Finite Group Databasebeta

← #33103667

Subgroups — #33103667

Order 18688.

766 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988622C211yesmaximal, characteristic
2#22612820C211yesmaximal, characteristic
3#22612812C211yesmaximal, characteristic
4#22610674C211yesmaximal, characteristic, complemented
5#22610672C211yesmaximal, characteristic, complemented
6#22610523C211yesmaximal, characteristic, complemented
7#22610521C211yesmaximal, characteristic, complemented
8#22610502C211yesmaximal, characteristic, complemented
9#22610500C211yesmaximal, characteristic, complemented
10#22610025C211yesmaximal, characteristic
11#22610024C211yesmaximal, characteristic
12#22609957C211yesmaximal, characteristic
13#22609831C211yesmaximal, characteristic, complemented
14#22609708C211yesmaximal, characteristic
15C73×#1776C211yesmaximal, characteristic, complemented
16C22×C8×D73C2211yescharacteristic
17C2×#14556549C2211yescharacteristic
18C2×#14556549C2211yescharacteristic
19#17988622C418yescomplemented
20C2×#14556525C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556503C2211yescharacteristic
25C2×#14556501C2211yescharacteristic
26#17988485C2211yescharacteristic
27#17988485C2211yescharacteristic
28#17988483C2211yescharacteristic
29#17988483C2211yescharacteristic
30#17988478—21no—
31#17988478—21no—
32#17988476—21no—
33#17988476—21no—
34#17988454—21no—
35#17988451—21no—
36#17988444C2211yescharacteristic
37#17988444C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4×#11924007—21no—
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×#106×C73C2211yescharacteristic
55C2×#97×C73C2211yescharacteristic
56C4×C4⋊C4×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×C4×D73C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C2×C8×D73—22no—
64C2×C8×D73—22no—
65C2×C8×D73C2×C414yes—
66#14556549—22no—
67#14556549—22no—
68#14556549C2×C414yes—
69#14556549C2×C414yes—
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×#11924005—21no—
79C2×#11924005—21no—
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×#11924005D411yescharacteristic
83C2×#11924005D411yescharacteristic
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73C2311yescharacteristic
89#14556525—22no—
90#14556524C2×C414yes—
91#14556524C2×C414yes—
92#14556523C2×C414yes—
93#14556503—22no—
94#14556501—22no—
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
105C22×C8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C42×C73—21no—
112C2×C42×C73—21no—
113C2×C42×C73C2311yescharacteristic
114#107×C73—22no—
115#106×C73C2×C414yes—
116#97×C73—22no—
117#95×C73—21no—
118#95×C73—21no—
119C23×D73C2×D411yescharacteristic
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73C2×D411yescharacteristic
125C22×Dic73C2×D411yescharacteristic
126C22×Dic73C2×D411yescharacteristic
127C2×C4×D73—21no—
128C2×C4×D73—21no—
129C2×C4×D73—21no—
130C2×C4×D73—21no—
131C2×C4×D73C22×C412yes—
132C2×C4×D73#4112yes—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831C22×C412yes—
139C8×D73—28no—
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—42no—
149#11924007—42no—
150#11924007—81no—
151#11924007—81no—
152#11924005—24no—
153#11924005—24no—
154#11924005—41no—
155#11924005—41no—
156#11924005—81no—
157#11924005—81no—
158C4×Dic73—21no—
159C4×Dic73—21no—
160C4×Dic73—24no—
161C4×Dic73—24no—
162C4×Dic73—42no—
163C4×Dic73—81no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2411yescharacteristic
171C2×C8×C73—21no—
172C2×C8×C73—21no—
173C2×C8×C73—21no—
174C2×C8×C73C22×C412yes—
175C2×C8×C73#4112yes—
176C2×C8×C73#4112yes—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73C22×C412yes—
184C4⋊C4×C73C22×C412yes—
185C42×C73—21no—
186C42×C73—21no—
187C42×C73—42no—
188C42×C73—81no—
189C23×C73C22×D411yescharacteristic
190C22×D73—21no—
191C22×D73—21no—
192C22×D73—21no—
193C22×D73—21no—
194C22×D73—21no—
195C22×D73—21no—
196C22×D73C4×D412yes—
197C2×C4×C73—22no—
198C2×C4×C73—22no—
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—41no—
206C2×C4×C73—41no—
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×C73C23×C411yescharacteristic
216C2×C4×C73C2×#4111yescharacteristic
217C2×C4×C73C2×#4111yescharacteristic
218C2×C4×C73C2×#4111yescharacteristic
219C2×C4×C732−1+411yescharacteristic
220C2×C4×C732−1+411yescharacteristic
221C2×Dic73—21no—
222C2×Dic73—21no—
223C2×Dic73—22no—
224C2×Dic73—22no—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—41no—
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×Dic73C4×D412yes—
242C4×D73—22no—
243C4×D73—22no—
244C4×D73—22no—
245C4×D73—22no—
246C8×C73—41no—
247C8×C73#11714yes—
248#10596831—22no—
249#10596831—41no—
250C22×C73C2×C4×D411yescharacteristic
251C22×C73#54711yescharacteristic
252C22×C73#54711yescharacteristic
253C22×C73#57511yescharacteristic
254C22×C73#57511yescharacteristic
255C22×C73#57611yescharacteristic
256C22×C73#57811yescharacteristic
257C2×D73—22no—
258C2×D73—22no—
259C2×D73—22no—
260C2×D73—41no—
261C2×D73—41no—
262C2×D73—41no—
263C2×D73—41no—
264C4×C73—41no—
265C4×C73—41no—
266C4×C73—42no—
267C4×C73—42no—
268C4×C73—81no—
269C4×C73—81no—
270C4×C73C2×#11711yescharacteristic
271C4×C73#51911yescharacteristic
272C4×C73#55211yescharacteristic
273C4×C73#55211yescharacteristic
274Dic73—22no—
275Dic73—41no—
276Dic73—41no—
277Dic73—42no—
278Dic73—42no—
279Dic73—81no—
280Dic73—81no—
281#22616—731nomaximal, Sylow
282C2×C73#234411yescharacteristic
283C2×C73#294311yescharacteristic
284C2×C73#295111yescharacteristic
285C2×C73#334711yescharacteristic
286C2×C73#334711yescharacteristic
287C2×C73#334911yescharacteristic
288C2×C73#334911yescharacteristic
289D73—42no—
290C2×#427—731no—
291#2347—731no—
292#2346—731no—
293#1944—731no—
294#1944—731no—
295#1921—731no—
296#1921—731no—
297#1918—731no—
298#1918—731no—
299#1825—731no—
300#1825—731no—
301#1808—731no—
302#1776—731no—
303#1776D7311yescharacteristic, complemented
304#1752—731no—
305C73#2261611yescharacteristic, complemented, Sylow
306C23×C8—731no—
307C2×#117—731no—
308C2×#117—731no—
309#427—738no—
310C2×#107—731no—
311C2×#107—731no—
312C2×#107—731no—
313C2×#107C2×D7311yescharacteristic
314C2×#106—731no—
315C2×#106C2×D7311yescharacteristic
316C2×C22⋊C8—731no—
317C2×#97—731no—
318C2×#97C2×D7311yescharacteristic
319#389—731no—
320#389—731no—
321#389—731no—
322#389—731no—
323#388—1461no—
324#388—1461no—
325#388—1461no—
326#388—1461no—
327#383—1461no—
328#382—1461no—
329#380—731no—
330#380—731no—
331C4×C4⋊C4—731no—
332C4×C4⋊C4—731no—
333C4×C4⋊C4—1461no—
334C4×C4⋊C4C2×D7311yescharacteristic
335C4×C4⋊C4C2×D7311yescharacteristic
336C4×C22⋊C4—731no—
337#376—1461no—
338#340—731no—
339#340—731no—
340#340—731no—
341#340—731no—
342#340—731no—
343#340—731no—
344#340—731no—
345#340—731no—
346#340C2×D7311yescharacteristic
347#340C2×D7311yescharacteristic
348#336—731no—
349#336—731no—
350C23×C4—731no—
351C22×C8—731no—
352C22×C8—734no—
353C22×C8—731no—
354C22×C8—1462no—
355C22×C8—1462no—
356C22×C8C22×D7311yescharacteristic
357C22×C8C22×D7311yescharacteristic
358#117—734no—
359#117—734no—
360#117—1462no—
361#117—1462no—
362C2×C4⋊C4—731no—
363C2×C4⋊C4—731no—
364C2×C4⋊C4—731no—
365C2×C4⋊C4—731no—
366C2×C4⋊C4—1461no—
367C2×C4⋊C4—1461no—
368C2×C4⋊C4—1461no—
369C2×C4⋊C4—1461no—
370C2×C4⋊C4C22×D7311yescharacteristic
371C2×C4⋊C4C22×D7311yescharacteristic
372C2×C4⋊C4C22×D7311yescharacteristic
373C2×C4⋊C4C22×D7311yescharacteristic
374C2×C22⋊C4—731no—
375C2×C22⋊C4—731no—
376C2×C22⋊C4—1461no—
377C2×C22⋊C4—1461no—
378C2×C22⋊C4—1461no—
379C2×C22⋊C4—1461no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—731no—
383C2×C42—1461no—
384C2×C42—1461no—
385C2×C42—1461no—
386C2×C42—1461no—
387C2×C42C22×D7311yescharacteristic
388#107—22no—
389#107—734no—
390#107—734no—
391#107—1462no—
392#106—734no—
393#106C4×D7314yes—
394C22⋊C8—1462no—
395#97—22no—
396#97—1462no—
397#95—21no—
398#95—21no—
399#95—731no—
400#95—731no—
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—731no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—731no—
415C22×C4—732no—
416C22×C4—732no—
417C22×C4—731no—
418C22×C4—731no—
419C22×C4—1461no—
420C22×C4—1461no—
421C22×C4—1461no—
422C22×C4—1461no—
423C22×C4—1461no—
424C22×C4—1461no—
425C22×C4—1461no—
426C22×C4—1461no—
427C22×C4D4×D7311yescharacteristic
428C22×C4D4×D7311yescharacteristic
429C22×C4C23×D7311yescharacteristic
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—732no—
434C2×C8—1461no—
435C2×C8—1461no—
436C2×C8—1461no—
437C2×C8—1461no—
438C2×C8—1461no—
439C2×C8—1468no—
440C2×C8C2×C4×D7312yes—
441C2×C8#1192402812yes—
442C2×C8#1192402812yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—2922no—
450C4⋊C4—2922no—
451C4⋊C4—5841no—
452C4⋊C4—5841no—
453C4⋊C4—732no—
454C4⋊C4—732no—
455C4⋊C4—1461no—
456C4⋊C4—1461no—
457C4⋊C4—1461no—
458C4⋊C4—1461no—
459C4⋊C4—1461no—
460C4⋊C4—1461no—
461C4⋊C4C2×C4×D7312yes—
462C4⋊C4C2×C4×D7312yes—
463C22⋊C4—2921no—
464C22⋊C4—2921no—
465C22⋊C4—5841no—
466C22⋊C4—5841no—
467C22⋊C4—1464no—
468C22⋊C4—1464no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42—81no—
473C42—2922no—
474C42—5841no—
475C42—1464no—
476C42—1464no—
477C42—1461no—
478C42—1461no—
479C23—732no—
480C23—1461no—
481C23—1461no—
482C23—1461no—
483C23—1461no—
484C23—1461no—
485C23—1461no—
486C23C2×D4×D7311yescharacteristic
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—2921no—
506C2×C4—2921no—
507C2×C4—2921no—
508C2×C4—2921no—
509C2×C4—2921no—
510C2×C4—2921no—
511C2×C4—2921no—
512C2×C4—2921no—
513C2×C4—2921no—
514C2×C4—2921no—
515C2×C4—732no—
516C2×C4—1461no—
517C2×C4—1461no—
518C2×C4—1461no—
519C2×C4—1461no—
520C2×C4—1461no—
521C2×C4—1461no—
522C2×C4—1462no—
523C2×C4—1462no—
524C2×C4—1461no—
525C2×C4—1461no—
526C2×C4—1462no—
527C2×C4—1462no—
528C2×C4—1462no—
529C2×C4—1462no—
530C2×C4C22×C4×D7311yescharacteristic
531C2×C4#41×D7311yescharacteristic
532C2×C4C2×#1192402811yescharacteristic
533C2×C4C2×#1192402811yescharacteristic
534C2×C4#1455663811yescharacteristic
535C2×C4#1455663911yescharacteristic
536C8—41no—
537C8—2921no—
538C8—1462no—
539C8#1455654814yes—
540C22—2921no—
541C22—2921no—
542C22—2921no—
543C22—2921no—
544C22—1462no—
545C22—1462no—
546C22—1462no—
547C22C4×D4×D7311yescharacteristic
548C22#1798923811yescharacteristic
549C22#1798924211yescharacteristic
550C22#1798938311yescharacteristic
551C22#1798938611yescharacteristic
552C22#1798939111yescharacteristic
553C22#1798940311yescharacteristic
554C4—42no—
555C4—42no—
556C4—41no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—2921no—
561C4—2922no—
562C4—2922no—
563C4—2921no—
564C4—5841no—
565C4—5841no—
566C4—1462no—
567C4C2×#1455654811yescharacteristic
568C4#1798906311yescharacteristic
569C4#1798927611yescharacteristic
570C4#1798927811yescharacteristic
571C2—2922no—
572C2#2261279611yescharacteristic
573C2#2261872611yescharacteristic
574C2#2261880311yescharacteristic
575C2#2262243211yescharacteristic
576C2#2262243611yescharacteristic
577C2#2262246211yescharacteristic
578C2#2262246611yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1776C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1776D73
hypocenterC73#22616
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #22610674C2semidirect
5. #22610672C2semidirect
6. #22610523C2semidirect
7. #22610521C2semidirect
8. #22610502C2semidirect
9. #22610500C2semidirect
13. #22609831C2semidirect
15. C73×#1776C2semidirect
19. #17988622C4semidirect
303. #1776D73semidirect
305. C73#22616semidirect

Maximal subgroups

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

subgroupindex
C2×#179886222
#226128202
#226128122
#226106742
#226106722
#226105232
#226105212
#226105022
#226105002
#226100252
#226100242
#226099572
#226098312
#226097082
C73×#17762
#2261673