Finite Group Databasebeta

← C67×#14328 #30693579

Subgroups — C67×#14328 #30693579

Order 17152.

1282 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#398×C67C211yesmaximal, characteristic
2C2×#389×C67C211yesmaximal, characteristic
3C2×#387×C67C211yesmaximal, characteristic
4C67×#1445C211yesmaximal, characteristic
5C67×#1444C212yesmaximal
6C67×#1444C211yesmaximal, characteristic
7C67×#1441C211yesmaximal, characteristic
8C67×#1441C212yesmaximal
9C67×C4⋊#95C212yesmaximal
10C67×#1434C212yesmaximal
11C4×#95×C67C211yesmaximal, characteristic
12C22×C4⋊C4×C67C2211yescharacteristic
13C22×C4⋊C4×C67C2211yescharacteristic
14C22×C4⋊C4×C67C2211yescharacteristic
15C22×C4⋊C4×C67C2211yescharacteristic
16C22×C4⋊C4×C67C2212yes—
17C22×C4⋊C4×C67C2212yes—
18C22×C4⋊C4×C67C2212yes—
19C22×C42×C67C2211yescharacteristic
20C22×C42×C67C2212yes—
21#398×C67C418yescomplemented
22#389×C67—24no—
23#387×C67—24no—
24C2×#95×C67C2211yescharacteristic
25C2×#95×C67C2211yescharacteristic
26C2×#95×C67C2211yescharacteristic
27C2×#95×C67C2211yescharacteristic
28C2×#95×C67C2212yes—
29C2×#95×C67C2212yes—
30C2×#95×C67C2211yescharacteristic
31C2×#95×C67C2212yes—
32C2×#95×C67C2212yes—
33C2×#95×C67C2212yes—
34C2×#95×C67C2212yes—
35C2×#95×C67C2212yes—
36C2×#95×C67C2211yescharacteristic
37C2×#95×C67C2212yes—
38C23×C4×C67C2312yes—
39C23×C4×C67C2311yescharacteristic
40C23×C4×C67C2311yescharacteristic
41C23×C4×C67C2312yes—
42C23×C4×C67C2311yescharacteristic
43C23×C4×C67C2311yescharacteristic
44C23×C4×C67C2311yescharacteristic
45C23×C4×C67C2312yes—
46C23×C4×C67C2311yescharacteristic
47C23×C4×C67C2312yes—
48C23×C4×C67C2311yescharacteristic
49C2×C4⋊C4×C67—22no—
50C2×C4⋊C4×C67—22no—
51C2×C4⋊C4×C67—22no—
52C2×C4⋊C4×C67—22no—
53C2×C4⋊C4×C67—22no—
54C2×C4⋊C4×C67—24no—
55C2×C4⋊C4×C67—22no—
56C2×C4⋊C4×C67—24no—
57C2×C4⋊C4×C67—42no—
58C2×C4⋊C4×C67—42no—
59C2×C4⋊C4×C67—42no—
60C2×C4⋊C4×C67—42no—
61C2×C4⋊C4×C67—42no—
62C2×C4⋊C4×C67—42no—
63C2×C4⋊C4×C67—42no—
64C2×C4⋊C4×C67—42no—
65C2×C4⋊C4×C67C2×C414yes—
66C2×C4⋊C4×C67C2×C414yes—
67C2×C4⋊C4×C67C2×C418yes—
68C2×C42×C67—42no—
69C2×C42×C67—42no—
70C2×C42×C67—42no—
71C2×C42×C67D414yes—
72C2×C42×C67Q818yes—
73#95×C67—22no—
74#95×C67—22no—
75#95×C67—22no—
76#95×C67—22no—
77#95×C67—24no—
78#95×C67—22no—
79#95×C67—24no—
80#95×C67—24no—
81#95×C67—42no—
82#95×C67—42no—
83#95×C67—42no—
84#95×C67—42no—
85#95×C67C2×C418yes—
86#95×C67C2×C414yes—
87C24×C67C2411yescharacteristic
88C22×C4×C67—22no—
89C22×C4×C67—21no—
90C22×C4×C67—22no—
91C22×C4×C67—22no—
92C22×C4×C67—22no—
93C22×C4×C67—22no—
94C22×C4×C67—22no—
95C22×C4×C67—22no—
96C22×C4×C67—21no—
97C22×C4×C67—21no—
98C22×C4×C67—22no—
99C22×C4×C67—21no—
100C22×C4×C67—21no—
101C22×C4×C67—22no—
102C22×C4×C67—21no—
103C22×C4×C67—22no—
104C22×C4×C67—22no—
105C22×C4×C67—22no—
106C22×C4×C67—22no—
107C22×C4×C67—22no—
108C22×C4×C67—22no—
109C22×C4×C67—21no—
110C22×C4×C67—21no—
111C22×C4×C67—22no—
112C22×C4×C67—21no—
113C22×C4×C67—21no—
114C22×C4×C67—22no—
115C22×C4×C67—22no—
116C22×C4×C67—22no—
117C22×C4×C67—21no—
118C22×C4×C67—22no—
119C22×C4×C67—22no—
120C22×C4×C67—22no—
121C22×C4×C67—21no—
122C22×C4×C67—22no—
123C22×C4×C67—22no—
124C22×C4×C67—21no—
125C22×C4×C67—21no—
126C22×C4×C67—22no—
127C22×C4×C67—22no—
128C22×C4×C67—22no—
129C22×C4×C67—22no—
130C22×C4×C67—22no—
131C22×C4×C67—22no—
132C22×C4×C67—22no—
133C22×C4×C67—21no—
134C22×C4×C67—22no—
135C22×C4×C67—22no—
136C22×C4×C67—21no—
137C22×C4×C67—21no—
138C22×C4×C67—22no—
139C22×C4×C67—21no—
140C22×C4×C67C22×C412yes—
141C22×C4×C67C22×C414yes—
142C22×C4×C67C22×C412yes—
143C22×C4×C67C22×C412yes—
144C22×C4×C67C22×C414yes—
145C22×C4×C67C2×D412yes—
146C22×C4×C67C2×D412yes—
147C22×C4×C67C2×D412yes—
148C22×C4×C67C2×Q814yes—
149C22×C4×C67C2×Q814yes—
150C22×C4×C67C2×Q814yes—
151C22×C4×C67#4112yes—
152C22×C4×C67#4114yes—
153C4⋊C4×C67—28no—
154C4⋊C4×C67—28no—
155C4⋊C4×C67—48no—
156C4⋊C4×C67—44no—
157C4⋊C4×C67—48no—
158C4⋊C4×C67—44no—
159C4⋊C4×C67—48no—
160C42×C67—44no—
161C42×C67—48no—
162C23×C67C23×C411yescharacteristic
163C23×C67C22×D411yescharacteristic
164C23×C67C22×Q812yes—
165C23×C67C2×#4111yescharacteristic
166C23×C67C2×#4112yes—
167C23×C67H(5,𝔽2)11yescharacteristic
168C23×C67H(5,𝔽2)11yescharacteristic
169C23×C67H(5,𝔽2)12yes—
170C23×C67H(5,𝔽2)11yescharacteristic
171C23×C672−1+412yes—
172C23×C672−1+411yescharacteristic
173C2×C4×C67—24no—
174C2×C4×C67—24no—
175C2×C4×C67—24no—
176C2×C4×C67—22no—
177C2×C4×C67—22no—
178C2×C4×C67—22no—
179C2×C4×C67—22no—
180C2×C4×C67—24no—
181C2×C4×C67—24no—
182C2×C4×C67—24no—
183C2×C4×C67—24no—
184C2×C4×C67—24no—
185C2×C4×C67—24no—
186C2×C4×C67—24no—
187C2×C4×C67—22no—
188C2×C4×C67—24no—
189C2×C4×C67—22no—
190C2×C4×C67—22no—
191C2×C4×C67—24no—
192C2×C4×C67—24no—
193C2×C4×C67—24no—
194C2×C4×C67—22no—
195C2×C4×C67—22no—
196C2×C4×C67—24no—
197C2×C4×C67—22no—
198C2×C4×C67—22no—
199C2×C4×C67—24no—
200C2×C4×C67—24no—
201C2×C4×C67—41no—
202C2×C4×C67—42no—
203C2×C4×C67—41no—
204C2×C4×C67—41no—
205C2×C4×C67—42no—
206C2×C4×C67—41no—
207C2×C4×C67—42no—
208C2×C4×C67—42no—
209C2×C4×C67—42no—
210C2×C4×C67—42no—
211C2×C4×C67—41no—
212C2×C4×C67—42no—
213C2×C4×C67—42no—
214C2×C4×C67—42no—
215C2×C4×C67—42no—
216C2×C4×C67—41no—
217C2×C4×C67—42no—
218C2×C4×C67—42no—
219C2×C4×C67—41no—
220C2×C4×C67—41no—
221C2×C4×C67—42no—
222C2×C4×C67—41no—
223C2×C4×C67—42no—
224C2×C4×C67—41no—
225C2×C4×C67—41no—
226C2×C4×C67—41no—
227C2×C4×C67—42no—
228C2×C4×C67—42no—
229C2×C4×C67—42no—
230C2×C4×C67—42no—
231C2×C4×C67—42no—
232C2×C4×C67—42no—
233C2×C4×C67—42no—
234C2×C4×C67—42no—
235C2×C4×C67C4×D414yes—
236C2×C4×C67C4×Q818yes—
237C2×C4×C67C4⋊Q814yes—
238C22×C67C2×C4×D411yescharacteristic
239C22×C67C2×C4×Q812yes—
240C22×C67#51811yescharacteristic
241C22×C67#52012yes—
242C22×C67#52011yescharacteristic
243C22×C67C2×C4⋊Q811yescharacteristic
244C22×C67#53211yescharacteristic
245C22×C67#53312yes—
246C22×C67#53411yescharacteristic
247C22×C67#53511yescharacteristic
248C22×C67#53611yescharacteristic
249C22×C67#53711yescharacteristic
250C22×C67#53812yes—
251C22×C67#53912yes—
252C22×C67#53912yes—
253C22×C67#54011yescharacteristic
254C22×C67#54012yes—
255C22×C67#54112yes—
256C22×C67#54211yescharacteristic
257C22×C67#54312yes—
258C22×C67#54412yes—
259C22×C67#54412yes—
260C22×C67#54412yes—
261C4×C67—24no—
262C4×C67—28no—
263C4×C67—24no—
264C4×C67—42no—
265C4×C67—42no—
266C4×C67—44no—
267C4×C67—42no—
268C4×C67—44no—
269C4×C67—82no—
270C4×C67—81no—
271C4×C67—81no—
272#14328C6711yesmaximal, characteristic, direct product factor, Sylow
273C2×C67C4×C4⋊Q811yescharacteristic
274C2×C67#232211yescharacteristic
275C2×C67#232411yescharacteristic
276C2×C67#232912yes—
277C2×C67#233012yes—
278C2×C67#265911yescharacteristic
279C2×C67#266011yescharacteristic
280C2×C67#266111yescharacteristic
281C2×C67#266411yescharacteristic
282C2×C67#266612yes—
283C2×C67#266712yes—
284C2×#398C2×C6711yescharacteristic
285C2×#389C2×C6711yescharacteristic
286C2×#387C2×C6711yescharacteristic
287#1445C2×C6711yescharacteristic
288#1444C2×C6711yescharacteristic
289#1444C2×C6712yes—
290#1441C2×C6711yescharacteristic
291#1441C2×C6712yes—
292C4⋊#95C2×C6712yes—
293#1434C2×C6712yes—
294C4×#95C2×C6711yescharacteristic
295C67#1432811yescharacteristic, direct product factor, Sylow
296C22×C4⋊C4C22×C6711yescharacteristic
297C22×C4⋊C4C22×C6711yescharacteristic
298C22×C4⋊C4C22×C6712yes—
299C22×C4⋊C4C22×C6711yescharacteristic
300C22×C4⋊C4C22×C6712yes—
301C22×C4⋊C4C22×C6711yescharacteristic
302C22×C4⋊C4C22×C6712yes—
303C22×C42C22×C6711yescharacteristic
304C22×C42C22×C6712yes—
305#398C4×C6718yescomplemented
306#389—24no—
307#387—24no—
308C2×#95C22×C6712yes—
309C2×#95C22×C6711yescharacteristic
310C2×#95C22×C6711yescharacteristic
311C2×#95C22×C6712yes—
312C2×#95C22×C6712yes—
313C2×#95C22×C6711yescharacteristic
314C2×#95C22×C6711yescharacteristic
315C2×#95C22×C6711yescharacteristic
316C2×#95C22×C6712yes—
317C2×#95C22×C6712yes—
318C2×#95C22×C6712yes—
319C2×#95C22×C6712yes—
320C2×#95C22×C6711yescharacteristic
321C2×#95C22×C6712yes—
322C23×C4C23×C6711yescharacteristic
323C23×C4C23×C6712yes—
324C23×C4C23×C6711yescharacteristic
325C23×C4C23×C6711yescharacteristic
326C23×C4C23×C6712yes—
327C23×C4C23×C6711yescharacteristic
328C23×C4C23×C6711yescharacteristic
329C23×C4C23×C6712yes—
330C23×C4C23×C6711yescharacteristic
331C23×C4C23×C6712yes—
332C23×C4C23×C6711yescharacteristic
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—42no—
342C2×C4⋊C4—42no—
343C2×C4⋊C4—42no—
344C2×C4⋊C4—42no—
345C2×C4⋊C4—42no—
346C2×C4⋊C4—42no—
347C2×C4⋊C4—42no—
348C2×C4⋊C4—42no—
349C2×C4⋊C4C2×C4×C6714yes—
350C2×C4⋊C4C2×C4×C6714yes—
351C2×C4⋊C4C2×C4×C6718yes—
352C2×C42—42no—
353C2×C42—42no—
354C2×C42—42no—
355C2×C42D4×C6714yes—
356C2×C42Q8×C6718yes—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—42no—
366#95—42no—
367#95—42no—
368#95—42no—
369#95C2×C4×C6718yes—
370#95C2×C4×C6714yes—
371C24C24×C6711yescharacteristic
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4C22×C4×C6712yes—
425C22×C4C22×C4×C6714yes—
426C22×C4C22×C4×C6712yes—
427C22×C4C22×C4×C6712yes—
428C22×C4C22×C4×C6714yes—
429C22×C4C2×D4×C6712yes—
430C22×C4C2×D4×C6712yes—
431C22×C4C2×D4×C6712yes—
432C22×C4C2×Q8×C6714yes—
433C22×C4C2×Q8×C6714yes—
434C22×C4C2×Q8×C6714yes—
435C22×C4#41×C6712yes—
436C22×C4#41×C6714yes—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C42—44no—
445C42—48no—
446C23C23×C4×C6711yescharacteristic
447C23C22×D4×C6711yescharacteristic
448C23C22×Q8×C6712yes—
449C23C2×#41×C6711yescharacteristic
450C23C2×#41×C6712yes—
451C23H(5,𝔽2)×C6711yescharacteristic
452C23H(5,𝔽2)×C6711yescharacteristic
453C23H(5,𝔽2)×C6712yes—
454C23H(5,𝔽2)×C6711yescharacteristic
455C232−1+4×C6712yes—
456C232−1+4×C6711yescharacteristic
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4C4×D4×C6714yes—
520C2×C4C4×Q8×C6718yes—
521C2×C4C4⋊Q8×C6714yes—
522C22C2×C4×D4×C6711yescharacteristic
523C22C2×C4×Q8×C6712yes—
524C22#518×C6711yescharacteristic
525C22#520×C6712yes—
526C22#520×C6711yescharacteristic
527C22C2×C4⋊Q8×C6711yescharacteristic
528C22#532×C6711yescharacteristic
529C22#533×C6712yes—
530C22#534×C6711yescharacteristic
531C22#535×C6711yescharacteristic
532C22#536×C6711yescharacteristic
533C22#537×C6711yescharacteristic
534C22#538×C6712yes—
535C22#539×C6712yes—
536C22#539×C6712yes—
537C22#540×C6711yescharacteristic
538C22#540×C6712yes—
539C22#541×C6712yes—
540C22#542×C6711yescharacteristic
541C22#543×C6712yes—
542C22#544×C6712yes—
543C22#544×C6712yes—
544C22#544×C6712yes—
545C4—24no—
546C4—28no—
547C4—24no—
548C4—42no—
549C4—42no—
550C4—44no—
551C4—42no—
552C4—44no—
553C4—82no—
554C4—81no—
555C4—81no—
556C2C4×C4⋊Q8×C6711yescharacteristic
557C2C67×#232211yescharacteristic
558C2C67×#232411yescharacteristic
559C2C67×#232912yes—
560C2C67×#233012yes—
561C2C67×#265911yescharacteristic
562C2C67×#266011yescharacteristic
563C2C67×#266111yescharacteristic
564C2C67×#266411yescharacteristic
565C2C67×#266612yes—
566C2C67×#266712yes—
subgroupG/H
center, socle, prime radicalC24×C67C24
commutatorC23C23×C4×C67
Frattini, cosocleC24C24×C67
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C67 ◁ G

Complements

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

NH ≅ G/Ntype
21. #398×C67C4semidirect
272. #14328C67direct (N × H)
295. C67#14328direct (N × H)
305. #398C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#398×C672
C2×#389×C672
C2×#387×C672
C67×#14452
C67×#14442
C67×#14412
C67×C4⋊#952
C67×#14342
C4×#95×C672
#1432867