Finite Group Databasebeta

← #33955527

Subgroups — #33955527

Order 18688.

3386 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989390C214yesmaximal, complemented
2#22623709C211yesmaximal, characteristic, complemented
3#22620414C2112yesmaximal, complemented
4#22620384C213yesmaximal, complemented
5#22620381C213yesmaximal, complemented
6#22620357C213yesmaximal, complemented
7C2×#17988963C213yesmaximal, complemented
8C4×#14556598C211yesmaximal, characteristic, complemented
9C73×#3498C211yesmaximal, characteristic, complemented
10C2×#14556637C2212yes—
11#17989390—232no—
12C2×#14556617C22112yescomplemented
13C2×#14556616C22112yescomplemented
14C2×#14556611C22112yescomplemented
15C22×#11924013C22112yescomplemented
16C2×#14556598C2212yes—
17#17989259—22no—
18D4×D292—26no—
19#17989156C2216yescomplemented
20#17989137C2213yescomplemented
21C2×#14556588C2213yescomplemented
22#17988977C22112yescomplemented
23#17988974C22112yescomplemented
24#17988969C22112yescomplemented
25#17988966C2211yescharacteristic, complemented
26#17988963C2213yescomplemented
27#17988963C2216yescomplemented
28#17988963C22112yescomplemented
29#17988961C2213yescomplemented
30#17988959C2213yescomplemented
31#17988543C2216yescomplemented
32C4×#11924012C2213yescomplemented
33C2×C4×#10596831C2213yescomplemented
34C2×H(5,𝔽2)×C73C2212yes—
35#550×C73—22no—
36D42×C73—26no—
37#535×C73C2216yescomplemented
38#532×C73C2213yescomplemented
39C2×#127×C73C2213yescomplemented
40C4×#41×C73C2211yescharacteristic
41#14556637—216no—
42C2×#11924029—22no—
43C2×#11924028C2316yes—
44C22×D292—26no—
45C22×D292C2316yes—
46C2×D4×D73—26no—
47#14556617—248no—
48#14556616—248no—
49#14556611—248no—
50C2×#11924018C2314yes—
51C2×#11924016C23112yes—
52C2×#11924015C23112yes—
53C2×#11924013—248no—
54C2×#11924013C23112yes—
55C2×#11924013C23124yes—
56#14556598D4116yescomplemented
57C2×#11924012C2316yes—
58C22×#10596831C2316yes—
59#14556591C2311yescharacteristic
60#14556588—26no—
61#14556588C2313yes—
62#14556588C2316yes—
63#14556579C2316yes—
64#14556566—212no—
65#14556565—26no—
66#14556561—212no—
67C4×D292—26no—
68#14556502C2316yes—
69C4×#10596831C2313yes—
70C4×#10596831C2311yescharacteristic
71C4×#10596831C2316yes—
72H(5,𝔽2)×C73—216no—
73C2×#41×C73—22no—
74C2×#41×C73C2316yes—
75C2×#41×C73C2311yescharacteristic
76C22×D4×C73—26no—
77C22×D4×C73—26no—
78C22×D4×C73C2316yes—
79C4⋊Q8×C73C2311yescharacteristic
80#127×C73—26no—
81#127×C73C2313yes—
82#127×C73C2316yes—
83#124×C73C2316yes—
84#121×C73—26no—
85#121×C73—212no—
86#120×C73—212no—
87C4×Q8×C73D412yes—
88C4×D4×C73—26no—
89C4×D4×C73D416yes—
90#117×C73C2313yes—
91C2×C42×C73C2313yes—
92C23×D73—26no—
93#11924029—48no—
94#11924028—224no—
95C2×Dic146C2412yes—
96C2×D292—224no—
97C2×D292—26no—
98C2×D292—26no—
99C2×D292—212no—
100C2×D292—812no—
101C2×D292C2416yes—
102C2×C22⋊D73—26no—
103D4×D73—424no—
104C2×C4×D73—26no—
105#11924018—28no—
106#11924016—224no—
107#11924015—224no—
108#11924013—224no—
109#11924013—248no—
110#11924012C2×D4124yescomplemented
111C2×#10596831C2×D4124yescomplemented
112C2×#10596831C2412yes—
113C2×#10596831C2416yes—
114C4⋊Dic73—22no—
115#11924005—212no—
116C24×C73—26no—
117#41×C73—224no—
118#41×C73—48no—
119#41×C73C2×D418yes—
120C2×Q8×C73C2×D411yescharacteristic
121C2×Q8×C73C2412yes—
122C2×D4×C73—26no—
123C2×D4×C73—224no—
124C2×D4×C73—26no—
125C2×D4×C73—26no—
126C2×D4×C73—212no—
127C2×D4×C73—424no—
128C2×D4×C73—812no—
129C2×D4×C73C2×D413yes—
130C2×D4×C73C2416yes—
131C22×C4×C73—26no—
132C22×C4×C73C2×D416yes—
133C22×C4×C73C2413yes—
134C4⋊C4×C73—22no—
135C4⋊C4×C73C2×D416yes—
136C22⋊C4×C73—212no—
137C22⋊C4×C73C2×D416yes—
138C42×C73C2×D416yes—
139C42×C73C2411yescharacteristic
140C42×C73C2413yes—
141C23×C73—26no—
142C23×C73—412no—
143C23×C73—812no—
144C23×C73C22×D413yes—
145C22×D73—26no—
146C22×D73—412no—
147C22×D73—812no—
148Q8×C73—24no—
149Q8×C73#12714yes—
150D4×C73—212no—
151D4×C73—412no—
152D4×C73—412no—
153D4×C73—412no—
154D4×C73—812no—
155D4×C73#127112yes—
156C2×C4×C73—212no—
157C2×C4×C73—22no—
158C2×C4×C73—412no—
159C2×C4×C73C22×D4112yes—
160C2×C4×C73C22×D416yes—
161C2×C4×C73C22×D413yes—
162C2×C4×C73C22×D412yes—
163C2×C4×C73C2511yescharacteristic
164C22⋊D73—412no—
165C2×Dic73—22no—
166D292—212no—
167D292—412no—
168D292—412no—
169D292—812no—
170C4×D73—412no—
171Dic146—24no—
172#10596831C22×D414yes—
173#10596831C22×D4112yes—
174C22×C73—26no—
175C22×C73—46no—
176C22×C73—412no—
177C22×C73—1612no—
178C22×C73C2×#12716yes—
179C22×C73C23×D411yescharacteristic
180C2×D73—46no—
181C2×D73—412no—
182C2×D73—1612no—
183C4×C73—26no—
184C4×C73—42no—
185C4×C73C2×#12716yes—
186C4×C73#57614yescomplemented
187C4×C73C23×D412yes—
188Dic73—42no—
189#60647—731nomaximal, Sylow
190C2×C73—26no—
191C2×C73—86no—
192C2×C73C22×#12711yescharacteristic
193C2×C73C2×#57612yes—
194D73—86no—
195C2×#576—734no—
196#3498—731no—
197#3498D7311yescharacteristic, complemented
198#3155—7312no—
199#3151—733no—
200#3151—733no—
201#3147—733no—
202C2×#493—733no—
203C4×#131—731no—
204C73#6064711yescharacteristic, complemented, Sylow
205C2×H(5,𝔽2)—732no—
206C2×H(5,𝔽2)C2×D7312yes—
207#576—14632no—
208C2×AGL(1,ℤ/8)—7312no—
209C2×AGL(1,ℤ/8)—7312no—
210C2×#135—7312no—
211C22×D8—7312no—
212C2×#131—732no—
213#550—22no—
214#550—1462no—
215D42—26no—
216D42—1466no—
217#535—736no—
218#535C2×D7316yescomplemented
219#532—733no—
220#532C2×D7313yescomplemented
221C2×#127—733no—
222C2×#127C2×D7313yescomplemented
223C4×#41C2×D7311yescharacteristic
224#496—7312no—
225#496—7312no—
226#495—7312no—
227#494—731no—
228#493—733no—
229#493—736no—
230#493—7312no—
231#492—733no—
232#492—733no—
233#405—736no—
234C4×M16—733no—
235C2×C4×C8—733no—
236H(5,𝔽2)—216no—
237H(5,𝔽2)—14616no—
238C2×#41—22no—
239C2×#41—736no—
240C2×#41—1462no—
241C2×#41C22×D7316yes—
242C2×#41C22×D7311yescharacteristic
243C22×D4—26no—
244C22×D4—26no—
245C22×D4—736no—
246C22×D4—1466no—
247C22×D4—1466no—
248C22×D4C22×D7316yes—
249AGL(1,ℤ/8)—14648no—
250AGL(1,ℤ/8)—14648no—
251#135—14648no—
252C2×Q16—734no—
253C2×SD16—7312no—
254C2×SD16—7312no—
255C2×D8—7312no—
256C2×D8—7324no—
257C2×D8—14648no—
258#131—7316no—
259C2×M16—736no—
260C22×C8—736no—
261C4⋊Q8—731no—
262C4⋊Q8C22×D7311yescharacteristic
263#127—26no—
264#127—733no—
265#127—736no—
266#127—1466no—
267#127C22×D7313yes—
268#127C22×D7316yes—
269#124—736no—
270#124C22×D7316yes—
271#121—26no—
272#121—212no—
273#121—1466no—
274#121—14612no—
275#120—212no—
276#120—14612no—
277C4×Q8C22⋊D7312yes—
278C4×D4—26no—
279C4×D4—1466no—
280C4×D4C22⋊D7316yes—
281#117C22×D7313yes—
282C2×C42C22×D7313yes—
283#97—736no—
284C4×C8—731no—
285C4×C8—733no—
286C4×C8—736no—
287C24—26no—
288C24—1466no—
289#41—224no—
290#41—48no—
291#41—2928no—
292#41—14624no—
293#41D4×D7318yes—
294C2×Q8—732no—
295C2×Q8C2×C22⋊D7311yescharacteristic
296C2×Q8C23×D7312yes—
297C2×D4—26no—
298C2×D4—224no—
299C2×D4—26no—
300C2×D4—26no—
301C2×D4—212no—
302C2×D4—424no—
303C2×D4—812no—
304C2×D4—29224no—
305C2×D4—58412no—
306C2×D4—736no—
307C2×D4—1466no—
308C2×D4—14624no—
309C2×D4—1466no—
310C2×D4—1466no—
311C2×D4—14612no—
312C2×D4C2×C22⋊D7313yes—
313C2×D4C23×D7316yes—
314C22×C4—26no—
315C22×C4—1466no—
316C22×C4C2×C22⋊D7316yes—
317C22×C4C23×D7313yes—
318Q16—1468no—
319SD16—14624no—
320SD16—14624no—
321D8—14624no—
322D8—14648no—
323M16—7324no—
324C2×C8—7324no—
325C2×C8—732no—
326C2×C8—736no—
327C4⋊C4—22no—
328C4⋊C4—1462no—
329C4⋊C4C2×C22⋊D7316yes—
330C22⋊C4—212no—
331C22⋊C4—14612no—
332C22⋊C4C2×C22⋊D7316yes—
333C42C2×C22⋊D7316yes—
334C42C23×D7311yescharacteristic
335C42C23×D7313yes—
336C23—26no—
337C23—412no—
338C23—812no—
339C23—29212no—
340C23—58412no—
341C23—1466no—
342C23C22×C22⋊D7313yes—
343Q8—24no—
344Q8—1464no—
345Q8#1455658714yes—
346D4—212no—
347D4—412no—
348D4—412no—
349D4—412no—
350D4—812no—
351D4—29212no—
352D4—29212no—
353D4—29212no—
354D4—58412no—
355D4—14612no—
356D4#14556587112yes—
357C2×C4—212no—
358C2×C4—22no—
359C2×C4—412no—
360C2×C4—29212no—
361C2×C4—1462no—
362C2×C4C2×D4×D73112yes—
363C2×C4C22×C22⋊D7316yes—
364C2×C4C22×C22⋊D7313yes—
365C2×C4C22×C22⋊D7312yes—
366C2×C4C24×D7311yescharacteristic
367C8—734no—
368C8—7312no—
369C22—26no—
370C22—412no—
371C22—46no—
372C22—1612no—
373C22—29212no—
374C22—2926no—
375C22—116812no—
376C22C2×#1455658716yes—
377C22C23×C22⋊D7311yescharacteristic
378C4—26no—
379C4—42no—
380C4—2922no—
381C4C2×#1455658716yes—
382C4#1798939014yescomplemented
383C4C22×D4×D7312yes—
384C2—26no—
385C2—86no—
386C2—5846no—
387C2C22×#1455658711yescharacteristic
388C2C2×#1798939012yes—
subgroupG/H
centerC22C23×C22⋊D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC73×#3498C2
socleC22×C73C23×D4
hypercenter#3498D73
hypocenterC73#60647
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3498

Complements

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

NH ≅ G/Ntype
1. C2×#17989390C2semidirect
2. #22623709C2semidirect
3. #22620414C2semidirect
4. #22620384C2semidirect
5. #22620381C2semidirect
6. #22620357C2semidirect
7. C2×#17988963C2semidirect
8. C4×#14556598C2semidirect
9. C73×#3498C2semidirect
12. C2×#14556617C22semidirect
13. C2×#14556616C22semidirect
14. C2×#14556611C22semidirect
15. C22×#11924013C22semidirect
19. #17989156C22semidirect
20. #17989137C22semidirect
21. C2×#14556588C22semidirect
22. #17988977C22semidirect
23. #17988974C22semidirect
24. #17988969C22semidirect
25. #17988966C22semidirect
26. #17988963C22semidirect
27. #17988963C22semidirect
28. #17988963C22semidirect
29. #17988961C22semidirect
30. #17988959C22semidirect
31. #17988543C22semidirect
32. C4×#11924012C22semidirect
33. C2×C4×#10596831C22semidirect
37. #535×C73C22semidirect
38. #532×C73C22semidirect
39. C2×#127×C73C22semidirect
56. #14556598D4semidirect
110. #11924012C2×D4semidirect
111. C2×#10596831C2×D4semidirect
186. C4×C73#576semidirect
197. #3498D73semidirect
204. C73#60647semidirect
218. #535C2×D73semidirect
220. #532C2×D73semidirect
222. C2×#127C2×D73semidirect
382. C4#17989390semidirect

Maximal subgroups

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

subgroupindex
C2×#179893902
#226237092
#226204142
#226203842
#226203812
#226203572
C2×#179889632
C4×#145565982
C73×#34982
#6064773