Finite Group Databasebeta

← #33097299

Subgroups — #33097299

Order 18688.

920 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612615C211yesmaximal, characteristic
2#22612517C211yesmaximal, characteristic
3#22610803C211yesmaximal, characteristic, complemented
4#22610793C211yesmaximal, characteristic, complemented
5#22610623C211yesmaximal, characteristic, complemented
6#22610609C211yesmaximal, characteristic, complemented
7#22610574C211yesmaximal, characteristic, complemented
8#22610567C211yesmaximal, characteristic, complemented
9#22610274C211yesmaximal, characteristic, complemented
10#22610274C211yesmaximal, characteristic, complemented
11#22610244C211yesmaximal, characteristic
12#22610132C211yesmaximal, characteristic
13#22609866C211yesmaximal, characteristic
14#22609699C211yesmaximal, characteristic
15C2×#451×C73C211yesmaximal, characteristic
16C2×Q8×Dic73C2211yescharacteristic
17C2×#14556550C2211yescharacteristic
18C2×#14556550C2211yescharacteristic
19C2×#14556525C2211yescharacteristic
20C2×#14556523C2211yescharacteristic
21C2×#14556515C2211yescharacteristic
22C2×#14556515C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×C22⋊#10596831C2211yescharacteristic
28C2×#14556501C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30#17988494—21no—
31#17988494—21no—
32#17988494C2211yescharacteristic
33#17988494C2211yescharacteristic
34#17988494C2211yescharacteristic
35#17988494C2211yescharacteristic
36#17988494C2211yescharacteristic
37#17988486—21no—
38#17988486C2211yescharacteristic
39#17988480—21no—
40#17988480—21no—
41#17988473—21no—
42#17988450—21no—
43#17988450—21no—
44#17988447C2211yescharacteristic
45C4⋊C4×Dic73C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C23×Q8×C73C2211yescharacteristic
53C22×Q16×C73C2211yescharacteristic
54C22×Q16×C73C2211yescharacteristic
55C2×C22⋊Q8×C73C2211yescharacteristic
56#451×C73C418yescomplemented
57C2×#103×C73C2211yescharacteristic
58C2×#103×C73C2211yescharacteristic
59C2×C22⋊C8×C73C2211yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62Q8×Dic73—41no—
63Q8×Dic73—41no—
64#14556550—41no—
65#14556550—41no—
66#14556550—41no—
67#14556550—41no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73D411yescharacteristic
71C2×C4⋊Dic73D411yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C22⋊Dic73—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73D411yescharacteristic
79C2×C22⋊Dic73D411yescharacteristic
80C2×C4×Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73C2311yescharacteristic
84C2×C4×Dic73C2311yescharacteristic
85#14556525—41no—
86#14556523—41no—
87#14556515—41no—
88#14556515—41no—
89#14556515—41no—
90#14556515—41no—
91#14556515—41no—
92#14556515—41no—
93C22⋊#10596831—22no—
94#14556501—41no—
95C8×Dic73—41no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109C22×Q8×C73—21no—
110C22×Q8×C73—22no—
111C22×Q8×C73—41no—
112C22×Q8×C73—41no—
113C22×Q8×C73—41no—
114C22×Q8×C73—41no—
115C22×Q8×C73C2×C414yes—
116C22×Q8×C73C2311yescharacteristic
117C22×Q8×C73C2311yescharacteristic
118C22×Q8×C73C2311yescharacteristic
119C23×C4×C73—21no—
120C23×C4×C73C2311yescharacteristic
121C2×Q16×C73—41no—
122C2×Q16×C73—41no—
123C2×Q16×C73—41no—
124C2×Q16×C73—41no—
125C2×Q16×C73C2×C414yes—
126C2×Q16×C73C2×C414yes—
127C22×C8×C73C2311yescharacteristic
128C22×C8×C73C2311yescharacteristic
129C22⋊Q8×C73—22no—
130C22⋊Q8×C73C2×C414yes—
131C2×C4⋊C4×C73—21no—
132C2×C4⋊C4×C73C2311yescharacteristic
133C2×C22⋊C4×C73—21no—
134#103×C73C2×C414yes—
135#103×C73C2×C414yes—
136C22⋊C8×C73C2×C414yes—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73C2×D411yescharacteristic
142C22×Dic73C2×D411yescharacteristic
143C22×Dic73C2×D411yescharacteristic
144C22×Dic73C2×D411yescharacteristic
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—41no—
160C4⋊Dic73—41no—
161C4⋊Dic73—42no—
162C4⋊Dic73—81no—
163C22⋊Dic73—41no—
164C22⋊Dic73—41no—
165C22⋊Dic73—42no—
166C22⋊Dic73—81no—
167C4×Dic73—21no—
168C4×Dic73—21no—
169C4×Dic73—21no—
170C4×Dic73—21no—
171C4×Dic73—42no—
172C4×Dic73—42no—
173C4×Dic73—81no—
174C24×C73C2×D411yescharacteristic
175C2×Q8×C73—24no—
176C2×Q8×C73—21no—
177C2×Q8×C73—21no—
178C2×Q8×C73—21no—
179C2×Q8×C73—21no—
180C2×Q8×C73—21no—
181C2×Q8×C73—21no—
182C2×Q8×C73—21no—
183C2×Q8×C73—21no—
184C2×Q8×C73—21no—
185C2×Q8×C73—21no—
186C2×Q8×C73—21no—
187C2×Q8×C73—21no—
188C2×Q8×C73—21no—
189C2×Q8×C73—21no—
190C2×Q8×C73—21no—
191C2×Q8×C73—44no—
192C2×Q8×C73—44no—
193C2×Q8×C73—42no—
194C2×Q8×C73—42no—
195C2×Q8×C73—44no—
196C2×Q8×C73—44no—
197C2×Q8×C73C22×C412yes—
198C2×Q8×C73C22×C412yes—
199C2×Q8×C73C22×C412yes—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73—21no—
204C22×C4×C73—24no—
205C22×C4×C73—21no—
206C22×C4×C73—21no—
207C22×C4×C73—21no—
208C22×C4×C73—21no—
209C22×C4×C73—42no—
210C22×C4×C73—42no—
211C22×C4×C73C22×C412yes—
212C22×C4×C73C2×D411yescharacteristic
213C22×C4×C73#4112yes—
214C22×C4×C73C2411yescharacteristic
215Q16×C73—44no—
216Q16×C73—44no—
217C2×C8×C73—21no—
218C2×C8×C73—21no—
219C2×C8×C73—21no—
220C2×C8×C73—21no—
221C2×C8×C73C22×C412yes—
222C2×C8×C73C22×C412yes—
223C4⋊C4×C73—24no—
224C4⋊C4×C73—21no—
225C4⋊C4×C73C22×C412yes—
226C22⋊C4×C73—24no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—21no—
231C23×C73—21no—
232C23×C73—21no—
233C23×C73C4×D412yes—
234C23×C73C22×D411yescharacteristic
235Q8×C73—22no—
236Q8×C73—22no—
237Q8×C73—22no—
238Q8×C73—22no—
239Q8×C73—22no—
240Q8×C73—22no—
241Q8×C73—48no—
242Q8×C73—41no—
243Q8×C73—41no—
244Q8×C73—41no—
245Q8×C73—41no—
246Q8×C73—41no—
247Q8×C73—41no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—22no—
251C2×C4×C73—22no—
252C2×C4×C73—22no—
253C2×C4×C73—22no—
254C2×C4×C73—22no—
255C2×C4×C73—22no—
256C2×C4×C73—22no—
257C2×C4×C73—22no—
258C2×C4×C73—48no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—41no—
262C2×C4×C73—41no—
263C2×C4×C73—41no—
264C2×C4×C73—41no—
265C2×C4×C73—41no—
266C2×C4×C73—41no—
267C2×C4×C73C4×D412yes—
268C2×C4×C73C23×C411yescharacteristic
269C2×C4×C73C2×#4111yescharacteristic
270C2×C4×C73H(5,𝔽2)11yescharacteristic
271C2×C4×C73H(5,𝔽2)11yescharacteristic
272C2×C4×C73H(5,𝔽2)11yescharacteristic
273C2×C4×C73H(5,𝔽2)11yescharacteristic
274C2×Dic73—21no—
275C2×Dic73—21no—
276C2×Dic73—21no—
277C2×Dic73—21no—
278C2×Dic73—21no—
279C2×Dic73—21no—
280C2×Dic73—21no—
281C2×Dic73—21no—
282C2×Dic73—21no—
283C2×Dic73—21no—
284C2×Dic73—21no—
285C2×Dic73—21no—
286C2×Dic73—41no—
287C2×Dic73—41no—
288C2×Dic73—41no—
289C2×Dic73—41no—
290C2×Dic73—41no—
291C2×Dic73—41no—
292C2×Dic73—41no—
293C2×Dic73—41no—
294C2×Dic73—41no—
295C2×Dic73—41no—
296C2×Dic73—41no—
297C2×Dic73—41no—
298C8×C73—22no—
299C8×C73—22no—
300#10596831—41no—
301#10596831—41no—
302C22×C73—22no—
303C22×C73—22no—
304C22×C73—22no—
305C22×C73—41no—
306C22×C73—41no—
307C22×C73—41no—
308C22×C73—41no—
309C22×C73C2×C4×D411yescharacteristic
310C22×C73#53411yescharacteristic
311C22×C73#53511yescharacteristic
312C22×C73#57711yescharacteristic
313C22×C73#57811yescharacteristic
314C22×C73#57811yescharacteristic
315C22×C73#57811yescharacteristic
316C4×C73—22no—
317C4×C73—42no—
318C4×C73—42no—
319C4×C73—42no—
320C4×C73—42no—
321C4×C73#51811yescharacteristic
322C4×C73#51811yescharacteristic
323C4×C73#53811yescharacteristic
324C4×C73#53811yescharacteristic
325Dic73—41no—
326Dic73—41no—
327Dic73—41no—
328Dic73—41no—
329Dic73—81no—
330Dic73—81no—
331Dic73—81no—
332Dic73—81no—
333#22054—731nomaximal, Sylow
334C2×C73—42no—
335C2×C73#232111yescharacteristic
336C2×C73#294711yescharacteristic
337C2×C73#295011yescharacteristic
338C2×C73#320111yescharacteristic
339C2×C73#320211yescharacteristic
340C2×C73#322311yescharacteristic
341C2×C73#322511yescharacteristic
342C2×#451D7311yescharacteristic
343#2323—731no—
344#2310—731no—
345#1971—731no—
346#1969—731no—
347#1935—731no—
348#1933—731no—
349#1928—731no—
350#1927—731no—
351#1878—731no—
352#1878—731no—
353#1873—731no—
354#1854—731no—
355#1787—731no—
356#1750—731no—
357C73#2205411yescharacteristic, complemented, Sylow
358C23×Q8C2×D7311yescharacteristic
359C22×Q16C2×D7311yescharacteristic
360C22×Q16C2×D7311yescharacteristic
361C2×C22⋊Q8C2×D7311yescharacteristic
362C2×C4×Q8—731no—
363C2×#117—731no—
364C2×#117—731no—
365#451Dic7318yescomplemented
366C2×#107—731no—
367C2×#106—731no—
368C2×#103—731no—
369C2×#103—731no—
370C2×#103—731no—
371C2×#103—731no—
372C2×#103—731no—
373C2×#103—731no—
374C2×#103C2×D7311yescharacteristic
375C2×#103C2×D7311yescharacteristic
376C2×C22⋊C8—731no—
377C2×C22⋊C8C2×D7311yescharacteristic
378C2×#97—731no—
379C2×C4×C8—731no—
380#391—731no—
381#391—731no—
382#391—731no—
383#391—731no—
384#391—731no—
385#391—1461no—
386#391—1461no—
387#389—731no—
388#389—1461no—
389#388—1461no—
390#388—1461no—
391#387—1461no—
392#381—1461no—
393#381—1461no—
394#380—731no—
395C4×C4⋊C4—731no—
396#340—731no—
397#340—731no—
398#340—731no—
399#340—731no—
400#336—731no—
401#336—731no—
402C22×Q8—21no—
403C22×Q8—22no—
404C22×Q8—41no—
405C22×Q8—41no—
406C22×Q8—41no—
407C22×Q8—41no—
408C22×Q8C2×Dic7314yes—
409C22×Q8C22×D7311yescharacteristic
410C22×Q8C22×D7311yescharacteristic
411C22×Q8C22×D7311yescharacteristic
412C23×C4—21no—
413C23×C4C22×D7311yescharacteristic
414C2×Q16—41no—
415C2×Q16—41no—
416C2×Q16—41no—
417C2×Q16—41no—
418C2×Q16C2×Dic7314yes—
419C2×Q16C2×Dic7314yes—
420C22×C8—731no—
421C22×C8—731no—
422C22×C8C22×D7311yescharacteristic
423C22×C8C22×D7311yescharacteristic
424C22⋊Q8—22no—
425C22⋊Q8C2×Dic7314yes—
426C4×Q8—2921no—
427C4×Q8—2921no—
428#117—2921no—
429#117—2921no—
430#117—2921no—
431#117—2921no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—731no—
434C2×C4⋊C4—731no—
435C2×C4⋊C4—731no—
436C2×C4⋊C4—731no—
437C2×C4⋊C4—731no—
438C2×C4⋊C4—731no—
439C2×C4⋊C4—1461no—
440C2×C4⋊C4—1461no—
441C2×C4⋊C4C22×D7311yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—731no—
444C2×C22⋊C4—731no—
445C2×C22⋊C4—1461no—
446C2×C22⋊C4—1461no—
447C2×C42—731no—
448C2×C42—731no—
449C2×C42—1461no—
450C2×C42—1461no—
451C2×C42—1461no—
452#107—2921no—
453#106—2921no—
454#103—2921no—
455#103—2921no—
456#103—2921no—
457#103—2921no—
458#103—2921no—
459#103—2921no—
460#103C2×Dic7314yes—
461#103C2×Dic7314yes—
462C22⋊C8—1462no—
463C22⋊C8C2×Dic7314yes—
464#97—2921no—
465C4×C8—2921no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470#95—1461no—
471#95—1461no—
472#95—1461no—
473#95—1461no—
474#95—1461no—
475#95—1461no—
476#95—1461no—
477#95—1461no—
478#95—1461no—
479C24D4×D7311yescharacteristic
480C2×Q8—24no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—44no—
497C2×Q8—44no—
498C2×Q8—42no—
499C2×Q8—42no—
500C2×Q8—44no—
501C2×Q8—44no—
502C2×Q8C22×Dic7312yes—
503C2×Q8C22×Dic7312yes—
504C2×Q8C22×Dic7312yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—24no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—731no—
517C22×C4—731no—
518C22×C4—731no—
519C22×C4—731no—
520C22×C4—1461no—
521C22×C4—1461no—
522C22×C4—1461no—
523C22×C4—1461no—
524C22×C4C22×Dic7312yes—
525C22×C4D4×D7311yescharacteristic
526C22×C4#1192402712yes—
527C22×C4C23×D7311yescharacteristic
528Q16—44no—
529Q16—44no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—1461no—
535C2×C8—1461no—
536C2×C8—1461no—
537C2×C8—1461no—
538C2×C8—1461no—
539C2×C8—1461no—
540C2×C8C22×Dic7312yes—
541C2×C8C22×Dic7312yes—
542C4⋊C4—24no—
543C4⋊C4—21no—
544C4⋊C4—2921no—
545C4⋊C4—2921no—
546C4⋊C4—2922no—
547C4⋊C4—5841no—
548C4⋊C4—1461no—
549C4⋊C4—1461no—
550C4⋊C4—1461no—
551C4⋊C4—1461no—
552C4⋊C4—1461no—
553C4⋊C4—1461no—
554C4⋊C4—1461no—
555C4⋊C4—1461no—
556C4⋊C4C22×Dic7312yes—
557C22⋊C4—24no—
558C22⋊C4—2922no—
559C22⋊C4—2921no—
560C22⋊C4—2921no—
561C22⋊C4—5841no—
562C42—2922no—
563C42—2922no—
564C42—5841no—
565C42—1461no—
566C42—1461no—
567C42—1461no—
568C42—1461no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23D4×Dic7312yes—
576C23C2×D4×D7311yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—48no—
584Q8—41no—
585Q8—41no—
586Q8—41no—
587Q8—41no—
588Q8—41no—
589Q8—41no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—48no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
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—2921no—
619C2×C4—2921no—
620C2×C4—2921no—
621C2×C4—1461no—
622C2×C4—1461no—
623C2×C4—1461no—
624C2×C4—1461no—
625C2×C4—1461no—
626C2×C4—1461no—
627C2×C4—1461no—
628C2×C4—1461no—
629C2×C4—1461no—
630C2×C4—1461no—
631C2×C4—1461no—
632C2×C4—1461no—
633C2×C4D4×Dic7312yes—
634C2×C4C23×Dic7311yescharacteristic
635C2×C4C2×#1192402711yescharacteristic
636C2×C4#1455663611yescharacteristic
637C2×C4#1455663611yescharacteristic
638C2×C4#1455663611yescharacteristic
639C2×C4#1455663611yescharacteristic
640C8—22no—
641C8—22no—
642C8—2921no—
643C8—2921no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22C2×D4×Dic7311yescharacteristic
652C22#1798914711yescharacteristic
653C22#1798915211yescharacteristic
654C22#1798939611yescharacteristic
655C22#1798940011yescharacteristic
656C22#1798940011yescharacteristic
657C22#1798940311yescharacteristic
658C4—22no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—2921no—
664C4—2921no—
665C4—2921no—
666C4—2921no—
667C4—5841no—
668C4—5841no—
669C4—5841no—
670C4—5841no—
671C4#1798906211yescharacteristic
672C4#1798906211yescharacteristic
673C4#1798917211yescharacteristic
674C4#1798917211yescharacteristic
675C2—42no—
676C2#2261259211yescharacteristic
677C2#2261877511yescharacteristic
678C2#2261879911yescharacteristic
679C2#2262081711yescharacteristic
680C2#2262082411yescharacteristic
681C2#2262101011yescharacteristic
682C2#2262104611yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#451×C73C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenterC2×#451D73
hypocenterC73#22054
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
3. #22610803C2semidirect
4. #22610793C2semidirect
5. #22610623C2semidirect
6. #22610609C2semidirect
7. #22610574C2semidirect
8. #22610567C2semidirect
9. #22610274C2semidirect
10. #22610274C2semidirect
56. #451×C73C4semidirect
357. C73#22054semidirect
365. #451Dic73semidirect

Maximal subgroups

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

subgroupindex
#226126152
#226125172
#226108032
#226107932
#226106232
#226106092
#226105742
#226105672
#226102742
#226102442
#226101322
#226098662
#226096992
C2×#451×C732
#2205473