Finite Group Databasebeta

← #33140350

Subgroups — #33140350

Order 18688.

1032 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988990C211yesmaximal, characteristic, complemented
2C2×#17988946C211yesmaximal, characteristic, complemented
3#22615752C211yesmaximal, characteristic, complemented
4#22611131C212yesmaximal, complemented
5#22611110C212yesmaximal
6#22610905C211yesmaximal, characteristic, complemented
7#22610743C211yesmaximal, characteristic, complemented
8#22610185C212yesmaximal, complemented
9#22610178C212yesmaximal, complemented
10#22609697C211yesmaximal, characteristic, complemented
11C2×#542×C73C211yesmaximal, characteristic
12C2×#14556591C2211yescharacteristic
13#17988990—24no—
14#17988946—24no—
15C2×#14556524C2212yes—
16C2×#14556524C2212yes—
17C2×#14556515C2212yes—
18C2×#14556515C2212yes—
19C2×#14556512C2212yes—
20C2×#14556512C2212yes—
21C2×C22⋊#10596831C2212yes—
22C2×#14556502C2211yescharacteristic
23C2×C4×#10596831C2211yescharacteristic
24#17988526—21no—
25#17988523—21no—
26#17988507—21no—
27#17988499—21no—
28#17988485C2212yes—
29#17988453C2211yescharacteristic
30#17988447C2212yes—
31#17988349C2212yes—
32#17988349C2212yes—
33#17988342C2212yes—
34#542×C73—24no—
35#542×C73—24no—
36C2×C4⋊Q8×C73C2211yescharacteristic
37C2×#126×C73—21no—
38C2×#124×C73—21no—
39C2×#124×C73C2211yescharacteristic
40C2×#123×C73—21no—
41C2×C22⋊Q8×C73—21no—
42C2×C22⋊Q8×C73C2211yescharacteristic
43C2×#121×C73C2211yescharacteristic
44C2×C4×Q8×C73C2211yescharacteristic
45C2×C4×D4×C73C2211yescharacteristic
46C2×#117×C73C2211yescharacteristic
47C22×Dic146—21no—
48C22×#10596831C2312yes—
49C22×#10596831C2312yes—
50#14556591—22no—
51#14556591—22no—
52C2×C4⋊Dic73—21no—
53C2×C4⋊Dic73C2312yes—
54C2×C4⋊Dic73C2312yes—
55C2×#11924007—22no—
56C2×C22⋊Dic73—22no—
57#14556524—24no—
58#14556524—24no—
59#14556515—24no—
60#14556515—24no—
61#14556512—24no—
62#14556512—24no—
63C22⋊#10596831—24no—
64#14556502—22no—
65C4×#10596831—22no—
66#14556498—22no—
67#14556498—21no—
68#14556498—21no—
69#14556498—22no—
70C22×Q8×C73—21no—
71C22×Q8×C73C2311yescharacteristic
72C22×D4×C73—21no—
73C22×D4×C73C2311yescharacteristic
74C23×C4×C73—21no—
75C23×C4×C73C2311yescharacteristic
76C4⋊Q8×C73—22no—
77C4⋊Q8×C73D414yes—
78#126×C73—28no—
79#124×C73—28no—
80#124×C73—22no—
81#124×C73D414yes—
82#123×C73—28no—
83C22⋊Q8×C73—28no—
84C22⋊Q8×C73—22no—
85C22⋊Q8×C73—22no—
86#121×C73—22no—
87#121×C73—22no—
88C4×Q8×C73—22no—
89C4×Q8×C73—22no—
90C4×D4×C73—22no—
91C4×D4×C73—22no—
92#117×C73—22no—
93#117×C73—22no—
94C2×C4⋊C4×C73—21no—
95C2×C4⋊C4×C73—21no—
96C2×C4⋊C4×C73—21no—
97C2×C4⋊C4×C73D411yescharacteristic
98C2×C4⋊C4×C73D411yescharacteristic
99C2×C4⋊C4×C73C2311yescharacteristic
100C2×C4⋊C4×C73C2311yescharacteristic
101C2×C22⋊C4×C73—21no—
102C2×C22⋊C4×C73—21no—
103C2×C22⋊C4×C73—21no—
104C2×C22⋊C4×C73—21no—
105C2×C22⋊C4×C73—21no—
106C2×C22⋊C4×C73D411yescharacteristic
107C2×C22⋊C4×C73D411yescharacteristic
108C2×C42×C73—21no—
109C2×C42×C73C2311yescharacteristic
110C2×C42×C73C2311yescharacteristic
111C2×Dic146—42no—
112C2×Dic146—82no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C2×#10596831—22no—
116C2×#10596831—22no—
117C2×#10596831—22no—
118C2×#10596831—22no—
119C2×#10596831—22no—
120C2×#10596831#4114yes—
121C4⋊Dic73—22no—
122C4⋊Dic73—22no—
123C4⋊Dic73—22no—
124C4⋊Dic73—22no—
125C4⋊Dic73—42no—
126#11924007—82no—
127C22⋊Dic73—82no—
128C24×C73—21no—
129C24×C73C2×D411yescharacteristic
130C2×Q8×C73—24no—
131C2×Q8×C73—21no—
132C2×Q8×C73—22no—
133C2×Q8×C73—22no—
134C2×Q8×C73—82no—
135C2×Q8×C73C2×D412yes—
136C2×D4×C73—24no—
137C2×D4×C73—21no—
138C2×D4×C73—22no—
139C2×D4×C73—22no—
140C2×D4×C73—82no—
141C2×D4×C73C2×D412yes—
142C22×C4×C73—21no—
143C22×C4×C73—21no—
144C22×C4×C73—21no—
145C22×C4×C73—24no—
146C22×C4×C73—21no—
147C22×C4×C73—21no—
148C22×C4×C73—22no—
149C22×C4×C73—22no—
150C22×C4×C73—82no—
151C22×C4×C73C2×D411yescharacteristic
152C22×C4×C73C2×D411yescharacteristic
153C22×C4×C73C2×D411yescharacteristic
154C22×C4×C73C2×D411yescharacteristic
155C22×C4×C73C2×D411yescharacteristic
156C22×C4×C73C2411yescharacteristic
157C4⋊C4×C73—24no—
158C4⋊C4×C73—24no—
159C4⋊C4×C73—24no—
160C4⋊C4×C73—22no—
161C4⋊C4×C73—22no—
162C4⋊C4×C73—21no—
163C4⋊C4×C73—21no—
164C4⋊C4×C73C2×D412yes—
165C4⋊C4×C73C2×D412yes—
166C22⋊C4×C73—24no—
167C22⋊C4×C73—24no—
168C22⋊C4×C73—24no—
169C22⋊C4×C73—24no—
170C22⋊C4×C73—24no—
171C22⋊C4×C73—22no—
172C22⋊C4×C73—22no—
173C42×C73—24no—
174C42×C73—21no—
175C42×C73—21no—
176C42×C73C2×D412yes—
177C42×C73C2×D412yes—
178C23×C73—21no—
179C23×C73—22no—
180C23×C73—21no—
181C23×C73—21no—
182C23×C73—22no—
183C23×C73—22no—
184C23×C73—41no—
185C23×C73—41no—
186C23×C73—42no—
187C23×C73—42no—
188C23×C73C22×D411yescharacteristic
189Q8×C73—24no—
190Q8×C73—42no—
191Q8×C73—84no—
192D4×C73—24no—
193D4×C73—42no—
194D4×C73—84no—
195C2×C4×C73—22no—
196C2×C4×C73—22no—
197C2×C4×C73—21no—
198C2×C4×C73—22no—
199C2×C4×C73—21no—
200C2×C4×C73—22no—
201C2×C4×C73—21no—
202C2×C4×C73—22no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
207C2×C4×C73—41no—
208C2×C4×C73—41no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—42no—
213C2×C4×C73—84no—
214C2×C4×C73#12012yes—
215C2×C4×C73#12012yes—
216C2×C4×C73C22×D411yescharacteristic
217C2×C4×C73C22×D411yescharacteristic
218C2×C4×C73C2×#4112yes—
219C2×C4×C732−1+412yes—
220C2×Dic73—42no—
221C2×Dic73—42no—
222C2×Dic73—42no—
223C2×Dic73—42no—
224C2×Dic73—42no—
225C2×Dic73—42no—
226Dic146—84no—
227#10596831—24no—
228#10596831—42no—
229C22×C73—42no—
230C22×C73—41no—
231C22×C73—42no—
232C22×C73—42no—
233C22×C73—42no—
234C22×C73—41no—
235C22×C73—41no—
236C22×C73—42no—
237C22×C73—82no—
238C22×C73—82no—
239C22×C73C2×#12011yescharacteristic
240C22×C73#53211yescharacteristic
241C22×C73#53611yescharacteristic
242C22×C73#57612yes—
243C22×C73#57812yes—
244C4×C73—22no—
245C4×C73—22no—
246C4×C73—41no—
247C4×C73—42no—
248C4×C73—41no—
249C4×C73—42no—
250C4×C73—42no—
251C4×C73—41no—
252C4×C73#54712yes—
253C4×C73#54812yes—
254Dic73—82no—
255Dic73—82no—
256#25681—731nomaximal, Sylow
257C2×C73—81no—
258C2×C73—82no—
259C2×C73#243211yescharacteristic
260C2×C73#300811yescharacteristic
261C2×C73#301111yescharacteristic
262C2×C73#317512yes—
263C2×C73#324312yes—
264C2×#542D7311yescharacteristic
265C2×#500—731no—
266C2×#489—731no—
267#2628—731no—
268#2036—732no—
269#2033—732no—
270#1993—731no—
271#1958—731no—
272#1863—732no—
273#1862—732no—
274#1750—731no—
275C73#2568111yescharacteristic, complemented, Sylow
276#542—24no—
277#542—24no—
278C2×C4⋊Q8—731no—
279C2×C4⋊Q8C2×D7311yescharacteristic
280C2×#126—21no—
281C2×#124—21no—
282C2×#124C2×D7311yescharacteristic
283C2×#123—21no—
284C2×C22⋊Q8—21no—
285C2×C22⋊Q8C2×D7311yescharacteristic
286C2×#121C2×D7311yescharacteristic
287C2×C4×Q8C2×D7311yescharacteristic
288C2×C4×D4C2×D7311yescharacteristic
289C2×#117C2×D7311yescharacteristic
290#500—1464no—
291#489—1464no—
292C2×#107—732no—
293C2×#107—732no—
294C2×#103—732no—
295C2×#103—732no—
296C2×#102—732no—
297C2×#102—732no—
298C2×C22⋊C8—732no—
299C2×#97—731no—
300C2×C4×C8—731no—
301#400—1461no—
302#399—1461no—
303#395—1461no—
304#393—1461no—
305#389—732no—
306#382—731no—
307#380—732no—
308#340—732no—
309#340—732no—
310#336—732no—
311C22×Q8—21no—
312C22×Q8—1461no—
313C22×Q8C22×D7311yescharacteristic
314C22×D4—21no—
315C22×D4C22×D7311yescharacteristic
316C23×C4—21no—
317C23×C4C22×D7311yescharacteristic
318C22×C8—732no—
319C22×C8—732no—
320C4⋊Q8—22no—
321C4⋊Q8—1462no—
322C4⋊Q8—1462no—
323C4⋊Q8C22⋊D7314yes—
324#126—28no—
325#124—28no—
326#124—22no—
327#124C22⋊D7314yes—
328#123—28no—
329C22⋊Q8—28no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332#121—22no—
333#121—22no—
334C4×Q8—22no—
335C4×Q8—22no—
336C4×D4—22no—
337C4×D4—22no—
338#117—22no—
339#117—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—732no—
344C2×C4⋊C4—732no—
345C2×C4⋊C4—1461no—
346C2×C4⋊C4—1462no—
347C2×C4⋊C4C22⋊D7311yescharacteristic
348C2×C4⋊C4C22⋊D7311yescharacteristic
349C2×C4⋊C4C22×D7311yescharacteristic
350C2×C4⋊C4C22×D7311yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—1462no—
357C2×C22⋊C4C22⋊D7311yescharacteristic
358C2×C22⋊C4C22⋊D7311yescharacteristic
359C2×C42—21no—
360C2×C42C22×D7311yescharacteristic
361C2×C42C22×D7311yescharacteristic
362#107—1464no—
363#107—1464no—
364#103—1464no—
365#103—1464no—
366#102—1464no—
367#102—1464no—
368C22⋊C8—1464no—
369#97—1462no—
370C4×C8—1462no—
371#95—1462no—
372#95—1461no—
373#95—1461no—
374#95—1462no—
375C24—21no—
376C24C2×C22⋊D7311yescharacteristic
377C2×Q8—24no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—82no—
382C2×Q8—2922no—
383C2×Q8—5842no—
384C2×Q8C2×C22⋊D7312yes—
385C2×D4—24no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—82no—
390C2×D4C2×C22⋊D7312yes—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—82no—
400C22×C4—1462no—
401C22×C4—1462no—
402C22×C4C2×C22⋊D7311yescharacteristic
403C22×C4C2×C22⋊D7311yescharacteristic
404C22×C4C2×C22⋊D7311yescharacteristic
405C22×C4C2×C22⋊D7311yescharacteristic
406C22×C4C2×C22⋊D7311yescharacteristic
407C22×C4C23×D7311yescharacteristic
408C2×C8—734no—
409C2×C8—1462no—
410C2×C8—1462no—
411C2×C8—1462no—
412C2×C8—1462no—
413C2×C8—1462no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—2922no—
422C4⋊C4—5842no—
423C4⋊C4—1462no—
424C4⋊C4—1462no—
425C4⋊C4—1462no—
426C4⋊C4—1462no—
427C4⋊C4C2×C22⋊D7312yes—
428C4⋊C4C2×C22⋊D7312yes—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—5842no—
437C42—24no—
438C42—21no—
439C42—21no—
440C42C2×C22⋊D7312yes—
441C42C2×C22⋊D7312yes—
442C23—21no—
443C23—22no—
444C23—22no—
445C23—21no—
446C23—21no—
447C23—22no—
448C23—41no—
449C23—42no—
450C23—42no—
451C23—41no—
452C23C22×C22⋊D7311yescharacteristic
453Q8—24no—
454Q8—42no—
455Q8—84no—
456Q8—5844no—
457D4—24no—
458D4—42no—
459D4—84no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—84no—
479C2×C4—2922no—
480C2×C4—2922no—
481C2×C4—2922no—
482C2×C4—2922no—
483C2×C4—2922no—
484C2×C4—2922no—
485C2×C4#1455656012yes—
486C2×C4#1455656012yes—
487C2×C4C22×C22⋊D7311yescharacteristic
488C2×C4C22×C22⋊D7311yescharacteristic
489C2×C4#41×D7312yes—
490C2×C4#1455663812yes—
491C8—2922no—
492C8—1464no—
493C22—42no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—42no—
500C22—42no—
501C22—82no—
502C22—82no—
503C22C2×#1455656011yescharacteristic
504C22#1798913311yescharacteristic
505C22#1798915811yescharacteristic
506C22#1798939012yes—
507C22#1798940212yes—
508C4—22no—
509C4—22no—
510C4—41no—
511C4—42no—
512C4—41no—
513C4—42no—
514C4—42no—
515C4—41no—
516C4—5842no—
517C4—5842no—
518C4#1798923612yes—
519C4#1798924412yes—
520C2—81no—
521C2—82no—
522C2#2261355511yescharacteristic
523C2#2261919911yescharacteristic
524C2#2261922611yescharacteristic
525C2#2262056412yes—
526C2#2262120812yes—
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#542×C73C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenterC2×#542D73
hypocenterC73#25681
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
1. C2×#17988990C2semidirect
2. C2×#17988946C2semidirect
3. #22615752C2semidirect
4. #22611131C2semidirect
6. #22610905C2semidirect
7. #22610743C2semidirect
8. #22610185C2semidirect
9. #22610178C2semidirect
10. #22609697C2semidirect
275. C73#25681semidirect

Maximal subgroups

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

subgroupindex
C2×#179889902
C2×#179889462
#226157522
#226111312
#226111102
#226109052
#226107432
#226101852
#226101782
#226096972
C2×#542×C732
#2568173