Finite Group Databasebeta

← #33008377

Subgroups — #33008377

Order 18688.

2166 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D73C211yesmaximal, characteristic
2C2×#17988446C211yesmaximal, characteristic
3C2×C4×#11924005C211yesmaximal, characteristic
4#22608183C212yesmaximal, complemented
5#22608182C212yesmaximal, complemented
6#22608166C211yesmaximal, characteristic, complemented
7#22608164C211yesmaximal, characteristic, complemented
8#22608163C211yesmaximal, characteristic, complemented
9#22608152C212yesmaximal
10#22608152C212yesmaximal
11C73×#1436C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C23×C4×D73C2211yescharacteristic
14C23×C4×D73C2211yescharacteristic
15C22×C4⋊Dic73C2211yescharacteristic
16C22×#11924007C2211yescharacteristic
17C22×#11924007C2211yescharacteristic
18C22×#11924007C2212yes—
19C2×C4⋊C4×D73—24no—
20C2×C4⋊C4×D73—24no—
21C2×C4⋊C4×D73C418yescomplemented
22C22×#11924005C2212yes—
23C22×#11924005C2212yes—
24C22×C4×Dic73C2211yescharacteristic
25#17988446—24no—
26C4×#11924005—24no—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2211yescharacteristic
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C2×#14556498C2212yes—
35C22×C4⋊C4×C73C2211yescharacteristic
36C22×C42×C73C2211yescharacteristic
37C2×#95×C73C2212yes—
38C2×#95×C73C2211yescharacteristic
39C2×#95×C73C2212yes—
40C24×D73C2311yescharacteristic
41C23×Dic73C2312yes—
42C23×Dic73C2311yescharacteristic
43C23×Dic73C2312yes—
44C23×Dic73C2311yescharacteristic
45C23×Dic73C2311yescharacteristic
46C22×C4×D73—22no—
47C22×C4×D73—22no—
48C22×C4×D73—24no—
49C22×C4×D73—22no—
50C22×C4×D73—22no—
51C22×C4×D73—24no—
52C22×C4×D73—42no—
53C22×C4×D73—42no—
54C22×C4×D73—42no—
55C22×C4×D73—42no—
56C22×C4×D73C2×C414yes—
57C22×C4×D73C2×C414yes—
58C22×C4×D73C2×C414yes—
59C22×C4×D73D414yes—
60C22×C4×D73D414yes—
61C2×C4⋊Dic73—22no—
62C2×C4⋊Dic73—22no—
63C2×C4⋊Dic73C2×C414yes—
64C2×#11924007—22no—
65C2×#11924007—22no—
66C2×#11924007—24no—
67C2×#11924007—22no—
68C2×#11924007—22no—
69C2×#11924007—42no—
70C2×#11924007—42no—
71C2×#11924007C2×C414yes—
72C2×#11924007C2×C414yes—
73C4⋊C4×D73—232no—
74C2×#11924005—24no—
75C2×#11924005—24no—
76C2×#11924005—24no—
77C2×#11924005—24no—
78C2×#11924005—24no—
79C2×#11924005Q818yes—
80C2×C4×Dic73—22no—
81C2×C4×Dic73—42no—
82#14556498—24no—
83#14556498—24no—
84#14556498—24no—
85#14556498—22no—
86#14556498—24no—
87#14556498—42no—
88#14556498—42no—
89#14556498—42no—
90C23×C4×C73C2312yes—
91C23×C4×C73C2311yescharacteristic
92C23×C4×C73C2312yes—
93C23×C4×C73C2311yescharacteristic
94C23×C4×C73C2311yescharacteristic
95C2×C4⋊C4×C73—22no—
96C2×C4⋊C4×C73—22no—
97C2×C4⋊C4×C73C2×C414yes—
98C2×C42×C73—22no—
99C2×C42×C73—42no—
100#95×C73—22no—
101#95×C73—24no—
102#95×C73—42no—
103C23×D73—22no—
104C23×D73—22no—
105C23×D73—22no—
106C23×D73—22no—
107C23×D73C22×C412yes—
108C23×D73C2×D412yes—
109C23×D73C2×D412yes—
110C23×D73C2×Q814yes—
111C23×D73#4114yes—
112C22×Dic73—21no—
113C22×Dic73—21no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—21no—
119C22×Dic73—21no—
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×Dic73—22no—
130C22×Dic73—22no—
131C22×Dic73—22no—
132C22×Dic73—22no—
133C22×Dic73—22no—
134C22×Dic73C22×C412yes—
135C22×Dic73C22×C412yes—
136C22×Dic73C22×C412yes—
137C22×Dic73C2×D412yes—
138C22×Dic73C2×D412yes—
139C22×Dic73C2×Q814yes—
140C22×Dic73#4114yes—
141C2×C4×D73—216no—
142C2×C4×D73—216no—
143C2×C4×D73—28no—
144C2×C4×D73—28no—
145C2×C4×D73—28no—
146C2×C4×D73—28no—
147C2×C4×D73—48no—
148C2×C4×D73—48no—
149C2×C4×D73—48no—
150C2×C4×D73—44no—
151C2×C4×D73—44no—
152C2×C4×D73—48no—
153C2×C4×D73—44no—
154C2×C4×D73—44no—
155C2×C4×D73C22⋊C4116yes—
156C4⋊Dic73—28no—
157#11924007—28no—
158#11924007—28no—
159#11924007—48no—
160#11924005—216no—
161#11924005—48no—
162C4×Dic73—44no—
163C24×C73C2411yescharacteristic
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73—22no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
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×C73—22no—
182C22×C4×C73—22no—
183C22×C4×C73—22no—
184C22×C4×C73—22no—
185C22×C4×C73—22no—
186C22×C4×C73C22×C412yes—
187C22×C4×C73C22×C412yes—
188C22×C4×C73C22×C412yes—
189C22×C4×C73C2×D412yes—
190C22×C4×C73C2×D412yes—
191C22×C4×C73C2×Q814yes—
192C22×C4×C73#4114yes—
193C4⋊C4×C73—28no—
194C42×C73—44no—
195C23×C73C23×C411yescharacteristic
196C23×C73C22×D411yescharacteristic
197C23×C73C22×D411yescharacteristic
198C23×C73C22×Q812yes—
199C23×C73C2×#4112yes—
200C23×C73C2×#4112yes—
201C23×C73C2×#4112yes—
202C23×C73H(5,𝔽2)12yes—
203C23×C73H(5,𝔽2)12yes—
204C22×D73—22no—
205C22×D73—22no—
206C22×D73—24no—
207C22×D73—24no—
208C22×D73—24no—
209C22×D73—24no—
210C22×D73—22no—
211C22×D73—22no—
212C22×D73—24no—
213C22×D73—24no—
214C22×D73—24no—
215C22×D73—24no—
216C22×D73—24no—
217C22×D73—24no—
218C22×D73—24no—
219C22×D73—24no—
220C22×D73C2×C22⋊C414yes—
221C22×D73C4×Q818yes—
222C22×D73C22⋊Q818yes—
223C22×D73#12414yes—
224C22×D73C4⋊Q814yes—
225C2×C4×C73—22no—
226C2×C4×C73—22no—
227C2×C4×C73—24no—
228C2×C4×C73—24no—
229C2×C4×C73—24no—
230C2×C4×C73—22no—
231C2×C4×C73—22no—
232C2×C4×C73—24no—
233C2×C4×C73—22no—
234C2×C4×C73—24no—
235C2×C4×C73—24no—
236C2×C4×C73—24no—
237C2×C4×C73—24no—
238C2×C4×C73—42no—
239C2×C4×C73—42no—
240C2×C4×C73—42no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—42no—
244C2×C4×C73—42no—
245C2×C4×C73—42no—
246C2×C4×C73—42no—
247C2×C4×C73—42no—
248C2×C4×C73—42no—
249C2×C4×C73—42no—
250C2×C4×C73—42no—
251C2×C4×C73—42no—
252C2×C4×C73C2×C22⋊C414yes—
253C2×C4×C73#11714yes—
254C2×Dic73—22no—
255C2×Dic73—22no—
256C2×Dic73—24no—
257C2×Dic73—24no—
258C2×Dic73—24no—
259C2×Dic73—24no—
260C2×Dic73—22no—
261C2×Dic73—22no—
262C2×Dic73—22no—
263C2×Dic73—24no—
264C2×Dic73—24no—
265C2×Dic73—24no—
266C2×Dic73—24no—
267C2×Dic73—42no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—42no—
273C2×Dic73—42no—
274C2×Dic73—42no—
275C2×Dic73—42no—
276C2×Dic73—42no—
277C2×Dic73—42no—
278C2×Dic73—42no—
279C2×Dic73—42no—
280C2×Dic73—42no—
281C2×Dic73C2×C22⋊C414yes—
282C2×Dic73#11714yes—
283C4×D73—232no—
284C4×D73—416no—
285C4×D73—416no—
286C22×C73C22×C22⋊C411yescharacteristic
287C22×C73C2×#11711yescharacteristic
288C22×C73C2×#11711yescharacteristic
289C22×C73C2×C4×Q812yes—
290C22×C73#51812yes—
291C22×C73C2×C22⋊Q812yes—
292C22×C73#52512yes—
293C22×C73#52512yes—
294C22×C73C2×#12411yescharacteristic
295C22×C73C2×C4⋊Q811yescharacteristic
296C22×C73#53412yes—
297C22×C73#53411yescharacteristic
298C22×C73#53411yescharacteristic
299C22×C73#55312yes—
300C22×C73#55312yes—
301C22×C73#55312yes—
302C22×C73#55312yes—
303C22×C73#55412yes—
304C22×C73#55412yes—
305C22×C73#55412yes—
306C22×C73#55412yes—
307C2×D73—24no—
308C2×D73—28no—
309C2×D73—28no—
310C2×D73—28no—
311C2×D73—28no—
312C2×D73—28no—
313C2×D73—28no—
314C2×D73—24no—
315C2×D73#39118yes—
316C4×C73—24no—
317C4×C73—24no—
318C4×C73—42no—
319C4×C73—44no—
320C4×C73—82no—
321Dic73—24no—
322Dic73—24no—
323Dic73—42no—
324Dic73—44no—
325Dic73—82no—
326#13923—731nomaximal, Sylow
327C2×C73#228011yescharacteristic
328C2×C73C2×#39111yescharacteristic
329C2×C73#230811yescharacteristic
330C2×C73#234512yes—
331C2×C73#234512yes—
332C2×C73#257912yes—
333C2×C73#257912yes—
334C2×C73#258312yes—
335C2×C73#259112yes—
336D73—28no—
337C23×C4⋊C4—731no—
338C2×#380—731no—
339C2×C4×C22⋊C4—731no—
340#1441—732no—
341#1441—732no—
342#1436—731no—
343#1436—731no—
344#1436—731no—
345#1436D7311yescharacteristic, complemented
346C22⋊#95—732no—
347C22⋊#95—732no—
348C73#1392311yescharacteristic, complemented, Sylow
349C24×C4—731no—
350C24×C4—731no—
351C24×C4—731no—
352C22×C4⋊C4—732no—
353C22×C4⋊C4—731no—
354C22×C4⋊C4—731no—
355C22×C4⋊C4—731no—
356C22×C4⋊C4—738no—
357C22×C4⋊C4—1464no—
358C22×C4⋊C4—1464no—
359C22×C4⋊C4C2×D7311yescharacteristic
360C22×C22⋊C4—732no—
361C22×C22⋊C4—732no—
362C22×C42—731no—
363C22×C42C2×D7311yescharacteristic
364#380—1464no—
365C4×C22⋊C4—1464no—
366C2×#95—732no—
367C2×#95—732no—
368C2×#95—732no—
369C2×#95—732no—
370C2×#95—732no—
371C2×#95—732no—
372C2×#95—731no—
373C2×#95—732no—
374C2×#95C2×D7312yes—
375C2×#95C2×D7311yescharacteristic
376C2×#95C2×D7312yes—
377C25—731no—
378C23×C4—2922no—
379C23×C4—2922no—
380C23×C4—2922no—
381C23×C4—2922no—
382C23×C4—734no—
383C23×C4—734no—
384C23×C4—734no—
385C23×C4—734no—
386C23×C4—731no—
387C23×C4—731no—
388C23×C4—732no—
389C23×C4—731no—
390C23×C4—734no—
391C23×C4—732no—
392C23×C4—1462no—
393C23×C4—1462no—
394C23×C4—1464no—
395C23×C4—1462no—
396C23×C4—1462no—
397C23×C4—1464no—
398C23×C4C22×D7312yes—
399C23×C4C22×D7311yescharacteristic
400C23×C4C22×D7312yes—
401C23×C4C22×D7311yescharacteristic
402C23×C4C22×D7311yescharacteristic
403C2×C4⋊C4—22no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—2922no—
406C2×C4⋊C4—2922no—
407C2×C4⋊C4—734no—
408C2×C4⋊C4—734no—
409C2×C4⋊C4—734no—
410C2×C4⋊C4—14632no—
411C2×C4⋊C4—1464no—
412C2×C4⋊C4—1462no—
413C2×C4⋊C4—1462no—
414C2×C4⋊C4—1462no—
415C2×C4⋊C4—1462no—
416C2×C4⋊C4—1462no—
417C2×C4⋊C4—1462no—
418C2×C4⋊C4C4×D7314yes—
419C2×C22⋊C4—738no—
420C2×C22⋊C4—1464no—
421C2×C22⋊C4—1464no—
422C2×C22⋊C4—1464no—
423C2×C22⋊C4—1464no—
424C2×C22⋊C4—1464no—
425C2×C42—22no—
426C2×C42—42no—
427C2×C42—2922no—
428C2×C42—1462no—
429#95—22no—
430#95—24no—
431#95—42no—
432#95—2922no—
433#95—2922no—
434#95—2922no—
435#95—1464no—
436#95—1464no—
437#95—1464no—
438#95—1462no—
439#95—1464no—
440C24—732no—
441C24—732no—
442C24—732no—
443C24—734no—
444C24—734no—
445C24—1462no—
446C24—1462no—
447C24—1462no—
448C24—1462no—
449C24C23×D7311yescharacteristic
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—2928no—
473C22×C4—2924no—
474C22×C4—2924no—
475C22×C4—2928no—
476C22×C4—2928no—
477C22×C4—2928no—
478C22×C4—2924no—
479C22×C4—2924no—
480C22×C4—7316no—
481C22×C4—732no—
482C22×C4—732no—
483C22×C4—732no—
484C22×C4—732no—
485C22×C4—734no—
486C22×C4—732no—
487C22×C4—734no—
488C22×C4—1468no—
489C22×C4—1468no—
490C22×C4—1468no—
491C22×C4—14616no—
492C22×C4—14616no—
493C22×C4—1462no—
494C22×C4—1462no—
495C22×C4—1462no—
496C22×C4—1462no—
497C22×C4—1462no—
498C22×C4—1462no—
499C22×C4—1462no—
500C22×C4—1462no—
501C22×C4—1461no—
502C22×C4—1461no—
503C22×C4—1462no—
504C22×C4—1462no—
505C22×C4—1462no—
506C22×C4—1462no—
507C22×C4—1462no—
508C22×C4—1462no—
509C22×C4—1462no—
510C22×C4—1461no—
511C22×C4—1461no—
512C22×C4—1462no—
513C22×C4—1462no—
514C22×C4—1462no—
515C22×C4—1468no—
516C22×C4C2×C4×D7312yes—
517C22×C4C2×C4×D7312yes—
518C22×C4C2×C4×D7312yes—
519C22×C4D4×D7312yes—
520C22×C4D4×D7312yes—
521C22×C4Q8×D7314yes—
522C22×C4#1192402814yes—
523C4⋊C4—28no—
524C4⋊C4—2928no—
525C4⋊C4—1468no—
526C4⋊C4—1468no—
527C4⋊C4—1468no—
528C22⋊C4—2928no—
529C22⋊C4—14616no—
530C42—44no—
531C42—2924no—
532C23—734no—
533C23—734no—
534C23—738no—
535C23—738no—
536C23—734no—
537C23—1464no—
538C23—1462no—
539C23—1462no—
540C23—1464no—
541C23—1464no—
542C23—1464no—
543C23—1464no—
544C23—1464no—
545C23—1464no—
546C23—1464no—
547C23—1462no—
548C23—1462no—
549C23—1464no—
550C23—1464no—
551C23—1464no—
552C23—1464no—
553C23C22×C4×D7311yescharacteristic
554C23C2×D4×D7311yescharacteristic
555C23C2×D4×D7311yescharacteristic
556C23C2×Q8×D7312yes—
557C23#41×D7312yes—
558C23#41×D7312yes—
559C23C2×#1192402812yes—
560C23#1455663612yes—
561C23#1455663712yes—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—29216no—
590C2×C4—29216no—
591C2×C4—2922no—
592C2×C4—2921no—
593C2×C4—2922no—
594C2×C4—2921no—
595C2×C4—2922no—
596C2×C4—2922no—
597C2×C4—2922no—
598C2×C4—2922no—
599C2×C4—2922no—
600C2×C4—2922no—
601C2×C4—2922no—
602C2×C4—2922no—
603C2×C4—2922no—
604C2×C4—2922no—
605C2×C4—734no—
606C2×C4—734no—
607C2×C4—14632no—
608C2×C4—1462no—
609C2×C4—1464no—
610C2×C4—1464no—
611C2×C4—1464no—
612C2×C4—1464no—
613C2×C4—1464no—
614C2×C4—1462no—
615C2×C4—1462no—
616C2×C4—1462no—
617C2×C4—1462no—
618C2×C4—1464no—
619C2×C4—1464no—
620C2×C4—1464no—
621C2×C4C22⋊C4×D7314yes—
622C2×C4#1455654814yes—
623C22—738no—
624C22—1468no—
625C22—1464no—
626C22—1468no—
627C22—1468no—
628C22—1464no—
629C22—1468no—
630C22—1468no—
631C22—1468no—
632C22C2×C22⋊C4×D7311yescharacteristic
633C22#117×D7311yescharacteristic
634C22C2×#1455654811yescharacteristic
635C22C4×Q8×D7312yes—
636C22#1798905912yes—
637C22C22⋊Q8×D7312yes—
638C22#1798909612yes—
639C22#1798909612yes—
640C22#124×D7311yescharacteristic
641C22C4⋊Q8×D7311yescharacteristic
642C22#1798914512yes—
643C22#1798914711yescharacteristic
644C22#1798914811yescharacteristic
645C22#1798928112yes—
646C22#1798928212yes—
647C22#1798928712yes—
648C22#1798928812yes—
649C22#1798929112yes—
650C22#1798929212yes—
651C22#1798929312yes—
652C22#1798929712yes—
653C4—24no—
654C4—24no—
655C4—42no—
656C4—44no—
657C4—82no—
658C4—2922no—
659C4—2924no—
660C4—5842no—
661C4—1464no—
662C4—1464no—
663C2—1468no—
664C2#2261231211yescharacteristic
665C2#391×D7311yescharacteristic
666C2#2261250511yescharacteristic
667C2#2261280412yes—
668C2#2261280512yes—
669C2#2261527612yes—
670C2#2261528212yes—
671C2#2261530912yes—
672C2#2261540912yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1436C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenter#1436D73
hypocenterC73#13923
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
4. #22608183C2semidirect
5. #22608182C2semidirect
6. #22608166C2semidirect
7. #22608164C2semidirect
8. #22608163C2semidirect
11. C73×#1436C2semidirect
21. C2×C4⋊C4×D73C4semidirect
345. #1436D73semidirect
348. C73#13923semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D732
C2×#179884462
C2×C4×#119240052
#226081832
#226081822
#226081662
#226081642
#226081632
#226081522
C73×#14362
#1392373