Finite Group Databasebeta

← C2×C67×#1618 #30697371

Subgroups — C2×C67×#1618 #30697371

Order 17152.

1200 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#538×C67C211yesmaximal, characteristic
2C2×#484×C67C211yesmaximal, characteristic, complemented
3C2×#481×C67C211yesmaximal, characteristic, complemented
4C2×#399×C67C211yesmaximal, characteristic, complemented
5C67×#1618C218yesmaximal, direct product factor
6C2×#328×C67C211yesmaximal, characteristic, complemented
7C2×#327×C67C211yesmaximal, characteristic, complemented
8C2×#323×C67C211yesmaximal, characteristic, complemented
9#538×C67—22no—
10#538×C67—22no—
11#538×C67—22no—
12#538×C67C2214yes—
13C2×#126×C67—21no—
14C2×#124×C67—21no—
15C2×#123×C67—21no—
16C2×C22⋊Q8×C67C2211yescharacteristic
17C2×#121×C67—21no—
18C2×#121×C67C2211yescharacteristic
19C2×#120×C67—21no—
20C2×C4×D4×C67—21no—
21C22×C4⋊C4×C67—21no—
22C22×C4⋊C4×C67C2211yescharacteristic
23C22×C22⋊C4×C67—21no—
24C22×C22⋊C4×C67C2211yescharacteristic
25#484×C67—22no—
26#484×C67C2214yescomplemented
27#481×C67—22no—
28#481×C67C2214yescomplemented
29C2×#107×C67—21no—
30C2×#106×C67—21no—
31C2×#103×C67—21no—
32C2×#102×C67—21no—
33C2×C22⋊C8×C67C2211yescharacteristic
34C2×C22⋊C8×C67C2211yescharacteristic
35#399×C67—22no—
36#399×C67C2214yescomplemented
37C2×#95×C67C2211yescharacteristic
38#328×C67C2214yescomplemented
39#327×C67C2214yescomplemented
40#323×C67C2214yescomplemented
41C25×C67D411yescharacteristic
42C22×Q8×C67D411yescharacteristic
43C22×D4×C67—21no—
44C22×D4×C67—21no—
45C22×D4×C67—21no—
46C22×D4×C67D411yescharacteristic
47C23×C4×C67—21no—
48C23×C4×C67—21no—
49C23×C4×C67—21no—
50C23×C4×C67D411yescharacteristic
51C23×C4×C67C2311yescharacteristic
52C22×C8×C67—21no—
53C22×C8×C67—21no—
54#126×C67—24no—
55#126×C67—24no—
56#124×C67—24no—
57#124×C67—24no—
58#123×C67—24no—
59#123×C67—24no—
60C22⋊Q8×C67—21no—
61C22⋊Q8×C67—21no—
62C22⋊Q8×C67D412yes—
63C22⋊Q8×C67C2312yes—
64#121×C67—24no—
65#121×C67—24no—
66#121×C67—21no—
67#121×C67—21no—
68#121×C67D412yes—
69#121×C67C2312yes—
70#120×C67—24no—
71#120×C67—24no—
72C4×D4×C67—24no—
73C4×D4×C67—24no—
74C2×C4⋊C4×C67—21no—
75C2×C4⋊C4×C67—21no—
76C2×C4⋊C4×C67—21no—
77C2×C4⋊C4×C67—21no—
78C2×C4⋊C4×C67—21no—
79C2×C4⋊C4×C67—21no—
80C2×C4⋊C4×C67—24no—
81C2×C4⋊C4×C67—21no—
82C2×C4⋊C4×C67—44no—
83C2×C4⋊C4×C67D411yescharacteristic
84C2×C4⋊C4×C67D411yescharacteristic
85C2×C4⋊C4×C67C2312yes—
86C2×C22⋊C4×C67—21no—
87C2×C22⋊C4×C67—21no—
88C2×C22⋊C4×C67—21no—
89C2×C22⋊C4×C67—21no—
90C2×C22⋊C4×C67—21no—
91C2×C22⋊C4×C67—21no—
92C2×C22⋊C4×C67—21no—
93C2×C22⋊C4×C67—21no—
94C2×C22⋊C4×C67—21no—
95C2×C22⋊C4×C67—24no—
96C2×C22⋊C4×C67—44no—
97C2×C22⋊C4×C67D412yes—
98C2×C22⋊C4×C67C2312yes—
99C2×C42×C67—21no—
100#107×C67—24no—
101#106×C67—24no—
102#103×C67—24no—
103#102×C67—24no—
104C22⋊C8×C67—21no—
105C22⋊C8×C67—21no—
106C22⋊C8×C67C2312yes—
107C22⋊C8×C67C2312yes—
108#95×C67—21no—
109#95×C67C2312yes—
110C24×C67—21no—
111C24×C67—21no—
112C24×C67—22no—
113C24×C67—21no—
114C24×C67—22no—
115C24×C67—42no—
116C24×C67—42no—
117C24×C67C2×D411yescharacteristic
118C24×C67C2×D412yes—
119C2×Q8×C67—21no—
120C2×Q8×C67—42no—
121C2×Q8×C67C2×D412yes—
122C2×D4×C67—22no—
123C2×D4×C67—22no—
124C2×D4×C67—24no—
125C2×D4×C67—24no—
126C2×D4×C67—21no—
127C2×D4×C67—42no—
128C2×D4×C67—42no—
129C2×D4×C67—42no—
130C2×D4×C67—82no—
131C2×D4×C67—82no—
132C2×D4×C67C2×D412yes—
133C22×C4×C67—24no—
134C22×C4×C67—21no—
135C22×C4×C67—21no—
136C22×C4×C67—21no—
137C22×C4×C67—21no—
138C22×C4×C67—22no—
139C22×C4×C67—24no—
140C22×C4×C67—21no—
141C22×C4×C67—22no—
142C22×C4×C67—42no—
143C22×C4×C67—42no—
144C22×C4×C67—42no—
145C22×C4×C67—42no—
146C22×C4×C67—42no—
147C22×C4×C67—42no—
148C22×C4×C67—41no—
149C22×C4×C67—41no—
150C22×C4×C67—82no—
151C22×C4×C67C2×D411yescharacteristic
152C22×C4×C67C2×D411yescharacteristic
153C22×C4×C67C2×D412yes—
154C22×C4×C67C2×D411yescharacteristic
155C22×C4×C67C2×D411yescharacteristic
156C22×C4×C67C2×D411yescharacteristic
157C22×C4×C67C2411yescharacteristic
158C2×C8×C67—22no—
159C2×C8×C67—22no—
160C2×C8×C67—42no—
161C2×C8×C67—42no—
162C4⋊C4×C67—24no—
163C4⋊C4×C67—22no—
164C4⋊C4×C67—21no—
165C4⋊C4×C67—21no—
166C4⋊C4×C67—22no—
167C4⋊C4×C67—22no—
168C4⋊C4×C67—22no—
169C4⋊C4×C67—48no—
170C4⋊C4×C67—41no—
171C4⋊C4×C67—41no—
172C4⋊C4×C67C2×D412yes—
173C4⋊C4×C67C2×D412yes—
174C22⋊C4×C67—24no—
175C22⋊C4×C67—24no—
176C22⋊C4×C67—24no—
177C22⋊C4×C67—22no—
178C22⋊C4×C67—22no—
179C22⋊C4×C67—22no—
180C22⋊C4×C67—22no—
181C22⋊C4×C67—22no—
182C22⋊C4×C67—22no—
183C22⋊C4×C67—22no—
184C22⋊C4×C67—22no—
185C22⋊C4×C67—48no—
186C42×C67—24no—
187C23×C67—22no—
188C23×C67—22no—
189C23×C67—22no—
190C23×C67—22no—
191C23×C67—22no—
192C23×C67—22no—
193C23×C67—42no—
194C23×C67—42no—
195C23×C67—48no—
196C23×C67—42no—
197C23×C67—42no—
198C23×C67—42no—
199C23×C67—42no—
200C23×C67—42no—
201C23×C67—42no—
202C23×C67—42no—
203C23×C67—81no—
204C23×C67—81no—
205C23×C67—84no—
206C23×C67#12011yescharacteristic
207C23×C67#12011yescharacteristic
208C23×C67#13512yes—
209C23×C67#13712yes—
210C23×C67C22×D411yescharacteristic
211Q8×C67—42no—
212Q8×C67—81no—
213D4×C67—42no—
214D4×C67—42no—
215D4×C67—42no—
216D4×C67—44no—
217D4×C67—81no—
218D4×C67—84no—
219D4×C67—84no—
220C2×C4×C67—22no—
221C2×C4×C67—22no—
222C2×C4×C67—24no—
223C2×C4×C67—22no—
224C2×C4×C67—22no—
225C2×C4×C67—22no—
226C2×C4×C67—22no—
227C2×C4×C67—41no—
228C2×C4×C67—41no—
229C2×C4×C67—41no—
230C2×C4×C67—41no—
231C2×C4×C67—48no—
232C2×C4×C67—42no—
233C2×C4×C67—42no—
234C2×C4×C67—42no—
235C2×C4×C67—42no—
236C2×C4×C67—42no—
237C2×C4×C67—42no—
238C2×C4×C67—42no—
239C2×C4×C67—42no—
240C2×C4×C67—42no—
241C2×C4×C67—42no—
242C2×C4×C67—81no—
243C2×C4×C67—84no—
244C2×C4×C67—82no—
245C2×C4×C67—82no—
246C2×C4×C67#12011yescharacteristic
247C2×C4×C67#12011yescharacteristic
248C2×C4×C67C22×D411yescharacteristic
249C2×C4×C67C22×D411yescharacteristic
250C8×C67—42no—
251C8×C67—42no—
252C22×C67—24no—
253C22×C67—24no—
254C22×C67—21no—
255C22×C67—21no—
256C22×C67—21no—
257C22×C67—21no—
258C22×C67—44no—
259C22×C67—42no—
260C22×C67—42no—
261C22×C67—42no—
262C22×C67—42no—
263C22×C67—42no—
264C22×C67—42no—
265C22×C67—82no—
266C22×C67—84no—
267C22×C67—84no—
268C22×C67—84no—
269C22×C67#45212yes—
270C22×C67#45212yes—
271C22×C67#45312yes—
272C22×C67#45812yes—
273C22×C67C2×#12011yescharacteristic
274C22×C67C2×#13511yescharacteristic
275C22×C67C2×#13711yescharacteristic
276C4×C67—21no—
277C4×C67—21no—
278C4×C67—21no—
279C4×C67—21no—
280C4×C67—44no—
281C4×C67—42no—
282C4×C67—42no—
283C4×C67—42no—
284C4×C67—42no—
285C4×C67—82no—
286C2×#1618C6711yesmaximal, characteristic, direct product factor, Sylow
287C2×C67—21no—
288C2×C67—21no—
289C2×C67—22no—
290C2×C67—82no—
291C2×C67—82no—
292C2×C67#161814yesdirect product factor
293C2×C67C2×#45211yescharacteristic
294C2×C67C2×#45311yescharacteristic
295C2×C67C2×#45811yescharacteristic
296C2×#538C2×C6711yescharacteristic
297C2×#484C2×C6711yescharacteristic, complemented
298C2×#481C2×C6711yescharacteristic, complemented
299C2×#399C2×C6711yescharacteristic, complemented
300#1618C2×C6718yesdirect product factor
301C2×#328C2×C6711yescharacteristic, complemented
302C2×#327C2×C6711yescharacteristic, complemented
303C2×#323C2×C6711yescharacteristic, complemented
304C67C2×#161811yescharacteristic, direct product factor, Sylow
305#538—22no—
306#538—22no—
307#538—22no—
308#538C22×C6714yes—
309C2×#126—21no—
310C2×#124—21no—
311C2×#123—21no—
312C2×C22⋊Q8C22×C6711yescharacteristic
313C2×#121—21no—
314C2×#121C22×C6711yescharacteristic
315C2×#120—21no—
316C2×C4×D4—21no—
317C22×C4⋊C4—21no—
318C22×C4⋊C4C22×C6711yescharacteristic
319C22×C22⋊C4—21no—
320C22×C22⋊C4C22×C6711yescharacteristic
321#484—22no—
322#484C22×C6714yescomplemented
323#481—22no—
324#481C22×C6714yescomplemented
325C2×#107—21no—
326C2×#106—21no—
327C2×#103—21no—
328C2×#102—21no—
329C2×C22⋊C8C22×C6711yescharacteristic
330C2×C22⋊C8C22×C6711yescharacteristic
331#399—22no—
332#399C22×C6714yescomplemented
333C2×#95C22×C6711yescharacteristic
334#328C22×C6714yescomplemented
335#327C22×C6714yescomplemented
336#323C22×C6714yescomplemented
337C25D4×C6711yescharacteristic
338C22×Q8D4×C6711yescharacteristic
339C22×D4—21no—
340C22×D4—21no—
341C22×D4—21no—
342C22×D4D4×C6711yescharacteristic
343C23×C4—21no—
344C23×C4—21no—
345C23×C4—21no—
346C23×C4D4×C6711yescharacteristic
347C23×C4C23×C6711yescharacteristic
348C22×C8—21no—
349C22×C8—21no—
350#126—24no—
351#126—24no—
352#124—24no—
353#124—24no—
354#123—24no—
355#123—24no—
356C22⋊Q8—21no—
357C22⋊Q8—21no—
358C22⋊Q8D4×C6712yes—
359C22⋊Q8C23×C6712yes—
360#121—24no—
361#121—24no—
362#121—21no—
363#121—21no—
364#121D4×C6712yes—
365#121C23×C6712yes—
366#120—24no—
367#120—24no—
368C4×D4—24no—
369C4×D4—24no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—44no—
379C2×C4⋊C4D4×C6711yescharacteristic
380C2×C4⋊C4D4×C6711yescharacteristic
381C2×C4⋊C4C23×C6712yes—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—44no—
393C2×C22⋊C4D4×C6712yes—
394C2×C22⋊C4C23×C6712yes—
395C2×C42—21no—
396#107—24no—
397#106—24no—
398#103—24no—
399#102—24no—
400C22⋊C8—21no—
401C22⋊C8—21no—
402C22⋊C8C23×C6712yes—
403C22⋊C8C23×C6712yes—
404#95—21no—
405#95C23×C6712yes—
406C24—21no—
407C24—21no—
408C24—22no—
409C24—21no—
410C24—22no—
411C24—42no—
412C24—42no—
413C24C2×D4×C6711yescharacteristic
414C24C2×D4×C6712yes—
415C2×Q8—21no—
416C2×Q8—42no—
417C2×Q8C2×D4×C6712yes—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—21no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4C2×D4×C6712yes—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—82no—
447C22×C4C2×D4×C6711yescharacteristic
448C22×C4C2×D4×C6711yescharacteristic
449C22×C4C2×D4×C6712yes—
450C22×C4C2×D4×C6711yescharacteristic
451C22×C4C2×D4×C6711yescharacteristic
452C22×C4C2×D4×C6711yescharacteristic
453C22×C4C24×C6711yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—42no—
457C2×C8—42no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—48no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4C2×D4×C6712yes—
469C4⋊C4C2×D4×C6712yes—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—48no—
482C42—24no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—42no—
490C23—42no—
491C23—48no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—81no—
500C23—81no—
501C23—84no—
502C23#120×C6711yescharacteristic
503C23#120×C6711yescharacteristic
504C23#135×C6712yes—
505C23#137×C6712yes—
506C23C22×D4×C6711yescharacteristic
507Q8—42no—
508Q8—81no—
509D4—42no—
510D4—44no—
511D4—42no—
512D4—42no—
513D4—81no—
514D4—84no—
515D4—84no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—81no—
539C2×C4—84no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4#120×C6711yescharacteristic
543C2×C4#120×C6711yescharacteristic
544C2×C4C22×D4×C6711yescharacteristic
545C2×C4C22×D4×C6711yescharacteristic
546C8—42no—
547C8—42no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—24no—
553C22—24no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—82no—
562C22—84no—
563C22—84no—
564C22—84no—
565C22#452×C6712yes—
566C22#452×C6712yes—
567C22#453×C6712yes—
568C22#458×C6712yes—
569C22C2×#120×C6711yescharacteristic
570C22C2×#135×C6711yescharacteristic
571C22C2×#137×C6711yescharacteristic
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—44no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—82no—
582C2—21no—
583C2—21no—
584C2—22no—
585C2—82no—
586C2—82no—
587C2C67×#161814yesdirect product factor
588C2C2×#452×C6711yescharacteristic
589C2C2×#453×C6711yescharacteristic
590C2C2×#458×C6711yescharacteristic
subgroupG/H
center, socleC23×C67#120
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC2×#538×C67C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#484×C67C2semidirect
3. C2×#481×C67C2semidirect
4. C2×#399×C67C2semidirect
5. C67×#1618C2direct (N × H)
6. C2×#328×C67C2semidirect
7. C2×#327×C67C2semidirect
8. C2×#323×C67C2semidirect
26. #484×C67C22semidirect
28. #481×C67C22semidirect
36. #399×C67C22semidirect
38. #328×C67C22semidirect
39. #327×C67C22semidirect
40. #323×C67C22semidirect
286. C2×#1618C67direct (N × H)
292. C2×C67#1618direct (N × H)
297. C2×#484C2×C67semidirect
298. C2×#481C2×C67semidirect
299. C2×#399C2×C67semidirect
300. #1618C2×C67direct (N × H)
301. C2×#328C2×C67semidirect
302. C2×#327C2×C67semidirect
303. C2×#323C2×C67semidirect
304. C67C2×#1618direct (N × H)
322. #484C22×C67semidirect
324. #481C22×C67semidirect
332. #399C22×C67semidirect
334. #328C22×C67semidirect
335. #327C22×C67semidirect
336. #323C22×C67semidirect
587. C2C67×#1618direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#538×C672
C2×#484×C672
C2×#481×C672
C2×#399×C672
C67×#16182
C2×#328×C672
C2×#327×C672
C2×#323×C672
C2×#161867