Finite Group Databasebeta

← #33102658

Subgroups — #33102658

Order 18688.

1066 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989293C211yesmaximal, characteristic
2C2×#17988716C211yesmaximal, characteristic, complemented
3#22612886C211yesmaximal, characteristic, complemented
4#22610902C211yesmaximal, characteristic, complemented
5#22610901C211yesmaximal, characteristic, complemented
6#22610894C211yesmaximal, characteristic, complemented
7#22610887C211yesmaximal, characteristic, complemented
8#22610881C211yesmaximal, characteristic, complemented
9#22609940C211yesmaximal, characteristic
10#22609940C211yesmaximal, characteristic
11#22609932C211yesmaximal, characteristic
12#22609759C211yesmaximal, characteristic
13#22609755C211yesmaximal, characteristic
14#22609745C211yesmaximal, characteristic
15C73×#1989C211yesmaximal, characteristic, complemented
16#17989293—24no—
17#17989293C418yescomplemented
18C2×#14556590C2211yescharacteristic
19C2×#14556582C2211yescharacteristic
20C2×#14556571—21no—
21C2×#14556569—21no—
22C2×#14556569—21no—
23C2×C4×Dic146C2211yescharacteristic
24C2×C4×D292C2211yescharacteristic
25C2×#14556553C2211yescharacteristic
26C2×#14556553C2211yescharacteristic
27C2×C4⋊C4×D73—21no—
28#17988716—24no—
29C2×#14556520C2211yescharacteristic
30C2×#14556520C2211yescharacteristic
31C2×#14556513C2211yescharacteristic
32C2×#14556513C2211yescharacteristic
33C2×#14556511C2211yescharacteristic
34C2×#14556511C2211yescharacteristic
35C2×#14556502C2211yescharacteristic
36C2×#14556501C2211yescharacteristic
37C2×C8×Dic73C2211yescharacteristic
38#17988493—21no—
39#17988474—21no—
40#17988474—21no—
41#17988457C2211yescharacteristic
42#17988452C2211yescharacteristic
43#17988452C2211yescharacteristic
44C4⋊C4×Dic73C2211yescharacteristic
45C4⋊C4×Dic73C2211yescharacteristic
46C4×C4⋊Dic73—21no—
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988349C2211yescharacteristic
55C2×#125×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×Dic146D411yescharacteristic
64C22×D292C2311yescharacteristic
65C22×C4×D73—21no—
66C22×C4×D73—21no—
67C22×C4×D73—21no—
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70#14556590—22no—
71#14556590C2×C414yes—
72#14556582—22no—
73#14556582C2×C414yes—
74#14556571—28no—
75#14556569—28no—
76#14556569—28no—
77C4×Dic146C2×C414yes—
78C4×Dic146D414yes—
79C4×D292—22no—
80C4×D292C2×C414yes—
81#14556553—22no—
82#14556553—22no—
83#14556553C2×C414yes—
84#14556553C2×C414yes—
85C2×C4⋊Dic73—21no—
86C2×C4⋊Dic73—21no—
87C2×C4⋊Dic73—21no—
88C2×C4⋊Dic73D411yescharacteristic
89C2×C4⋊Dic73C2311yescharacteristic
90C2×C4⋊Dic73C2311yescharacteristic
91C2×C4⋊Dic73C2311yescharacteristic
92C2×#11924007—21no—
93C2×#11924007—21no—
94C2×#11924007D411yescharacteristic
95C2×#11924007D411yescharacteristic
96C4⋊C4×D73—28no—
97C2×#11924005—21no—
98C2×#11924005—21no—
99C2×#11924005—21no—
100C2×C4×Dic73—21no—
101C2×C4×Dic73—21no—
102C2×C4×Dic73C2311yescharacteristic
103C2×C4×Dic73C2311yescharacteristic
104#14556520—22no—
105#14556520—22no—
106#14556513—22no—
107#14556513—22no—
108#14556511—22no—
109#14556511Q814yescomplemented
110#14556502—22no—
111#14556501—22no—
112C8×Dic73—22no—
113#14556498—21no—
114#14556498—21no—
115#14556498—21no—
116#14556498—21no—
117#14556498—21no—
118#14556498—21no—
119C22×C8×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121#125×C73—22no—
122#125×C73C2×C414yes—
123C2×C4⋊C4×C73—21no—
124C2×C4⋊C4×C73—21no—
125C2×C4⋊C4×C73C2311yescharacteristic
126C2×C4⋊C4×C73C2311yescharacteristic
127C2×C4⋊C4×C73C2311yescharacteristic
128C2×C4⋊C4×C73C2311yescharacteristic
129C2×C42×C73—21no—
130C2×C42×C73C2311yescharacteristic
131C4⋊C8×C73—22no—
132#95×C73—21no—
133C23×D73—21no—
134C2×Dic146—24no—
135C2×Dic146—41no—
136C2×Dic146—41no—
137C2×Dic146—81no—
138C2×Dic146—81no—
139C2×Dic146C22⋊C414yes—
140C2×D292—21no—
141C2×D292—21no—
142C2×D292—21no—
143C2×D292C22×C412yes—
144C2×D292C2×Q812yes—
145C2×D292#4112yes—
146C22×Dic73—21no—
147C22×Dic73—21no—
148C22×Dic73—21no—
149C22×Dic73C2×D411yescharacteristic
150C22×Dic73C2×D411yescharacteristic
151C22×Dic73C2×D411yescharacteristic
152C22×Dic73C2×D411yescharacteristic
153C2×C4×D73—24no—
154C2×C4×D73—24no—
155C2×C4×D73—24no—
156C2×C4×D73—42no—
157C2×C4×D73—42no—
158C2×C4×D73—81no—
159C2×C4×D73—81no—
160C2×C4×D73—81no—
161C2×C4×D73—81no—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831C2×Q812yes—
168C4⋊Dic73—21no—
169C4⋊Dic73—24no—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73—21no—
174C4⋊Dic73—42no—
175C4⋊Dic73—81no—
176C4⋊Dic73C22⋊C414yes—
177C4⋊Dic73C22×C412yes—
178#11924007—24no—
179#11924007—24no—
180#11924007C22⋊C414yes—
181#11924007C22⋊C414yes—
182#11924005—24no—
183#11924005—24no—
184#11924005—24no—
185C4×Dic73—81no—
186C4×Dic73—81no—
187C4×Dic73C22×C412yes—
188C4×Dic73C22×C412yes—
189C4×Dic73C2×D412yes—
190C4×Dic73C2×D412yes—
191C22×C4×C73—21no—
192C22×C4×C73—21no—
193C22×C4×C73—21no—
194C22×C4×C73—21no—
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2411yescharacteristic
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C4⋊C4×C73—24no—
205C4⋊C4×C73—24no—
206C4⋊C4×C73—21no—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—21no—
211C4⋊C4×C73C22×C412yes—
212C4⋊C4×C73C22×C412yes—
213C4⋊C4×C73C2×Q812yes—
214C42×C73—42no—
215C42×C73C22×C412yes—
216C42×C73C2×D412yes—
217C23×C73C22×D411yescharacteristic
218C22×D73—22no—
219C22×D73—22no—
220C22×D73—22no—
221C22×D73—41no—
222C22×D73—41no—
223C22×D73—41no—
224C22×D73—41no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—22no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73C2×C22⋊C412yes—
242C2×C4×C73C2×C22⋊C412yes—
243C2×C4×C73C23×C411yescharacteristic
244C2×C4×C73C22×D411yescharacteristic
245C2×C4×C73C22×Q811yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×C4×C732−1+411yescharacteristic
248C2×C4×C732−1+411yescharacteristic
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—22no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73C2×C22⋊C412yes—
267C2×Dic73C2×C22⋊C412yes—
268C2×Dic73C2×C22⋊C412yes—
269C2×Dic73C2×C22⋊C412yes—
270D292—22no—
271D292—22no—
272D292—22no—
273D292C4×Q814yes—
274C4×D73—48no—
275C4×D73—84no—
276C4×D73—84no—
277Dic146—44no—
278Dic146—84no—
279C8×C73—41no—
280C8×C73—41no—
281#10596831—22no—
282#10596831—41no—
283C22×C73C22×C22⋊C411yescharacteristic
284C22×C73#54811yescharacteristic
285C22×C73D4×Q811yescharacteristic
286C22×C73#57511yescharacteristic
287C22×C73#57511yescharacteristic
288C22×C73#57611yescharacteristic
289C22×C73#57711yescharacteristic
290C2×D73—24no—
291C2×D73—42no—
292C2×D73—42no—
293C2×D73—42no—
294C2×D73—42no—
295C2×D73—42no—
296C2×D73—42no—
297C4×C73—22no—
298C4×C73—22no—
299C4×C73—42no—
300C4×C73—42no—
301C4×C73—42no—
302C4×C73—81no—
303C4×C73C2×C4×Q811yescharacteristic
304C4×C73#51911yescharacteristic
305C4×C73#54811yescharacteristic
306C4×C73D4×Q811yescharacteristic
307Dic73—22no—
308Dic73—22no—
309Dic73—22no—
310Dic73—22no—
311Dic73—42no—
312Dic73—81no—
313Dic73—81no—
314#22526—731nomaximal, Sylow
315C2×C73Q8×C22⋊C411yescharacteristic
316C2×C73#289611yescharacteristic
317C2×C73#289711yescharacteristic
318C2×C73#337011yescharacteristic
319C2×C73#337611yescharacteristic
320C2×C73#338811yescharacteristic
321C2×C73#338911yescharacteristic
322D73—44no—
323C2×#554—731no—
324C2×#446—731no—
325#2355—731no—
326#1993—731no—
327#1992—731no—
328#1991—731no—
329#1989—731no—
330#1989D7311yescharacteristic, complemented
331#1988—731no—
332#1804—731no—
333#1804—731no—
334#1802—731no—
335#1762—731no—
336#1762—731no—
337C4×#102—731no—
338C73#2252611yescharacteristic, complemented, Sylow
339#554—738no—
340#554—1464no—
341C2×C4⋊Q8—731no—
342C2×#125—731no—
343C2×#125C2×D7311yescharacteristic
344C2×C22⋊Q8—1461no—
345C2×C22⋊Q8—1461no—
346C2×C22⋊Q8—1461no—
347C2×C4×Q8—731no—
348C2×C4×D4—731no—
349C2×C4×D4—731no—
350C2×C4×D4—731no—
351C22×C4⋊C4—1461no—
352#446—1464no—
353C2×C4⋊C8—731no—
354C2×C4⋊C8—731no—
355C2×C4⋊C8C2×D7311yescharacteristic
356C2×#102—731no—
357C2×#102—731no—
358C2×#102—731no—
359C2×#102—731no—
360C2×#97—731no—
361C2×#97—731no—
362C2×C4×C8—731no—
363#391—1461no—
364#387—1461no—
365#387—1461no—
366#384—731no—
367#382—731no—
368#382—731no—
369C4×C4⋊C4—731no—
370C4×C4⋊C4—731no—
371C4×C4⋊C4—1461no—
372C4×C4⋊C4C2×D7311yescharacteristic
373#340—731no—
374#340—731no—
375#340—731no—
376#340—731no—
377#340—731no—
378#340—731no—
379#340—731no—
380#340—731no—
381#340C2×D7311yescharacteristic
382#340C2×D7311yescharacteristic
383#340C2×D7311yescharacteristic
384#340C2×D7311yescharacteristic
385C22×Q8—731no—
386C22×Q8—1461no—
387C22×D4—731no—
388C23×C4—1461no—
389C23×C4—1461no—
390C23×C4—1461no—
391C22×C8—731no—
392C22×C8—731no—
393C22×C8C22×D7311yescharacteristic
394C22×C8C22×D7311yescharacteristic
395C4⋊Q8—734no—
396C4⋊Q8—1462no—
397#125—22no—
398#125—734no—
399#125—1462no—
400#125C4×D7314yes—
401C22⋊Q8—1468no—
402C22⋊Q8—1468no—
403C22⋊Q8—1468no—
404C4×Q8—734no—
405C4×Q8—734no—
406C4×D4—734no—
407C4×D4—734no—
408C4×D4—734no—
409C4×D4—1462no—
410C4×D4—1462no—
411C4×D4—1462no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—731no—
416C2×C4⋊C4—731no—
417C2×C4⋊C4—731no—
418C2×C4⋊C4—731no—
419C2×C4⋊C4—731no—
420C2×C4⋊C4—1468no—
421C2×C4⋊C4—1461no—
422C2×C4⋊C4—1461no—
423C2×C4⋊C4—1461no—
424C2×C4⋊C4—1461no—
425C2×C4⋊C4—1461no—
426C2×C4⋊C4C22×D7311yescharacteristic
427C2×C4⋊C4C22×D7311yescharacteristic
428C2×C4⋊C4C22×D7311yescharacteristic
429C2×C4⋊C4C22×D7311yescharacteristic
430C2×C22⋊C4—1461no—
431C2×C22⋊C4—1461no—
432C2×C22⋊C4—1461no—
433C2×C42—21no—
434C2×C42—731no—
435C2×C42—731no—
436C2×C42—1461no—
437C2×C42—1461no—
438C2×C42C22×D7311yescharacteristic
439C4⋊C8—22no—
440C4⋊C8—1462no—
441C4⋊C8—1462no—
442#102—734no—
443#102—1462no—
444#102—1462no—
445#102—1462no—
446#97—1462no—
447#97—1462no—
448C4×C8—1462no—
449#95—21no—
450#95—1461no—
451#95—1461no—
452#95—1461no—
453#95—1461no—
454#95—1461no—
455#95—1461no—
456C24—1461no—
457C2×Q8—2921no—
458C2×Q8—2921no—
459C2×Q8—5841no—
460C2×Q8—5841no—
461C2×Q8—734no—
462C2×Q8—1464no—
463C2×D4—732no—
464C2×D4—732no—
465C2×D4—732no—
466C2×D4—1461no—
467C2×D4—1461no—
468C2×D4—1461no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—2922no—
474C22×C4—2922no—
475C22×C4—5841no—
476C22×C4—5841no—
477C22×C4—5841no—
478C22×C4—5841no—
479C22×C4—731no—
480C22×C4—731no—
481C22×C4—731no—
482C22×C4—731no—
483C22×C4—1464no—
484C22×C4—1464no—
485C22×C4—1461no—
486C22×C4—1461no—
487C22×C4—1461no—
488C22×C4—1464no—
489C22×C4D4×D7311yescharacteristic
490C22×C4D4×D7311yescharacteristic
491C22×C4C23×D7311yescharacteristic
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—732no—
499C2×C8—1461no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—2922no—
512C4⋊C4—5841no—
513C4⋊C4—734no—
514C4⋊C4—734no—
515C4⋊C4—734no—
516C4⋊C4—732no—
517C4⋊C4—1464no—
518C4⋊C4—1464no—
519C4⋊C4—1464no—
520C4⋊C4—1461no—
521C4⋊C4—1461no—
522C4⋊C4—1461no—
523C4⋊C4—1461no—
524C4⋊C4—1461no—
525C4⋊C4C2×C4×D7312yes—
526C4⋊C4C2×C4×D7312yes—
527C4⋊C4Q8×D7312yes—
528C22⋊C4—1464no—
529C22⋊C4—1464no—
530C22⋊C4—1464no—
531C42—42no—
532C42—5841no—
533C42—5841no—
534C42—732no—
535C42—732no—
536C42—732no—
537C42—732no—
538C42C2×C4×D7312yes—
539C42D4×D7312yes—
540C23—2921no—
541C23—2921no—
542C23—2921no—
543C23—2921no—
544C23—1462no—
545C23—1462no—
546C23—1462no—
547C23C2×D4×D7311yescharacteristic
548Q8—2924no—
549Q8—5844no—
550D4—734no—
551D4—1462no—
552D4—1462no—
553D4—1462no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—2928no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—5844no—
580C2×C4—5844no—
581C2×C4—732no—
582C2×C4—732no—
583C2×C4—732no—
584C2×C4—732no—
585C2×C4—1462no—
586C2×C4—1461no—
587C2×C4—1461no—
588C2×C4—1461no—
589C2×C4—1461no—
590C2×C4—1461no—
591C2×C4—1461no—
592C2×C4—1461no—
593C2×C4—1461no—
594C2×C4C22⋊C4×D7312yes—
595C2×C4C22⋊C4×D7312yes—
596C2×C4C22×C4×D7311yescharacteristic
597C2×C4C2×D4×D7311yescharacteristic
598C2×C4C2×Q8×D7311yescharacteristic
599C2×C4#41×D7311yescharacteristic
600C2×C4#1455663811yescharacteristic
601C2×C4#1455663911yescharacteristic
602C8—41no—
603C8—41no—
604C8—2921no—
605C8—1462no—
606C22—2922no—
607C22—2922no—
608C22—2922no—
609C22—2922no—
610C22—2922no—
611C22—2922no—
612C22—1464no—
613C22C2×C22⋊C4×D7311yescharacteristic
614C22#1798924711yescharacteristic
615C22#1798925011yescharacteristic
616C22#1798938311yescharacteristic
617C22#1798938611yescharacteristic
618C22#1798938911yescharacteristic
619C22#1798939411yescharacteristic
620C4—22no—
621C4—22no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—81no—
626C4—2922no—
627C4—5841no—
628C4—5841no—
629C4—1462no—
630C4—1462no—
631C4—1462no—
632C4—1462no—
633C4C4×Q8×D7311yescharacteristic
634C4#1798906311yescharacteristic
635C4#1798924511yescharacteristic
636C4#1798925211yescharacteristic
637C2—2924no—
638C2#2261276511yescharacteristic
639C2#2261844711yescharacteristic
640C2#2261845311yescharacteristic
641C2#2262268711yescharacteristic
642C2#2262276111yescharacteristic
643C2#2262287911yescharacteristic
644C2#2262288811yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1989C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1989D73
hypocenterC73#22526
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
2. C2×#17988716C2semidirect
3. #22612886C2semidirect
4. #22610902C2semidirect
5. #22610901C2semidirect
6. #22610894C2semidirect
7. #22610887C2semidirect
8. #22610881C2semidirect
15. C73×#1989C2semidirect
17. #17989293C4semidirect
109. #14556511Q8semidirect
330. #1989D73semidirect
338. C73#22526semidirect

Maximal subgroups

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

subgroupindex
C2×#179892932
C2×#179887162
#226128862
#226109022
#226109012
#226108942
#226108872
#226108812
#226099402
#226099322
#226097592
#226097552
#226097452
C73×#19892
#2252673