Finite Group Databasebeta

← #33007231

Subgroups — #33007231

Order 18688.

2102 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556547C211yesmaximal, characteristic
2C2×#17988445C212yesmaximal
3C2×C4×#11924007C211yesmaximal, characteristic, complemented
4C2×#17988438C211yesmaximal, characteristic, complemented
5#22608168C212yesmaximal
6#22608168C211yesmaximal, characteristic
7#22608166C211yesmaximal, characteristic, complemented
8#22608164C211yesmaximal, characteristic, complemented
9#22608162C212yesmaximal, complemented
10#22608159C212yesmaximal, complemented
11C73×#1437C211yesmaximal, characteristic
12C24×Dic73C2211yescharacteristic
13C2×#14556547—24no—
14C2×#14556547—24no—
15C2×#14556547C418yescomplemented
16C22×#11924007C2212yes—
17C22×#11924007C2212yes—
18C22×#11924007C2211yescharacteristic
19C22×#11924007C2211yescharacteristic
20C22×C22⋊Dic73C2211yescharacteristic
21C22×C22⋊Dic73C2212yes—
22C22×C4×Dic73C2211yescharacteristic
23C22×C4×Dic73C2212yes—
24C22×C4×Dic73C2211yescharacteristic
25C22×C4×Dic73C2211yescharacteristic
26#17988445C4116yescomplemented
27C4×#11924007—24no—
28#17988438—24no—
29C2×#14556498C2212yes—
30C2×#14556498C2211yescharacteristic
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C2×#14556498C2212yes—
35C2×#14556498C2212yes—
36C22×C22⋊C4×C73C2211yescharacteristic
37C22×C22⋊C4×C73C2212yes—
38C22×C42×C73C2211yescharacteristic
39C2×#95×C73C2211yescharacteristic
40C2×#95×C73C2212yes—
41C23×Dic73—22no—
42C23×Dic73—22no—
43C23×Dic73—24no—
44C23×Dic73C2×C418yes—
45C23×Dic73C2×C414yes—
46C23×Dic73C2312yes—
47C23×Dic73C2312yes—
48C23×Dic73C2311yescharacteristic
49C23×Dic73C2311yescharacteristic
50C23×Dic73C2311yescharacteristic
51C23×Dic73C2311yescharacteristic
52#14556547—232no—
53C2×#11924007—24no—
54C2×#11924007—24no—
55C2×#11924007—22no—
56C2×#11924007—22no—
57C2×#11924007—24no—
58C2×#11924007—24no—
59C2×#11924007—22no—
60C2×#11924007—22no—
61C2×#11924007C2×C418yes—
62C2×#11924007C2×C418yes—
63C2×#11924007C2×C414yes—
64C2×#11924007C2×C414yes—
65C2×C22⋊Dic73—22no—
66C2×C22⋊Dic73—22no—
67C2×C22⋊Dic73—42no—
68C2×C22⋊Dic73—42no—
69C2×C22⋊Dic73C2×C414yes—
70C2×C22⋊Dic73C2×C418yes—
71C2×C4×Dic73—22no—
72C2×C4×Dic73—22no—
73C2×C4×Dic73—24no—
74C2×C4×Dic73—22no—
75C2×C4×Dic73—42no—
76C2×C4×Dic73—42no—
77C2×C4×Dic73—42no—
78C2×C4×Dic73C2×C414yes—
79C2×C4×Dic73C2×C414yes—
80C2×C4×Dic73D414yes—
81C2×C4×Dic73D414yes—
82#14556498—24no—
83#14556498—22no—
84#14556498—42no—
85#14556498—42no—
86#14556498—42no—
87#14556498C2×C418yes—
88#14556498C2×C418yes—
89C25×C73C2311yescharacteristic
90C23×C4×C73C2312yes—
91C23×C4×C73C2312yes—
92C23×C4×C73C2311yescharacteristic
93C23×C4×C73C2311yescharacteristic
94C2×C22⋊C4×C73—22no—
95C2×C22⋊C4×C73—22no—
96C2×C22⋊C4×C73—42no—
97C2×C22⋊C4×C73—42no—
98C2×C22⋊C4×C73C2×C414yes—
99C2×C22⋊C4×C73C2×C418yes—
100C2×C42×C73—22no—
101C2×C42×C73—42no—
102#95×C73—22no—
103#95×C73—42no—
104C22×Dic73—21no—
105C22×Dic73—21no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—28no—
110C22×Dic73—28no—
111C22×Dic73—216no—
112C22×Dic73—216no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—21no—
116C22×Dic73—21no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73—22no—
128C22×Dic73—22no—
129C22×Dic73C42116yescomplemented
130C22×Dic73C22×C414yes—
131C22×Dic73C22×C414yes—
132C22×Dic73C22×C412yes—
133C22×Dic73C22×C414yes—
134C22×Dic73C22×C412yes—
135C22×Dic73C22×C412yes—
136C22×Dic73C22×C412yes—
137C22×Dic73C22×C414yes—
138C22×Dic73C2×D412yes—
139C22×Dic73C2×D412yes—
140C22×Dic73C2×D412yes—
141C22×Dic73C2×D412yes—
142C22×Dic73#4114yes—
143#11924007—28no—
144#11924007—28no—
145#11924007—216no—
146#11924007—216no—
147C22⋊Dic73—28no—
148C22⋊Dic73—48no—
149C4×Dic73—28no—
150C4×Dic73—48no—
151C4×Dic73—44no—
152C4×Dic73C22⋊C4116yescomplemented
153C24×C73—21no—
154C24×C73—21no—
155C24×C73—22no—
156C24×C73—22no—
157C24×C73—22no—
158C24×C73—22no—
159C24×C73—22no—
160C24×C73C22×C412yes—
161C24×C73C22×C414yes—
162C24×C73C2411yescharacteristic
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—22no—
170C22×C4×C73—22no—
171C22×C4×C73—22no—
172C22×C4×C73—22no—
173C22×C4×C73—22no—
174C22×C4×C73—22no—
175C22×C4×C73—22no—
176C22×C4×C73—22no—
177C22×C4×C73—22no—
178C22×C4×C73—22no—
179C22×C4×C73—22no—
180C22×C4×C73—22no—
181C22×C4×C73C22×C414yes—
182C22×C4×C73C22×C414yes—
183C22×C4×C73C22×C412yes—
184C22×C4×C73C22×C412yes—
185C22×C4×C73C2×D412yes—
186C22×C4×C73C2×D412yes—
187C22⋊C4×C73—28no—
188C22⋊C4×C73—48no—
189C42×C73—44no—
190C23×C73—22no—
191C23×C73—22no—
192C23×C73—24no—
193C23×C73—24no—
194C23×C73—24no—
195C23×C73—24no—
196C23×C73—24no—
197C23×C73—24no—
198C23×C73—24no—
199C23×C73—24no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73—42no—
205C23×C73—42no—
206C23×C73—42no—
207C23×C73—42no—
208C23×C73—42no—
209C23×C73—42no—
210C23×C73C2×C4214yes—
211C23×C73C23×C411yescharacteristic
212C23×C73C23×C412yes—
213C23×C73C22×D411yescharacteristic
214C23×C73C22×D411yescharacteristic
215C23×C73C2×#4112yes—
216C23×C73H(5,𝔽2)12yes—
217C23×C73H(5,𝔽2)12yes—
218C23×C732−1+412yes—
219C23×C732−1+412yes—
220C2×C4×C73—24no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—24no—
224C2×C4×C73—22no—
225C2×C4×C73—24no—
226C2×C4×C73—24no—
227C2×C4×C73—24no—
228C2×C4×C73—24no—
229C2×C4×C73—24no—
230C2×C4×C73—24no—
231C2×C4×C73—24no—
232C2×C4×C73—42no—
233C2×C4×C73—41no—
234C2×C4×C73—42no—
235C2×C4×C73—41no—
236C2×C4×C73—42no—
237C2×C4×C73—42no—
238C2×C4×C73—42no—
239C2×C4×C73—42no—
240C2×C4×C73—42no—
241C2×C4×C73—42no—
242C2×C4×C73—42no—
243C2×C4×C73C2×C22⋊C414yes—
244C2×Dic73—22no—
245C2×Dic73—22no—
246C2×Dic73—24no—
247C2×Dic73—24no—
248C2×Dic73—24no—
249C2×Dic73—22no—
250C2×Dic73—22no—
251C2×Dic73—24no—
252C2×Dic73—232no—
253C2×Dic73—22no—
254C2×Dic73—24no—
255C2×Dic73—24no—
256C2×Dic73—24no—
257C2×Dic73—24no—
258C2×Dic73—24no—
259C2×Dic73—24no—
260C2×Dic73—24no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—42no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73C2×C4214yes—
273C2×Dic73C2×C4214yes—
274C2×Dic73C2×C22⋊C414yes—
275C2×Dic73C2×C22⋊C414yes—
276C2×Dic73#11714yes—
277C2×Dic73C4×D418yes—
278C2×Dic73C4×D418yes—
279C22×C73—24no—
280C22×C73—28no—
281C22×C73—42no—
282C22×C73—42no—
283C22×C73—44no—
284C22×C73—44no—
285C22×C73—44no—
286C22×C73—44no—
287C22×C73—44no—
288C22×C73C22×C4211yescharacteristic
289C22×C73C22×C22⋊C411yescharacteristic
290C22×C73C2×#11711yescharacteristic
291C22×C73C2×C4×D412yes—
292C22×C73C2×C4×D412yes—
293C22×C73#51812yes—
294C22×C73#51912yes—
295C22×C73#52012yes—
296C22×C73#52012yes—
297C22×C73#52012yes—
298C22×C73#52012yes—
299C22×C73#53511yescharacteristic
300C22×C73#53511yescharacteristic
301C22×C73#53611yescharacteristic
302C22×C73#53611yescharacteristic
303C22×C73#53712yes—
304C22×C73#53712yes—
305C22×C73#54012yes—
306C22×C73#54112yes—
307C22×C73#54212yes—
308C22×C73#54212yes—
309C4×C73—24no—
310C4×C73—42no—
311C4×C73—44no—
312C4×C73—44no—
313Dic73—24no—
314Dic73—24no—
315Dic73—42no—
316Dic73—44no—
317Dic73—44no—
318Dic73C4×C22⋊C418yes—
319#13777—731nomaximal, Sylow
320C2×C73—44no—
321C2×C73C2×C4×C22⋊C411yescharacteristic
322C2×C73#227412yes—
323C2×C73#231111yescharacteristic
324C2×C73#231111yescharacteristic
325C2×C73#231412yes—
326C2×C73#232312yes—
327C2×C73#232312yes—
328C2×C73#232412yes—
329C2×C73#232412yes—
330C22×#117—731no—
331C2×#380—732no—
332C2×C4×C4⋊C4—731no—
333C2×C4×C4⋊C4—731no—
334#1437—731no—
335#1437—732no—
336#1437D7311yescharacteristic
337#1436—731no—
338#1436—732no—
339#1436—731no—
340C4⋊#95—732no—
341C73#1377711yescharacteristic, complemented, Sylow
342C24×C4—731no—
343C2×#117—738no—
344C2×#117—1464no—
345C2×#117—1464no—
346C22×C4⋊C4—731no—
347C22×C4⋊C4—731no—
348C22×C4⋊C4—732no—
349C22×C4⋊C4—732no—
350C22×C22⋊C4—732no—
351C22×C22⋊C4—731no—
352C22×C22⋊C4C2×D7311yescharacteristic
353C22×C22⋊C4C2×D7312yes—
354C22×C42—732no—
355C22×C42—731no—
356C22×C42—731no—
357C22×C42—731no—
358C22×C42C2×D7311yescharacteristic
359#380—7316no—
360C4×C4⋊C4—1464no—
361C4×C4⋊C4—1464no—
362C2×#95—732no—
363C2×#95—732no—
364C2×#95—732no—
365C2×#95—732no—
366C2×#95—732no—
367C2×#95—732no—
368C2×#95—731no—
369C2×#95C2×D7311yescharacteristic
370C2×#95C2×D7312yes—
371C25C22×D7311yescharacteristic
372C23×C4—731no—
373C23×C4—734no—
374C23×C4—738no—
375C23×C4—731no—
376C23×C4—731no—
377C23×C4—732no—
378C23×C4—732no—
379C23×C4—731no—
380C23×C4—1462no—
381C23×C4—1462no—
382C23×C4—1464no—
383C23×C4C22×D7312yes—
384C23×C4C22×D7312yes—
385C23×C4C22×D7311yescharacteristic
386C23×C4C22×D7311yescharacteristic
387#117—14632no—
388C2×C4⋊C4—734no—
389C2×C4⋊C4—734no—
390C2×C4⋊C4—738no—
391C2×C4⋊C4—738no—
392C2×C4⋊C4—1462no—
393C2×C4⋊C4—1462no—
394C2×C4⋊C4—1464no—
395C2×C4⋊C4—1464no—
396C2×C4⋊C4—1464no—
397C2×C4⋊C4—1464no—
398C2×C4⋊C4—1462no—
399C2×C4⋊C4—1462no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—42no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—2922no—
405C2×C22⋊C4—2922no—
406C2×C22⋊C4—738no—
407C2×C22⋊C4—734no—
408C2×C22⋊C4—1462no—
409C2×C22⋊C4—1462no—
410C2×C22⋊C4C4×D7314yes—
411C2×C22⋊C4C4×D7318yes—
412C2×C42—22no—
413C2×C42—42no—
414C2×C42—2922no—
415C2×C42—2922no—
416C2×C42—2922no—
417C2×C42—734no—
418C2×C42—734no—
419C2×C42—734no—
420C2×C42—734no—
421C2×C42—1464no—
422C2×C42—1462no—
423C2×C42—1462no—
424C2×C42—1462no—
425#95—22no—
426#95—42no—
427#95—2922no—
428#95—2922no—
429#95—2922no—
430#95—738no—
431#95—738no—
432#95—1464no—
433#95—1462no—
434C24—21no—
435C24—21no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24C2×C4×D7312yes—
442C24C2×C4×D7314yes—
443C24C23×D7311yescharacteristic
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—732no—
463C22×C4—734no—
464C22×C4—732no—
465C22×C4—732no—
466C22×C4—734no—
467C22×C4—7316no—
468C22×C4—734no—
469C22×C4—732no—
470C22×C4—732no—
471C22×C4—732no—
472C22×C4—732no—
473C22×C4—734no—
474C22×C4—734no—
475C22×C4—732no—
476C22×C4—1462no—
477C22×C4—1462no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1462no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4—1462no—
491C22×C4—1462no—
492C22×C4—1468no—
493C22×C4—1468no—
494C22×C4—14616no—
495C22×C4—14616no—
496C22×C4—1461no—
497C22×C4—1461no—
498C22×C4—1462no—
499C22×C4—1462no—
500C22×C4—1462no—
501C22×C4C2×C4×D7314yes—
502C22×C4C2×C4×D7314yes—
503C22×C4C2×C4×D7312yes—
504C22×C4C2×C4×D7312yes—
505C22×C4D4×D7312yes—
506C22×C4D4×D7312yes—
507C4⋊C4—14616no—
508C4⋊C4—14616no—
509C4⋊C4—1468no—
510C4⋊C4—1468no—
511C22⋊C4—28no—
512C22⋊C4—48no—
513C22⋊C4—2928no—
514C22⋊C4—1468no—
515C42—44no—
516C42—2924no—
517C42—2928no—
518C42—7316no—
519C42—1468no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23C42×D7314yes—
541C23C22×C4×D7311yescharacteristic
542C23C22×C4×D7312yes—
543C23C2×D4×D7311yescharacteristic
544C23C2×D4×D7311yescharacteristic
545C23#41×D7312yes—
546C23#1455663612yes—
547C23#1455663612yes—
548C23#1455663812yes—
549C23#1455663812yes—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—2922no—
574C2×C4—2922no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2922no—
578C2×C4—2922no—
579C2×C4—2922no—
580C2×C4—2922no—
581C2×C4—2922no—
582C2×C4—2922no—
583C2×C4—2922no—
584C2×C4—734no—
585C2×C4—734no—
586C2×C4—734no—
587C2×C4—738no—
588C2×C4—738no—
589C2×C4—734no—
590C2×C4—734no—
591C2×C4—1462no—
592C2×C4—1464no—
593C2×C4—1464no—
594C2×C4—1464no—
595C2×C4—1464no—
596C2×C4—1464no—
597C2×C4—1464no—
598C2×C4—1464no—
599C2×C4—14632no—
600C2×C4—1462no—
601C2×C4—1462no—
602C2×C4—1464no—
603C2×C4—1462no—
604C2×C4—1462no—
605C2×C4—1464no—
606C2×C4—1464no—
607C2×C4—1464no—
608C2×C4C22⋊C4×D7314yes—
609C22—24no—
610C22—28no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22C2×C42×D7311yescharacteristic
619C22C2×C22⋊C4×D7311yescharacteristic
620C22#117×D7311yescharacteristic
621C22C4×D4×D7312yes—
622C22C4×D4×D7312yes—
623C22#1798906012yes—
624C22#1798906412yes—
625C22#1798906612yes—
626C22#1798906612yes—
627C22#1798906612yes—
628C22#1798906612yes—
629C22#1798915011yescharacteristic
630C22#1798915211yescharacteristic
631C22#1798915711yescharacteristic
632C22#1798915911yescharacteristic
633C22#1798916412yes—
634C22#1798916612yes—
635C22#1798918712yes—
636C22#1798919412yes—
637C22#1798920112yes—
638C22#1798920512yes—
639C4—24no—
640C4—42no—
641C4—44no—
642C4—44no—
643C4—2922no—
644C4—2924no—
645C4—2924no—
646C4—738no—
647C4—1464no—
648C4—1464no—
649C2—44no—
650C2C4×C22⋊C4×D7311yescharacteristic
651C2#2261228612yes—
652C2#2261251811yescharacteristic
653C2#2261252111yescharacteristic
654C2#2261253512yes—
655C2#2261260612yes—
656C2#2261261012yes—
657C2#2261261712yes—
658C2#2261262112yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC22×C73C22×C42
FittingC73×#1437C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenter#1437D73
hypocenterC73#13777
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
3. C2×C4×#11924007C2semidirect
4. C2×#17988438C2semidirect
7. #22608166C2semidirect
8. #22608164C2semidirect
9. #22608162C2semidirect
10. #22608159C2semidirect
15. C2×#14556547C4semidirect
26. #17988445C4semidirect
129. C22×Dic73C42semidirect
152. C4×Dic73C22⋊C4semidirect
341. C73#13777semidirect

Maximal subgroups

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

subgroupindex
C22×#145565472
C2×#179884452
C2×C4×#119240072
C2×#179884382
#226081682
#226081662
#226081642
#226081622
#226081592
C73×#14372
#1377773