Finite Group Databasebeta

← #33016081

Subgroups — #33016081

Order 18688.

1706 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988499C211yesmaximal, characteristic
2C2×#17988491C212yesmaximal, complemented
3C2×#17988459C212yesmaximal
4#22608194C212yesmaximal, complemented
5#22608186C212yesmaximal, complemented
6#22608168C212yesmaximal
7#22608155C212yesmaximal, complemented
8#22608153C211yesmaximal, characteristic
9C73×#1439C211yesmaximal, characteristic
10C23×Dic146C2211yescharacteristic
11C24×Dic73C2211yescharacteristic
12C22×#11924007C2212yes—
13C22×#11924007C2212yes—
14C22×C22⋊Dic73C2212yes—
15C22×C22⋊Dic73C2211yescharacteristic
16C22×C4×Dic73C2212yes—
17#17988499C418yescomplemented
18#17988491—28no—
19#17988459—28no—
20C2×#14556498C2212yes—
21C2×#14556498C2212yes—
22C2×#14556498C2212yes—
23C2×#14556498C2212yes—
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2211yescharacteristic
28C2×#14556498C2212yes—
29C22×C4⋊C4×C73C2211yescharacteristic
30C22×C22⋊C4×C73C2212yes—
31C22×C22⋊C4×C73C2211yescharacteristic
32C2×#95×C73C2212yes—
33C2×#95×C73C2211yescharacteristic
34C22×Dic146—24no—
35C22×Dic146—24no—
36C22×Dic146—41no—
37C22×Dic146—41no—
38C22×Dic146C2×C414yescomplemented
39C23×Dic73—24no—
40C23×Dic73—22no—
41C23×Dic73—42no—
42C23×Dic73—41no—
43C23×Dic73—41no—
44C23×Dic73C2312yes—
45C23×Dic73C2312yes—
46C23×Dic73C2312yes—
47C23×Dic73C2312yes—
48C2×#11924007—24no—
49C2×#11924007—24no—
50C2×#11924007—24no—
51C2×#11924007—42no—
52C2×#11924007—42no—
53C2×#11924007—42no—
54C2×C22⋊Dic73—24no—
55C2×C22⋊Dic73—24no—
56C2×C22⋊Dic73—41no—
57C2×C22⋊Dic73—41no—
58C2×C22⋊Dic73C2×C418yes—
59C2×C22⋊Dic73Q814yescomplemented
60C2×C4×Dic73—24no—
61C2×C4×Dic73—24no—
62C2×C4×Dic73—42no—
63#14556498—24no—
64#14556498—24no—
65#14556498—24no—
66#14556498—42no—
67#14556498—42no—
68#14556498—42no—
69#14556498C2×C418yescomplemented
70#14556498C2×C414yescomplemented
71#14556498D418yes—
72C25×C73C2311yescharacteristic
73C23×C4×C73C2312yes—
74C23×C4×C73C2311yescharacteristic
75C23×C4×C73C2311yescharacteristic
76C23×C4×C73C2312yes—
77C2×C4⋊C4×C73—41no—
78C2×C4⋊C4×C73—42no—
79C2×C22⋊C4×C73—24no—
80C2×C22⋊C4×C73—24no—
81C2×C22⋊C4×C73—41no—
82C2×C22⋊C4×C73—41no—
83C2×C22⋊C4×C73—42no—
84C2×C22⋊C4×C73C2×C414yes—
85#95×C73—24no—
86#95×C73—41no—
87C2×Dic146—216no—
88C2×Dic146—44no—
89C2×Dic146—44no—
90C2×Dic146—48no—
91C2×Dic146—44no—
92C22×Dic73—22no—
93C22×Dic73—22no—
94C22×Dic73—22no—
95C22×Dic73—22no—
96C22×Dic73—22no—
97C22×Dic73—22no—
98C22×Dic73—22no—
99C22×Dic73—22no—
100C22×Dic73—22no—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—44no—
115C22×Dic73—48no—
116C22×Dic73—48no—
117C22×Dic73—44no—
118C22×Dic73—44no—
119C22×Dic73C22×C414yes—
120C22×Dic73C22×C414yes—
121C22×Dic73C2×D414yes—
122C22×Dic73C2×D414yes—
123C22×Dic73C2×Q814yes—
124C22×Dic73#4114yes—
125#11924007—48no—
126#11924007—48no—
127C22⋊Dic73—216no—
128C22⋊Dic73—44no—
129C4×Dic73—48no—
130C24×C73—22no—
131C24×C73—22no—
132C24×C73—21no—
133C24×C73—21no—
134C24×C73—22no—
135C24×C73—21no—
136C24×C73—21no—
137C24×C73—22no—
138C24×C73C22×C412yes—
139C24×C73C2×Q812yes—
140C24×C73#4112yes—
141C24×C73C2411yescharacteristic
142C22×C4×C73—22no—
143C22×C4×C73—22no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—21no—
147C22×C4×C73—22no—
148C22×C4×C73—21no—
149C22×C4×C73—22no—
150C22×C4×C73—21no—
151C22×C4×C73—21no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73C22×C412yes—
164C22×C4×C73C22×C412yes—
165C22×C4×C73C2×D414yes—
166C4⋊C4×C73—44no—
167C22⋊C4×C73—44no—
168C22⋊C4×C73—48no—
169C23×C73—24no—
170C23×C73—22no—
171C23×C73—24no—
172C23×C73—22no—
173C23×C73—22no—
174C23×C73—22no—
175C23×C73—22no—
176C23×C73—22no—
177C23×C73—24no—
178C23×C73—24no—
179C23×C73—24no—
180C23×C73—24no—
181C23×C73—42no—
182C23×C73—42no—
183C23×C73—42no—
184C23×C73—42no—
185C23×C73—41no—
186C23×C73—42no—
187C23×C73—41no—
188C23×C73—41no—
189C23×C73—42no—
190C23×C73—41no—
191C23×C73C4×Q814yes—
192C23×C73C23×C411yescharacteristic
193C23×C73C22×D412yes—
194C23×C73C22×Q811yescharacteristic
195C23×C73C2×#4112yes—
196C23×C73C2×#4111yescharacteristic
197C23×C73H(5,𝔽2)11yescharacteristic
198C23×C73H(5,𝔽2)11yescharacteristic
199C23×C73H(5,𝔽2)12yes—
200C23×C73H(5,𝔽2)11yescharacteristic
201C23×C732−1+411yescharacteristic
202C23×C732−1+412yes—
203C2×C4×C73—24no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—24no—
207C2×C4×C73—24no—
208C2×C4×C73—24no—
209C2×C4×C73—24no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—41no—
213C2×C4×C73—42no—
214C2×C4×C73—41no—
215C2×C4×C73—42no—
216C2×C4×C73—41no—
217C2×C4×C73—42no—
218C2×C4×C73—42no—
219C2×C4×C73—41no—
220C2×C4×C73—42no—
221C2×C4×C73—42no—
222C2×C4×C73—42no—
223C2×C4×C73—42no—
224C2×C4×C73—42no—
225C2×C4×C73—42no—
226C2×C4×C73—42no—
227C2×Dic73—24no—
228C2×Dic73—24no—
229C2×Dic73—24no—
230C2×Dic73—24no—
231C2×Dic73—24no—
232C2×Dic73—24no—
233C2×Dic73—24no—
234C2×Dic73—24no—
235C2×Dic73—24no—
236C2×Dic73—24no—
237C2×Dic73—24no—
238C2×Dic73—24no—
239C2×Dic73—24no—
240C2×Dic73—24no—
241C2×Dic73—24no—
242C2×Dic73—416no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—42no—
249C2×Dic73—42no—
250C2×Dic73—42no—
251C2×Dic73—42no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—42no—
255C2×Dic73C4×D418yes—
256Dic146—416no—
257C22×C73—24no—
258C22×C73—28no—
259C22×C73—44no—
260C22×C73—42no—
261C22×C73—42no—
262C22×C73—42no—
263C22×C73—42no—
264C22×C73—44no—
265C22×C73—44no—
266C22×C73—42no—
267C22×C73—42no—
268C22×C73C2×C4×D412yes—
269C22×C73C2×C4×Q811yescharacteristic
270C22×C73#51811yescharacteristic
271C22×C73#52012yes—
272C22×C73#52011yescharacteristic
273C22×C73#53512yes—
274C22×C73#53612yes—
275C22×C73#53811yescharacteristic
276C22×C73#53911yescharacteristic
277C22×C73#54212yes—
278C22×C73#54212yes—
279C22×C73#54311yescharacteristic
280C22×C73#54411yescharacteristic
281C22×C73D4211yescharacteristic
282C22×C73#54612yes—
283C22×C73#54612yes—
284C22×C73D4×Q812yes—
285C22×C73#55111yescharacteristic
286C22×C73#55212yes—
287C22×C73#55412yes—
288C22×C73H(3,𝔽4)11yescharacteristic
289C22×C73#56212yes—
290C22×C73#56311yescharacteristic
291C4×C73—42no—
292C4×C73—42no—
293C4×C73—44no—
294C4×C73—82no—
295Dic73—28no—
296Dic73—44no—
297Dic73—44no—
298Dic73—44no—
299#14575—731nomaximal, Sylow
300C2×C73—44no—
301C2×C73#232612yes—
302C2×C73#232911yescharacteristic
303C2×C73#235811yescharacteristic
304C2×C73#236012yes—
305C2×C73#237911yescharacteristic
306C2×C73#267011yescharacteristic
307C2×C73#267711yescharacteristic
308C2×C73#268712yes—
309C2×C73#269212yes—
310C2×C73#276311yescharacteristic
311C2×C73#276411yescharacteristic
312C2×#393—731no—
313C2×#391—732no—
314C2×#385—732no—
315#1444—732no—
316#1442—732no—
317#1439D7311yescharacteristic
318#1437—732no—
319#1434—732no—
320C22⋊#95—731no—
321C73#1457511yescharacteristic, complemented, Sylow
322C23×Q8—731no—
323C24×C4—731no—
324C22×C4⋊C4—732no—
325C22×C4⋊C4—732no—
326C22×C4⋊C4C2×D7311yescharacteristic
327C22×C22⋊C4—731no—
328C22×C22⋊C4—732no—
329C22×C22⋊C4C2×D7312yes—
330C22×C22⋊C4C2×D7311yescharacteristic
331C22×C42—732no—
332#393—738no—
333#391—1468no—
334#385—1468no—
335C2×#95—732no—
336C2×#95—732no—
337C2×#95—732no—
338C2×#95—732no—
339C2×#95—731no—
340C2×#95—732no—
341C2×#95—732no—
342C2×#95—732no—
343C2×#95—732no—
344C2×#95C2×D7312yes—
345C2×#95C2×D7311yescharacteristic
346C25C22×D7311yescharacteristic
347C22×Q8—2921no—
348C22×Q8—2921no—
349C22×Q8—734no—
350C22×Q8—1464no—
351C22×Q8—1464no—
352C23×C4—2921no—
353C23×C4—2921no—
354C23×C4—2922no—
355C23×C4—732no—
356C23×C4—732no—
357C23×C4—732no—
358C23×C4—732no—
359C23×C4—1464no—
360C23×C4—1462no—
361C23×C4C22×D7312yes—
362C23×C4C22×D7311yescharacteristic
363C23×C4C22×D7311yescharacteristic
364C23×C4C22×D7312yes—
365C2×C4⋊C4—42no—
366C2×C4⋊C4—41no—
367C2×C4⋊C4—2922no—
368C2×C4⋊C4—2922no—
369C2×C4⋊C4—2922no—
370C2×C4⋊C4—1464no—
371C2×C4⋊C4—1464no—
372C2×C4⋊C4—1464no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—41no—
376C2×C22⋊C4—41no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—2921no—
379C2×C22⋊C4—2921no—
380C2×C22⋊C4—734no—
381C2×C22⋊C4—738no—
382C2×C22⋊C4—1464no—
383C2×C22⋊C4—1464no—
384C2×C22⋊C4C4×D7314yes—
385C2×C42—2922no—
386C2×C42—1464no—
387C2×C42—1464no—
388#95—24no—
389#95—41no—
390#95—2922no—
391#95—2922no—
392#95—2922no—
393#95—738no—
394#95—734no—
395#95—738no—
396#95—1464no—
397#95—1464no—
398#95—1464no—
399C24—22no—
400C24—22no—
401C24—21no—
402C24—21no—
403C24—22no—
404C24—21no—
405C24—21no—
406C24—22no—
407C24C2×C4×D7312yes—
408C24Q8×D7312yes—
409C24#1192402712yes—
410C24C23×D7311yescharacteristic
411C2×Q8—2924no—
412C2×Q8—2924no—
413C2×Q8—2928no—
414C2×Q8—2924no—
415C2×Q8—14616no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—2924no—
438C22×C4—2924no—
439C22×C4—2928no—
440C22×C4—2928no—
441C22×C4—2924no—
442C22×C4—734no—
443C22×C4—734no—
444C22×C4—734no—
445C22×C4—734no—
446C22×C4—734no—
447C22×C4—734no—
448C22×C4—1462no—
449C22×C4—1462no—
450C22×C4—1462no—
451C22×C4—1462no—
452C22×C4—1462no—
453C22×C4—1462no—
454C22×C4—1462no—
455C22×C4—1462no—
456C22×C4—1462no—
457C22×C4—1462no—
458C22×C4—1462no—
459C22×C4—1462no—
460C22×C4—1462no—
461C22×C4—1462no—
462C22×C4—1462no—
463C22×C4—1462no—
464C22×C4—1462no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4—1462no—
470C22×C4C2×C4×D7312yes—
471C22×C4C2×C4×D7312yes—
472C22×C4D4×D7314yes—
473C4⋊C4—44no—
474C4⋊C4—2928no—
475C4⋊C4—2928no—
476C22⋊C4—44no—
477C22⋊C4—48no—
478C22⋊C4—2924no—
479C22⋊C4—14616no—
480C42—2928no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—24no—
492C23—22no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—41no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—41no—
503C23#1455655614yes—
504C23C22×C4×D7311yescharacteristic
505C23C2×D4×D7312yes—
506C23C2×Q8×D7311yescharacteristic
507C23#41×D7312yes—
508C23C2×#1192402711yescharacteristic
509C23#1455663612yes—
510C23#1455663611yescharacteristic
511C23#1455663611yescharacteristic
512C23#1455663711yescharacteristic
513C23#1455663811yescharacteristic
514C23#1455663812yes—
515Q8—29216no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—29216no—
541C2×C4—2922no—
542C2×C4—2922no—
543C2×C4—2922no—
544C2×C4—2922no—
545C2×C4—2922no—
546C2×C4—2922no—
547C2×C4—2922no—
548C2×C4—2922no—
549C2×C4—2922no—
550C2×C4—2922no—
551C2×C4—2922no—
552C2×C4—2922no—
553C2×C4—738no—
554C2×C4—1464no—
555C2×C4—1464no—
556C2×C4—1464no—
557C2×C4—1464no—
558C2×C4—1464no—
559C2×C4—1464no—
560C2×C4—1464no—
561C2×C4—1464no—
562C2×C4—1464no—
563C2×C4—1464no—
564C2×C4—1464no—
565C2×C4—1464no—
566C2×C4—1464no—
567C2×C4—1464no—
568C2×C4—1464no—
569C22—24no—
570C22—28no—
571C22—42no—
572C22—42no—
573C22—44no—
574C22—42no—
575C22—42no—
576C22—44no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22C4×D4×D7312yes—
581C22C2×#1455655611yescharacteristic
582C22#1798906011yescharacteristic
583C22#1798906612yes—
584C22#1798906711yescharacteristic
585C22#1798915312yes—
586C22#1798915912yes—
587C22#1798917411yescharacteristic
588C22#1798918211yescharacteristic
589C22#1798920112yes—
590C22#1798920312yes—
591C22#1798921211yescharacteristic
592C22#1798921611yescharacteristic
593C22#1798922211yescharacteristic
594C22#1798922412yes—
595C22#1798922612yes—
596C22#1798925012yes—
597C22#1798926711yescharacteristic
598C22#1798927412yes—
599C22#1798929112yes—
600C22#1798933011yescharacteristic
601C22#1798933212yes—
602C22#1798934011yescharacteristic
603C4—44no—
604C4—42no—
605C4—42no—
606C4—82no—
607C4—2924no—
608C4—2924no—
609C4—2924no—
610C4—1468no—
611C2—44no—
612C2#2261264012yes—
613C2#2261266111yescharacteristic
614C2#2261291211yescharacteristic
615C2#2261293112yes—
616C2#2261310111yescharacteristic
617C2#2261615811yescharacteristic
618C2#2261624711yescharacteristic
619C2#2261635812yes—
620C2#2261643312yes—
621C2#2261734911yescharacteristic
622C2#2261735811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1439C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenter#1439D73
hypocenterC73#14575
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#17988491C2semidirect
4. #22608194C2semidirect
5. #22608186C2semidirect
7. #22608155C2semidirect
17. #17988499C4semidirect
38. C22×Dic146C2×C4semidirect
59. C2×C22⋊Dic73Q8semidirect
69. #14556498C2×C4semidirect
70. #14556498C2×C4semidirect
321. C73#14575semidirect

Maximal subgroups

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

subgroupindex
C2×#179884992
C2×#179884912
C2×#179884592
#226081942
#226081862
#226081682
#226081552
#226081532
C73×#14392
#1457573