Finite Group Databasebeta

← #33014530

Subgroups — #33014530

Order 18688.

1450 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988447C211yesmaximal, characteristic
2#22608191C212yesmaximal, complemented
3#22608183C212yesmaximal, complemented
4#22608174C212yesmaximal
5#22608168C212yesmaximal
6#22608166C212yesmaximal, complemented
7#22608165C212yesmaximal, complemented
8#22608154C211yesmaximal, characteristic
9C73×C22⋊#95C211yesmaximal, characteristic
10C22×C4⋊Dic73C2211yescharacteristic
11C22×C4⋊Dic73C2211yescharacteristic
12C22×#11924007C2212yes—
13C22×C22⋊Dic73C2211yescharacteristic
14C22×C22⋊Dic73C2212yes—
15C22×C22⋊Dic73C2211yescharacteristic
16C22×C4×Dic73C2212yes—
17#17988447C418yescomplemented
18C2×#14556498C2212yes—
19C2×#14556498C2212yes—
20C2×#14556498C2212yes—
21C2×#14556498C2212yes—
22C2×#14556498C2212yes—
23C2×#14556498C2212yes—
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C24×C4×C73C2211yescharacteristic
28C22×C22⋊C4×C73C2212yes—
29C2×#95×C73C2212yes—
30C2×#95×C73C2211yescharacteristic
31C2×#95×C73C2211yescharacteristic
32C23×Dic73C2312yes—
33C23×Dic73C2312yes—
34C23×Dic73C2312yes—
35C23×Dic73C2312yes—
36C2×C4⋊Dic73—41no—
37C2×C4⋊Dic73—42no—
38C2×C4⋊Dic73—41no—
39C2×C4⋊Dic73C2×C414yes—
40C2×C4⋊Dic73C2×C414yes—
41C2×#11924007—24no—
42C2×#11924007—42no—
43C2×#11924007—42no—
44C2×C22⋊Dic73—22no—
45C2×C22⋊Dic73—41no—
46C2×C22⋊Dic73—41no—
47C2×C22⋊Dic73—42no—
48C2×C22⋊Dic73—42no—
49C2×C22⋊Dic73—41no—
50C2×C22⋊Dic73—41no—
51C2×C22⋊Dic73C2×C418yes—
52C2×C22⋊Dic73Q814yes—
53C2×C4×Dic73—42no—
54C2×C4×Dic73—42no—
55C2×C4×Dic73—42no—
56#14556498—24no—
57#14556498—24no—
58#14556498—42no—
59#14556498—42no—
60#14556498—42no—
61#14556498—42no—
62#14556498—42no—
63#14556498—42no—
64#14556498C2×C418yes—
65C25×C73C2311yescharacteristic
66C23×C4×C73—24no—
67C23×C4×C73—41no—
68C23×C4×C73—41no—
69C23×C4×C73—42no—
70C23×C4×C73C2×C414yes—
71C23×C4×C73C2312yes—
72C23×C4×C73C2311yescharacteristic
73C23×C4×C73C2312yes—
74C23×C4×C73C2311yescharacteristic
75C2×C22⋊C4×C73—24no—
76C2×C22⋊C4×C73—24no—
77C2×C22⋊C4×C73—42no—
78#95×C73—24no—
79#95×C73—41no—
80#95×C73—41no—
81C22×Dic73—22no—
82C22×Dic73—22no—
83C22×Dic73—22no—
84C22×Dic73—22no—
85C22×Dic73—22no—
86C22×Dic73—22no—
87C22×Dic73—22no—
88C22×Dic73—22no—
89C22×Dic73—22no—
90C22×Dic73—22no—
91C22×Dic73—22no—
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×Dic73C22×C414yes—
107C22×Dic73C22×C414yes—
108C22×Dic73C2×Q814yes—
109C4⋊Dic73—44no—
110C4⋊Dic73—44no—
111#11924007—48no—
112C22⋊Dic73—48no—
113C22⋊Dic73—44no—
114C22⋊Dic73—44no—
115C4×Dic73—48no—
116C24×C73—22no—
117C24×C73—22no—
118C24×C73—22no—
119C24×C73—21no—
120C24×C73—21no—
121C24×C73—22no—
122C24×C73—21no—
123C24×C73—21no—
124C24×C73C22×C412yes—
125C24×C73C2×Q812yes—
126C24×C73#4112yes—
127C24×C73C2411yescharacteristic
128C22×C4×C73—21no—
129C22×C4×C73—21no—
130C22×C4×C73—22no—
131C22×C4×C73—22no—
132C22×C4×C73—22no—
133C22×C4×C73—22no—
134C22×C4×C73—21no—
135C22×C4×C73—21no—
136C22×C4×C73—22no—
137C22×C4×C73—22no—
138C22×C4×C73—22no—
139C22×C4×C73—22no—
140C22×C4×C73—22no—
141C22×C4×C73—28no—
142C22×C4×C73—22no—
143C22×C4×C73—22no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73—44no—
150C22×C4×C73—44no—
151C22×C4×C73—48no—
152C22×C4×C73—48no—
153C22×C4×C73C22×C412yes—
154C22×C4×C73C22×C412yes—
155C22×C4×C73#4114yes—
156C22×C4×C73#4114yes—
157C22⋊C4×C73—48no—
158C23×C73—22no—
159C23×C73—22no—
160C23×C73—22no—
161C23×C73—22no—
162C23×C73—22no—
163C23×C73—22no—
164C23×C73—22no—
165C23×C73—22no—
166C23×C73—22no—
167C23×C73—22no—
168C23×C73—24no—
169C23×C73—24no—
170C23×C73—24no—
171C23×C73—24no—
172C23×C73—42no—
173C23×C73—42no—
174C23×C73—42no—
175C23×C73—42no—
176C23×C73—42no—
177C23×C73—42no—
178C23×C73—42no—
179C23×C73—42no—
180C23×C73C4×Q814yes—
181C23×C73C23×C411yescharacteristic
182C23×C73C22×Q811yescharacteristic
183C23×C73C2×#4112yes—
184C23×C73C2×#4111yescharacteristic
185C23×C73C2×#4112yes—
186C23×C73H(5,𝔽2)11yescharacteristic
187C23×C73H(5,𝔽2)11yescharacteristic
188C23×C73H(5,𝔽2)12yes—
189C23×C73H(5,𝔽2)12yes—
190C23×C73H(5,𝔽2)11yescharacteristic
191C23×C732−1+411yescharacteristic
192C2×C4×C73—22no—
193C2×C4×C73—22no—
194C2×C4×C73—24no—
195C2×C4×C73—24no—
196C2×C4×C73—22no—
197C2×C4×C73—22no—
198C2×C4×C73—22no—
199C2×C4×C73—24no—
200C2×C4×C73—24no—
201C2×C4×C73—24no—
202C2×C4×C73—416no—
203C2×C4×C73—42no—
204C2×C4×C73—42no—
205C2×C4×C73—42no—
206C2×C4×C73—42no—
207C2×C4×C73—42no—
208C2×C4×C73—42no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—42no—
213C2×C4×C73—42no—
214C2×C4×C73—42no—
215C2×C4×C73—42no—
216C2×C4×C73#11714yes—
217C2×Dic73—24no—
218C2×Dic73—24no—
219C2×Dic73—24no—
220C2×Dic73—24no—
221C2×Dic73—24no—
222C2×Dic73—24no—
223C2×Dic73—24no—
224C2×Dic73—24no—
225C2×Dic73—24no—
226C2×Dic73—42no—
227C2×Dic73—42no—
228C2×Dic73—42no—
229C2×Dic73—42no—
230C2×Dic73—42no—
231C2×Dic73—42no—
232C2×Dic73—42no—
233C2×Dic73—42no—
234C2×Dic73—42no—
235C2×Dic73—42no—
236C2×Dic73—42no—
237C2×Dic73—42no—
238C2×Dic73—42no—
239C2×Dic73—42no—
240C2×Dic73—42no—
241C2×Dic73—42no—
242C2×Dic73—42no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C22×C73—24no—
246C22×C73—24no—
247C22×C73—24no—
248C22×C73—42no—
249C22×C73—42no—
250C22×C73—42no—
251C22×C73—42no—
252C22×C73—44no—
253C22×C73—44no—
254C22×C73—44no—
255C22×C73—44no—
256C22×C73C2×#11711yescharacteristic
257C22×C73C2×C4×Q811yescharacteristic
258C22×C73C4×#4111yescharacteristic
259C22×C73#51811yescharacteristic
260C22×C73#51811yescharacteristic
261C22×C73#51811yescharacteristic
262C22×C73#52011yescharacteristic
263C22×C73#53812yes—
264C22×C73#53812yes—
265C22×C73#53812yes—
266C22×C73#53811yescharacteristic
267C22×C73#53911yescharacteristic
268C22×C73#54012yes—
269C22×C73#54311yescharacteristic
270C22×C73#54411yescharacteristic
271C22×C73#55112yes—
272C22×C73#55312yes—
273C22×C73#55312yes—
274C22×C73#55412yes—
275C22×C73#55412yes—
276C22×C73#55612yes—
277C22×C73#56012yes—
278C22×C73#56212yes—
279C4×C73—24no—
280C4×C73—42no—
281C4×C73—44no—
282C4×C73—82no—
283Dic73—44no—
284Dic73—44no—
285Dic73—44no—
286Dic73—82no—
287#14453—731nomaximal, Sylow
288C2×C73—44no—
289C2×C73#231211yescharacteristic
290C2×C73#232911yescharacteristic
291C2×C73#233311yescharacteristic
292C2×C73#234511yescharacteristic
293C2×C73#234711yescharacteristic
294C2×C73#236811yescharacteristic
295C2×C73#237711yescharacteristic
296C2×C73#277812yes—
297C2×C73#278012yes—
298C2×C73#278612yes—
299C2×C73#278812yes—
300C2×#380—731no—
301#1443—732no—
302#1441—732no—
303#1439—732no—
304#1437—732no—
305#1436—732no—
306#1436—732no—
307C22⋊#95—731no—
308C22⋊#95D7311yescharacteristic
309C73#1445311yescharacteristic, complemented, Sylow
310C24×C4C2×D7311yescharacteristic
311C22×C4⋊C4—731no—
312C22×C4⋊C4—731no—
313C22×C4⋊C4—732no—
314C22×C22⋊C4—731no—
315C22×C22⋊C4—732no—
316C22×C22⋊C4—731no—
317C22×C22⋊C4C2×D7312yes—
318C22×C42—732no—
319#380—738no—
320C2×#95—732no—
321C2×#95—732no—
322C2×#95—732no—
323C2×#95—732no—
324C2×#95—732no—
325C2×#95—732no—
326C2×#95—732no—
327C2×#95—732no—
328C2×#95—732no—
329C2×#95C2×D7312yes—
330C2×#95C2×D7311yescharacteristic
331C2×#95C2×D7311yescharacteristic
332C25C22×D7311yescharacteristic
333C23×C4—24no—
334C23×C4—41no—
335C23×C4—41no—
336C23×C4—42no—
337C23×C4—732no—
338C23×C4—732no—
339C23×C4—732no—
340C23×C4—732no—
341C23×C4C4×D7314yes—
342C23×C4C22×D7312yes—
343C23×C4C22×D7311yescharacteristic
344C23×C4C22×D7312yes—
345C23×C4C22×D7311yescharacteristic
346C2×C4⋊C4—2922no—
347C2×C4⋊C4—2922no—
348C2×C4⋊C4—2921no—
349C2×C4⋊C4—2922no—
350C2×C4⋊C4—2921no—
351C2×C4⋊C4—734no—
352C2×C4⋊C4—734no—
353C2×C4⋊C4—1464no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—42no—
357C2×C22⋊C4—2921no—
358C2×C22⋊C4—2921no—
359C2×C22⋊C4—2921no—
360C2×C22⋊C4—2922no—
361C2×C22⋊C4—2922no—
362C2×C22⋊C4—2921no—
363C2×C22⋊C4—738no—
364C2×C22⋊C4—734no—
365C2×C22⋊C4—1462no—
366C2×C42—2922no—
367C2×C42—2922no—
368C2×C42—2922no—
369#95—24no—
370#95—41no—
371#95—41no—
372#95—2922no—
373#95—2922no—
374#95—2922no—
375#95—2922no—
376#95—2922no—
377#95—2922no—
378#95—738no—
379#95—1464no—
380#95—1464no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—21no—
385C24—21no—
386C24—21no—
387C24—22no—
388C24—21no—
389C24C2×C4×D7312yes—
390C24Q8×D7312yes—
391C24#1192402712yes—
392C24C23×D7311yescharacteristic
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—734no—
419C22×C4—734no—
420C22×C4—734no—
421C22×C4—1462no—
422C22×C4—1462no—
423C22×C4—1462no—
424C22×C4—1462no—
425C22×C4—1462no—
426C22×C4—1462no—
427C22×C4—1462no—
428C22×C4—1462no—
429C22×C4—1462no—
430C22×C4—1462no—
431C22×C4—1462no—
432C22×C4—1462no—
433C22×C4—1462no—
434C22×C4—1462no—
435C22×C4—1462no—
436C22×C4—1462no—
437C22×C4—1462no—
438C22×C4—1462no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4—1462no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4C2×C4×D7312yes—
447C22×C4C2×C4×D7312yes—
448C22×C4#1192402714yes—
449C22×C4#1192402814yes—
450C4⋊C4—2928no—
451C4⋊C4—2924no—
452C4⋊C4—2924no—
453C22⋊C4—48no—
454C22⋊C4—2924no—
455C22⋊C4—2924no—
456C22⋊C4—2928no—
457C42—2928no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23#1455655614yes—
481C23C22×C4×D7311yescharacteristic
482C23C2×Q8×D7311yescharacteristic
483C23C2×#1192402712yes—
484C23C2×#1192402711yescharacteristic
485C23C2×#1192402812yes—
486C23#1455663611yescharacteristic
487C23#1455663611yescharacteristic
488C23#1455663612yes—
489C23#1455663712yes—
490C23#1455663711yescharacteristic
491C23#1455663811yescharacteristic
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—416no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—2922no—
517C2×C4—2922no—
518C2×C4—2922no—
519C2×C4—2922no—
520C2×C4—2922no—
521C2×C4—2922no—
522C2×C4—2922no—
523C2×C4—2922no—
524C2×C4—2922no—
525C2×C4—2922no—
526C2×C4—2922no—
527C2×C4—2922no—
528C2×C4—2922no—
529C2×C4—2922no—
530C2×C4—2922no—
531C2×C4—2922no—
532C2×C4—2922no—
533C2×C4—2922no—
534C2×C4—2922no—
535C2×C4—1464no—
536C2×C4—1464no—
537C2×C4—1464no—
538C2×C4—1464no—
539C2×C4—1464no—
540C2×C4—1464no—
541C2×C4—1464no—
542C2×C4—1464no—
543C2×C4—1464no—
544C2×C4#1455654714yes—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—42no—
549C22—44no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—42no—
555C22—44no—
556C22C2×#1455654711yescharacteristic
557C22C2×#1455655611yescharacteristic
558C22C4×#1192402811yescharacteristic
559C22#1798906011yescharacteristic
560C22#1798906011yescharacteristic
561C22#1798906111yescharacteristic
562C22#1798906711yescharacteristic
563C22#1798917012yes—
564C22#1798917212yes—
565C22#1798917212yes—
566C22#1798917411yescharacteristic
567C22#1798918211yescharacteristic
568C22#1798918812yes—
569C22#1798921211yescharacteristic
570C22#1798921611yescharacteristic
571C22#1798926312yes—
572C22#1798928312yes—
573C22#1798928912yes—
574C22#1798929212yes—
575C22#1798929512yes—
576C22#1798930712yes—
577C22#1798932712yes—
578C22#1798933412yes—
579C4—24no—
580C4—42no—
581C4—44no—
582C4—82no—
583C4—2924no—
584C4—2924no—
585C4—2924no—
586C4—5842no—
587C2—44no—
588C2#2261252711yescharacteristic
589C2#2261266111yescharacteristic
590C2#2261269611yescharacteristic
591C2#2261281011yescharacteristic
592C2#2261282211yescharacteristic
593C2#2261301111yescharacteristic
594C2#2261309011yescharacteristic
595C2#2261754912yes—
596C2#2261757912yes—
597C2#2261764712yes—
598C2#2261767712yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×C22⋊#95C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC22⋊#95D73
hypocenterC73#14453
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. #22608191C2semidirect
3. #22608183C2semidirect
6. #22608166C2semidirect
7. #22608165C2semidirect
17. #17988447C4semidirect
309. C73#14453semidirect

Maximal subgroups

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

subgroupindex
C2×#179884472
#226081912
#226081832
#226081742
#226081682
#226081662
#226081652
#226081542
C73×C22⋊#952
#1445373