Finite Group Databasebeta

← #33103755

Subgroups — #33103755

Order 18688.

810 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
1C2×#17988628C211yesmaximal, characteristic
2#22612812C211yesmaximal, characteristic
3#22612796C211yesmaximal, characteristic
4#22610672C211yesmaximal, characteristic, complemented
5#22610659C211yesmaximal, characteristic, complemented
6#22610521C211yesmaximal, characteristic, complemented
7#22610516C211yesmaximal, characteristic, complemented
8#22610507C211yesmaximal, characteristic, complemented
9#22610502C211yesmaximal, characteristic, complemented
10#22610025C211yesmaximal, characteristic
11#22610020C211yesmaximal, characteristic
12#22609965C211yesmaximal, characteristic
13#22609830C211yesmaximal, characteristic, complemented
14#22609714C211yesmaximal, characteristic
15C4×#107×C73C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C2×#14556549C2211yescharacteristic
18C2×#14556548—21no—
19C2×C4⋊C4×D73C2211yescharacteristic
20#17988628C418yescomplemented
21C2×#14556524C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556503C2211yescharacteristic
26C2×#14556501C2211yescharacteristic
27#17988485C2211yescharacteristic
28#17988485C2211yescharacteristic
29#17988483C2211yescharacteristic
30#17988483C2211yescharacteristic
31#17988478—21no—
32#17988476—21no—
33#17988471—21no—
34#17988469—21no—
35#17988451—21no—
36#17988449—21no—
37#17988444C2211yescharacteristic
38#17988444C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C4×#11924005C2211yescharacteristic
42#17988433—21no—
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988342C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54C2×#107×C73C2211yescharacteristic
55C2×C4×C8×C73C2211yescharacteristic
56C4×C4⋊C4×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×C4×D73—21no—
61C22×C4×D73C2311yescharacteristic
62C2×#11924011—22no—
63C2×#11924011—22no—
64C2×#11924011C2×C414yes—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556549—22no—
68#14556549C2×C414yes—
69#14556548—44no—
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—
79C4⋊C4×D73—22no—
80C4⋊C4×D73C2×C414yes—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×#11924005—21no—
84C2×#11924005D411yescharacteristic
85C2×#11924005D411yescharacteristic
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73C2311yescharacteristic
90#14556524C2×C414yes—
91#14556523—22no—
92#14556523C2×C414yes—
93#14556522C2×C414yes—
94#14556503—22no—
95#14556501—22no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498D411yescharacteristic
105#14556498D411yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C42×C73—21no—
113C2×C42×C73—21no—
114C2×C42×C73C2311yescharacteristic
115#107×C73—22no—
116#107×C73C2×C414yes—
117C4×C8×C73—22no—
118#95×C73—21no—
119#95×C73—21no—
120C23×D73C2×D411yescharacteristic
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73C2×D411yescharacteristic
126C22×Dic73C2×D411yescharacteristic
127C22×Dic73C2×D411yescharacteristic
128C2×C4×D73—21no—
129C2×C4×D73—21no—
130C2×C4×D73—21no—
131C2×C4×D73—21no—
132C2×C4×D73—24no—
133C2×C4×D73—42no—
134C2×C4×D73—42no—
135C2×C4×D73C22×C412yes—
136C2×C4×D73#4112yes—
137#11924011—28no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831C22×C412yes—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73C22×C412yes—
151C4⋊Dic73C22×C412yes—
152#11924007—24no—
153#11924007—42no—
154#11924007—42no—
155#11924007—81no—
156#11924007—81no—
157#11924005—24no—
158#11924005—22no—
159#11924005—41no—
160#11924005—42no—
161#11924005—81no—
162C4×Dic73—21no—
163C4×Dic73—21no—
164C4×Dic73—24no—
165C4×Dic73—42no—
166C4×Dic73—42no—
167C22×C4×C73—21no—
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×C73C22×C412yes—
178C2×C8×C73#4112yes—
179C2×C8×C73#4112yes—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73C22×C412yes—
187C4⋊C4×C73C22×C412yes—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—42no—
191C42×C73—42no—
192C23×C73C22×D411yescharacteristic
193C22×D73—21no—
194C22×D73—21no—
195C22×D73—21no—
196C22×D73—21no—
197C22×D73—21no—
198C22×D73—21no—
199C22×D73C4×D412yes—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—22no—
203C2×C4×C73—22no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—41no—
208C2×C4×C73—41no—
209C2×C4×C73—41no—
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×C73AGL(1,ℤ/8)12yes—
218C2×C4×C73C23×C411yescharacteristic
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—22no—
227C2×Dic73—22no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
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×Dic73C4×D412yes—
245C4×D73—22no—
246C4×D73—22no—
247C4×D73—22no—
248C4×D73—22no—
249C4×D73—48no—
250C8×C73—41no—
251C8×C73#11714yescomplemented
252#10596831—22no—
253#10596831—41no—
254C22×C73C2×C4×D411yescharacteristic
255C22×C73#54711yescharacteristic
256C22×C73#54711yescharacteristic
257C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C73#57511yescharacteristic
259C22×C73#57711yescharacteristic
260C22×C73#57811yescharacteristic
261C2×D73—22no—
262C2×D73—22no—
263C2×D73—22no—
264C2×D73—41no—
265C2×D73—41no—
266C2×D73—41no—
267C2×D73—41no—
268C4×C73—22no—
269C4×C73—41no—
270C4×C73—42no—
271C4×C73—42no—
272C4×C73—81no—
273C4×C73—81no—
274C4×C73C2×#11711yescharacteristic
275C4×C73#51911yescharacteristic
276C4×C73#55211yescharacteristic
277C4×C73#55211yescharacteristic
278Dic73—22no—
279Dic73—41no—
280Dic73—41no—
281Dic73—42no—
282Dic73—42no—
283Dic73—81no—
284Dic73—81no—
285#22624—731nomaximal, Sylow
286C2×C73#234411yescharacteristic
287C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
288C2×C73#294711yescharacteristic
289C2×C73#334311yescharacteristic
290C2×C73#334811yescharacteristic
291C2×C73#335211yescharacteristic
292C2×C73#335411yescharacteristic
293D73—42no—
294C2×#428—731no—
295#2346—731no—
296#2344—731no—
297#1944—731no—
298#1941—731no—
299#1921—731no—
300#1920—731no—
301#1919—731no—
302#1918—731no—
303#1825—731no—
304#1824—731no—
305#1809—731no—
306#1776—731no—
307C4×#107D7311yescharacteristic, complemented
308#1753—731no—
309C73#2262411yescharacteristic, complemented, Sylow
310C22×M16—731no—
311C2×#117—731no—
312C2×#117—1461no—
313C22×C4⋊C4—731no—
314#428—738no—
315C2×#107—731no—
316C2×#107C2×D7311yescharacteristic
317C2×#107C2×D7311yescharacteristic
318C2×#106—731no—
319C2×#106—731no—
320C2×#106—731no—
321C2×C22⋊C8—731no—
322C2×#97—731no—
323C2×C4×C8C2×D7311yescharacteristic
324#389—731no—
325#389—731no—
326#389—731no—
327#389—731no—
328#388—1461no—
329#388—1461no—
330#387—1461no—
331#387—1461no—
332#382—1461no—
333#381—1461no—
334#380—731no—
335#380—731no—
336C4×C4⋊C4—731no—
337C4×C4⋊C4—731no—
338C4×C4⋊C4C2×D7311yescharacteristic
339C4×C4⋊C4C2×D7311yescharacteristic
340C4×C22⋊C4—731no—
341#376—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#336—731no—
354C23×C4—731no—
355C23×C4—1461no—
356C2×M16—734no—
357C2×M16—1462no—
358C2×M16—1462no—
359C22×C8—731no—
360C22×C8—731no—
361C22×C8C22×D7311yescharacteristic
362C22×C8C22×D7311yescharacteristic
363#117—2924no—
364#117—734no—
365#117—1462no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—734no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—1462no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C4⋊C4C22×D7311yescharacteristic
380C2×C4⋊C4C22×D7311yescharacteristic
381C2×C22⋊C4—731no—
382C2×C22⋊C4—731no—
383C2×C22⋊C4—1461no—
384C2×C22⋊C4—1461no—
385C2×C22⋊C4—1461no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—731no—
389C2×C42—1461no—
390C2×C42—1461no—
391C2×C42—1461no—
392C2×C42C22×D7311yescharacteristic
393#107—22no—
394#107—734no—
395#107C4×D7314yes—
396#106—734no—
397#106—734no—
398#106—1462no—
399C22⋊C8—1462no—
400#97—1462no—
401C4×C8—22no—
402#95—21no—
403#95—21no—
404#95—731no—
405#95—731no—
406#95—1461no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412#95—1461no—
413#95—1461no—
414C24—731no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—2922no—
420C22×C4—2922no—
421C22×C4—731no—
422C22×C4—732no—
423C22×C4—732no—
424C22×C4—731no—
425C22×C4—731no—
426C22×C4—1464no—
427C22×C4—1461no—
428C22×C4—1461no—
429C22×C4—1461no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4—1461no—
433C22×C4—1461no—
434C22×C4—1461no—
435C22×C4D4×D7311yescharacteristic
436C22×C4D4×D7311yescharacteristic
437C22×C4C23×D7311yescharacteristic
438M16—1468no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—732no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8C2×C4×D7312yes—
449C2×C8#1192402812yes—
450C2×C8#1192402812yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—2922no—
458C4⋊C4—2922no—
459C4⋊C4—5841no—
460C4⋊C4—5841no—
461C4⋊C4—732no—
462C4⋊C4—732no—
463C4⋊C4—1464no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4C2×C4×D7312yes—
471C4⋊C4C2×C4×D7312yes—
472C22⋊C4—2921no—
473C22⋊C4—2922no—
474C22⋊C4—5841no—
475C22⋊C4—1464no—
476C22⋊C4—1462no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42—42no—
481C42—2922no—
482C42—2922no—
483C42—1464no—
484C42—1461no—
485C42—1461no—
486C23—732no—
487C23—1461no—
488C23—1461no—
489C23—1461no—
490C23—1461no—
491C23—1461no—
492C23—1461no—
493C23C2×D4×D7311yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—2928no—
512C2×C4—2921no—
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—732no—
523C2×C4—1461no—
524C2×C4—1461no—
525C2×C4—1461no—
526C2×C4—1461no—
527C2×C4—1461no—
528C2×C4—1462no—
529C2×C4—1462no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1462no—
534C2×C4—1462no—
535C2×C4—1462no—
536C2×C4—1462no—
537C2×C4#1455661512yes—
538C2×C4C22×C4×D7311yescharacteristic
539C2×C4#41×D7311yescharacteristic
540C2×C4C2×#1192402811yescharacteristic
541C2×C4C2×#1192402811yescharacteristic
542C2×C4#1455663811yescharacteristic
543C2×C4#1455663911yescharacteristic
544C8—41no—
545C8—2921no—
546C8—1462no—
547C8#1455654814yescomplemented
548C22—2921no—
549C22—2921no—
550C22—2921no—
551C22—2921no—
552C22—1462no—
553C22—1462no—
554C22—1462no—
555C22C4×D4×D7311yescharacteristic
556C22#1798923811yescharacteristic
557C22#1798924211yescharacteristic
558C22C2×#1455661511yescharacteristic
559C22#1798938411yescharacteristic
560C22#1798939411yescharacteristic
561C22#1798940011yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—2921no—
569C4—2921no—
570C4—2922no—
571C4—2922no—
572C4—5841no—
573C4—5841no—
574C4—1462no—
575C4C2×#1455654811yescharacteristic
576C4#1798906311yescharacteristic
577C4#1798927611yescharacteristic
578C4#1798927811yescharacteristic
579C2—2922no—
580C2#2261279611yescharacteristic
581C2C4×#1455661511yescharacteristic
582C2#2261876911yescharacteristic
583C2#2262236711yescharacteristic
584C2#2262244211yescharacteristic
585C2#2262250711yescharacteristic
586C2#2262252811yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC4×#107×C73C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenterC4×#107D73
hypocenterC73#22624
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#107

Complements

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

NH ≅ G/Ntype
4. #22610672C2semidirect
5. #22610659C2semidirect
6. #22610521C2semidirect
7. #22610516C2semidirect
8. #22610507C2semidirect
9. #22610502C2semidirect
13. #22609830C2semidirect
15. C4×#107×C73C2semidirect
20. #17988628C4semidirect
251. C8×C73#117semidirect
307. C4×#107D73semidirect
309. C73#22624semidirect
547. C8#14556548semidirect

Maximal subgroups

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

subgroupindex
C2×#179886282
#226128122
#226127962
#226106722
#226106592
#226105212
#226105162
#226105072
#226105022
#226100252
#226100202
#226099652
#226098302
#226097142
C4×#107×C732
#2262473