Finite Group Databasebeta

← #33144151

Subgroups — #33144151

Order 18688.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612902C211yesmaximal, characteristic
2#22611316C211yesmaximal, characteristic
3#22611302C211yesmaximal, characteristic
4#22611074C211yesmaximal, characteristic
5#22611064C211yesmaximal, characteristic
6#22610889C211yesmaximal, characteristic, complemented
7#22610884C211yesmaximal, characteristic, complemented
8#22610699C211yesmaximal, characteristic, complemented
9#22610650C211yesmaximal, characteristic, complemented
10#22610520C211yesmaximal, characteristic, complemented
11#22610513C211yesmaximal, characteristic, complemented
12#22610061C211yesmaximal, characteristic, complemented
13#22609938C211yesmaximal, characteristic
14C4×#14556511C211yesmaximal, characteristic
15C73×#2550C211yesmaximal, characteristic, complemented
16C2×C4×D292C2211yescharacteristic
17C2×#14556524C2211yescharacteristic
18C2×#14556522C2211yescharacteristic
19C2×C4⋊#10596831C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×#14556511C2211yescharacteristic
23C2×#14556511C2211yescharacteristic
24C2×#14556511C2211yescharacteristic
25C2×#14556511C2211yescharacteristic
26C2×C4×#10596831C2211yescharacteristic
27#17988489C2211yescharacteristic
28#17988489C2211yescharacteristic
29#17988477—21no—
30#17988477—21no—
31#17988453C2211yescharacteristic
32#17988449—21no—
33C4×C4⋊Dic73C2211yescharacteristic
34C4×C4⋊Dic73C2211yescharacteristic
35#17988434—21no—
36#17988349C2211yescharacteristic
37#17988349C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×C4⋊Q8×C73C2211yescharacteristic
49C2×#125×C73C2211yescharacteristic
50#398×C73—21no—
51#395×C73—21no—
52#391×C73—21no—
53#389×C73—21no—
54#389×C73C2211yescharacteristic
55#389×C73C2211yescharacteristic
56#382×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C4⋊C4×C73C2211yescharacteristic
59C22×D292C2311yescharacteristic
60C22×C4×D73—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C4×D292—22no—
66C4×D292Q814yescomplemented
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×#11924005—21no—
71C2×#11924005—21no—
72C2×#11924005—21no—
73C2×#11924005—21no—
74C2×#11924005—21no—
75C2×C4×Dic73—21no—
76C2×C4×Dic73—21no—
77#14556524—22no—
78#14556522—22no—
79C4⋊#10596831—22no—
80C4⋊#10596831—41no—
81C4⋊#10596831—41no—
82#14556511—22no—
83#14556511—22no—
84#14556511—41no—
85#14556511—41no—
86C4×#10596831—22no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91C22×Q8×C73—21no—
92C4⋊Q8×C73—22no—
93C4⋊Q8×C73—41no—
94#125×C73—22no—
95#125×C73—41no—
96C2×C4⋊C4×C73—21no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73D411yescharacteristic
102C2×C4⋊C4×C73D411yescharacteristic
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C4⋊C4×C73C2311yescharacteristic
106C2×C4⋊C4×C73C2311yescharacteristic
107C2×C42×C73—21no—
108C2×C42×C73C2311yescharacteristic
109C2×C42×C73C2311yescharacteristic
110C2×C42×C73C2311yescharacteristic
111#95×C73—21no—
112#95×C73—21no—
113#95×C73—21no—
114#95×C73—21no—
115#95×C73—21no—
116#95×C73D411yescharacteristic
117#95×C73D411yescharacteristic
118C23×D73—21no—
119C2×D292—21no—
120C2×D292—21no—
121C2×D292—21no—
122C2×D292—21no—
123C2×D292—21no—
124C2×D292C2×Q812yes—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C2×C4×D73—24no—
129C2×C4×D73—81no—
130C2×C4×D73—81no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73C2×Q812yes—
149#11924005—24no—
150#11924005—42no—
151#11924005—42no—
152#11924005—81no—
153#11924005—81no—
154C4×Dic73—81no—
155C4×Dic73—81no—
156C2×Q8×C73—42no—
157C2×Q8×C73—42no—
158C2×Q8×C73—81no—
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73C2×D411yescharacteristic
164C22×C4×C73C2×D411yescharacteristic
165C22×C4×C73C2×D411yescharacteristic
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2411yescharacteristic
170C4⋊C4×C73—22no—
171C4⋊C4×C73—21no—
172C4⋊C4×C73—21no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—21no—
177C4⋊C4×C73—41no—
178C4⋊C4×C73—42no—
179C4⋊C4×C73—42no—
180C4⋊C4×C73—42no—
181C4⋊C4×C73—42no—
182C4⋊C4×C73—81no—
183C4⋊C4×C73#4112yes—
184C4⋊C4×C73#4112yes—
185C42×C73—21no—
186C42×C73—21no—
187C42×C73—21no—
188C42×C73—21no—
189C42×C73—81no—
190C42×C73C2×Q812yes—
191C42×C73#4112yes—
192C23×C73C22×D411yescharacteristic
193C22×D73—22no—
194C22×D73—41no—
195C22×D73—41no—
196C22×D73—41no—
197C22×D73—41no—
198C22×D73—41no—
199C22×D73—41no—
200Q8×C73—84no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—22no—
211C2×C4×C73—22no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73C22⋊Q812yes—
229C2×C4×C73C22⋊Q812yes—
230C2×C4×C73#13512yes—
231C2×C4×C73C22×Q811yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73C2×#4111yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C2×C4×C73H(5,𝔽2)11yescharacteristic
236C2×C4×C732−1+411yescharacteristic
237C2×Dic73—22no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246D292—22no—
247D292—22no—
248D292—41no—
249D292—41no—
250C4×D73—84no—
251#10596831—41no—
252#10596831—41no—
253#10596831—41no—
254#10596831—41no—
255C22×C73C2×C22⋊Q811yescharacteristic
256C22×C73#54611yescharacteristic
257C22×C73#54811yescharacteristic
258C22×C73C2×#13511yescharacteristic
259C22×C73#57511yescharacteristic
260C22×C73#57611yescharacteristic
261C22×C73#57711yescharacteristic
262C2×D73—42no—
263C2×D73—42no—
264C2×D73—42no—
265C2×D73—81no—
266C2×D73—81no—
267C2×D73—81no—
268C2×D73—81no—
269C4×C73—22no—
270C4×C73—22no—
271C4×C73—22no—
272C4×C73—42no—
273C4×C73—41no—
274C4×C73—42no—
275C4×C73—41no—
276C4×C73—41no—
277C4×C73—81no—
278C4×C73—81no—
279C4×C73#53311yescharacteristic
280C4×C73#54411yescharacteristic
281C4×C73#55511yescharacteristic
282C4×C73#55611yescharacteristic
283Dic73—42no—
284Dic73—81no—
285Dic73—81no—
286#26028—731nomaximal, Sylow
287C2×C73#243911yescharacteristic
288C2×C73#307611yescharacteristic
289C2×C73#308111yescharacteristic
290C2×C73#329711yescharacteristic
291C2×C73#332611yescharacteristic
292C2×C73#336511yescharacteristic
293C2×C73#337011yescharacteristic
294D73—82no—
295#2550D7311yescharacteristic, complemented
296#2357—731no—
297#2067—731no—
298#2065—731no—
299#2025—731no—
300#2023—731no—
301#1989—731no—
302#1988—731no—
303#1949—731no—
304#1939—731no—
305#1921—731no—
306#1920—731no—
307#1835—731no—
308#1804—731no—
309C4×#102—731no—
310C73#2602811yescharacteristic, complemented, Sylow
311C2×C4⋊Q8C2×D7311yescharacteristic
312C2×#125C2×D7311yescharacteristic
313C2×C4×D4—731no—
314C2×#107—731no—
315C2×#106—731no—
316C2×C4⋊C8—731no—
317C2×C4⋊C8—731no—
318C2×C4⋊C8—731no—
319C2×#102—731no—
320C2×#102—731no—
321C2×#102—731no—
322C2×#102—731no—
323C2×C4×C8—731no—
324#398—21no—
325#395—21no—
326#391—21no—
327#390—731no—
328#390—731no—
329#389—21no—
330#389C2×D7311yescharacteristic
331#389C2×D7311yescharacteristic
332#388—1461no—
333#388—1461no—
334#382—731no—
335#382C2×D7311yescharacteristic
336#381—1461no—
337C4×C4⋊C4—731no—
338C4×C4⋊C4—731no—
339C4×C4⋊C4C2×D7311yescharacteristic
340C4×C4⋊C4C2×D7311yescharacteristic
341C4×C22⋊C4—1461no—
342#340—731no—
343#340—731no—
344#340—731no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#336—731no—
353#336—731no—
354C22×Q8—21no—
355C22×D4—731no—
356C23×C4—1461no—
357C22×C8—731no—
358C22×C8—731no—
359C22×C8—731no—
360C22×C8—731no—
361C4⋊Q8—22no—
362C4⋊Q8—41no—
363#125—22no—
364#125—41no—
365C4×D4—734no—
366C4×D4—1462no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4C22⋊D7311yescharacteristic
376C2×C4⋊C4C22⋊D7311yescharacteristic
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C4⋊C4C22×D7311yescharacteristic
380C2×C4⋊C4C22×D7311yescharacteristic
381C2×C22⋊C4—1461no—
382C2×C22⋊C4—1461no—
383C2×C22⋊C4—1461no—
384C2×C22⋊C4—1461no—
385C2×C22⋊C4—1461no—
386C2×C42—21no—
387C2×C42—1461no—
388C2×C42—1461no—
389C2×C42C22×D7311yescharacteristic
390C2×C42C22×D7311yescharacteristic
391C2×C42C22×D7311yescharacteristic
392#107—1462no—
393#106—1462no—
394C4⋊C8—2921no—
395C4⋊C8—2921no—
396C4⋊C8—1462no—
397#102—2921no—
398#102—2921no—
399#102—1462no—
400#102—1462no—
401C4×C8—1462no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95C22⋊D7311yescharacteristic
412#95C22⋊D7311yescharacteristic
413C24—1461no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—81no—
417C2×D4—732no—
418C2×D4—1461no—
419C2×D4—1461no—
420C2×D4—1461no—
421C2×D4—1461no—
422C2×D4—1461no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—5841no—
428C22×C4—5841no—
429C22×C4—1461no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4—1464no—
433C22×C4C2×C22⋊D7311yescharacteristic
434C22×C4C2×C22⋊D7311yescharacteristic
435C22×C4C2×C22⋊D7311yescharacteristic
436C22×C4C2×C22⋊D7311yescharacteristic
437C22×C4C2×C22⋊D7311yescharacteristic
438C22×C4C2×C22⋊D7311yescharacteristic
439C22×C4C23×D7311yescharacteristic
440C2×C8—1461no—
441C2×C8—1461no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8—1461no—
449C2×C8—1461no—
450C2×C8—1461no—
451C2×C8—1461no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—41no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—81no—
465C4⋊C4—732no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4—1461no—
471C4⋊C4#1192402712yes—
472C4⋊C4#1192402912yes—
473C22⋊C4—2922no—
474C22⋊C4—2922no—
475C22⋊C4—5841no—
476C22⋊C4—5841no—
477C22⋊C4—1464no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—81no—
483C42—5841no—
484C42—5841no—
485C42Q8×D7312yes—
486C42#1192402912yes—
487C23—2921no—
488C23—2921no—
489C23—2921no—
490C23—2921no—
491C23—2921no—
492C23—2921no—
493C23—1462no—
494C23C22×C22⋊D7311yescharacteristic
495Q8—84no—
496D4—2921no—
497D4—2921no—
498D4—1462no—
499D4—1462no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—2921no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—5844no—
536C2×C4—1462no—
537C2×C4#1455656712yes—
538C2×C4#1455656712yes—
539C2×C4#1455661112yes—
540C2×C4C2×Q8×D7311yescharacteristic
541C2×C4C2×#1192402711yescharacteristic
542C2×C4C2×#1192402911yescharacteristic
543C2×C4C2×#1192402911yescharacteristic
544C2×C4#1455663611yescharacteristic
545C2×C4#1455663911yescharacteristic
546C8—2921no—
547C8—2921no—
548C8—2921no—
549C8—2921no—
550C22—2922no—
551C22—2922no—
552C22—2922no—
553C22—5841no—
554C22—5841no—
555C22—5841no—
556C22—5841no—
557C22C2×#1455656711yescharacteristic
558C22#1798923111yescharacteristic
559C22#1798924811yescharacteristic
560C22C2×#1455661111yescharacteristic
561C22#1798938511yescharacteristic
562C22#1798939011yescharacteristic
563C22#1798939511yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—41no—
568C4—42no—
569C4—42no—
570C4—41no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—2922no—
575C4—5841no—
576C4—5841no—
577C4#1798914211yescharacteristic
578C4#1798921411yescharacteristic
579C4#1798930311yescharacteristic
580C4#1798931111yescharacteristic
581C2—5842no—
582C2#2261361811yescharacteristic
583C2#2261971311yescharacteristic
584C2#2261975611yescharacteristic
585C2#2262178111yescharacteristic
586C2#2262218411yescharacteristic
587C2#2262262711yescharacteristic
588C2#2262268411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2550C2
socleC23×C73C22×D4
prime radicalC22×D292C23
hypercenter#2550D73
hypocenterC73#26028
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2550

Complements

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

NH ≅ G/Ntype
6. #22610889C2semidirect
7. #22610884C2semidirect
8. #22610699C2semidirect
9. #22610650C2semidirect
10. #22610520C2semidirect
11. #22610513C2semidirect
12. #22610061C2semidirect
15. C73×#2550C2semidirect
66. C4×D292Q8semidirect
295. #2550D73semidirect
310. C73#26028semidirect

Maximal subgroups

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

subgroupindex
#226129022
#226113162
#226113022
#226110742
#226110642
#226108892
#226108842
#226106992
#226106502
#226105202
#226105132
#226100612
#226099382
C4×#145565112
C73×#25502
#2602873