Finite Group Databasebeta

← #33096200

Subgroups — #33096200

Order 18688.

648 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612428C211yesmaximal, characteristic
2#22610545C211yesmaximal, characteristic
3#22610545C211yesmaximal, characteristic
4#22610538C211yesmaximal, characteristic
5#22610538C211yesmaximal, characteristic
6#22610498C211yesmaximal, characteristic, complemented
7#22610498C211yesmaximal, characteristic, complemented
8#22610491C211yesmaximal, characteristic, complemented
9#22610491C211yesmaximal, characteristic, complemented
10#22610083C211yesmaximal, characteristic, complemented
11#22609912C211yesmaximal, characteristic
12#22609912C211yesmaximal, characteristic
13#22609790C211yesmaximal, characteristic, complemented
14#22609790C211yesmaximal, characteristic, complemented
15C73×#1831C211yesmaximal, characteristic, complemented
16C2×#14556548C2211yescharacteristic
17C2×C4⋊#10596831C2211yescharacteristic
18C2×C4⋊#10596831C2211yescharacteristic
19C2×C4⋊#10596831C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×#14556503C2211yescharacteristic
24C2×#14556503C2211yescharacteristic
25C2×#14556503C2211yescharacteristic
26C2×#14556503C2211yescharacteristic
27C2×#14556501C2211yescharacteristic
28C2×#14556501C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30C2×C8×Dic73C2211yescharacteristic
31#17988478C411yescharacteristic
32#17988478C411yescharacteristic
33#17988478C411yescharacteristic
34#17988478C411yescharacteristic
35#17988471C411yescharacteristic
36#17988471C411yescharacteristic
37#17988471C411yescharacteristic
38#17988471C411yescharacteristic
39C4×#11924007C2211yescharacteristic
40C4×#11924007C2211yescharacteristic
41C4×#11924005C2211yescharacteristic
42C4×#11924005C2211yescharacteristic
43#17988433C2211yescharacteristic
44#17988341C2211yescharacteristic
45#17988341C2211yescharacteristic
46#17988341C2211yescharacteristic
47#17988341C2211yescharacteristic
48#17988341C2211yescharacteristic
49#17988341C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×C4⋊C8×C73C2211yescharacteristic
53C2×C4⋊C8×C73C2211yescharacteristic
54C43×C73C2211yescharacteristic
55#336×C73C2211yescharacteristic
56#336×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×C4×D73C2311yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64#14556548—22no—
65#14556548—22no—
66C2×#11924007C2×C411yescharacteristic
67C2×#11924007C2×C411yescharacteristic
68C2×#11924007C2×C411yescharacteristic
69C2×#11924007C2×C411yescharacteristic
70C2×#11924007C2×C411yescharacteristic
71C2×#11924007C2×C411yescharacteristic
72C2×#11924005C2×C411yescharacteristic
73C2×#11924005C2×C411yescharacteristic
74C2×#11924005C2×C411yescharacteristic
75C2×#11924005C2×C411yescharacteristic
76C2×#11924005C2×C411yescharacteristic
77C2×#11924005C2×C411yescharacteristic
78C2×C4×Dic73C2311yescharacteristic
79C2×C4×Dic73C2311yescharacteristic
80C2×C4×Dic73C2311yescharacteristic
81C4⋊#10596831—22no—
82C4⋊#10596831—22no—
83C4⋊#10596831—41no—
84C4⋊#10596831—41no—
85C4⋊#10596831—41no—
86C4⋊#10596831—41no—
87#14556503—41no—
88#14556503—41no—
89#14556503—41no—
90#14556503—41no—
91#14556501—41no—
92#14556501—41no—
93C8×Dic73—41no—
94C8×Dic73—41no—
95#14556498C2×C411yescharacteristic
96#14556498C2×C411yescharacteristic
97#14556498C2×C411yescharacteristic
98#14556498C2×C411yescharacteristic
99#14556498C2×C411yescharacteristic
100#14556498C2×C411yescharacteristic
101#14556498C2×C411yescharacteristic
102#14556498C2×C411yescharacteristic
103#14556498C2×C411yescharacteristic
104#14556498C2×C411yescharacteristic
105#14556498C2×C411yescharacteristic
106#14556498C2×C411yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109C22×C8×C73C2311yescharacteristic
110C22×C8×C73C2311yescharacteristic
111C2×C42×C73C2×C411yescharacteristic
112C2×C42×C73C2×C411yescharacteristic
113C2×C42×C73C2×C411yescharacteristic
114C2×C42×C73C2×C411yescharacteristic
115C2×C42×C73C2311yescharacteristic
116C2×C42×C73C2311yescharacteristic
117C2×C42×C73C2311yescharacteristic
118C4⋊C8×C73—22no—
119C4⋊C8×C73—22no—
120C23×D73C22×C411yescharacteristic
121C22×Dic73C22×C411yescharacteristic
122C22×Dic73C22×C411yescharacteristic
123C22×Dic73C22×C411yescharacteristic
124C22×Dic73C22×C411yescharacteristic
125C22×Dic73C22×C411yescharacteristic
126C22×Dic73C22×C411yescharacteristic
127C22×Dic73C22×C411yescharacteristic
128C2×C4×D73—21no—
129C2×C4×D73—21no—
130C2×C4×D73—21no—
131C2×C4×D73—21no—
132C2×C4×D73—21no—
133C2×C4×D73—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146#11924007—22no—
147#11924007—22no—
148#11924007—22no—
149#11924007—22no—
150#11924007—41no—
151#11924007—41no—
152#11924005—22no—
153#11924005—22no—
154#11924005—22no—
155#11924005—22no—
156#11924005—41no—
157#11924005—41no—
158C4×Dic73—21no—
159C4×Dic73—21no—
160C4×Dic73—21no—
161C4×Dic73—21no—
162C4×Dic73—21no—
163C4×Dic73—21no—
164C22×C4×C73C22×C411yescharacteristic
165C22×C4×C73C22×C411yescharacteristic
166C22×C4×C73C22×C411yescharacteristic
167C22×C4×C73C22×C411yescharacteristic
168C22×C4×C73C22×C411yescharacteristic
169C22×C4×C73C22×C411yescharacteristic
170C22×C4×C73C2411yescharacteristic
171C2×C8×C73—21no—
172C2×C8×C73—21no—
173C2×C8×C73—21no—
174C2×C8×C73—21no—
175C2×C8×C73—21no—
176C2×C8×C73—21no—
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C42×C73—22no—
184C42×C73—22no—
185C42×C73—21no—
186C42×C73—21no—
187C42×C73—21no—
188C42×C73—21no—
189C42×C73—41no—
190C42×C73—41no—
191C42×C73#4112yes—
192C42×C73#4112yes—
193C23×C73C23×C411yescharacteristic
194C22×D73—21no—
195C22×D73—21no—
196C22×D73—21no—
197C22×D73—21no—
198C22×D73—21no—
199C22×D73—21no—
200C22×D73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
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#11712yes—
216C2×C4×C73#11712yes—
217C2×C4×C73#13112yes—
218C2×C4×C73#13112yes—
219C2×C4×C73C2×#4111yescharacteristic
220C2×C4×C73C2×#4111yescharacteristic
221C2×C4×C73H(5,𝔽2)11yescharacteristic
222C2×C4×C73H(5,𝔽2)11yescharacteristic
223C2×C4×C732−1+411yescharacteristic
224C2×C4×C732−1+411yescharacteristic
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
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—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C4×D73—41no—
247C4×D73—41no—
248C4×D73—41no—
249C4×D73—41no—
250C8×C73—41no—
251C8×C73—41no—
252C8×C73—41no—
253C8×C73—41no—
254#10596831—41no—
255#10596831—41no—
256#10596831—41no—
257#10596831—41no—
258C22×C73C2×#11711yescharacteristic
259C22×C73#52011yescharacteristic
260C22×C73#52011yescharacteristic
261C22×C73C2×#13111yescharacteristic
262C22×C73C2×#13111yescharacteristic
263C22×C73#56811yescharacteristic
264C22×C73#56811yescharacteristic
265C2×D73—41no—
266C2×D73—41no—
267C2×D73—41no—
268C2×D73—41no—
269C2×D73—41no—
270C2×D73—41no—
271C2×D73—41no—
272C4×C73—22no—
273C4×C73—22no—
274C4×C73—22no—
275C4×C73—22no—
276C4×C73—41no—
277C4×C73—41no—
278C4×C73#54011yescharacteristic
279C4×C73#54111yescharacteristic
280C4×C73#54211yescharacteristic
281C4×C73#54211yescharacteristic
282Dic73—41no—
283Dic73—41no—
284Dic73—41no—
285Dic73—41no—
286Dic73—41no—
287Dic73—41no—
288Dic73—41no—
289#21929—731nomaximal, Sylow
290C2×C73#231411yescharacteristic
291C2×C73#292511yescharacteristic
292C2×C73#292511yescharacteristic
293C2×C73#299211yescharacteristic
294C2×C73#299311yescharacteristic
295C2×C73#299411yescharacteristic
296C2×C73#299511yescharacteristic
297D73—81no—
298#2298—731no—
299#1924—731no—
300#1924—731no—
301#1923—731no—
302#1923—731no—
303#1917—731no—
304#1917—731no—
305#1916—731no—
306#1916—731no—
307#1842—731no—
308#1831D7311yescharacteristic, complemented
309#1798—731no—
310#1798—731no—
311#1768—731no—
312#1768—731no—
313C73#2192911yescharacteristic, complemented, Sylow
314C2×#117—731no—
315C2×C4⋊C8—731no—
316C2×C4⋊C8—731no—
317C2×C4⋊C8—731no—
318C2×C4⋊C8—731no—
319C2×C4⋊C8—731no—
320C2×C4⋊C8—731no—
321C2×C4⋊C8C2×D7311yescharacteristic
322C2×C4⋊C8C2×D7311yescharacteristic
323C2×C22⋊C8—731no—
324C2×C22⋊C8—731no—
325C2×C22⋊C8—731no—
326C2×C22⋊C8—731no—
327C2×#97—731no—
328C2×#97—731no—
329C2×C4×C8—731no—
330C2×C4×C8—731no—
331#388—731no—
332#388—731no—
333#388—731no—
334#388—731no—
335#387—731no—
336#387—731no—
337#387—731no—
338#387—731no—
339C4×C4⋊C4—731no—
340C4×C4⋊C4—731no—
341C4×C22⋊C4—731no—
342C4×C22⋊C4—731no—
343#376—731no—
344C43C2×D7311yescharacteristic
345#336—731no—
346#336—731no—
347#336—731no—
348#336—731no—
349#336—731no—
350#336—731no—
351#336—731no—
352#336—731no—
353#336C2×D7311yescharacteristic
354#336C2×D7311yescharacteristic
355#336C2×D7311yescharacteristic
356#336C2×D7311yescharacteristic
357C23×C4—731no—
358C22×C8—731no—
359C22×C8—731no—
360C22×C8—731no—
361C22×C8—731no—
362C22×C8C22×D7311yescharacteristic
363C22×C8C22×D7311yescharacteristic
364C22×C8C22×D7311yescharacteristic
365C22×C8C22×D7311yescharacteristic
366#117—1462no—
367#117—1462no—
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×C22⋊C4—731no—
375C2×C22⋊C4—731no—
376C2×C22⋊C4—731no—
377C2×C22⋊C4—731no—
378C2×C22⋊C4—731no—
379C2×C22⋊C4—731no—
380C2×C42—731no—
381C2×C42—731no—
382C2×C42—731no—
383C2×C42C4×D7311yescharacteristic
384C2×C42C4×D7311yescharacteristic
385C2×C42C4×D7311yescharacteristic
386C2×C42C4×D7311yescharacteristic
387C2×C42C22×D7311yescharacteristic
388C2×C42C22×D7311yescharacteristic
389C2×C42C22×D7311yescharacteristic
390C4⋊C8—22no—
391C4⋊C8—22no—
392C4⋊C8—2921no—
393C4⋊C8—2921no—
394C4⋊C8—2921no—
395C4⋊C8—2921no—
396C4⋊C8—1462no—
397C4⋊C8—1462no—
398C22⋊C8—2921no—
399C22⋊C8—2921no—
400C22⋊C8—2921no—
401C22⋊C8—2921no—
402#97—2921no—
403#97—2921no—
404C4×C8—2921no—
405C4×C8—2921no—
406#95—731no—
407#95—731no—
408#95—731no—
409#95—731no—
410#95—731no—
411#95—731no—
412#95—731no—
413#95—731no—
414#95—731no—
415#95—731no—
416#95—731no—
417#95—731no—
418C24—731no—
419C22×C4—731no—
420C22×C4—731no—
421C22×C4—731no—
422C22×C4—731no—
423C22×C4—731no—
424C22×C4—731no—
425C22×C4—731no—
426C22×C4—1461no—
427C22×C4—1461no—
428C22×C4—1461no—
429C22×C4—1461no—
430C22×C4—1461no—
431C22×C4—1461no—
432C22×C4C2×C4×D7311yescharacteristic
433C22×C4C2×C4×D7311yescharacteristic
434C22×C4C2×C4×D7311yescharacteristic
435C22×C4C2×C4×D7311yescharacteristic
436C22×C4C2×C4×D7311yescharacteristic
437C22×C4C2×C4×D7311yescharacteristic
438C22×C4C23×D7311yescharacteristic
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C2×C8—1461no—
462C2×C8—1461no—
463C4⋊C4—2921no—
464C4⋊C4—2921no—
465C4⋊C4—1462no—
466C4⋊C4—1462no—
467C4⋊C4—1462no—
468C4⋊C4—1462no—
469C22⋊C4—2921no—
470C22⋊C4—2921no—
471C22⋊C4—1462no—
472C22⋊C4—1462no—
473C22⋊C4—1462no—
474C22⋊C4—1462no—
475C42—22no—
476C42—22no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42—41no—
483C42—1461no—
484C42—1461no—
485C42—1461no—
486C42—1461no—
487C42—1461no—
488C42—1461no—
489C42#1192402712yes—
490C42#1192402912yes—
491C23—1461no—
492C23—1461no—
493C23—1461no—
494C23—1461no—
495C23—1461no—
496C23—1461no—
497C23—1461no—
498C23C22×C4×D7311yescharacteristic
499C2×C4—21no—
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—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—2921no—
514C2×C4—2921no—
515C2×C4—2921no—
516C2×C4—2921no—
517C2×C4—1461no—
518C2×C4—1461no—
519C2×C4—1461no—
520C2×C4—1461no—
521C2×C4—1461no—
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—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4#1455654912yes—
539C2×C4#1455654912yes—
540C2×C4#1455659712yes—
541C2×C4#1455659912yes—
542C2×C4C2×#1192402711yescharacteristic
543C2×C4C2×#1192402911yescharacteristic
544C2×C4#1455663711yescharacteristic
545C2×C4#1455663711yescharacteristic
546C2×C4#1455663811yescharacteristic
547C2×C4#1455663811yescharacteristic
548C8—41no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—2921no—
553C8—2921no—
554C8—2921no—
555C8—2921no—
556C22—2921no—
557C22—2921no—
558C22—2921no—
559C22—2921no—
560C22—2921no—
561C22—2921no—
562C22—2921no—
563C22C2×#1455654911yescharacteristic
564C22#1798906711yescharacteristic
565C22#1798906711yescharacteristic
566C22C2×#1455659711yescharacteristic
567C22C2×#1455659911yescharacteristic
568C22#1798935311yescharacteristic
569C22#1798935311yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—41no—
575C4—41no—
576C4—2921no—
577C4—2921no—
578C4—2921no—
579C4—2921no—
580C4—2921no—
581C4—2921no—
582C4—2921no—
583C4#1798919311yescharacteristic
584C4#1798920011yescharacteristic
585C4#1798920611yescharacteristic
586C4#1798920611yescharacteristic
587C2—5841no—
588C2#2261254111yescharacteristic
589C2#2261859611yescharacteristic
590C2#2261859911yescharacteristic
591C2#2261907211yescharacteristic
592C2#2261908211yescharacteristic
593C2#2261909311yescharacteristic
594C2#2261910311yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC73×#1831C2
prime radicalC23×D73C22×C4
cosocleC22×C4×C73C24
hypercenter#1831D73
hypocenterC73#21929
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #1831

Complements

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

NH ≅ G/Ntype
6. #22610498C2semidirect
7. #22610498C2semidirect
8. #22610491C2semidirect
9. #22610491C2semidirect
10. #22610083C2semidirect
13. #22609790C2semidirect
14. #22609790C2semidirect
15. C73×#1831C2semidirect
308. #1831D73semidirect
313. C73#21929semidirect

Maximal subgroups

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

subgroupindex
#226124282
#226105452
#226105382
#226104982
#226104912
#226100832
#226099122
#226097902
C73×#18312
#2192973