Finite Group Databasebeta

← #33089009

Subgroups — #33089009

Order 18688.

1042 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988697C211yesmaximal, characteristic
2C42×D292C211yesmaximal, characteristic, complemented
3#22610555C211yesmaximal, characteristic, complemented
4#22610548C211yesmaximal, characteristic, complemented
5#22610513C211yesmaximal, characteristic, complemented
6#22610506C211yesmaximal, characteristic, complemented
7#22610282C211yesmaximal, characteristic
8#22610282C211yesmaximal, characteristic
9#22610191C211yesmaximal, characteristic, complemented
10#22610177C211yesmaximal, characteristic, complemented
11#22609829C211yesmaximal, characteristic, complemented
12#22609759C211yesmaximal, characteristic, complemented
13#22609756C211yesmaximal, characteristic, complemented
14#22609680C211yesmaximal, characteristic, complemented
15C73×#2293C211yesmaximal, characteristic, complemented
16C22×#11924013C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19C2×C4×D292C2211yescharacteristic
20C2×C42×D73—21no—
21#17988697C418yescomplemented
22C2×#14556524C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×#14556511C2211yescharacteristic
29C2×#14556511C2211yescharacteristic
30C2×#14556511C2211yescharacteristic
31C2×#14556502C2211yescharacteristic
32C2×#14556502C2211yescharacteristic
33C4×C4⋊Dic73C2211yescharacteristic
34C4×C4⋊Dic73C2211yescharacteristic
35C4×C4⋊Dic73C2211yescharacteristic
36C4×#11924005—21no—
37C4×#11924005—21no—
38C4×#11924005—21no—
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988342C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49C2×C4×D4×C73C2211yescharacteristic
50C2×#117×C73—21no—
51#390×C73C2211yescharacteristic
52#390×C73C2211yescharacteristic
53#389×C73C2211yescharacteristic
54#389×C73C2211yescharacteristic
55#388×C73—21no—
56#388×C73—21no—
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73—21no—
59C43×C73C2211yescharacteristic
60C22×D292C2311yescharacteristic
61C22×C4×D73—21no—
62C22×C4×D73—21no—
63C22×C4×D73—21no—
64C2×#11924013—22no—
65C2×#11924013—22no—
66C2×#11924013C2×C414yes—
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C4×D292—22no—
72C4×D292—22no—
73C4×D292—22no—
74C4×D292—22no—
75C4×D292—22no—
76C4×D292C2×C414yes—
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C42×D73—44no—
87#14556524—22no—
88#14556522—22no—
89#14556522C2×C414yescomplemented
90#14556512—22no—
91#14556512—22no—
92#14556512C2×C414yescomplemented
93#14556511—22no—
94#14556511—22no—
95#14556511C2×C414yes—
96#14556502—22no—
97#14556502C2×C414yescomplemented
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101C22×D4×C73C2311yescharacteristic
102C23×C4×C73—21no—
103C4×D4×C73—22no—
104C4×D4×C73C2×C414yescomplemented
105#117×C73—44no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115C2×C22⋊C4×C73—21no—
116C2×C42×C73—21no—
117C2×C42×C73D411yescharacteristic
118C2×C42×C73D411yescharacteristic
119C2×C42×C73D411yescharacteristic
120C2×C42×C73D411yescharacteristic
121C2×C42×C73C2311yescharacteristic
122C2×C42×C73C2311yescharacteristic
123C2×C42×C73C2311yescharacteristic
124#95×C73—21no—
125#95×C73—21no—
126#95×C73—21no—
127C23×D73—21no—
128C2×D292—21no—
129C2×D292—21no—
130C2×D292—21no—
131C2×D292C22×C412yes—
132C2×D292#4112yes—
133C2×D292#4112yes—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C2×C4×D73—24no—
138C2×C4×D73—24no—
139C2×C4×D73—24no—
140C2×C4×D73—42no—
141C2×C4×D73—42no—
142C2×C4×D73—42no—
143C2×C4×D73—42no—
144C2×C4×D73—42no—
145C2×C4×D73—42no—
146#11924013—28no—
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—
157C2×#10596831C22×C412yes—
158C2×#10596831C22×C412yes—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73C22×C412yes—
163C4⋊Dic73#4112yes—
164C4⋊Dic73#4112yes—
165#11924005—24no—
166#11924005—24no—
167#11924005—24no—
168C4×Dic73—42no—
169C4×Dic73—42no—
170C4×Dic73—42no—
171C24×C73—21no—
172C2×D4×C73—21no—
173C2×D4×C73—21no—
174C2×D4×C73—21no—
175C2×D4×C73—21no—
176C2×D4×C73—21no—
177C2×D4×C73C22×C412yes—
178C22×C4×C73—21no—
179C22×C4×C73—24no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—42no—
183C22×C4×C73—42no—
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2411yescharacteristic
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—42no—
197C4⋊C4×C73—42no—
198C4⋊C4×C73C22×C412yes—
199C22⋊C4×C73—24no—
200C22⋊C4×C73—42no—
201C22⋊C4×C73—42no—
202C22⋊C4×C73—42no—
203C22⋊C4×C73—42no—
204C42×C73—22no—
205C42×C73—22no—
206C42×C73—21no—
207C42×C73—21no—
208C42×C73—21no—
209C42×C73—21no—
210C42×C73—41no—
211C42×C73—41no—
212C42×C73—42no—
213C42×C73C22×C412yes—
214C42×C73#4112yes—
215C23×C73—22no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—41no—
219C23×C73—41no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73C22×D411yescharacteristic
223C22×D73—22no—
224C22×D73—22no—
225C22×D73—22no—
226C22×D73—41no—
227C22×D73—41no—
228C22×D73—41no—
229C22×D73—41no—
230D4×C73—22no—
231D4×C73—22no—
232D4×C73—41no—
233D4×C73—41no—
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—22no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—48no—
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×C73C4×D412yes—
259C2×C4×C73C4×D412yes—
260C2×C4×C73AGL(1,ℤ/8)12yes—
261C2×C4×C73AGL(1,ℤ/8)12yes—
262C2×C4×C73C23×C411yescharacteristic
263C2×C4×C73C2×#4111yescharacteristic
264C2×C4×C73C2×#4111yescharacteristic
265C2×C4×C73C2×#4111yescharacteristic
266C2×C4×C73C2×#4111yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×Dic73—22no—
269C2×Dic73—22no—
270C2×Dic73—22no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277D292—22no—
278D292—22no—
279D292—22no—
280D292#11714yescomplemented
281C4×D73—48no—
282C4×D73—48no—
283C4×D73—48no—
284#10596831—22no—
285#10596831—22no—
286#10596831—41no—
287#10596831—41no—
288C22×C73—42no—
289C22×C73—42no—
290C22×C73—42no—
291C22×C73—81no—
292C22×C73—81no—
293C22×C73—81no—
294C22×C73—81no—
295C22×C73C2×C4×D411yescharacteristic
296C22×C73#52511yescharacteristic
297C22×C73#52511yescharacteristic
298C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C73#57511yescharacteristic
301C22×C73#57511yescharacteristic
302C2×D73—24no—
303C2×D73—42no—
304C2×D73—42no—
305C2×D73—42no—
306C2×D73—42no—
307C2×D73—42no—
308C2×D73—42no—
309C4×C73—22no—
310C4×C73—22no—
311C4×C73—22no—
312C4×C73—22no—
313C4×C73—41no—
314C4×C73—41no—
315C4×C73—42no—
316C4×C73—81no—
317C4×C73—81no—
318C4×C73C2×#11711yescharacteristic
319C4×C73C4×#4111yescharacteristic
320C4×C73#52911yescharacteristic
321C4×C73#52911yescharacteristic
322Dic73—42no—
323Dic73—42no—
324Dic73—42no—
325#21122—731nomaximal, Sylow
326C2×C73—82no—
327C2×C73#229011yescharacteristic
328C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
329C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
330C2×C73#310911yescharacteristic
331C2×C73#311111yescharacteristic
332C2×C73#311411yescharacteristic
333C2×C73#311511yescharacteristic
334D73—44no—
335C2×#442—731no—
336#2293D7311yescharacteristic, complemented
337C42×D4—731no—
338#1926—731no—
339#1925—731no—
340#1920—731no—
341#1919—731no—
342#1879—731no—
343#1879—731no—
344#1864—731no—
345#1862—731no—
346#1776—731no—
347#1762—731no—
348#1762—731no—
349#1745—731no—
350C73#2112211yescharacteristic, complemented, Sylow
351C22×D8—731no—
352C2×C4×D4—731no—
353C2×C4×D4—731no—
354C2×C4×D4—731no—
355C2×C4×D4C2×D7311yescharacteristic
356C2×#117—21no—
357C22×C42—1461no—
358#442—738no—
359C2×#107—731no—
360C2×#106—731no—
361C2×#106—731no—
362C2×#102—731no—
363C2×#102—731no—
364C2×#102—731no—
365C2×#102—731no—
366C2×#102—731no—
367C2×#102—731no—
368C2×#97—731no—
369C2×#97—731no—
370#390C2×D7311yescharacteristic
371#390C2×D7311yescharacteristic
372#389C2×D7311yescharacteristic
373#389C2×D7311yescharacteristic
374#388—21no—
375#388—21no—
376C4×C4⋊C4—731no—
377C4×C4⋊C4—731no—
378C4×C4⋊C4—731no—
379C4×C4⋊C4C2×D7311yescharacteristic
380C4×C22⋊C4—21no—
381C4×C22⋊C4—1461no—
382C4×C22⋊C4—1461no—
383C4×C22⋊C4—1461no—
384C43C2×D7311yescharacteristic
385#340—731no—
386#340—731no—
387#340—731no—
388#340—731no—
389#340—731no—
390#340—731no—
391#336—731no—
392#336—731no—
393#336—731no—
394#336—731no—
395C22×D4—731no—
396C22×D4C22×D7311yescharacteristic
397C23×C4—21no—
398C23×C4—1461no—
399C23×C4—1461no—
400C23×C4—1461no—
401C2×D8—734no—
402C2×D8—1462no—
403C2×D8—1462no—
404C22×C8—731no—
405C22×C8—731no—
406C22×C8—731no—
407C22×C8—731no—
408C4×D4—22no—
409C4×D4—734no—
410C4×D4—1462no—
411C4×D4—1462no—
412C4×D4—1462no—
413C4×D4—1462no—
414C4×D4—1462no—
415C4×D4C4×D7314yescomplemented
416#117—44no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—731no—
420C2×C4⋊C4—731no—
421C2×C4⋊C4—731no—
422C2×C4⋊C4C22×D7311yescharacteristic
423C2×C4⋊C4C22×D7311yescharacteristic
424C2×C4⋊C4C22×D7311yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—1461no—
431C2×C22⋊C4—1461no—
432C2×C22⋊C4—1461no—
433C2×C42—21no—
434C2×C42—2924no—
435C2×C42—1461no—
436C2×C42—1461no—
437C2×C42—1461no—
438C2×C42C22⋊D7311yescharacteristic
439C2×C42C22⋊D7311yescharacteristic
440C2×C42C22⋊D7311yescharacteristic
441C2×C42C22⋊D7311yescharacteristic
442C2×C42C22×D7311yescharacteristic
443C2×C42C22×D7311yescharacteristic
444C2×C42C22×D7311yescharacteristic
445#107—1462no—
446#106—734no—
447#106—1462no—
448#102—734no—
449#102—734no—
450#102—1462no—
451#102—1462no—
452#102—1462no—
453#102—1462no—
454#97—734no—
455#97—1462no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—1461no—
460#95—1461no—
461#95—1461no—
462C24—21no—
463C24—1461no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—732no—
470C2×D4—732no—
471C2×D4—732no—
472C2×D4—1461no—
473C2×D4—1461no—
474C2×D4—1461no—
475C2×D4C2×C4×D7312yes—
476C22×C4—21no—
477C22×C4—24no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—2922no—
483C22×C4—2922no—
484C22×C4—2922no—
485C22×C4—2922no—
486C22×C4—2922no—
487C22×C4—2922no—
488C22×C4—1464no—
489C22×C4—1464no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4—1464no—
494C22×C4C2×C22⋊D7311yescharacteristic
495C22×C4C2×C22⋊D7311yescharacteristic
496C22×C4C2×C22⋊D7311yescharacteristic
497C22×C4C2×C22⋊D7311yescharacteristic
498C22×C4C2×C22⋊D7311yescharacteristic
499C22×C4C2×C22⋊D7311yescharacteristic
500C22×C4C23×D7311yescharacteristic
501D8—1468no—
502C2×C8—732no—
503C2×C8—732no—
504C2×C8—1461no—
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—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—732no—
522C4⋊C4—732no—
523C4⋊C4—732no—
524C4⋊C4—1461no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4C2×C4×D7312yes—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—1464no—
534C22⋊C4—1464no—
535C22⋊C4—1464no—
536C42—22no—
537C42—22no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—41no—
543C42—41no—
544C42—42no—
545C42—2922no—
546C42—2922no—
547C42—2922no—
548C42C2×C4×D7312yes—
549C42#1192402812yes—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—2921no—
558C23—2921no—
559C23—2921no—
560C23—2921no—
561C23—1462no—
562C23—1462no—
563C23—1462no—
564C23C22×C22⋊D7311yescharacteristic
565D4—22no—
566D4—22no—
567D4—41no—
568D4—41no—
569D4—734no—
570D4—1462no—
571D4—1462no—
572D4—1462no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—48no—
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—2928no—
598C2×C4—2928no—
599C2×C4—2928no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—2921no—
603C2×C4—2921no—
604C2×C4—2921no—
605C2×C4—2921no—
606C2×C4—1462no—
607C2×C4—1462no—
608C2×C4—1462no—
609C2×C4C4×C22⋊D7312yes—
610C2×C4C4×C22⋊D7312yes—
611C2×C4#1455661612yes—
612C2×C4#1455661712yes—
613C2×C4C22×C4×D7311yescharacteristic
614C2×C4#41×D7311yescharacteristic
615C2×C4#41×D7311yescharacteristic
616C2×C4#41×D7311yescharacteristic
617C2×C4#41×D7311yescharacteristic
618C2×C4C2×#1192402811yescharacteristic
619C8—2921no—
620C8—2921no—
621C8—1462no—
622C8—1462no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—2922no—
631C22—2922no—
632C22—2922no—
633C22—2922no—
634C22—2922no—
635C22—2922no—
636C22—1464no—
637C22C2×C4×C22⋊D7311yescharacteristic
638C22#1798909411yescharacteristic
639C22#1798909411yescharacteristic
640C22C2×#1455661611yescharacteristic
641C22C2×#1455661711yescharacteristic
642C22#1798938511yescharacteristic
643C22#1798938511yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—41no—
650C4—41no—
651C4—81no—
652C4—81no—
653C4—2922no—
654C4—2922no—
655C4—2922no—
656C4#117×D7311yescharacteristic
657C4#1798905511yescharacteristic
658C4#1798911511yescharacteristic
659C4#1798911511yescharacteristic
660C2—82no—
661C2—2924no—
662C2#2261236811yescharacteristic
663C2C4×#1455661611yescharacteristic
664C2C4×#1455661711yescharacteristic
665C2#2262001811yescharacteristic
666C2#2262003411yescharacteristic
667C2#2262006511yescharacteristic
668C2#2262007611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2293C2
socleC23×C73C22×D4
prime radicalC22×#11924013C22
cosocleC22×C4×C73C24
hypercenter#2293D73
hypocenterC73#21122
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. C42×D292C2semidirect
3. #22610555C2semidirect
4. #22610548C2semidirect
5. #22610513C2semidirect
6. #22610506C2semidirect
9. #22610191C2semidirect
10. #22610177C2semidirect
11. #22609829C2semidirect
12. #22609759C2semidirect
13. #22609756C2semidirect
14. #22609680C2semidirect
15. C73×#2293C2semidirect
21. #17988697C4semidirect
89. #14556522C2×C4semidirect
92. #14556512C2×C4semidirect
97. #14556502C2×C4semidirect
104. C4×D4×C73C2×C4semidirect
280. D292#117semidirect
336. #2293D73semidirect
350. C73#21122semidirect
415. C4×D4C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886972
C42×D2922
#226105552
#226105482
#226105132
#226105062
#226102822
#226101912
#226101772
#226098292
#226097592
#226097562
#226096802
C73×#22932
#2112273