Finite Group Databasebeta

← #44560831

Subgroups — #44560831

Order 32128.

572 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 165 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×#22548748C212yesmaximal, complemented
2C2×#22548743C212yesmaximal, complemented
3#30632237C211yesmaximal, characteristic, complemented
4C2×#22548703C211yesmaximal, characteristic, complemented
5#30632081C212yesmaximal, complemented
6#30632076C212yesmaximal, complemented
7#30632071C214yesmaximal, complemented
8C4×M16×C251C211yesmaximal, characteristic, complemented
9C2×#16860810C2212yes—
10C22×Dic502C2211yescharacteristic
11C22×D1004C2211yescharacteristic
12#22548748—28no—
13#22548743—28no—
14C2×Dic1004C2214yescomplemented
15C2×#16860799C2214yes—
16C2×#16860799C2214yescomplemented
17C2×D2008C2214yes—
18#22548715C2211yescharacteristic, complemented
19#22548712C2211yescharacteristic, complemented
20#22548703—22no—
21#22548703C2212yes—
22#22548703C2212yes—
23#22548703C2212yescomplemented
24#22548689—22no—
25C4×D1004—22no—
26C2×(C2×D251)⋊C4—22no—
27C2×M16×C251C2212yes—
28C2×C42×C251C2211yescharacteristic
29#97×C251C2212yescomplemented
30C4×C8×C251C2212yescomplemented
31C23×D251—21no—
32#16860810—24no—
33#16860810—24no—
34C2×Dic502—21no—
35C2×Dic502—24no—
36C2×Dic502C2312yes—
37C2×Dic502C2312yes—
38C2×D1004—24no—
39C2×D1004—21no—
40C2×D1004—22no—
41C2×D1004C2312yes—
42C2×D1004C2312yes—
43C2×C22⋊D251—22no—
44C22×Dic251—21no—
45C2×C4×D251—22no—
46Dic1004—28no—
47#16860799—28no—
48#16860799—28no—
49D2008—28no—
50C4⋊Dic251—22no—
51(C2×D251)⋊C4—24no—
52(C2×D251)⋊C4—24no—
53(C2×D251)⋊C4—24no—
54C22×C4×C251D412yes—
55C22×C4×C251C2311yescharacteristic
56M16×C251D418yescomplemented
57C2×C8×C251C2314yes—
58C42×C251D412yes—
59C42×C251C2311yescharacteristic
60C42×C251C2311yescharacteristic
61C22×D251—22no—
62C22×D251—22no—
63C22×D251—44no—
64C22×D251—42no—
65Dic502—22no—
66Dic502—24no—
67Dic502—22no—
68Dic502—42no—
69D1004—22no—
70D1004—24no—
71D1004—22no—
72D1004—44no—
73D1004—42no—
74C22⋊D251—44no—
75C2×Dic251—22no—
76C2×Dic251—22no—
77C2×Dic251—42no—
78C4×D251—44no—
79C23×C251C2×D411yescharacteristic
80C2×C4×C251—42no—
81C2×C4×C251C2×D412yes—
82C2×C4×C251C2×D412yes—
83C2×C4×C251C2×D411yescharacteristic
84C2×C4×C251C2×D414yes—
85C2×C4×C251C2411yescharacteristic
86C8×C251C2×D418yes—
87C22×C251—22no—
88C22×C251#12712yes—
89C22×C251C22×D411yescharacteristic
90C2×D251—42no—
91C2×D251—42no—
92C2×D251—44no—
93C2×D251—44no—
94C2×D251—84no—
95C4×C251—22no—
96C4×C251—22no—
97C4×C251#12712yes—
98C4×C251C22×D412yes—
99Dic251—42no—
100Dic251—42no—
101C2×C251—22no—
102C2×C251C2×#12711yescharacteristic
103C2×C251#57512yes—
104D251—82no—
105D251—82no—
106C251#315311yescharacteristic, complemented, Sylow
107#3153—2511nomaximal, Sylow
108C2×#137—2512no—
109C2×AGL(1,ℤ/8)—2512no—
110#532—2511no—
111C2×#124—2511no—
112#497—2512no—
113#496—2512no—
114#495—2514no—
115C4×M16D25111yescharacteristic, complemented
116C2×#41—2512no—
117C22×Q8—2511no—
118C22×D4—2511no—
119#137—5028no—
120AGL(1,ℤ/8)—5028no—
121C2×Q16—2514no—
122C2×SD16—2514no—
123C2×SD16—2514no—
124C2×D8—2514no—
125C2×M16C2×D25112yes—
126C4⋊Q8—2511no—
127#127—2511no—
128#124—5022no—
129#124—2512no—
130#124—2512no—
131#124—2512no—
132#121—5022no—
133C4×D4—5022no—
134C2×C22⋊C4—5022no—
135C2×C42C2×D25111yescharacteristic
136#97C2×D25112yescomplemented
137C4×C8C2×D25112yescomplemented
138C24—5021no—
139#41—5024no—
140#41—5024no—
141C2×Q8—5024no—
142C2×Q8—5021no—
143C2×Q8—2512no—
144C2×Q8—2512no—
145C2×D4—5022no—
146C2×D4—5022no—
147C2×D4—5021no—
148C2×D4—5024no—
149C2×D4—2512no—
150C2×D4—2512no—
151C22×C4—5021no—
152C22×C4—5022no—
153C22×C4D100412yes—
154C22×C4C22×D25111yescharacteristic
155Q16—5028no—
156SD16—5028no—
157SD16—5028no—
158D8—5028no—
159M16D100418yescomplemented
160C2×C8C22×D25114yes—
161C4⋊C4—5022no—
162C22⋊C4—5024no—
163C22⋊C4—5024no—
164C22⋊C4—5024no—
165C42D100412yes—
166C42C22×D25111yescharacteristic
167C42C22×D25111yescharacteristic
168C23—10044no—
169C23—10042no—
170C23—5022no—
171C23—5022no—
172C23C2×D100411yescharacteristic
173Q8—10042no—
174Q8—5022no—
175Q8—5024no—
176Q8—5022no—
177D4—10044no—
178D4—10044no—
179D4—10042no—
180D4—5022no—
181D4—5024no—
182D4—5022no—
183C2×C4—42no—
184C2×C4—10042no—
185C2×C4—10044no—
186C2×C4—5022no—
187C2×C4—5022no—
188C2×C4C2×D100412yes—
189C2×C4C2×D100412yes—
190C2×C4C2×D100414yes—
191C2×C4C2×D100411yescharacteristic
192C2×C4C23×D25111yescharacteristic
193C8C2×D100418yes—
194C22—22no—
195C22—20084no—
196C22—10042no—
197C22—10042no—
198C22—10044no—
199C22—10044no—
200C22#2254871212yes—
201C22C22×D100411yescharacteristic
202C4—22no—
203C4—22no—
204C4—10042no—
205C4—10042no—
206C4#2254871212yes—
207C4C22×D100412yes—
208C2—22no—
209C2—20082no—
210C2—20082no—
211C2C2×#2254871211yescharacteristic
212C2#3063248712yes—
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC4×M16×C251C2
socleC22×C251C22×D4
hypercenterC4×M16D251
hypocenterC251#3153
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#22548748C2semidirect
2. C2×#22548743C2semidirect
3. #30632237C2semidirect
4. C2×#22548703C2semidirect
5. #30632081C2semidirect
6. #30632076C2semidirect
7. #30632071C2semidirect
8. C4×M16×C251C2semidirect
14. C2×Dic1004C22semidirect
16. C2×#16860799C22semidirect
18. #22548715C22semidirect
19. #22548712C22semidirect
23. #22548703C22semidirect
29. #97×C251C22semidirect
30. C4×C8×C251C22semidirect
56. M16×C251D4semidirect
106. C251#3153semidirect
115. C4×M16D251semidirect
136. #97C2×D251semidirect
137. C4×C8C2×D251semidirect
159. M16D1004semidirect

Maximal subgroups

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

subgroupindex
C2×#225487482
C2×#225487432
#306322372
C2×#225487032
#306320812
#306320762
#306320712
C4×M16×C2512
#3153251