Finite Group Databasebeta

← #27181615

Subgroups — #27181615

Order 12928.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692281C211yesmaximal, characteristic, complemented
2C2×#15692280C211yesmaximal, characteristic, complemented
3C2×#15692275C211yesmaximal, characteristic, complemented
4C22×#13108059C211yesmaximal, characteristic, complemented
5#20265746C211yesmaximal, characteristic, complemented
6#20265567C212yesmaximal, complemented
7#20265564C212yesmaximal, complemented
8#20265559C212yesmaximal, complemented
9#20265553C212yesmaximal, complemented
10#20265133C211yesmaximal, characteristic, complemented
11#535×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13C22×D404C2211yescharacteristic
14#15692281—24no—
15#15692280—24no—
16#15692275—24no—
17C2×#13108064C2211yescharacteristic, complemented
18C2×#13108062C2212yescomplemented
19C2×#13108062C2211yescharacteristic, complemented
20C2×#13108061C2212yescomplemented
21C2×#13108061C2211yescharacteristic, complemented
22C2×#13108059—24no—
23C2×#13108059C2212yes—
24C2×#13108059C2212yes—
25C2×#13108059C2211yescharacteristic, complemented
26C2×#13108059C2212yescomplemented
27C2×#13108059C2212yescomplemented
28C2×#13108058C2211yescharacteristic
29C22×#11717579C2211yescharacteristic
30#15692252C2212yescomplemented
31#15692243C2212yescomplemented
32#15692230—22no—
33#15692225—22no—
34#15692166C2212yescomplemented
35C4×#11717579C2212yescomplemented
36C2×#41×C101C2211yescharacteristic
37C22×D4×C101C2211yescharacteristic
38#127×C101C2212yescomplemented
39#124×C101C2212yescomplemented
40#121×C101—22no—
41#120×C101—22no—
42#117×C101C2211yescharacteristic
43C23×D101—21no—
44#13108074—22no—
45#13108074—22no—
46C2×Dic202C2311yescharacteristic
47C2×D404—22no—
48C2×D404—22no—
49C2×D404—21no—
50C2×D404—24no—
51C2×D404C2312yes—
52C2×D404C2311yescharacteristic
53C2×C22⋊D101—21no—
54C2×C4×D101—21no—
55#13108064—22no—
56#13108062—22no—
57#13108062—24no—
58#13108061—22no—
59#13108061—24no—
60#13108059—24no—
61#13108059—24no—
62#13108059—22no—
63#13108059—24no—
64#13108059—24no—
65#13108058D414yescomplemented
66C2×#11717579D414yescomplemented
67C2×#11717579C2311yescharacteristic
68C2×#11717579C2312yes—
69C2×#11717579C2311yescharacteristic
70#13108051—24no—
71C24×C101—21no—
72#41×C101—22no—
73#41×C101—22no—
74C2×Q8×C101C2311yescharacteristic
75C2×D4×C101—21no—
76C2×D4×C101—22no—
77C2×D4×C101—22no—
78C2×D4×C101—21no—
79C2×D4×C101—24no—
80C2×D4×C101C2312yes—
81C2×D4×C101C2311yescharacteristic
82C22×C4×C101—21no—
83C22×C4×C101C2311yescharacteristic
84C4⋊C4×C101D412yes—
85C22⋊C4×C101—24no—
86C22⋊C4×C101D412yes—
87C42×C101C2312yes—
88C23×C101—22no—
89C23×C101—21no—
90C23×C101—42no—
91C23×C101—82no—
92C23×C101C2×D411yescharacteristic
93C22×D101—22no—
94C22×D101—21no—
95C22×D101—42no—
96C22×D101—82no—
97Q8×C101—21no—
98Q8×C101—21no—
99D4×C101—22no—
100D4×C101—22no—
101D4×C101—21no—
102D4×C101—21no—
103D4×C101—42no—
104D4×C101—42no—
105D4×C101—84no—
106C2×C4×C101—21no—
107C2×C4×C101—42no—
108C2×C4×C101C2×D412yes—
109C2×C4×C101C2×D412yes—
110C2×C4×C101C2×D414yes—
111C2×C4×C101C2×D411yescharacteristic
112C2×C4×C101C2411yescharacteristic
113C22⋊D101—42no—
114C2×Dic101—21no—
115D404—22no—
116D404—21no—
117D404—22no—
118D404—21no—
119D404—42no—
120D404—84no—
121C4×D101—42no—
122Dic202—21no—
123Dic202—21no—
124#11717579C2×D412yes—
125#11717579C2×D412yes—
126#11717579C2×D414yes—
127C22×C101—22no—
128C22×C101—41no—
129C22×C101—42no—
130C22×C101—42no—
131C22×C101—44no—
132C22×C101—84no—
133C22×C101#12712yes—
134C22×C101C22×D411yescharacteristic
135C2×D101—41no—
136C2×D101—42no—
137C2×D101—42no—
138C2×D101—44no—
139C2×D101—84no—
140C4×C101—24no—
141C4×C101—41no—
142C4×C101#12712yes—
143C4×C101C22×D411yescharacteristic
144C4×C101C22×D411yescharacteristic
145Dic101—41no—
146C2×C101—22no—
147C2×C101—81no—
148C2×C101—82no—
149C2×C101C2×#12711yescharacteristic
150C2×C101#57612yes—
151D101—81no—
152D101—82no—
153#3155—1011nomaximal, Sylow
154C101#315511yescharacteristic, complemented, Sylow
155C2×AGL(1,ℤ/8)—1011no—
156C2×AGL(1,ℤ/8)—1011no—
157C2×#135—1011no—
158C22×D8—1011no—
159#535—1011no—
160#535D10111yescharacteristic, complemented
161#496—1012no—
162#496—1012no—
163#495—1012no—
164#493—1012no—
165#405—1011no—
166C2×#41—1011no—
167C2×#41C2×D10111yescharacteristic
168C22×D4—1011no—
169C22×D4C2×D10111yescharacteristic
170AGL(1,ℤ/8)—2024no—
171AGL(1,ℤ/8)—2024no—
172#135—2024no—
173C2×Q16—1011no—
174C2×SD16—1011no—
175C2×SD16—1012no—
176C2×SD16—1011no—
177C2×SD16—1012no—
178C2×D8—1012no—
179C2×D8—1012no—
180C2×D8—1011no—
181C2×D8—1012no—
182C2×D8—1012no—
183C2×D8—2024no—
184C2×M16—1011no—
185C22×C8—1011no—
186#127—1012no—
187#127C2×D10112yescomplemented
188#124—1012no—
189#124C2×D10112yescomplemented
190#121—22no—
191#121—2022no—
192#120—22no—
193#120—2022no—
194#117C2×D10111yescharacteristic
195#97—1012no—
196C4×C8—1012no—
197C24—21no—
198C24—2021no—
199#41—22no—
200#41—22no—
201#41—2022no—
202#41—2022no—
203C2×Q8—1011no—
204C2×Q8C22×D10111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—24no—
210C2×D4—1012no—
211C2×D4—1011no—
212C2×D4—2021no—
213C2×D4—2022no—
214C2×D4—2022no—
215C2×D4—2024no—
216C2×D4—2021no—
217C2×D4C22×D10112yes—
218C2×D4C22×D10111yescharacteristic
219C22×C4—21no—
220C22×C4—2021no—
221C22×C4C22×D10111yescharacteristic
222Q16—2022no—
223SD16—2022no—
224SD16—2024no—
225SD16—2022no—
226SD16—2024no—
227D8—2024no—
228D8—2024no—
229D8—2022no—
230D8—2024no—
231D8—2024no—
232M16—1014no—
233C2×C8—1014no—
234C2×C8—1011no—
235C2×C8—1012no—
236C2×C8—1011no—
237C4⋊C4C22⋊D10112yes—
238C22⋊C4—24no—
239C22⋊C4—2024no—
240C22⋊C4C22⋊D10112yes—
241C42C22×D10112yes—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—8082no—
247C23—4042no—
248C23—2022no—
249C23—2021no—
250C23C2×C22⋊D10111yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—2021no—
254Q8—2021no—
255D4—22no—
256D4—21no—
257D4—22no—
258D4—21no—
259D4—42no—
260D4—42no—
261D4—84no—
262D4—8084no—
263D4—4042no—
264D4—4042no—
265D4—2022no—
266D4—2021no—
267D4—2022no—
268D4—2021no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—4042no—
272C2×C4—2021no—
273C2×C4D4×D10112yes—
274C2×C4D4×D10112yes—
275C2×C4C2×C22⋊D10114yes—
276C2×C4C2×C22⋊D10111yescharacteristic
277C2×C4C23×D10111yescharacteristic
278C8—1012no—
279C8—1012no—
280C8—1014no—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—8084no—
288C22—4041no—
289C22—4042no—
290C22—4042no—
291C22—4044no—
292C22#1569225112yes—
293C22C22×C22⋊D10111yescharacteristic
294C4—24no—
295C4—41no—
296C4—4041no—
297C4#1569225112yes—
298C4C2×D4×D10111yescharacteristic
299C4C2×D4×D10111yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—8082no—
304C2—8081no—
305C2C2×#1569225111yescharacteristic
306C2#2026598012yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#535×C101C2
socleC22×C101C22×D4
hypercenter#535D101
hypocenterC101#3155
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#15692281C2semidirect
2. C2×#15692280C2semidirect
3. C2×#15692275C2semidirect
4. C22×#13108059C2semidirect
5. #20265746C2semidirect
6. #20265567C2semidirect
7. #20265564C2semidirect
8. #20265559C2semidirect
9. #20265553C2semidirect
10. #20265133C2semidirect
11. #535×C101C2semidirect
17. C2×#13108064C22semidirect
18. C2×#13108062C22semidirect
19. C2×#13108062C22semidirect
20. C2×#13108061C22semidirect
21. C2×#13108061C22semidirect
25. C2×#13108059C22semidirect
26. C2×#13108059C22semidirect
27. C2×#13108059C22semidirect
30. #15692252C22semidirect
31. #15692243C22semidirect
34. #15692166C22semidirect
35. C4×#11717579C22semidirect
38. #127×C101C22semidirect
39. #124×C101C22semidirect
65. #13108058D4semidirect
66. C2×#11717579D4semidirect
154. C101#3155semidirect
160. #535D101semidirect
187. #127C2×D101semidirect
189. #124C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922812
C2×#156922802
C2×#156922752
C22×#131080592
#202657462
#202655672
#202655642
#202655592
#202655532
#202651332
#535×C1012
#3155101