Finite Group Databasebeta

← #27184544

Subgroups — #27184544

Order 12928.

1160 conjugacy classes of subgroups; only the 471 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 471 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
1#20265776C216yesmaximal, complemented
2#20265767C216yesmaximal, complemented
3#20265739C213yesmaximal, complemented
4#20265713C216yesmaximal, complemented
5#20265690C216yesmaximal, complemented
6C2×#15692215C213yesmaximal, complemented
7#543×C101C211yesmaximal, characteristic, complemented
8C2×#13108075C2212yescomplemented
9C22×C22⋊D101C2213yes—
10C23×Dic101C2211yescharacteristic
11C22×C4×D101C2213yescomplemented
12#15692250C22112yescomplemented
13#15692246C2216yescomplemented
14#15692244C2216yescomplemented
15#15692240C22112yescomplemented
16#15692239C22112yescomplemented
17#15692231C2216yescomplemented
18#15692230C2216yescomplemented
19#15692229C2216yes—
20#15692229C2216yes—
21#15692225C22112yescomplemented
22Q8×Dic101C2212yescomplemented
23#15692217C2216yescomplemented
24#15692215C22112yescomplemented
25#15692215C22112yescomplemented
26#15692213C2216yescomplemented
27C2×#13108053C2213yescomplemented
28C2×#13108051C2213yescomplemented
29C2×C4×Dic101C2213yescomplemented
30C22⋊Q8×C101C2216yescomplemented
31C22⋊Q8×C101C2216yescomplemented
32C2×C22⋊C4×C101C2213yes—
33C23×D101C2313yes—
34#13108075—44no—
35C2×D404C2316yes—
36C2×D404C2316yes—
37C2×C22⋊D101—212no—
38C2×C22⋊D101C23112yes—
39C22×Dic101—42no—
40C22×Dic101C2313yes—
41C22×Dic101C2313yes—
42C22×Dic101C2313yes—
43C2×C4×D101—46no—
44C2×C4×D101C2316yes—
45C2×C4×D101C2316yes—
46C4⋊Dic101C2316yes—
47C4⋊Dic101C2316yes—
48#13108053C23112yes—
49#13108051C23112yes—
50#13108051C23112yes—
51#13108051C23112yes—
52C4×Dic101C2316yes—
53C4×Dic101C2316yes—
54C24×C101C2311yescharacteristic
55C2×Q8×C101C2312yes—
56C2×Q8×C101C2312yes—
57C22×C4×C101C2313yes—
58C22×C4×C101C2313yes—
59C4⋊C4×C101C2316yes—
60C4⋊C4×C101C2316yes—
61C22⋊C4×C101C23112yes—
62C23×C101—41no—
63C23×C101—41no—
64C23×C101#4114yescomplemented
65C23×C101C2413yes—
66C22×D101—26no—
67C22×D101—43no—
68C22×D101—43no—
69C22×D101C2416yes—
70Q8×C101—42no—
71Q8×C101—42no—
72C2×C4×C101—43no—
73C2×C4×C101—43no—
74C2×C4×C101C2416yes—
75C2×C4×C101C2416yes—
76C22⋊D101—224no—
77C2×Dic101—26no—
78C2×Dic101—26no—
79C2×Dic101—43no—
80C2×Dic101C2413yes—
81C2×Dic101C2416yes—
82C2×Dic101C2411yescharacteristic
83D404—46no—
84D404—46no—
85C4×D101—212no—
86C4×D101—46no—
87C22×C101—23no—
88C22×C101—23no—
89C22×C101—44no—
90C22×C101C2×#4116yes—
91C22×C101C2511yescharacteristic
92C2×D101—26no—
93C2×D101—26no—
94C2×D101—26no—
95C2×D101—412no—
96C4×C101—26no—
97C4×C101—26no—
98Dic101—26no—
99Dic101—23no—
100Dic101H(5,𝔽2)12yes—
101C2×C101—26no—
102C2×C101C22×#4111yescharacteristic
103C2×C101C2×H(5,𝔽2)11yescharacteristic
104C2×C101#58511yescharacteristic
105D101—46no—
106#3459—1011nomaximal, Sylow
107C101#345911yescharacteristic, complemented, Sylow
108#543D10111yescharacteristic, complemented
109#539—1016no—
110#538—1016no—
111#534—1013no—
112#529—1016no—
113#525—1016no—
114C2×C4×D4—1013no—
115C2×#41—1012no—
116C22×D4—1013no—
117C23×C4—1011no—
118C23×C4—1013no—
119#126—10112no—
120#125—1016no—
121#124—1016no—
122#123—10112no—
123#123—10112no—
124C22⋊Q8—1016no—
125C22⋊Q8C2×D10116yescomplemented
126C22⋊Q8C2×D10116yescomplemented
127#121—1016no—
128#121—1016no—
129#121—1016no—
130#120—10112no—
131C4×Q8—1012no—
132C4×D4—1016no—
133C4×D4—10112no—
134C4×D4—10112no—
135#117—1016no—
136C2×C4⋊C4—1013no—
137C2×C22⋊C4—1013no—
138C2×C22⋊C4C2×D10113yes—
139C2×C42—1013no—
140C24—1013no—
141C24C22×D10111yescharacteristic
142#41—4044no—
143C2×Q8C22×D10112yes—
144C2×Q8C22×D10112yes—
145C2×D4—10112no—
146C2×D4—1016no—
147C2×D4—1016no—
148C2×D4—20212no—
149C22×C4—1013no—
150C22×C4—1013no—
151C22×C4—1013no—
152C22×C4—1016no—
153C22×C4—1016no—
154C22×C4—4046no—
155C22×C4—4042no—
156C22×C4C22×D10113yes—
157C22×C4C22×D10113yes—
158C4⋊C4—10112no—
159C4⋊C4—1016no—
160C4⋊C4—1016no—
161C4⋊C4C22×D10116yes—
162C4⋊C4C22×D10116yes—
163C22⋊C4—10112no—
164C22⋊C4—10112no—
165C22⋊C4—10112no—
166C22⋊C4C22×D101112yes—
167C42—1016no—
168C42—1016no—
169C23—41no—
170C23—41no—
171C23—1016no—
172C23—4043no—
173C23—4043no—
174C23—2026no—
175C23#1310807514yescomplemented
176C23C23×D10113yes—
177Q8—42no—
178Q8—42no—
179D4—4046no—
180D4—4046no—
181D4—20224no—
182C2×C4—43no—
183C2×C4—43no—
184C2×C4—1011no—
185C2×C4—1016no—
186C2×C4—1013no—
187C2×C4—4046no—
188C2×C4—4043no—
189C2×C4—20212no—
190C2×C4—2026no—
191C2×C4—2026no—
192C2×C4C23×D10116yes—
193C2×C4C23×D10116yes—
194C22—23no—
195C22—23no—
196C22—44no—
197C22—40412no—
198C22—2026no—
199C22—2026no—
200C22—2026no—
201C22C2×#1310807516yes—
202C22C24×D10111yescharacteristic
203C4—26no—
204C4—26no—
205C4—1012no—
206C4—2026no—
207C4—2023no—
208C2—26no—
209C2—4046no—
210C2C22×#1310807511yescharacteristic
211C2H(5,𝔽2)×D10111yescharacteristic
212C2#2026601211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D101
commutator, socle, cosocleC22×C101C25
Fitting#543×C101C2
hypercenter#543D101
hypocenterC101#3459
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #20265776C2semidirect
2. #20265767C2semidirect
3. #20265739C2semidirect
4. #20265713C2semidirect
5. #20265690C2semidirect
6. C2×#15692215C2semidirect
7. #543×C101C2semidirect
8. C2×#13108075C22semidirect
11. C22×C4×D101C22semidirect
12. #15692250C22semidirect
13. #15692246C22semidirect
14. #15692244C22semidirect
15. #15692240C22semidirect
16. #15692239C22semidirect
17. #15692231C22semidirect
18. #15692230C22semidirect
21. #15692225C22semidirect
22. Q8×Dic101C22semidirect
23. #15692217C22semidirect
24. #15692215C22semidirect
25. #15692215C22semidirect
26. #15692213C22semidirect
27. C2×#13108053C22semidirect
28. C2×#13108051C22semidirect
29. C2×C4×Dic101C22semidirect
30. C22⋊Q8×C101C22semidirect
31. C22⋊Q8×C101C22semidirect
64. C23×C101#41semidirect
107. C101#3459semidirect
108. #543D101semidirect
125. C22⋊Q8C2×D101semidirect
126. C22⋊Q8C2×D101semidirect
175. C23#13108075semidirect

Maximal subgroups

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

subgroupindex
#202657762
#202657672
#202657392
#202657132
#202656902
C2×#156922152
#543×C1012
#3459101