Finite Group Databasebeta

← #33008688

Subgroups — #33008688

Order 18688.

2078 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556547C211yesmaximal, characteristic
2C2×#17988448C211yesmaximal, characteristic
3C22×#14556498C211yesmaximal, characteristic
4#22608197C212yesmaximal, complemented
5#22608197C212yesmaximal, complemented
6#22608157C212yesmaximal, complemented
7#22608156C212yesmaximal, complemented
8#22608154C211yesmaximal, characteristic
9#22608153C211yesmaximal, characteristic
10#22608153C211yesmaximal, characteristic
11C73×C22⋊#95C211yesmaximal, characteristic
12C24×Dic73C2211yescharacteristic
13C24×Dic73C2211yescharacteristic
14C2×#14556547—24no—
15C2×#14556547—24no—
16C2×#14556547C418yescomplemented
17C22×#11924007C2211yescharacteristic
18C22×#11924007C2211yescharacteristic
19C22×C22⋊Dic73C2211yescharacteristic
20C22×C22⋊Dic73C2211yescharacteristic
21C22×C4×Dic73C2211yescharacteristic
22C22×C4×Dic73C2211yescharacteristic
23#17988448—24no—
24C2×#14556498—24no—
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C2×#14556498C2212yes—
35C24×C4×C73C2211yescharacteristic
36C22×C22⋊C4×C73C2211yescharacteristic
37C22×C22⋊C4×C73C2211yescharacteristic
38C2×#95×C73C2212yes—
39C2×#95×C73C2212yes—
40C23×Dic73—24no—
41C23×Dic73—22no—
42C23×Dic73—22no—
43C23×Dic73—22no—
44C23×Dic73—22no—
45C23×Dic73—22no—
46C23×Dic73—22no—
47C23×Dic73—22no—
48C23×Dic73—42no—
49C23×Dic73C2×C414yes—
50C23×Dic73D414yes—
51C23×Dic73D414yes—
52C23×Dic73C2312yes—
53C23×Dic73C2311yescharacteristic
54C23×Dic73C2311yescharacteristic
55C23×Dic73C2312yes—
56C23×Dic73C2311yescharacteristic
57C23×Dic73C2311yescharacteristic
58#14556547—232no—
59C2×#11924007—22no—
60C2×#11924007—22no—
61C2×#11924007—22no—
62C2×#11924007—22no—
63C2×#11924007C2×C414yescomplemented
64C2×#11924007C2×C414yescomplemented
65C2×C22⋊Dic73—22no—
66C2×C22⋊Dic73—22no—
67C2×C22⋊Dic73—22no—
68C2×C22⋊Dic73—22no—
69C2×C22⋊Dic73—22no—
70C2×C22⋊Dic73C2×C414yes—
71C2×C4×Dic73—22no—
72C2×C4×Dic73—22no—
73C2×C4×Dic73—22no—
74C2×C4×Dic73—22no—
75C2×C4×Dic73C2×C414yescomplemented
76C2×C4×Dic73C2×C414yescomplemented
77#14556498—24no—
78#14556498—24no—
79#14556498—24no—
80#14556498—24no—
81#14556498—24no—
82#14556498—24no—
83#14556498—24no—
84#14556498—42no—
85#14556498—42no—
86#14556498—42no—
87C25×C73C2311yescharacteristic
88C23×C4×C73—22no—
89C23×C4×C73—24no—
90C23×C4×C73—22no—
91C23×C4×C73—22no—
92C23×C4×C73—42no—
93C23×C4×C73C2312yes—
94C23×C4×C73C2311yescharacteristic
95C23×C4×C73C2311yescharacteristic
96C23×C4×C73C2312yes—
97C2×C22⋊C4×C73—22no—
98C2×C22⋊C4×C73—22no—
99C2×C22⋊C4×C73—22no—
100C2×C22⋊C4×C73—22no—
101C2×C22⋊C4×C73—22no—
102C2×C22⋊C4×C73C2×C414yes—
103#95×C73—24no—
104#95×C73—42no—
105C22×Dic73—28no—
106C22×Dic73—28no—
107C22×Dic73—28no—
108C22×Dic73—28no—
109C22×Dic73—28no—
110C22×Dic73—28no—
111C22×Dic73—21no—
112C22×Dic73—21no—
113C22×Dic73—21no—
114C22×Dic73—21no—
115C22×Dic73—21no—
116C22×Dic73—21no—
117C22×Dic73—21no—
118C22×Dic73—21no—
119C22×Dic73—28no—
120C22×Dic73—22no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73—22no—
128C22×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73—22no—
131C22×Dic73—22no—
132C22×Dic73—22no—
133C22×Dic73—22no—
134C22×Dic73—22no—
135C22×Dic73—22no—
136C22×Dic73—22no—
137C22×Dic73—22no—
138C22×Dic73—22no—
139C22×Dic73—22no—
140C22×Dic73—22no—
141C22×Dic73—48no—
142C22×Dic73—48no—
143C22×Dic73—44no—
144C22×Dic73—44no—
145C22×Dic73C22⋊C4116yes—
146C22×Dic73C22×C412yes—
147C22×Dic73C22×C412yes—
148C22×Dic73C22×C412yes—
149C22×Dic73C22×C412yes—
150C22×Dic73C2×D412yes—
151C22×Dic73C2×D412yes—
152C22×Dic73C2×D412yes—
153C22×Dic73C2×D412yes—
154C22×Dic73#4114yes—
155#11924007—28no—
156#11924007—28no—
157C22⋊Dic73—28no—
158C22⋊Dic73—44no—
159C4×Dic73—28no—
160C4×Dic73—28no—
161C24×C73—22no—
162C24×C73—22no—
163C24×C73—22no—
164C24×C73—22no—
165C24×C73C22×C412yes—
166C24×C73C2×D412yes—
167C24×C73C2×D412yes—
168C24×C73#4112yes—
169C24×C73#4112yes—
170C24×C73#4112yes—
171C24×C73#4112yes—
172C24×C73C2411yescharacteristic
173C22×C4×C73—28no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—22no—
179C22×C4×C73—22no—
180C22×C4×C73—22no—
181C22×C4×C73—22no—
182C22×C4×C73—22no—
183C22×C4×C73—22no—
184C22×C4×C73—22no—
185C22×C4×C73—22no—
186C22×C4×C73—22no—
187C22×C4×C73—22no—
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—48no—
196C22×C4×C73—48no—
197C22×C4×C73—44no—
198C22×C4×C73—44no—
199C22×C4×C73C22×C412yes—
200C22×C4×C73C22×C412yes—
201C22×C4×C73#4114yes—
202C22⋊C4×C73—28no—
203C22⋊C4×C73—44no—
204C23×C73—24no—
205C23×C73—24no—
206C23×C73—22no—
207C23×C73—22no—
208C23×C73—24no—
209C23×C73—22no—
210C23×C73—22no—
211C23×C73—24no—
212C23×C73—24no—
213C23×C73—24no—
214C23×C73—24no—
215C23×C73—24no—
216C23×C73—24no—
217C23×C73—24no—
218C23×C73—24no—
219C23×C73—24no—
220C23×C73C2×C22⋊C414yes—
221C23×C73#11714yes—
222C23×C73#11714yes—
223C23×C73#12314yes—
224C23×C73#12314yes—
225C23×C73#12314yes—
226C23×C73#12314yes—
227C23×C73C23×C411yescharacteristic
228C23×C73C22×D411yescharacteristic
229C23×C73C22×D411yescharacteristic
230C23×C73C2×#4112yes—
231C23×C73C2×#4111yescharacteristic
232C23×C73C2×#4112yes—
233C23×C73C2×#4111yescharacteristic
234C23×C73C2×#4111yescharacteristic
235C23×C73C2×#4111yescharacteristic
236C23×C73H(5,𝔽2)12yes—
237C23×C732−1+412yes—
238C2×C4×C73—22no—
239C2×C4×C73—22no—
240C2×C4×C73—24no—
241C2×C4×C73—24no—
242C2×C4×C73—24no—
243C2×C4×C73—24no—
244C2×C4×C73—24no—
245C2×C4×C73—416no—
246C2×C4×C73—42no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—42no—
250C2×C4×C73—41no—
251C2×C4×C73—42no—
252C2×C4×C73—41no—
253C2×C4×C73—42no—
254C2×C4×C73—42no—
255C2×C4×C73—42no—
256C2×C4×C73—42no—
257C2×C4×C73—42no—
258C2×C4×C73—42no—
259C2×C4×C73—42no—
260C2×C4×C73—42no—
261C2×C4×C73—42no—
262C2×C4×C73—42no—
263C2×Dic73—22no—
264C2×Dic73—22no—
265C2×Dic73—22no—
266C2×Dic73—22no—
267C2×Dic73—22no—
268C2×Dic73—22no—
269C2×Dic73—22no—
270C2×Dic73—22no—
271C2×Dic73—22no—
272C2×Dic73—22no—
273C2×Dic73—22no—
274C2×Dic73—22no—
275C2×Dic73—232no—
276C2×Dic73—22no—
277C2×Dic73—22no—
278C2×Dic73—24no—
279C2×Dic73—24no—
280C2×Dic73—24no—
281C2×Dic73—24no—
282C2×Dic73—24no—
283C2×Dic73—416no—
284C2×Dic73—42no—
285C2×Dic73—41no—
286C2×Dic73—41no—
287C2×Dic73—42no—
288C2×Dic73—41no—
289C2×Dic73—42no—
290C2×Dic73—41no—
291C2×Dic73—42no—
292C2×Dic73—42no—
293C2×Dic73—42no—
294C2×Dic73—42no—
295C2×Dic73—42no—
296C2×Dic73—42no—
297C2×Dic73—42no—
298C2×Dic73—42no—
299C2×Dic73—42no—
300C2×Dic73—42no—
301C2×Dic73C2×C22⋊C414yes—
302C2×Dic73C2×C22⋊C414yes—
303C22×C73—24no—
304C22×C73—28no—
305C22×C73—28no—
306C22×C73—24no—
307C22×C73—24no—
308C22×C73—24no—
309C22×C73—24no—
310C22×C73—28no—
311C22×C73—28no—
312C22×C73—24no—
313C22×C73#38118yes—
314C22×C73C22×C22⋊C411yescharacteristic
315C22×C73C2×#11711yescharacteristic
316C22×C73C2×#11711yescharacteristic
317C22×C73C4×#4112yes—
318C22×C73#52012yes—
319C22×C73C2×#12311yescharacteristic
320C22×C73C2×#12311yescharacteristic
321C22×C73C2×#12311yescharacteristic
322C22×C73C2×#12311yescharacteristic
323C22×C73#52511yescharacteristic
324C22×C73#52511yescharacteristic
325C22×C73#53212yes—
326C22×C73#53511yescharacteristic
327C22×C73#53611yescharacteristic
328C22×C73#53712yes—
329C22×C73#55112yes—
330C22×C73#55112yes—
331C22×C73#55112yes—
332C22×C73#55112yes—
333C22×C73#55212yes—
334C22×C73#55212yes—
335C22×C73#55212yes—
336C22×C73#55212yes—
337C4×C73—42no—
338C4×C73—44no—
339C4×C73—42no—
340C4×C73—82no—
341Dic73—24no—
342Dic73—24no—
343Dic73—42no—
344Dic73—44no—
345Dic73—42no—
346Dic73—82no—
347#13958—731nomaximal, Sylow
348C2×C73—28no—
349C2×C73C2×#38111yescharacteristic
350C2×C73#229711yescharacteristic
351C2×C73#231111yescharacteristic
352C2×C73#234312yes—
353C2×C73#234312yes—
354C2×C73#248211yescharacteristic
355C2×C73#248211yescharacteristic
356C2×C73#248411yescharacteristic
357C2×C73#248411yescharacteristic
358C2×C73#249812yes—
359C2×C73#249812yes—
360C22×#117—731no—
361C2×#381—731no—
362C22×#95—731no—
363#1445—732no—
364#1445—732no—
365#1434—732no—
366#1434—732no—
367C22⋊#95—731no—
368C22⋊#95—731no—
369C22⋊#95—731no—
370C22⋊#95D7311yescharacteristic
371C73#1395811yescharacteristic, complemented, Sylow
372C24×C4—731no—
373C24×C4—731no—
374C24×C4C2×D7311yescharacteristic
375C2×#117—738no—
376C2×#117—1464no—
377C2×#117—1464no—
378C22×C4⋊C4—731no—
379C22×C4⋊C4—731no—
380C22×C22⋊C4—731no—
381C22×C22⋊C4—731no—
382C22×C22⋊C4C2×D7311yescharacteristic
383C22×C22⋊C4C2×D7311yescharacteristic
384C22×C42—731no—
385C22×C42—731no—
386#381—1464no—
387C2×#95—732no—
388C2×#95—732no—
389C2×#95—732no—
390C2×#95—732no—
391C2×#95—732no—
392C2×#95—732no—
393C2×#95—732no—
394C2×#95—732no—
395C2×#95—732no—
396C2×#95—732no—
397C2×#95—1464no—
398C2×#95C2×D7312yes—
399C2×#95C2×D7312yes—
400C25C22×D7311yescharacteristic
401C23×C4—22no—
402C23×C4—24no—
403C23×C4—22no—
404C23×C4—22no—
405C23×C4—42no—
406C23×C4—2922no—
407C23×C4—734no—
408C23×C4—734no—
409C23×C4—734no—
410C23×C4—731no—
411C23×C4—731no—
412C23×C4—731no—
413C23×C4—731no—
414C23×C4—732no—
415C23×C4—732no—
416C23×C4—1462no—
417C23×C4—1462no—
418C23×C4—1464no—
419C23×C4—1462no—
420C23×C4—1462no—
421C23×C4—1462no—
422C23×C4—1462no—
423C23×C4—1462no—
424C23×C4C22×D7312yes—
425C23×C4C22×D7311yescharacteristic
426C23×C4C22×D7311yescharacteristic
427C23×C4C22×D7312yes—
428#117—14632no—
429C2×C4⋊C4—734no—
430C2×C4⋊C4—734no—
431C2×C4⋊C4—1462no—
432C2×C4⋊C4—1462no—
433C2×C4⋊C4—1462no—
434C2×C4⋊C4—1462no—
435C2×C22⋊C4—22no—
436C2×C22⋊C4—22no—
437C2×C22⋊C4—22no—
438C2×C22⋊C4—22no—
439C2×C22⋊C4—22no—
440C2×C22⋊C4—734no—
441C2×C22⋊C4—1462no—
442C2×C22⋊C4—1462no—
443C2×C22⋊C4—1462no—
444C2×C22⋊C4—1462no—
445C2×C22⋊C4—1462no—
446C2×C22⋊C4C4×D7314yes—
447C2×C42—734no—
448C2×C42—734no—
449C2×C42—1462no—
450C2×C42—1462no—
451C2×C42—1462no—
452C2×C42—1462no—
453#95—24no—
454#95—42no—
455#95—2922no—
456#95—2922no—
457#95—2922no—
458#95—1464no—
459#95—1464no—
460#95—1464no—
461#95—1464no—
462#95—1464no—
463#95—1464no—
464#95—1464no—
465C24—22no—
466C24—22no—
467C24—22no—
468C24—22no—
469C24C2×C4×D7312yes—
470C24D4×D7312yes—
471C24D4×D7312yes—
472C24#1192402712yes—
473C24#1192402812yes—
474C24#1192402812yes—
475C24#1192402912yes—
476C24C23×D7311yescharacteristic
477C22×C4—28no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—2924no—
504C22×C4—2924no—
505C22×C4—2928no—
506C22×C4—2928no—
507C22×C4—7316no—
508C22×C4—732no—
509C22×C4—732no—
510C22×C4—732no—
511C22×C4—732no—
512C22×C4—732no—
513C22×C4—732no—
514C22×C4—732no—
515C22×C4—732no—
516C22×C4—734no—
517C22×C4—1462no—
518C22×C4—1462no—
519C22×C4—1462no—
520C22×C4—1462no—
521C22×C4—1462no—
522C22×C4—1462no—
523C22×C4—1462no—
524C22×C4—1468no—
525C22×C4—1462no—
526C22×C4—1461no—
527C22×C4—1461no—
528C22×C4—1461no—
529C22×C4—1461no—
530C22×C4—1462no—
531C22×C4—1462no—
532C22×C4—1462no—
533C22×C4—1462no—
534C22×C4—1462no—
535C22×C4—1462no—
536C22×C4—1462no—
537C22×C4—1462no—
538C22×C4—1462no—
539C22×C4—1468no—
540C22×C4—1468no—
541C22×C4—1468no—
542C22×C4—1468no—
543C22×C4—1468no—
544C22×C4—1468no—
545C22×C4—1461no—
546C22×C4—1461no—
547C22×C4—1461no—
548C22×C4—1461no—
549C22×C4—1461no—
550C22×C4—1461no—
551C22×C4—1461no—
552C22×C4—1461no—
553C22×C4C2×C4×D7312yes—
554C22×C4C2×C4×D7312yes—
555C22×C4#1192402714yes—
556C4⋊C4—1468no—
557C4⋊C4—1468no—
558C22⋊C4—28no—
559C22⋊C4—44no—
560C22⋊C4—2924no—
561C22⋊C4—1468no—
562C42—1468no—
563C42—1468no—
564C23—24no—
565C23—22no—
566C23—22no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—22no—
579C23—22no—
580C23C22⋊C4×D7314yes—
581C23#1455654814yes—
582C23#1455654914yes—
583C23#1455657314yes—
584C23#1455657314yes—
585C23#1455657614yes—
586C23#1455657614yes—
587C23C22×C4×D7311yescharacteristic
588C23C2×D4×D7311yescharacteristic
589C23C2×D4×D7311yescharacteristic
590C23#41×D7312yes—
591C23C2×#1192402712yes—
592C23C2×#1192402711yescharacteristic
593C23C2×#1192402811yescharacteristic
594C23C2×#1192402811yescharacteristic
595C23C2×#1192402911yescharacteristic
596C23#1455663612yes—
597C23#1455663812yes—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—416no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—29216no—
624C2×C4—2922no—
625C2×C4—2922no—
626C2×C4—2921no—
627C2×C4—2921no—
628C2×C4—2922no—
629C2×C4—2922no—
630C2×C4—2922no—
631C2×C4—2922no—
632C2×C4—2921no—
633C2×C4—2921no—
634C2×C4—2922no—
635C2×C4—2922no—
636C2×C4—2922no—
637C2×C4—2922no—
638C2×C4—2922no—
639C2×C4—2922no—
640C2×C4—2922no—
641C2×C4—734no—
642C2×C4—734no—
643C2×C4—1464no—
644C2×C4—1464no—
645C2×C4—1462no—
646C2×C4—1462no—
647C2×C4—1464no—
648C2×C4—1464no—
649C2×C4—1464no—
650C2×C4—14632no—
651C2×C4—1462no—
652C2×C4—1462no—
653C2×C4—1462no—
654C2×C4—1462no—
655C2×C4—1462no—
656C2×C4—1462no—
657C2×C4—1462no—
658C2×C4—1462no—
659C2×C4—1462no—
660C2×C4—1462no—
661C2×C4—1462no—
662C2×C4—1462no—
663C22—24no—
664C22—28no—
665C22—28no—
666C22—28no—
667C22—28no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22#1798844918yes—
674C22C2×C22⋊C4×D7311yescharacteristic
675C22C2×#1455654811yescharacteristic
676C22C2×#1455654911yescharacteristic
677C22C4×#1192402712yes—
678C22#1798906612yes—
679C22C2×#1455657311yescharacteristic
680C22C2×#1455657311yescharacteristic
681C22C2×#1455657611yescharacteristic
682C22C2×#1455657611yescharacteristic
683C22#1798909311yescharacteristic
684C22#1798909511yescharacteristic
685C22#1798913212yes—
686C22#1798915211yescharacteristic
687C22#1798915911yescharacteristic
688C22#1798916412yes—
689C22#1798926312yes—
690C22#1798926412yes—
691C22#1798926612yes—
692C22#1798926812yes—
693C22#1798927012yes—
694C22#1798927412yes—
695C22#1798927712yes—
696C22#1798927912yes—
697C4—42no—
698C4—44no—
699C4—42no—
700C4—82no—
701C4—2922no—
702C4—2924no—
703C4—2922no—
704C4—5842no—
705C4—1464no—
706C4—1464no—
707C2—28no—
708C2C2×#1798844911yescharacteristic
709C2#2261241611yescharacteristic
710C2#2261251811yescharacteristic
711C2#2261278512yes—
712C2#2261278912yes—
713C2#2261420211yescharacteristic
714C2#2261420411yescharacteristic
715C2#2261421511yescharacteristic
716C2#2261421711yescharacteristic
717C2#2261437112yes—
718C2#2261437312yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×C22⋊#95C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC22⋊#95D73
hypocenterC73#13958
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
4. #22608197C2semidirect
5. #22608197C2semidirect
6. #22608157C2semidirect
7. #22608156C2semidirect
16. C2×#14556547C4semidirect
63. C2×#11924007C2×C4semidirect
64. C2×#11924007C2×C4semidirect
75. C2×C4×Dic73C2×C4semidirect
76. C2×C4×Dic73C2×C4semidirect
371. C73#13958semidirect

Maximal subgroups

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

subgroupindex
C22×#145565472
C2×#179884482
C22×#145564982
#226081972
#226081572
#226081562
#226081542
#226081532
C73×C22⋊#952
#1395873