Finite Group Databasebeta

← #33131139

Subgroups — #33131139

Order 18688.

706 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614851C211yesmaximal, characteristic
2#22612889C211yesmaximal, characteristic, complemented
3#22611504C211yesmaximal, characteristic, complemented
4#22611500C211yesmaximal, characteristic, complemented
5#22611347C211yesmaximal, characteristic
6#22611344C211yesmaximal, characteristic
7#22610865C211yesmaximal, characteristic, complemented
8#22610691C211yesmaximal, characteristic, complemented
9#22610561C211yesmaximal, characteristic
10#22610518C211yesmaximal, characteristic, complemented
11#22610516C211yesmaximal, characteristic, complemented
12#22610057C211yesmaximal, characteristic, complemented
13#22609929C211yesmaximal, characteristic
14#102×Dic73C211yesmaximal, characteristic
15C73×#1926C211yesmaximal, characteristic
16C2×#14556568—21no—
17C2×C4×Dic146C2211yescharacteristic
18C2×D4×Dic73C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×#14556520C2211yescharacteristic
21C2×#14556512C2211yescharacteristic
22C2×#14556512C2211yescharacteristic
23C2×C8×Dic73C2211yescharacteristic
24#17988510—21no—
25#17988510—21no—
26#17988491—21no—
27#17988490C2211yescharacteristic
28#17988486C2211yescharacteristic
29#17988485C2211yescharacteristic
30#17988471—21no—
31#17988470—21no—
32#17988456C2211yescharacteristic
33#17988456C2211yescharacteristic
34#17988453C2211yescharacteristic
35#17988452C2211yescharacteristic
36#17988445—21no—
37C4⋊C4×Dic73C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4×#11924007—21no—
40#17988350C2211yescharacteristic
41#17988350C2211yescharacteristic
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988342C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×#106×C73C2211yescharacteristic
53C2×#102×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55#390×C73C2211yescharacteristic
56C4×C4⋊C4×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×Dic146D411yescharacteristic
60C23×Dic73—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63#14556568—44no—
64C4×Dic146—22no—
65C4×Dic146—22no—
66D4×Dic73—22no—
67D4×Dic73—22no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73D411yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007D411yescharacteristic
80C2×#11924007D411yescharacteristic
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C22⋊Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73C2311yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88#14556522—22no—
89#14556520—22no—
90#14556512—22no—
91#14556512—22no—
92C8×Dic73—22no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101C22×D4×C73C2311yescharacteristic
102C22×C8×C73C2311yescharacteristic
103C22×C8×C73C2311yescharacteristic
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C4⋊C4×C73C2311yescharacteristic
106C2×C4⋊C4×C73C2311yescharacteristic
107C2×C22⋊C4×C73—21no—
108C2×C22⋊C4×C73—21no—
109C2×C42×C73—21no—
110C2×C42×C73C2311yescharacteristic
111#106×C73—22no—
112#102×C73—22no—
113#102×C73—22no—
114#95×C73—21no—
115C2×Dic146—22no—
116C2×Dic146—22no—
117C2×Dic146—22no—
118C22×Dic73—21no—
119C22×Dic73—21no—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—42no—
123C22×Dic73—42no—
124C22×Dic73—42no—
125C22×Dic73C2×D411yescharacteristic
126C22×Dic73C2×D411yescharacteristic
127C22×Dic73C2×D411yescharacteristic
128C22×Dic73C2×D411yescharacteristic
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C4⋊Dic73—22no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—42no—
145#11924007—22no—
146#11924007—22no—
147#11924007—42no—
148#11924007—42no—
149#11924007—42no—
150#11924007—81no—
151#11924007—81no—
152C22⋊Dic73—42no—
153C22⋊Dic73—42no—
154C22⋊Dic73—81no—
155C4×Dic73—21no—
156C4×Dic73—21no—
157C4×Dic73—42no—
158C4×Dic73—81no—
159C4×Dic73#4112yes—
160C4×Dic73#4112yes—
161C24×C73—21no—
162C2×D4×C73—21no—
163C2×D4×C73—21no—
164C2×D4×C73—21no—
165C2×D4×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73#4112yes—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2411yescharacteristic
174C2×C8×C73—21no—
175C2×C8×C73—21no—
176C2×C8×C73—21no—
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73#4112yes—
186C22⋊C4×C73—42no—
187C22⋊C4×C73—81no—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—81no—
191C23×C73—22no—
192C23×C73—41no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73—41no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73C22×D411yescharacteristic
199D4×C73—22no—
200D4×C73—22no—
201D4×C73—41no—
202D4×C73—41no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—22no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—41no—
211C2×C4×C73—41no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73C2×#4111yescharacteristic
219C2×C4×C73C2×#4111yescharacteristic
220C2×C4×C73C2×#4111yescharacteristic
221C2×C4×C73C2×#4111yescharacteristic
222C2×C4×C732−1+411yescharacteristic
223C2×C4×C732−1+411yescharacteristic
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73—41no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—84no—
245C2×Dic73#12312yes—
246C2×Dic73#12312yes—
247C2×Dic73#13512yes—
248C2×Dic73AGL(1,ℤ/8)12yes—
249Dic146—44no—
250C8×C73—41no—
251C8×C73—41no—
252#10596831—41no—
253#10596831—41no—
254C22×C73—42no—
255C22×C73—42no—
256C22×C73—42no—
257C22×C73—81no—
258C22×C73—81no—
259C22×C73—81no—
260C22×C73—81no—
261C22×C73C2×#12311yescharacteristic
262C22×C73#54811yescharacteristic
263C22×C73#54811yescharacteristic
264C22×C73C2×#13511yescharacteristic
265C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C73#57711yescharacteristic
267C22×C73#57711yescharacteristic
268C4×C73—41no—
269C4×C73—41no—
270C4×C73—42no—
271C4×C73—81no—
272C4×C73—81no—
273C4×C73#52911yescharacteristic
274C4×C73#54111yescharacteristic
275C4×C73#55211yescharacteristic
276C4×C73#55611yescharacteristic
277Dic73—22no—
278Dic73—41no—
279Dic73—41no—
280Dic73—81no—
281Dic73—81no—
282Dic73—81no—
283Dic73—81no—
284Dic73#48214yes—
285#24991—731nomaximal, Sylow
286C2×C73—82no—
287C2×C73#242111yescharacteristic
288C2×C73C2×#48211yescharacteristic
289C2×C73#309911yescharacteristic
290C2×C73#336011yescharacteristic
291C2×C73#336011yescharacteristic
292C2×C73#336811yescharacteristic
293C2×C73#337311yescharacteristic
294#2538—731no—
295#2355—731no—
296#2101—731no—
297#2101—731no—
298#2073—731no—
299#2073—731no—
300#1984—731no—
301#1948—731no—
302#1926—731no—
303#1926D7311yescharacteristic
304#1920—731no—
305#1920—731no—
306#1834—731no—
307#1802—731no—
308C4×#102—731no—
309C73#2499111yescharacteristic, complemented, Sylow
310C2×C22⋊Q8—1461no—
311C2×C4×Q8—731no—
312C2×C4×D4—731no—
313C2×#106—731no—
314C2×#106C2×D7311yescharacteristic
315C2×C4⋊C8—731no—
316C2×#102—731no—
317C2×#102—731no—
318C2×#102C2×D7311yescharacteristic
319C2×#102C2×D7311yescharacteristic
320C2×C4×C8—731no—
321#396—1461no—
322#396—1461no—
323#391—1461no—
324#390—731no—
325#390C2×D7311yescharacteristic
326#389—731no—
327#389—731no—
328#387—1461no—
329#387—1461no—
330#384—731no—
331#384—731no—
332#382—731no—
333#382—731no—
334#380—1461no—
335C4×C4⋊C4—731no—
336C4×C4⋊C4—731no—
337C4×C4⋊C4—1461no—
338C4×C4⋊C4C2×D7311yescharacteristic
339#340—731no—
340#340—731no—
341#340—731no—
342#340—731no—
343#340—731no—
344#340—731no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340C2×D7311yescharacteristic
349#336—731no—
350#336—731no—
351#336—731no—
352#336C2×D7311yescharacteristic
353C22×Q8—731no—
354C22×D4C22×D7311yescharacteristic
355C23×C4—1461no—
356C22×C8—731no—
357C22×C8—731no—
358C22×C8C22×D7311yescharacteristic
359C22×C8C22×D7311yescharacteristic
360C22⋊Q8—2924no—
361C4×Q8—1462no—
362C4×Q8—1462no—
363C4×D4—1462no—
364C4×D4—1462no—
365C2×C4⋊C4—731no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C4⋊C4C22×D7311yescharacteristic
380C2×C4⋊C4C22×D7311yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1461no—
384C2×C22⋊C4—1461no—
385C2×C22⋊C4—1461no—
386C2×C42—21no—
387C2×C42—731no—
388C2×C42—731no—
389C2×C42—1461no—
390C2×C42—1461no—
391C2×C42C22×D7311yescharacteristic
392#106—22no—
393#106—1462no—
394C4⋊C8—1462no—
395#102—22no—
396#102—22no—
397#102—1462no—
398#102—1462no—
399C4×C8—1462no—
400#95—21no—
401#95—1461no—
402#95—1461no—
403#95—1461no—
404#95—1461no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408#95—1461no—
409C24—21no—
410C2×Q8—1462no—
411C2×Q8—1462no—
412C2×Q8—1462no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4#1192402812yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—2922no—
423C22×C4—2922no—
424C22×C4—2922no—
425C22×C4—731no—
426C22×C4—731no—
427C22×C4—731no—
428C22×C4—731no—
429C22×C4—1461no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4—1461no—
433C22×C4D4×D7311yescharacteristic
434C22×C4D4×D7311yescharacteristic
435C22×C4C23×D7311yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—2922no—
454C4⋊C4—2922no—
455C4⋊C4—2922no—
456C4⋊C4—2922no—
457C4⋊C4—5841no—
458C4⋊C4—5841no—
459C4⋊C4—1462no—
460C4⋊C4—1462no—
461C4⋊C4—1462no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4#1192402712yes—
471C22⋊C4—42no—
472C22⋊C4—81no—
473C22⋊C4—2922no—
474C22⋊C4—2922no—
475C22⋊C4—5841no—
476C42—21no—
477C42—21no—
478C42—81no—
479C42—2922no—
480C42—5841no—
481C42—732no—
482C42—732no—
483C42—1461no—
484C42—1461no—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C2×D4×D7311yescharacteristic
493Q8—2924no—
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—2921no—
514C2×C4—2921no—
515C2×C4—2921no—
516C2×C4—2921no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—5844no—
526C2×C4—732no—
527C2×C4—732no—
528C2×C4—732no—
529C2×C4—732no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4#41×D7311yescharacteristic
539C2×C4#41×D7311yescharacteristic
540C2×C4C2×#1192402711yescharacteristic
541C2×C4C2×#1192402811yescharacteristic
542C2×C4#1455663811yescharacteristic
543C2×C4#1455663811yescharacteristic
544C8—41no—
545C8—41no—
546C8—2921no—
547C8—2921no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22#123×D7311yescharacteristic
556C22#1798924311yescharacteristic
557C22#1798924611yescharacteristic
558C22#135×D7311yescharacteristic
559C22AGL(1,ℤ/8)×D7311yescharacteristic
560C22#1798939311yescharacteristic
561C22#1798939711yescharacteristic
562C4—41no—
563C4—42no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—2921no—
568C4—2921no—
569C4—5841no—
570C4—5841no—
571C4—5841no—
572C4—5841no—
573C4—734no—
574C4—1462no—
575C4#1798911711yescharacteristic
576C4#1798919411yescharacteristic
577C4#1798927011yescharacteristic
578C4#1798930611yescharacteristic
579C2—82no—
580C2#2261343111yescharacteristic
581C2#482×D7311yescharacteristic
582C2#2261992111yescharacteristic
583C2#2262257511yescharacteristic
584C2#2262258111yescharacteristic
585C2#2262265511yescharacteristic
586C2#2262272411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1926C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#1926D73
hypocenterC73#24991
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1926

Complements

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

NH ≅ G/Ntype
2. #22612889C2semidirect
3. #22611504C2semidirect
4. #22611500C2semidirect
7. #22610865C2semidirect
8. #22610691C2semidirect
10. #22610518C2semidirect
11. #22610516C2semidirect
12. #22610057C2semidirect
309. C73#24991semidirect

Maximal subgroups

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

subgroupindex
#226148512
#226128892
#226115042
#226115002
#226113472
#226113442
#226108652
#226106912
#226105612
#226105182
#226105162
#226100572
#226099292
#102×Dic732
C73×#19262
#2499173