321C2×M16C23×C6712yes—
322C2×M16C23×C6712yes—
323C22×C8—22no—
324C22×C8C23×C6712yes—
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8—21no—
329#124—21no—
330#124—21no—
331#124—21no—
332#124—21no—
333C4×Q8D4×C6711yescharacteristic
334C4×Q8D4×C6711yescharacteristic
335C4×D4—21no—
336C4×D4—21no—
337C4×D4D4×C6711yescharacteristic
338C4×D4D4×C6711yescharacteristic
339#117—21no—
340#117C23×C6711yescharacteristic
341C2×C4⋊C4—21no—
342C2×C4⋊C4C23×C6712yes—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C42—21no—
346C2×C42C23×C6711yescharacteristic
347#107—22no—
348#106—22no—
349C4⋊C8—21no—
350C4⋊C8—21no—
351C4⋊C8—21no—
352C4⋊C8—21no—
353#103—22no—
354#103—22no—
355#103—22no—
356#103—21no—
357#103—21no—
358#102—22no—
359#102—22no—
360#102—22no—
361#102—21no—
362#102—21no—
363#101—24no—
364#100—24no—
365#97—22no—
366#97—21no—
367#97D4×C6711yescharacteristic
368#97D4×C6711yescharacteristic
369#97D4×C6712yes—
370C4×C8—22no—
371C4×C8—21no—
372C24—21no—
373#41—22no—
374#41#41×C6714yes—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—82no—
382C2×Q8C2×D4×C6711yescharacteristic
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4C2×D4×C6711yescharacteristic
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4#41×C6711yescharacteristic
395C22×C4#41×C6711yescharacteristic
396C22×C4C24×C6711yescharacteristic
397Q16—42no—
398SD16—42no—
399SD16—42no—
400D8—42no—
401M16—28no—
402M16—22no—
403M16—22no—
404M16—22no—
405M16—22no—
406M16—22no—
407M16#41×C6714yes—
408C2×C8—28no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—21no—
413C2×C8—22no—
414C2×C8—21no—
415C2×C8C2×D4×C6712yes—
416C2×C8C2×D4×C6712yes—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4C2×D4×C6711yescharacteristic
424C4⋊C4C2×D4×C6711yescharacteristic
425C22⋊C4—21no—
426C22⋊C4—21no—
427C22⋊C4—21no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42C2×D4×C6711yescharacteristic
436C42C2×D4×C6711yescharacteristic
437C42C2×D4×C6711yescharacteristic
438C42C2×D4×C6711yescharacteristic
439C23—21no—
440C23—41no—
441C23—42no—
442C23—81no—
443C23—81no—
444C23C2×#41×C6711yescharacteristic
445Q8—21no—
446Q8—42no—
447Q8—41no—
448Q8—41no—
449Q8—82no—
450Q8—82no—
451Q8#124×C6712yes—
452D4—24no—
453D4—21no—
454D4—42no—
455D4—41no—
456D4—41no—
457D4#124×C6712yes—
458C2×C4—21no—
459C2×C4—24no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4#121×C6711yescharacteristic
471C2×C4#121×C6711yescharacteristic
472C2×C4#121×C6711yescharacteristic
473C2×C4#121×C6711yescharacteristic
474C2×C4C22×D4×C6711yescharacteristic
475C2×C4C22×D4×C6711yescharacteristic
476C2×C4C2×#41×C6712yes—
477C2×C4C2×#41×C6712yes—
478C8—22no—
479C8—24no—
480C8—22no—
481C8—41no—
482C8—41no—
483C8—42no—
484C22—21no—
485C22—21no—
486C22—22no—
487C22—41no—
488C22—41no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—162no—
493C22C2×#121×C6711yescharacteristic
494C22#529×C6712yes—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—22no—
500C4—41no—
501C4—41no—
502C4—82no—
503C4—81no—
504C4#525×C6712yes—
505C4C2×#124×C6711yescharacteristic
506C4#532×C6711yescharacteristic
507C2—22no—
508C2—42no—
509C2—161no—
510C2C67×#285011yescharacteristic
511C2C67×#306811yescharacteristic
512C2C67×#306911yescharacteristic
subgroupG/H
center, socleC22×C67C2×#121
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC2×AGL(1,ℤ/8)×C67C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#489×C67C2semidirect
2. C67×#3079C2semidirect
4. C67×#2875C2semidirect
5. C67×#1997C2semidirect
6. C67×#1967C2semidirect
7. C67×#1931C2semidirect
8. C67×#1930C2semidirect
9. C67×#1908C2semidirect
11. C67×#1838C2semidirect
250. #33063C67direct (N × H)
257. C2×#489C2×C67semidirect
258. #3079C2×C67semidirect
260. #2875C2×C67semidirect
261. #1997C2×C67semidirect
262. #1967C2×C67semidirect
263. #1931C2×C67semidirect
264. #1930C2×C67semidirect
265. #1908C2×C67semidirect
267. #1838C2×C67semidirect
268. C67#33063direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#489×C672
C67×#30792
C67×#30402
C67×#28752
C67×#19972
C67×#19672
C67×#19312
C67×#19302
C67×#19082
C67×#19072
C67×#18382
#3306367