Finite Group Databasebeta

← #33112089

Subgroups — #33112089

Order 18688.

758 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
1C4×#14556574C211yesmaximal, characteristic
2#22610727C211yesmaximal, characteristic
3#22610648C211yesmaximal, characteristic, complemented
4#22610646C211yesmaximal, characteristic, complemented
5#22610540C211yesmaximal, characteristic, complemented
6#22610538C211yesmaximal, characteristic, complemented
7#22610536C211yesmaximal, characteristic, complemented
8#22610534C211yesmaximal, characteristic, complemented
9#22610498C211yesmaximal, characteristic, complemented
10#22610492C211yesmaximal, characteristic, complemented
11#22610225C211yesmaximal, characteristic
12#22610115C211yesmaximal, characteristic, complemented
13#22610112C211yesmaximal, characteristic, complemented
14#22609911C211yesmaximal, characteristic, complemented
15C73×#1797C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C2×#14556574C411yescharacteristic
18C2×C4×C22⋊D73C2211yescharacteristic
19C2×C4⋊#10596831C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556520C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556520C2211yescharacteristic
24C2×C22⋊#10596831C2211yescharacteristic
25C2×C22⋊#10596831C2211yescharacteristic
26C2×#14556503C2211yescharacteristic
27C2×#14556503C2211yescharacteristic
28C2×#14556503C2211yescharacteristic
29C2×#14556501C2211yescharacteristic
30C2×C8×Dic73C2211yescharacteristic
31C2×C8×Dic73C2211yescharacteristic
32#17988478C411yescharacteristic
33#17988478C411yescharacteristic
34#17988471C411yescharacteristic
35#17988471C411yescharacteristic
36#17988466C411yescharacteristic
37#17988460C411yescharacteristic
38#17988450C411yescharacteristic
39C4×#11924007C2211yescharacteristic
40C4×#11924007C2211yescharacteristic
41C4×#11924005C2211yescharacteristic
42C4×#11924005C2211yescharacteristic
43C4×C22⋊Dic73C2211yescharacteristic
44#17988342C2211yescharacteristic
45#17988342C2211yescharacteristic
46#17988341C2211yescharacteristic
47#17988341C2211yescharacteristic
48#17988341C2211yescharacteristic
49#17988341C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C22×C42×C73C2211yescharacteristic
53C2×C4⋊C8×C73C2211yescharacteristic
54C2×C4⋊C8×C73C2211yescharacteristic
55C2×C22⋊C8×C73C2211yescharacteristic
56C2×C22⋊C8×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×C22⋊D73C2×C411yescharacteristic
60C22×C4×D73C2311yescharacteristic
61C2×#11924011—41no—
62C2×#11924011—41no—
63C2×#11924011—41no—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556574—24no—
69C4×C22⋊D73—41no—
70C4×C22⋊D73—41no—
71C2×#11924007C2×C411yescharacteristic
72C2×#11924007C2×C411yescharacteristic
73C2×#11924007C2×C411yescharacteristic
74C2×#11924007C2×C411yescharacteristic
75C2×#11924007C2×C411yescharacteristic
76C2×C22⋊Dic73C2×C411yescharacteristic
77C2×C22⋊Dic73C2×C411yescharacteristic
78C2×C22⋊Dic73C2×C411yescharacteristic
79C2×#11924005C2×C411yescharacteristic
80C2×#11924005C2×C411yescharacteristic
81C2×#11924005C2×C411yescharacteristic
82C2×#11924005C2×C411yescharacteristic
83C2×#11924005C2×C411yescharacteristic
84C2×C4×Dic73C2311yescharacteristic
85C2×C4×Dic73C2311yescharacteristic
86C2×C4×Dic73C2311yescharacteristic
87C4⋊#10596831—41no—
88C4⋊#10596831—41no—
89#14556520—41no—
90#14556520—41no—
91#14556520—41no—
92C22⋊#10596831—41no—
93C22⋊#10596831—41no—
94#14556503—41no—
95#14556503—41no—
96#14556503—41no—
97#14556501—41no—
98C8×Dic73—41no—
99C8×Dic73—41no—
100#14556498C2×C411yescharacteristic
101#14556498C2×C411yescharacteristic
102#14556498C2×C411yescharacteristic
103#14556498C2×C411yescharacteristic
104#14556498C2×C411yescharacteristic
105#14556498C2×C411yescharacteristic
106#14556498C2×C411yescharacteristic
107#14556498C2×C411yescharacteristic
108#14556498C2×C411yescharacteristic
109#14556498C2×C411yescharacteristic
110C23×C4×C73C2×C411yescharacteristic
111C23×C4×C73C2×C411yescharacteristic
112C23×C4×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C22×C8×C73C2311yescharacteristic
116C22×C8×C73C2311yescharacteristic
117C2×C42×C73—41no—
118C2×C42×C73—41no—
119C2×C42×C73C2×C411yescharacteristic
120C2×C42×C73C2×C411yescharacteristic
121C2×C42×C73C2311yescharacteristic
122C2×C42×C73C2311yescharacteristic
123C4⋊C8×C73—41no—
124C4⋊C8×C73—41no—
125C22⋊C8×C73—41no—
126C22⋊C8×C73—41no—
127C23×D73C22×C411yescharacteristic
128C2×C22⋊D73—22no—
129C2×C22⋊D73—41no—
130C2×C22⋊D73—41no—
131C22×Dic73C22×C411yescharacteristic
132C22×Dic73C22×C411yescharacteristic
133C22×Dic73C22×C411yescharacteristic
134C22×Dic73C22×C411yescharacteristic
135C22×Dic73C22×C411yescharacteristic
136C22×Dic73C22×C411yescharacteristic
137C22×Dic73C22×C411yescharacteristic
138C2×C4×D73—21no—
139C2×C4×D73—21no—
140C2×C4×D73—21no—
141C2×C4×D73—21no—
142C2×C4×D73—21no—
143C2×C4×D73—21no—
144#11924011—44no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157#11924007—22no—
158#11924007—22no—
159#11924007—22no—
160#11924007—22no—
161#11924007—41no—
162C22⋊Dic73—41no—
163C22⋊Dic73—41no—
164C22⋊Dic73M1614yescomplemented
165#11924005—22no—
166#11924005—22no—
167#11924005—22no—
168#11924005—22no—
169#11924005—41no—
170C4×Dic73—21no—
171C4×Dic73—21no—
172C4×Dic73—21no—
173C4×Dic73—21no—
174C4×Dic73—21no—
175C4×Dic73—21no—
176C24×C73C22×C411yescharacteristic
177C22×C4×C73—22no—
178C22×C4×C73—22no—
179C22×C4×C73—22no—
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—41no—
187C22×C4×C73—41no—
188C22×C4×C73—41no—
189C22×C4×C73C22×C411yescharacteristic
190C22×C4×C73C22×C411yescharacteristic
191C22×C4×C73C22×C411yescharacteristic
192C22×C4×C73C22×C411yescharacteristic
193C22×C4×C73C22×C411yescharacteristic
194C22×C4×C73C2411yescharacteristic
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73—21no—
206C2×C8×C73—21no—
207C42×C73—21no—
208C42×C73—21no—
209C42×C73—21no—
210C42×C73—21no—
211C42×C73—41no—
212C42×C73—41no—
213C23×C73—21no—
214C23×C73—21no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73C2×M1612yes—
220C23×C73C23×C411yescharacteristic
221C22×D73—21no—
222C22×D73—21no—
223C22×D73—21no—
224C22×D73—21no—
225C22×D73—21no—
226C22×D73—21no—
227C22×D73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
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—44no—
242C2×C4×C73—44no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73#13112yes—
248C2×C4×C73#13112yes—
249C2×C4×C73H(5,𝔽2)11yescharacteristic
250C2×C4×C73H(5,𝔽2)11yescharacteristic
251C2×C4×C73H(5,𝔽2)11yescharacteristic
252C2×C4×C73H(5,𝔽2)11yescharacteristic
253C2×C4×C732−1+411yescharacteristic
254C2×C4×C732−1+411yescharacteristic
255C22⋊D73—44no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—21no—
266C2×Dic73—21no—
267C2×Dic73—21no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—21no—
271C2×Dic73—21no—
272C2×Dic73—21no—
273C2×Dic73—21no—
274C2×Dic73—21no—
275C2×Dic73C2×M1612yes—
276C2×Dic73C2×M1612yes—
277C4×D73—41no—
278C4×D73—41no—
279C4×D73—41no—
280C4×D73—41no—
281C8×C73—41no—
282C8×C73—41no—
283C8×C73—41no—
284C8×C73—41no—
285#10596831—41no—
286#10596831—41no—
287#10596831—41no—
288#10596831—41no—
289C22×C73—22no—
290C22×C73—22no—
291C22×C73—22no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73—41no—
295C22×C73—41no—
296C22×C73#51811yescharacteristic
297C22×C73#52011yescharacteristic
298C22×C73#52011yescharacteristic
299C22×C73C22×M1611yescharacteristic
300C22×C73C2×#13111yescharacteristic
301C22×C73C2×#13111yescharacteristic
302C22×C73#56811yescharacteristic
303C2×D73—41no—
304C2×D73—41no—
305C2×D73—41no—
306C2×D73—41no—
307C2×D73—41no—
308C2×D73—41no—
309C2×D73—41no—
310C4×C73—22no—
311C4×C73—22no—
312C4×C73—41no—
313C4×C73—41no—
314C4×C73—41no—
315C4×C73#56011yescharacteristic
316C4×C73#56211yescharacteristic
317C4×C73#56211yescharacteristic
318C4×C73#56311yescharacteristic
319Dic73—22no—
320Dic73—22no—
321Dic73—41no—
322Dic73—41no—
323Dic73—41no—
324Dic73—41no—
325Dic73—41no—
326#23316—731nomaximal, Sylow
327C2×C73—42no—
328C2×C73#237611yescharacteristic
329C2×C73#297511yescharacteristic
330C2×C73#297711yescharacteristic
331C2×C73#299011yescharacteristic
332C2×C73#299011yescharacteristic
333C2×C73#299311yescharacteristic
334C2×C73#299311yescharacteristic
335D73—81no—
336C4×#123—731no—
337#1955—731no—
338#1939—731no—
339#1939—731no—
340#1923—731no—
341#1923—731no—
342#1923—731no—
343#1923—731no—
344#1917—731no—
345#1917—731no—
346#1870—731no—
347#1851—731no—
348#1851—731no—
349#1797—731no—
350#1797D7311yescharacteristic, complemented
351C73#2331611yescharacteristic, complemented, Sylow
352C22×M16—731no—
353C2×#123—731no—
354C2×C4×D4—731no—
355C22×C42C2×D7311yescharacteristic
356C2×C4⋊C8—731no—
357C2×C4⋊C8—731no—
358C2×C4⋊C8—731no—
359C2×C4⋊C8—731no—
360C2×C4⋊C8—731no—
361C2×C4⋊C8C2×D7311yescharacteristic
362C2×C4⋊C8C2×D7311yescharacteristic
363C2×C22⋊C8—731no—
364C2×C22⋊C8—731no—
365C2×C22⋊C8—731no—
366C2×C22⋊C8—731no—
367C2×C22⋊C8—731no—
368C2×C22⋊C8C2×D7311yescharacteristic
369C2×C22⋊C8C2×D7311yescharacteristic
370C2×#97—731no—
371C2×C4×C8—731no—
372C2×C4×C8—731no—
373#388—731no—
374#388—731no—
375#387—731no—
376#387—731no—
377#386—731no—
378#385—731no—
379#381—731no—
380C4×C4⋊C4—731no—
381C4×C4⋊C4—731no—
382C4×C22⋊C4—731no—
383C4×C22⋊C4—731no—
384C4×C22⋊C4—731no—
385#336—731no—
386#336—731no—
387#336—731no—
388#336—731no—
389#336—731no—
390#336—731no—
391#336—731no—
392#336—731no—
393#336C2×D7311yescharacteristic
394#336C2×D7311yescharacteristic
395C22×D4—731no—
396C23×C4—731no—
397C23×C4C4×D7311yescharacteristic
398C23×C4C4×D7311yescharacteristic
399C23×C4C22×D7311yescharacteristic
400C2×M16—2921no—
401C2×M16—2921no—
402C2×M16—2921no—
403C22×C8—731no—
404C22×C8—731no—
405C22×C8—731no—
406C22×C8—731no—
407C22×C8C22×D7311yescharacteristic
408C22×C8C22×D7311yescharacteristic
409C22×C8C22×D7311yescharacteristic
410C22×C8C22×D7311yescharacteristic
411#123—1464no—
412C4×D4—2921no—
413C4×D4—2921no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—731no—
416C2×C4⋊C4—731no—
417C2×C4⋊C4—731no—
418C2×C4⋊C4—731no—
419C2×C22⋊C4—731no—
420C2×C22⋊C4—731no—
421C2×C22⋊C4—731no—
422C2×C22⋊C4—731no—
423C2×C22⋊C4—731no—
424C2×C22⋊C4—731no—
425C2×C22⋊C4—731no—
426C2×C22⋊C4—731no—
427C2×C42—41no—
428C2×C42—41no—
429C2×C42—731no—
430C2×C42—731no—
431C2×C42—731no—
432C2×C42C4×D7311yescharacteristic
433C2×C42C4×D7311yescharacteristic
434C2×C42C22×D7311yescharacteristic
435C2×C42C22×D7311yescharacteristic
436C4⋊C8—41no—
437C4⋊C8—41no—
438C4⋊C8—2921no—
439C4⋊C8—2921no—
440C4⋊C8—2921no—
441C4⋊C8—2921no—
442C4⋊C8—2921no—
443C22⋊C8—41no—
444C22⋊C8—41no—
445C22⋊C8—2921no—
446C22⋊C8—2921no—
447C22⋊C8—2921no—
448C22⋊C8—2921no—
449C22⋊C8—2921no—
450#97—2921no—
451C4×C8—2921no—
452C4×C8—2921no—
453#95—731no—
454#95—731no—
455#95—731no—
456#95—731no—
457#95—731no—
458#95—731no—
459#95—731no—
460#95—731no—
461#95—731no—
462#95—731no—
463C24—731no—
464C24C2×C4×D7311yescharacteristic
465C2×D4—2921no—
466C2×D4—2921no—
467C2×D4—1462no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—731no—
481C22×C4—731no—
482C22×C4—731no—
483C22×C4—731no—
484C22×C4—731no—
485C22×C4—731no—
486C22×C4—731no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4—1461no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4C2×C4×D7311yescharacteristic
494C22×C4C2×C4×D7311yescharacteristic
495C22×C4C2×C4×D7311yescharacteristic
496C22×C4C2×C4×D7311yescharacteristic
497C22×C4C2×C4×D7311yescharacteristic
498C22×C4C23×D7311yescharacteristic
499M16—2924no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
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—
521C2×C8—1461no—
522C2×C8—1461no—
523C2×C8—1461no—
524C4⋊C4—2921no—
525C4⋊C4—1462no—
526C4⋊C4—1462no—
527C4⋊C4—1462no—
528C4⋊C4—1462no—
529C22⋊C4—2921no—
530C22⋊C4—2921no—
531C22⋊C4—2921no—
532C22⋊C4—734no—
533C22⋊C4—1462no—
534C22⋊C4—1462no—
535C22⋊C4—1462no—
536C22⋊C4—1462no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—41no—
542C42—41no—
543C42—1461no—
544C42—1461no—
545C42—1461no—
546C42—1461no—
547C42—1461no—
548C42—1461no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—1461no—
556C23—1461no—
557C23—1461no—
558C23—1461no—
559C23—1461no—
560C23—1461no—
561C23—1461no—
562C23M16×D7312yes—
563C23C22×C4×D7311yescharacteristic
564D4—2924no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—732no—
589C2×C4—732no—
590C2×C4—1461no—
591C2×C4—1461no—
592C2×C4—1461no—
593C2×C4—1461no—
594C2×C4—1461no—
595C2×C4—1461no—
596C2×C4—1461no—
597C2×C4—1461no—
598C2×C4—1461no—
599C2×C4—1461no—
600C2×C4—1461no—
601C2×C4—1461no—
602C2×C4—1461no—
603C2×C4—1461no—
604C2×C4—1461no—
605C2×C4—1461no—
606C2×C4—1461no—
607C2×C4—1461no—
608C2×C4—1461no—
609C2×C4#1455659712yes—
610C2×C4#1455659912yes—
611C2×C4#1455663611yescharacteristic
612C2×C4#1455663611yescharacteristic
613C2×C4#1455663611yescharacteristic
614C2×C4#1455663711yescharacteristic
615C2×C4#1455663811yescharacteristic
616C2×C4#1455663911yescharacteristic
617C8—41no—
618C8—41no—
619C8—41no—
620C8—41no—
621C8—2921no—
622C8—2921no—
623C8—2921no—
624C8—2921no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—2921no—
633C22—2921no—
634C22—2921no—
635C22—2921no—
636C22—2921no—
637C22—2921no—
638C22—2921no—
639C22#1798906011yescharacteristic
640C22#1798906711yescharacteristic
641C22#1798906811yescharacteristic
642C22C2×M16×D7311yescharacteristic
643C22C2×#1455659711yescharacteristic
644C22C2×#1455659911yescharacteristic
645C22#1798935311yescharacteristic
646C4—22no—
647C4—22no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—2921no—
652C4—2921no—
653C4—2921no—
654C4—2921no—
655C4—2921no—
656C4—1462no—
657C4—1462no—
658C4#1798932511yescharacteristic
659C4#1798933211yescharacteristic
660C4#1798933311yescharacteristic
661C4#1798933811yescharacteristic
662C2—42no—
663C2—5841no—
664C2#2261307511yescharacteristic
665C2#2261897011yescharacteristic
666C2#2261898511yescharacteristic
667C2#2261905311yescharacteristic
668C2#2261905811yescharacteristic
669C2#2261907611yescharacteristic
670C2#2261908211yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC73×#1797C2
prime radicalC22×C22⋊D73C2×C4
cosocleC22×C4×C73C24
hypercenter#1797D73
hypocenterC73#23316
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #22610648C2semidirect
4. #22610646C2semidirect
5. #22610540C2semidirect
6. #22610538C2semidirect
7. #22610536C2semidirect
8. #22610534C2semidirect
9. #22610498C2semidirect
10. #22610492C2semidirect
12. #22610115C2semidirect
13. #22610112C2semidirect
14. #22609911C2semidirect
15. C73×#1797C2semidirect
164. C22⋊Dic73M16semidirect
350. #1797D73semidirect
351. C73#23316semidirect

Maximal subgroups

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

subgroupindex
C4×#145565742
#226107272
#226106482
#226106462
#226105402
#226105382
#226105362
#226105342
#226104982
#226104922
#226102252
#226101152
#226101122
#226099112
C73×#17972
#2331673