Finite Group Databasebeta

← #33090528

Subgroups — #33090528

Order 18688.

1142 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989214C211yesmaximal, characteristic
2C2×#17988612C211yesmaximal, characteristic
3#22612285C211yesmaximal, characteristic
4#22610888C211yesmaximal, characteristic, complemented
5#22610887C211yesmaximal, characteristic, complemented
6#22610886C211yesmaximal, characteristic, complemented
7#22610885C211yesmaximal, characteristic, complemented
8#22610881C211yesmaximal, characteristic, complemented
9#22610880C211yesmaximal, characteristic, complemented
10#22610877C211yesmaximal, characteristic, complemented
11#22609649C211yesmaximal, characteristic
12#22609649C211yesmaximal, characteristic
13#22609646C211yesmaximal, characteristic
14#340×D73C211yesmaximal, characteristic
15C73×#1988C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C22×C8×D73C2211yescharacteristic
18#17989214—24no—
19#17989214C418yescomplemented
20C2×#14556590C2211yescharacteristic
21C2×#14556589—21no—
22C2×#14556583C2211yescharacteristic
23C2×#14556582C2211yescharacteristic
24C2×#14556581—21no—
25C2×#14556569—21no—
26C2×#14556567—21no—
27C2×#14556549C2211yescharacteristic
28C2×#14556549C2211yescharacteristic
29C2×#14556548C2211yescharacteristic
30C2×C4⋊C4×D73C2211yescharacteristic
31C2×C4⋊C4×D73C2211yescharacteristic
32#17988612—24no—
33C2×C4⋊#10596831C2211yescharacteristic
34C2×#14556520C2211yescharacteristic
35C2×#14556520C2211yescharacteristic
36C4⋊C4×Dic73C2211yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4×#11924007—21no—
39C4×C4⋊Dic73C2211yescharacteristic
40#17988438—21no—
41C4×#11924005—21no—
42#17988434—21no—
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988349C2211yescharacteristic
55C2×C4⋊Q8×C73C2211yescharacteristic
56C2×C4⋊C8×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#340×C73C2211yescharacteristic
62C22×Dic146—21no—
63C22×C4×D73—21no—
64C22×C4×D73—21no—
65C22×C4×D73C2311yescharacteristic
66C2×#11924011—22no—
67C2×#11924011—22no—
68C2×#11924011—22no—
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C2×C8×D73—22no—
72C2×C8×D73—22no—
73C2×C8×D73—22no—
74#14556590—22no—
75#14556590C2×C414yescomplemented
76#14556589—28no—
77#14556583—22no—
78#14556583C2×C414yescomplemented
79#14556582—22no—
80#14556582C2×C414yescomplemented
81#14556581—28no—
82#14556569—28no—
83#14556567—28no—
84#14556549—22no—
85#14556549—22no—
86#14556549C2×C414yes—
87#14556549Q814yescomplemented
88#14556548C2×C414yes—
89#14556548D414yes—
90C2×C4⋊Dic73—21no—
91C2×C4⋊Dic73C2311yescharacteristic
92C2×C4⋊Dic73C2311yescharacteristic
93C2×C4⋊Dic73C2311yescharacteristic
94C2×C4⋊Dic73C2311yescharacteristic
95C2×#11924007—21no—
96C2×#11924007—21no—
97C2×#11924007—21no—
98C2×#11924007—21no—
99C2×#11924007—21no—
100C2×#11924007—21no—
101C2×#11924007D411yescharacteristic
102C2×#11924007D411yescharacteristic
103C4⋊C4×D73—22no—
104C4⋊C4×D73—22no—
105C4⋊C4×D73C2×C414yes—
106C4⋊C4×D73Q814yescomplemented
107C2×#11924005—21no—
108C2×#11924005—21no—
109C2×#11924005D411yescharacteristic
110C2×#11924005D411yescharacteristic
111C2×C4×Dic73—21no—
112C2×C4×Dic73—21no—
113C2×C4×Dic73—21no—
114C2×C4×Dic73—21no—
115C2×C4×Dic73—21no—
116C2×C4×Dic73C2311yescharacteristic
117C4⋊#10596831—22no—
118#14556520—22no—
119#14556520—22no—
120#14556498—21no—
121#14556498—21no—
122#14556498—21no—
123C22×Q8×C73—21no—
124C22×C8×C73C2311yescharacteristic
125C22×C8×C73C2311yescharacteristic
126C4⋊Q8×C73—22no—
127C4⋊Q8×C73C2×C414yescomplemented
128C2×C4⋊C4×C73—21no—
129C2×C4⋊C4×C73C2311yescharacteristic
130C2×C4⋊C4×C73C2311yescharacteristic
131C2×C4⋊C4×C73C2311yescharacteristic
132C2×C4⋊C4×C73C2311yescharacteristic
133C2×C42×C73—21no—
134C2×C42×C73C2311yescharacteristic
135C4⋊C8×C73—22no—
136#95×C73—21no—
137C23×D73C2×D411yescharacteristic
138C2×Dic146—24no—
139C2×Dic146—81no—
140C2×Dic146—81no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C22×Dic73—21no—
145C22×Dic73C2×D411yescharacteristic
146C22×Dic73C2×D411yescharacteristic
147C22×Dic73C2×D411yescharacteristic
148C2×C4×D73—24no—
149C2×C4×D73—24no—
150C2×C4×D73—81no—
151C2×C4×D73—81no—
152C2×C4×D73—81no—
153C2×C4×D73—81no—
154C2×C4×D73C22×C412yes—
155C2×C4×D73C2×D412yes—
156C2×C4×D73C2×Q812yes—
157C2×C4×D73C2×Q812yes—
158C2×C4×D73#4112yes—
159C2×C4×D73#4112yes—
160#11924011—44no—
161C2×#10596831—21no—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C8×D73—44no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73—24no—
173C4⋊Dic73C22×C412yes—
174C4⋊Dic73C22×C412yes—
175C4⋊Dic73C2×Q812yes—
176C4⋊Dic73C2×Q812yes—
177#11924007—24no—
178#11924007—24no—
179#11924007—24no—
180#11924007—24no—
181#11924007—24no—
182#11924007—24no—
183#11924007C22⋊C414yes—
184#11924007C22⋊C414yes—
185#11924005—24no—
186#11924005—24no—
187#11924005C22⋊C414yes—
188#11924005C22⋊C414yes—
189C4×Dic73—24no—
190C4×Dic73—24no—
191C4×Dic73—81no—
192C4×Dic73—81no—
193C4×Dic73—81no—
194C4×Dic73C22×C412yes—
195C4×Dic73C2×D412yes—
196C2×Q8×C73—24no—
197C2×Q8×C73—81no—
198C2×Q8×C73—81no—
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73C2×D411yescharacteristic
204C22×C4×C73C2×D411yescharacteristic
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—24no—
213C4⋊C4×C73—21no—
214C4⋊C4×C73—21no—
215C4⋊C4×C73—21no—
216C4⋊C4×C73—21no—
217C4⋊C4×C73C22×C412yes—
218C4⋊C4×C73C22×C412yes—
219C4⋊C4×C73C2×Q812yes—
220C4⋊C4×C73C2×Q812yes—
221C42×C73—81no—
222C42×C73C22×C412yes—
223C42×C73C2×D412yes—
224C23×C73C22×D411yescharacteristic
225C22×D73—21no—
226C22×D73—21no—
227C22×D73—21no—
228C22×D73—21no—
229C22×D73C2×C22⋊C412yes—
230C22×D73C22⋊Q812yes—
231C22×D73C22⋊Q812yes—
232Q8×C73—84no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
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—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73C2×C22⋊C412yes—
250C2×C4×C73C2×C22⋊C412yes—
251C2×C4×C73C23×C411yescharacteristic
252C2×C4×C73C22×D411yescharacteristic
253C2×C4×C73C22×Q811yescharacteristic
254C2×C4×C73C22×Q811yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—22no—
262C2×Dic73—22no—
263C2×Dic73—22no—
264C2×Dic73—22no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73C2×C22⋊C412yes—
274C2×Dic73C2×C22⋊C412yes—
275C2×Dic73C2×C22⋊C412yes—
276C2×Dic73C22⋊Q812yes—
277C2×Dic73C22⋊Q812yes—
278C4×D73—84no—
279C4×D73—84no—
280C4×D73C4×Q814yes—
281C4×D73C4×Q814yes—
282C4×D73#12414yes—
283C4×D73C4⋊Q814yes—
284Dic146—84no—
285C8×C73—41no—
286C8×C73—41no—
287#10596831—41no—
288#10596831—41no—
289C22×C73C22×C22⋊C411yescharacteristic
290C22×C73C2×C22⋊Q811yescharacteristic
291C22×C73C2×C22⋊Q811yescharacteristic
292C22×C73#57611yescharacteristic
293C22×C73#57711yescharacteristic
294C22×C73#57711yescharacteristic
295C22×C73#57811yescharacteristic
296C2×D73—22no—
297C2×D73—22no—
298C2×D73—22no—
299C2×D73—22no—
300C2×D73—22no—
301C2×D73—22no—
302C2×D73#39114yes—
303C4×C73—22no—
304C4×C73—22no—
305C4×C73—42no—
306C4×C73—42no—
307C4×C73—42no—
308C4×C73—42no—
309C4×C73C2×C4×Q811yescharacteristic
310C4×C73C2×C4×Q811yescharacteristic
311C4×C73C2×#12411yescharacteristic
312C4×C73C2×C4⋊Q811yescharacteristic
313Dic73—22no—
314Dic73—22no—
315Dic73—42no—
316Dic73—42no—
317Dic73—42no—
318Dic73—42no—
319Dic73#39114yes—
320#21315—731nomaximal, Sylow
321C2×C73C2×#39111yescharacteristic
322C2×C73#289711yescharacteristic
323C2×C73#289811yescharacteristic
324C2×C73#308111yescharacteristic
325C2×C73#308211yescharacteristic
326C2×C73#308311yescharacteristic
327C2×C73#308411yescharacteristic
328D73—24no—
329C2×#544—731no—
330C2×#424—731no—
331#2274—731no—
332#1989—731no—
333#1989—731no—
334#1989—731no—
335#1989—731no—
336#1988—731no—
337#1988D7311yescharacteristic, complemented
338#1987—731no—
339#1987—731no—
340#1736—731no—
341#1736—731no—
342#1735—731no—
343C2×#340—731no—
344C73#2131511yescharacteristic, complemented, Sylow
345C22×M16—731no—
346C23×C8—731no—
347#544—738no—
348#544—1464no—
349C2×C4⋊Q8—731no—
350C2×C4⋊Q8—1461no—
351C2×C4⋊Q8C2×D7311yescharacteristic
352C2×#125—731no—
353C2×#125—731no—
354C2×#125—1461no—
355C2×C22⋊Q8—1461no—
356C2×C22⋊Q8—1461no—
357C2×#117—731no—
358C2×#117—731no—
359C2×#117—731no—
360C22×C4⋊C4—731no—
361C22×C4⋊C4—731no—
362#424—1464no—
363C2×C4⋊C8—731no—
364C2×C4⋊C8—731no—
365C2×C4⋊C8—731no—
366C2×C4⋊C8C2×D7311yescharacteristic
367C4×C4⋊C4—731no—
368C4×C4⋊C4—731no—
369C4×C4⋊C4—731no—
370C4×C4⋊C4—1461no—
371C4×C4⋊C4—1461no—
372C4×C4⋊C4C2×D7311yescharacteristic
373C4×C22⋊C4—1461no—
374C4×C22⋊C4—1461no—
375#340—731no—
376#340—731no—
377#340—731no—
378#340—731no—
379#340—731no—
380#340—731no—
381#340—731no—
382#340—731no—
383#340—731no—
384#340—731no—
385#340—731no—
386#340—731no—
387#340C2×D7311yescharacteristic
388#340C2×D7311yescharacteristic
389#340C2×D7311yescharacteristic
390#340C2×D7311yescharacteristic
391C22×Q8—21no—
392C22×Q8—1461no—
393C23×C4—731no—
394C23×C4—1461no—
395C23×C4—1461no—
396C2×M16—1462no—
397C2×M16—1462no—
398C2×M16—1462no—
399C22×C8—731no—
400C22×C8—731no—
401C22×C8—1462no—
402C22×C8—1462no—
403C22×C8—1462no—
404C22×C8C22×D7311yescharacteristic
405C22×C8C22×D7311yescharacteristic
406C4⋊Q8—22no—
407C4⋊Q8—734no—
408C4⋊Q8—1462no—
409C4⋊Q8—1468no—
410C4⋊Q8C4×D7314yescomplemented
411#125—734no—
412#125—734no—
413#125—1462no—
414#125—1468no—
415#125—1462no—
416C22⋊Q8—1468no—
417C22⋊Q8—1468no—
418#117—734no—
419#117—734no—
420#117—734no—
421#117—734no—
422#117—1462no—
423#117—1462no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—731no—
427C2×C4⋊C4—731no—
428C2×C4⋊C4—731no—
429C2×C4⋊C4—731no—
430C2×C4⋊C4—731no—
431C2×C4⋊C4—734no—
432C2×C4⋊C4—734no—
433C2×C4⋊C4—1462no—
434C2×C4⋊C4—1462no—
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⋊C4C22×D7311yescharacteristic
443C2×C4⋊C4C22×D7311yescharacteristic
444C2×C4⋊C4C22×D7311yescharacteristic
445C2×C4⋊C4C22×D7311yescharacteristic
446C2×C22⋊C4—731no—
447C2×C22⋊C4—731no—
448C2×C22⋊C4—1461no—
449C2×C22⋊C4—1461no—
450C2×C42—21no—
451C2×C42—731no—
452C2×C42—1461no—
453C2×C42—1461no—
454C2×C42—1461no—
455C2×C42—1461no—
456C2×C42—1461no—
457C2×C42C22×D7311yescharacteristic
458C4⋊C8—22no—
459C4⋊C8—1462no—
460C4⋊C8—1462no—
461C4⋊C8—1462no—
462#95—21no—
463#95—1461no—
464#95—1461no—
465#95—1461no—
466C24—731no—
467C2×Q8—24no—
468C2×Q8—81no—
469C2×Q8—81no—
470C2×Q8—5841no—
471C2×Q8—5841no—
472C2×Q8—1464no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—5841no—
478C22×C4—5841no—
479C22×C4—5841no—
480C22×C4—5841no—
481C22×C4—731no—
482C22×C4—732no—
483C22×C4—732no—
484C22×C4—732no—
485C22×C4—732no—
486C22×C4—732no—
487C22×C4—732no—
488C22×C4—731no—
489C22×C4—731no—
490C22×C4—1464no—
491C22×C4—1464no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4—1461no—
495C22×C4—1461no—
496C22×C4D4×D7311yescharacteristic
497C22×C4D4×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—2924no—
507C2×C8—1461no—
508C2×C8—1461no—
509C2×C8—1461no—
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8—1461no—
513C4⋊C4—24no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—734no—
519C4⋊C4—734no—
520C4⋊C4—732no—
521C4⋊C4—732no—
522C4⋊C4—732no—
523C4⋊C4—732no—
524C4⋊C4—1464no—
525C4⋊C4—1464no—
526C4⋊C4—1464no—
527C4⋊C4—1464no—
528C4⋊C4—1464no—
529C4⋊C4—1464no—
530C4⋊C4—1464no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4C2×C4×D7312yes—
536C4⋊C4C2×C4×D7312yes—
537C4⋊C4Q8×D7312yes—
538C4⋊C4Q8×D7312yes—
539C22⋊C4—734no—
540C22⋊C4—734no—
541C22⋊C4—1464no—
542C22⋊C4—1464no—
543C42—81no—
544C42—5841no—
545C42—5841no—
546C42—5841no—
547C42—732no—
548C42—732no—
549C42—1464no—
550C42—1464no—
551C42C2×C4×D7312yes—
552C42D4×D7312yes—
553C23—732no—
554C23—732no—
555C23—732no—
556C23—1461no—
557C23—1461no—
558C23—1461no—
559C23—1461no—
560C23C2×D4×D7311yescharacteristic
561Q8—84no—
562Q8—5844no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—5844no—
588C2×C4—5844no—
589C2×C4—732no—
590C2×C4—732no—
591C2×C4—732no—
592C2×C4—734no—
593C2×C4—734no—
594C2×C4—734no—
595C2×C4—734no—
596C2×C4—732no—
597C2×C4—732no—
598C2×C4—1462no—
599C2×C4—1462no—
600C2×C4—1462no—
601C2×C4—1462no—
602C2×C4—1461no—
603C2×C4—1461no—
604C2×C4—1461no—
605C2×C4—1461no—
606C2×C4C22⋊C4×D7312yes—
607C2×C4C22⋊C4×D7312yes—
608C2×C4C22×C4×D7311yescharacteristic
609C2×C4C2×D4×D7311yescharacteristic
610C2×C4C2×Q8×D7311yescharacteristic
611C2×C4C2×Q8×D7311yescharacteristic
612C2×C4#41×D7311yescharacteristic
613C2×C4#41×D7311yescharacteristic
614C8—41no—
615C8—41no—
616C8—2921no—
617C8—2921no—
618C22—734no—
619C22—1462no—
620C22—1462no—
621C22—1462no—
622C22—1462no—
623C22—1462no—
624C22—1462no—
625C22C2×C22⋊C4×D7311yescharacteristic
626C22C22⋊Q8×D7311yescharacteristic
627C22C22⋊Q8×D7311yescharacteristic
628C22#1798938811yescharacteristic
629C22#1798939311yescharacteristic
630C22#1798939611yescharacteristic
631C22#1798940311yescharacteristic
632C4—22no—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—2922no—
639C4—2922no—
640C4—2922no—
641C4—2922no—
642C4—734no—
643C4—1462no—
644C4—1462no—
645C4C4×Q8×D7311yescharacteristic
646C4C4×Q8×D7311yescharacteristic
647C4#124×D7311yescharacteristic
648C4C4⋊Q8×D7311yescharacteristic
649C2—1464no—
650C2#391×D7311yescharacteristic
651C2#2261845211yescharacteristic
652C2#2261846311yescharacteristic
653C2#2261975511yescharacteristic
654C2#2261976611yescharacteristic
655C2#2261977711yescharacteristic
656C2#2261979011yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1988C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1988D73
hypocenterC73#21315
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
4. #22610888C2semidirect
5. #22610887C2semidirect
6. #22610886C2semidirect
7. #22610885C2semidirect
8. #22610881C2semidirect
9. #22610880C2semidirect
10. #22610877C2semidirect
15. C73×#1988C2semidirect
19. #17989214C4semidirect
75. #14556590C2×C4semidirect
78. #14556583C2×C4semidirect
80. #14556582C2×C4semidirect
87. #14556549Q8semidirect
106. C4⋊C4×D73Q8semidirect
127. C4⋊Q8×C73C2×C4semidirect
337. #1988D73semidirect
344. C73#21315semidirect
410. C4⋊Q8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179892142
C2×#179886122
#226122852
#226108882
#226108872
#226108862
#226108852
#226108812
#226108802
#226108772
#226096492
#226096462
#340×D732
C73×#19882
#2131573