Finite Group Databasebeta

← #44560817

Subgroups — #44560817

Order 32128.

656 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 173 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×#22548743C214yesmaximal, complemented
2#30632237C211yesmaximal, characteristic, complemented
3C2×#22548712C211yesmaximal, characteristic, complemented
4#30632076C214yesmaximal, complemented
5#30632064C212yesmaximal, complemented
6#30632060C212yesmaximal, complemented
7C4×M16×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2212yes—
9C22×D1004—22no—
10C22×D1004C2212yes—
11#22548743—216no—
12C2×#16860799C2218yescomplemented
13C2×D2008C2218yescomplemented
14#22548715C2211yescharacteristic, complemented
15#22548712—22no—
16#22548712C2212yes—
17#22548712C2212yes—
18#22548712C2211yescharacteristic, complemented
19#22548703C2212yescomplemented
20#22548689—22no—
21C4×D1004—22no—
22C2×M16×C251C2212yes—
23C2×C42×C251C2211yescharacteristic
24#97×C251C2212yescomplemented
25C4×C8×C251C2212yescomplemented
26C23×D251—22no—
27#16860810—28no—
28C2×Dic502C2312yes—
29C2×D1004—28no—
30C2×D1004—24no—
31C2×D1004—24no—
32C2×D1004—22no—
33C2×D1004—22no—
34C2×D1004—48no—
35C2×D1004C2314yes—
36C2×D1004C2312yes—
37C2×C22⋊D251—22no—
38C2×C4×D251—22no—
39#16860799—216no—
40D2008—216no—
41C4⋊Dic251—22no—
42(C2×D251)⋊C4—24no—
43C22×C4×C251D412yes—
44C22×C4×C251C2311yescharacteristic
45M16×C251D418yescomplemented
46C2×C8×C251C2314yes—
47C42×C251D412yes—
48C42×C251C2311yescharacteristic
49C42×C251C2311yescharacteristic
50C22×D251—22no—
51C22×D251—24no—
52C22×D251—48no—
53C22×D251—44no—
54Dic502—24no—
55D1004—24no—
56D1004—28no—
57D1004—48no—
58D1004—44no—
59D1004—44no—
60D1004—48no—
61C22⋊D251—44no—
62C2×Dic251—22no—
63C4×D251—44no—
64C23×C251C2×D411yescharacteristic
65C2×C4×C251—24no—
66C2×C4×C251C2×D412yes—
67C2×C4×C251C2×D411yescharacteristic
68C2×C4×C251C2×D414yes—
69C2×C4×C251C2×D412yes—
70C2×C4×C251C2411yescharacteristic
71C8×C251C2×D418yescomplemented
72C22×C251—22no—
73C22×C251#12712yes—
74C22×C251C22×D411yescharacteristic
75C2×D251—42no—
76C2×D251—44no—
77C2×D251—48no—
78C2×D251—44no—
79C2×D251—88no—
80C4×C251—22no—
81C4×C251#12712yes—
82C4×C251AGL(1,ℤ/8)14yescomplemented
83C4×C251C22×D412yes—
84Dic251—42no—
85C2×C251—22no—
86C2×C251C2×#12711yescharacteristic
87C2×C251C2×AGL(1,ℤ/8)12yes—
88D251—82no—
89D251—84no—
90C251#315111yescharacteristic, complemented, Sylow
91#3151—2511nomaximal, Sylow
92C2×AGL(1,ℤ/8)—2514no—
93#532—2511no—
94C2×#127—2511no—
95#496—2514no—
96#493—2512no—
97#492—2512no—
98C4×M16D25111yescharacteristic, complemented
99C2×#41—2512no—
100C22×D4—5022no—
101C22×D4—2512no—
102AGL(1,ℤ/8)—50216no—
103C2×SD16—2518no—
104C2×D8—2518no—
105C2×M16C2×D25112yes—
106C4⋊Q8—2511no—
107#127—5022no—
108#127—2512no—
109#127—2512no—
110#127—2511no—
111#124—2512no—
112#121—5022no—
113C4×D4—5022no—
114C2×C42C2×D25111yescharacteristic
115#97C2×D25112yescomplemented
116C4×C8C2×D25112yescomplemented
117C24—5022no—
118#41—5028no—
119C2×Q8—2512no—
120C2×D4—10048no—
121C2×D4—5022no—
122C2×D4—5022no—
123C2×D4—5022no—
124C2×D4—5024no—
125C2×D4—5024no—
126C2×D4—5028no—
127C2×D4—2512no—
128C2×D4—2514no—
129C22×C4—5022no—
130C22×C4D100412yes—
131C22×C4C22×D25111yescharacteristic
132SD16—50216no—
133D8—50216no—
134M16D100418yescomplemented
135C2×C8C22×D25114yes—
136C4⋊C4—5022no—
137C22⋊C4—5024no—
138C42D100412yes—
139C42C22×D25111yescharacteristic
140C42C22×D25111yescharacteristic
141C23—10048no—
142C23—10044no—
143C23—5022no—
144C23—5024no—
145C23C2×D100411yescharacteristic
146Q8—5024no—
147D4—10048no—
148D4—10044no—
149D4—10048no—
150D4—10044no—
151D4—10044no—
152D4—5024no—
153D4—5028no—
154C2×C4—24no—
155C2×C4—10044no—
156C2×C4—5022no—
157C2×C4C2×D100412yes—
158C2×C4C2×D100412yes—
159C2×C4C2×D100414yes—
160C2×C4C2×D100411yescharacteristic
161C2×C4C23×D25111yescharacteristic
162C8C2×D100418yescomplemented
163C22—22no—
164C22—20088no—
165C22—10048no—
166C22—10044no—
167C22—10042no—
168C22—10044no—
169C22#2254871212yes—
170C22C22×D100411yescharacteristic
171C4—22no—
172C4—10042no—
173C4#2254871212yes—
174C4#2254874314yescomplemented
175C4C22×D100412yes—
176C2—22no—
177C2—20084no—
178C2—20082no—
179C2C2×#2254871211yescharacteristic
180C2C2×#2254874312yes—
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC4×M16×C251C2
socleC22×C251C22×D4
hypercenterC4×M16D251
hypocenterC251#3151
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×#22548743C2semidirect
2. #30632237C2semidirect
3. C2×#22548712C2semidirect
4. #30632076C2semidirect
5. #30632064C2semidirect
6. #30632060C2semidirect
7. C4×M16×C251C2semidirect
12. C2×#16860799C22semidirect
13. C2×D2008C22semidirect
14. #22548715C22semidirect
18. #22548712C22semidirect
19. #22548703C22semidirect
24. #97×C251C22semidirect
25. C4×C8×C251C22semidirect
45. M16×C251D4semidirect
71. C8×C251C2×D4semidirect
82. C4×C251AGL(1,ℤ/8)semidirect
90. C251#3151semidirect
98. C4×M16D251semidirect
115. #97C2×D251semidirect
116. C4×C8C2×D251semidirect
134. M16D1004semidirect
162. C8C2×D1004semidirect
174. C4#22548743semidirect

Maximal subgroups

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

subgroupindex
C2×#225487432
#306322372
C2×#225487122
#306320762
#306320642
#306320602
C4×M16×C2512
#3151251