Finite Group Databasebeta

← C67×#18023 #30697274

Subgroups — C67×#18023 #30697274

Order 17152.

592 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3108C211yesmaximal, characteristic
2C67×#2994C211yesmaximal, characteristic
3C67×#2984C211yesmaximal, characteristic, complemented
4C67×#1593C211yesmaximal, characteristic, complemented
5C67×#1592C211yesmaximal, characteristic, complemented
6C67×#1584C211yesmaximal, characteristic, complemented
7C67×#1584C211yesmaximal, characteristic, complemented
8C67×#1583C211yesmaximal, characteristic, complemented
9C67×#1582C211yesmaximal, characteristic, complemented
10C67×#1579C211yesmaximal, characteristic, complemented
11C67×SD16⋊C8C211yesmaximal, characteristic, complemented
12C67×#1573C211yesmaximal, characteristic, complemented
13C67×#1494C211yesmaximal, characteristic
14C2×Q8⋊C8×C67C211yesmaximal, characteristic, complemented
15C67×#1452C211yesmaximal, characteristic, complemented
16#529×C67C2211yescharacteristic
17C2×C4×Q8×C67C2211yescharacteristic
18#483×C67C411yescharacteristic
19#480×C67C411yescharacteristic
20#479×C67C411yescharacteristic
21#479×C67C411yescharacteristic
22#464×C67C411yescharacteristic
23#464×C67C411yescharacteristic
24#451×C67C411yescharacteristic
25#448×C67C411yescharacteristic
26#446×C67—21no—
27#446×C67—21no—
28#446×C67C2211yescharacteristic
29#446×C67C2211yescharacteristic
30#446×C67C2211yescharacteristic
31C8×Q8×C67C2211yescharacteristic
32C8×Q8×C67C2211yescharacteristic
33C8×Q8×C67C2211yescharacteristic
34C4×Q16×C67C2211yescharacteristic
35C4×Q16×C67C2211yescharacteristic
36C4×SD16×C67C2211yescharacteristic
37C4×SD16×C67C2211yescharacteristic
38#436×C67C2211yescharacteristic
39#435×C67—21no—
40C8×D4×C67—21no—
41C8×D4×C67C2211yescharacteristic
42#433×C67—21no—
43#432×C67—21no—
44#432×C67C2211yescharacteristic
45#431×C67—21no—
46#431×C67C2211yescharacteristic
47#431×C67C2211yescharacteristic
48#424×C67—21no—
49C2×C4⋊C8×C67C2211yescharacteristic
50C2×C4×C8×C67C2211yescharacteristic
51#335×C67C2211yescharacteristic
52#335×C67C2211yescharacteristic
53#334×C67C2211yescharacteristic
54#334×C67C2211yescharacteristic
55Q8⋊C8×C67C2211yescharacteristic
56Q8⋊C8×C67C2211yescharacteristic
57Q8⋊C8×C67C2211yescharacteristic
58Q8⋊C8×C67C2211yescharacteristic
59Q8⋊C8×C67C2211yescharacteristic
60Q8⋊C8×C67C2211yescharacteristic
61D4⋊C8×C67C2211yescharacteristic
62D4⋊C8×C67C2211yescharacteristic
63#322×C67C2211yescharacteristic
64#322×C67C2211yescharacteristic
65#322×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C22×Q8×C67C2×C411yescharacteristic
68C2×Q16×C67C2×C411yescharacteristic
69C2×Q16×C67C2×C411yescharacteristic
70C2×SD16×C67C2×C411yescharacteristic
71C2×SD16×C67C2×C411yescharacteristic
72C2×M16×C67—21no—
73C22×C8×C67—21no—
74C22×C8×C67—21no—
75C22×C8×C67—21no—
76#126×C67—21no—
77#125×C67C2×C411yescharacteristic
78#124×C67C2×C411yescharacteristic
79#123×C67—21no—
80C22⋊Q8×C67C2×C411yescharacteristic
81#121×C67C2×C411yescharacteristic
82C4×Q8×C67—21no—
83C4×Q8×C67—21no—
84C4×Q8×C67C2×C411yescharacteristic
85C4×Q8×C67C2×C411yescharacteristic
86C4×Q8×C67C2311yescharacteristic
87C4×Q8×C67C2311yescharacteristic
88C4×Q8×C67C2311yescharacteristic
89C4×D4×C67—21no—
90C4×D4×C67C2311yescharacteristic
91#117×C67—21no—
92C2×C4⋊C4×C67—21no—
93C2×C4⋊C4×C67C2×C411yescharacteristic
94C2×C42×C67—21no—
95C2×C42×C67C2311yescharacteristic
96#107×C67C2×C411yescharacteristic
97#107×C67C2×C411yescharacteristic
98#106×C67C2×C411yescharacteristic
99#106×C67C2×C411yescharacteristic
100C4⋊C8×C67—21no—
101C4⋊C8×C67—21no—
102C4⋊C8×C67—21no—
103C4⋊C8×C67—21no—
104C4⋊C8×C67—21no—
105C4⋊C8×C67—21no—
106C4⋊C8×C67—21no—
107C4⋊C8×C67C2×C411yescharacteristic
108C4⋊C8×C67C2×C411yescharacteristic
109C4⋊C8×C67D411yescharacteristic
110C4⋊C8×C67D411yescharacteristic
111C4⋊C8×C67C2311yescharacteristic
112C4⋊C8×C67C2311yescharacteristic
113C4⋊C8×C67C2311yescharacteristic
114C4⋊C8×C67C2311yescharacteristic
115#103×C67C2×C411yescharacteristic
116#103×C67C2×C411yescharacteristic
117#103×C67C2×C411yescharacteristic
118#103×C67C2×C411yescharacteristic
119#103×C67C2×C411yescharacteristic
120#103×C67C2×C411yescharacteristic
121#102×C67C2×C411yescharacteristic
122#102×C67C2×C411yescharacteristic
123C22⋊C8×C67—21no—
124C22⋊C8×C67—21no—
125C22⋊C8×C67—21no—
126C22⋊C8×C67C2×C411yescharacteristic
127C22⋊C8×C67C2×C411yescharacteristic
128C22⋊C8×C67D411yescharacteristic
129C22⋊C8×C67D411yescharacteristic
130#97×C67—21no—
131#97×C67—21no—
132#97×C67—21no—
133#97×C67—21no—
134C4×C8×C67—21no—
135C4×C8×C67—21no—
136C4×C8×C67—21no—
137C4×C8×C67—21no—
138C4×C8×C67C2311yescharacteristic
139C4×C8×C67C2311yescharacteristic
140C4×C8×C67C2311yescharacteristic
141C4×C8×C67C2311yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C2×Q8×C67—21no—
145C2×Q8×C67—22no—
146C2×Q8×C67—41no—
147C2×Q8×C67C22×C411yescharacteristic
148C2×Q8×C67C22×C411yescharacteristic
149C2×Q8×C67C22×C411yescharacteristic
150C2×D4×C67—21no—
151C2×D4×C67C22×C411yescharacteristic
152C22×C4×C67—21no—
153C22×C4×C67—21no—
154C22×C4×C67—21no—
155C22×C4×C67C22×C411yescharacteristic
156C22×C4×C67C2×D411yescharacteristic
157C22×C4×C67#4111yescharacteristic
158Q16×C67—22no—
159Q16×C67—41no—
160SD16×C67—22no—
161SD16×C67—41no—
162M16×C67—81no—
163C2×C8×C67—21no—
164C2×C8×C67—21no—
165C2×C8×C67—21no—
166C2×C8×C67—21no—
167C2×C8×C67—21no—
168C2×C8×C67—21no—
169C2×C8×C67—42no—
170C2×C8×C67—81no—
171C2×C8×C67—81no—
172C2×C8×C67C22×C411yescharacteristic
173C2×C8×C67C22×C411yescharacteristic
174C2×C8×C67C22×C411yescharacteristic
175C2×C8×C67C22×C411yescharacteristic
176C2×C8×C67C2×D411yescharacteristic
177C2×C8×C67C2×D411yescharacteristic
178C2×C8×C67C2×D411yescharacteristic
179C2×C8×C67C2×D411yescharacteristic
180C4⋊C4×C67—21no—
181C4⋊C4×C67—21no—
182C4⋊C4×C67—21no—
183C4⋊C4×C67—21no—
184C4⋊C4×C67—21no—
185C4⋊C4×C67—21no—
186C4⋊C4×C67—21no—
187C4⋊C4×C67—21no—
188C4⋊C4×C67—21no—
189C4⋊C4×C67C22×C411yescharacteristic
190C4⋊C4×C67C22×C411yescharacteristic
191C4⋊C4×C67C22×C411yescharacteristic
192C4⋊C4×C67C22×C411yescharacteristic
193C22⋊C4×C67—21no—
194C22⋊C4×C67—21no—
195C22⋊C4×C67—21no—
196C22⋊C4×C67—21no—
197C22⋊C4×C67—21no—
198C42×C67—21no—
199C42×C67—21no—
200C42×C67—21no—
201C42×C67—21no—
202C42×C67—21no—
203C42×C67C22×C411yescharacteristic
204C42×C67C2×D411yescharacteristic
205C42×C67#4111yescharacteristic
206C42×C67C2411yescharacteristic
207C23×C67—21no—
208C23×C67C4×D411yescharacteristic
209Q8×C67—21no—
210Q8×C67—21no—
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—42no—
215Q8×C67#13112yes—
216D4×C67—21no—
217D4×C67—21no—
218D4×C67—42no—
219C2×C4×C67—21no—
220C2×C4×C67—22no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
223C2×C4×C67—21no—
224C2×C4×C67—21no—
225C2×C4×C67—21no—
226C2×C4×C67—21no—
227C2×C4×C67—21no—
228C2×C4×C67—21no—
229C2×C4×C67—21no—
230C2×C4×C67—21no—
231C2×C4×C67—42no—
232C2×C4×C67—42no—
233C2×C4×C67—41no—
234C2×C4×C67—41no—
235C2×C4×C67C4×D411yescharacteristic
236C2×C4×C67C4×D411yescharacteristic
237C2×C4×C67C4×D411yescharacteristic
238C2×C4×C67C23×C411yescharacteristic
239C2×C4×C67C22×D411yescharacteristic
240C2×C4×C67C2×#4111yescharacteristic
241C8×C67—21no—
242C8×C67—21no—
243C8×C67—21no—
244C8×C67—21no—
245C8×C67—21no—
246C8×C67—21no—
247C8×C67—21no—
248C8×C67—21no—
249C8×C67—41no—
250C8×C67—41no—
251C8×C67—41no—
252C8×C67—41no—
253C8×C67—41no—
254C8×C67—41no—
255C22×C67—21no—
256C22×C67—21no—
257C22×C67—41no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67#44412yes—
261C22×C67C2×C4×D411yescharacteristic
262C4×C67—21no—
263C4×C67—22no—
264C4×C67—21no—
265C4×C67—21no—
266C4×C67—21no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67—41no—
271C4×C67—41no—
272C4×C67—41no—
273C4×C67C2×#13111yescharacteristic
274C4×C67#56811yescharacteristic
275C4×C67#57711yescharacteristic
276C4×C67#57811yescharacteristic
277#18023C6711yesmaximal, characteristic, direct product factor, Sylow
278C2×C67—22no—
279C2×C67—81no—
280C2×C67#293311yescharacteristic
281C2×C67#294711yescharacteristic
282C2×C67C2×#44411yescharacteristic
283#3108C2×C6711yescharacteristic
284#2994C2×C6711yescharacteristic
285#2984C2×C6711yescharacteristic, complemented
286#1593C2×C6711yescharacteristic, complemented
287#1592C2×C6711yescharacteristic, complemented
288#1584C2×C6711yescharacteristic, complemented
289#1584C2×C6711yescharacteristic, complemented
290#1583C2×C6711yescharacteristic, complemented
291#1582C2×C6711yescharacteristic, complemented
292#1579C2×C6711yescharacteristic, complemented
293SD16⋊C8C2×C6711yescharacteristic, complemented
294#1573C2×C6711yescharacteristic, complemented
295#1494C2×C6711yescharacteristic
296C2×Q8⋊C8C2×C6711yescharacteristic, complemented
297#1452C2×C6711yescharacteristic, complemented
298C67#1802311yescharacteristic, direct product factor, Sylow
299#529C22×C6711yescharacteristic
300C2×C4×Q8C22×C6711yescharacteristic
301#483C4×C6711yescharacteristic
302#480C4×C6711yescharacteristic
303#479C4×C6711yescharacteristic
304#479C4×C6711yescharacteristic
305#464C4×C6711yescharacteristic
306#464C4×C6711yescharacteristic
307#451C4×C6711yescharacteristic
308#448C4×C6711yescharacteristic
309#446—21no—
310#446—21no—
311#446C22×C6711yescharacteristic
312#446C22×C6711yescharacteristic
313#446C22×C6711yescharacteristic
314C8×Q8C22×C6711yescharacteristic
315C8×Q8C22×C6711yescharacteristic
316C8×Q8C22×C6711yescharacteristic
317C4×Q16C22×C6711yescharacteristic
318C4×Q16C22×C6711yescharacteristic
319C4×SD16C22×C6711yescharacteristic
320C4×SD16C22×C6711yescharacteristic
321#436C22×C6711yescharacteristic
322#435—21no—
323C8×D4—21no—
324C8×D4C22×C6711yescharacteristic
325#433—21no—
326#432—21no—
327#432C22×C6711yescharacteristic
328#431—21no—
329#431C22×C6711yescharacteristic
330#431C22×C6711yescharacteristic
331#424—21no—
332C2×C4⋊C8C22×C6711yescharacteristic
333C2×C4×C8C22×C6711yescharacteristic
334#335C22×C6711yescharacteristic
335#335C22×C6711yescharacteristic
336#334C22×C6711yescharacteristic
337#334C22×C6711yescharacteristic
338Q8⋊C8C22×C6711yescharacteristic
339Q8⋊C8C22×C6711yescharacteristic
340Q8⋊C8C22×C6711yescharacteristic
341Q8⋊C8C22×C6711yescharacteristic
342Q8⋊C8C22×C6711yescharacteristic
343Q8⋊C8C22×C6711yescharacteristic
344D4⋊C8C22×C6711yescharacteristic
345D4⋊C8C22×C6711yescharacteristic
346#322C22×C6711yescharacteristic
347#322C22×C6711yescharacteristic
348#322C22×C6711yescharacteristic
349#322C22×C6711yescharacteristic
350C22×Q8C2×C4×C6711yescharacteristic
351C2×Q16C2×C4×C6711yescharacteristic
352C2×Q16C2×C4×C6711yescharacteristic
353C2×SD16C2×C4×C6711yescharacteristic
354C2×SD16C2×C4×C6711yescharacteristic
355C2×M16—21no—
356C22×C8—21no—
357C22×C8—21no—
358C22×C8—21no—
359#126—21no—
360#125C2×C4×C6711yescharacteristic
361#124C2×C4×C6711yescharacteristic
362#123—21no—
363C22⋊Q8C2×C4×C6711yescharacteristic
364#121C2×C4×C6711yescharacteristic
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8C2×C4×C6711yescharacteristic
368C4×Q8C2×C4×C6711yescharacteristic
369C4×Q8C23×C6711yescharacteristic
370C4×Q8C23×C6711yescharacteristic
371C4×Q8C23×C6711yescharacteristic
372C4×D4—21no—
373C4×D4C23×C6711yescharacteristic
374#117—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4C2×C4×C6711yescharacteristic
377C2×C42—21no—
378C2×C42C23×C6711yescharacteristic
379#107C2×C4×C6711yescharacteristic
380#107C2×C4×C6711yescharacteristic
381#106C2×C4×C6711yescharacteristic
382#106C2×C4×C6711yescharacteristic
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8C2×C4×C6711yescharacteristic
391C4⋊C8C2×C4×C6711yescharacteristic
392C4⋊C8D4×C6711yescharacteristic
393C4⋊C8D4×C6711yescharacteristic
394C4⋊C8C23×C6711yescharacteristic
395C4⋊C8C23×C6711yescharacteristic
396C4⋊C8C23×C6711yescharacteristic
397C4⋊C8C23×C6711yescharacteristic
398#103C2×C4×C6711yescharacteristic
399#103C2×C4×C6711yescharacteristic
400#103C2×C4×C6711yescharacteristic
401#103C2×C4×C6711yescharacteristic
402#103C2×C4×C6711yescharacteristic
403#103C2×C4×C6711yescharacteristic
404#102C2×C4×C6711yescharacteristic
405#102C2×C4×C6711yescharacteristic
406C22⋊C8—21no—
407C22⋊C8—21no—
408C22⋊C8—21no—
409C22⋊C8C2×C4×C6711yescharacteristic
410C22⋊C8C2×C4×C6711yescharacteristic
411C22⋊C8D4×C6711yescharacteristic
412C22⋊C8D4×C6711yescharacteristic
413#97—21no—
414#97—21no—
415#97—21no—
416#97—21no—
417C4×C8—21no—
418C4×C8—21no—
419C4×C8—21no—
420C4×C8—21no—
421C4×C8C23×C6711yescharacteristic
422C4×C8C23×C6711yescharacteristic
423C4×C8C23×C6711yescharacteristic
424C4×C8C23×C6711yescharacteristic
425C4×C8C23×C6711yescharacteristic
426C4×C8C23×C6711yescharacteristic
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—41no—
430C2×Q8C22×C4×C6711yescharacteristic
431C2×Q8C22×C4×C6711yescharacteristic
432C2×Q8C22×C4×C6711yescharacteristic
433C2×D4—21no—
434C2×D4C22×C4×C6711yescharacteristic
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4C22×C4×C6711yescharacteristic
439C22×C4C2×D4×C6711yescharacteristic
440C22×C4#41×C6711yescharacteristic
441Q16—22no—
442Q16—41no—
443SD16—22no—
444SD16—41no—
445M16—81no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—42no—
453C2×C8—81no—
454C2×C8—81no—
455C2×C8C22×C4×C6711yescharacteristic
456C2×C8C22×C4×C6711yescharacteristic
457C2×C8C22×C4×C6711yescharacteristic
458C2×C8C22×C4×C6711yescharacteristic
459C2×C8C2×D4×C6711yescharacteristic
460C2×C8C2×D4×C6711yescharacteristic
461C2×C8C2×D4×C6711yescharacteristic
462C2×C8C2×D4×C6711yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4C22×C4×C6711yescharacteristic
473C4⋊C4C22×C4×C6711yescharacteristic
474C4⋊C4C22×C4×C6711yescharacteristic
475C4⋊C4C22×C4×C6711yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42C22×C4×C6711yescharacteristic
487C42C2×D4×C6711yescharacteristic
488C42#41×C6711yescharacteristic
489C42C24×C6711yescharacteristic
490C23—21no—
491C23C4×D4×C6711yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—42no—
498Q8#131×C6712yes—
499D4—21no—
500D4—21no—
501D4—42no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4C4×D4×C6711yescharacteristic
519C2×C4C4×D4×C6711yescharacteristic
520C2×C4C4×D4×C6711yescharacteristic
521C2×C4C23×C4×C6711yescharacteristic
522C2×C4C22×D4×C6711yescharacteristic
523C2×C4C2×#41×C6711yescharacteristic
524C8—21no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—21no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—41no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22#444×C6712yes—
544C22C2×C4×D4×C6711yescharacteristic
545C4—21no—
546C4—22no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4C2×#131×C6711yescharacteristic
557C4#568×C6711yescharacteristic
558C4#577×C6711yescharacteristic
559C4#578×C6711yescharacteristic
560C2—22no—
561C2—81no—
562C2C67×#293311yescharacteristic
563C2C67×#294711yescharacteristic
564C2C2×#444×C6711yescharacteristic
subgroupG/H
centerC2×C4×C67C22×D4
commutatorC2×C4C23×C4×C67
Frattini, cosocleC42C24×C67
socleC22×C67C2×C4×D4
prime radical#121×C67C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ #431×C67 ◁ G

Complements

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

NH ≅ G/Ntype
3. C67×#2984C2semidirect
4. C67×#1593C2semidirect
5. C67×#1592C2semidirect
6. C67×#1584C2semidirect
7. C67×#1584C2semidirect
8. C67×#1583C2semidirect
9. C67×#1582C2semidirect
10. C67×#1579C2semidirect
11. C67×SD16⋊C8C2semidirect
12. C67×#1573C2semidirect
14. C2×Q8⋊C8×C67C2semidirect
15. C67×#1452C2semidirect
277. #18023C67direct (N × H)
285. #2984C2×C67semidirect
286. #1593C2×C67semidirect
287. #1592C2×C67semidirect
288. #1584C2×C67semidirect
289. #1584C2×C67semidirect
290. #1583C2×C67semidirect
291. #1582C2×C67semidirect
292. #1579C2×C67semidirect
293. SD16⋊C8C2×C67semidirect
294. #1573C2×C67semidirect
296. C2×Q8⋊C8C2×C67semidirect
297. #1452C2×C67semidirect
298. C67#18023direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#31082
C67×#29942
C67×#29842
C67×#15932
C67×#15922
C67×#15842
C67×#15832
C67×#15822
C67×#15792
C67×SD16⋊C82
C67×#15732
C67×#14942
C2×Q8⋊C8×C672
C67×#14522
#1802367