Finite Group Databasebeta

← C67×#10405 #30689656

Subgroups — C67×#10405 #30689656

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2639C211yesmaximal, characteristic
2C67×#1874C211yesmaximal, characteristic
3C67×#1872C211yesmaximal, characteristic, complemented
4C2×#328×C67C211yesmaximal, characteristic, complemented
5C2×#327×C67C211yesmaximal, characteristic
6C67×#1444C211yesmaximal, characteristic, complemented
7C67×#1280C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C67C2211yescharacteristic
9C2×#121×C67C2211yescharacteristic
10C22×C4⋊C4×C67—21no—
11C22×C4⋊C4×C67C2211yescharacteristic
12C2×#103×C67—21no—
13C2×#102×C67—21no—
14C2×C22⋊C8×C67C2211yescharacteristic
15C2×C22⋊C8×C67C2211yescharacteristic
16#400×C67—21no—
17#397×C67—21no—
18#394×C67—21no—
19#393×C67—21no—
20#386×C67—21no—
21#386×C67—21no—
22C2×#95×C67—21no—
23C2×#95×C67C2211yescharacteristic
24C2×#95×C67C2211yescharacteristic
25#340×C67—21no—
26#340×C67—21no—
27#328×C67—22no—
28#327×C67—22no—
29C22×Q8×C67D411yescharacteristic
30C22×D4×C67—21no—
31C22×D4×C67D411yescharacteristic
32C23×C4×C67—21no—
33C23×C4×C67—21no—
34C23×C4×C67D411yescharacteristic
35C23×C4×C67D411yescharacteristic
36C23×C4×C67C2311yescharacteristic
37C22×C8×C67—21no—
38C22×C8×C67—21no—
39C22⋊Q8×C67—21no—
40C22⋊Q8×C67—21no—
41C22⋊Q8×C67—21no—
42C22⋊Q8×C67—21no—
43#121×C67—21no—
44#121×C67—21no—
45#121×C67—21no—
46#121×C67—21no—
47C2×C4⋊C4×C67—21no—
48C2×C4⋊C4×C67—21no—
49C2×C4⋊C4×C67—21no—
50C2×C4⋊C4×C67—42no—
51C2×C4⋊C4×C67—42no—
52C2×C4⋊C4×C67—42no—
53C2×C4⋊C4×C67—42no—
54C2×C4⋊C4×C67D411yescharacteristic
55C2×C4⋊C4×C67D411yescharacteristic
56C2×C22⋊C4×C67—21no—
57C2×C22⋊C4×C67—21no—
58C2×C22⋊C4×C67—21no—
59C2×C22⋊C4×C67—21no—
60C2×C22⋊C4×C67—21no—
61C2×C22⋊C4×C67—21no—
62#103×C67—42no—
63#102×C67—42no—
64C22⋊C8×C67—21no—
65C22⋊C8×C67—21no—
66C22⋊C8×C67—21no—
67C22⋊C8×C67—21no—
68#95×C67—21no—
69#95×C67—21no—
70#95×C67—21no—
71#95×C67—21no—
72#95×C67—21no—
73#95×C67—21no—
74#95×C67—21no—
75#95×C67—42no—
76#95×C67D412yes—
77C24×C67—21no—
78C24×C67C2×D411yescharacteristic
79C2×Q8×C67—21no—
80C2×Q8×C67—21no—
81C2×Q8×C67—41no—
82C2×Q8×C67—41no—
83C2×D4×C67—21no—
84C2×D4×C67—21no—
85C2×D4×C67—41no—
86C2×D4×C67—41no—
87C2×D4×C67—41no—
88C2×D4×C67—41no—
89C2×D4×C67—41no—
90C2×D4×C67—41no—
91C2×D4×C67—41no—
92C2×D4×C67—41no—
93C22×C4×C67—22no—
94C22×C4×C67—22no—
95C22×C4×C67—22no—
96C22×C4×C67—22no—
97C22×C4×C67—21no—
98C22×C4×C67—21no—
99C22×C4×C67—21no—
100C22×C4×C67—21no—
101C22×C4×C67—21no—
102C22×C4×C67—21no—
103C22×C4×C67—21no—
104C22×C4×C67—21no—
105C22×C4×C67—21no—
106C22×C4×C67—41no—
107C22×C4×C67—41no—
108C22×C4×C67—41no—
109C22×C4×C67—41no—
110C22×C4×C67—41no—
111C22×C4×C67—41no—
112C22×C4×C67—41no—
113C22×C4×C67—41no—
114C22×C4×C67—41no—
115C22×C4×C67—41no—
116C22×C4×C67—41no—
117C22×C4×C67—41no—
118C22×C4×C67—41no—
119C22×C4×C67—41no—
120C22×C4×C67—41no—
121C22×C4×C67—41no—
122C22×C4×C67C2×D411yescharacteristic
123C22×C4×C67C2×D411yescharacteristic
124C22×C4×C67C2×D411yescharacteristic
125C22×C4×C67#4111yescharacteristic
126C22×C4×C67#4111yescharacteristic
127C22×C4×C67#4111yescharacteristic
128C2×C8×C67—41no—
129C2×C8×C67—41no—
130C2×C8×C67—41no—
131C2×C8×C67—41no—
132C2×C8×C67—41no—
133C2×C8×C67—41no—
134C4⋊C4×C67—22no—
135C4⋊C4×C67—22no—
136C4⋊C4×C67—21no—
137C4⋊C4×C67—21no—
138C4⋊C4×C67—21no—
139C4⋊C4×C67—21no—
140C4⋊C4×C67—41no—
141C4⋊C4×C67—41no—
142C4⋊C4×C67—48no—
143C4⋊C4×C67—41no—
144C4⋊C4×C67—41no—
145C22⋊C4×C67—41no—
146C22⋊C4×C67—41no—
147C22⋊C4×C67—41no—
148C22⋊C4×C67—41no—
149C22⋊C4×C67—81no—
150C22⋊C4×C67—81no—
151C22⋊C4×C67—81no—
152C22⋊C4×C67—81no—
153C23×C67—21no—
154C23×C67—21no—
155C23×C67—21no—
156C23×C67—21no—
157C23×C67—41no—
158C23×C67—41no—
159C23×C67—41no—
160C23×C67—41no—
161C23×C67—41no—
162C23×C67—41no—
163C23×C67—41no—
164C23×C67#12011yescharacteristic
165C23×C67#12111yescharacteristic
166C23×C67#12411yescharacteristic
167C23×C67#13511yescharacteristic
168C23×C67#13511yescharacteristic
169C23×C67AGL(1,ℤ/8)11yescharacteristic
170C23×C67#13711yescharacteristic
171Q8×C67—42no—
172Q8×C67—81no—
173D4×C67—42no—
174D4×C67—81no—
175D4×C67—81no—
176D4×C67—81no—
177D4×C67—81no—
178D4×C67—81no—
179C2×C4×C67—21no—
180C2×C4×C67—21no—
181C2×C4×C67—21no—
182C2×C4×C67—21no—
183C2×C4×C67—24no—
184C2×C4×C67—41no—
185C2×C4×C67—41no—
186C2×C4×C67—41no—
187C2×C4×C67—41no—
188C2×C4×C67—42no—
189C2×C4×C67—42no—
190C2×C4×C67—42no—
191C2×C4×C67—42no—
192C2×C4×C67—42no—
193C2×C4×C67—42no—
194C2×C4×C67—42no—
195C2×C4×C67—42no—
196C2×C4×C67—42no—
197C2×C4×C67—42no—
198C2×C4×C67—41no—
199C2×C4×C67—41no—
200C2×C4×C67—41no—
201C2×C4×C67—41no—
202C2×C4×C67—41no—
203C2×C4×C67—41no—
204C2×C4×C67—41no—
205C2×C4×C67—41no—
206C2×C4×C67—41no—
207C2×C4×C67—41no—
208C2×C4×C67—41no—
209C2×C4×C67—41no—
210C2×C4×C67—41no—
211C2×C4×C67—42no—
212C2×C4×C67—41no—
213C2×C4×C67—41no—
214C2×C4×C67—81no—
215C2×C4×C67—81no—
216C2×C4×C67—81no—
217C2×C4×C67—81no—
218C2×C4×C67—81no—
219C2×C4×C67—81no—
220C2×C4×C67#12111yescharacteristic
221C2×C4×C67#12111yescharacteristic
222C2×C4×C67#12311yescharacteristic
223C2×C4×C67#12311yescharacteristic
224C8×C67—81no—
225C8×C67—81no—
226C22×C67—21no—
227C22×C67—21no—
228C22×C67—21no—
229C22×C67—21no—
230C22×C67—21no—
231C22×C67—21no—
232C22×C67—21no—
233C22×C67—21no—
234C22×C67—21no—
235C22×C67—21no—
236C22×C67—21no—
237C22×C67—21no—
238C22×C67—81no—
239C22×C67—81no—
240C22×C67—81no—
241C22×C67—81no—
242C22×C67—81no—
243C22×C67—81no—
244C22×C67—81no—
245C22×C67#39411yescharacteristic
246C22×C67#44911yescharacteristic
247C22×C67#45211yescharacteristic
248C22×C67#45311yescharacteristic
249C22×C67#45611yescharacteristic
250C22×C67B(4,𝔽2)11yescharacteristic
251C22×C67#45811yescharacteristic
252C22×C67#46311yescharacteristic
253C22×C67#46411yescharacteristic
254C22×C67#48811yescharacteristic
255C22×C67#48911yescharacteristic
256C4×C67—21no—
257C4×C67—21no—
258C4×C67—21no—
259C4×C67—21no—
260C4×C67—44no—
261C4×C67—81no—
262C4×C67—82no—
263C4×C67—81no—
264C4×C67—81no—
265C4×C67—81no—
266C4×C67—81no—
267#10405C6711yesmaximal, characteristic, direct product factor, Sylow
268C2×C67—21no—
269C2×C67—21no—
270C2×C67—21no—
271C2×C67—21no—
272C2×C67—161no—
273C2×C67#160011yescharacteristic
274C2×C67#160311yescharacteristic
275C2×C67#160911yescharacteristic
276C2×C67#161811yescharacteristic
277C2×C67#203911yescharacteristic
278C2×C67#204211yescharacteristic
279C2×C67#204411yescharacteristic
280#2639C2×C6711yescharacteristic
281#1874C2×C6711yescharacteristic
282#1872C2×C6711yescharacteristic, complemented
283C2×#328C2×C6711yescharacteristic, complemented
284C2×#327C2×C6711yescharacteristic
285#1444C2×C6711yescharacteristic, complemented
286#1280C2×C6711yescharacteristic, complemented
287C67#1040511yescharacteristic, direct product factor, Sylow
288C2×C22⋊Q8C22×C6711yescharacteristic
289C2×#121C22×C6711yescharacteristic
290C22×C4⋊C4—21no—
291C22×C4⋊C4C22×C6711yescharacteristic
292C2×#103—21no—
293C2×#102—21no—
294C2×C22⋊C8C22×C6711yescharacteristic
295C2×C22⋊C8C22×C6711yescharacteristic
296#400—21no—
297#397—21no—
298#394—21no—
299#393—21no—
300#386—21no—
301#386—21no—
302C2×#95—21no—
303C2×#95C22×C6711yescharacteristic
304C2×#95C22×C6711yescharacteristic
305#340—21no—
306#340—21no—
307#328—22no—
308#327—22no—
309C22×Q8D4×C6711yescharacteristic
310C22×D4—21no—
311C22×D4D4×C6711yescharacteristic
312C23×C4—21no—
313C23×C4—21no—
314C23×C4D4×C6711yescharacteristic
315C23×C4D4×C6711yescharacteristic
316C23×C4C23×C6711yescharacteristic
317C22×C8—21no—
318C22×C8—21no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322C22⋊Q8—21no—
323#121—21no—
324#121—21no—
325#121—21no—
326#121—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4D4×C6711yescharacteristic
335C2×C4⋊C4D4×C6711yescharacteristic
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342#103—42no—
343#102—42no—
344C22⋊C8—21no—
345C22⋊C8—21no—
346C22⋊C8—21no—
347C22⋊C8—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—42no—
356#95D4×C6712yes—
357C24—21no—
358C24C2×D4×C6711yescharacteristic
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4C2×D4×C6711yescharacteristic
403C22×C4C2×D4×C6711yescharacteristic
404C22×C4C2×D4×C6711yescharacteristic
405C22×C4#41×C6711yescharacteristic
406C22×C4#41×C6711yescharacteristic
407C22×C4#41×C6711yescharacteristic
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—48no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—81no—
430C22⋊C4—81no—
431C22⋊C4—81no—
432C22⋊C4—81no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23#120×C6711yescharacteristic
445C23#121×C6711yescharacteristic
446C23#124×C6711yescharacteristic
447C23#135×C6711yescharacteristic
448C23#135×C6711yescharacteristic
449C23AGL(1,ℤ/8)×C6711yescharacteristic
450C23#137×C6711yescharacteristic
451Q8—42no—
452Q8—81no—
453D4—42no—
454D4—81no—
455D4—81no—
456D4—81no—
457D4—81no—
458D4—81no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4#121×C6711yescharacteristic
501C2×C4#121×C6711yescharacteristic
502C2×C4#123×C6711yescharacteristic
503C2×C4#123×C6711yescharacteristic
504C8—81no—
505C8—81no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22#394×C6711yescharacteristic
526C22#449×C6711yescharacteristic
527C22#452×C6711yescharacteristic
528C22#453×C6711yescharacteristic
529C22#456×C6711yescharacteristic
530C22B(4,𝔽2)×C6711yescharacteristic
531C22#458×C6711yescharacteristic
532C22#463×C6711yescharacteristic
533C22#464×C6711yescharacteristic
534C22#488×C6711yescharacteristic
535C22#489×C6711yescharacteristic
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—44no—
541C4—81no—
542C4—81no—
543C4—81no—
544C4—81no—
545C4—82no—
546C4—81no—
547C2—21no—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—161no—
552C2C67×#160011yescharacteristic
553C2C67×#160311yescharacteristic
554C2C67×#160911yescharacteristic
555C2C67×#161811yescharacteristic
556C2C67×#203911yescharacteristic
557C2C67×#204211yescharacteristic
558C2C67×#204411yescharacteristic
subgroupG/H
center, socleC23×C67#120
commutator, Frattini, cosocleC23×C4C23×C67
prime radicalC2×#121×C67C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
3. C67×#1872C2semidirect
4. C2×#328×C67C2semidirect
6. C67×#1444C2semidirect
7. C67×#1280C2semidirect
267. #10405C67direct (N × H)
282. #1872C2×C67semidirect
283. C2×#328C2×C67semidirect
285. #1444C2×C67semidirect
286. #1280C2×C67semidirect
287. C67#10405direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#26392
C67×#18742
C67×#18722
C2×#328×C672
C2×#327×C672
C67×#14442
C67×#12802
#1040567