Finite Group Databasebeta

← C2×#22610245 #33087181

Subgroups — C2×#22610245 #33087181

Order 18688.

2614 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556562C211yesmaximal, characteristic
2C22×C4⋊C4×D73C211yesmaximal, characteristic, complemented
3C22×#14556512C211yesmaximal, characteristic, complemented
4C22×#14556503C211yesmaximal, characteristic, complemented
5#22610245C218yesmaximal, direct product factor
6C2×#17988350C211yesmaximal, characteristic, complemented
7C2×#17988349C211yesmaximal, characteristic, complemented
8C22×#102×C73C211yesmaximal, characteristic, complemented
9C23×C22⋊D73—21no—
10C23×C4×D73—21no—
11C23×C4×D73C2211yescharacteristic
12C23×#10596831C2211yescharacteristic
13C2×#14556562—24no—
14C2×#14556562—22no—
15C2×#14556562C418yescomplemented
16C2×#14556562C2214yes—
17C22×C4⋊Dic73C2211yescharacteristic
18C22×C4⋊Dic73C2211yescharacteristic
19C22×#11924007—21no—
20C2×C4⋊C4×D73—22no—
21C2×C4⋊C4×D73—24no—
22C2×C4⋊C4×D73—24no—
23C2×C4⋊C4×D73C2214yescomplemented
24C22×C22⋊Dic73—21no—
25C2×#14556512—24no—
26C2×#14556512C2214yescomplemented
27C2×#14556503—24no—
28C2×#14556503C2214yescomplemented
29#17988350C2214yescomplemented
30#17988349C2214yescomplemented
31C23×D4×C73C2211yescharacteristic
32C23×C8×C73C2211yescharacteristic
33C22×C4⋊C4×C73C2211yescharacteristic
34C2×#102×C73—24no—
35C2×#102×C73C2214yescomplemented
36C24×D73D411yescharacteristic
37C22×C22⋊D73—28no—
38C22×C22⋊D73—24no—
39C22×C22⋊D73—44no—
40C22×C22⋊D73—44no—
41C23×Dic73—21no—
42C23×Dic73—21no—
43C23×Dic73D411yescharacteristic
44C22×C4×D73—28no—
45C22×C4×D73—24no—
46C22×C4×D73—22no—
47C22×C4×D73—22no—
48C22×C4×D73—22no—
49C22×C4×D73—22no—
50C22×C4×D73—44no—
51C22×C4×D73—44no—
52C22×C4×D73C2×C414yes—
53C22×C4×D73D412yes—
54C22×C4×D73D414yes—
55C22×C4×D73C2312yes—
56C22×#10596831—22no—
57C22×#10596831—22no—
58C22×#10596831—22no—
59C22×#10596831C2312yes—
60#14556562—216no—
61#14556562—28no—
62#14556562C2×C4116yescomplemented
63C2×C4⋊Dic73—21no—
64C2×C4⋊Dic73—22no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73—22no—
67C2×C4⋊Dic73C2×C414yescomplemented
68C2×C4⋊Dic73D414yescomplemented
69C2×C4⋊Dic73C2312yes—
70C2×C4⋊Dic73C2312yes—
71C2×#11924007—28no—
72C2×#11924007—24no—
73C4⋊C4×D73—216no—
74C4⋊C4×D73—28no—
75C4⋊C4×D73—28no—
76C2×C22⋊Dic73—28no—
77C2×C22⋊Dic73—24no—
78#14556512—28no—
79#14556503—28no—
80C25×C73—21no—
81C22×D4×C73—22no—
82C22×D4×C73—22no—
83C22×D4×C73—21no—
84C22×D4×C73—22no—
85C22×D4×C73—22no—
86C22×D4×C73—22no—
87C22×D4×C73C2×C414yescomplemented
88C22×D4×C73C2312yes—
89C23×C4×C73—21no—
90C23×C4×C73C2311yescharacteristic
91C22×C8×C73—22no—
92C22×C8×C73—22no—
93C22×C8×C73—22no—
94C22×C8×C73C2312yes—
95C2×C4⋊C4×C73—21no—
96C2×C4⋊C4×C73—22no—
97C2×C4⋊C4×C73D414yescomplemented
98C2×C4⋊C4×C73C2312yes—
99#102×C73—28no—
100C23×D73—22no—
101C23×D73—22no—
102C23×D73—22no—
103C23×D73—22no—
104C23×D73C22⋊C414yes—
105C23×D73D814yes—
106C23×D73SD1614yes—
107C23×D73C2×D412yes—
108C2×C22⋊D73—216no—
109C2×C22⋊D73—432no—
110C2×C22⋊D73—48no—
111C2×C22⋊D73—48no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—24no—
115C22×Dic73—24no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—42no—
119C22×Dic73—42no—
120C22×Dic73—42no—
121C22×Dic73—42no—
122C22×Dic73C22⋊C414yes—
123C22×Dic73C2×D412yes—
124C2×C4×D73—28no—
125C2×C4×D73—28no—
126C2×C4×D73—24no—
127C2×C4×D73—28no—
128C2×C4×D73—28no—
129C2×C4×D73—24no—
130C2×C4×D73—22no—
131C2×C4×D73—22no—
132C2×C4×D73—22no—
133C2×C4×D73—22no—
134C2×C4×D73—216no—
135C2×C4×D73—48no—
136C2×C4×D73—432no—
137C2×C4×D73—48no—
138C2×C4×D73C22⋊C418yes—
139C2×C4×D73C22×C414yes—
140C2×C4×D73C2×D414yes—
141C2×#10596831—22no—
142C2×#10596831—22no—
143C2×#10596831—22no—
144C2×#10596831—44no—
145C4⋊Dic73—24no—
146C4⋊Dic73—24no—
147C4⋊Dic73—22no—
148C4⋊Dic73—22no—
149C4⋊Dic73C22×C414yescomplemented
150C4⋊Dic73C2×D414yescomplemented
151#11924007—216no—
152C22⋊Dic73—216no—
153C24×C73—24no—
154C24×C73—22no—
155C24×C73—42no—
156C24×C73—42no—
157C24×C73—42no—
158C24×C73—42no—
159C24×C73—42no—
160C24×C73—42no—
161C24×C73C2×D411yescharacteristic
162C2×D4×C73—24no—
163C2×D4×C73—28no—
164C2×D4×C73—28no—
165C2×D4×C73—22no—
166C2×D4×C73—22no—
167C2×D4×C73—22no—
168C2×D4×C73—22no—
169C2×D4×C73—22no—
170C2×D4×C73—42no—
171C2×D4×C73—44no—
172C2×D4×C73—44no—
173C2×D4×C73—42no—
174C2×D4×C73C22×C414yescomplemented
175C22×C4×C73—24no—
176C22×C4×C73—22no—
177C22×C4×C73—42no—
178C22×C4×C73—42no—
179C22×C4×C73C22×C412yes—
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D412yes—
182C22×C4×C73C2×D412yes—
183C22×C4×C73C2×D412yes—
184C22×C4×C73#4112yes—
185C22×C4×C73#4112yes—
186C22×C4×C73C2411yescharacteristic
187C2×C8×C73—22no—
188C2×C8×C73—22no—
189C2×C8×C73—22no—
190C2×C8×C73—44no—
191C4⋊C4×C73—24no—
192C4⋊C4×C73—22no—
193C4⋊C4×C73C2×D414yescomplemented
194C23×C73—24no—
195C23×C73—42no—
196C23×C73—42no—
197C23×C73—42no—
198C23×C73—42no—
199C23×C73—42no—
200C23×C73—42no—
201C23×C73—48no—
202C23×C73—48no—
203C23×C73—48no—
204C23×C73—84no—
205C23×C73—84no—
206C23×C73—84no—
207C23×C73—84no—
208C23×C73C2×C22⋊C412yes—
209C23×C73#12012yes—
210C23×C73#12312yes—
211C23×C73C2×D812yes—
212C23×C73C2×SD1612yes—
213C23×C73AGL(1,ℤ/8)12yes—
214C23×C73#13712yes—
215C23×C73C22×D411yescharacteristic
216C22×D73—22no—
217C22×D73—22no—
218C22×D73—22no—
219C22×D73—22no—
220C22×D73—28no—
221C22×D73—28no—
222C22×D73—28no—
223C22×D73—28no—
224C22×D73—28no—
225C22×D73—28no—
226C22×D73#102116yescomplemented
227C22×D73C2×C22⋊C414yes—
228C22×D73C2×D814yes—
229C22×D73C2×SD1614yes—
230D4×C73—24no—
231D4×C73—24no—
232D4×C73—42no—
233D4×C73—42no—
234D4×C73—48no—
235C2×C4×C73—24no—
236C2×C4×C73—42no—
237C2×C4×C73—42no—
238C2×C4×C73—48no—
239C2×C4×C73C2×C22⋊C412yes—
240C2×C4×C73C4×D414yes—
241C2×C4×C73C4×D414yes—
242C2×C4×C73#12012yes—
243C2×C4×C73#12114yes—
244C2×C4×C73#12114yes—
245C2×C4×C73#12312yes—
246C2×C4×C73C23×C411yescharacteristic
247C2×C4×C73C22×D411yescharacteristic
248C2×C4×C73C22×D411yescharacteristic
249C2×C4×C73C22×D411yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73C2×#4111yescharacteristic
252C22⋊D73—432no—
253C2×Dic73—22no—
254C2×Dic73—22no—
255C2×Dic73—28no—
256C2×Dic73—24no—
257C2×Dic73—24no—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—48no—
263C2×Dic73—48no—
264C2×Dic73C2×C22⋊C414yes—
265C4×D73—24no—
266C4×D73—24no—
267C4×D73—24no—
268C4×D73—24no—
269C4×D73—216no—
270C4×D73—432no—
271C8×C73—42no—
272#10596831—42no—
273C22×C73—44no—
274C22×C73—44no—
275C22×C73—44no—
276C22×C73—82no—
277C22×C73—82no—
278C22×C73—816no—
279C22×C73—82no—
280C22×C73—82no—
281C22×C73#38614yes—
282C22×C73C2×#10214yes—
283C22×C73#41714yes—
284C22×C73#44714yes—
285C22×C73#44814yes—
286C22×C73#48014yes—
287C22×C73#48114yes—
288C22×C73C22×C22⋊C411yescharacteristic
289C22×C73C2×#12011yescharacteristic
290C22×C73C2×#12311yescharacteristic
291C22×C73C22×D811yescharacteristic
292C22×C73C22×SD1611yescharacteristic
293C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C73C2×#13711yescharacteristic
295C2×D73—24no—
296C2×D73—24no—
297C2×D73—24no—
298C2×D73—232no—
299C2×D73—24no—
300C2×D73—24no—
301C2×D73—24no—
302C2×D73C2×#10218yescomplemented
303C4×C73—44no—
304C4×C73#38614yes—
305C4×C73C2×C4×D411yescharacteristic
306C4×C73C2×C4×D411yescharacteristic
307C4×C73C2×#12111yescharacteristic
308C4×C73C2×#12111yescharacteristic
309Dic73—24no—
310Dic73—44no—
311Dic73—44no—
312C2×#1874—731nomaximal, Sylow
313C2×C73—84no—
314C2×C73#187418yescomplemented
315C2×C73C2×#38611yescharacteristic
316C2×C73C22×#10211yescharacteristic
317C2×C73C2×#41711yescharacteristic
318C2×C73C2×#44711yescharacteristic
319C2×C73C2×#44811yescharacteristic
320C2×C73C2×#48011yescharacteristic
321C2×C73C2×#48111yescharacteristic
322D73—28no—
323C22×#121—731no—
324C23×C4⋊C4—731no—
325C22×#102—731no—
326C22×#102D7311yescharacteristic, complemented
327C22×C22⋊C8—731no—
328#1874—738no—
329C2×#340—731no—
330C2×#340—731no—
331C73C2×#187411yescharacteristic, complemented, Sylow
332C23×D4—1461no—
333C23×D4C2×D7311yescharacteristic
334C24×C4—731no—
335C24×C4—1461no—
336C23×C8—731no—
337C23×C8C2×D7311yescharacteristic
338C2×#121—738no—
339C2×#121—734no—
340C2×#121—1462no—
341C2×#121—1464no—
342C22×C4⋊C4—731no—
343C22×C4⋊C4—731no—
344C22×C4⋊C4—734no—
345C22×C4⋊C4—1464no—
346C22×C4⋊C4—1462no—
347C22×C4⋊C4—1461no—
348C22×C4⋊C4—1464no—
349C22×C4⋊C4C2×D7311yescharacteristic
350C22×C22⋊C4—1461no—
351C2×#102—24no—
352C2×#102—734no—
353C2×#102—1464no—
354C2×#102C2×D7314yescomplemented
355C2×C22⋊C8—734no—
356C2×C22⋊C8—1464no—
357#340—734no—
358#340—734no—
359C25—21no—
360C25—731no—
361C22×D4—22no—
362C22×D4—21no—
363C22×D4—22no—
364C22×D4—22no—
365C22×D4—22no—
366C22×D4—22no—
367C22×D4—2924no—
368C22×D4—2924no—
369C22×D4—1468no—
370C22×D4—1464no—
371C22×D4C4×D7314yescomplemented
372C22×D4C22×D7312yes—
373C23×C4—21no—
374C23×C4—2924no—
375C23×C4—2924no—
376C23×C4—731no—
377C23×C4—734no—
378C23×C4—732no—
379C23×C4—734no—
380C23×C4—732no—
381C23×C4—1464no—
382C23×C4—1468no—
383C23×C4—1461no—
384C23×C4—1461no—
385C23×C4—1462no—
386C23×C4—1462no—
387C23×C4—1462no—
388C23×C4—1462no—
389C23×C4C22×D7311yescharacteristic
390C22×C8—22no—
391C22×C8—22no—
392C22×C8—22no—
393C22×C8—732no—
394C22×C8—1462no—
395C22×C8—1462no—
396C22×C8—1462no—
397C22×C8C22×D7312yes—
398#121—7316no—
399#121—14616no—
400#121—1468no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—734no—
404C2×C4⋊C4—734no—
405C2×C4⋊C4—732no—
406C2×C4⋊C4—732no—
407C2×C4⋊C4—1468no—
408C2×C4⋊C4—14616no—
409C2×C4⋊C4—1464no—
410C2×C4⋊C4—1468no—
411C2×C4⋊C4—1461no—
412C2×C4⋊C4—1461no—
413C2×C4⋊C4—1462no—
414C2×C4⋊C4—1462no—
415C2×C4⋊C4—1468no—
416C2×C4⋊C4C22⋊D7314yescomplemented
417C2×C4⋊C4C22×D7312yes—
418C2×C22⋊C4—1468no—
419C2×C22⋊C4—1464no—
420#102—28no—
421#102—1468no—
422C22⋊C8—1468no—
423C24—24no—
424C24—22no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24—42no—
429C24—42no—
430C24—42no—
431C24—734no—
432C24—734no—
433C24—734no—
434C24—732no—
435C24—1462no—
436C24—1462no—
437C24—1462no—
438C24—1462no—
439C24D4×D7311yescharacteristic
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—42no—
452C2×D4—29232no—
453C2×D4—2928no—
454C2×D4—2928no—
455C2×D4—14616no—
456C2×D4C2×C4×D7314yescomplemented
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—2928no—
462C22×C4—29232no—
463C22×C4—2928no—
464C22×C4—2922no—
465C22×C4—2922no—
466C22×C4—2922no—
467C22×C4—2922no—
468C22×C4—734no—
469C22×C4—732no—
470C22×C4—738no—
471C22×C4—734no—
472C22×C4—734no—
473C22×C4—14616no—
474C22×C4—1464no—
475C22×C4—1464no—
476C22×C4—1462no—
477C22×C4—1462no—
478C22×C4—1462no—
479C22×C4—1462no—
480C22×C4—1468no—
481C22×C4—1468no—
482C22×C4—1464no—
483C22×C4—1468no—
484C22×C4—1468no—
485C22×C4—1464no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4C2×C4×D7312yes—
491C22×C4D4×D7311yescharacteristic
492C22×C4D4×D7312yes—
493C22×C4D4×D7312yes—
494C22×C4C2×C22⋊D7312yes—
495C22×C4#1192402712yes—
496C22×C4#1192402812yes—
497C22×C4C23×D7311yescharacteristic
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—44no—
502C2×C8—2924no—
503C2×C8—1462no—
504C2×C8—1462no—
505C2×C8—1462no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—734no—
509C4⋊C4—734no—
510C4⋊C4—14616no—
511C4⋊C4—1464no—
512C4⋊C4—1464no—
513C4⋊C4—1462no—
514C4⋊C4—1462no—
515C4⋊C4C2×C22⋊D7314yescomplemented
516C22⋊C4—14616no—
517C23—24no—
518C23—42no—
519C23—48no—
520C23—42no—
521C23—48no—
522C23—42no—
523C23—48no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23—734no—
532C23—7316no—
533C23—734no—
534C23—734no—
535C23—1462no—
536C23—1468no—
537C23—1468no—
538C23—1468no—
539C23—1468no—
540C23—1468no—
541C23—1462no—
542C23—1462no—
543C23—1468no—
544C23—1462no—
545C23C22⋊C4×D7312yes—
546C23#1455655912yes—
547C23#1455657312yes—
548C23D8×D7312yes—
549C23SD16×D7312yes—
550C23#1455661512yes—
551C23#1455662012yes—
552C23C2×D4×D7311yescharacteristic
553D4—24no—
554D4—24no—
555D4—48no—
556D4—42no—
557D4—42no—
558D4—29232no—
559C2×C4—24no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—29232no—
564C2×C4—2928no—
565C2×C4—2928no—
566C2×C4—2922no—
567C2×C4—2922no—
568C2×C4—2922no—
569C2×C4—2922no—
570C2×C4—734no—
571C2×C4—1464no—
572C2×C4—1464no—
573C2×C4—1468no—
574C2×C4—1462no—
575C2×C4—1462no—
576C2×C4—14616no—
577C2×C4—1464no—
578C2×C4—1464no—
579C2×C4—1464no—
580C2×C4—1464no—
581C2×C4C22⋊C4×D7312yes—
582C2×C4#1455655114yes—
583C2×C4C4×C22⋊D7314yes—
584C2×C4#1455655912yes—
585C2×C4#1455656214yes—
586C2×C4#1455656414yes—
587C2×C4#1455657312yes—
588C2×C4C22×C4×D7311yescharacteristic
589C2×C4C2×D4×D7311yescharacteristic
590C2×C4C2×D4×D7311yescharacteristic
591C2×C4C22×C22⋊D7311yescharacteristic
592C2×C4C2×#1192402711yescharacteristic
593C2×C4C2×#1192402811yescharacteristic
594C8—42no—
595C8—2922no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—816no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—738no—
605C22—14632no—
606C22—1464no—
607C22—1464no—
608C22—1464no—
609C22—1464no—
610C22—1464no—
611C22—1464no—
612C22#1798846314yes—
613C22#102×D7314yes—
614C22#1798858514yes—
615C22#1798871914yes—
616C22#1798872414yes—
617C22#1798889914yes—
618C22#1798890414yes—
619C22C2×C22⋊C4×D7311yescharacteristic
620C22C2×#1455655911yescharacteristic
621C22C2×#1455657311yescharacteristic
622C22C2×D8×D7311yescharacteristic
623C22C2×SD16×D7311yescharacteristic
624C22C2×#1455661511yescharacteristic
625C22C2×#1455662011yescharacteristic
626C4—44no—
627C4—2924no—
628C4—2924no—
629C4—1464no—
630C4#1798846314yes—
631C4C2×#1455655111yescharacteristic
632C4C2×C4×C22⋊D7311yescharacteristic
633C4C2×#1455656211yescharacteristic
634C4C2×#1455656411yescharacteristic
635C2—84no—
636C2—1468no—
637C2#2261024518yesdirect product factor
638C2C2×#1798846311yescharacteristic
639C2C2×#102×D7311yescharacteristic
640C2C2×#1798858511yescharacteristic
641C2C2×#1798871911yescharacteristic
642C2C2×#1798872411yescharacteristic
643C2C2×#1798889911yescharacteristic
644C2C2×#1798890411yescharacteristic
subgroupG/H
centerC24D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC22×#102×C73C2
socleC24×C73C2×D4
prime radicalC22×#14556562C2
cosocleC22×C4×C73C24
hypercenterC22×#102D73
hypocenterC73C2×#1874
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D73C2semidirect
3. C22×#14556512C2semidirect
4. C22×#14556503C2semidirect
5. #22610245C2direct (N × H)
6. C2×#17988350C2semidirect
7. C2×#17988349C2semidirect
8. C22×#102×C73C2semidirect
15. C2×#14556562C4semidirect
23. C2×C4⋊C4×D73C22semidirect
26. C2×#14556512C22semidirect
28. C2×#14556503C22semidirect
29. #17988350C22semidirect
30. #17988349C22semidirect
35. C2×#102×C73C22semidirect
62. #14556562C2×C4semidirect
67. C2×C4⋊Dic73C2×C4semidirect
68. C2×C4⋊Dic73D4semidirect
87. C22×D4×C73C2×C4semidirect
97. C2×C4⋊C4×C73D4semidirect
149. C4⋊Dic73C22×C4semidirect
150. C4⋊Dic73C2×D4semidirect
174. C2×D4×C73C22×C4semidirect
193. C4⋊C4×C73C2×D4semidirect
226. C22×D73#102semidirect
302. C2×D73C2×#102semidirect
314. C2×C73#1874semidirect
326. C22×#102D73semidirect
331. C73C2×#1874semidirect
354. C2×#102C2×D73semidirect
371. C22×D4C4×D73semidirect
416. C2×C4⋊C4C22⋊D73semidirect
456. C2×D4C2×C4×D73semidirect
515. C4⋊C4C2×C22⋊D73semidirect
637. C2#22610245direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145565622
C22×C4⋊C4×D732
C22×#145565122
C22×#145565032
#226102452
C2×#179883502
C2×#179883492
C22×#102×C732
C2×#187473