Finite Group Databasebeta

← #33154071

Subgroups — #33154071

Order 18688.

826 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612934C211yesmaximal, characteristic, complemented
2#22611248C211yesmaximal, characteristic, complemented
3#22611127C211yesmaximal, characteristic, complemented
4#22611069C211yesmaximal, characteristic, complemented
5#22610934C211yesmaximal, characteristic
6#22610884C211yesmaximal, characteristic, complemented
7#22610617C211yesmaximal, characteristic, complemented
8#22610555C211yesmaximal, characteristic, complemented
9#22610539C211yesmaximal, characteristic, complemented
10#22610516C211yesmaximal, characteristic, complemented
11#22610289C211yesmaximal, characteristic, complemented
12#22610249C211yesmaximal, characteristic
13#22610019C211yesmaximal, characteristic, complemented
14C4×#14556514C211yesmaximal, characteristic, complemented
15C73×#2491C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×#14556570C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C22×C4⋊Dic73C2211yescharacteristic
20C2×#14556522C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×#14556514C2211yescharacteristic
23C2×#14556514C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×C22⋊#10596831C2211yescharacteristic
29C2×C22⋊#10596831C2211yescharacteristic
30C2×C4×#10596831C2211yescharacteristic
31#17988493C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988484—21no—
34#17988474—21no—
35#17988459—21no—
36C4×C4⋊Dic73C2211yescharacteristic
37C22⋊C4×Dic73—21no—
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×C4⋊Q8×C73C2211yescharacteristic
49C2×#124×C73—21no—
50C2×#121×C73C2211yescharacteristic
51#394×C73—21no—
52#393×C73—21no—
53#390×C73C2211yescharacteristic
54#390×C73C2211yescharacteristic
55#388×C73—21no—
56#380×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×Dic146C2311yescharacteristic
60C23×Dic73—21no—
61C2×#11924016—22no—
62C2×#11924016—41no—
63C2×#11924016—41no—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556570—22no—
69#14556570—22no—
70C4×Dic146—22no—
71C4×Dic146—41no—
72C2×C4⋊Dic73—22no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73—41no—
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×C22⋊Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84#14556522—22no—
85C4⋊#10596831—41no—
86#14556514—22no—
87#14556514—22no—
88#14556514—41no—
89#14556512—22no—
90#14556512—22no—
91#14556512—41no—
92C22⋊#10596831—41no—
93C22⋊#10596831—41no—
94C4×#10596831—41no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99C22×Q8×C73—21no—
100C22×D4×C73—21no—
101C22×D4×C73C2311yescharacteristic
102C23×C4×C73C2311yescharacteristic
103C4⋊Q8×C73—22no—
104C4⋊Q8×C73—41no—
105#124×C73—44no—
106#121×C73—22no—
107#121×C73—41no—
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115C2×C22⋊C4×C73—21no—
116C2×C22⋊C4×C73—21no—
117C2×C22⋊C4×C73—21no—
118C2×C22⋊C4×C73D411yescharacteristic
119C2×C22⋊C4×C73D411yescharacteristic
120C2×C42×C73—21no—
121C2×C42×C73C2311yescharacteristic
122C2×C42×C73C2311yescharacteristic
123#95×C73—21no—
124#95×C73—21no—
125#95×C73D411yescharacteristic
126#95×C73D411yescharacteristic
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146—21no—
132C2×Dic146#4112yes—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—42no—
138C22×Dic73—42no—
139C22×Dic73—81no—
140#11924016—44no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—42no—
158C4⋊Dic73#4112yes—
159C4⋊Dic73#4112yes—
160#11924007—24no—
161#11924007—42no—
162#11924007—81no—
163C22⋊Dic73—24no—
164C4×Dic73—42no—
165C4×Dic73—81no—
166C24×C73—21no—
167C24×C73C2×D411yescharacteristic
168C2×Q8×C73—42no—
169C2×Q8×C73—42no—
170C2×Q8×C73—81no—
171C2×D4×C73—21no—
172C2×D4×C73—21no—
173C2×D4×C73—21no—
174C2×D4×C73—21no—
175C2×D4×C73—21no—
176C2×D4×C73—21no—
177C2×D4×C73—42no—
178C2×D4×C73—42no—
179C2×D4×C73—81no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2411yescharacteristic
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—42no—
201C4⋊C4×C73#4112yes—
202C22⋊C4×C73—41no—
203C22⋊C4×C73—42no—
204C22⋊C4×C73—42no—
205C22⋊C4×C73—42no—
206C22⋊C4×C73—42no—
207C22⋊C4×C73—42no—
208C22⋊C4×C73—81no—
209C22⋊C4×C73SD1614yescomplemented
210C42×C73—21no—
211C42×C73—21no—
212C42×C73—21no—
213C42×C73—21no—
214C42×C73—42no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—41no—
222C23×C73—41no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73—41no—
226C23×C73—41no—
227C23×C73—41no—
228C23×C73C2×SD1612yes—
229C23×C73C22×D411yescharacteristic
230Q8×C73—84no—
231D4×C73—41no—
232D4×C73—41no—
233D4×C73—41no—
234D4×C73—41no—
235D4×C73—84no—
236C2×C4×C73—21no—
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—22no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—41no—
262C2×C4×C73C2×SD1612yes—
263C2×C4×C73C2×SD1612yes—
264C2×C4×C73C2×#4111yescharacteristic
265C2×C4×C73C2×#4111yescharacteristic
266C2×C4×C73C2×#4111yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×C4×C73H(5,𝔽2)11yescharacteristic
269C2×C4×C73H(5,𝔽2)11yescharacteristic
270C2×Dic73—22no—
271C2×Dic73—22no—
272C2×Dic73—22no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—84no—
283Dic146—22no—
284Dic146—22no—
285Dic146—41no—
286Dic146—41no—
287#10596831—41no—
288#10596831—41no—
289#10596831—41no—
290#10596831—41no—
291C22×C73—22no—
292C22×C73—22no—
293C22×C73—22no—
294C22×C73—41no—
295C22×C73—41no—
296C22×C73—41no—
297C22×C73—41no—
298C22×C73—81no—
299C22×C73—81no—
300C22×C73—81no—
301C22×C73—81no—
302C22×C73—81no—
303C22×C73—81no—
304C22×C73—81no—
305C22×C73#52511yescharacteristic
306C22×C73#54611yescharacteristic
307C22×C73#54611yescharacteristic
308C22×C73C22×SD1611yescharacteristic
309C22×C73#57511yescharacteristic
310C22×C73#57611yescharacteristic
311C22×C73#57811yescharacteristic
312C4×C73—22no—
313C4×C73—22no—
314C4×C73—41no—
315C4×C73—42no—
316C4×C73—41no—
317C4×C73—41no—
318C4×C73—81no—
319C4×C73—81no—
320C4×C73#52911yescharacteristic
321C4×C73#53811yescharacteristic
322C4×C73#55511yescharacteristic
323C4×C73#55911yescharacteristic
324Dic73—42no—
325Dic73—42no—
326Dic73—42no—
327Dic73—81no—
328#26724—731nomaximal, Sylow
329C2×C73—42no—
330C2×C73—161no—
331C2×C73#245911yescharacteristic
332C2×C73#310311yescharacteristic
333C2×C73#312411yescharacteristic
334C2×C73#329511yescharacteristic
335C2×C73#329811yescharacteristic
336C2×C73#331911yescharacteristic
337C2×C73#332711yescharacteristic
338#2491D7311yescharacteristic
339#2360—731no—
340#2055—731no—
341#2035—731no—
342#2024—731no—
343#2000—731no—
344#1988—731no—
345#1934—731no—
346#1926—731no—
347#1923—731no—
348#1920—731no—
349#1880—731no—
350#1874—731no—
351#1824—731no—
352C4×#103—731no—
353C73#2672411yescharacteristic, complemented, Sylow
354C22×SD16—731no—
355C2×C4⋊Q8C2×D7311yescharacteristic
356C2×#124—21no—
357C2×C22⋊Q8—731no—
358C2×#121C2×D7311yescharacteristic
359C2×C4×Q8—731no—
360C22×C4⋊C4—731no—
361C2×#106—731no—
362C2×C4⋊C8—731no—
363C2×#103—731no—
364C2×#103—731no—
365C2×#103—731no—
366C2×#102—731no—
367C2×#102—731no—
368C2×#102—731no—
369C2×C22⋊C8—731no—
370C2×C22⋊C8—731no—
371C2×C4×C8—731no—
372#394—21no—
373#393—21no—
374#391—731no—
375#390C2×D7311yescharacteristic
376#390C2×D7311yescharacteristic
377#389—731no—
378#389—1461no—
379#388—21no—
380#387—1461no—
381#385—1461no—
382#380C2×D7311yescharacteristic
383C4×C4⋊C4—731no—
384C4×C4⋊C4C2×D7311yescharacteristic
385C4×C22⋊C4—1461no—
386C4×C22⋊C4C2×D7311yescharacteristic
387#340—731no—
388#340—731no—
389#340—731no—
390#340—731no—
391#340—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#336—731no—
396#336—731no—
397C22×Q8—21no—
398C22×Q8—731no—
399C22×D4—21no—
400C22×D4C22×D7311yescharacteristic
401C23×C4—1461no—
402C23×C4C22×D7311yescharacteristic
403C2×SD16—2921no—
404C2×SD16—2921no—
405C2×SD16—1462no—
406C22×C8—731no—
407C22×C8—731no—
408C22×C8—731no—
409C22×C8—731no—
410C4⋊Q8—22no—
411C4⋊Q8—41no—
412#124—44no—
413C22⋊Q8—1462no—
414C22⋊Q8—1462no—
415#121—22no—
416#121—41no—
417C4×Q8—2921no—
418C4×Q8—1462no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—2921no—
421C2×C4⋊C4—731no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—1461no—
425C2×C4⋊C4—1461no—
426C2×C4⋊C4—1462no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4C22×D7311yescharacteristic
430C2×C4⋊C4C22×D7311yescharacteristic
431C2×C4⋊C4C22×D7311yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4C22⋊D7311yescharacteristic
440C2×C22⋊C4C22⋊D7311yescharacteristic
441C2×C42—21no—
442C2×C42—1461no—
443C2×C42—1461no—
444C2×C42C22×D7311yescharacteristic
445C2×C42C22×D7311yescharacteristic
446#106—1462no—
447C4⋊C8—2921no—
448#103—2921no—
449#103—1462no—
450#103—1462no—
451#102—2921no—
452#102—1462no—
453#102—1462no—
454C22⋊C8—2921no—
455C22⋊C8—2921no—
456C4×C8—2921no—
457#95—21no—
458#95—21no—
459#95—1461no—
460#95—1461no—
461#95—1461no—
462#95—1461no—
463#95C22⋊D7311yescharacteristic
464#95C22⋊D7311yescharacteristic
465C24—21no—
466C24C2×C22⋊D7311yescharacteristic
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—81no—
470C2×Q8—732no—
471C2×Q8—1461no—
472C2×Q8—1461no—
473C2×Q8—1461no—
474C2×Q8—1461no—
475C2×Q8—1461no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—81no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—2922no—
495C22×C4—2922no—
496C22×C4—5841no—
497C22×C4—1461no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4C2×C22⋊D7311yescharacteristic
502C22×C4C2×C22⋊D7311yescharacteristic
503C22×C4C2×C22⋊D7311yescharacteristic
504C22×C4C2×C22⋊D7311yescharacteristic
505C22×C4C2×C22⋊D7311yescharacteristic
506C22×C4C23×D7311yescharacteristic
507SD16—2924no—
508C2×C8—1461no—
509C2×C8—1461no—
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—42no—
526C4⋊C4—2922no—
527C4⋊C4—2922no—
528C4⋊C4—5841no—
529C4⋊C4—732no—
530C4⋊C4—732no—
531C4⋊C4—1464no—
532C4⋊C4—1461no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4—1461no—
536C4⋊C4#1192402712yes—
537C22⋊C4—41no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—81no—
544C22⋊C4—1464no—
545C22⋊C4#1192401614yescomplemented
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—42no—
551C42—2922no—
552C42—5841no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23C2×#1192401612yes—
567C23C22×C22⋊D7311yescharacteristic
568Q8—84no—
569Q8—2921no—
570Q8—2921no—
571Q8—1462no—
572Q8—1462no—
573D4—41no—
574D4—41no—
575D4—41no—
576D4—41no—
577D4—84no—
578C2×C4—21no—
579C2×C4—21no—
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—22no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—2921no—
605C2×C4—2921no—
606C2×C4—2921no—
607C2×C4—2921no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—2921no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—5844no—
614C2×C4—1462no—
615C2×C4—1462no—
616C2×C4—1462no—
617C2×C4C2×#1192401612yes—
618C2×C4C2×#1192401612yes—
619C2×C4#41×D7311yescharacteristic
620C2×C4#41×D7311yescharacteristic
621C2×C4#41×D7311yescharacteristic
622C2×C4C2×#1192402711yescharacteristic
623C2×C4#1455663611yescharacteristic
624C2×C4#1455663711yescharacteristic
625C8—2921no—
626C8—2921no—
627C8—2921no—
628C8—2921no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22#1798909411yescharacteristic
644C22#1798922511yescharacteristic
645C22#1798923111yescharacteristic
646C22C22×#1192401611yescharacteristic
647C22#1798938511yescharacteristic
648C22#1798939011yescharacteristic
649C22#1798940211yescharacteristic
650C4—22no—
651C4—22no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—81no—
657C4—81no—
658C4—2922no—
659C4—2922no—
660C4—2922no—
661C4—5841no—
662C4#1798911611yescharacteristic
663C4#1798917111yescharacteristic
664C4#1798929911yescharacteristic
665C4#1798931911yescharacteristic
666C2—42no—
667C2—161no—
668C2#2261386411yescharacteristic
669C2#2261995811yescharacteristic
670C2#2262018711yescharacteristic
671C2#2262175511yescharacteristic
672C2#2262179011yescharacteristic
673C2#2262208111yescharacteristic
674C2#2262219911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2491C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenter#2491D73
hypocenterC73#26724
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2491

Complements

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

NH ≅ G/Ntype
1. #22612934C2semidirect
2. #22611248C2semidirect
3. #22611127C2semidirect
4. #22611069C2semidirect
6. #22610884C2semidirect
7. #22610617C2semidirect
8. #22610555C2semidirect
9. #22610539C2semidirect
10. #22610516C2semidirect
11. #22610289C2semidirect
13. #22610019C2semidirect
14. C4×#14556514C2semidirect
209. C22⋊C4×C73SD16semidirect
353. C73#26724semidirect
545. C22⋊C4#11924016semidirect

Maximal subgroups

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

subgroupindex
#226129342
#226112482
#226111272
#226110692
#226109342
#226108842
#226106172
#226105552
#226105392
#226105162
#226102892
#226102492
#226100192
C4×#145565142
C73×#24912
#2672473