Finite Group Databasebeta

← #33103262

Subgroups — #33103262

Order 18688.

934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988596C211yesmaximal, characteristic
2#22612826C211yesmaximal, characteristic
3Q8×C22⋊Dic73C211yesmaximal, characteristic
4#22610618C211yesmaximal, characteristic, complemented
5#22610604C211yesmaximal, characteristic, complemented
6#22610575C211yesmaximal, characteristic, complemented
7#22610568C211yesmaximal, characteristic, complemented
8#22610197C211yesmaximal, characteristic, complemented
9#22610183C211yesmaximal, characteristic, complemented
10#22610027C211yesmaximal, characteristic
11#22609962C211yesmaximal, characteristic
12#22609763C211yesmaximal, characteristic, complemented
13#103×Dic73C211yesmaximal, characteristic, complemented
14#22609717C211yesmaximal, characteristic
15C73×#1787C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×Q8×Dic73C2211yescharacteristic
19C2×#14556550C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22#17988596C418yescomplemented
23C2×#14556515C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556501C2211yescharacteristic
28C2×C8×Dic73C2211yescharacteristic
29#17988494—21no—
30#17988494—21no—
31#17988494C2211yescharacteristic
32#17988494C2211yescharacteristic
33#17988483C2211yescharacteristic
34#17988483C2211yescharacteristic
35#17988475—21no—
36#17988475—21no—
37#17988454—21no—
38#17988447—21no—
39#17988447C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42#17988438—21no—
43C4×C22⋊Dic73—21no—
44C4×C22⋊Dic73C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C23×Q8×C73C2211yescharacteristic
54C2×#103×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×#103×C73C2211yescharacteristic
57C2×#103×C73C2211yescharacteristic
58C2×C22⋊C8×C73C2211yescharacteristic
59#380×C73C2211yescharacteristic
60C2×#11924012—41no—
61C2×#11924012—41no—
62C2×#11924012C2×C414yes—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65Q8×Dic73—22no—
66Q8×Dic73—22no—
67Q8×Dic73—41no—
68Q8×Dic73—41no—
69#14556550—22no—
70#14556550C2×C414yes—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73—21no—
81C2×C22⋊Dic73D411yescharacteristic
82C2×C22⋊Dic73D411yescharacteristic
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73C2311yescharacteristic
89C2×C4×Dic73C2311yescharacteristic
90#14556524—41no—
91#14556522—41no—
92#14556515C2×C414yes—
93#14556515C2×C414yes—
94#14556515C2×C414yes—
95#14556515C2×C414yes—
96#14556501—41no—
97C8×Dic73—41no—
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#14556498D411yescharacteristic
108#14556498D411yescharacteristic
109C22×Q8×C73—21no—
110C22×Q8×C73—22no—
111C22×Q8×C73—22no—
112C22×Q8×C73—22no—
113C22×Q8×C73—41no—
114C22×Q8×C73—41no—
115C22×Q8×C73C2×C414yes—
116C22×Q8×C73C2311yescharacteristic
117C22×Q8×C73C2311yescharacteristic
118C23×C4×C73—21no—
119C23×C4×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121C22×C8×C73C2311yescharacteristic
122C2×C4⋊C4×C73C2311yescharacteristic
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C22⋊C4×C73—21no—
125#103×C73—22no—
126#103×C73—22no—
127#103×C73—41no—
128#103×C73—41no—
129C22⋊C8×C73—22no—
130#95×C73—21no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73C2×D411yescharacteristic
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139#11924012—44no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831C22×C412yes—
145C2×#10596831C22×C412yes—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—42no—
153C4⋊Dic73—42no—
154C4⋊Dic73C22×C412yes—
155C4⋊Dic73C22×C412yes—
156#11924007—42no—
157#11924007—42no—
158C22⋊Dic73—24no—
159C22⋊Dic73—22no—
160C22⋊Dic73—41no—
161C22⋊Dic73—42no—
162C4×Dic73—24no—
163C4×Dic73—21no—
164C4×Dic73—21no—
165C4×Dic73—21no—
166C4×Dic73—21no—
167C4×Dic73—42no—
168C4×Dic73—42no—
169C4×Dic73—81no—
170C4×Dic73—81no—
171C24×C73C2×D411yescharacteristic
172C2×Q8×C73—24no—
173C2×Q8×C73—21no—
174C2×Q8×C73—21no—
175C2×Q8×C73—21no—
176C2×Q8×C73—21no—
177C2×Q8×C73—28no—
178C2×Q8×C73—21no—
179C2×Q8×C73—21no—
180C2×Q8×C73—21no—
181C2×Q8×C73—21no—
182C2×Q8×C73—42no—
183C2×Q8×C73—42no—
184C2×Q8×C73—44no—
185C2×Q8×C73—44no—
186C2×Q8×C73—44no—
187C2×Q8×C73C22×C412yes—
188C2×Q8×C73C22×C412yes—
189C2×Q8×C73#4112yes—
190C2×Q8×C73#4112yes—
191C22×C4×C73—21no—
192C22×C4×C73—21no—
193C22×C4×C73—21no—
194C22×C4×C73—21no—
195C22×C4×C73—21no—
196C22×C4×C73—24no—
197C22×C4×C73—21no—
198C22×C4×C73—21no—
199C22×C4×C73—21no—
200C22×C4×C73—42no—
201C22×C4×C73—42no—
202C22×C4×C73C22×C412yes—
203C22×C4×C73C2×D411yescharacteristic
204C22×C4×C73#4112yes—
205C22×C4×C73C2411yescharacteristic
206C2×C8×C73—21no—
207C2×C8×C73—21no—
208C2×C8×C73—21no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73—21no—
212C4⋊C4×C73—21no—
213C4⋊C4×C73—21no—
214C4⋊C4×C73—21no—
215C4⋊C4×C73—21no—
216C22⋊C4×C73—81no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—21no—
222C23×C73—21no—
223C23×C73C4×D412yes—
224C23×C73C22×D411yescharacteristic
225Q8×C73—22no—
226Q8×C73—22no—
227Q8×C73—22no—
228Q8×C73—22no—
229Q8×C73—22no—
230Q8×C73—48no—
231Q8×C73—41no—
232Q8×C73—41no—
233Q8×C73#11714yes—
234C2×C4×C73—22no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—22no—
238C2×C4×C73—22no—
239C2×C4×C73—22no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—22no—
244C2×C4×C73—48no—
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×C73C4×D412yes—
254C2×C4×C73C23×C411yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73H(5,𝔽2)11yescharacteristic
259C2×C4×C73H(5,𝔽2)11yescharacteristic
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—22no—
263C2×Dic73—22no—
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—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#13712yes—
284C8×C73—41no—
285C8×C73—41no—
286#10596831—22no—
287#10596831—22no—
288C22×C73—22no—
289C22×C73—22no—
290C22×C73—22no—
291C22×C73—41no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73—41no—
295C22×C73C2×C4×D411yescharacteristic
296C22×C73#54611yescharacteristic
297C22×C73#54611yescharacteristic
298C22×C73C2×#13711yescharacteristic
299C22×C73#57511yescharacteristic
300C22×C73#57711yescharacteristic
301C22×C73#57811yescharacteristic
302C4×C73—22no—
303C4×C73—24no—
304C4×C73—42no—
305C4×C73—81no—
306C4×C73—81no—
307C4×C73C2×#11711yescharacteristic
308C4×C73#51811yescharacteristic
309C4×C73#55111yescharacteristic
310C4×C73#55111yescharacteristic
311Dic73—22no—
312Dic73—41no—
313Dic73—42no—
314Dic73—42no—
315Dic73—41no—
316Dic73—41no—
317Dic73—81no—
318Dic73—81no—
319#22581—731nomaximal, Sylow
320C2×C73—42no—
321C2×C73#234211yescharacteristic
322C2×C73C4×#13711yescharacteristic
323C2×C73#294711yescharacteristic
324C2×C73#331311yescharacteristic
325C2×C73#332011yescharacteristic
326C2×C73#332611yescharacteristic
327C2×C73#332811yescharacteristic
328C2×#419—731no—
329#2347—731no—
330Q8×C22⋊C4—731no—
331#1935—731no—
332#1933—731no—
333#1928—731no—
334#1927—731no—
335#1865—731no—
336#1863—731no—
337#1825—731no—
338#1809—731no—
339#1787D7311yescharacteristic
340#1763—731no—
341C4×#103—731no—
342#1753—731no—
343C73#2258111yescharacteristic, complemented, Sylow
344C23×Q8C2×D7311yescharacteristic
345C22×M16—731no—
346C2×C4×Q8—731no—
347C2×C4×Q8—731no—
348C2×#117—731no—
349C2×#107—731no—
350C2×#106—731no—
351#419—738no—
352C2×#103—731no—
353C2×#103—731no—
354C2×#103—731no—
355C2×#103—731no—
356C2×#103C2×D7311yescharacteristic
357C2×#103C2×D7311yescharacteristic
358C2×#103C2×D7311yescharacteristic
359C2×#103C2×D7311yescharacteristic
360C2×C22⋊C8C2×D7311yescharacteristic
361C2×#97—731no—
362C2×C4×C8—731no—
363#391—731no—
364#391—731no—
365#391—1461no—
366#391—1461no—
367#389—731no—
368#389—731no—
369#388—1461no—
370#388—1461no—
371#383—1461no—
372#380—731no—
373#380—1461no—
374#380C2×D7311yescharacteristic
375C4×C4⋊C4—731no—
376C4×C4⋊C4—731no—
377C4×C4⋊C4—1461no—
378C4×C22⋊C4—731no—
379C4×C22⋊C4—1461no—
380#340—731no—
381#340—731no—
382#340—731no—
383#340—731no—
384#340—731no—
385#340—731no—
386#336—731no—
387#336—731no—
388C22×Q8—21no—
389C22×Q8—22no—
390C22×Q8—22no—
391C22×Q8—22no—
392C22×Q8—41no—
393C22×Q8—41no—
394C22×Q8C4×D7314yes—
395C22×Q8C22×D7311yescharacteristic
396C22×Q8C22×D7311yescharacteristic
397C23×C4—21no—
398C23×C4C22×D7311yescharacteristic
399C2×M16—2921no—
400C2×M16—2921no—
401C2×M16—734no—
402C22×C8—731no—
403C22×C8—731no—
404C22×C8C22×D7311yescharacteristic
405C22×C8C22×D7311yescharacteristic
406C4×Q8—2921no—
407C4×Q8—2921no—
408C4×Q8—1462no—
409C4×Q8—1462no—
410#117—734no—
411#117—1462no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—731no—
416C2×C4⋊C4—1461no—
417C2×C4⋊C4—1461no—
418C2×C4⋊C4—1461no—
419C2×C4⋊C4—1461no—
420C2×C4⋊C4C22×D7311yescharacteristic
421C2×C4⋊C4C22×D7311yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—731no—
424C2×C22⋊C4—731no—
425C2×C22⋊C4—1461no—
426C2×C22⋊C4—1461no—
427C2×C42—731no—
428C2×C42—731no—
429C2×C42—1461no—
430C2×C42—1461no—
431C2×C42—1461no—
432C2×C42—1461no—
433C2×C42—1461no—
434#107—2921no—
435#106—2921no—
436#103—22no—
437#103—22no—
438#103—41no—
439#103—41no—
440#103—734no—
441#103—734no—
442#103—734no—
443#103—734no—
444C22⋊C8—22no—
445#97—2921no—
446C4×C8—2921no—
447#95—21no—
448#95—731no—
449#95—731no—
450#95—1461no—
451#95—1461no—
452#95—1461no—
453#95—1461no—
454#95—1461no—
455#95—1461no—
456#95—1461no—
457#95—1461no—
458#95—1461no—
459C24D4×D7311yescharacteristic
460C2×Q8—24no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—28no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8C2×C4×D7312yes—
476C2×Q8C2×C4×D7312yes—
477C2×Q8#1192402712yes—
478C2×Q8#1192402712yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—24no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—731no—
491C22×C4—731no—
492C22×C4—731no—
493C22×C4—731no—
494C22×C4—1461no—
495C22×C4—1461no—
496C22×C4—1461no—
497C22×C4—1461no—
498C22×C4C2×C4×D7312yes—
499C22×C4D4×D7311yescharacteristic
500C22×C4#1192402712yes—
501C22×C4C23×D7311yescharacteristic
502M16—2924no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—732no—
510C2×C8—732no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—2922no—
520C4⋊C4—2922no—
521C4⋊C4—2922no—
522C4⋊C4—2922no—
523C4⋊C4—732no—
524C4⋊C4—732no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C22⋊C4—81no—
532C22⋊C4—2922no—
533C22⋊C4—2921no—
534C22⋊C4—1462no—
535C22⋊C4—1464no—
536C42—2922no—
537C42—2922no—
538C42—5841no—
539C42—5841no—
540C42—1464no—
541C42—1461no—
542C42—1461no—
543C42—1461no—
544C42—1461no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23#1455655112yes—
552C23C2×D4×D7311yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—48no—
559Q8—41no—
560Q8—41no—
561Q8#1455654714yes—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—732no—
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—1462no—
602C2×C4—1462no—
603C2×C4—1461no—
604C2×C4—1461no—
605C2×C4#1455655112yes—
606C2×C4C22×C4×D7311yescharacteristic
607C2×C4C2×#1192402711yescharacteristic
608C2×C4C2×#1192402711yescharacteristic
609C2×C4C2×#1192402711yescharacteristic
610C2×C4#1455663611yescharacteristic
611C2×C4#1455663611yescharacteristic
612C8—41no—
613C8—41no—
614C8—1462no—
615C8—1462no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×#1455655111yescharacteristic
624C22#1798922411yescharacteristic
625C22#1798922411yescharacteristic
626C22#137×D7311yescharacteristic
627C22#1798938611yescharacteristic
628C22#1798939311yescharacteristic
629C22#1798940011yescharacteristic
630C4—22no—
631C4—24no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—2921no—
636C4—2922no—
637C4—2922no—
638C4—2921no—
639C4—2921no—
640C4—5841no—
641C4—5841no—
642C4—1462no—
643C4C2×#1455654711yescharacteristic
644C4#1798906011yescharacteristic
645C4#1798926811yescharacteristic
646C4#1798926811yescharacteristic
647C2—42no—
648C2#2261278211yescharacteristic
649C2#2261874611yescharacteristic
650C2#2261876811yescharacteristic
651C2#2262199011yescharacteristic
652C2#2262209511yescharacteristic
653C2#2262218811yescharacteristic
654C2#2262220711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1787C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#1787D73
hypocenterC73#22581
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
4. #22610618C2semidirect
5. #22610604C2semidirect
6. #22610575C2semidirect
7. #22610568C2semidirect
8. #22610197C2semidirect
9. #22610183C2semidirect
12. #22609763C2semidirect
13. #103×Dic73C2semidirect
22. #17988596C4semidirect
343. C73#22581semidirect

Maximal subgroups

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

subgroupindex
C2×#179885962
#226128262
Q8×C22⋊Dic732
#226106182
#226106042
#226105752
#226105682
#226101972
#226101832
#226100272
#226099622
#226097632
#103×Dic732
#226097172
C73×#17872
#2258173