Finite Group Databasebeta

← #33118216

Subgroups — #33118216

Order 18688.

1240 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988742C211yesmaximal, characteristic, complemented
2C22×#14556514C211yesmaximal, characteristic, complemented
3#22613296C211yesmaximal, characteristic
4#22611079C212yesmaximal, complemented
5#22610939C212yesmaximal, complemented
6#22610355C211yesmaximal, characteristic, complemented
7#22610272C212yesmaximal, complemented
8#22610243C212yesmaximal, complemented
9#22610224C211yesmaximal, characteristic
10#22610017C211yesmaximal, characteristic, complemented
11C2×#552×C73C211yesmaximal, characteristic
12C23×Dic146C2211yescharacteristic
13C22×#11924018C2212yes—
14C23×#10596831C2211yescharacteristic
15C2×#14556570C2212yes—
16#17988742—24no—
17C2×#14556515C2212yes—
18C2×#14556514—24no—
19C2×#14556514C2212yes—
20C2×#14556514C2212yes—
21C2×#14556514C2212yes—
22C2×C22⋊#10596831C2211yescharacteristic
23C2×C22⋊#10596831C2212yes—
24#17988507—21no—
25#17988499—21no—
26#17988493C2212yes—
27#17988467—21no—
28#17988459—21no—
29#17988447C2211yescharacteristic
30#17988349C2212yes—
31#17988349C2212yes—
32#17988349C2212yes—
33#17988342C2212yes—
34#552×C73—24no—
35#552×C73—24no—
36C2×#126×C73—21no—
37C2×#125×C73—21no—
38C2×#125×C73C2211yescharacteristic
39C2×#123×C73—21no—
40C2×C22⋊Q8×C73C2211yescharacteristic
41C2×C22⋊Q8×C73C2211yescharacteristic
42C2×C4×Q8×C73C2211yescharacteristic
43C2×C4×D4×C73C2211yescharacteristic
44C2×#117×C73—21no—
45C2×#117×C73C2211yescharacteristic
46C22×C4⋊C4×C73C2211yescharacteristic
47C22×Dic146—24no—
48C22×Dic146—21no—
49C22×Dic146—22no—
50C22×Dic146—22no—
51C22×Dic146—42no—
52C22×Dic146C2312yes—
53C23×Dic73—21no—
54C2×#11924018—24no—
55C2×#11924018—24no—
56C2×#11924018—42no—
57C22×#10596831—22no—
58C22×#10596831—42no—
59C22×#10596831C2312yes—
60C22×#10596831C2312yes—
61#14556570—24no—
62#14556570—42no—
63C2×C4⋊Dic73C2312yes—
64C2×#11924007—22no—
65C2×C22⋊Dic73—21no—
66C2×C22⋊Dic73—22no—
67#14556515—42no—
68#14556514—24no—
69#14556514—24no—
70#14556514—24no—
71C22⋊#10596831—22no—
72C22⋊#10596831—42no—
73#14556498—21no—
74#14556498—22no—
75#14556498—22no—
76C22×Q8×C73C2311yescharacteristic
77C22×D4×C73D411yescharacteristic
78C23×C4×C73—21no—
79C23×C4×C73C2311yescharacteristic
80C23×C4×C73C2311yescharacteristic
81#126×C73—28no—
82#125×C73—28no—
83#125×C73—22no—
84#125×C73—22no—
85#123×C73—28no—
86C22⋊Q8×C73—22no—
87C22⋊Q8×C73—22no—
88C22⋊Q8×C73—22no—
89C22⋊Q8×C73D414yes—
90C4×Q8×C73—22no—
91C4×Q8×C73—22no—
92C4×D4×C73—22no—
93C4×D4×C73—22no—
94#117×C73—28no—
95#117×C73—22no—
96#117×C73—22no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73—22no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73D411yescharacteristic
105C2×C4⋊C4×C73D414yes—
106C2×C4⋊C4×C73C2311yescharacteristic
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C22⋊C4×C73—21no—
110C2×C22⋊C4×C73—21no—
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73D411yescharacteristic
113C2×C22⋊C4×C73D411yescharacteristic
114C2×C42×C73—21no—
115C2×C42×C73—21no—
116C2×C42×C73C2311yescharacteristic
117C2×Dic146—28no—
118C2×Dic146—22no—
119C2×Dic146—22no—
120C2×Dic146—22no—
121C2×Dic146—22no—
122C2×Dic146—22no—
123C2×Dic146—48no—
124C2×Dic146—44no—
125C2×Dic146—44no—
126C2×Dic146—42no—
127C2×Dic146#4114yes—
128C22×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73—44no—
131C22×Dic73—42no—
132#11924018—48no—
133C2×#10596831—22no—
134C2×#10596831—22no—
135C2×#10596831—22no—
136C2×#10596831—22no—
137C2×#10596831—22no—
138C2×#10596831—22no—
139C2×#10596831—44no—
140C4⋊Dic73—22no—
141C4⋊Dic73—22no—
142#11924007—44no—
143C22⋊Dic73—44no—
144C22⋊Dic73—42no—
145C24×C73C2×D411yescharacteristic
146C24×C73C2×D411yescharacteristic
147C2×Q8×C73—21no—
148C2×Q8×C73—22no—
149C2×Q8×C73—22no—
150C2×Q8×C73C2×D412yes—
151C2×D4×C73—22no—
152C2×D4×C73—22no—
153C2×D4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—24no—
157C22×C4×C73—21no—
158C22×C4×C73—21no—
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—42no—
166C22×C4×C73—42no—
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D412yes—
172C22×C4×C73C2×D412yes—
173C22×C4×C73C2411yescharacteristic
174C4⋊C4×C73—24no—
175C4⋊C4×C73—24no—
176C4⋊C4×C73—24no—
177C4⋊C4×C73—24no—
178C4⋊C4×C73—24no—
179C4⋊C4×C73—24no—
180C4⋊C4×C73—22no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73C2×D412yes—
185C4⋊C4×C73C2×D412yes—
186C4⋊C4×C73C2×D412yes—
187C22⋊C4×C73—24no—
188C22⋊C4×C73—24no—
189C22⋊C4×C73—24no—
190C22⋊C4×C73—22no—
191C22⋊C4×C73—22no—
192C42×C73—24no—
193C42×C73—24no—
194C42×C73—21no—
195C42×C73—21no—
196C23×C73—22no—
197C23×C73—22no—
198C23×C73—21no—
199C23×C73—22no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73—22no—
203C23×C73#12012yes—
204C23×C73#13512yes—
205C23×C73#13712yes—
206C23×C73C22×D411yescharacteristic
207Q8×C73—24no—
208Q8×C73—41no—
209Q8×C73—41no—
210D4×C73—44no—
211C2×C4×C73—22no—
212C2×C4×C73—22no—
213C2×C4×C73—22no—
214C2×C4×C73—24no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—21no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—22no—
226C2×C4×C73—42no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—42no—
232C2×C4×C73—48no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—42no—
236C2×C4×C73#12012yes—
237C2×C4×C73C22×D411yescharacteristic
238C2×C4×C73C22×D411yescharacteristic
239C2×C4×C73C2×#4112yes—
240C2×C4×C73H(5,𝔽2)12yes—
241C2×Dic73—24no—
242C2×Dic73—48no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248Dic146—24no—
249Dic146—24no—
250Dic146—48no—
251Dic146—42no—
252Dic146—42no—
253#10596831—42no—
254#10596831—42no—
255C22×C73—22no—
256C22×C73—22no—
257C22×C73—24no—
258C22×C73—24no—
259C22×C73—41no—
260C22×C73—41no—
261C22×C73—41no—
262C22×C73—42no—
263C22×C73—42no—
264C22×C73#45214yes—
265C22×C73C2×#12011yescharacteristic
266C22×C73#54612yes—
267C22×C73C2×#13511yescharacteristic
268C22×C73C2×#13711yescharacteristic
269C22×C73#57812yes—
270C4×C73—22no—
271C4×C73—41no—
272C4×C73—42no—
273C4×C73—41no—
274C4×C73—42no—
275C4×C73—42no—
276C4×C73—41no—
277C4×C73—42no—
278C4×C73#52511yescharacteristic
279C4×C73#53511yescharacteristic
280C4×C73#54612yes—
281Dic73—44no—
282Dic73—82no—
283#23967—731nomaximal, Sylow
284C2×C73—24no—
285C2×C73—81no—
286C2×C73#240411yescharacteristic
287C2×C73C2×#45211yescharacteristic
288C2×C73#301111yescharacteristic
289C2×C73#330012yes—
290C2×C73#331212yes—
291C2×#552D7311yescharacteristic
292C2×#451—731no—
293C22×#103—731no—
294#2409—731no—
295#2026—732no—
296#2001—732no—
297#1892—731no—
298#1878—732no—
299#1873—732no—
300#1869—731no—
301#1823—731no—
302C73#2396711yescharacteristic, complemented, Sylow
303C23×Q8—731no—
304C22×Q16—732no—
305C23×C8—731no—
306#552—24no—
307#552—24no—
308C2×#126—21no—
309C2×#125—21no—
310C2×#125C2×D7311yescharacteristic
311C2×#123—21no—
312C2×C22⋊Q8—732no—
313C2×C22⋊Q8C2×D7311yescharacteristic
314C2×C22⋊Q8C2×D7311yescharacteristic
315C2×C4×Q8C2×D7311yescharacteristic
316C2×C4×D4C2×D7311yescharacteristic
317C2×#117—21no—
318C2×#117C2×D7311yescharacteristic
319C22×C4⋊C4C2×D7311yescharacteristic
320#451—1464no—
321C2×#103—732no—
322C2×#103—732no—
323C2×#103—732no—
324C2×#103—732no—
325C2×#103—1464no—
326C2×C22⋊C8—732no—
327C2×C22⋊C8—731no—
328#395—1461no—
329#393—1461no—
330#391—732no—
331#386—1461no—
332#385—1461no—
333#380—731no—
334#340—732no—
335#340—732no—
336#340—732no—
337#336—732no—
338C22×Q8—2922no—
339C22×Q8—732no—
340C22×Q8—1464no—
341C22×Q8—1462no—
342C22×Q8—1462no—
343C22×Q8—1461no—
344C22×Q8C22×D7311yescharacteristic
345C22×D4C22⋊D7311yescharacteristic
346C23×C4—21no—
347C23×C4—1461no—
348C23×C4C22×D7311yescharacteristic
349C23×C4C22×D7311yescharacteristic
350C2×Q16—2922no—
351C2×Q16—1464no—
352C2×Q16—1464no—
353C22×C8—2922no—
354C22×C8—732no—
355C22×C8—732no—
356C22×C8—1462no—
357#126—28no—
358#125—28no—
359#125—22no—
360#125—22no—
361#123—28no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—2922no—
366C22⋊Q8—1464no—
367C22⋊Q8C22⋊D7314yes—
368C4×Q8—22no—
369C4×Q8—22no—
370C4×D4—22no—
371C4×D4—22no—
372#117—28no—
373#117—22no—
374#117—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—732no—
383C2×C4⋊C4—1462no—
384C2×C4⋊C4C22⋊D7311yescharacteristic
385C2×C4⋊C4C22⋊D7314yes—
386C2×C4⋊C4C22×D7311yescharacteristic
387C2×C4⋊C4C22×D7311yescharacteristic
388C2×C4⋊C4C22×D7311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1462no—
393C2×C22⋊C4—1461no—
394C2×C22⋊C4C22⋊D7311yescharacteristic
395C2×C22⋊C4C22⋊D7311yescharacteristic
396C2×C42—21no—
397C2×C42—21no—
398C2×C42C22×D7311yescharacteristic
399#103—2922no—
400#103—1464no—
401#103—1464no—
402#103—1464no—
403C22⋊C8—2922no—
404C22⋊C8—1462no—
405#95—1461no—
406#95—1462no—
407#95—1462no—
408C24C2×C22⋊D7311yescharacteristic
409C24C2×C22⋊D7311yescharacteristic
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—2924no—
414C2×Q8—2928no—
415C2×Q8—2924no—
416C2×Q8—2922no—
417C2×Q8—734no—
418C2×Q8—1468no—
419C2×Q8—1462no—
420C2×Q8—1462no—
421C2×Q8—1462no—
422C2×Q8—1462no—
423C2×Q8—1462no—
424C2×Q8C2×C22⋊D7312yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—2922no—
442C22×C4—2924no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4C2×C22⋊D7311yescharacteristic
446C22×C4C2×C22⋊D7311yescharacteristic
447C22×C4C2×C22⋊D7311yescharacteristic
448C22×C4C2×C22⋊D7311yescharacteristic
449C22×C4C2×C22⋊D7312yes—
450C22×C4C2×C22⋊D7312yes—
451C22×C4C23×D7311yescharacteristic
452Q16—2928no—
453C2×C8—2924no—
454C2×C8—1462no—
455C2×C8—1462no—
456C2×C8—1462no—
457C2×C8—1462no—
458C2×C8—1462no—
459C2×C8—1462no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—2924no—
471C4⋊C4—1462no—
472C4⋊C4—1462no—
473C4⋊C4C2×C22⋊D7312yes—
474C4⋊C4C2×C22⋊D7312yes—
475C4⋊C4C2×C22⋊D7312yes—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—2922no—
482C22⋊C4—2924no—
483C42—24no—
484C42—24no—
485C42—21no—
486C42—21no—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—22no—
491C23—22no—
492C23—21no—
493C23—21no—
494C23#1455656012yes—
495C23#1455661112yes—
496C23#1455662112yes—
497C23C22×C22⋊D7311yescharacteristic
498Q8—24no—
499Q8—41no—
500Q8—41no—
501Q8—2928no—
502Q8—2922no—
503Q8—2922no—
504Q8—1464no—
505Q8—1464no—
506D4—44no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—48no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—2928no—
533C2×C4—2922no—
534C2×C4—2922no—
535C2×C4—2922no—
536C2×C4—2922no—
537C2×C4—2922no—
538C2×C4—1464no—
539C2×C4#1455656012yes—
540C2×C4C22×C22⋊D7311yescharacteristic
541C2×C4C22×C22⋊D7311yescharacteristic
542C2×C4#41×D7312yes—
543C2×C4#1455663712yes—
544C8—2922no—
545C8—2922no—
546C22—24no—
547C22—22no—
548C22—24no—
549C22—22no—
550C22—41no—
551C22—42no—
552C22—41no—
553C22—42no—
554C22—41no—
555C22#1798874914yes—
556C22C2×#1455656011yescharacteristic
557C22#1798922512yes—
558C22C2×#1455661111yescharacteristic
559C22C2×#1455662111yescharacteristic
560C22#1798940212yes—
561C4—22no—
562C4—41no—
563C4—42no—
564C4—42no—
565C4—41no—
566C4—42no—
567C4—41no—
568C4—42no—
569C4—2924no—
570C4—5842no—
571C4#1798909411yescharacteristic
572C4#1798915111yescharacteristic
573C4#1798922512yes—
574C2—24no—
575C2—81no—
576C2#2261324611yescharacteristic
577C2C2×#1798874911yescharacteristic
578C2#2261922611yescharacteristic
579C2#2262181212yes—
580C2#2262197612yes—
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#552×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73D4
hypercenterC2×#552D73
hypocenterC73#23967
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×#17988742C2semidirect
2. C22×#14556514C2semidirect
4. #22611079C2semidirect
5. #22610939C2semidirect
6. #22610355C2semidirect
7. #22610272C2semidirect
8. #22610243C2semidirect
10. #22610017C2semidirect
302. C73#23967semidirect

Maximal subgroups

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

subgroupindex
C2×#179887422
C22×#145565142
#226132962
#226110792
#226109392
#226103552
#226102722
#226102432
#226102242
#226100172
C2×#552×C732
#2396773