Finite Group Databasebeta

← #33091261

Subgroups — #33091261

Order 18688.

1070 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×#17989197C211yesmaximal, characteristic
2C2×#17988542C211yesmaximal, characteristic
3#22612538C211yesmaximal, characteristic
4#22610898C211yesmaximal, characteristic, complemented
5#22610894C211yesmaximal, characteristic, complemented
6#22610749C211yesmaximal, characteristic, complemented
7#22610731C211yesmaximal, characteristic, complemented
8#22610365C211yesmaximal, characteristic
9#22610362C211yesmaximal, characteristic
10#22610354C211yesmaximal, characteristic
11#22609763C211yesmaximal, characteristic, complemented
12#22609760C211yesmaximal, characteristic, complemented
13#103×Dic73C211yesmaximal, characteristic, complemented
14#22609750C211yesmaximal, characteristic, complemented
15C73×#1892C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17#17989197—24no—
18#17989197C418yescomplemented
19C2×#14556589C2211yescharacteristic
20C2×#14556584—21no—
21C2×#14556584—21no—
22C2×#14556582—21no—
23C2×#14556582C2211yescharacteristic
24C2×#14556581—21no—
25C2×#14556570C2211yescharacteristic
26C2×Q8×Dic73C2211yescharacteristic
27C2×#14556556C2211yescharacteristic
28C2×#14556550C2211yescharacteristic
29C2×#14556516C2211yescharacteristic
30C2×#14556516C2211yescharacteristic
31C2×#14556515C2211yescharacteristic
32C2×#14556515C2211yescharacteristic
33C2×#14556514C2211yescharacteristic
34C2×#14556514C2211yescharacteristic
35#17988542—24no—
36C2×#14556501C2211yescharacteristic
37C2×#14556501C2211yescharacteristic
38C2×C8×Dic73C2211yescharacteristic
39C2×C8×Dic73C2211yescharacteristic
40#17988481—21no—
41#17988475—21no—
42#17988472—21no—
43#17988470—21no—
44#17988456C2211yescharacteristic
45#17988452C2211yescharacteristic
46#17988447C2211yescharacteristic
47C4⋊C4×Dic73C2211yescharacteristic
48C4⋊C4×Dic73C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988349C2211yescharacteristic
55C23×C8×C73C2211yescharacteristic
56C2×C22⋊Q8×C73C2211yescharacteristic
57C2×#103×C73C2211yescharacteristic
58C2×#103×C73C2211yescharacteristic
59#380×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#340×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C2×#11924012—22no—
64C2×#11924012—22no—
65C2×#11924012—22no—
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556589—22no—
69#14556589C2×C414yes—
70#14556584—28no—
71#14556584—28no—
72#14556582—28no—
73#14556582—22no—
74#14556582C2×C414yes—
75#14556581—28no—
76#14556570—22no—
77#14556570C2×C414yes—
78Q8×Dic73—22no—
79Q8×Dic73C2×C414yes—
80#14556556—22no—
81#14556556C2×C414yes—
82#14556550C2×C414yes—
83#14556550D414yes—
84C2×C4⋊Dic73—21no—
85C2×C4⋊Dic73—21no—
86C2×C4⋊Dic73D411yescharacteristic
87C2×C4⋊Dic73D411yescharacteristic
88C2×C4⋊Dic73C2311yescharacteristic
89C2×C4⋊Dic73C2311yescharacteristic
90C2×C4⋊Dic73C2311yescharacteristic
91C2×#11924007—21no—
92C2×#11924007—21no—
93C2×#11924007—21no—
94C2×#11924007—21no—
95C2×#11924007—21no—
96C2×#11924007—21no—
97C2×#11924007—21no—
98C2×C22⋊Dic73—21no—
99C2×C22⋊Dic73—21no—
100C2×C22⋊Dic73D411yescharacteristic
101C2×C22⋊Dic73D411yescharacteristic
102C2×C4×Dic73—21no—
103C2×C4×Dic73—21no—
104C2×C4×Dic73—21no—
105C2×C4×Dic73—21no—
106C2×C4×Dic73C2311yescharacteristic
107C2×C4×Dic73C2311yescharacteristic
108#14556516—22no—
109#14556516—22no—
110#14556515—22no—
111#14556515—22no—
112#14556514—22no—
113#14556514—22no—
114#14556501—22no—
115#14556501—22no—
116C8×Dic73—22no—
117C8×Dic73—22no—
118#14556498—21no—
119#14556498—21no—
120#14556498—21no—
121#14556498—21no—
122#14556498—21no—
123C22×Q8×C73C2311yescharacteristic
124C23×C4×C73C2311yescharacteristic
125C22×C8×C73—22no—
126C22×C8×C73—22no—
127C22×C8×C73—22no—
128C22×C8×C73C2311yescharacteristic
129C22×C8×C73C2311yescharacteristic
130C22⋊Q8×C73—22no—
131C22⋊Q8×C73C2×C414yes—
132C2×C4⋊C4×C73—21no—
133C2×C4⋊C4×C73C2311yescharacteristic
134C2×C4⋊C4×C73C2311yescharacteristic
135C2×C4⋊C4×C73C2311yescharacteristic
136C2×C22⋊C4×C73—21no—
137C2×C22⋊C4×C73—21no—
138#103×C73—22no—
139#103×C73—22no—
140#95×C73—21no—
141C2×Dic146—21no—
142C2×Dic146—21no—
143C2×Dic146—21no—
144C2×Dic146—21no—
145C2×Dic146—21no—
146C2×Dic146C22×C412yes—
147C22×Dic73—21no—
148C22×Dic73—21no—
149C22×Dic73—21no—
150C22×Dic73—21no—
151C22×Dic73C2×D411yescharacteristic
152C22×Dic73C2×D411yescharacteristic
153C22×Dic73C2×D411yescharacteristic
154C22×Dic73C2×D411yescharacteristic
155#11924012—28no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831#4112yes—
161C2×#10596831#4112yes—
162C4⋊Dic73—21no—
163C4⋊Dic73—24no—
164C4⋊Dic73—24no—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73C22⋊C414yes—
170C4⋊Dic73C22⋊C414yes—
171C4⋊Dic73C22×C412yes—
172#11924007—24no—
173#11924007—24no—
174#11924007—24no—
175#11924007—24no—
176#11924007—24no—
177#11924007—81no—
178#11924007—81no—
179C22⋊Dic73—24no—
180C22⋊Dic73—81no—
181C22⋊Dic73C22⋊C414yes—
182C22⋊Dic73C22⋊C414yes—
183C4×Dic73—24no—
184C4×Dic73—24no—
185C4×Dic73—81no—
186C4×Dic73—81no—
187C4×Dic73C22×C412yes—
188C4×Dic73C22×C412yes—
189C4×Dic73C2×D412yes—
190C4×Dic73C2×D412yes—
191C24×C73C2×D411yescharacteristic
192C2×Q8×C73—21no—
193C2×Q8×C73—21no—
194C2×Q8×C73—21no—
195C2×Q8×C73—21no—
196C2×Q8×C73—21no—
197C2×Q8×C73C22×C412yes—
198C22×C4×C73—21no—
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73—21no—
204C22×C4×C73—21no—
205C22×C4×C73—21no—
206C22×C4×C73C22×C412yes—
207C22×C4×C73C2×D411yescharacteristic
208C22×C4×C73C2×D412yes—
209C22×C4×C73C2411yescharacteristic
210C2×C8×C73—28no—
211C2×C8×C73—21no—
212C2×C8×C73—21no—
213C2×C8×C73—21no—
214C2×C8×C73—21no—
215C2×C8×C73#4112yes—
216C2×C8×C73#4112yes—
217C4⋊C4×C73—24no—
218C4⋊C4×C73—21no—
219C4⋊C4×C73—21no—
220C4⋊C4×C73—21no—
221C4⋊C4×C73—21no—
222C4⋊C4×C73—21no—
223C4⋊C4×C73C22×C412yes—
224C22⋊C4×C73—24no—
225C22⋊C4×C73—81no—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—21no—
231C23×C73—21no—
232C23×C73C2×C22⋊C412yes—
233C23×C73C22×D411yescharacteristic
234Q8×C73—22no—
235Q8×C73—22no—
236Q8×C73—41no—
237Q8×C73—41no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—22no—
244C2×C4×C73—22no—
245C2×C4×C73—22no—
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×C73C2×C22⋊C412yes—
257C2×C4×C73C23×C411yescharacteristic
258C2×C4×C73C22×D411yescharacteristic
259C2×C4×C73C2×#4111yescharacteristic
260C2×C4×C73C2×#4111yescharacteristic
261C2×C4×C73C2×#4111yescharacteristic
262C2×C4×C73C2×#4111yescharacteristic
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—22no—
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×Dic73C2×C22⋊C412yes—
284C2×Dic73C2×C22⋊C412yes—
285C2×Dic73C2×C22⋊C412yes—
286C2×Dic73C2×C22⋊C412yes—
287Dic146—22no—
288Dic146—22no—
289Dic146—41no—
290Dic146—41no—
291C8×C73—22no—
292C8×C73—22no—
293#10596831—22no—
294#10596831—22no—
295C22×C73—22no—
296C22×C73—22no—
297C22×C73—22no—
298C22×C73—41no—
299C22×C73—41no—
300C22×C73—41no—
301C22×C73—41no—
302C22×C73C22×C22⋊C411yescharacteristic
303C22×C73#53211yescharacteristic
304C22×C73#53211yescharacteristic
305C22×C73#57711yescharacteristic
306C22×C73#57711yescharacteristic
307C22×C73#57811yescharacteristic
308C22×C73#57811yescharacteristic
309C4×C73—22no—
310C4×C73—42no—
311C4×C73—42no—
312C4×C73—81no—
313C4×C73—81no—
314C4×C73C4×#4111yescharacteristic
315C4×C73C4×#4111yescharacteristic
316C4×C73#52511yescharacteristic
317C4×C73#52511yescharacteristic
318Dic73—22no—
319Dic73—22no—
320Dic73—22no—
321Dic73—22no—
322Dic73—42no—
323Dic73—42no—
324Dic73—81no—
325Dic73—81no—
326#21413—731nomaximal, Sylow
327C2×C73—42no—
328C2×C73#229711yescharacteristic
329C2×C73#289811yescharacteristic
330C2×C73#289811yescharacteristic
331C2×C73#317411yescharacteristic
332C2×C73#317711yescharacteristic
333C2×C73#317811yescharacteristic
334C2×C73#317811yescharacteristic
335C2×#541—731no—
336C2×#405—731no—
337#2314—731no—
338#1992—731no—
339#1991—731no—
340#1960—731no—
341#1956—731no—
342#1894—731no—
343#1894—731no—
344#1892—731no—
345#1892D7311yescharacteristic
346#1763—731no—
347#1763—731no—
348C4×#103—731no—
349C4×#103—731no—
350C73#2141311yescharacteristic, complemented, Sylow
351C22×M16—731no—
352C23×C8C2×D7311yescharacteristic
353#541—738no—
354#541—1464no—
355C2×C4⋊Q8—731no—
356C2×#126—1461no—
357C2×#126—1461no—
358C2×#125—731no—
359C2×#125—1461no—
360C2×#125—1461no—
361C2×C22⋊Q8—731no—
362C2×C22⋊Q8C2×D7311yescharacteristic
363C2×C4×Q8—731no—
364C2×C4×Q8—731no—
365C2×#117—731no—
366C2×#103—731no—
367C2×#103—731no—
368C2×#103—731no—
369C2×#103—731no—
370C2×#103—731no—
371C2×#103—731no—
372C2×#103C2×D7311yescharacteristic
373C2×#103C2×D7311yescharacteristic
374#405—1464no—
375C2×#97—731no—
376C2×#97—731no—
377C2×C4×C8—731no—
378C2×C4×C8—731no—
379#388—1461no—
380#388—1461no—
381#387—1461no—
382#387—1461no—
383#384—731no—
384#382—731no—
385#380—731no—
386#380C2×D7311yescharacteristic
387C4×C4⋊C4—731no—
388C4×C4⋊C4—731no—
389#340—731no—
390#340—731no—
391#340—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#340C2×D7311yescharacteristic
396#340C2×D7311yescharacteristic
397C22×Q8—731no—
398C22×Q8C22×D7311yescharacteristic
399C23×C4C22×D7311yescharacteristic
400C2×M16—1462no—
401C2×M16—1462no—
402C2×M16—1462no—
403C22×C8—22no—
404C22×C8—22no—
405C22×C8—22no—
406C22×C8—731no—
407C22×C8—731no—
408C22×C8C22×D7311yescharacteristic
409C22×C8C22×D7311yescharacteristic
410C4⋊Q8—734no—
411C4⋊Q8—1462no—
412#126—1468no—
413#126—1468no—
414#125—734no—
415#125—1462no—
416#125—1468no—
417#125—1468no—
418C22⋊Q8—22no—
419C22⋊Q8—734no—
420C22⋊Q8—1462no—
421C22⋊Q8C4×D7314yes—
422C4×Q8—734no—
423C4×Q8—734no—
424C4×Q8—1462no—
425C4×Q8—1462no—
426#117—734no—
427#117—734no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—731no—
430C2×C4⋊C4—731no—
431C2×C4⋊C4—731no—
432C2×C4⋊C4—731no—
433C2×C4⋊C4—731no—
434C2×C4⋊C4—1461no—
435C2×C4⋊C4—1461no—
436C2×C4⋊C4—1461no—
437C2×C4⋊C4—1461no—
438C2×C4⋊C4—1461no—
439C2×C4⋊C4—1461no—
440C2×C4⋊C4—1461no—
441C2×C4⋊C4—1461no—
442C2×C4⋊C4—1461no—
443C2×C4⋊C4C22×D7311yescharacteristic
444C2×C4⋊C4C22×D7311yescharacteristic
445C2×C4⋊C4C22×D7311yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—731no—
449C2×C22⋊C4—731no—
450C2×C22⋊C4—1461no—
451C2×C22⋊C4—1461no—
452C2×C42—731no—
453C2×C42—731no—
454C2×C42—1461no—
455C2×C42—1461no—
456C2×C42—1461no—
457C2×C42—1461no—
458#103—22no—
459#103—22no—
460#103—1462no—
461#103—1462no—
462#103—1462no—
463#103—1462no—
464#103—1462no—
465#103—1462no—
466#97—1462no—
467#97—1462no—
468C4×C8—1462no—
469C4×C8—1462no—
470#95—21no—
471#95—1461no—
472#95—1461no—
473#95—1461no—
474#95—1461no—
475#95—1461no—
476C24D4×D7311yescharacteristic
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—732no—
483C2×Q8—1461no—
484C2×Q8—1461no—
485C2×Q8—1461no—
486C2×Q8—1461no—
487C2×Q8—1461no—
488C2×Q8C2×C4×D7312yes—
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—731no—
498C22×C4—731no—
499C22×C4—731no—
500C22×C4—731no—
501C22×C4—1461no—
502C22×C4—1461no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4C2×C4×D7312yes—
506C22×C4D4×D7311yescharacteristic
507C22×C4D4×D7312yes—
508C22×C4C23×D7311yescharacteristic
509M16—1468no—
510C2×C8—28no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—732no—
516C2×C8—732no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8#1192402712yes—
522C2×C8#1192402712yes—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—5841no—
530C4⋊C4—5841no—
531C4⋊C4—734no—
532C4⋊C4—734no—
533C4⋊C4—732no—
534C4⋊C4—1464no—
535C4⋊C4—1464no—
536C4⋊C4—1464no—
537C4⋊C4—1464no—
538C4⋊C4—1464no—
539C4⋊C4—1464no—
540C4⋊C4—1461no—
541C4⋊C4—1461no—
542C4⋊C4—1461no—
543C4⋊C4—1461no—
544C4⋊C4—1464no—
545C4⋊C4—1461no—
546C4⋊C4C2×C4×D7312yes—
547C22⋊C4—24no—
548C22⋊C4—81no—
549C22⋊C4—5841no—
550C22⋊C4—734no—
551C22⋊C4—734no—
552C22⋊C4—1464no—
553C42—5841no—
554C42—5841no—
555C42—732no—
556C42—732no—
557C42—732no—
558C42—732no—
559C42—1464no—
560C42—1464no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23C22⋊C4×D7312yes—
568C23C2×D4×D7311yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573Q8—2921no—
574Q8—2921no—
575Q8—1462no—
576Q8—1462no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
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—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—2921no—
603C2×C4—2921no—
604C2×C4—2921no—
605C2×C4—732no—
606C2×C4—732no—
607C2×C4—732no—
608C2×C4—732no—
609C2×C4—1462no—
610C2×C4—1462no—
611C2×C4—1461no—
612C2×C4—1461no—
613C2×C4—1461no—
614C2×C4—1461no—
615C2×C4—1461no—
616C2×C4—1461no—
617C2×C4—1461no—
618C2×C4—1461no—
619C2×C4C22⋊C4×D7312yes—
620C2×C4C22×C4×D7311yescharacteristic
621C2×C4C2×D4×D7311yescharacteristic
622C2×C4#41×D7311yescharacteristic
623C2×C4#41×D7311yescharacteristic
624C2×C4C2×#1192402711yescharacteristic
625C2×C4C2×#1192402711yescharacteristic
626C8—22no—
627C8—22no—
628C8—1462no—
629C8—1462no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×C22⋊C4×D7311yescharacteristic
638C22#1798913211yescharacteristic
639C22#1798913211yescharacteristic
640C22#1798939311yescharacteristic
641C22#1798939411yescharacteristic
642C22#1798940011yescharacteristic
643C22#1798940111yescharacteristic
644C4—22no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—2922no—
650C4—2922no—
651C4—5841no—
652C4—5841no—
653C4—1462no—
654C4—1462no—
655C4—1462no—
656C4—1462no—
657C4C4×#1192402711yescharacteristic
658C4C4×#1192402711yescharacteristic
659C4#1798909311yescharacteristic
660C4#1798909511yescharacteristic
661C2—42no—
662C2#2261241611yescharacteristic
663C2#2261845911yescharacteristic
664C2#2261846011yescharacteristic
665C2#2262055211yescharacteristic
666C2#2262058311yescharacteristic
667C2#2262059411yescharacteristic
668C2#2262060211yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1892C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#1892D73
hypocenterC73#21413
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
4. #22610898C2semidirect
5. #22610894C2semidirect
6. #22610749C2semidirect
7. #22610731C2semidirect
11. #22609763C2semidirect
12. #22609760C2semidirect
13. #103×Dic73C2semidirect
14. #22609750C2semidirect
18. #17989197C4semidirect
350. C73#21413semidirect

Maximal subgroups

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

subgroupindex
C2×#179891972
C2×#179885422
#226125382
#226108982
#226108942
#226107492
#226107312
#226103652
#226103622
#226103542
#226097632
#226097602
#103×Dic732
#226097502
C73×#18922
#2141373