Finite Group Databasebeta

← #33008737

Subgroups — #33008737

Order 18688.

2150 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988461C212yesmaximal
2C2×#17988458C212yesmaximal
3C2×#17988444C211yesmaximal, characteristic
4C2×#17988434C211yesmaximal, characteristic
5C2×#95×D73C211yesmaximal, characteristic
6#22608196C211yesmaximal, characteristic, complemented
7#22608195C211yesmaximal, characteristic, complemented
8#22608194C211yesmaximal, characteristic, complemented
9#22608160C212yesmaximal, complemented
10#22608159C212yesmaximal, complemented
11C73×#1444C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C23×C4×D73C2211yescharacteristic
14C23×C4×D73C2211yescharacteristic
15C22×C4⋊Dic73C2212yes—
16C22×C4⋊Dic73C2211yescharacteristic
17C22×#11924007C2212yes—
18C22×#11924007C2212yes—
19C22×#11924005C2212yes—
20C22×#11924005C2212yes—
21C22×C4×Dic73C2212yes—
22#17988461—28no—
23#17988458—28no—
24#17988444—24no—
25#17988434C418yescomplemented
26C2×#14556498C2211yescharacteristic
27C2×#14556498C2211yescharacteristic
28C2×#14556498C2211yescharacteristic
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2211yescharacteristic
34#95×D73—24no—
35C22×C4⋊C4×C73C2212yes—
36C22×C4⋊C4×C73C2211yescharacteristic
37C2×#95×C73C2211yescharacteristic
38C2×#95×C73C2212yes—
39C2×#95×C73C2211yescharacteristic
40C24×D73C2311yescharacteristic
41C23×Dic73C2312yes—
42C23×Dic73C2311yescharacteristic
43C23×Dic73C2311yescharacteristic
44C23×Dic73C2311yescharacteristic
45C23×Dic73C2312yes—
46C22×C4×D73—22no—
47C22×C4×D73—22no—
48C22×C4×D73—22no—
49C22×C4×D73—22no—
50C22×C4×D73—24no—
51C22×C4×D73—24no—
52C22×C4×D73—22no—
53C22×C4×D73—22no—
54C22×C4×D73—42no—
55C22×C4×D73—42no—
56C22×C4×D73—42no—
57C22×C4×D73—42no—
58C22×C4×D73C2×C414yes—
59C22×C4×D73D414yes—
60C22×C4×D73Q814yes—
61C2×C4⋊Dic73—24no—
62C2×C4⋊Dic73—22no—
63C2×C4⋊Dic73—24no—
64C2×C4⋊Dic73—42no—
65C2×C4⋊Dic73—42no—
66C2×#11924007—24no—
67C2×#11924007—24no—
68C2×#11924007—24no—
69C2×#11924007—24no—
70C2×#11924007—42no—
71C2×#11924007—42no—
72C2×#11924005—24no—
73C2×#11924005—24no—
74C2×#11924005—24no—
75C2×#11924005C2×C418yes—
76C2×#11924005D418yes—
77C2×#11924005Q818yes—
78C2×C4×Dic73—42no—
79C2×C4×Dic73—42no—
80C2×C4×Dic73C2×C418yescomplemented
81#14556498—22no—
82#14556498—22no—
83#14556498—24no—
84#14556498—24no—
85#14556498—24no—
86#14556498—22no—
87#14556498—24no—
88#14556498C2×C414yescomplemented
89C23×C4×C73C2312yes—
90C23×C4×C73C2311yescharacteristic
91C23×C4×C73C2311yescharacteristic
92C23×C4×C73C2311yescharacteristic
93C23×C4×C73C2312yes—
94C2×C4⋊C4×C73—22no—
95C2×C4⋊C4×C73—24no—
96C2×C4⋊C4×C73—24no—
97C2×C4⋊C4×C73—42no—
98C2×C4⋊C4×C73—42no—
99#95×C73—22no—
100#95×C73—24no—
101#95×C73C2×C414yescomplemented
102C23×D73—21no—
103C23×D73—21no—
104C23×D73—21no—
105C23×D73—21no—
106C23×D73—22no—
107C23×D73—22no—
108C23×D73C22×C412yes—
109C23×D73C2×D414yes—
110C23×D73C2×D412yes—
111C23×D73C2×Q814yes—
112C23×D73C2×Q812yes—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—21no—
120C22×Dic73—21no—
121C22×Dic73—22no—
122C22×Dic73—22no—
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×Dic73C22×C414yes—
135C22×Dic73C22×C412yes—
136C22×Dic73C2×D412yes—
137C22×Dic73C2×D414yes—
138C22×Dic73C2×Q814yes—
139C22×Dic73C2×Q812yes—
140C22×Dic73#4112yes—
141C22×Dic73#4112yes—
142C2×C4×D73—28no—
143C2×C4×D73—28no—
144C2×C4×D73—48no—
145C2×C4×D73—44no—
146C2×C4×D73—44no—
147C2×C4×D73—48no—
148C2×C4×D73—48no—
149C2×C4×D73—48no—
150C2×C4×D73—48no—
151C2×C4×D73—48no—
152C2×C4×D73—44no—
153C2×C4×D73—44no—
154C2×C4×D73—44no—
155C2×C4×D73—44no—
156C2×C4×D73C4⋊C4116yescomplemented
157C4⋊Dic73—48no—
158C4⋊Dic73—44no—
159#11924007—48no—
160#11924007—48no—
161#11924005—48no—
162#11924005C4⋊C4132yescomplemented
163C4×Dic73—48no—
164C24×C73C2411yescharacteristic
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73—22no—
170C22×C4×C73—22no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—22no—
174C22×C4×C73—22no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
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×C414yes—
188C22×C4×C73C2×D414yes—
189C22×C4×C73C2×D412yes—
190C22×C4×C73C2×Q814yes—
191C22×C4×C73C2×Q812yes—
192C22×C4×C73#4112yes—
193C22×C4×C73#4112yes—
194C4⋊C4×C73—48no—
195C4⋊C4×C73—44no—
196C23×C73C23×C411yescharacteristic
197C23×C73C22×D411yescharacteristic
198C23×C73C22×D412yes—
199C23×C73C22×Q811yescharacteristic
200C23×C73C22×Q812yes—
201C23×C73C2×#4111yescharacteristic
202C23×C73C2×#4111yescharacteristic
203C23×C73C2×#4111yescharacteristic
204C23×C73C2×#4111yescharacteristic
205C23×C73H(5,𝔽2)12yes—
206C23×C73H(5,𝔽2)12yes—
207C22×D73—22no—
208C22×D73—22no—
209C22×D73—24no—
210C22×D73—22no—
211C22×D73—22no—
212C22×D73—22no—
213C22×D73—22no—
214C22×D73—22no—
215C22×D73—22no—
216C22×D73—24no—
217C22×D73—24no—
218C22×D73—24no—
219C22×D73—24no—
220C22×D73—24no—
221C22×D73—24no—
222C22×D73—24no—
223C22×D73—24no—
224C22×D73—24no—
225C22×D73C2×C4⋊C418yes—
226C22×D73C2×C4⋊C414yes—
227C22×D73#12714yes—
228C22×D73C4⋊Q814yes—
229C22×D73C4⋊Q818yes—
230C2×C4×C73—24no—
231C2×C4×C73—24no—
232C2×C4×C73—24no—
233C2×C4×C73—22no—
234C2×C4×C73—22no—
235C2×C4×C73—24no—
236C2×C4×C73—24no—
237C2×C4×C73—24no—
238C2×C4×C73—24no—
239C2×C4×C73—24no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—24no—
243C2×C4×C73—24no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—42no—
249C2×C4×C73—42no—
250C2×C4×C73—42no—
251C2×C4×C73—42no—
252C2×C4×C73—42no—
253C2×C4×C73—42no—
254C2×C4×C73—42no—
255C2×C4×C73—42no—
256C2×C4×C73—42no—
257C2×C4×C73C2×C4⋊C418yes—
258C2×C4×C73C2×C4⋊C414yes—
259C2×Dic73—24no—
260C2×Dic73—24no—
261C2×Dic73—24no—
262C2×Dic73—22no—
263C2×Dic73—22no—
264C2×Dic73—24no—
265C2×Dic73—24no—
266C2×Dic73—24no—
267C2×Dic73—24no—
268C2×Dic73—24no—
269C2×Dic73—22no—
270C2×Dic73—22no—
271C2×Dic73—24no—
272C2×Dic73—24no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—42no—
278C2×Dic73—42no—
279C2×Dic73—42no—
280C2×Dic73—42no—
281C2×Dic73—42no—
282C2×Dic73—42no—
283C2×Dic73—42no—
284C2×Dic73—42no—
285C2×Dic73—42no—
286C2×Dic73C2×C4⋊C418yes—
287C2×Dic73C2×C4⋊C414yes—
288C4×D73—416no—
289C4×D73—416no—
290C4×D73—416no—
291C22×C73C22×C4⋊C411yescharacteristic
292C22×C73C22×C4⋊C412yes—
293C22×C73C4×#4111yescharacteristic
294C22×C73C4×#4111yescharacteristic
295C22×C73#51812yes—
296C22×C73#52511yescharacteristic
297C22×C73#52511yescharacteristic
298C22×C73C2×#12711yescharacteristic
299C22×C73C2×C4⋊Q811yescharacteristic
300C22×C73C2×C4⋊Q812yes—
301C22×C73#53311yescharacteristic
302C22×C73#53311yescharacteristic
303C22×C73#53411yescharacteristic
304C22×C73#53411yescharacteristic
305C22×C73#54312yes—
306C22×C73#54612yes—
307C22×C73#54612yes—
308C22×C73#54612yes—
309C22×C73#54612yes—
310C22×C73#55412yes—
311C22×C73#55412yes—
312C22×C73#55412yes—
313C22×C73#55412yes—
314C2×D73—24no—
315C2×D73—24no—
316C2×D73—24no—
317C2×D73—28no—
318C2×D73—28no—
319C2×D73—24no—
320C2×D73—24no—
321C2×D73—28no—
322C2×D73—28no—
323C2×D73—24no—
324C2×D73#38418yes—
325C4×C73—24no—
326C4×C73—28no—
327C4×C73—42no—
328C4×C73—42no—
329C4×C73—82no—
330Dic73—24no—
331Dic73—28no—
332Dic73—42no—
333Dic73—42no—
334Dic73—82no—
335#13965—731nomaximal, Sylow
336C2×C73C2×#38411yescharacteristic
337C2×C73#229611yescharacteristic
338C2×C73#231511yescharacteristic
339C2×C73#234912yes—
340C2×C73#234912yes—
341C2×C73#253911yescharacteristic
342C2×C73#253911yescharacteristic
343C2×C73#254912yes—
344C2×C73#254912yes—
345C2×C73#258311yescharacteristic
346C2×C73#258311yescharacteristic
347D73—28no—
348C2×#385—732no—
349C2×#385—732no—
350C2×#380—731no—
351C2×C4×C22⋊C4—731no—
352C22×#95—731no—
353#1444—731no—
354#1444—731no—
355#1444—731no—
356#1444D7311yescharacteristic, complemented
357C4⋊#95—732no—
358C4⋊#95—732no—
359C73#1396511yescharacteristic, complemented, Sylow
360C24×C4—731no—
361C24×C4—731no—
362C24×C4—731no—
363C22×C4⋊C4—731no—
364C22×C4⋊C4—732no—
365C22×C4⋊C4—732no—
366C22×C4⋊C4—732no—
367C22×C4⋊C4C2×D7312yes—
368C22×C4⋊C4C2×D7311yescharacteristic
369C22×C22⋊C4—732no—
370C22×C22⋊C4—732no—
371C22×C42—732no—
372#385—1468no—
373#385—1468no—
374#380—1464no—
375C4×C22⋊C4—738no—
376C2×#95—731no—
377C2×#95—731no—
378C2×#95—731no—
379C2×#95—732no—
380C2×#95—732no—
381C2×#95—732no—
382C2×#95—732no—
383C2×#95—731no—
384C2×#95—1464no—
385C2×#95C2×D7312yes—
386C2×#95C2×D7311yescharacteristic
387C2×#95C2×D7311yescharacteristic
388C25—731no—
389C23×C4—2922no—
390C23×C4—2922no—
391C23×C4—2922no—
392C23×C4—2922no—
393C23×C4—734no—
394C23×C4—734no—
395C23×C4—734no—
396C23×C4—732no—
397C23×C4—731no—
398C23×C4—731no—
399C23×C4—731no—
400C23×C4—732no—
401C23×C4—1464no—
402C23×C4—1464no—
403C23×C4—1462no—
404C23×C4—1462no—
405C23×C4—1462no—
406C23×C4—1462no—
407C23×C4—1462no—
408C23×C4—1462no—
409C23×C4C22×D7312yes—
410C23×C4C22×D7311yescharacteristic
411C23×C4C22×D7311yescharacteristic
412C23×C4C22×D7312yes—
413C23×C4C22×D7311yescharacteristic
414C2×C4⋊C4—22no—
415C2×C4⋊C4—24no—
416C2×C4⋊C4—24no—
417C2×C4⋊C4—42no—
418C2×C4⋊C4—42no—
419C2×C4⋊C4—2922no—
420C2×C4⋊C4—2922no—
421C2×C4⋊C4—2922no—
422C2×C4⋊C4—2922no—
423C2×C4⋊C4—1464no—
424C2×C4⋊C4—1464no—
425C2×C4⋊C4—1464no—
426C2×C4⋊C4—1464no—
427C2×C4⋊C4—1462no—
428C2×C4⋊C4—1464no—
429C2×C4⋊C4—1464no—
430C2×C22⋊C4—738no—
431C2×C22⋊C4—738no—
432C2×C22⋊C4—738no—
433C2×C22⋊C4—1464no—
434C2×C22⋊C4—1464no—
435C2×C22⋊C4—1464no—
436C2×C42—2922no—
437C2×C42—2922no—
438C2×C42—738no—
439#95—22no—
440#95—24no—
441#95—734no—
442#95—1462no—
443#95—1462no—
444#95—1464no—
445#95—1464no—
446#95—1464no—
447#95—1462no—
448#95—1464no—
449#95C4×D7314yescomplemented
450C24—732no—
451C24—734no—
452C24—732no—
453C24—734no—
454C24—732no—
455C24—1461no—
456C24—1461no—
457C24—1461no—
458C24—1461no—
459C24—1462no—
460C24—1462no—
461C24C23×D7311yescharacteristic
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—2928no—
484C22×C4—2928no—
485C22×C4—2924no—
486C22×C4—2924no—
487C22×C4—2928no—
488C22×C4—2928no—
489C22×C4—2924no—
490C22×C4—2924no—
491C22×C4—2924no—
492C22×C4—2924no—
493C22×C4—2928no—
494C22×C4—2928no—
495C22×C4—7316no—
496C22×C4—734no—
497C22×C4—732no—
498C22×C4—732no—
499C22×C4—734no—
500C22×C4—734no—
501C22×C4—732no—
502C22×C4—732no—
503C22×C4—732no—
504C22×C4—1468no—
505C22×C4—1468no—
506C22×C4—1462no—
507C22×C4—1462no—
508C22×C4—1462no—
509C22×C4—1462no—
510C22×C4—1462no—
511C22×C4—1462no—
512C22×C4—1462no—
513C22×C4—1461no—
514C22×C4—1461no—
515C22×C4—1462no—
516C22×C4—1462no—
517C22×C4—1461no—
518C22×C4—1461no—
519C22×C4—1462no—
520C22×C4—1462no—
521C22×C4—1462no—
522C22×C4—1462no—
523C22×C4—1462no—
524C22×C4—1462no—
525C22×C4—1462no—
526C22×C4—1462no—
527C22×C4C2×C4×D7312yes—
528C22×C4C2×C4×D7314yes—
529C22×C4D4×D7314yes—
530C22×C4D4×D7312yes—
531C22×C4Q8×D7314yes—
532C22×C4Q8×D7312yes—
533C22×C4#1192402712yes—
534C22×C4#1192402912yes—
535C4⋊C4—48no—
536C4⋊C4—44no—
537C4⋊C4—2924no—
538C4⋊C4—2928no—
539C4⋊C4—2928no—
540C4⋊C4—2928no—
541C22⋊C4—2928no—
542C22⋊C4—7332no—
543C42—2928no—
544C23—734no—
545C23—734no—
546C23—738no—
547C23—738no—
548C23—734no—
549C23—1462no—
550C23—1462no—
551C23—1462no—
552C23—1462no—
553C23—1464no—
554C23—1464no—
555C23—1464no—
556C23—1464no—
557C23—1464no—
558C23—1464no—
559C23—1464no—
560C23—1464no—
561C23—1464no—
562C23—1462no—
563C23—1462no—
564C23—1464no—
565C23—1462no—
566C23—1462no—
567C23C22×C4×D7311yescharacteristic
568C23C2×D4×D7311yescharacteristic
569C23C2×D4×D7312yes—
570C23C2×Q8×D7311yescharacteristic
571C23C2×Q8×D7312yes—
572C23#41×D7311yescharacteristic
573C23#41×D7311yescharacteristic
574C23C2×#1192402711yescharacteristic
575C23C2×#1192402911yescharacteristic
576C23#1455663612yes—
577C23#1455663712yes—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—29216no—
606C2×C4—29216no—
607C2×C4—29216no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—2921no—
611C2×C4—2921no—
612C2×C4—2922no—
613C2×C4—2922no—
614C2×C4—2922no—
615C2×C4—2922no—
616C2×C4—2922no—
617C2×C4—2922no—
618C2×C4—2922no—
619C2×C4—2922no—
620C2×C4—2922no—
621C2×C4—738no—
622C2×C4—734no—
623C2×C4—1464no—
624C2×C4—1464no—
625C2×C4—1462no—
626C2×C4—1462no—
627C2×C4—1462no—
628C2×C4—1462no—
629C2×C4—1464no—
630C2×C4—1464no—
631C2×C4—1464no—
632C2×C4—1464no—
633C2×C4—1464no—
634C2×C4—1464no—
635C2×C4—1464no—
636C2×C4—1464no—
637C2×C4C4⋊C4×D7318yes—
638C2×C4C4⋊C4×D7314yes—
639C22—738no—
640C22—1464no—
641C22—1464no—
642C22—1464no—
643C22—1464no—
644C22—1464no—
645C22—1468no—
646C22—1468no—
647C22—1468no—
648C22—1468no—
649C22—1464no—
650C22C2×C4⋊C4×D7311yescharacteristic
651C22C2×C4⋊C4×D7312yes—
652C22C4×#1192402711yescharacteristic
653C22C4×#1192402911yescharacteristic
654C22#1798905912yes—
655C22#1798909711yescharacteristic
656C22#1798910011yescharacteristic
657C22#127×D7311yescharacteristic
658C22C4⋊Q8×D7311yescharacteristic
659C22C4⋊Q8×D7312yes—
660C22#1798914211yescharacteristic
661C22#1798914311yescharacteristic
662C22#1798914711yescharacteristic
663C22#1798914811yescharacteristic
664C22#1798921212yes—
665C22#1798922412yes—
666C22#1798922812yes—
667C22#1798922912yes—
668C22#1798923412yes—
669C22#1798929112yes—
670C22#1798929312yes—
671C22#1798929512yes—
672C22#1798929812yes—
673C4—24no—
674C4—28no—
675C4—42no—
676C4—42no—
677C4—82no—
678C4—2922no—
679C4—2922no—
680C4—5842no—
681C4—1464no—
682C4—1468no—
683C2—1468no—
684C2#384×D7311yescharacteristic
685C2#2261241211yescharacteristic
686C2#2261254211yescharacteristic
687C2#2261283712yes—
688C2#2261284012yes—
689C2#2261485611yescharacteristic
690C2#2261485811yescharacteristic
691C2#2261493612yes—
692C2#2261493812yes—
693C2#2261530511yescharacteristic
694C2#2261530711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1444C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenter#1444D73
hypocenterC73#13965
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
6. #22608196C2semidirect
7. #22608195C2semidirect
8. #22608194C2semidirect
9. #22608160C2semidirect
10. #22608159C2semidirect
11. C73×#1444C2semidirect
25. #17988434C4semidirect
80. C2×C4×Dic73C2×C4semidirect
88. #14556498C2×C4semidirect
101. #95×C73C2×C4semidirect
156. C2×C4×D73C4⋊C4semidirect
162. #11924005C4⋊C4semidirect
356. #1444D73semidirect
359. C73#13965semidirect
449. #95C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179884612
C2×#179884582
C2×#179884442
C2×#179884342
C2×#95×D732
#226081962
#226081952
#226081942
#226081602
#226081592
C73×#14442
#1396573