Finite Group Databasebeta

← #33103751

Subgroups — #33103751

Order 18688.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988630C211yesmaximal, characteristic
2#22612806C211yesmaximal, characteristic
3#22610671C211yesmaximal, characteristic, complemented
4#22610656C211yesmaximal, characteristic, complemented
5#22610523C211yesmaximal, characteristic, complemented
6#22610513C211yesmaximal, characteristic, complemented
7#22610506C211yesmaximal, characteristic, complemented
8#22610502C211yesmaximal, characteristic, complemented
9#22610026C211yesmaximal, characteristic
10#22610015C211yesmaximal, characteristic
11#22609962C211yesmaximal, characteristic
12#22609829C211yesmaximal, characteristic, complemented
13C4×#14556524C211yesmaximal, characteristic, complemented
14#22609715C211yesmaximal, characteristic
15C73×#2347C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C22×C4⋊Dic73C2211yescharacteristic
18#17988630C418yescomplemented
19C2×#14556524C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×C22⋊#10596831C2211yescharacteristic
26C2×#14556502C2211yescharacteristic
27C2×C4×#10596831C2211yescharacteristic
28#17988485C2211yescharacteristic
29#17988485C2211yescharacteristic
30#17988470—21no—
31#17988470—21no—
32#17988447C2211yescharacteristic
33#17988445—21no—
34C4×C4⋊Dic73C2211yescharacteristic
35C4×C4⋊Dic73C2211yescharacteristic
36C22⋊C4×Dic73—21no—
37#17988349C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49C2×#117×C73C2211yescharacteristic
50#389×C73C2211yescharacteristic
51#389×C73C2211yescharacteristic
52#388×C73—21no—
53#388×C73—21no—
54#383×C73—21no—
55#380×C73C2211yescharacteristic
56C4×C4⋊C4×C73—21no—
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C4⋊C4×C73C2211yescharacteristic
59C4×C22⋊C4×C73C2211yescharacteristic
60C23×Dic73—21no—
61C2×#11924012—22no—
62C2×#11924012—22no—
63C2×#11924012C2×C414yes—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—22no—
70C2×C4⋊Dic73C2×C414yes—
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×C22⋊Dic73—21no—
78C2×C4×Dic73—21no—
79C2×C4×Dic73—21no—
80#14556524—22no—
81#14556524C2×C414yes—
82#14556524C2×C414yes—
83#14556522—22no—
84#14556522C2×C414yes—
85#14556522C2×C414yes—
86C22⋊#10596831—22no—
87#14556502—22no—
88C4×#10596831—22no—
89#14556498—21no—
90#14556498—21no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94C23×C4×C73C2311yescharacteristic
95#117×C73—22no—
96#117×C73C2×C414yes—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73C2311yescharacteristic
100C2×C4⋊C4×C73C2311yescharacteristic
101C2×C4⋊C4×C73C2311yescharacteristic
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C22⋊C4×C73—21no—
104C2×C22⋊C4×C73—21no—
105C2×C22⋊C4×C73D411yescharacteristic
106C2×C22⋊C4×C73D411yescharacteristic
107C2×C42×C73—21no—
108C2×C42×C73—21no—
109C2×C42×C73—21no—
110C2×C42×C73C2311yescharacteristic
111C2×C42×C73C2311yescharacteristic
112#95×C73—21no—
113#95×C73—21no—
114#95×C73—21no—
115#95×C73—21no—
116#95×C73—21no—
117#95×C73D411yescharacteristic
118#95×C73D411yescharacteristic
119C22×Dic73—21no—
120C22×Dic73—21no—
121C22×Dic73—24no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—81no—
125C22×Dic73—81no—
126#11924012—28no—
127C2×#10596831—21no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831C22×C412yes—
136C2×#10596831C22×C412yes—
137C2×#10596831#4112yes—
138C2×#10596831#4112yes—
139C4⋊Dic73—24no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73C22×C412yes—
147C4⋊Dic73C22×C412yes—
148#11924007—81no—
149#11924007—81no—
150C22⋊Dic73—42no—
151C4×Dic73—81no—
152C4×Dic73—81no—
153C24×C73C2×D411yescharacteristic
154C22×C4×C73—21no—
155C22×C4×C73—21no—
156C22×C4×C73—21no—
157C22×C4×C73—21no—
158C22×C4×C73—21no—
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73C22×C412yes—
163C22×C4×C73C2×D411yescharacteristic
164C22×C4×C73C2×D411yescharacteristic
165C22×C4×C73C2×D411yescharacteristic
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73#4112yes—
169C22×C4×C73C2411yescharacteristic
170C4⋊C4×C73—21no—
171C4⋊C4×C73—21no—
172C4⋊C4×C73—21no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—42no—
177C4⋊C4×C73—42no—
178C4⋊C4×C73C22×C412yes—
179C4⋊C4×C73C22×C412yes—
180C22⋊C4×C73—24no—
181C22⋊C4×C73—22no—
182C22⋊C4×C73—41no—
183C22⋊C4×C73—81no—
184C42×C73—24no—
185C42×C73—21no—
186C42×C73—21no—
187C42×C73—21no—
188C42×C73—21no—
189C42×C73—81no—
190C42×C73—81no—
191C23×C73—21no—
192C23×C73—21no—
193C23×C73—21no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73—21no—
197C23×C73C4×D412yes—
198C23×C73C22×D411yescharacteristic
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—22no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
208C2×C4×C73—22no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
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×C73C4×D412yes—
229C2×C4×C73#13712yes—
230C2×C4×C73C23×C411yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73C2×#4111yescharacteristic
234C2×C4×C732−1+411yescharacteristic
235C2×C4×C732−1+411yescharacteristic
236C2×Dic73—22no—
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—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—84no—
249#10596831—22no—
250#10596831—41no—
251#10596831—41no—
252#10596831#11714yes—
253C22×C73—22no—
254C22×C73—22no—
255C22×C73—22no—
256C22×C73—41no—
257C22×C73—41no—
258C22×C73—41no—
259C22×C73—41no—
260C22×C73C2×C4×D411yescharacteristic
261C22×C73#54711yescharacteristic
262C22×C73#54711yescharacteristic
263C22×C73C2×#13711yescharacteristic
264C22×C73#57511yescharacteristic
265C22×C73#57611yescharacteristic
266C22×C73#57711yescharacteristic
267C4×C73—22no—
268C4×C73—22no—
269C4×C73—41no—
270C4×C73—41no—
271C4×C73—41no—
272C4×C73—42no—
273C4×C73—42no—
274C4×C73—81no—
275C4×C73—81no—
276C4×C73C2×#11711yescharacteristic
277C4×C73#51911yescharacteristic
278C4×C73#55211yescharacteristic
279C4×C73#55211yescharacteristic
280Dic73—42no—
281Dic73—42no—
282Dic73—81no—
283Dic73—81no—
284#22623—731nomaximal, Sylow
285C2×C73—42no—
286C2×C73#234411yescharacteristic
287C2×C73C4×#13711yescharacteristic
288C2×C73#294611yescharacteristic
289C2×C73#334611yescharacteristic
290C2×C73#335011yescharacteristic
291C2×C73#335511yescharacteristic
292C2×C73#335611yescharacteristic
293C2×#428—731no—
294#2347D7311yescharacteristic
295#2345—731no—
296#1944—731no—
297#1941—731no—
298#1921—731no—
299#1920—731no—
300#1919—731no—
301#1918—731no—
302#1825—731no—
303#1823—731no—
304#1809—731no—
305#1776—731no—
306C4×#107—731no—
307#1753—731no—
308C73#2262311yescharacteristic, complemented, Sylow
309C22×M16—731no—
310C2×#117C2×D7311yescharacteristic
311C22×C4⋊C4—731no—
312#428—738no—
313C2×#107—731no—
314C2×#107—731no—
315C2×#107—731no—
316C2×#106—731no—
317C2×#106—731no—
318C2×#106—731no—
319C2×C22⋊C8—731no—
320C2×#97—731no—
321C2×C4×C8—731no—
322#389—731no—
323#389—731no—
324#389C2×D7311yescharacteristic
325#389C2×D7311yescharacteristic
326#388—21no—
327#388—21no—
328#387—1461no—
329#387—1461no—
330#383—21no—
331#380—731no—
332#380—1461no—
333#380C2×D7311yescharacteristic
334C4×C4⋊C4—21no—
335C4×C4⋊C4—731no—
336C4×C4⋊C4—731no—
337C4×C4⋊C4C2×D7311yescharacteristic
338C4×C4⋊C4C2×D7311yescharacteristic
339C4×C22⋊C4—1461no—
340C4×C22⋊C4C2×D7311yescharacteristic
341#340—731no—
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#336—731no—
352#336—731no—
353C23×C4—1461no—
354C23×C4C22×D7311yescharacteristic
355C2×M16—734no—
356C2×M16—1462no—
357C2×M16—1462no—
358C22×C8—731no—
359C22×C8—731no—
360C22×C8—731no—
361C22×C8—731no—
362#117—22no—
363#117C4×D7314yes—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—734no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1462no—
375C2×C4⋊C4C22×D7311yescharacteristic
376C2×C4⋊C4C22×D7311yescharacteristic
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—1461no—
382C2×C22⋊C4C22⋊D7311yescharacteristic
383C2×C22⋊C4C22⋊D7311yescharacteristic
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—1461no—
388C2×C42—1461no—
389C2×C42C22×D7311yescharacteristic
390C2×C42C22×D7311yescharacteristic
391#107—734no—
392#107—734no—
393#107—1462no—
394#106—734no—
395#106—734no—
396#106—1462no—
397C22⋊C8—1462no—
398#97—1462no—
399C4×C8—1462no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95C22⋊D7311yescharacteristic
411#95C22⋊D7311yescharacteristic
412C24C2×C22⋊D7311yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—5841no—
422C22×C4—5841no—
423C22×C4—1461no—
424C22×C4—1461no—
425C22×C4—1461no—
426C22×C4—1461no—
427C22×C4—1464no—
428C22×C4C2×C4×D7312yes—
429C22×C4C2×C22⋊D7311yescharacteristic
430C22×C4C2×C22⋊D7311yescharacteristic
431C22×C4C2×C22⋊D7311yescharacteristic
432C22×C4C2×C22⋊D7311yescharacteristic
433C22×C4C2×C22⋊D7311yescharacteristic
434C22×C4#1192402812yes—
435C22×C4C23×D7311yescharacteristic
436M16—1468no—
437C2×C8—732no—
438C2×C8—732no—
439C2×C8—732no—
440C2×C8—732no—
441C2×C8—1461no—
442C2×C8—1461no—
443C2×C8—1461no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8—1461no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—5841no—
458C4⋊C4—5841no—
459C4⋊C4—732no—
460C4⋊C4—732no—
461C4⋊C4—1461no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1464no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4C2×C4×D7312yes—
469C4⋊C4C2×C4×D7312yes—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—41no—
473C22⋊C4—81no—
474C22⋊C4—2922no—
475C42—24no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—81no—
481C42—81no—
482C42—5841no—
483C42—5841no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23C4×C22⋊D7312yes—
491C23C22×C22⋊D7311yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
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—22no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2921no—
531C2×C4—5844no—
532C2×C4—1462no—
533C2×C4—1462no—
534C2×C4C4×C22⋊D7312yes—
535C2×C4#1455662112yes—
536C2×C4C22×C4×D7311yescharacteristic
537C2×C4#41×D7311yescharacteristic
538C2×C4#41×D7311yescharacteristic
539C2×C4C2×#1192402811yescharacteristic
540C2×C4#1455663811yescharacteristic
541C2×C4#1455663811yescharacteristic
542C8—2921no—
543C8—2921no—
544C8—734no—
545C8—1462no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22C2×C4×C22⋊D7311yescharacteristic
554C22#1798923611yescharacteristic
555C22#1798923611yescharacteristic
556C22C2×#1455662111yescharacteristic
557C22#1798938511yescharacteristic
558C22#1798939011yescharacteristic
559C22#1798939511yescharacteristic
560C4—22no—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—42no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—2922no—
570C4—2922no—
571C4—5841no—
572C4—5841no—
573C4#117×D7311yescharacteristic
574C4#1798906411yescharacteristic
575C4#1798927011yescharacteristic
576C4#1798927011yescharacteristic
577C2—42no—
578C2#2261280111yescharacteristic
579C2C4×#1455662111yescharacteristic
580C2#2261875911yescharacteristic
581C2#2262241611yescharacteristic
582C2#2262247611yescharacteristic
583C2#2262253111yescharacteristic
584C2#2262254011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2347C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#2347D73
hypocenterC73#22623
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2347

Complements

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

NH ≅ G/Ntype
3. #22610671C2semidirect
4. #22610656C2semidirect
5. #22610523C2semidirect
6. #22610513C2semidirect
7. #22610506C2semidirect
8. #22610502C2semidirect
12. #22609829C2semidirect
13. C4×#14556524C2semidirect
18. #17988630C4semidirect
308. C73#22623semidirect

Maximal subgroups

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

subgroupindex
C2×#179886302
#226128062
#226106712
#226106562
#226105232
#226105132
#226105062
#226105022
#226100262
#226100152
#226099622
#226098292
C4×#145565242
#226097152
C73×#23472
#2262373