Finite Group Databasebeta

← #33103588

Subgroups — #33103588

Order 18688.

692 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618585C211yesmaximal, characteristic
2C2×#17988623C211yesmaximal, characteristic
3#22610681C212yesmaximal, complemented
4#22610530C212yesmaximal, complemented
5#22610527C212yesmaximal, complemented
6#22610030C211yesmaximal, characteristic
7#22610028C211yesmaximal, characteristic
8#22609967C211yesmaximal, characteristic
9#22609816C212yesmaximal, complemented
10#22609710C211yesmaximal, characteristic
11C73×#2924C211yesmaximal, characteristic
12C23×#10596831C2211yescharacteristic
13C2×#14556550C2211yescharacteristic
14#17988645—22no—
15#17988641—21no—
16#17988637—21no—
17C2×#14556526C2212yes—
18#17988623C411yescharacteristic, complemented
19#17988623C411yescharacteristic, complemented
20#17988623C411yescharacteristic, complemented
21#17988623C411yescharacteristic, complemented
22#17988623C2211yescharacteristic
23#17988623C2211yescharacteristic
24#17988623C2211yescharacteristic
25#17988623C2211yescharacteristic
26C2×#14556524C411yescharacteristic, complemented
27C2×#14556524C411yescharacteristic, complemented
28C2×#14556522C411yescharacteristic, complemented
29C2×#14556522C411yescharacteristic, complemented
30#17988613C2212yes—
31#17988551C2211yescharacteristic
32#17988542C2212yes—
33C2×#14556501C2212yes—
34C4×C22⋊Dic73C2211yescharacteristic
35#17988351C2212yes—
36#17988348C2212yes—
37#17988348C2212yes—
38#17988346C2212yes—
39#17988346C2212yes—
40#17988341C2212yes—
41C2×#117×C73C2211yescharacteristic
42#433×C73—22no—
43#432×C73—21no—
44#431×C73—21no—
45#424×C73C2212yes—
46#407×C73C2211yescharacteristic
47C2×C22⋊C8×C73C2211yescharacteristic
48#405×C73C2212yes—
49C2×#11924012C2312yes—
50C22×#10596831—24no—
51C22×#10596831C2×C411yescharacteristic
52C22×#10596831C2×C411yescharacteristic
53C22×#10596831C2×C411yescharacteristic
54C22×#10596831C2×C411yescharacteristic
55C22×#10596831C2311yescharacteristic
56C22×#10596831C2311yescharacteristic
57#14556550—21no—
58#14556550—21no—
59#14556550C2×C411yescharacteristic
60#14556550C2×C411yescharacteristic
61#14556550C2311yescharacteristic
62#14556550C2311yescharacteristic
63C2×C4⋊Dic73C2×C411yescharacteristic
64C2×C4⋊Dic73C2×C411yescharacteristic
65C2×C22⋊Dic73—21no—
66C2×C22⋊Dic73—21no—
67C2×C4×Dic73—21no—
68C2×C4×Dic73C2312yes—
69#14556526—24no—
70#14556524C2×C411yescharacteristic
71#14556524C2×C411yescharacteristic
72#14556524C2×C411yescharacteristic
73#14556524C2×C411yescharacteristic
74#14556524C2×C411yescharacteristic
75#14556524C2×C411yescharacteristic
76#14556524C2×C411yescharacteristic
77#14556524C2×C411yescharacteristic
78#14556522C2×C411yescharacteristic
79#14556522C2×C411yescharacteristic
80#14556522C2×C411yescharacteristic
81#14556522C2×C411yescharacteristic
82#14556522C2×C411yescharacteristic
83#14556522C2×C411yescharacteristic
84#14556522C2×C411yescharacteristic
85#14556522C2×C411yescharacteristic
86#14556520—22no—
87#14556520—22no—
88C22⋊#10596831—21no—
89C22⋊#10596831—21no—
90#14556501—24no—
91#14556501—22no—
92C8×Dic73—22no—
93#14556498—21no—
94C23×C4×C73C2311yescharacteristic
95C2×M16×C73C2312yes—
96C22×C8×C73C2312yes—
97#117×C73—21no—
98#117×C73—21no—
99#117×C73C2×C411yescharacteristic
100#117×C73C2×C411yescharacteristic
101#117×C73C2311yescharacteristic
102#117×C73C2311yescharacteristic
103C2×C4⋊C4×C73C2×C411yescharacteristic
104C2×C4⋊C4×C73C2×C411yescharacteristic
105C2×C22⋊C4×C73—21no—
106C2×C42×C73—21no—
107C4⋊C8×C73—22no—
108C4⋊C8×C73—22no—
109C22⋊C8×C73—21no—
110C22⋊C8×C73—21no—
111C22⋊C8×C73D411yescharacteristic
112C22⋊C8×C73D411yescharacteristic
113C22⋊C8×C73D412yes—
114#97×C73—22no—
115C4×C8×C73—22no—
116C22×Dic73—21no—
117C22×Dic73—21no—
118C22×Dic73—22no—
119#11924012—22no—
120#11924012—22no—
121C2×#10596831—22no—
122C2×#10596831—22no—
123C2×#10596831—22no—
124C2×#10596831—22no—
125C2×#10596831—22no—
126C2×#10596831—22no—
127C2×#10596831C22×C411yescharacteristic
128C2×#10596831C22×C411yescharacteristic
129C2×#10596831C22×C411yescharacteristic
130C2×#10596831C22×C411yescharacteristic
131C2×#10596831#4114yes—
132C4⋊Dic73—21no—
133C4⋊Dic73—21no—
134C4⋊Dic73C22×C411yescharacteristic
135C4⋊Dic73C22×C411yescharacteristic
136C4⋊Dic73C22×C411yescharacteristic
137C4⋊Dic73C22×C411yescharacteristic
138C22⋊Dic73—21no—
139C22⋊Dic73—21no—
140C22⋊Dic73—21no—
141C22⋊Dic73—21no—
142C22⋊Dic73—42no—
143C4×Dic73—21no—
144C4×Dic73—21no—
145C4×Dic73—21no—
146C4×Dic73—21no—
147C4×Dic73—22no—
148C4×Dic73—22no—
149C24×C73#4111yescharacteristic
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—21no—
153C22×C4×C73—21no—
154C22×C4×C73C22×C411yescharacteristic
155C22×C4×C73C22×C411yescharacteristic
156C22×C4×C73C2×D411yescharacteristic
157C22×C4×C73C2×D411yescharacteristic
158C22×C4×C73#4111yescharacteristic
159C22×C4×C73C2411yescharacteristic
160M16×C73—22no—
161M16×C73—22no—
162C2×C8×C73—22no—
163C2×C8×C73—22no—
164C2×C8×C73—22no—
165C2×C8×C73C2×D412yes—
166C2×C8×C73C2×D412yes—
167C4⋊C4×C73—21no—
168C4⋊C4×C73—21no—
169C4⋊C4×C73C22×C411yescharacteristic
170C4⋊C4×C73C22×C411yescharacteristic
171C4⋊C4×C73C22×C411yescharacteristic
172C4⋊C4×C73C22×C411yescharacteristic
173C22⋊C4×C73—21no—
174C22⋊C4×C73—21no—
175C22⋊C4×C73—21no—
176C22⋊C4×C73—21no—
177C42×C73—21no—
178C42×C73—21no—
179C42×C73—21no—
180C42×C73—21no—
181C23×C73—22no—
182C23×C73—22no—
183C23×C73—41no—
184C23×C73C4×D411yescharacteristic
185C23×C73C4×D411yescharacteristic
186C23×C73C2×#4111yescharacteristic
187C2×C4×C73—22no—
188C2×C4×C73—21no—
189C2×C4×C73—21no—
190C2×C4×C73—21no—
191C2×C4×C73—21no—
192C2×C4×C73—22no—
193C2×C4×C73—22no—
194C2×C4×C73—21no—
195C2×C4×C73—21no—
196C2×C4×C73—21no—
197C2×C4×C73—21no—
198C2×C4×C73—42no—
199C2×C4×C73—81no—
200C2×C4×C73C4×D411yescharacteristic
201C2×C4×C73C4×D411yescharacteristic
202C2×C4×C73C23×C411yescharacteristic
203C2×C4×C73C22×D411yescharacteristic
204C2×C4×C73C2×#4112yes—
205C2×C4×C732−1+412yes—
206C2×Dic73—21no—
207C2×Dic73—21no—
208C2×Dic73—21no—
209C2×Dic73—21no—
210C2×Dic73—42no—
211C2×Dic73—42no—
212C2×Dic73—42no—
213C2×Dic73—42no—
214C2×Dic73—81no—
215C8×C73—22no—
216C8×C73—22no—
217C8×C73—42no—
218#10596831—21no—
219#10596831—21no—
220#10596831—42no—
221#10596831#11712yes—
222#10596831#11712yes—
223C22×C73—21no—
224C22×C73—21no—
225C22×C73—21no—
226C22×C73—21no—
227C22×C73—21no—
228C22×C73—21no—
229C22×C73—21no—
230C22×C73—44no—
231C22×C73#44412yes—
232C22×C73C2×C4×D411yescharacteristic
233C22×C73#55212yes—
234C4×C73—21no—
235C4×C73—21no—
236C4×C73—41no—
237C4×C73—41no—
238C4×C73—41no—
239C4×C73—41no—
240C4×C73C2×#11711yescharacteristic
241C4×C73#51911yescharacteristic
242C4×C73#54712yes—
243Dic73—41no—
244Dic73—41no—
245Dic73—41no—
246Dic73—41no—
247Dic73—82no—
248#22607—731nomaximal, Sylow
249C2×C73—22no—
250C2×C73—22no—
251C2×C73—41no—
252C2×C73#234411yescharacteristic
253C2×C73C2×#44411yescharacteristic
254C2×C73#295711yescharacteristic
255#2924—731no—
256#2924D7311yescharacteristic
257C2×#427—731no—
258#1946—732no—
259#1922—732no—
260#1922—732no—
261#1826—731no—
262#1826—731no—
263#1810—731no—
264#1773—732no—
265#1752—731no—
266C73#2260711yescharacteristic, complemented, Sylow
267C23×C8—731no—
268C2×#117—731no—
269C2×#117C2×D7311yescharacteristic
270#433—22no—
271#433—1462no—
272#432—21no—
273#432—1461no—
274#431—21no—
275#431—1461no—
276C2×#108—732no—
277#427—731no—
278#427—731no—
279#427—731no—
280#427—731no—
281#427—731no—
282#427—731no—
283#427—731no—
284#427—731no—
285C2×#107—731no—
286C2×#107—731no—
287C2×#106—731no—
288C2×#106—731no—
289#424—732no—
290#424C2×D7312yes—
291#407—731no—
292#407C2×D7311yescharacteristic
293C2×C22⋊C8C2×D7311yescharacteristic
294#405—732no—
295#405C2×D7312yes—
296C2×#97—732no—
297C4×C22⋊C4—731no—
298#341—732no—
299#339—732no—
300#339—732no—
301#339—732no—
302#339—732no—
303#336—732no—
304C23×C4C22×D7311yescharacteristic
305C2×M16—732no—
306C2×M16C22×D7312yes—
307C22×C8—731no—
308C22×C8—731no—
309C22×C8—731no—
310C22×C8—731no—
311C22×C8—731no—
312C22×C8—731no—
313C22×C8—1464no—
314C22×C8C22×D7312yes—
315#117—21no—
316#117—21no—
317#117—731no—
318#117—731no—
319#117—731no—
320#117—731no—
321#117—1461no—
322#117—1461no—
323#117C4×D7311yescharacteristic
324#117C4×D7311yescharacteristic
325#117C22×D7311yescharacteristic
326#117C22×D7311yescharacteristic
327C2×C4⋊C4—731no—
328C2×C4⋊C4—731no—
329C2×C4⋊C4C4×D7311yescharacteristic
330C2×C4⋊C4C4×D7311yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—1461no—
333C2×C22⋊C4—1461no—
334C2×C42—21no—
335C2×C42—732no—
336C2×C42—1461no—
337#108—1464no—
338#107—731no—
339#107—731no—
340#107—731no—
341#107—731no—
342#107—731no—
343#107—731no—
344#107—731no—
345#107—731no—
346#106—731no—
347#106—731no—
348#106—731no—
349#106—731no—
350#106—731no—
351#106—731no—
352#106—731no—
353#106—731no—
354C4⋊C8—22no—
355C4⋊C8—22no—
356C4⋊C8—1462no—
357C4⋊C8—1462no—
358C22⋊C8—21no—
359C22⋊C8—21no—
360C22⋊C8—1461no—
361C22⋊C8—1461no—
362C22⋊C8C22⋊D7311yescharacteristic
363C22⋊C8C22⋊D7311yescharacteristic
364C22⋊C8C22⋊D7312yes—
365#97—22no—
366#97—1464no—
367#97—1462no—
368C4×C8—22no—
369C4×C8—1462no—
370#95—1461no—
371C24#1192402811yescharacteristic
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—1462no—
377C22×C4—1461no—
378C22×C4—1461no—
379C22×C4C2×C4×D7311yescharacteristic
380C22×C4C2×C4×D7311yescharacteristic
381C22×C4C2×C22⋊D7311yescharacteristic
382C22×C4C2×C22⋊D7311yescharacteristic
383C22×C4#1192402811yescharacteristic
384C22×C4C23×D7311yescharacteristic
385M16—22no—
386M16—22no—
387M16—1462no—
388M16—1462no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—731no—
393C2×C8—731no—
394C2×C8—734no—
395C2×C8—731no—
396C2×C8—731no—
397C2×C8—1462no—
398C2×C8—1462no—
399C2×C8—1462no—
400C2×C8—1462no—
401C2×C8—1462no—
402C2×C8—1462no—
403C2×C8C2×C22⋊D7312yes—
404C2×C8C2×C22⋊D7312yes—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—731no—
408C4⋊C4—731no—
409C4⋊C4—731no—
410C4⋊C4—731no—
411C4⋊C4—1461no—
412C4⋊C4—1461no—
413C4⋊C4C2×C4×D7311yescharacteristic
414C4⋊C4C2×C4×D7311yescharacteristic
415C4⋊C4C2×C4×D7311yescharacteristic
416C4⋊C4C2×C4×D7311yescharacteristic
417C22⋊C4—21no—
418C22⋊C4—21no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—2922no—
422C22⋊C4—1461no—
423C22⋊C4—1461no—
424C22⋊C4—1461no—
425C22⋊C4—1461no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—1462no—
431C42—1462no—
432C42—1461no—
433C42—1461no—
434C42—1461no—
435C42—1461no—
436C23—22no—
437C23—22no—
438C23—41no—
439C23C4×C22⋊D7311yescharacteristic
440C23C4×C22⋊D7311yescharacteristic
441C23C2×#1192402811yescharacteristic
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—42no—
454C2×C4—81no—
455C2×C4—2922no—
456C2×C4—2922no—
457C2×C4—2922no—
458C2×C4—2922no—
459C2×C4—5841no—
460C2×C4—1461no—
461C2×C4—1461no—
462C2×C4—1461no—
463C2×C4—1461no—
464C2×C4C4×C22⋊D7311yescharacteristic
465C2×C4C4×C22⋊D7311yescharacteristic
466C2×C4C22×C4×D7311yescharacteristic
467C2×C4C22×C22⋊D7311yescharacteristic
468C2×C4#41×D7312yes—
469C2×C4#1455663812yes—
470C8—22no—
471C8—22no—
472C8—42no—
473C8—2922no—
474C8—732no—
475C8—732no—
476C8—1461no—
477C8—1461no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—44no—
486C22#1798870712yes—
487C22C2×C4×C22⋊D7311yescharacteristic
488C22#1798927012yes—
489C4—21no—
490C4—21no—
491C4—41no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—2921no—
496C4—2921no—
497C4—2921no—
498C4—2921no—
499C4—5842no—
500C4#117×D7311yescharacteristic
501C4#1798906411yescharacteristic
502C4#1798923612yes—
503C2—22no—
504C2—22no—
505C2—41no—
506C2#2261280111yescharacteristic
507C2C2×#1798870711yescharacteristic
508C2#2261884611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2924C2
socleC22×C73C2×C4×D4
prime radicalC24×C73#41
cosocleC22×C4×C73C24
hypercenter#2924D73
hypocenterC73#22607
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ #2924

Complements

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

NH ≅ G/Ntype
3. #22610681C2semidirect
4. #22610530C2semidirect
5. #22610527C2semidirect
9. #22609816C2semidirect
18. #17988623C4semidirect
19. #17988623C4semidirect
20. #17988623C4semidirect
21. #17988623C4semidirect
26. C2×#14556524C4semidirect
27. C2×#14556524C4semidirect
28. C2×#14556522C4semidirect
29. C2×#14556522C4semidirect
266. C73#22607semidirect

Maximal subgroups

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

subgroupindex
#226185852
C2×#179886232
#226106812
#226105302
#226105272
#226100302
#226100282
#226099672
#226098162
#226097102
C73×#29242
#2260773