Finite Group Databasebeta

← #33149608

Subgroups — #33149608

Order 18688.

746 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22613622C211yesmaximal, characteristic
2#22612956C211yesmaximal, characteristic
3#22611500C211yesmaximal, characteristic, complemented
4#22611424C211yesmaximal, characteristic, complemented
5#22611325C211yesmaximal, characteristic, complemented
6#22611073C211yesmaximal, characteristic, complemented
7#22610882C211yesmaximal, characteristic, complemented
8#22610517C211yesmaximal, characteristic, complemented
9#22610354C211yesmaximal, characteristic
10#22610241C211yesmaximal, characteristic
11#22610202C211yesmaximal, characteristic, complemented
12#22610132C211yesmaximal, characteristic
13#22609947C211yesmaximal, characteristic, complemented
14#22609648C211yesmaximal, characteristic
15C73×#1850C211yesmaximal, characteristic
16C2×#14556591C2211yescharacteristic
17C2×#14556570C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C2×#14556550C2211yescharacteristic
20C2×#14556550C2211yescharacteristic
21C2×#14556525C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556516C2211yescharacteristic
24C2×#14556516C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27#17988510—21no—
28#17988499—21no—
29#17988493C2211yescharacteristic
30#17988491—21no—
31#17988485C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988481—21no—
35#17988475—21no—
36#17988474—21no—
37#17988469—21no—
38#17988453C2211yescharacteristic
39#17988447C2211yescharacteristic
40C4×C4⋊Dic73C2211yescharacteristic
41#17988438—21no—
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52C23×C8×C73C2211yescharacteristic
53C2×C4⋊C8×C73C2211yescharacteristic
54C2×C4⋊C8×C73C2211yescharacteristic
55C2×C22⋊C8×C73C2211yescharacteristic
56C4×C22⋊C4×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×Dic146—21no—
60C22×Dic146C2311yescharacteristic
61#14556591—22no—
62#14556591—41no—
63#14556570—22no—
64#14556570Q814yescomplemented
65C4×Dic146—22no—
66C4×Dic146—41no—
67#14556550—22no—
68#14556550—22no—
69#14556550—41no—
70#14556550—41no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×C22⋊Dic73—21no—
85C2×C22⋊Dic73—21no—
86C2×C22⋊Dic73—21no—
87C2×C22⋊Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92#14556525—41no—
93#14556523—22no—
94#14556516—22no—
95#14556516—41no—
96#14556516—41no—
97#14556516—41no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107C23×C4×C73C2311yescharacteristic
108C22×C8×C73—22no—
109C22×C8×C73—41no—
110C22×C8×C73—41no—
111C22×C8×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C2×C22⋊C4×C73D411yescharacteristic
116C2×C22⋊C4×C73D411yescharacteristic
117C2×C42×C73C2311yescharacteristic
118C2×C42×C73C2311yescharacteristic
119C4⋊C8×C73—41no—
120C4⋊C8×C73—41no—
121C22⋊C8×C73—22no—
122#95×C73D411yescharacteristic
123#95×C73D411yescharacteristic
124C2×Dic146—21no—
125C2×Dic146—21no—
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—42no—
130C2×Dic146—42no—
131C2×Dic146—81no—
132C2×Dic146C2×Q812yes—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—42no—
153C4⋊Dic73C2×Q812yes—
154C4⋊Dic73#4112yes—
155C4⋊Dic73#4112yes—
156#11924007—24no—
157#11924007—42no—
158#11924007—42no—
159#11924007—42no—
160#11924007—81no—
161C22⋊Dic73—24no—
162C22⋊Dic73—42no—
163C22⋊Dic73—42no—
164C22⋊Dic73—81no—
165C4×Dic73—42no—
166C4×Dic73—42no—
167C4×Dic73—42no—
168C4×Dic73—42no—
169C24×C73C2×D411yescharacteristic
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×Q812yes—
180C22×C4×C73#4112yes—
181C22×C4×C73C2411yescharacteristic
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—44no—
195C22⋊C4×C73—22no—
196C22⋊C4×C73—41no—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—21no—
201C23×C73—21no—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73C22⋊Q812yes—
208C23×C73C22×D411yescharacteristic
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73C22⋊Q812yes—
227C2×C4×C73#13512yes—
228C2×C4×C73C22×Q811yescharacteristic
229C2×C4×C73C2×#4111yescharacteristic
230C2×C4×C73C2×#4111yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C2×C4×C73H(5,𝔽2)11yescharacteristic
233C2×C4×C73H(5,𝔽2)11yescharacteristic
234C2×Dic73—22no—
235C2×Dic73—22no—
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—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258Dic146—22no—
259Dic146—22no—
260Dic146—41no—
261Dic146—41no—
262Dic146—84no—
263C8×C73—41no—
264C8×C73—41no—
265C8×C73—41no—
266C8×C73—41no—
267C22×C73—22no—
268C22×C73—22no—
269C22×C73—22no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73C2×C22⋊Q811yescharacteristic
275C22×C73#54611yescharacteristic
276C22×C73#54611yescharacteristic
277C22×C73C2×#13511yescharacteristic
278C22×C73#57511yescharacteristic
279C22×C73#57711yescharacteristic
280C22×C73#57811yescharacteristic
281C4×C73—22no—
282C4×C73—22no—
283C4×C73—41no—
284C4×C73—41no—
285C4×C73—41no—
286C4×C73#52911yescharacteristic
287C4×C73#53811yescharacteristic
288C4×C73#55411yescharacteristic
289C4×C73#55411yescharacteristic
290Dic73—42no—
291Dic73—42no—
292Dic73—42no—
293Dic73—42no—
294Dic73—81no—
295Dic73—81no—
296Dic73—81no—
297Dic73—81no—
298#26413—731nomaximal, Sylow
299C2×C73—42no—
300C2×C73#245011yescharacteristic
301C2×C73#307611yescharacteristic
302C2×C73#308211yescharacteristic
303C2×C73#329611yescharacteristic
304C2×C73#330211yescharacteristic
305C2×C73#331311yescharacteristic
306C2×C73#332111yescharacteristic
307#2439—731no—
308#2362—731no—
309#2101—731no—
310#2087—731no—
311#2068—731no—
312#2024—731no—
313#1988—731no—
314#1920—731no—
315#1892—731no—
316#1873—731no—
317#1865—731no—
318#1854—731no—
319#1850D7311yescharacteristic
320#1805—731no—
321#1735—731no—
322C73#2641311yescharacteristic, complemented, Sylow
323C23×C8C2×D7311yescharacteristic
324C2×C4⋊Q8—731no—
325C2×C22⋊Q8—731no—
326C2×C4×Q8—731no—
327C2×#117—731no—
328C2×#117—731no—
329C2×#107—731no—
330C2×#106—731no—
331C2×C4⋊C8C2×D7311yescharacteristic
332C2×C4⋊C8C2×D7311yescharacteristic
333C2×#103—731no—
334C2×#103—731no—
335C2×#103—731no—
336C2×#103—731no—
337C2×C22⋊C8C2×D7311yescharacteristic
338#396—1461no—
339#393—1461no—
340#391—731no—
341#391—1461no—
342#389—731no—
343#389—731no—
344#389—731no—
345#388—1461no—
346#388—1461no—
347#387—1461no—
348#387—1461no—
349#382—731no—
350#380—731no—
351C4×C4⋊C4—731no—
352C4×C4⋊C4—1461no—
353C4×C22⋊C4C2×D7311yescharacteristic
354#340—731no—
355#340—731no—
356#340—731no—
357#340—731no—
358#340—731no—
359#340—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#336C2×D7311yescharacteristic
365#336C2×D7311yescharacteristic
366C22×Q8—731no—
367C22×Q8—1461no—
368C23×C4C22×D7311yescharacteristic
369C22×C8—22no—
370C22×C8—41no—
371C22×C8—41no—
372C22×C8C22×D7311yescharacteristic
373C22×C8C22×D7311yescharacteristic
374C22×C8C22×D7311yescharacteristic
375C22×C8C22×D7311yescharacteristic
376C4⋊Q8—2921no—
377C4⋊Q8—1462no—
378C22⋊Q8—734no—
379C22⋊Q8—1462no—
380C4×Q8—2921no—
381C4×Q8—1462no—
382#117—2921no—
383#117—2921no—
384#117—1462no—
385#117—1462no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4—1461no—
398C2×C4⋊C4—1461no—
399C2×C22⋊C4—1461no—
400C2×C22⋊C4—1461no—
401C2×C22⋊C4—1461no—
402C2×C22⋊C4—1461no—
403C2×C22⋊C4D29211yescharacteristic
404C2×C22⋊C4D29211yescharacteristic
405C2×C42—1461no—
406C2×C42—1461no—
407C2×C42—1461no—
408C2×C42—1461no—
409C2×C42C22×D7311yescharacteristic
410C2×C42C22×D7311yescharacteristic
411#107—2921no—
412#106—1462no—
413C4⋊C8—41no—
414C4⋊C8—41no—
415#103—2921no—
416#103—2921no—
417#103—2921no—
418#103—1462no—
419C22⋊C8—22no—
420#95—1461no—
421#95—1461no—
422#95—1461no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426#95—1461no—
427#95—1461no—
428#95—1461no—
429#95D29211yescharacteristic
430#95D29211yescharacteristic
431C24C2×D29211yescharacteristic
432C2×Q8—2922no—
433C2×Q8—2922no—
434C2×Q8—5841no—
435C2×Q8—732no—
436C2×Q8—1461no—
437C2×Q8—1461no—
438C2×Q8—1461no—
439C2×Q8—1461no—
440C2×Q8—1461no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—1461no—
446C22×C4—1461no—
447C22×C4—1461no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4C2×D29211yescharacteristic
454C22×C4C2×D29211yescharacteristic
455C22×C4C2×D29211yescharacteristic
456C22×C4C2×D29211yescharacteristic
457C22×C4C2×D29211yescharacteristic
458C22×C4Q8×D7312yes—
459C22×C4#1192402712yes—
460C22×C4C23×D7311yescharacteristic
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—44no—
474C4⋊C4—2922no—
475C4⋊C4—2922no—
476C4⋊C4—2922no—
477C4⋊C4—2922no—
478C4⋊C4—5841no—
479C4⋊C4—732no—
480C4⋊C4—732no—
481C4⋊C4—732no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1464no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C22⋊C4—22no—
495C22⋊C4—41no—
496C22⋊C4—2922no—
497C22⋊C4—2922no—
498C22⋊C4—5841no—
499C22⋊C4—1464no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—2922no—
505C42—2922no—
506C42—2922no—
507C42—2922no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23#1455657112yes—
515C23C22×D29211yescharacteristic
516Q8—2921no—
517Q8—2921no—
518Q8—5844no—
519Q8—1462no—
520Q8—1462no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—1462no—
559C2×C4—1462no—
560C2×C4—1462no—
561C2×C4—1462no—
562C2×C4#1455657112yes—
563C2×C4#1455661312yes—
564C2×C4C2×Q8×D7311yescharacteristic
565C2×C4#41×D7311yescharacteristic
566C2×C4#41×D7311yescharacteristic
567C2×C4C2×#1192402711yescharacteristic
568C2×C4#1455663611yescharacteristic
569C2×C4#1455663611yescharacteristic
570C8—41no—
571C8—41no—
572C8—41no—
573C8—41no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C2×#1455657111yescharacteristic
582C22#1798923211yescharacteristic
583C22#1798923211yescharacteristic
584C22C2×#1455661311yescharacteristic
585C22#1798938711yescharacteristic
586C22#1798939911yescharacteristic
587C22#1798940411yescharacteristic
588C4—22no—
589C4—22no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—2922no—
594C4—2922no—
595C4—2922no—
596C4—2922no—
597C4—5841no—
598C4—5841no—
599C4—5841no—
600C4—5841no—
601C4#1798911411yescharacteristic
602C4#1798917211yescharacteristic
603C4#1798929111yescharacteristic
604C4#1798929111yescharacteristic
605C2—42no—
606C2#2261375311yescharacteristic
607C2#2261972111yescharacteristic
608C2#2261977411yescharacteristic
609C2#2262176811yescharacteristic
610C2#2262184911yescharacteristic
611C2#2262199911yescharacteristic
612C2#2262211911yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1850C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#1850D73
hypocenterC73#26413
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
3. #22611500C2semidirect
4. #22611424C2semidirect
5. #22611325C2semidirect
6. #22611073C2semidirect
7. #22610882C2semidirect
8. #22610517C2semidirect
11. #22610202C2semidirect
13. #22609947C2semidirect
64. #14556570Q8semidirect
322. C73#26413semidirect

Maximal subgroups

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

subgroupindex
#226136222
#226129562
#226115002
#226114242
#226113252
#226110732
#226108822
#226105172
#226103542
#226102412
#226102022
#226101322
#226099472
#226096482
C73×#18502
#2641373