Finite Group Databasebeta

← #33144211

Subgroups — #33144211

Order 18688.

736 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612878C211yesmaximal, characteristic
2#22611302C211yesmaximal, characteristic
3#22611302C211yesmaximal, characteristic
4#22611074C211yesmaximal, characteristic
5#22611064C211yesmaximal, characteristic
6#22610889C211yesmaximal, characteristic, complemented
7#22610884C211yesmaximal, characteristic, complemented
8#22610694C211yesmaximal, characteristic, complemented
9#22610655C211yesmaximal, characteristic, complemented
10#22610513C211yesmaximal, characteristic, complemented
11#22610506C211yesmaximal, characteristic, complemented
12#22610064C211yesmaximal, characteristic, complemented
13#22609938C211yesmaximal, characteristic
14#22609759C211yesmaximal, characteristic
15C73×#2550C211yesmaximal, characteristic, complemented
16C2×C4×D292C2211yescharacteristic
17C2×#14556549—21no—
18C2×#14556522C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×#14556511C2211yescharacteristic
24C2×#14556511C2211yescharacteristic
25C2×#14556511C2211yescharacteristic
26C2×#14556511C2211yescharacteristic
27C2×#14556502C2211yescharacteristic
28#17988489C2211yescharacteristic
29#17988489C2211yescharacteristic
30#17988479—21no—
31#17988479—21no—
32#17988457C2211yescharacteristic
33#17988444—21no—
34C4×C4⋊Dic73C2211yescharacteristic
35C4×C4⋊Dic73C2211yescharacteristic
36#17988349C2211yescharacteristic
37#17988349C2211yescharacteristic
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×#125×C73C2211yescharacteristic
50#398×C73—21no—
51#395×C73—21no—
52#391×C73—21no—
53#389×C73—21no—
54#389×C73C2211yescharacteristic
55#389×C73C2211yescharacteristic
56#382×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C4⋊C4×C73C2211yescharacteristic
59C22×D292C2311yescharacteristic
60C22×C4×D73—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C4×D292—22no—
66C4×D292Q814yescomplemented
67#14556549—44no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924005—21no—
74C2×#11924005—21no—
75C2×#11924005—21no—
76C2×#11924005—21no—
77C2×#11924005—21no—
78C2×C4×Dic73—21no—
79C2×C4×Dic73—21no—
80#14556522—22no—
81#14556522—22no—
82C4⋊#10596831—22no—
83C4⋊#10596831—41no—
84C4⋊#10596831—41no—
85#14556511—22no—
86#14556511—22no—
87#14556511—41no—
88#14556511—41no—
89#14556502—22no—
90#14556498—21no—
91#14556498—21no—
92C22×Q8×C73—21no—
93C4⋊Q8×C73—22no—
94C4⋊Q8×C73—41no—
95#125×C73—22no—
96#125×C73—41no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73D411yescharacteristic
103C2×C4⋊C4×C73D411yescharacteristic
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C4⋊C4×C73C2311yescharacteristic
106C2×C4⋊C4×C73C2311yescharacteristic
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C42×C73—21no—
109C2×C42×C73C2311yescharacteristic
110C2×C42×C73C2311yescharacteristic
111C2×C42×C73C2311yescharacteristic
112#95×C73—21no—
113#95×C73—21no—
114#95×C73—21no—
115#95×C73—21no—
116#95×C73—21no—
117#95×C73D411yescharacteristic
118#95×C73D411yescharacteristic
119C23×D73—21no—
120C2×D292—21no—
121C2×D292—21no—
122C2×D292—21no—
123C2×D292—21no—
124C2×D292—21no—
125C2×D292C2×Q812yes—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C2×C4×D73—24no—
130C2×C4×D73—42no—
131C2×C4×D73—42no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—42no—
150C4⋊Dic73—81no—
151C4⋊Dic73C2×Q812yes—
152#11924005—24no—
153#11924005—42no—
154#11924005—42no—
155#11924005—81no—
156#11924005—81no—
157C4×Dic73—42no—
158C4×Dic73—42no—
159C2×Q8×C73—42no—
160C2×Q8×C73—42no—
161C2×Q8×C73—81no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2411yescharacteristic
173C4⋊C4×C73—22no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—21no—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—41no—
181C4⋊C4×C73—42no—
182C4⋊C4×C73—42no—
183C4⋊C4×C73—42no—
184C4⋊C4×C73—42no—
185C4⋊C4×C73—81no—
186C4⋊C4×C73#4112yes—
187C4⋊C4×C73#4112yes—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—21no—
191C42×C73—21no—
192C42×C73—81no—
193C42×C73C2×Q812yes—
194C42×C73#4112yes—
195C23×C73C22×D411yescharacteristic
196C22×D73—22no—
197C22×D73—41no—
198C22×D73—41no—
199C22×D73—41no—
200C22×D73—41no—
201C22×D73—41no—
202C22×D73—41no—
203Q8×C73—84no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73C22⋊Q812yes—
232C2×C4×C73C22⋊Q812yes—
233C2×C4×C73AGL(1,ℤ/8)12yes—
234C2×C4×C73C22×Q811yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73H(5,𝔽2)11yescharacteristic
239C2×C4×C732−1+411yescharacteristic
240C2×Dic73—22no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249D292—22no—
250D292—22no—
251D292—41no—
252D292—41no—
253C4×D73—48no—
254#10596831—41no—
255#10596831—41no—
256#10596831—41no—
257#10596831—41no—
258C22×C73C2×C22⋊Q811yescharacteristic
259C22×C73#54611yescharacteristic
260C22×C73#54811yescharacteristic
261C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C73#57511yescharacteristic
263C22×C73#57711yescharacteristic
264C22×C73#57711yescharacteristic
265C2×D73—42no—
266C2×D73—42no—
267C2×D73—42no—
268C2×D73—81no—
269C2×D73—81no—
270C2×D73—81no—
271C2×D73—81no—
272C4×C73—22no—
273C4×C73—22no—
274C4×C73—22no—
275C4×C73—42no—
276C4×C73—41no—
277C4×C73—42no—
278C4×C73—41no—
279C4×C73—41no—
280C4×C73—81no—
281C4×C73—81no—
282C4×C73#53311yescharacteristic
283C4×C73#54411yescharacteristic
284C4×C73#55511yescharacteristic
285C4×C73#55611yescharacteristic
286Dic73—42no—
287Dic73—81no—
288Dic73—81no—
289#26032—731nomaximal, Sylow
290C2×C73#243911yescharacteristic
291C2×C73#307711yescharacteristic
292C2×C73#308411yescharacteristic
293C2×C73#330611yescharacteristic
294C2×C73#332111yescharacteristic
295C2×C73#336811yescharacteristic
296C2×C73#337611yescharacteristic
297D73—82no—
298#2550D7311yescharacteristic, complemented
299#2354—731no—
300#2065—731no—
301#2065—731no—
302#2025—731no—
303#2023—731no—
304#1989—731no—
305#1988—731no—
306#1948—731no—
307#1940—731no—
308#1920—731no—
309#1919—731no—
310#1836—731no—
311#1804—731no—
312#1762—731no—
313C73#2603211yescharacteristic, complemented, Sylow
314C2×C4⋊Q8C2×D7311yescharacteristic
315C2×#125C2×D7311yescharacteristic
316C2×C4×D4—731no—
317C2×#117—1461no—
318C2×#106—731no—
319C2×#106—731no—
320C2×C4⋊C8—731no—
321C2×C4⋊C8—731no—
322C2×C4⋊C8—731no—
323C2×#102—731no—
324C2×#102—731no—
325C2×#102—731no—
326C2×#102—731no—
327C2×#97—731no—
328#398—21no—
329#395—21no—
330#391—21no—
331#390—731no—
332#390—731no—
333#389—21no—
334#389C2×D7311yescharacteristic
335#389C2×D7311yescharacteristic
336#388—1461no—
337#388—1461no—
338#384—731no—
339#382C2×D7311yescharacteristic
340#380—1461no—
341C4×C4⋊C4—731no—
342C4×C4⋊C4—731no—
343C4×C4⋊C4C2×D7311yescharacteristic
344C4×C4⋊C4C2×D7311yescharacteristic
345#340—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#340—731no—
353#340—731no—
354#340—731no—
355#336—731no—
356#336—731no—
357C22×Q8—21no—
358C22×D4—731no—
359C23×C4—1461no—
360C22×C8—731no—
361C22×C8—731no—
362C22×C8—731no—
363C22×C8—731no—
364C4⋊Q8—22no—
365C4⋊Q8—41no—
366#125—22no—
367#125—41no—
368C4×D4—734no—
369C4×D4—1462no—
370#117—2924no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—731no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4C22⋊D7311yescharacteristic
382C2×C4⋊C4C22⋊D7311yescharacteristic
383C2×C4⋊C4C22×D7311yescharacteristic
384C2×C4⋊C4C22×D7311yescharacteristic
385C2×C4⋊C4C22×D7311yescharacteristic
386C2×C4⋊C4C22×D7311yescharacteristic
387C2×C22⋊C4—1461no—
388C2×C22⋊C4—1461no—
389C2×C22⋊C4—1461no—
390C2×C22⋊C4—1461no—
391C2×C22⋊C4—1461no—
392C2×C42—21no—
393C2×C42—1461no—
394C2×C42—1461no—
395C2×C42C22×D7311yescharacteristic
396C2×C42C22×D7311yescharacteristic
397C2×C42C22×D7311yescharacteristic
398#106—1462no—
399#106—1462no—
400C4⋊C8—2921no—
401C4⋊C8—2921no—
402C4⋊C8—1462no—
403#102—2921no—
404#102—2921no—
405#102—1462no—
406#102—1462no—
407#97—1462no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—1461no—
414#95—1461no—
415#95C22⋊D7311yescharacteristic
416#95C22⋊D7311yescharacteristic
417C24—1461no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—81no—
421C2×D4—732no—
422C2×D4—1461no—
423C2×D4—1461no—
424C2×D4—1461no—
425C2×D4—1461no—
426C2×D4—1461no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—2922no—
432C22×C4—2922no—
433C22×C4—1461no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1464no—
437C22×C4C2×C22⋊D7311yescharacteristic
438C22×C4C2×C22⋊D7311yescharacteristic
439C22×C4C2×C22⋊D7311yescharacteristic
440C22×C4C2×C22⋊D7311yescharacteristic
441C22×C4C2×C22⋊D7311yescharacteristic
442C22×C4C2×C22⋊D7311yescharacteristic
443C22×C4C23×D7311yescharacteristic
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8—1461no—
449C2×C8—1461no—
450C2×C8—1461no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—81no—
469C4⋊C4—2922no—
470C4⋊C4—5841no—
471C4⋊C4—732no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4#1192402712yes—
478C4⋊C4#1192402912yes—
479C22⋊C4—2922no—
480C22⋊C4—2922no—
481C22⋊C4—5841no—
482C22⋊C4—5841no—
483C22⋊C4—1464no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—81no—
489C42—2922no—
490C42—2922no—
491C42Q8×D7312yes—
492C42#1192402912yes—
493C23—2921no—
494C23—2921no—
495C23—2921no—
496C23—2921no—
497C23—2921no—
498C23—2921no—
499C23—1462no—
500C23C22×C22⋊D7311yescharacteristic
501Q8—84no—
502D4—2921no—
503D4—2921no—
504D4—1462no—
505D4—1462no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—2928no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—1462no—
543C2×C4#1455656712yes—
544C2×C4#1455656712yes—
545C2×C4#1455661612yes—
546C2×C4C2×Q8×D7311yescharacteristic
547C2×C4C2×#1192402711yescharacteristic
548C2×C4C2×#1192402911yescharacteristic
549C2×C4C2×#1192402911yescharacteristic
550C2×C4#1455663611yescharacteristic
551C2×C4#1455663911yescharacteristic
552C8—2921no—
553C8—2921no—
554C8—2921no—
555C8—2921no—
556C22—2922no—
557C22—2922no—
558C22—2922no—
559C22—5841no—
560C22—5841no—
561C22—5841no—
562C22—5841no—
563C22C2×#1455656711yescharacteristic
564C22#1798923111yescharacteristic
565C22#1798924811yescharacteristic
566C22C2×#1455661611yescharacteristic
567C22#1798938511yescharacteristic
568C22#1798939511yescharacteristic
569C22#1798939811yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—42no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—42no—
578C4—81no—
579C4—81no—
580C4—2922no—
581C4—5841no—
582C4—5841no—
583C4#1798914211yescharacteristic
584C4#1798921411yescharacteristic
585C4#1798930311yescharacteristic
586C4#1798931111yescharacteristic
587C2—5842no—
588C2#2261361811yescharacteristic
589C2#2261972411yescharacteristic
590C2#2261978911yescharacteristic
591C2#2262189211yescharacteristic
592C2#2262211711yescharacteristic
593C2#2262266011yescharacteristic
594C2#2262275811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2550C2
socleC23×C73C22×D4
prime radicalC22×D292C23
hypercenter#2550D73
hypocenterC73#26032
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2550

Complements

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

NH ≅ G/Ntype
6. #22610889C2semidirect
7. #22610884C2semidirect
8. #22610694C2semidirect
9. #22610655C2semidirect
10. #22610513C2semidirect
11. #22610506C2semidirect
12. #22610064C2semidirect
15. C73×#2550C2semidirect
66. C4×D292Q8semidirect
298. #2550D73semidirect
313. C73#26032semidirect

Maximal subgroups

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

subgroupindex
#226128782
#226113022
#226110742
#226110642
#226108892
#226108842
#226106942
#226106552
#226105132
#226105062
#226100642
#226099382
#226097592
C73×#25502
#2603273