Finite Group Databasebeta

← #34021116

Subgroups — #34021116

Order 18688.

1418 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989612C212yesmaximal, complemented
2#22625825C212yesmaximal, complemented
3#22625790C212yesmaximal, complemented
4#22625787C211yesmaximal, characteristic, complemented
5#22625785C211yesmaximal, characteristic, complemented
6#22625754C211yesmaximal, characteristic
7#22625754C211yesmaximal, characteristic
8#22625660C212yesmaximal, complemented
9#22625657C211yesmaximal, characteristic, complemented
10C4×C22⋊#10596833C211yesmaximal, characteristic, complemented
11C22×#14556549C211yesmaximal, characteristic
12#17989612—28no—
13C2×#14556658C2212yescomplemented
14C2×C4⋊#10596833C2212yescomplemented
15C2×#14556648C2212yes—
16C2×#14556648C2212yes—
17C2×#14556648C2212yes—
18C2×#14556648C2212yes—
19C2×C22⋊#10596833C2211yescharacteristic
20C2×C22⋊#10596833C2211yescharacteristic
21C2×C22⋊#10596833C2211yescharacteristic
22C2×C22⋊#10596833C2211yescharacteristic
23C2×#14556646C2212yescomplemented
24C2×C4×#10596833C2212yescomplemented
25#17989453C2212yescomplemented
26#17989453C2212yescomplemented
27#17989453C2212yescomplemented
28#17989453C2212yescomplemented
29C23×C4×D73C2211yescharacteristic
30C2×#14556549—28no—
31C2×#14556549C411yescharacteristic
32C2×#14556549C411yescharacteristic
33C2×#14556549C411yescharacteristic
34C2×#14556549C411yescharacteristic
35C2×#14556549C2211yescharacteristic
36C2×#14556549C2211yescharacteristic
37C2×#14556549C2211yescharacteristic
38C2×#14556549C2211yescharacteristic
39C22×C4⋊Dic73C411yescharacteristic
40C22×#11924005C411yescharacteristic
41C22×#11924005C411yescharacteristic
42C22×C4×Dic73C2211yescharacteristic
43C22×C4×Dic73C2211yescharacteristic
44C22×C4⋊C4×C73C411yescharacteristic
45C22×#10596833C2312yes—
46C22×#10596833C2312yes—
47C22×#10596833C2312yes—
48C22×#10596833C2312yes—
49#14556658—24no—
50C4⋊#10596833—24no—
51#14556648—24no—
52#14556648—24no—
53#14556648—24no—
54#14556648—24no—
55C22⋊#10596833—22no—
56C22⋊#10596833—22no—
57C22⋊#10596833—22no—
58C22⋊#10596833—22no—
59#14556646—24no—
60C4×#10596833—24no—
61C24×D73C2×C411yescharacteristic
62C23×Dic73C2×C411yescharacteristic
63C23×Dic73C2×C411yescharacteristic
64C23×Dic73C2311yescharacteristic
65C22×C4×D73—22no—
66C22×C4×D73—22no—
67C22×C4×D73—24no—
68C22×C4×D73—24no—
69C22×C4×D73C2×C411yescharacteristic
70C22×C4×D73C2×C411yescharacteristic
71C22×C4×D73C2311yescharacteristic
72C22×C4×D73C2311yescharacteristic
73#14556549—24no—
74#14556549—24no—
75#14556549—24no—
76#14556549—24no—
77#14556549—24no—
78#14556549—24no—
79#14556549—24no—
80#14556549D418yes—
81C2×C4⋊Dic73—24no—
82C2×C4⋊Dic73C2×C411yescharacteristic
83C2×C4⋊Dic73C2×C411yescharacteristic
84C2×C4⋊Dic73C2×C411yescharacteristic
85C2×C4⋊Dic73C2×C411yescharacteristic
86C2×#11924005—24no—
87C2×#11924005—24no—
88C2×#11924005C2×C411yescharacteristic
89C2×#11924005C2×C411yescharacteristic
90C2×#11924005C2×C411yescharacteristic
91C2×#11924005C2×C411yescharacteristic
92C2×#11924005C2×C411yescharacteristic
93C2×#11924005C2×C411yescharacteristic
94C2×#11924005C2×C411yescharacteristic
95C2×#11924005C2×C411yescharacteristic
96C2×C4×Dic73—24no—
97C2×C4×Dic73—24no—
98C2×C4×Dic73C2×C411yescharacteristic
99C2×C4×Dic73C2×C411yescharacteristic
100C2×C4×Dic73C2×C411yescharacteristic
101C2×C4×Dic73C2×C411yescharacteristic
102C2×C4×Dic73C2311yescharacteristic
103C2×C4×Dic73C2311yescharacteristic
104C2×C4×Dic73C2311yescharacteristic
105C2×C4×Dic73C2311yescharacteristic
106C23×C4×C73C2×C411yescharacteristic
107C23×C4×C73C2×C411yescharacteristic
108C23×C4×C73C2×C411yescharacteristic
109C2×C4⋊C4×C73—24no—
110C2×C4⋊C4×C73C2×C411yescharacteristic
111C2×C4⋊C4×C73C2×C411yescharacteristic
112C2×C4⋊C4×C73C2×C411yescharacteristic
113C2×C4⋊C4×C73C2×C411yescharacteristic
114C2×#10596833—22no—
115C2×#10596833—22no—
116C2×#10596833—22no—
117C2×#10596833—22no—
118C2×#10596833—22no—
119C2×#10596833—22no—
120C2×#10596833—22no—
121C2×#10596833—22no—
122C2×#10596833—22no—
123C2×#10596833—22no—
124C2×#10596833—22no—
125C2×#10596833—22no—
126C23×D73—22no—
127C23×D73—22no—
128C23×D73—22no—
129C23×D73—22no—
130C23×D73—22no—
131C23×D73—41no—
132C23×D73—41no—
133C23×D73C22×C411yescharacteristic
134C23×D73C22×C411yescharacteristic
135C22×Dic73—22no—
136C22×Dic73—22no—
137C22×Dic73—22no—
138C22×Dic73—22no—
139C22×Dic73—22no—
140C22×Dic73—22no—
141C22×Dic73—41no—
142C22×Dic73—41no—
143C22×Dic73C22×C411yescharacteristic
144C22×Dic73C22×C411yescharacteristic
145C22×Dic73C22×C411yescharacteristic
146C22×Dic73C22×C411yescharacteristic
147C22×Dic73C22×C411yescharacteristic
148C22×Dic73#4112yes—
149C22×Dic73#4112yes—
150C22×Dic73C2411yescharacteristic
151C2×C4×D73—216no—
152C2×C4×D73—216no—
153C2×C4×D73—22no—
154C2×C4×D73—22no—
155C2×C4×D73—22no—
156C2×C4×D73—22no—
157C2×C4×D73—22no—
158C2×C4×D73—22no—
159C2×C4×D73—22no—
160C2×C4×D73—21no—
161C2×C4×D73—21no—
162C2×C4×D73—22no—
163C2×C4×D73—21no—
164C2×C4×D73—21no—
165C2×C4×D73C2×D414yes—
166C2×C4×D73#4114yes—
167C4⋊Dic73—22no—
168C4⋊Dic73—22no—
169C4⋊Dic73—22no—
170C4⋊Dic73C22⋊C414yes—
171#11924005—22no—
172#11924005—22no—
173#11924005—22no—
174#11924005—22no—
175#11924005—22no—
176#11924005—22no—
177#11924005C22⋊C414yes—
178#11924005C22⋊C414yes—
179C4×Dic73—22no—
180C4×Dic73—22no—
181C4×Dic73—22no—
182C4×Dic73—22no—
183C4×Dic73—22no—
184C4×Dic73—22no—
185C4×Dic73C2×D414yes—
186C4×Dic73C2×D414yes—
187C24×C73C22×C411yescharacteristic
188C22×C4×C73—22no—
189C22×C4×C73—22no—
190C22×C4×C73—22no—
191C22×C4×C73—22no—
192C22×C4×C73—22no—
193C22×C4×C73—22no—
194C22×C4×C73—22no—
195C22×C4×C73—41no—
196C22×C4×C73—41no—
197C22×C4×C73C22×C411yescharacteristic
198C22×C4×C73C22×C411yescharacteristic
199C22×C4×C73C22×C411yescharacteristic
200C22×C4×C73C22×C411yescharacteristic
201C22×C4×C73C22×C411yescharacteristic
202C22×C4×C73C22×C411yescharacteristic
203C4⋊C4×C73—22no—
204C4⋊C4×C73—22no—
205C4⋊C4×C73—22no—
206C4⋊C4×C73C22⋊C414yes—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73#11712yes—
212C23×C73#13112yes—
213C23×C73#13112yes—
214C23×C73C23×C411yescharacteristic
215C22×D73—28no—
216C22×D73—21no—
217C22×D73—21no—
218C22×D73—21no—
219C22×D73—21no—
220C22×D73—21no—
221C22×D73—21no—
222C22×D73—21no—
223C22×D73—21no—
224C22×D73—21no—
225C22×D73—21no—
226C22×D73—21no—
227C22×D73—21no—
228C22×D73—44no—
229C22×D73—44no—
230C22×D73—44no—
231C22×D73—44no—
232C22×D73—44no—
233C22×D73—44no—
234C22×D73C2×C22⋊C412yes—
235C22×D73#11712yes—
236C2×C4×C73—28no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
251C2×C4×C73—44no—
252C2×C4×C73—44no—
253C2×C4×C73C2×C22⋊C412yes—
254C2×C4×C73C2×C22⋊C412yes—
255C2×C4×C73C2×C22⋊C412yes—
256C2×C4×C73#11712yes—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—24no—
264C2×Dic73—24no—
265C2×Dic73—21no—
266C2×Dic73—21no—
267C2×Dic73—21no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—21no—
271C2×Dic73—44no—
272C2×Dic73—44no—
273C2×Dic73C2×C22⋊C412yes—
274C2×Dic73C2×C22⋊C412yes—
275C2×Dic73#11712yes—
276C2×Dic73C22×D412yes—
277C2×Dic73C2×#4111yescharacteristic
278C2×Dic73C2×#4111yescharacteristic
279C2×Dic73C2×#4112yes—
280C4×D73—28no—
281C4×D73—28no—
282C4×D73—24no—
283C4×D73—24no—
284C4×D73—24no—
285C4×D73—24no—
286#10596833—42no—
287#10596833—42no—
288#10596833—42no—
289#10596833—42no—
290C22×C73—22no—
291C22×C73—22no—
292C22×C73—22no—
293C22×C73—22no—
294C22×C73—22no—
295C22×C73—22no—
296C22×C73#43314yescomplemented
297C22×C73C22×C22⋊C411yescharacteristic
298C22×C73C2×#11711yescharacteristic
299C22×C73C2×#11711yescharacteristic
300C22×C73C2×#13111yescharacteristic
301C22×C73C2×#13111yescharacteristic
302C22×C73#56811yescharacteristic
303C22×C73#56811yescharacteristic
304C2×D73—22no—
305C2×D73—22no—
306C2×D73—22no—
307C2×D73—22no—
308C2×D73—22no—
309C2×D73—22no—
310C2×D73—416no—
311C2×D73—41no—
312C2×D73—41no—
313C2×D73—41no—
314C2×D73—41no—
315C2×D73—41no—
316C2×D73—41no—
317C2×D73—41no—
318C2×D73—41no—
319C4×C73—22no—
320C4×C73—22no—
321C4×C73—22no—
322C4×C73—22no—
323C4×C73—41no—
324C4×C73—41no—
325Dic73—22no—
326Dic73—22no—
327Dic73—22no—
328Dic73—41no—
329Dic73—41no—
330Dic73#52512yes—
331Dic73#52512yes—
332#20716—731nomaximal, Sylow
333C2×C73—24no—
334C2×C73#228011yescharacteristic
335C2×C73#288011yescharacteristic
336C2×C73#288011yescharacteristic
337C2×C73C2×#43311yescharacteristic
338C2×C73#292111yescharacteristic
339C2×C73#292411yescharacteristic
340C2×C73#292411yescharacteristic
341D73—42no—
342D73—42no—
343C22×#117—731no—
344C2×#433—732no—
345#1818—732no—
346#1798—732no—
347#1797—731no—
348#1797—731no—
349#1779—731no—
350#1779—731no—
351#1750—732no—
352#1749—731no—
353C4×C22⋊C8—731no—
354C73#2071611yescharacteristic, complemented, Sylow
355C24×C4—731no—
356C2×#117—731no—
357C2×#117—731no—
358C2×#117—731no—
359C2×#117—731no—
360C2×#117—731no—
361C2×#117—731no—
362C2×#117—731no—
363C2×#117—731no—
364C2×#117—1468no—
365C22×C4⋊C4—731no—
366C22×C4⋊C4#6440811yescharacteristic
367C22×C22⋊C4—731no—
368C22×C22⋊C4—731no—
369C22×C42—731no—
370C22×C42—731no—
371#433—1468no—
372C2×C4⋊C8—732no—
373C2×C4⋊C8—732no—
374C2×C22⋊C8—731no—
375C2×C22⋊C8—731no—
376C2×C22⋊C8—731no—
377C2×C22⋊C8—731no—
378C2×C22⋊C8—732no—
379C2×C22⋊C8—732no—
380C2×C22⋊C8—732no—
381C2×C22⋊C8—732no—
382C2×#97—732no—
383C2×C4×C8—732no—
384#336—732no—
385#336—732no—
386#336—732no—
387#336—732no—
388C25—731no—
389C23×C4—731no—
390C23×C4—731no—
391C23×C4—731no—
392C23×C4—731no—
393C23×C4—731no—
394C23×C4—731no—
395C23×C4—731no—
396C23×C4—1464no—
397C23×C4—1462no—
398C23×C4—1462no—
399C23×C4—1464no—
400C23×C4C2×#6440811yescharacteristic
401C23×C4C2×#6440811yescharacteristic
402C23×C4C2×#6440811yescharacteristic
403C22×C8—732no—
404C22×C8—732no—
405C22×C8—732no—
406C22×C8—732no—
407#117—738no—
408#117—1464no—
409#117—1464no—
410#117—1464no—
411#117—1464no—
412#117—1464no—
413#117—1464no—
414#117—1464no—
415C2×C4⋊C4—24no—
416C2×C4⋊C4—731no—
417C2×C4⋊C4—731no—
418C2×C4⋊C4—731no—
419C2×C4⋊C4—731no—
420C2×C4⋊C4—1464no—
421C2×C4⋊C4C2×#6440811yescharacteristic
422C2×C4⋊C4C2×#6440811yescharacteristic
423C2×C4⋊C4C2×#6440811yescharacteristic
424C2×C4⋊C4C2×#6440811yescharacteristic
425C2×C22⋊C4—731no—
426C2×C22⋊C4—731no—
427C2×C22⋊C4—731no—
428C2×C22⋊C4—731no—
429C2×C22⋊C4—731no—
430C2×C22⋊C4—731no—
431C2×C22⋊C4—731no—
432C2×C22⋊C4—731no—
433C2×C22⋊C4—1464no—
434C2×C22⋊C4—1464no—
435C2×C42—731no—
436C2×C42—731no—
437C2×C42—731no—
438C2×C42—731no—
439C2×C42—731no—
440C2×C42—731no—
441C2×C42—731no—
442C2×C42—731no—
443C2×C42—1464no—
444C2×C42—1464no—
445C4⋊C8—1464no—
446C4⋊C8—1464no—
447C22⋊C8—1464no—
448C22⋊C8—1464no—
449C22⋊C8—1462no—
450C22⋊C8—1462no—
451C22⋊C8—1462no—
452C22⋊C8—1462no—
453C22⋊C8—1464no—
454C22⋊C8—1464no—
455#97—1464no—
456C4×C8—1464no—
457C24—2921no—
458C24—2921no—
459C24—731no—
460C24—731no—
461C24—1462no—
462C24—1462no—
463C24—1462no—
464C24—1462no—
465C24—1462no—
466C24C22×#6440811yescharacteristic
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—2921no—
477C22×C4—2921no—
478C22×C4—731no—
479C22×C4—731no—
480C22×C4—731no—
481C22×C4—731no—
482C22×C4—732no—
483C22×C4—732no—
484C22×C4—734no—
485C22×C4—734no—
486C22×C4—731no—
487C22×C4—731no—
488C22×C4—1462no—
489C22×C4—1461no—
490C22×C4—1461no—
491C22×C4—1462no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4—1462no—
495C22×C4—1462no—
496C22×C4—1462no—
497C22×C4—1462no—
498C22×C4—14616no—
499C22×C4—14616no—
500C22×C4—1462no—
501C22×C4—1462no—
502C22×C4—1462no—
503C22×C4—1462no—
504C22×C4—1462no—
505C22×C4—1462no—
506C22×C4—1462no—
507C22×C4—1462no—
508C22×C4C22×#6440811yescharacteristic
509C22×C4C22×#6440811yescharacteristic
510C22×C4C22×#6440811yescharacteristic
511C22×C4C22×#6440811yescharacteristic
512C22×C4C22×#6440811yescharacteristic
513C22×C4C22×#6440811yescharacteristic
514C2×C8—1462no—
515C2×C8—1462no—
516C2×C8—1462no—
517C2×C8—1462no—
518C2×C8—1462no—
519C2×C8—1462no—
520C2×C8—1462no—
521C2×C8—1462no—
522C2×C8—1462no—
523C2×C8—1462no—
524C2×C8—1462no—
525C2×C8—1462no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—734no—
530C4⋊C4—1462no—
531C4⋊C4—1462no—
532C4⋊C4—1462no—
533C4⋊C4C22⋊#6440814yes—
534C22⋊C4—734no—
535C22⋊C4—734no—
536C22⋊C4—1462no—
537C22⋊C4—1462no—
538C22⋊C4—1462no—
539C22⋊C4—1462no—
540C22⋊C4—1462no—
541C22⋊C4—1462no—
542C42—734no—
543C42—734no—
544C42—1462no—
545C42—1462no—
546C42—1462no—
547C42—1462no—
548C42—1462no—
549C42—1462no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—2924no—
555C23—2924no—
556C23—2924no—
557C23—2924no—
558C23—2924no—
559C23—2924no—
560C23—732no—
561C23—732no—
562C23—1468no—
563C23—1461no—
564C23—1461no—
565C23—1461no—
566C23—1461no—
567C23—1461no—
568C23—1461no—
569C23—1461no—
570C23—1461no—
571C23—1461no—
572C23—1461no—
573C23—1461no—
574C23—1461no—
575C23#1455667012yes—
576C23#1455667812yes—
577C23#1455667912yes—
578C23C23×#6440811yescharacteristic
579C2×C4—28no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—2924no—
597C2×C4—2924no—
598C2×C4—732no—
599C2×C4—731no—
600C2×C4—731no—
601C2×C4—732no—
602C2×C4—732no—
603C2×C4—732no—
604C2×C4—732no—
605C2×C4—1464no—
606C2×C4—1464no—
607C2×C4—1464no—
608C2×C4—1464no—
609C2×C4—1468no—
610C2×C4—1468no—
611C2×C4—1461no—
612C2×C4—1461no—
613C2×C4—1461no—
614C2×C4—1461no—
615C2×C4—1461no—
616C2×C4—1461no—
617C2×C4—1464no—
618C2×C4—1464no—
619C2×C4—1461no—
620C2×C4—1461no—
621C2×C4—1461no—
622C2×C4—1461no—
623C2×C4—1461no—
624C2×C4—1461no—
625C2×C4C2×C22⋊#6440812yes—
626C2×C4C2×C22⋊#6440812yes—
627C2×C4C2×C22⋊#6440812yes—
628C2×C4#1455667012yes—
629C8—2922no—
630C8—2922no—
631C8—2922no—
632C8—2922no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—29216no—
640C22—2921no—
641C22—2921no—
642C22—2921no—
643C22—2921no—
644C22—2921no—
645C22—2921no—
646C22—2921no—
647C22—2921no—
648C22—1462no—
649C22—1462no—
650C22—1462no—
651C22—1462no—
652C22—1462no—
653C22—1462no—
654C22#1798961214yescomplemented
655C22C22×C22⋊#6440811yescharacteristic
656C22C2×#1455667011yescharacteristic
657C22C2×#1455667011yescharacteristic
658C22C2×#1455667811yescharacteristic
659C22C2×#1455667911yescharacteristic
660C22#1798966511yescharacteristic
661C22#1798966711yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—41no—
667C4—41no—
668C4—2921no—
669C4—2921no—
670C4—732no—
671C4—732no—
672C4—1462no—
673C4—1462no—
674C4—1462no—
675C2—24no—
676C2—2922no—
677C2—2922no—
678C2#2262637511yescharacteristic
679C2#2262651911yescharacteristic
680C2#2262652011yescharacteristic
681C2C2×#1798961211yescharacteristic
682C2#2262661311yescharacteristic
683C2#2262662111yescharacteristic
684C2#2262662211yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
FittingC22×C4⋊C4×C73C4
prime radicalC24×D73C2×C4
cosocleC22×Dic73C24
hypercenterC22×C4⋊C4#64408
hypocenterC73#20716
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#17989612C2semidirect
2. #22625825C2semidirect
3. #22625790C2semidirect
4. #22625787C2semidirect
5. #22625785C2semidirect
8. #22625660C2semidirect
9. #22625657C2semidirect
10. C4×C22⋊#10596833C2semidirect
13. C2×#14556658C22semidirect
14. C2×C4⋊#10596833C22semidirect
23. C2×#14556646C22semidirect
24. C2×C4×#10596833C22semidirect
25. #17989453C22semidirect
26. #17989453C22semidirect
27. #17989453C22semidirect
28. #17989453C22semidirect
296. C22×C73#433semidirect
354. C73#20716semidirect
654. C22#17989612semidirect

Maximal subgroups

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

subgroupindex
C2×#179896122
#226258252
#226257902
#226257872
#226257852
#226257542
#226256602
#226256572
C4×C22⋊#105968332
C22×#145565492
#2071673