Finite Group Databasebeta

← C4×#17988808 #33092217

Subgroups — C4×#17988808 #33092217

Order 18688.

1016 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×#17988808C211yesmaximal, characteristic
2C2×C4×Dic292C211yesmaximal, characteristic, complemented
3C2×C4×#11924016C211yesmaximal, characteristic
4C4×#14556578C211yesmaximal, characteristic
5C4×#14556556C211yesmaximal, characteristic, complemented
6#22610862C211yesmaximal, characteristic
7#22610702C211yesmaximal, characteristic, complemented
8#22610563C211yesmaximal, characteristic, complemented
9#22610553C211yesmaximal, characteristic
10#22610283C211yesmaximal, characteristic
11#22610273C211yesmaximal, characteristic, complemented
12#22609945C211yesmaximal, characteristic, complemented
13C4×#14556520C211yesmaximal, characteristic, complemented
14C4×#14556514C211yesmaximal, characteristic, complemented
15C4×#102×C73C211yesmaximal, characteristic
16C22×Dic292C2211yescharacteristic
17C22×#11924016C2211yescharacteristic
18C2×#14556578C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20C2×C4×Dic146C2211yescharacteristic
21C2×#14556556C2211yescharacteristic
22C2×#14556556C2211yescharacteristic
23#17988808C418yesdirect product factor
24C4×Dic292—24no—
25C4×#11924016—24no—
26C2×#14556525C2211yescharacteristic
27C2×#14556522C2211yescharacteristic
28C2×#14556520C2211yescharacteristic
29C2×#14556520C2211yescharacteristic
30C2×#14556516C2211yescharacteristic
31C2×#14556516C2211yescharacteristic
32C2×#14556514C2211yescharacteristic
33C2×#14556514C2211yescharacteristic
34C2×#14556514C2211yescharacteristic
35C2×#14556512C2211yescharacteristic
36C2×C4×#10596831C2211yescharacteristic
37#17988492C2211yescharacteristic
38#17988490C2211yescharacteristic
39#17988481—21no—
40#17988481—21no—
41#17988452C2211yescharacteristic
42C4⋊C4×Dic73C2211yescharacteristic
43C4⋊C4×Dic73C2211yescharacteristic
44C4×#11924007—21no—
45#17988438—21no—
46#17988438—21no—
47C4×C22⋊Dic73—21no—
48C4×C22⋊Dic73—21no—
49C42×Dic73—21no—
50C42×Dic73C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988341C2211yescharacteristic
54#17988341C2211yescharacteristic
55C2×C4×D4×C73C2211yescharacteristic
56C2×#102×C73C2211yescharacteristic
57C2×#102×C73C2211yescharacteristic
58C2×C4×C8×C73C2211yescharacteristic
59C4×C4⋊C4×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#340×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C22×Dic146C2311yescharacteristic
64C2×Dic292—22no—
65C2×Dic292—22no—
66C2×Dic292C2×C414yescomplemented
67C2×#11924016—22no—
68C2×#11924016—22no—
69C2×#11924016C2×C414yes—
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72#14556578—22no—
73#14556578C2×C414yes—
74C4×Dic146—22no—
75C4×Dic146—22no—
76C4×Dic146—22no—
77C4×Dic146D414yes—
78#14556556—22no—
79#14556556—22no—
80#14556556—22no—
81#14556556C2×C414yescomplemented
82C2×C4⋊Dic73C2311yescharacteristic
83C2×C4⋊Dic73C2311yescharacteristic
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×#11924007—21no—
87C2×#11924007—21no—
88C2×C22⋊Dic73—21no—
89C2×C22⋊Dic73—21no—
90C2×C22⋊Dic73—21no—
91C2×C22⋊Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73—21no—
95C2×C4×Dic73—21no—
96C2×C4×Dic73—21no—
97C2×C4×Dic73—21no—
98C2×C4×Dic73D411yescharacteristic
99C2×C4×Dic73D411yescharacteristic
100C2×C4×Dic73D411yescharacteristic
101C2×C4×Dic73D411yescharacteristic
102C2×C4×Dic73C2311yescharacteristic
103C2×C4×Dic73C2311yescharacteristic
104#14556525—22no—
105#14556522—22no—
106#14556520—22no—
107#14556520C2×C414yescomplemented
108#14556516—22no—
109#14556516—22no—
110#14556514—22no—
111#14556514—22no—
112#14556514C2×C414yescomplemented
113#14556512—22no—
114C4×#10596831—22no—
115#14556498—21no—
116#14556498—21no—
117#14556498—21no—
118#14556498—21no—
119#14556498—21no—
120#14556498—21no—
121C22×D4×C73C2311yescharacteristic
122C23×C4×C73—21no—
123C22×C8×C73C2311yescharacteristic
124C22×C8×C73C2311yescharacteristic
125C4×D4×C73—22no—
126C4×D4×C73—22no—
127C2×C4⋊C4×C73C2311yescharacteristic
128C2×C4⋊C4×C73C2311yescharacteristic
129C2×C4⋊C4×C73C2311yescharacteristic
130C2×C22⋊C4×C73—21no—
131C2×C42×C73—21no—
132C2×C42×C73C2311yescharacteristic
133#102×C73—22no—
134#102×C73C2×C414yes—
135C4×C8×C73—22no—
136#95×C73—21no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146—21no—
143C2×Dic146—21no—
144C2×Dic146—21no—
145C2×Dic146C22×C412yes—
146C2×Dic146C22×C412yes—
147C2×Dic146C2×D412yes—
148C2×Dic146#4112yes—
149C22×Dic73—21no—
150C22×Dic73—21no—
151C22×Dic73—21no—
152C22×Dic73—21no—
153C22×Dic73C2×D411yescharacteristic
154C22×Dic73C2×D411yescharacteristic
155C22×Dic73C2×D411yescharacteristic
156C22×Dic73C2×D411yescharacteristic
157Dic292—28no—
158#11924016—28no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C2×#10596831C22×C412yes—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73C2×D412yes—
169#11924007—24no—
170#11924007—24no—
171#11924007—24no—
172#11924007—42no—
173C22⋊Dic73—24no—
174C22⋊Dic73—24no—
175C22⋊Dic73—42no—
176C22⋊Dic73—42no—
177C4×Dic73—24no—
178C4×Dic73—24no—
179C4×Dic73—22no—
180C4×Dic73—22no—
181C4×Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73—24no—
184C4×Dic73—22no—
185C4×Dic73—41no—
186C4×Dic73—42no—
187C4×Dic73—42no—
188C4×Dic73—42no—
189C4×Dic73C22×C412yes—
190C4×Dic73#4112yes—
191C24×C73—21no—
192C2×D4×C73—21no—
193C2×D4×C73—21no—
194C2×D4×C73—21no—
195C2×D4×C73—21no—
196C2×D4×C73—21no—
197C2×D4×C73C22×C412yes—
198C22×C4×C73—21no—
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201C22×C4×C73—42no—
202C22×C4×C73—42no—
203C22×C4×C73—42no—
204C22×C4×C73C2×D411yescharacteristic
205C22×C4×C73C2×D411yescharacteristic
206C22×C4×C73C2411yescharacteristic
207C2×C8×C73—21no—
208C2×C8×C73—21no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73—21no—
212C2×C8×C73C22×C412yes—
213C4⋊C4×C73—21no—
214C4⋊C4×C73—21no—
215C4⋊C4×C73—21no—
216C4⋊C4×C73—21no—
217C4⋊C4×C73C22×C412yes—
218C4⋊C4×C73#4112yes—
219C22⋊C4×C73—42no—
220C42×C73—42no—
221C42×C73C2×D412yes—
222C42×C73#4112yes—
223C23×C73—22no—
224C23×C73—41no—
225C23×C73—41no—
226C23×C73—41no—
227C23×C73—41no—
228C23×C73—41no—
229C23×C73—41no—
230C23×C73C22×D411yescharacteristic
231D4×C73—22no—
232D4×C73—22no—
233D4×C73—41no—
234D4×C73—41no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
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—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—84no—
247C2×C4×C73#12112yes—
248C2×C4×C73#12112yes—
249C2×C4×C73#13512yes—
250C2×C4×C73#13712yes—
251C2×C4×C73C23×C411yescharacteristic
252C2×C4×C73C22×D411yescharacteristic
253C2×C4×C73C2×#4111yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×Dic73—22no—
258C2×Dic73—22no—
259C2×Dic73—21no—
260C2×Dic73—22no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—21no—
266C2×Dic73—22no—
267C2×Dic73—22no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—21no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73C4×D412yes—
279C2×Dic73C4×D412yes—
280C2×Dic73#13512yes—
281Dic146—22no—
282Dic146—22no—
283Dic146—22no—
284Dic146—22no—
285Dic146—22no—
286Dic146—41no—
287Dic146—41no—
288Dic146C4×D414yes—
289C8×C73—22no—
290C8×C73—41no—
291#10596831—22no—
292#10596831—41no—
293C22×C73—42no—
294C22×C73—42no—
295C22×C73—42no—
296C22×C73—81no—
297C22×C73—81no—
298C22×C73—81no—
299C22×C73—81no—
300C22×C73C2×C4×D411yescharacteristic
301C22×C73C2×#12111yescharacteristic
302C22×C73#52511yescharacteristic
303C22×C73C2×#13511yescharacteristic
304C22×C73C2×#13511yescharacteristic
305C22×C73C2×#13711yescharacteristic
306C22×C73#57511yescharacteristic
307C4×C73—22no—
308C4×C73—42no—
309C4×C73—42no—
310C4×C73—81no—
311C4×C73#46414yescomplemented
312C4×C73C2×C4×D411yescharacteristic
313C4×C73C4×#4111yescharacteristic
314C4×C73#52911yescharacteristic
315C4×C73#53211yescharacteristic
316Dic73—22no—
317Dic73—22no—
318Dic73—22no—
319Dic73—24no—
320Dic73—41no—
321Dic73—42no—
322Dic73—42no—
323Dic73—81no—
324C4×#464—731nomaximal, Sylow
325C2×C73—82no—
326C2×C73C4×#12111yescharacteristic
327C2×C73C4×#13511yescharacteristic
328C2×C73C4×#13711yescharacteristic
329C2×C73C2×#46411yescharacteristic
330C2×C73#303511yescharacteristic
331C2×C73#310211yescharacteristic
332C2×C73#310811yescharacteristic
333C2×#464—731no—
334C2×C4×Q16—731no—
335C2×C4×SD16—731no—
336C4×#124—731no—
337C42×Q8—731no—
338#1983—731no—
339#1949—731no—
340#1927—731no—
341#1925—731no—
342#1880—731no—
343#1878—731no—
344#1805—731no—
345C4×C4⋊C8—731no—
346C4×#103—731no—
347C4×#102D7311yescharacteristic
348C73C4×#46411yescharacteristic, complemented, Sylow
349C22×Q16—731no—
350C22×SD16—731no—
351C2×#124—731no—
352C2×C4×Q8—731no—
353C2×C4×Q8—731no—
354C2×C4×Q8—731no—
355C2×C4×Q8—731no—
356C2×C4×D4C2×D7311yescharacteristic
357#464—738no—
358C4×Q16—1464no—
359C4×SD16—1464no—
360C2×#107—731no—
361C2×#106—731no—
362C2×C4⋊C8—731no—
363C2×C4⋊C8—731no—
364C2×#103—731no—
365C2×#103—731no—
366C2×#103—731no—
367C2×#103—731no—
368C2×#103—731no—
369C2×#102—731no—
370C2×#102C2×D7311yescharacteristic
371C2×#102C2×D7311yescharacteristic
372C2×C4×C8—731no—
373C2×C4×C8C2×D7311yescharacteristic
374#391—731no—
375#390—731no—
376#388—1461no—
377#388—1461no—
378#382—731no—
379C4×C4⋊C4—731no—
380C4×C4⋊C4—731no—
381C4×C4⋊C4—1461no—
382C4×C4⋊C4—1461no—
383C4×C4⋊C4—1461no—
384C4×C4⋊C4C2×D7311yescharacteristic
385C4×C22⋊C4—1461no—
386C4×C22⋊C4—1461no—
387C43—731no—
388C43—1461no—
389#340—731no—
390#340—731no—
391#340C2×D7311yescharacteristic
392#340C2×D7311yescharacteristic
393#336—731no—
394#336—731no—
395C22×Q8—731no—
396C22×Q8—731no—
397C22×D4C22×D7311yescharacteristic
398C23×C4—21no—
399C2×Q16—734no—
400C2×Q16—1462no—
401C2×Q16—1462no—
402C2×SD16—734no—
403C2×SD16—1462no—
404C2×SD16—1462no—
405C22×C8—731no—
406C22×C8—731no—
407C22×C8C22×D7311yescharacteristic
408C22×C8C22×D7311yescharacteristic
409#124—734no—
410#124—1462no—
411C4×Q8—734no—
412C4×Q8—734no—
413C4×Q8—1462no—
414C4×Q8—1462no—
415C4×Q8—1462no—
416C4×Q8—1462no—
417C4×Q8—1462no—
418C4×Q8—1462no—
419C4×D4—22no—
420C4×D4—22no—
421C2×C4⋊C4—731no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—1461no—
424C2×C4⋊C4—1461no—
425C2×C4⋊C4—1461no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4C22×D7311yescharacteristic
428C2×C4⋊C4C22×D7311yescharacteristic
429C2×C4⋊C4C22×D7311yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1461no—
432C2×C22⋊C4—1461no—
433C2×C22⋊C4—1461no—
434C2×C22⋊C4—1461no—
435C2×C42—21no—
436C2×C42—731no—
437C2×C42—731no—
438C2×C42—731no—
439C2×C42—731no—
440C2×C42—731no—
441C2×C42—731no—
442C2×C42—1461no—
443C2×C42—1461no—
444C2×C42—1461no—
445C2×C42—1461no—
446C2×C42—1461no—
447C2×C42—1461no—
448C2×C42C22×D7311yescharacteristic
449#107—1462no—
450#106—1462no—
451C4⋊C8—734no—
452C4⋊C8—1462no—
453#103—734no—
454#103—1462no—
455#103—1462no—
456#103—1462no—
457#103—1462no—
458#102—22no—
459#102—1462no—
460#102C4×D7314yes—
461C4×C8—22no—
462C4×C8—1462no—
463#95—21no—
464#95—1461no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470C24—21no—
471C2×Q8—732no—
472C2×Q8—732no—
473C2×Q8—732no—
474C2×Q8—732no—
475C2×Q8—1461no—
476C2×Q8—1461no—
477C2×Q8—1461no—
478C2×Q8—1461no—
479C2×Q8—1461no—
480C2×Q8—1461no—
481C2×Q8—1461no—
482C2×Q8—1461no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C2×C4×D7312yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—731no—
496C22×C4—731no—
497C22×C4—731no—
498C22×C4—731no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4—1461no—
503C22×C4D4×D7311yescharacteristic
504C22×C4D4×D7311yescharacteristic
505C22×C4C23×D7311yescharacteristic
506Q16—1468no—
507SD16—1468no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—732no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8C2×C4×D7312yes—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—2922no—
525C4⋊C4—732no—
526C4⋊C4—1464no—
527C4⋊C4—1464no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1464no—
531C4⋊C4—1461no—
532C4⋊C4C2×C4×D7312yes—
533C4⋊C4#1192402812yes—
534C22⋊C4—42no—
535C22⋊C4—2922no—
536C22⋊C4—2922no—
537C22⋊C4—1464no—
538C22⋊C4—1464no—
539C42—42no—
540C42—2921no—
541C42—2922no—
542C42—2922no—
543C42—2922no—
544C42—732no—
545C42—732no—
546C42—1462no—
547C42—1464no—
548C42—1464no—
549C42—1464no—
550C42—1462no—
551C42—1462no—
552C42—1461no—
553C42—1461no—
554C42D4×D7312yes—
555C42#1192402812yes—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C2×D4×D7311yescharacteristic
564Q8—2921no—
565Q8—2921no—
566Q8—734no—
567Q8—1462no—
568Q8—1462no—
569Q8—1462no—
570Q8—1462no—
571Q8—1462no—
572D4—22no—
573D4—22no—
574D4—41no—
575D4—41no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—84no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—732no—
596C2×C4—732no—
597C2×C4—732no—
598C2×C4—1461no—
599C2×C4—1462no—
600C2×C4—1462no—
601C2×C4—1461no—
602C2×C4—1461no—
603C2×C4—1461no—
604C2×C4—1461no—
605C2×C4—1461no—
606C2×C4—1462no—
607C2×C4—1462no—
608C2×C4—1461no—
609C2×C4—1462no—
610C2×C4—1461no—
611C2×C4—1461no—
612C2×C4#1455656412yes—
613C2×C4#1455656412yes—
614C2×C4#1455661012yes—
615C2×C4#1455662012yes—
616C2×C4C22×C4×D7311yescharacteristic
617C2×C4C2×D4×D7311yescharacteristic
618C2×C4#41×D7311yescharacteristic
619C2×C4#41×D7311yescharacteristic
620C2×C4C2×#1192402811yescharacteristic
621C2×C4C2×#1192402811yescharacteristic
622C8—22no—
623C8—41no—
624C8—2921no—
625C8—1462no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C4×D4×D7311yescharacteristic
634C22C2×#1455656411yescharacteristic
635C22#1798909611yescharacteristic
636C22#135×D7311yescharacteristic
637C22C2×#1455661011yescharacteristic
638C22C2×#1455662011yescharacteristic
639C22#1798938411yescharacteristic
640C4—22no—
641C4—42no—
642C4—42no—
643C4—81no—
644C4—2921no—
645C4—2922no—
646C4—2922no—
647C4—5841no—
648C4—1462no—
649C4—1462no—
650C4—1462no—
651C4—1464no—
652C4#1798880814yesdirect product factor
653C4C4×D4×D7311yescharacteristic
654C4C4×#1192402811yescharacteristic
655C4#1798911511yescharacteristic
656C4#1798913411yescharacteristic
657C2—82no—
658C2C4×#1455656411yescharacteristic
659C2C4×#1455661011yescharacteristic
660C2C4×#1455662011yescharacteristic
661C2C2×#1798880811yescharacteristic
662C2#2261938411yescharacteristic
663C2#2261994511yescharacteristic
664C2#2262000911yescharacteristic
subgroupG/H
centerC22×C4D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC4×#102×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenterC4×#102D73
hypocenterC73C4×#464
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic292C2semidirect
5. C4×#14556556C2semidirect
7. #22610702C2semidirect
8. #22610563C2semidirect
11. #22610273C2semidirect
12. #22609945C2semidirect
13. C4×#14556520C2semidirect
14. C4×#14556514C2semidirect
23. #17988808C4direct (N × H)
66. C2×Dic292C2×C4semidirect
81. #14556556C2×C4semidirect
107. #14556520C2×C4semidirect
112. #14556514C2×C4semidirect
311. C4×C73#464semidirect
348. C73C4×#464semidirect
652. C4#17988808direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179888082
C2×C4×Dic2922
C2×C4×#119240162
C4×#145565782
C4×#145565562
#226108622
#226107022
#226105632
#226105532
#226102832
#226102732
#226099452
C4×#145565202
C4×#145565142
C4×#102×C732
C4×#46473