Finite Group Databasebeta

← #33103709

Subgroups — #33103709

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×#17988627C211yesmaximal, characteristic
2#22612818C211yesmaximal, characteristic
3#22612808C211yesmaximal, characteristic
4#22610668C211yesmaximal, characteristic, complemented
5#22610663C211yesmaximal, characteristic, complemented
6#22610521C211yesmaximal, characteristic, complemented
7#22610519C211yesmaximal, characteristic, complemented
8#22610512C211yesmaximal, characteristic, complemented
9#22610500C211yesmaximal, characteristic, complemented
10#22610027C211yesmaximal, characteristic
11#22609962C211yesmaximal, characteristic
12#22609827C211yesmaximal, characteristic, complemented
13#22609819C211yesmaximal, characteristic, complemented
14#22609717C211yesmaximal, characteristic
15C73×#1823C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C2×#14556550C2211yescharacteristic
18#17988627C418yescomplemented
19C2×#14556524C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556501C2211yescharacteristic
26C2×C8×Dic73C2211yescharacteristic
27#17988486C2211yescharacteristic
28#17988486C2211yescharacteristic
29#17988483C2211yescharacteristic
30#17988483C2211yescharacteristic
31#17988481—21no—
32#17988481—21no—
33#17988473—21no—
34#17988473—21no—
35#17988451—21no—
36#17988447—21no—
37#17988447C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4⋊C4×Dic73C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42C4×C22⋊Dic73—21no—
43C4×C22⋊Dic73C2211yescharacteristic
44#17988432—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C22×C4⋊C4×C73C2211yescharacteristic
54C2×C22⋊C8×C73C2211yescharacteristic
55#380×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C2×#11924012—22no—
61C2×#11924012—22no—
62C2×#11924012C2×C414yes—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556550—22no—
66#14556550C2×C414yes—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×C22⋊Dic73—21no—
76C2×C22⋊Dic73—21no—
77C2×C22⋊Dic73D411yescharacteristic
78C2×C22⋊Dic73D411yescharacteristic
79C2×C4×Dic73—21no—
80C2×C4×Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73C2311yescharacteristic
85C2×C4×Dic73C2311yescharacteristic
86#14556524—22no—
87#14556524C2×C414yes—
88#14556524C2×C414yes—
89#14556522—22no—
90#14556522C2×C414yes—
91#14556522C2×C414yes—
92#14556501—22no—
93C8×Dic73—22no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498D411yescharacteristic
104#14556498D411yescharacteristic
105C23×C4×C73—21no—
106C23×C4×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73—22no—
111C2×C4⋊C4×C73C2×C414yes—
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C22⋊C4×C73—21no—
117C22⋊C8×C73—22no—
118#95×C73—21no—
119C22×Dic73—21no—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73C2×D411yescharacteristic
124C22×Dic73C2×D411yescharacteristic
125C22×Dic73C2×D411yescharacteristic
126C22×Dic73C2×D411yescharacteristic
127#11924012—28no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831C22×C412yes—
131C2×#10596831C22×C412yes—
132C2×#10596831#4112yes—
133C2×#10596831#4112yes—
134C4⋊Dic73—21no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—81no—
141C4⋊Dic73—81no—
142C4⋊Dic73C22×C412yes—
143C4⋊Dic73C22×C412yes—
144#11924007—81no—
145#11924007—81no—
146C22⋊Dic73—24no—
147C22⋊Dic73—22no—
148C22⋊Dic73—41no—
149C22⋊Dic73—42no—
150C4×Dic73—24no—
151C4×Dic73—21no—
152C4×Dic73—21no—
153C4×Dic73—21no—
154C4×Dic73—21no—
155C4×Dic73—42no—
156C4×Dic73—42no—
157C4×Dic73—81no—
158C4×Dic73—81no—
159C24×C73C2×D411yescharacteristic
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—24no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—81no—
170C22×C4×C73—81no—
171C22×C4×C73C22×C412yes—
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73#4112yes—
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—24no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73C22×C412yes—
189C4⋊C4×C73C22×C412yes—
190C22⋊C4×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—22no—
202C2×C4×C73—22no—
203C2×C4×C73—22no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
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×C73—84no—
218C2×C4×C73C4×D412yes—
219C2×C4×C73C23×C411yescharacteristic
220C2×C4×C73C2×#4111yescharacteristic
221C2×C4×C73C2×#4111yescharacteristic
222C2×C4×C73C2×#4111yescharacteristic
223C2×C4×C732−1+411yescharacteristic
224C2×C4×C732−1+411yescharacteristic
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—22no—
228C2×Dic73—22no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
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#13712yes—
249C8×C73—41no—
250C8×C73—41no—
251#10596831—22no—
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—42no—
269C4×C73—42no—
270C4×C73—81no—
271C4×C73—81no—
272C4×C73C2×#11711yescharacteristic
273C4×C73#51911yescharacteristic
274C4×C73#55211yescharacteristic
275C4×C73#55211yescharacteristic
276Dic73—22no—
277Dic73—41no—
278Dic73—41no—
279Dic73—41no—
280Dic73—42no—
281Dic73—42no—
282Dic73—81no—
283Dic73—81no—
284#22621—731nomaximal, Sylow
285C2×C73—42no—
286C2×C73#234411yescharacteristic
287C2×C73C4×#13711yescharacteristic
288C2×C73#294611yescharacteristic
289C2×C73#334411yescharacteristic
290C2×C73#334711yescharacteristic
291C2×C73#335111yescharacteristic
292C2×C73#335711yescharacteristic
293C2×#428—731no—
294#2346—731no—
295#2345—731no—
296#1943—731no—
297#1942—731no—
298#1921—731no—
299#1920—731no—
300#1919—731no—
301#1918—731no—
302#1825—731no—
303#1823D7311yescharacteristic
304#1809—731no—
305#1776—731no—
306C4×#106—731no—
307#1753—731no—
308C73#2262111yescharacteristic, complemented, Sylow
309C22×M16—731no—
310C2×#117—731no—
311C22×C4⋊C4C2×D7311yescharacteristic
312#428—738no—
313C2×#107—731no—
314C2×#107—731no—
315C2×#107—731no—
316C2×#106—731no—
317C2×#106—731no—
318C2×#106—731no—
319C2×C22⋊C8C2×D7311yescharacteristic
320C2×#97—731no—
321C2×C4×C8—731no—
322#389—731no—
323#389—731no—
324#389—731no—
325#389—731no—
326#388—1461no—
327#388—1461no—
328#387—1461no—
329#387—1461no—
330#382—1461no—
331#380—731no—
332#380—1461no—
333#380C2×D7311yescharacteristic
334C4×C4⋊C4—731no—
335C4×C4⋊C4—731no—
336C4×C4⋊C4—731no—
337C4×C4⋊C4—731no—
338C4×C22⋊C4—731no—
339C4×C22⋊C4—1461no—
340#376—1461no—
341#340—731no—
342#340—731no—
343#340—731no—
344#340—731no—
345#340—731no—
346#340—731no—
347#340C2×D7311yescharacteristic
348#340C2×D7311yescharacteristic
349#340C2×D7311yescharacteristic
350#340C2×D7311yescharacteristic
351#336—731no—
352#336—731no—
353C23×C4—21no—
354C23×C4C22×D7311yescharacteristic
355C2×M16—734no—
356C2×M16—1462no—
357C2×M16—1462no—
358C22×C8—731no—
359C22×C8—731no—
360C22×C8C22×D7311yescharacteristic
361C22×C8C22×D7311yescharacteristic
362#117—734no—
363#117—1462no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—1461no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4C4×D7314yes—
375C2×C4⋊C4C22×D7311yescharacteristic
376C2×C4⋊C4C22×D7311yescharacteristic
377C2×C4⋊C4C22×D7311yescharacteristic
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—731no—
381C2×C22⋊C4—731no—
382C2×C22⋊C4—1461no—
383C2×C22⋊C4—1461no—
384C2×C42—731no—
385C2×C42—731no—
386C2×C42—1461no—
387C2×C42—1461no—
388C2×C42—1461no—
389C2×C42—1461no—
390C2×C42—1461no—
391#107—734no—
392#107—734no—
393#107—1462no—
394#106—734no—
395#106—734no—
396#106—1462no—
397C22⋊C8—22no—
398#97—1462no—
399C4×C8—1462no—
400#95—21no—
401#95—731no—
402#95—731no—
403#95—1461no—
404#95—1461no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412C24D4×D7311yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—731no—
425C22×C4—731no—
426C22×C4—731no—
427C22×C4—731no—
428C22×C4—1461no—
429C22×C4—1461no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4C2×C4×D7312yes—
433C22×C4D4×D7311yescharacteristic
434C22×C4#1192402712yes—
435C22×C4C23×D7311yescharacteristic
436M16—1468no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—732no—
444C2×C8—732no—
445C2×C8—732no—
446C2×C8—732no—
447C2×C8—1461no—
448C2×C8—1461no—
449C4⋊C4—24no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—5841no—
457C4⋊C4—5841no—
458C4⋊C4—5841no—
459C4⋊C4—5841no—
460C4⋊C4—732no—
461C4⋊C4—732no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4C2×C4×D7312yes—
469C4⋊C4C2×C4×D7312yes—
470C22⋊C4—81no—
471C22⋊C4—2922no—
472C22⋊C4—2921no—
473C22⋊C4—1462no—
474C22⋊C4—1464no—
475C42—2922no—
476C42—2922no—
477C42—5841no—
478C42—5841no—
479C42—1464no—
480C42—1461no—
481C42—1461no—
482C42—1461no—
483C42—1461no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23#1455655112yes—
491C23C2×D4×D7311yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—41no—
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—84no—
511C2×C4—2921no—
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—732no—
522C2×C4—1461no—
523C2×C4—1461no—
524C2×C4—1461no—
525C2×C4—1461no—
526C2×C4—1461no—
527C2×C4—1461no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1462no—
532C2×C4—1462no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4#1455655112yes—
536C2×C4C22×C4×D7311yescharacteristic
537C2×C4#41×D7311yescharacteristic
538C2×C4#41×D7311yescharacteristic
539C2×C4C2×#1192402711yescharacteristic
540C2×C4#1455663811yescharacteristic
541C2×C4#1455663811yescharacteristic
542C8—41no—
543C8—41no—
544C8—734no—
545C8—1462no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22C2×#1455655111yescharacteristic
554C22#1798923511yescharacteristic
555C22#1798923511yescharacteristic
556C22#137×D7311yescharacteristic
557C22#1798938611yescharacteristic
558C22#1798938811yescharacteristic
559C22#1798939311yescharacteristic
560C4—22no—
561C4—42no—
562C4—42no—
563C4—81no—
564C4—81no—
565C4—2921no—
566C4—2921no—
567C4—2922no—
568C4—2922no—
569C4—2921no—
570C4—5841no—
571C4—5841no—
572C4—1462no—
573C4#117×D7311yescharacteristic
574C4#1798906411yescharacteristic
575C4#1798927411yescharacteristic
576C4#1798927411yescharacteristic
577C2—42no—
578C2#2261279511yescharacteristic
579C2#2261874611yescharacteristic
580C2#2261875711yescharacteristic
581C2#2262238911yescharacteristic
582C2#2262243411yescharacteristic
583C2#2262249411yescharacteristic
584C2#2262255011yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1823C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#1823D73
hypocenterC73#22621
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
4. #22610668C2semidirect
5. #22610663C2semidirect
6. #22610521C2semidirect
7. #22610519C2semidirect
8. #22610512C2semidirect
9. #22610500C2semidirect
12. #22609827C2semidirect
13. #22609819C2semidirect
18. #17988627C4semidirect
308. C73#22621semidirect

Maximal subgroups

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

subgroupindex
C2×#179886272
#226128182
#226128082
#226106682
#226106632
#226105212
#226105192
#226105122
#226105002
#226100272
#226099622
#226098272
#226098192
#226097172
C73×#18232
#2262173