Finite Group Databasebeta

← C2×C67×#1867 #30700138

Subgroups — C2×C67×#1867 #30700138

Order 17152.

1994 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#518×C67C211yesmaximal, characteristic
2C22×#99×C67C211yesmaximal, characteristic
3C2×#386×C67C211yesmaximal, characteristic
4C2×#385×C67C212yesmaximal, complemented
5C67×#1867C218yesmaximal, direct product factor
6C2×#342×C67C212yesmaximal, complemented
7C23×D4×C67C2211yescharacteristic
8C24×C4×C67C2211yescharacteristic
9#518×C67—22no—
10#518×C67—24no—
11#518×C67C2214yes—
12C2×C4×D4×C67—22no—
13C2×C4×D4×C67—22no—
14C2×#117×C67—21no—
15C22×C4⋊C4×C67—22no—
16C22×C22⋊C4×C67—21no—
17C22×C22⋊C4×C67—21no—
18C22×C22⋊C4×C67C2211yescharacteristic
19C22×C22⋊C4×C67C2212yes—
20C22×C22⋊C4×C67C2212yes—
21C2×#99×C67C418yescomplemented
22C2×#99×C67C2214yes—
23#386×C67—22no—
24#386×C67C2214yes—
25#385×C67—24no—
26#385×C67C2218yescomplemented
27C2×#95×C67—22no—
28C2×#95×C67—21no—
29#342×C67C2218yescomplemented
30C25×C67D411yescharacteristic
31C25×C67C2311yescharacteristic
32C22×D4×C67—22no—
33C22×D4×C67—22no—
34C22×D4×C67—24no—
35C22×D4×C67C2×C414yes—
36C22×D4×C67D412yes—
37C22×D4×C67Q814yes—
38C22×D4×C67C2312yes—
39C23×C4×C67—21no—
40C23×C4×C67—22no—
41C23×C4×C67—22no—
42C23×C4×C67—21no—
43C23×C4×C67—21no—
44C23×C4×C67—21no—
45C23×C4×C67—22no—
46C23×C4×C67—21no—
47C23×C4×C67—44no—
48C23×C4×C67D411yescharacteristic
49C23×C4×C67D412yes—
50C23×C4×C67D412yes—
51C23×C4×C67C2312yes—
52C4×D4×C67—28no—
53C4×D4×C67—28no—
54C4×D4×C67—28no—
55C4×D4×C67—28no—
56#117×C67—24no—
57#117×C67—24no—
58C2×C4⋊C4×C67—21no—
59C2×C4⋊C4×C67—21no—
60C2×C4⋊C4×C67—28no—
61C2×C4⋊C4×C67—48no—
62C2×C22⋊C4×C67—22no—
63C2×C22⋊C4×C67—24no—
64C2×C22⋊C4×C67—22no—
65C2×C22⋊C4×C67—24no—
66C2×C22⋊C4×C67—22no—
67C2×C22⋊C4×C67—22no—
68C2×C22⋊C4×C67—22no—
69C2×C22⋊C4×C67—22no—
70C2×C22⋊C4×C67—24no—
71C2×C22⋊C4×C67—24no—
72C2×C22⋊C4×C67—24no—
73C2×C22⋊C4×C67—21no—
74C2×C22⋊C4×C67—44no—
75C2×C22⋊C4×C67C2×C418yescomplemented
76C2×C22⋊C4×C67D414yes—
77C2×C22⋊C4×C67C2312yes—
78C2×C22⋊C4×C67C2314yes—
79C2×C22⋊C4×C67C2314yes—
80C2×C42×C67—22no—
81#99×C67C2×C4116yescomplemented
82#95×C67—28no—
83#95×C67—24no—
84C24×C67—21no—
85C24×C67—21no—
86C24×C67—22no—
87C24×C67—42no—
88C24×C67—42no—
89C24×C67—42no—
90C24×C67—42no—
91C24×C67C4⋊C414yescomplemented
92C24×C67C22×C412yes—
93C24×C67C2×D411yescharacteristic
94C24×C67C2×D412yes—
95C24×C67C2×D412yes—
96C24×C67C2×D411yescharacteristic
97C24×C67C2×D412yes—
98C24×C67C2×Q812yes—
99C24×C67#4114yes—
100C24×C67C2411yescharacteristic
101C2×D4×C67—28no—
102C2×D4×C67—28no—
103C2×D4×C67—24no—
104C2×D4×C67—28no—
105C2×D4×C67—28no—
106C2×D4×C67—24no—
107C2×D4×C67—24no—
108C2×D4×C67—24no—
109C2×D4×C67C4⋊C418yescomplemented
110C2×D4×C67C22×C414yes—
111C2×D4×C67C2×Q814yes—
112C22×C4×C67—21no—
113C22×C4×C67—21no—
114C22×C4×C67—22no—
115C22×C4×C67—24no—
116C22×C4×C67—28no—
117C22×C4×C67—28no—
118C22×C4×C67—22no—
119C22×C4×C67—22no—
120C22×C4×C67—22no—
121C22×C4×C67—24no—
122C22×C4×C67—24no—
123C22×C4×C67—24no—
124C22×C4×C67—24no—
125C22×C4×C67—22no—
126C22×C4×C67—22no—
127C22×C4×C67—22no—
128C22×C4×C67—44no—
129C22×C4×C67—44no—
130C22×C4×C67—48no—
131C22×C4×C67—44no—
132C22×C4×C67—44no—
133C22×C4×C67—44no—
134C22×C4×C67—42no—
135C22×C4×C67—41no—
136C22×C4×C67—42no—
137C22×C4×C67—41no—
138C22×C4×C67—44no—
139C22×C4×C67—44no—
140C22×C4×C67—42no—
141C22×C4×C67—42no—
142C22×C4×C67—42no—
143C22×C4×C67—82no—
144C22×C4×C67—82no—
145C22×C4×C67C4⋊C414yescomplemented
146C22×C4×C67C2×D412yes—
147C22×C4×C67C2×D414yes—
148C22×C4×C67C2×D414yes—
149C4⋊C4×C67—24no—
150C4⋊C4×C67—24no—
151C4⋊C4×C67—416no—
152C22⋊C4×C67—28no—
153C22⋊C4×C67—28no—
154C22⋊C4×C67—24no—
155C22⋊C4×C67—24no—
156C22⋊C4×C67—24no—
157C22⋊C4×C67—24no—
158C22⋊C4×C67—24no—
159C22⋊C4×C67—24no—
160C22⋊C4×C67—48no—
161C22⋊C4×C67—42no—
162C22⋊C4×C67C22×C418yescomplemented
163C42×C67—28no—
164C23×C67—24no—
165C23×C67—22no—
166C23×C67—24no—
167C23×C67—22no—
168C23×C67—22no—
169C23×C67—24no—
170C23×C67—22no—
171C23×C67—24no—
172C23×C67—42no—
173C23×C67—41no—
174C23×C67—41no—
175C23×C67—44no—
176C23×C67—42no—
177C23×C67—41no—
178C23×C67—44no—
179C23×C67—48no—
180C23×C67—41no—
181C23×C67—42no—
182C23×C67—48no—
183C23×C67—42no—
184C23×C67—42no—
185C23×C67—42no—
186C23×C67—84no—
187C23×C67—81no—
188C23×C67—81no—
189C23×C67C2×C4⋊C412yes—
190C23×C67C2×C4⋊C412yes—
191C23×C67C2×C4⋊C414yescomplemented
192C23×C67C4×D414yes—
193C23×C67C4×D414yes—
194C23×C67#12011yescharacteristic
195C23×C67#12012yes—
196C23×C67#12011yescharacteristic
197C23×C67C22⋊Q814yes—
198C23×C67C22⋊Q814yes—
199C23×C67#12312yes—
200C23×C67#12312yes—
201C23×C67C23×C411yescharacteristic
202C23×C67C22×D411yescharacteristic
203C23×C67C22×D412yes—
204C23×C67C22×Q811yescharacteristic
205C23×C67C2×#4112yes—
206D4×C67—28no—
207D4×C67—28no—
208D4×C67—216no—
209C2×C4×C67—22no—
210C2×C4×C67—22no—
211C2×C4×C67—24no—
212C2×C4×C67—24no—
213C2×C4×C67—24no—
214C2×C4×C67—28no—
215C2×C4×C67—28no—
216C2×C4×C67—24no—
217C2×C4×C67—24no—
218C2×C4×C67—44no—
219C2×C4×C67—416no—
220C2×C4×C67—44no—
221C2×C4×C67—44no—
222C2×C4×C67—42no—
223C2×C4×C67—48no—
224C2×C4×C67—44no—
225C2×C4×C67—42no—
226C2×C4×C67—44no—
227C2×C4×C67—44no—
228C2×C4×C67—44no—
229C2×C4×C67—44no—
230C2×C4×C67—44no—
231C2×C4×C67—42no—
232C2×C4×C67—42no—
233C2×C4×C67—42no—
234C2×C4×C67—82no—
235C2×C4×C67—82no—
236C2×C4×C67—84no—
237C2×C4×C67—84no—
238C2×C4×C67—84no—
239C2×C4×C67C2×C4⋊C414yescomplemented
240C22×C67—28no—
241C22×C67—24no—
242C22×C67—24no—
243C22×C67—28no—
244C22×C67—22no—
245C22×C67—21no—
246C22×C67—22no—
247C22×C67—21no—
248C22×C67—22no—
249C22×C67—21no—
250C22×C67—22no—
251C22×C67—21no—
252C22×C67—24no—
253C22×C67—42no—
254C22×C67—44no—
255C22×C67—44no—
256C22×C67—42no—
257C22×C67—41no—
258C22×C67—41no—
259C22×C67—42no—
260C22×C67—44no—
261C22×C67—48no—
262C22×C67—42no—
263C22×C67—42no—
264C22×C67—82no—
265C22×C67—84no—
266C22×C67#38512yes—
267C22×C67#38514yes—
268C22×C67#38512yes—
269C22×C67B(4,𝔽2)12yes—
270C22×C67#45812yes—
271C22×C67C22×C4⋊C411yescharacteristic
272C22×C67C2×C4×D412yes—
273C22×C67C2×#12011yescharacteristic
274C22×C67C2×C22⋊Q812yes—
275C22×C67C2×#12311yescharacteristic
276C4×C67—24no—
277C4×C67—48no—
278C4×C67—44no—
279C4×C67—44no—
280C4×C67—44no—
281C4×C67—82no—
282C2×#1867C6711yesmaximal, characteristic, direct product factor, Sylow
283C2×C67—22no—
284C2×C67—21no—
285C2×C67—21no—
286C2×C67—22no—
287C2×C67—22no—
288C2×C67—24no—
289C2×C67—44no—
290C2×C67#186714yesdirect product factor
291C2×C67C2×#38511yescharacteristic
292C2×C67C2×B(4,𝔽2)11yescharacteristic
293C2×C67C2×#45811yescharacteristic
294C2×#518C2×C6711yescharacteristic
295C22×#99C2×C6711yescharacteristic
296C2×#386C2×C6711yescharacteristic
297C2×#385C2×C6712yescomplemented
298#1867C2×C6718yesdirect product factor
299C2×#342C2×C6712yescomplemented
300C67C2×#186711yescharacteristic, direct product factor, Sylow
301C23×D4C22×C6711yescharacteristic
302C24×C4C22×C6711yescharacteristic
303#518—22no—
304#518—24no—
305#518C22×C6714yes—
306C2×C4×D4—22no—
307C2×C4×D4—22no—
308C2×#117—21no—
309C22×C4⋊C4—22no—
310C22×C22⋊C4—21no—
311C22×C22⋊C4—21no—
312C22×C22⋊C4C22×C6711yescharacteristic
313C22×C22⋊C4C22×C6712yes—
314C22×C22⋊C4C22×C6712yes—
315C2×#99C4×C6718yescomplemented
316C2×#99C22×C6714yes—
317#386—22no—
318#386C22×C6714yes—
319#385—24no—
320#385C22×C6718yescomplemented
321C2×#95—22no—
322C2×#95—21no—
323#342C22×C6718yescomplemented
324C25D4×C6711yescharacteristic
325C25C23×C6711yescharacteristic
326C22×D4—22no—
327C22×D4—22no—
328C22×D4—24no—
329C22×D4C2×C4×C6714yes—
330C22×D4D4×C6712yes—
331C22×D4Q8×C6714yes—
332C22×D4C23×C6712yes—
333C23×C4—21no—
334C23×C4—22no—
335C23×C4—22no—
336C23×C4—21no—
337C23×C4—21no—
338C23×C4—21no—
339C23×C4—22no—
340C23×C4—21no—
341C23×C4—44no—
342C23×C4D4×C6711yescharacteristic
343C23×C4D4×C6712yes—
344C23×C4D4×C6712yes—
345C23×C4C23×C6712yes—
346C4×D4—28no—
347C4×D4—28no—
348C4×D4—28no—
349C4×D4—28no—
350#117—24no—
351#117—24no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—28no—
355C2×C4⋊C4—48no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—44no—
369C2×C22⋊C4C2×C4×C6718yescomplemented
370C2×C22⋊C4D4×C6714yes—
371C2×C22⋊C4C23×C6712yes—
372C2×C22⋊C4C23×C6714yes—
373C2×C22⋊C4C23×C6714yes—
374C2×C42—22no—
375#99C2×C4×C67116yescomplemented
376#95—28no—
377#95—24no—
378C24—21no—
379C24—21no—
380C24—22no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24C4⋊C4×C6714yescomplemented
386C24C22×C4×C6712yes—
387C24C2×D4×C6711yescharacteristic
388C24C2×D4×C6712yes—
389C24C2×D4×C6712yes—
390C24C2×D4×C6711yescharacteristic
391C24C2×D4×C6712yes—
392C24C2×Q8×C6712yes—
393C24#41×C6714yes—
394C24C24×C6711yescharacteristic
395C2×D4—28no—
396C2×D4—28no—
397C2×D4—24no—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4C4⋊C4×C6718yescomplemented
404C2×D4C22×C4×C6714yes—
405C2×D4C2×Q8×C6714yes—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—41no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4C4⋊C4×C6714yescomplemented
440C22×C4C2×D4×C6712yes—
441C22×C4C2×D4×C6714yes—
442C22×C4C2×D4×C6714yes—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—416no—
446C22⋊C4—28no—
447C22⋊C4—28no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—48no—
455C22⋊C4—42no—
456C22⋊C4C22×C4×C6718yescomplemented
457C42—28no—
458C23—24no—
459C23—22no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—22no—
465C23—24no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—44no—
470C23—42no—
471C23—41no—
472C23—44no—
473C23—48no—
474C23—41no—
475C23—42no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—81no—
481C23—81no—
482C23—84no—
483C23C2×C4⋊C4×C6712yes—
484C23C2×C4⋊C4×C6712yes—
485C23C2×C4⋊C4×C6714yescomplemented
486C23C4×D4×C6714yes—
487C23C4×D4×C6714yes—
488C23#120×C6711yescharacteristic
489C23#120×C6712yes—
490C23#120×C6711yescharacteristic
491C23C22⋊Q8×C6714yes—
492C23C22⋊Q8×C6714yes—
493C23#123×C6712yes—
494C23#123×C6712yes—
495C23C23×C4×C6711yescharacteristic
496C23C22×D4×C6711yescharacteristic
497C23C22×D4×C6712yes—
498C23C22×Q8×C6711yescharacteristic
499C23C2×#41×C6712yes—
500D4—28no—
501D4—28no—
502D4—216no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—48no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—416no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—84no—
531C2×C4—84no—
532C2×C4—84no—
533C2×C4C2×C4⋊C4×C6714yescomplemented
534C22—22no—
535C22—21no—
536C22—22no—
537C22—21no—
538C22—22no—
539C22—21no—
540C22—22no—
541C22—21no—
542C22—28no—
543C22—24no—
544C22—24no—
545C22—28no—
546C22—24no—
547C22—42no—
548C22—44no—
549C22—44no—
550C22—42no—
551C22—41no—
552C22—41no—
553C22—42no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—48no—
558C22—82no—
559C22—84no—
560C22#385×C6712yes—
561C22#385×C6714yes—
562C22#385×C6712yes—
563C22B(4,𝔽2)×C6712yes—
564C22#458×C6712yes—
565C22C22×C4⋊C4×C6711yescharacteristic
566C22C2×C4×D4×C6712yes—
567C22C2×#120×C6711yescharacteristic
568C22C2×C22⋊Q8×C6712yes—
569C22C2×#123×C6711yescharacteristic
570C4—24no—
571C4—48no—
572C4—44no—
573C4—44no—
574C4—44no—
575C4—82no—
576C2—22no—
577C2—21no—
578C2—21no—
579C2—22no—
580C2—22no—
581C2—24no—
582C2—44no—
583C2C67×#186714yesdirect product factor
584C2C2×#385×C6711yescharacteristic
585C2C2×B(4,𝔽2)×C6711yescharacteristic
586C2C2×#458×C6711yescharacteristic
subgroupG/H
center, socleC23×C67#120
commutatorC23C23×C4×C67
Frattini, cosocleC24C24×C67
prime radicalC23×D4×C67C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C67 ◁ C25×C67 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×#385×C67C2semidirect
5. C67×#1867C2direct (N × H)
6. C2×#342×C67C2semidirect
21. C2×#99×C67C4semidirect
26. #385×C67C22semidirect
29. #342×C67C22semidirect
75. C2×C22⋊C4×C67C2×C4semidirect
81. #99×C67C2×C4semidirect
91. C24×C67C4⋊C4semidirect
109. C2×D4×C67C4⋊C4semidirect
145. C22×C4×C67C4⋊C4semidirect
162. C22⋊C4×C67C22×C4semidirect
191. C23×C67C2×C4⋊C4semidirect
239. C2×C4×C67C2×C4⋊C4semidirect
282. C2×#1867C67direct (N × H)
290. C2×C67#1867direct (N × H)
297. C2×#385C2×C67semidirect
298. #1867C2×C67direct (N × H)
299. C2×#342C2×C67semidirect
300. C67C2×#1867direct (N × H)
315. C2×#99C4×C67semidirect
320. #385C22×C67semidirect
323. #342C22×C67semidirect
369. C2×C22⋊C4C2×C4×C67semidirect
375. #99C2×C4×C67semidirect
385. C24C4⋊C4×C67semidirect
403. C2×D4C4⋊C4×C67semidirect
439. C22×C4C4⋊C4×C67semidirect
456. C22⋊C4C22×C4×C67semidirect
485. C23C2×C4⋊C4×C67semidirect
533. C2×C4C2×C4⋊C4×C67semidirect
583. C2C67×#1867direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#518×C672
C22×#99×C672
C2×#386×C672
C2×#385×C672
C67×#18672
C2×#342×C672
C2×#186767