Finite Group Databasebeta

← #33136354

Subgroups — #33136354

Order 18688.

832 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988891C211yesmaximal, characteristic, complemented
2C2×#17988659C211yesmaximal, characteristic
3#22614853C211yesmaximal, characteristic
4#22611368C211yesmaximal, characteristic, complemented
5#22611362C211yesmaximal, characteristic, complemented
6#22611127C211yesmaximal, characteristic, complemented
7#22611127C211yesmaximal, characteristic, complemented
8#22610901C211yesmaximal, characteristic, complemented
9#22610694C211yesmaximal, characteristic, complemented
10#22610362C211yesmaximal, characteristic, complemented
11#22610352C211yesmaximal, characteristic, complemented
12#22609962C211yesmaximal, characteristic, complemented
13#22609954C211yesmaximal, characteristic, complemented
14#22609764C211yesmaximal, characteristic, complemented
15C73×#2562C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C23×#10596831C2211yescharacteristic
18C2×#14556570C2211yescharacteristic
19C2×#14556570C2211yescharacteristic
20C2×C4×Dic146C2211yescharacteristic
21#17988891—24no—
22#17988659—24no—
23C2×#14556524C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×#14556514C2211yescharacteristic
32C2×C22⋊#10596831C2211yescharacteristic
33C2×C22⋊#10596831C2211yescharacteristic
34C2×#14556502C2211yescharacteristic
35#17988510—21no—
36#17988510—21no—
37#17988490—21no—
38#17988483—21no—
39#17988457C2211yescharacteristic
40#17988447C2211yescharacteristic
41#17988447C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50C2×#125×C73C2211yescharacteristic
51C2×C4×D4×C73C2211yescharacteristic
52#400×C73—21no—
53#394×C73—21no—
54#388×C73—21no—
55#388×C73—21no—
56#380×C73C2211yescharacteristic
57#380×C73C2211yescharacteristic
58#380×C73C2211yescharacteristic
59#380×C73C2211yescharacteristic
60C4×C4⋊C4×C73C2211yescharacteristic
61C22×Dic146C2311yescharacteristic
62C2×#11924012—22no—
63C2×#11924012—22no—
64C2×#11924012—41no—
65C22×#10596831—22no—
66C22×#10596831—22no—
67C22×#10596831—41no—
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72#14556570—22no—
73#14556570—22no—
74#14556570—41no—
75#14556570—41no—
76C4×Dic146—22no—
77C4×Dic146—22no—
78C2×C4⋊Dic73—21no—
79C2×C4⋊Dic73—21no—
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×C22⋊Dic73—21no—
87C2×C22⋊Dic73—21no—
88C2×C22⋊Dic73—21no—
89C2×C22⋊Dic73—21no—
90C2×C4×Dic73—21no—
91#14556524—22no—
92#14556522—22no—
93#14556522—41no—
94#14556522—41no—
95C4⋊#10596831D414yes—
96#14556514—22no—
97#14556514—22no—
98#14556514—22no—
99#14556514—22no—
100C22⋊#10596831—22no—
101C22⋊#10596831—22no—
102#14556502—22no—
103#14556498—21no—
104#14556498—21no—
105C22×D4×C73D411yescharacteristic
106C23×C4×C73C2311yescharacteristic
107C23×C4×C73C2311yescharacteristic
108#125×C73—22no—
109#125×C73—22no—
110C4×D4×C73—22no—
111C4×D4×C73—22no—
112C2×C4⋊C4×C73—21no—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73D411yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C22⋊C4×C73—21no—
122C2×C22⋊C4×C73—21no—
123C2×C22⋊C4×C73D411yescharacteristic
124C2×C22⋊C4×C73D411yescharacteristic
125C2×C42×C73—21no—
126C2×C42×C73C2311yescharacteristic
127#95×C73—21no—
128#95×C73—21no—
129#95×C73—21no—
130#95×C73—21no—
131#95×C73—21no—
132C2×Dic146—21no—
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146—21no—
136C2×Dic146—21no—
137C2×Dic146#4112yes—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142#11924012—44no—
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—44no—
153C2×#10596831C2×D412yes—
154C2×#10596831C2×D412yes—
155C2×#10596831#4112yes—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—42no—
163C4⋊Dic73—81no—
164#11924007—42no—
165#11924007—42no—
166#11924007—42no—
167C22⋊Dic73—42no—
168C22⋊Dic73—42no—
169C22⋊Dic73—81no—
170C22⋊Dic73—81no—
171C4×Dic73—42no—
172C24×C73C2×D411yescharacteristic
173C24×C73C2×D411yescharacteristic
174C2×D4×C73—22no—
175C2×D4×C73—22no—
176C2×D4×C73—41no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
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×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73—21no—
192C22×C4×C73—21no—
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2411yescharacteristic
198C4⋊C4×C73—22no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73—21no—
205C4⋊C4×C73—21no—
206C4⋊C4×C73—21no—
207C4⋊C4×C73—42no—
208C4⋊C4×C73—42no—
209C22⋊C4×C73—22no—
210C22⋊C4×C73—22no—
211C22⋊C4×C73—42no—
212C22⋊C4×C73—42no—
213C22⋊C4×C73—81no—
214C22⋊C4×C73—81no—
215C42×C73—81no—
216C42×C73C2×D412yes—
217C42×C73#4112yes—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—21no—
222C23×C73—21no—
223C23×C73—21no—
224C23×C73—21no—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—21no—
231C23×C73#13712yes—
232C23×C73C22×D411yescharacteristic
233D4×C73—44no—
234C2×C4×C73—21no—
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—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
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×C73—41no—
263C2×C4×C73—41no—
264C2×C4×C73#12112yes—
265C2×C4×C73#12112yes—
266C2×C4×C73C22×D411yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×C4×C73C2×#4111yescharacteristic
269C2×C4×C73C2×#4111yescharacteristic
270C2×C4×C73H(5,𝔽2)11yescharacteristic
271C2×C4×C732−1+411yescharacteristic
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—41no—
283C2×Dic73—41no—
284Dic146—22no—
285Dic146—22no—
286Dic146—41no—
287Dic146—41no—
288#10596831—22no—
289#10596831—22no—
290#10596831—22no—
291#10596831—41no—
292C22×C73—22no—
293C22×C73—22no—
294C22×C73—22no—
295C22×C73—41no—
296C22×C73—41no—
297C22×C73—41no—
298C22×C73—41no—
299C22×C73—41no—
300C22×C73—41no—
301C22×C73—41no—
302C22×C73—41no—
303C22×C73—41no—
304C22×C73—41no—
305C22×C73—41no—
306C22×C73C2×#12111yescharacteristic
307C22×C73#54611yescharacteristic
308C22×C73#54711yescharacteristic
309C22×C73C2×#13711yescharacteristic
310C22×C73#57511yescharacteristic
311C22×C73#57711yescharacteristic
312C22×C73#57711yescharacteristic
313C4×C73—22no—
314C4×C73—22no—
315C4×C73—41no—
316C4×C73—41no—
317C4×C73—81no—
318C4×C73—81no—
319C4×C73—81no—
320C4×C73—81no—
321C4×C73#52911yescharacteristic
322C4×C73#54211yescharacteristic
323C4×C73#54611yescharacteristic
324C4×C73#54811yescharacteristic
325Dic73—42no—
326Dic73—81no—
327Dic73—81no—
328Dic73—81no—
329#25357—731nomaximal, Sylow
330C2×C73—42no—
331C2×C73—81no—
332C2×C73#242711yescharacteristic
333C2×C73#305611yescharacteristic
334C2×C73#306011yescharacteristic
335C2×C73#331711yescharacteristic
336C2×C73#332011yescharacteristic
337C2×C73#334411yescharacteristic
338C2×C73#335011yescharacteristic
339C2×#479—731no—
340C2×#435—731no—
341#2562D7311yescharacteristic
342#2538—731no—
343#2077—731no—
344#2076—731no—
345#2035—731no—
346#2035—731no—
347#1992—731no—
348#1948—731no—
349#1894—731no—
350#1892—731no—
351#1809—731no—
352#1807—731no—
353#1763—731no—
354C73#2535711yescharacteristic, complemented, Sylow
355C22×M16—731no—
356C23×C8—731no—
357C2×#125C2×D7311yescharacteristic
358C2×C22⋊Q8—731no—
359C2×C22⋊Q8—731no—
360C2×C4×Q8—731no—
361C2×C4×D4C2×D7311yescharacteristic
362#479—1464no—
363#435—1464no—
364C2×#107—731no—
365C2×#106—731no—
366C2×#106—731no—
367C2×#106—731no—
368C2×C4⋊C8—731no—
369C2×#103—731no—
370C2×#103—731no—
371C2×#103—731no—
372C2×#103—731no—
373C2×C22⋊C8—731no—
374C2×C22⋊C8—731no—
375C2×#97—731no—
376#400—21no—
377#396—1461no—
378#396—1461no—
379#394—21no—
380#390—1461no—
381#389—1461no—
382#388—21no—
383#388—21no—
384#384—731no—
385#380—731no—
386#380—731no—
387#380C2×D7311yescharacteristic
388#380C2×D7311yescharacteristic
389#380C2×D7311yescharacteristic
390#380C2×D7311yescharacteristic
391C4×C4⋊C4C2×D7311yescharacteristic
392#340—731no—
393#340—731no—
394#340—731no—
395#340—731no—
396#340—731no—
397#340—731no—
398#340—731no—
399#340—731no—
400C22×Q8—731no—
401C22×D4C22⋊D7311yescharacteristic
402C23×C4C22×D7311yescharacteristic
403C23×C4C22×D7311yescharacteristic
404C2×M16—2921no—
405C2×M16—1462no—
406C2×M16—1462no—
407C22×C8—2921no—
408C22×C8—731no—
409C22×C8—731no—
410C22×C8—731no—
411C22×C8—731no—
412C22×C8—1462no—
413C22×C8—1462no—
414#125—22no—
415#125—22no—
416C22⋊Q8—2921no—
417C22⋊Q8—2921no—
418C22⋊Q8—1462no—
419C22⋊Q8—1462no—
420C4×Q8—1462no—
421C4×Q8—1462no—
422C4×D4—22no—
423C4×D4—22no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—731no—
427C2×C4⋊C4—731no—
428C2×C4⋊C4—731no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4—1461no—
431C2×C4⋊C4—1461no—
432C2×C4⋊C4—1461no—
433C2×C4⋊C4—1461no—
434C2×C4⋊C4C22⋊D7311yescharacteristic
435C2×C4⋊C4C22×D7311yescharacteristic
436C2×C4⋊C4C22×D7311yescharacteristic
437C2×C4⋊C4C22×D7311yescharacteristic
438C2×C4⋊C4C22×D7311yescharacteristic
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—1461no—
444C2×C22⋊C4—1461no—
445C2×C22⋊C4—1461no—
446C2×C22⋊C4—1461no—
447C2×C22⋊C4C22⋊D7311yescharacteristic
448C2×C22⋊C4C22⋊D7311yescharacteristic
449C2×C42—21no—
450C2×C42—1461no—
451C2×C42C22×D7311yescharacteristic
452#107—1462no—
453#106—2921no—
454#106—2921no—
455#106—1462no—
456C4⋊C8—734no—
457#103—1462no—
458#103—1462no—
459#103—1462no—
460#103—1462no—
461C22⋊C8—1462no—
462C22⋊C8—1462no—
463#97—1462no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—1461no—
470#95—1461no—
471C24C2×C22⋊D7311yescharacteristic
472C24C2×C22⋊D7311yescharacteristic
473C2×Q8—732no—
474C2×Q8—1461no—
475C2×Q8—1461no—
476C2×Q8—1461no—
477C2×Q8—1461no—
478C2×Q8—1461no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—41no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
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—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4C2×C22⋊D7311yescharacteristic
503C22×C4C2×C22⋊D7311yescharacteristic
504C22×C4C2×C22⋊D7311yescharacteristic
505C22×C4C2×C22⋊D7311yescharacteristic
506C22×C4C23×D7311yescharacteristic
507M16—2924no—
508C2×C8—2924no—
509C2×C8—732no—
510C2×C8—732no—
511C2×C8—732no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—2922no—
533C4⋊C4—2922no—
534C4⋊C4—2922no—
535C4⋊C4—2922no—
536C4⋊C4—5841no—
537C4⋊C4—1461no—
538C4⋊C4—1461no—
539C4⋊C4—1461no—
540C4⋊C4—1461no—
541C4⋊C4—1461no—
542C4⋊C4—1461no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—81no—
548C22⋊C4—81no—
549C22⋊C4—2922no—
550C22⋊C4—2922no—
551C22⋊C4—5841no—
552C22⋊C4—5841no—
553C42—81no—
554C42—2922no—
555C42D4×D7312yes—
556C42#1192402712yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23#1455662112yes—
571C23C22×C22⋊D7311yescharacteristic
572Q8—2921no—
573Q8—2921no—
574Q8—1462no—
575Q8—1462no—
576D4—44no—
577C2×C4—21no—
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—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
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—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—2921no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—2921no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—2921no—
614C2×C4—2921no—
615C2×C4—2921no—
616C2×C4—2921no—
617C2×C4—2921no—
618C2×C4—1462no—
619C2×C4#1455656312yes—
620C2×C4#1455656312yes—
621C2×C4C2×D4×D7311yescharacteristic
622C2×C4#41×D7311yescharacteristic
623C2×C4#41×D7311yescharacteristic
624C2×C4C2×#1192402711yescharacteristic
625C2×C4#1455663711yescharacteristic
626C2×C4#1455663811yescharacteristic
627C8—2921no—
628C8—1462no—
629C8—1462no—
630C8—1462no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22C2×#1455656311yescharacteristic
646C22#1798922511yescharacteristic
647C22#1798923611yescharacteristic
648C22C2×#1455662111yescharacteristic
649C22#1798938511yescharacteristic
650C22#1798939511yescharacteristic
651C22#1798939811yescharacteristic
652C4—22no—
653C4—22no—
654C4—41no—
655C4—41no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—2922no—
661C4—5841no—
662C4—5841no—
663C4—5841no—
664C4#1798911611yescharacteristic
665C4#1798920611yescharacteristic
666C4#1798923011yescharacteristic
667C4#1798924511yescharacteristic
668C2—42no—
669C2—81no—
670C2#2261351111yescharacteristic
671C2#2261956311yescharacteristic
672C2#2261959011yescharacteristic
673C2#2262205111yescharacteristic
674C2#2262209611yescharacteristic
675C2#2262238611yescharacteristic
676C2#2262247011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2562C2
socleC23×C73C22×D4
prime radicalC22×D4×C73D4
hypercenter#2562D73
hypocenterC73#25357
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2562

Complements

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

NH ≅ G/Ntype
1. C2×#17988891C2semidirect
4. #22611368C2semidirect
5. #22611362C2semidirect
6. #22611127C2semidirect
7. #22611127C2semidirect
8. #22610901C2semidirect
9. #22610694C2semidirect
10. #22610362C2semidirect
11. #22610352C2semidirect
12. #22609962C2semidirect
13. #22609954C2semidirect
14. #22609764C2semidirect
354. C73#25357semidirect

Maximal subgroups

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

subgroupindex
C2×#179888912
C2×#179886592
#226148532
#226113682
#226113622
#226111272
#226109012
#226106942
#226103622
#226103522
#226099622
#226099542
#226097642
C73×#25622
#2535773