Finite Group Databasebeta

← C2×#22608434 #33053123

Subgroups — C2×#22608434 #33053123

Order 18688.

1834 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989100C211yesmaximal, characteristic
2C2×#17988549C211yesmaximal, characteristic, complemented
3C2×#95×D73C211yesmaximal, characteristic, complemented
4#22608434C218yesmaximal, direct product factor
5C2×#17988324C211yesmaximal, characteristic, complemented
6C2×#17988322C211yesmaximal, characteristic, complemented
7C2×#17988320C211yesmaximal, characteristic, complemented
8C2×#328×C73C211yesmaximal, characteristic, complemented
9C22×#11924029C411yescharacteristic, complemented
10C23×C4×D73—21no—
11C23×C4×D73C2211yescharacteristic
12C22×#11924011—21no—
13#17989100—22no—
14#17989100—22no—
15#17989100C414yescomplemented
16#17989100C2214yes—
17C2×#14556576—21no—
18C2×#14556567C411yescharacteristic, complemented
19C2×#14556566C411yescharacteristic, complemented
20C2×#14556565C2211yescharacteristic
21C2×#14556561—21no—
22C2×#14556553—21no—
23C2×#14556551—21no—
24C2×#14556549C411yescharacteristic, complemented
25#17988549—22no—
26#17988549C2214yescomplemented
27C2×C22⋊#10596831C2211yescharacteristic
28C2×#14556503—21no—
29C2×#14556498—21no—
30C2×#14556498C2211yescharacteristic
31#95×D73—22no—
32#95×D73C2214yescomplemented
33#17988324C2214yescomplemented
34#17988322C2214yescomplemented
35#17988320C2214yescomplemented
36C2×C22⋊Q8×C73C2211yescharacteristic
37C2×C22⋊C8×C73C2211yescharacteristic
38C2×#95×C73C2211yescharacteristic
39#328×C73C2214yescomplemented
40C24×D73D411yescharacteristic
41C2×#11924029—22no—
42C2×#11924029—44no—
43C2×#11924029C2×C414yescomplemented
44C22×D292—21no—
45C22×D292—21no—
46C22×D292C2×C411yescharacteristic, complemented
47C22×C22⋊D73—21no—
48C23×Dic73—21no—
49C23×Dic73D411yescharacteristic
50C22×C4×D73—28no—
51C22×C4×D73—24no—
52C22×C4×D73—21no—
53C22×C4×D73—44no—
54C22×C4×D73—44no—
55C22×C4×D73—44no—
56C22×C4×D73C2×C411yescharacteristic
57C22×C4×D73C2×C411yescharacteristic
58C22×C4×D73C2×C412yes—
59C22×C4×D73D411yescharacteristic
60C22×C4×D73D412yes—
61C22×C4×D73D411yescharacteristic
62C22×C4×D73D412yes—
63C22×C4×D73C2312yes—
64C2×#11924011—24no—
65C2×#11924011—44no—
66C22×#10596831—21no—
67#14556576—24no—
68#14556576—24no—
69#14556567—22no—
70#14556567C2×C414yescomplemented
71#14556566—22no—
72#14556566C2×C414yescomplemented
73#14556565—21no—
74#14556565—21no—
75#14556565C2×C412yescomplemented
76#14556565C2312yes—
77#14556561—24no—
78#14556561—24no—
79#14556553—24no—
80#14556553—24no—
81#14556551—24no—
82#14556551—24no—
83#14556549—22no—
84#14556549C2×C414yescomplemented
85C2×C4⋊Dic73C2×C411yescharacteristic, complemented
86C2×#11924007—21no—
87C2×#11924005—21no—
88C2×#11924005—21no—
89C2×#11924005—21no—
90C2×#11924005—21no—
91C2×C4×Dic73—21no—
92C22⋊#10596831—21no—
93C22⋊#10596831C2312yes—
94#14556503—24no—
95#14556498—24no—
96#14556498—21no—
97#14556498C2312yes—
98C22×Q8×C73C2×C411yescharacteristic, complemented
99C23×C4×C73—21no—
100C23×C4×C73C2311yescharacteristic
101C22×C8×C73—21no—
102C22⋊Q8×C73—21no—
103C22⋊Q8×C73—21no—
104C22⋊Q8×C73C2×C412yescomplemented
105C22⋊Q8×C73C2312yes—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73C2×C411yescharacteristic, complemented
108C2×C22⋊C4×C73—21no—
109C22⋊C8×C73—21no—
110C22⋊C8×C73C2312yes—
111#95×C73—21no—
112#95×C73C2312yes—
113C23×D73—22no—
114C23×D73—22no—
115C23×D73—24no—
116C23×D73—24no—
117C23×D73—21no—
118C23×D73C22⋊C412yes—
119C23×D73C22⋊C411yescharacteristic
120C23×D73C22⋊C411yescharacteristic
121C23×D73C2×D412yes—
122#11924029—48no—
123#11924029—48no—
124C2×D292—24no—
125C2×D292—24no—
126C2×D292—21no—
127C2×D292—42no—
128C2×D292—44no—
129C2×D292—44no—
130C2×D292C22×C412yescomplemented
131C2×C22⋊D73—22no—
132C2×C22⋊D73—22no—
133C2×C22⋊D73—42no—
134C2×C22⋊D73—42no—
135C22×Dic73—24no—
136C22×Dic73—21no—
137C22×Dic73—22no—
138C22×Dic73—42no—
139C22×Dic73—42no—
140C22×Dic73—42no—
141C22×Dic73C22⋊C411yescharacteristic
142C22×Dic73C22⋊C411yescharacteristic
143C22×Dic73C22⋊C412yes—
144C22×Dic73C2×D412yes—
145C2×C4×D73—24no—
146C2×C4×D73—24no—
147C2×C4×D73—216no—
148C2×C4×D73—24no—
149C2×C4×D73—44no—
150C2×C4×D73—44no—
151C2×C4×D73—44no—
152C2×C4×D73—44no—
153C2×C4×D73—42no—
154C2×C4×D73—42no—
155C2×C4×D73—44no—
156C2×C4×D73—432no—
157C2×C4×D73—48no—
158C2×C4×D73—48no—
159C2×C4×D73C22⋊C412yes—
160C2×C4×D73C22⋊C412yes—
161C2×C4×D73C22⋊C412yes—
162C2×C4×D73C22⋊C412yes—
163C2×C4×D73C22×C412yes—
164C2×C4×D73C22×C412yes—
165C2×C4×D73C2×D412yes—
166C2×C4×D73C2×D412yes—
167#11924011—48no—
168C2×#10596831—22no—
169C2×#10596831—42no—
170C4⋊Dic73—21no—
171C4⋊Dic73C22×C412yescomplemented
172#11924007—24no—
173#11924005—24no—
174#11924005—24no—
175#11924005—22no—
176#11924005—22no—
177#11924005—24no—
178C4×Dic73—24no—
179C24×C73C2×D411yescharacteristic
180C2×Q8×C73—21no—
181C2×Q8×C73—42no—
182C2×Q8×C73C22×C412yescomplemented
183C22×C4×C73—24no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—22no—
187C22×C4×C73—42no—
188C22×C4×C73—42no—
189C22×C4×C73—42no—
190C22×C4×C73C22×C411yescharacteristic
191C22×C4×C73C22×C411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2411yescharacteristic
196C2×C8×C73—22no—
197C2×C8×C73—42no—
198C4⋊C4×C73—22no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—22no—
201C4⋊C4×C73C22×C412yescomplemented
202C22⋊C4×C73—22no—
203C22⋊C4×C73—22no—
204C23×C73—22no—
205C23×C73—22no—
206C23×C73C2×C22⋊C411yescharacteristic
207C23×C73C2×C22⋊C411yescharacteristic
208C23×C73AGL(1,ℤ/8)12yes—
209C23×C73#13712yes—
210C23×C73C22×D411yescharacteristic
211C22×D73—216no—
212C22×D73—216no—
213C22×D73—22no—
214C22×D73—22no—
215C22×D73—22no—
216C22×D73—24no—
217C22×D73—22no—
218C22×D73—22no—
219C22×D73—22no—
220C22×D73—24no—
221C22×D73—24no—
222C22×D73—24no—
223C22×D73—24no—
224C22×D73—22no—
225C22×D73—42no—
226C22×D73—42no—
227C22×D73—42no—
228C22×D73#9914yes—
229C22×D73#10414yes—
230C22×D73C2×C22⋊C412yes—
231C22×D73C2×C22⋊C412yes—
232Q8×C73—42no—
233Q8×C73—42no—
234C2×C4×C73—22no—
235C2×C4×C73—22no—
236C2×C4×C73—24no—
237C2×C4×C73—22no—
238C2×C4×C73—22no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—42no—
244C2×C4×C73—48no—
245C2×C4×C73—42no—
246C2×C4×C73—42no—
247C2×C4×C73—42no—
248C2×C4×C73C2×C22⋊C411yescharacteristic
249C2×C4×C73C2×C22⋊C411yescharacteristic
250C2×C4×C73C23×C411yescharacteristic
251C2×C4×C73C22×D411yescharacteristic
252C22⋊D73—42no—
253C22⋊D73—42no—
254C22⋊D73—44no—
255C2×Dic73—22no—
256C2×Dic73—22no—
257C2×Dic73—24no—
258C2×Dic73—22no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—48no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73C2×C22⋊C412yes—
266C2×Dic73C2×C22⋊C412yes—
267D292—42no—
268D292—42no—
269D292—48no—
270D292—48no—
271C4×D73—24no—
272C4×D73—24no—
273C4×D73—24no—
274C4×D73—24no—
275C4×D73—42no—
276C4×D73—42no—
277C4×D73—42no—
278C4×D73—42no—
279C4×D73—48no—
280C4×D73—432no—
281C8×C73—42no—
282#10596831—42no—
283C22×C73—21no—
284C22×C73—21no—
285C22×C73—21no—
286C22×C73—21no—
287C22×C73—24no—
288C22×C73—24no—
289C22×C73C2×#9912yes—
290C22×C73#41712yes—
291C22×C73#41712yes—
292C22×C73C2×#10412yes—
293C22×C73C22×C22⋊C411yescharacteristic
294C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
295C22×C73C2×#13711yescharacteristic
296C2×D73—28no—
297C2×D73—28no—
298C2×D73—216no—
299C2×D73—22no—
300C2×D73—22no—
301C2×D73—22no—
302C2×D73—22no—
303C2×D73—28no—
304C2×D73—28no—
305C2×D73—42no—
306C2×D73—42no—
307C2×D73—42no—
308C2×D73—84no—
309C2×D73C2×#9914yes—
310C2×D73C2×#10414yes—
311C4×C73—21no—
312C4×C73—21no—
313C4×C73—21no—
314C4×C73—21no—
315C4×C73—44no—
316C4×C73—42no—
317C4×C73—42no—
318Dic73—22no—
319Dic73—22no—
320Dic73—44no—
321Dic73—42no—
322C2×#1504—731nomaximal, Sylow
323C2×C73—21no—
324C2×C73—21no—
325C2×C73—22no—
326C2×C73#150414yescomplemented
327C2×C73C22×#9911yescharacteristic
328C2×C73C2×#41711yescharacteristic
329C2×C73C22×#10411yescharacteristic
330D73—24no—
331D73—24no—
332D73—82no—
333C2×#525—731no—
334C2×#407—731no—
335C22×#95—731no—
336#1504—738no—
337C2×#328—731no—
338C2×#328D7311yescharacteristic, complemented
339C2×#327—731no—
340C2×#327—731no—
341C73C2×#150411yescharacteristic, complemented, Sylow
342C22×#41—731no—
343C24×C4—731no—
344C24×C4—1461no—
345C22×M16—1461no—
346#525—734no—
347#525—734no—
348#525—1462no—
349#525—1462no—
350C2×#123—1461no—
351C2×C22⋊Q8—731no—
352C2×C22⋊Q8C2×D7311yescharacteristic
353C2×#121—731no—
354C2×#121—731no—
355C2×#120—1461no—
356C2×C4×D4—1461no—
357C2×C4×D4—1461no—
358C2×#117—731no—
359#407—734no—
360#407—1462no—
361C2×C22⋊C8—731no—
362C2×C22⋊C8—1461no—
363C2×C22⋊C8C2×D7311yescharacteristic
364C2×#95—731no—
365C2×#95—734no—
366C2×#95—1462no—
367C2×#95—1461no—
368C2×#95C2×D7311yescharacteristic
369#328—734no—
370#328C2×D7314yescomplemented
371#327—734no—
372#327—734no—
373C25—731no—
374C2×#41—2924no—
375C2×#41—734no—
376C2×#41—1462no—
377C22×Q8C4×D7311yescharacteristic, complemented
378C22×D4—731no—
379C22×D4—1461no—
380C22×D4—1461no—
381C22×D4—1461no—
382C23×C4—21no—
383C23×C4—2924no—
384C23×C4—2924no—
385C23×C4—2924no—
386C23×C4—731no—
387C23×C4—731no—
388C23×C4—732no—
389C23×C4—731no—
390C23×C4—732no—
391C23×C4—732no—
392C23×C4—732no—
393C23×C4—731no—
394C23×C4—731no—
395C23×C4—1461no—
396C23×C4—1464no—
397C23×C4—1468no—
398C23×C4—1461no—
399C23×C4C22×D7311yescharacteristic
400C2×M16—2924no—
401C2×M16—1464no—
402C22×C8—21no—
403C22×C8—1461no—
404#123—1464no—
405#123—1464no—
406C22⋊Q8—21no—
407C22⋊Q8—21no—
408C22⋊Q8—734no—
409C22⋊Q8—1462no—
410C22⋊Q8C4×D7312yescomplemented
411C22⋊Q8C22×D7312yes—
412#121—732no—
413#121—732no—
414#121—734no—
415#121—1462no—
416#121—1461no—
417#121—1461no—
418#120—1464no—
419#120—1464no—
420C4×D4—1464no—
421C4×D4—1464no—
422C4×D4—1464no—
423C4×D4—1464no—
424#117—734no—
425#117—1462no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—731no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4C4×D7311yescharacteristic, complemented
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1461no—
432C2×C22⋊C4—1461no—
433C2×C22⋊C4—1461no—
434C2×C22⋊C4—1461no—
435C2×C42—1461no—
436C22⋊C8—21no—
437C22⋊C8—732no—
438C22⋊C8—1464no—
439C22⋊C8—1461no—
440C22⋊C8C22×D7312yes—
441#95—21no—
442#95—732no—
443#95—1464no—
444#95—1461no—
445#95C22×D7312yes—
446C24—731no—
447C24—731no—
448C24—732no—
449C24—732no—
450C24—1464no—
451C24—1464no—
452C24—1461no—
453C24—1462no—
454C24—1462no—
455C24D4×D7311yescharacteristic
456#41—2928no—
457#41—2928no—
458C2×Q8—21no—
459C2×Q8—42no—
460C2×Q8C2×C4×D7312yescomplemented
461C2×D4—2922no—
462C2×D4—2924no—
463C2×D4—2924no—
464C2×D4—2922no—
465C2×D4—2922no—
466C2×D4—732no—
467C2×D4—1464no—
468C2×D4—1464no—
469C2×D4—1462no—
470C2×D4—1462no—
471C2×D4—1461no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—2922no—
480C22×C4—2922no—
481C22×C4—2928no—
482C22×C4—29232no—
483C22×C4—2928no—
484C22×C4—2924no—
485C22×C4—2924no—
486C22×C4—2924no—
487C22×C4—2924no—
488C22×C4—2924no—
489C22×C4—2922no—
490C22×C4—2922no—
491C22×C4—2922no—
492C22×C4—732no—
493C22×C4—732no—
494C22×C4—732no—
495C22×C4—732no—
496C22×C4—732no—
497C22×C4—732no—
498C22×C4—731no—
499C22×C4—731no—
500C22×C4—732no—
501C22×C4—732no—
502C22×C4—732no—
503C22×C4—732no—
504C22×C4—1464no—
505C22×C4—14616no—
506C22×C4—1464no—
507C22×C4—1464no—
508C22×C4—1464no—
509C22×C4—1461no—
510C22×C4—1462no—
511C22×C4C2×C4×D7311yescharacteristic
512C22×C4C2×C4×D7311yescharacteristic
513C22×C4D4×D7311yescharacteristic
514C22×C4D4×D7311yescharacteristic
515C22×C4D4×D7311yescharacteristic
516C22×C4C23×D7311yescharacteristic
517M16—2928no—
518C2×C8—22no—
519C2×C8—42no—
520C2×C8—2922no—
521C2×C8—1462no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—732no—
526C4⋊C4—1464no—
527C4⋊C4—1461no—
528C4⋊C4C2×C4×D7312yescomplemented
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—1464no—
532C22⋊C4—1464no—
533C22⋊C4—1464no—
534C22⋊C4—1462no—
535C22⋊C4—1462no—
536C42—1464no—
537C23—22no—
538C23—22no—
539C23—2922no—
540C23—2922no—
541C23—2922no—
542C23—732no—
543C23—732no—
544C23—734no—
545C23—734no—
546C23—1464no—
547C23—1464no—
548C23—1464no—
549C23—1464no—
550C23—14616no—
551C23—14616no—
552C23—1462no—
553C23—1462no—
554C23—1462no—
555C23—1464no—
556C23—1462no—
557C23—1462no—
558C23—1462no—
559C23—1462no—
560C23C22⋊C4×D7311yescharacteristic
561C23C22⋊C4×D7311yescharacteristic
562C23#1455661812yes—
563C23#1455662312yes—
564C23C2×D4×D7311yescharacteristic
565Q8—42no—
566Q8—42no—
567D4—2922no—
568D4—2922no—
569D4—2928no—
570D4—2928no—
571D4—2922no—
572D4—2924no—
573D4—2922no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—2922no—
589C2×C4—2922no—
590C2×C4—2922no—
591C2×C4—2922no—
592C2×C4—29232no—
593C2×C4—2928no—
594C2×C4—2922no—
595C2×C4—2928no—
596C2×C4—2922no—
597C2×C4—2922no—
598C2×C4—2922no—
599C2×C4—2922no—
600C2×C4—732no—
601C2×C4—732no—
602C2×C4—1464no—
603C2×C4—1464no—
604C2×C4—1464no—
605C2×C4—1464no—
606C2×C4—1464no—
607C2×C4—1462no—
608C2×C4—1462no—
609C2×C4—1462no—
610C2×C4C22⋊C4×D7311yescharacteristic
611C2×C4C22⋊C4×D7311yescharacteristic
612C2×C4C22×C4×D7311yescharacteristic
613C2×C4C2×D4×D7311yescharacteristic
614C8—42no—
615C8—2922no—
616C22—21no—
617C22—21no—
618C22—24no—
619C22—24no—
620C22—21no—
621C22—21no—
622C22—2922no—
623C22—2922no—
624C22—2922no—
625C22—5844no—
626C22—734no—
627C22—734no—
628C22—1468no—
629C22—1468no—
630C22—14616no—
631C22—1462no—
632C22—1468no—
633C22—1468no—
634C22—1462no—
635C22—1462no—
636C22—1462no—
637C22#99×D7312yes—
638C22#1798858712yes—
639C22#1798858712yes—
640C22#104×D7312yes—
641C22C2×C22⋊C4×D7311yescharacteristic
642C22C2×#1455661811yescharacteristic
643C22C2×#1455662311yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—2924no—
652C4—2922no—
653C4—1462no—
654C4—1462no—
655C2—21no—
656C2—21no—
657C2—22no—
658C2—5842no—
659C2—1464no—
660C2—1464no—
661C2#2260843414yesdirect product factor
662C2C2×#99×D7311yescharacteristic
663C2C2×#1798858711yescharacteristic
664C2C2×#104×D7311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#328×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC2×#17989100C2
cosocleC22×C4×C73C24
hypercenterC2×#328D73
hypocenterC73C2×#1504
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
2. C2×#17988549C2semidirect
3. C2×#95×D73C2semidirect
4. #22608434C2direct (N × H)
5. C2×#17988324C2semidirect
6. C2×#17988322C2semidirect
7. C2×#17988320C2semidirect
8. C2×#328×C73C2semidirect
9. C22×#11924029C4semidirect
15. #17989100C4semidirect
18. C2×#14556567C4semidirect
19. C2×#14556566C4semidirect
24. C2×#14556549C4semidirect
26. #17988549C22semidirect
32. #95×D73C22semidirect
33. #17988324C22semidirect
34. #17988322C22semidirect
35. #17988320C22semidirect
39. #328×C73C22semidirect
43. C2×#11924029C2×C4semidirect
46. C22×D292C2×C4semidirect
70. #14556567C2×C4semidirect
72. #14556566C2×C4semidirect
75. #14556565C2×C4semidirect
84. #14556549C2×C4semidirect
85. C2×C4⋊Dic73C2×C4semidirect
98. C22×Q8×C73C2×C4semidirect
104. C22⋊Q8×C73C2×C4semidirect
107. C2×C4⋊C4×C73C2×C4semidirect
130. C2×D292C22×C4semidirect
171. C4⋊Dic73C22×C4semidirect
182. C2×Q8×C73C22×C4semidirect
201. C4⋊C4×C73C22×C4semidirect
326. C2×C73#1504semidirect
338. C2×#328D73semidirect
341. C73C2×#1504semidirect
370. #328C2×D73semidirect
377. C22×Q8C4×D73semidirect
410. C22⋊Q8C4×D73semidirect
429. C2×C4⋊C4C4×D73semidirect
460. C2×Q8C2×C4×D73semidirect
528. C4⋊C4C2×C4×D73semidirect
661. C2#22608434direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179891002
C2×#179885492
C2×#95×D732
#226084342
C2×#179883242
C2×#179883222
C2×#179883202
C2×#328×C732
C2×#150473