Finite Group Databasebeta

← #33144178

Subgroups — #33144178

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#22614948C211yesmaximal, characteristic, complemented
2#22612903C211yesmaximal, characteristic
3#22611319C211yesmaximal, characteristic
4#22611303C211yesmaximal, characteristic
5#22611075C211yesmaximal, characteristic
6#22610885C211yesmaximal, characteristic, complemented
7#22610880C211yesmaximal, characteristic, complemented
8#22610698C211yesmaximal, characteristic, complemented
9#22610648C211yesmaximal, characteristic, complemented
10#22610510C211yesmaximal, characteristic, complemented
11#22610500C211yesmaximal, characteristic, complemented
12#22610060C211yesmaximal, characteristic, complemented
13#22609939C211yesmaximal, characteristic
14#102×Dic73C211yesmaximal, characteristic
15C73×#2023C211yesmaximal, characteristic
16C2×#14556582C2211yescharacteristic
17C2×D4×Dic73C2211yescharacteristic
18C2×#14556524C2211yescharacteristic
19C2×#14556523C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556520C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×C8×Dic73C2211yescharacteristic
26#17988517—21no—
27#17988506—21no—
28#17988494—21no—
29#17988490C2211yescharacteristic
30#17988485C2211yescharacteristic
31#17988483C2211yescharacteristic
32#17988482—21no—
33#17988481—21no—
34#17988475—21no—
35#17988452C2211yescharacteristic
36#17988452C2211yescharacteristic
37#17988448—21no—
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4×#11924007C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42C22⋊C4×Dic73—21no—
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×C4⋊Q8×C73C2211yescharacteristic
53C2×#102×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55#390×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C23×Dic73—21no—
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62#14556582—22no—
63#14556582—41no—
64D4×Dic73—22no—
65D4×Dic73Q814yescomplemented
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×#11924007D411yescharacteristic
75C2×#11924007D411yescharacteristic
76C2×C22⋊Dic73—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C4×Dic73—21no—
80C2×C4×Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73C2311yescharacteristic
83C2×C4×Dic73C2311yescharacteristic
84#14556524—22no—
85#14556523—22no—
86C4⋊#10596831—41no—
87#14556520—22no—
88#14556520—41no—
89#14556512—22no—
90#14556512—41no—
91C8×Dic73—22no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498D411yescharacteristic
102#14556498D411yescharacteristic
103C22×Q8×C73—21no—
104C22×D4×C73C2311yescharacteristic
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C4⋊Q8×C73—22no—
108C4⋊Q8×C73—41no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115C2×C42×C73C2311yescharacteristic
116#102×C73—22no—
117#102×C73—41no—
118C22×Dic73—21no—
119C22×Dic73—21no—
120C22×Dic73—24no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—81no—
124C22×Dic73—81no—
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—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—42no—
142C4⋊Dic73C2×Q812yes—
143C4⋊Dic73#4112yes—
144#11924007—22no—
145#11924007—41no—
146#11924007—42no—
147#11924007—42no—
148#11924007—81no—
149C22⋊Dic73—24no—
150C22⋊Dic73—42no—
151C22⋊Dic73—81no—
152C4×Dic73—21no—
153C4×Dic73—21no—
154C4×Dic73—81no—
155C4×Dic73—81no—
156C4×Dic73—81no—
157C4×Dic73C2×Q812yes—
158C4×Dic73#4112yes—
159C24×C73—21no—
160C2×Q8×C73—42no—
161C2×Q8×C73—42no—
162C2×Q8×C73—81no—
163C2×D4×C73—21no—
164C2×D4×C73—21no—
165C2×D4×C73—21no—
166C2×D4×C73—21no—
167C2×D4×C73—21no—
168C2×D4×C73C2×Q812yes—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2411yescharacteristic
175C2×C8×C73—21no—
176C2×C8×C73—21no—
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—42no—
187C4⋊C4×C73#4112yes—
188C22⋊C4×C73—42no—
189C22⋊C4×C73—81no—
190C42×C73—21no—
191C42×C73—21no—
192C23×C73—22no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73—41no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73C22×D411yescharacteristic
200Q8×C73—84no—
201D4×C73—22no—
202D4×C73—22no—
203D4×C73—41no—
204D4×C73—41no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—22no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
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×C73—41no—
219C2×C4×C73#13512yes—
220C2×C4×C73C22×Q811yescharacteristic
221C2×C4×C73C2×#4111yescharacteristic
222C2×C4×C73C2×#4111yescharacteristic
223C2×C4×C73C2×#4111yescharacteristic
224C2×C4×C73H(5,𝔽2)11yescharacteristic
225C2×C4×C732−1+411yescharacteristic
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—22no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—22no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—84no—
249C2×Dic73C22⋊Q812yes—
250C2×Dic73C22⋊Q812yes—
251C8×C73—41no—
252C8×C73—41no—
253#10596831—41no—
254#10596831—41no—
255C22×C73—42no—
256C22×C73—42no—
257C22×C73—42no—
258C22×C73—81no—
259C22×C73—81no—
260C22×C73—81no—
261C22×C73—81no—
262C22×C73C2×C22⋊Q811yescharacteristic
263C22×C73#54611yescharacteristic
264C22×C73#54811yescharacteristic
265C22×C73C2×#13511yescharacteristic
266C22×C73#57511yescharacteristic
267C22×C73#57611yescharacteristic
268C22×C73#57711yescharacteristic
269C4×C73—22no—
270C4×C73—41no—
271C4×C73—42no—
272C4×C73—81no—
273C4×C73—81no—
274C4×C73#53311yescharacteristic
275C4×C73#54411yescharacteristic
276C4×C73#55511yescharacteristic
277C4×C73#55611yescharacteristic
278Dic73—22no—
279Dic73—22no—
280Dic73—42no—
281Dic73—42no—
282Dic73—41no—
283Dic73—41no—
284Dic73—81no—
285Dic73—81no—
286#26030—731nomaximal, Sylow
287C2×C73—82no—
288C2×C73#243911yescharacteristic
289C2×C73#307611yescharacteristic
290C2×C73#308111yescharacteristic
291C2×C73#330111yescharacteristic
292C2×C73#332211yescharacteristic
293C2×C73#336111yescharacteristic
294C2×C73#337111yescharacteristic
295#2550—731no—
296#2357—731no—
297#2067—731no—
298#2065—731no—
299#2025—731no—
300#2023D7311yescharacteristic
301#1989—731no—
302#1987—731no—
303#1949—731no—
304#1939—731no—
305#1919—731no—
306#1918—731no—
307#1835—731no—
308#1804—731no—
309C4×#102—731no—
310C73#2603011yescharacteristic, complemented, Sylow
311C2×C4⋊Q8C2×D7311yescharacteristic
312C2×#125—731no—
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×#102C2×D7311yescharacteristic
322C2×#102C2×D7311yescharacteristic
323C2×C4×C8—731no—
324#398—1461no—
325#395—1461no—
326#391—1461no—
327#390—731no—
328#390C2×D7311yescharacteristic
329#389—731no—
330#389—731no—
331#389—1461no—
332#388—1461no—
333#388—1461no—
334#382—731no—
335#382—731no—
336#381—1461no—
337C4×C4⋊C4—731no—
338C4×C4⋊C4—731no—
339C4×C4⋊C4—731no—
340C4×C4⋊C4—731no—
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#340C2×D7311yescharacteristic
351#340C2×D7311yescharacteristic
352#336—731no—
353#336C2×D7311yescharacteristic
354C22×Q8—21no—
355C22×D4C22×D7311yescharacteristic
356C23×C4—1461no—
357C22×C8—731no—
358C22×C8—731no—
359C22×C8C22×D7311yescharacteristic
360C22×C8C22×D7311yescharacteristic
361C4⋊Q8—22no—
362C4⋊Q8—41no—
363#125—2921no—
364#125—1462no—
365C4×D4—734no—
366C4×D4—1462no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
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—731no—
387C2×C42—731no—
388C2×C42—1461no—
389C2×C42—1461no—
390C2×C42—1461no—
391C2×C42C22×D7311yescharacteristic
392#107—1462no—
393#106—1462no—
394C4⋊C8—2921no—
395C4⋊C8—2921no—
396C4⋊C8—1462no—
397#102—22no—
398#102—41no—
399#102—2921no—
400#102—1462no—
401C4×C8—1462no—
402#95—731no—
403#95—731no—
404#95—1461no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412#95—1461no—
413C24—21no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—81no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4Q8×D7312yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—5841no—
427C22×C4—5841no—
428C22×C4—731no—
429C22×C4—731no—
430C22×C4—731no—
431C22×C4—731no—
432C22×C4—1461no—
433C22×C4—1461no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1464no—
437C22×C4D4×D7311yescharacteristic
438C22×C4D4×D7311yescharacteristic
439C22×C4C23×D7311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8—1461no—
449C2×C8—1461no—
450C2×C8—1461no—
451C2×C8—1461no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—2922no—
459C4⋊C4—2922no—
460C4⋊C4—2921no—
461C4⋊C4—2922no—
462C4⋊C4—5841no—
463C4⋊C4—732no—
464C4⋊C4—732no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1462no—
470C4⋊C4—1461no—
471C4⋊C4—1461no—
472C4⋊C4#1192402712yes—
473C22⋊C4—42no—
474C22⋊C4—81no—
475C22⋊C4—2922no—
476C22⋊C4—5841no—
477C22⋊C4—1464no—
478C42—21no—
479C42—21no—
480C42—5841no—
481C42—5841no—
482C42—5841no—
483C42—732no—
484C42—732no—
485C42—1461no—
486C42—1461no—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C2×D4×D7311yescharacteristic
495Q8—84no—
496D4—22no—
497D4—22no—
498D4—41no—
499D4—41no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
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—5844no—
525C2×C4—732no—
526C2×C4—732no—
527C2×C4—1461no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1462no—
533C2×C4—1462no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4—1461no—
539C2×C4#1455661212yes—
540C2×C4C2×Q8×D7311yescharacteristic
541C2×C4#41×D7311yescharacteristic
542C2×C4#41×D7311yescharacteristic
543C2×C4C2×#1192402711yescharacteristic
544C2×C4#1455663611yescharacteristic
545C2×C4#1455663811yescharacteristic
546C8—41no—
547C8—41no—
548C8—2921no—
549C8—2921no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C22⋊Q8×D7311yescharacteristic
558C22#1798922411yescharacteristic
559C22#1798924511yescharacteristic
560C22C2×#1455661211yescharacteristic
561C22#1798938311yescharacteristic
562C22#1798938811yescharacteristic
563C22#1798939611yescharacteristic
564C4—22no—
565C4—41no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—2922no—
570C4—2922no—
571C4—2921no—
572C4—2921no—
573C4—5841no—
574C4—5841no—
575C4—1462no—
576C4—1462no—
577C4#1798913911yescharacteristic
578C4#1798921611yescharacteristic
579C4#1798929911yescharacteristic
580C4#1798930611yescharacteristic
581C2—82no—
582C2#2261361511yescharacteristic
583C2#2261972011yescharacteristic
584C2#2261975511yescharacteristic
585C2#2262183711yescharacteristic
586C2#2262213611yescharacteristic
587C2#2262259611yescharacteristic
588C2#2262269811yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2023C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2023D73
hypocenterC73#26030
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. #22614948C2semidirect
6. #22610885C2semidirect
7. #22610880C2semidirect
8. #22610698C2semidirect
9. #22610648C2semidirect
10. #22610510C2semidirect
11. #22610500C2semidirect
12. #22610060C2semidirect
65. D4×Dic73Q8semidirect
310. C73#26030semidirect

Maximal subgroups

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

subgroupindex
#226149482
#226129032
#226113192
#226113032
#226110752
#226108852
#226108802
#226106982
#226106482
#226105102
#226105002
#226100602
#226099392
#102×Dic732
C73×#20232
#2603073