Finite Group Databasebeta

← #33149529

Subgroups — #33149529

Order 18688.

902 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22613633C211yesmaximal, characteristic, complemented
2#22611489C211yesmaximal, characteristic, complemented
3#22611422C211yesmaximal, characteristic, complemented
4#22611319C211yesmaximal, characteristic, complemented
5#22611067C211yesmaximal, characteristic, complemented
6#22610883C211yesmaximal, characteristic, complemented
7#22610502C211yesmaximal, characteristic, complemented
8#22610360C211yesmaximal, characteristic
9#22610249C211yesmaximal, characteristic
10#22610178C211yesmaximal, characteristic, complemented
11#22610119C211yesmaximal, characteristic, complemented
12#22610111C211yesmaximal, characteristic, complemented
13#22609931C211yesmaximal, characteristic, complemented
14C2×#17988349C211yesmaximal, characteristic, complemented
15C73×#2361C211yesmaximal, characteristic
16C23×#10596831C2211yescharacteristic
17C2×#14556591C2211yescharacteristic
18C2×#14556568—21no—
19C22×C4⋊Dic73C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×C22⋊#10596831C2211yescharacteristic
29#17988519—21no—
30#17988508—21no—
31#17988485C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988462—21no—
34#17988457C2211yescharacteristic
35#17988447C2211yescharacteristic
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×#121×C73C2211yescharacteristic
49C2×C4×D4×C73—21no—
50C2×C4×D4×C73C2211yescharacteristic
51C22×C4⋊C4×C73C2211yescharacteristic
52#390×C73C2211yescharacteristic
53#389×C73C2211yescharacteristic
54#388×C73—21no—
55#386×C73—21no—
56#385×C73—21no—
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×Dic146—21no—
60C23×Dic73—21no—
61C22×#10596831—22no—
62C22×#10596831—41no—
63C22×#10596831—41no—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556591—22no—
69#14556591—41no—
70#14556568—44no—
71C2×C4⋊Dic73—22no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—41no—
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×C22⋊Dic73—21no—
84#14556524—22no—
85#14556524—41no—
86C4⋊#10596831—41no—
87C4⋊#10596831—41no—
88#14556512—22no—
89#14556512—41no—
90#14556512—41no—
91#14556512—41no—
92C22⋊#10596831—22no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96C22×D4×C73—21no—
97C22×D4×C73C2311yescharacteristic
98C23×C4×C73—21no—
99C23×C4×C73—21no—
100C23×C4×C73C2311yescharacteristic
101#121×C73—22no—
102#121×C73Q814yescomplemented
103C4×D4×C73—22no—
104C4×D4×C73—41no—
105C4×D4×C73—44no—
106C2×C4⋊C4×C73—22no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73—41no—
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115C2×C22⋊C4×C73—21no—
116C2×C22⋊C4×C73—21no—
117C2×C22⋊C4×C73D411yescharacteristic
118C2×C22⋊C4×C73D411yescharacteristic
119C2×C42×C73—21no—
120C2×C42×C73C2311yescharacteristic
121C2×C42×C73C2311yescharacteristic
122#95×C73—21no—
123#95×C73—21no—
124#95×C73D411yescharacteristic
125#95×C73D411yescharacteristic
126C2×Dic146—42no—
127C2×Dic146—42no—
128C2×Dic146—81no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—42no—
134C22×Dic73—42no—
135C22×Dic73—81no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—44no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—42no—
157C4⋊Dic73—42no—
158C4⋊Dic73—42no—
159C4⋊Dic73—81no—
160C4⋊Dic73#4112yes—
161#11924007—42no—
162#11924007—42no—
163C22⋊Dic73—42no—
164C24×C73—21no—
165C24×C73C2×D411yescharacteristic
166C2×D4×C73—24no—
167C2×D4×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73—21no—
172C2×D4×C73—42no—
173C2×D4×C73—42no—
174C2×D4×C73C2×Q812yes—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—42no—
183C22×C4×C73—42no—
184C22×C4×C73—42no—
185C22×C4×C73—42no—
186C22×C4×C73—81no—
187C22×C4×C73—81no—
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×Q812yes—
194C22×C4×C73#4112yes—
195C22×C4×C73C2411yescharacteristic
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—42no—
201C4⋊C4×C73—42no—
202C4⋊C4×C73C2×Q812yes—
203C4⋊C4×C73#4112yes—
204C22⋊C4×C73—22no—
205C22⋊C4×C73—24no—
206C22⋊C4×C73—42no—
207C22⋊C4×C73—42no—
208C22⋊C4×C73—81no—
209C22⋊C4×C73D814yes—
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—22no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73—41no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73—41no—
226C23×C73C22⋊Q812yes—
227C23×C73C2×D812yes—
228C23×C73#13712yes—
229C23×C73C22×D411yescharacteristic
230D4×C73—22no—
231D4×C73—22no—
232D4×C73—48no—
233D4×C73—41no—
234D4×C73—41no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—22no—
244C2×C4×C73—22no—
245C2×C4×C73—22no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—22no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
251C2×C4×C73—21no—
252C2×C4×C73—21no—
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—84no—
261C2×C4×C73—84no—
262C2×C4×C73C22⋊Q812yes—
263C2×C4×C73C2×D812yes—
264C2×C4×C73C2×D812yes—
265C2×C4×C73C22×Q811yescharacteristic
266C2×C4×C73C2×#4111yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×C4×C73C2×#4111yescharacteristic
269C2×C4×C73H(5,𝔽2)11yescharacteristic
270C2×C4×C73H(5,𝔽2)11yescharacteristic
271C2×Dic73—22no—
272C2×Dic73—41no—
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—41no—
283C2×Dic73—84no—
284Dic146—84no—
285#10596831—41no—
286#10596831—41no—
287#10596831—41no—
288#10596831—41no—
289C22×C73—22no—
290C22×C73—22no—
291C22×C73—22no—
292C22×C73—22no—
293C22×C73—22no—
294C22×C73—22no—
295C22×C73—42no—
296C22×C73—42no—
297C22×C73—42no—
298C22×C73—81no—
299C22×C73—81no—
300C22×C73—81no—
301C22×C73—81no—
302C22×C73#47414yes—
303C22×C73C2×C22⋊Q811yescharacteristic
304C22×C73#54611yescharacteristic
305C22×C73#54611yescharacteristic
306C22×C73C22×D811yescharacteristic
307C22×C73C2×#13711yescharacteristic
308C22×C73#57711yescharacteristic
309C22×C73#57711yescharacteristic
310C4×C73—22no—
311C4×C73—22no—
312C4×C73—22no—
313C4×C73—42no—
314C4×C73—41no—
315C4×C73—41no—
316C4×C73—42no—
317C4×C73—81no—
318C4×C73#52911yescharacteristic
319C4×C73#53811yescharacteristic
320C4×C73#55411yescharacteristic
321C4×C73#55411yescharacteristic
322Dic73—42no—
323Dic73—81no—
324Dic73—81no—
325Dic73—81no—
326#26408—731nomaximal, Sylow
327C2×C73—24no—
328C2×C73—82no—
329C2×C73#245011yescharacteristic
330C2×C73C2×#47411yescharacteristic
331C2×C73#308411yescharacteristic
332C2×C73#329411yescharacteristic
333C2×C73#329411yescharacteristic
334C2×C73#331611yescharacteristic
335C2×C73#332011yescharacteristic
336#2440—731no—
337#2361D7311yescharacteristic
338#2099—731no—
339#2086—731no—
340#2067—731no—
341#2023—731no—
342#1988—731no—
343#1918—731no—
344#1893—731no—
345#1874—731no—
346#1862—731no—
347#1852—731no—
348#1850—731no—
349#1802—731no—
350C2×#340—731no—
351C73#2640811yescharacteristic, complemented, Sylow
352C23×C8—731no—
353C2×C4⋊Q8—731no—
354C2×C22⋊Q8—1461no—
355C2×#121C2×D7311yescharacteristic
356C2×C4×D4—21no—
357C2×C4×D4C2×D7311yescharacteristic
358C22×C4⋊C4—731no—
359C22×C4⋊C4C2×D7311yescharacteristic
360C2×#107—731no—
361C2×#107—731no—
362C2×C4⋊C8—731no—
363C2×C4⋊C8—731no—
364C2×#102—731no—
365C2×#102—731no—
366C2×#102—731no—
367C2×#102—731no—
368C2×C22⋊C8—731no—
369#398—1461no—
370#395—1461no—
371#390C2×D7311yescharacteristic
372#389—731no—
373#389—731no—
374#389C2×D7311yescharacteristic
375#388—21no—
376#386—21no—
377#385—21no—
378#385—1461no—
379#384—731no—
380#380—731no—
381C4×C4⋊C4C2×D7311yescharacteristic
382C4×C22⋊C4C2×D7311yescharacteristic
383#340—731no—
384#340—731no—
385#340—731no—
386#340—731no—
387#340—731no—
388#340—731no—
389#340—731no—
390#340—731no—
391#340—731no—
392#340—731no—
393#336—731no—
394#336—731no—
395C22×Q8—1461no—
396C22×D4—21no—
397C22×D4C22×D7311yescharacteristic
398C23×C4—21no—
399C23×C4—21no—
400C23×C4—1461no—
401C23×C4C22×D7311yescharacteristic
402C22×C8—2921no—
403C22×C8—2921no—
404C22×C8—731no—
405C22×C8—731no—
406C22×C8—731no—
407C22×C8—731no—
408C22×C8—1462no—
409C4⋊Q8—2921no—
410C4⋊Q8—1462no—
411C22⋊Q8—2924no—
412#121—22no—
413#121Dic14614yescomplemented
414C4×D4—22no—
415C4×D4—41no—
416C4×D4—44no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—41no—
421C2×C4⋊C4—2921no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4—1461no—
431C2×C4⋊C4—1462no—
432C2×C4⋊C4—1461no—
433C2×C4⋊C4C22×D7311yescharacteristic
434C2×C4⋊C4C22×D7311yescharacteristic
435C2×C4⋊C4C22×D7311yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—1461no—
441C2×C22⋊C4C22⋊D7311yescharacteristic
442C2×C22⋊C4C22⋊D7311yescharacteristic
443C2×C42—21no—
444C2×C42C22×D7311yescharacteristic
445C2×C42C22×D7311yescharacteristic
446#107—2921no—
447#107—1462no—
448C4⋊C8—2921no—
449C4⋊C8—2921no—
450#102—2921no—
451#102—2921no—
452#102—2921no—
453#102—1462no—
454C22⋊C8—1462no—
455#95—21no—
456#95—21no—
457#95—1461no—
458#95—1461no—
459#95—1461no—
460#95C22⋊D7311yescharacteristic
461#95C22⋊D7311yescharacteristic
462C24—21no—
463C24C2×C22⋊D7311yescharacteristic
464C2×Q8—2922no—
465C2×Q8—2922no—
466C2×Q8—5841no—
467C2×D4—24no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4C2×Dic14612yes—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—81no—
488C22×C4—81no—
489C22×C4—2922no—
490C22×C4—2922no—
491C22×C4—5841no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4—1461no—
495C22×C4—1461no—
496C22×C4C2×C22⋊D7311yescharacteristic
497C22×C4C2×C22⋊D7311yescharacteristic
498C22×C4C2×C22⋊D7311yescharacteristic
499C22×C4C2×C22⋊D7311yescharacteristic
500C22×C4C2×C22⋊D7311yescharacteristic
501C22×C4C2×Dic14612yes—
502C22×C4#1192402812yes—
503C22×C4C23×D7311yescharacteristic
504C2×C8—2924no—
505C2×C8—1461no—
506C2×C8—1461no—
507C2×C8—1461no—
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—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—2922no—
524C4⋊C4—2922no—
525C4⋊C4—2922no—
526C4⋊C4—2922no—
527C4⋊C4—2922no—
528C4⋊C4—5841no—
529C4⋊C4—732no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4—1461no—
536C4⋊C4—1461no—
537C4⋊C4C2×Dic14612yes—
538C4⋊C4#1192402712yes—
539C22⋊C4—22no—
540C22⋊C4—24no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—81no—
544C22⋊C4—2922no—
545C22⋊C4#1192401314yes—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—42no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23#1455657012yes—
563C23C2×#1192401312yes—
564C23#1455662112yes—
565C23C22×C22⋊D7311yescharacteristic
566Q8—5844no—
567D4—22no—
568D4—22no—
569D4—48no—
570D4—41no—
571D4—41no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—2921no—
603C2×C4—2921no—
604C2×C4—2921no—
605C2×C4—2921no—
606C2×C4—2921no—
607C2×C4—2921no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—5844no—
611C2×C4—1462no—
612C2×C4#1455657012yes—
613C2×C4C2×#1192401312yes—
614C2×C4C2×#1192401312yes—
615C2×C4C22×Dic14611yescharacteristic
616C2×C4#41×D7311yescharacteristic
617C2×C4C2×#1192402711yescharacteristic
618C2×C4C2×#1192402811yescharacteristic
619C2×C4#1455663611yescharacteristic
620C2×C4#1455663711yescharacteristic
621C8—2921no—
622C8—2921no—
623C8—2921no—
624C8—2921no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22#1798886714yes—
639C22C2×#1455657011yescharacteristic
640C22#1798922511yescharacteristic
641C22#1798923111yescharacteristic
642C22C22×#1192401311yescharacteristic
643C22C2×#1455662111yescharacteristic
644C22#1798939511yescharacteristic
645C22#1798939811yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—42no—
651C4—41no—
652C4—41no—
653C4—81no—
654C4—2922no—
655C4—5841no—
656C4—5841no—
657C4—5841no—
658C4#1798911711yescharacteristic
659C4#1798917011yescharacteristic
660C4#1798929411yescharacteristic
661C4#1798929611yescharacteristic
662C2—24no—
663C2—82no—
664C2#2261374911yescharacteristic
665C2C2×#1798886711yescharacteristic
666C2#2261979311yescharacteristic
667C2#2262173811yescharacteristic
668C2#2262174411yescharacteristic
669C2#2262204211yescharacteristic
670C2#2262209611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2361C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenter#2361D73
hypocenterC73#26408
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. #22613633C2semidirect
2. #22611489C2semidirect
3. #22611422C2semidirect
4. #22611319C2semidirect
5. #22611067C2semidirect
6. #22610883C2semidirect
7. #22610502C2semidirect
10. #22610178C2semidirect
11. #22610119C2semidirect
12. #22610111C2semidirect
13. #22609931C2semidirect
14. C2×#17988349C2semidirect
102. #121×C73Q8semidirect
351. C73#26408semidirect
413. #121Dic146semidirect

Maximal subgroups

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

subgroupindex
#226136332
#226114892
#226114222
#226113192
#226110672
#226108832
#226105022
#226103602
#226102492
#226101782
#226101192
#226101112
#226099312
C2×#179883492
C73×#23612
#2640873