Finite Group Databasebeta

← #45666274

Subgroups — #45666274

Order 32896.

220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 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#30657539C211yesmaximal, characteristic, complemented
2C2×#22551533C211yesmaximal, characteristic, complemented
3#30657004C211yesmaximal, characteristic, complemented
4#30656551C211yesmaximal, characteristic, complemented
5#30656544C211yesmaximal, characteristic, complemented
6#30656522C211yesmaximal, characteristic, complemented
7#509×C257C211yesmaximal, characteristic, complemented
8#22551638—21no—
9#22551619—22no—
10#22551618—21no—
11#22551617—21no—
12#22551615—21no—
13C2×D2056C2211yescharacteristic
14D8×D257—21no—
15#22551598C2211yescharacteristic
16#22551533—22no—
17C2×#16861593C2211yescharacteristic, complemented
18#22551528C2211yescharacteristic, complemented
19#22551509—21no—
20#135×C257C2211yescharacteristic
21C2×D8×C257C2211yescharacteristic, complemented
22SD32×C257—21no—
23D16×C257—21no—
24M32×C257C2211yescharacteristic, complemented
25#16861619—22no—
26#16861618—41no—
27#16861618D411yescharacteristic
28C2×D1028—21no—
29C2×D1028—41no—
30D4×D257—22no—
31D4×D257—42no—
32#16861606—41no—
33#16861605—22no—
34D2056—21no—
35D2056D412yes—
36#16861603—22no—
37#16861603—41no—
38#16861602—21no—
39#16861602D411yescharacteristic
40#16861601—21no—
41C8×D257—21no—
42#16861593—21no—
43#41×C257—21no—
44C2×D4×C257—21no—
45Q16×C257D411yescharacteristic
46SD16×C257—21no—
47D8×C257—21no—
48D8×C257—21no—
49D8×C257D411yescharacteristic
50C2×C8×C257C2311yescharacteristic
51C16×C257—21no—
52C22×D257—42no—
53C22×D257—41no—
54Dic514D811yescharacteristic
55D1028—22no—
56D1028—42no—
57D1028—41no—
58D1028D811yescharacteristic
59C22⋊D257—21no—
60C22⋊D257—42no—
61C4×D257—21no—
62C4×D257—42no—
63#14523339—21no—
64#14523339—41no—
65C23×C257—41no—
66Q8×C257—21no—
67D4×C257—21no—
68D4×C257—41no—
69D4×C257—41no—
70D4×C257—41no—
71C2×C4×C257—41no—
72C2×C4×C257C2×D411yescharacteristic
73C8×C257C2×D411yescharacteristic
74C8×C257C2×D411yescharacteristic
75C22×C257—41no—
76C22×C257—81no—
77C22×C257—161no—
78C22×C257AGL(1,ℤ/8)11yescharacteristic
79C2×D257—21no—
80C2×D257—42no—
81C2×D257—162no—
82C2×D257—161no—
83C4×C257—41no—
84C4×C257#12011yescharacteristic
85C4×C257C2×D811yescharacteristic
86Dic257—21no—
87C2×C257—21no—
88C2×C257—81no—
89C2×C257—161no—
90C2×C257#44711yescharacteristic
91D257—41no—
92D257—82no—
93C257#219211yescharacteristic, complemented, Sylow
94#2192—2571nomaximal, Sylow
95#576—2571no—
96#509D25711yescharacteristic, complemented
97C2×D16—2571no—
98#472—2571no—
99#361—2571no—
100#359—2571no—
101#350—2571no—
102H(5,𝔽2)—5141no—
103AGL(1,ℤ/8)—5141no—
104AGL(1,ℤ/8)—5142no—
105AGL(1,ℤ/8)—5141no—
106#135—5141no—
107#135C2×D25711yescharacteristic
108C2×D8—2571no—
109C2×D8—5141no—
110C2×D8C2×D25711yescharacteristic, complemented
111#131—2571no—
112SD32—21no—
113D16—21no—
114D16—5142no—
115M32C2×D25711yescharacteristic, complemented
116C2×C16—2571no—
117#108—2571no—
118#100—5141no—
119#41—2571no—
120#41—21no—
121#41—5142no—
122#41—10281no—
123C2×D4—21no—
124C2×D4—5142no—
125C2×D4—5141no—
126C2×D4—10281no—
127C2×D4—10282no—
128Q16C22⋊D25711yescharacteristic
129SD16—21no—
130SD16—5142no—
131SD16—10281no—
132D8—2572no—
133D8—21no—
134D8—21no—
135D8—5142no—
136D8—5141no—
137D8—10281no—
138D8C22⋊D25711yescharacteristic
139M16—2571no—
140M16—5141no—
141M16—5141no—
142C2×C8—5141no—
143C2×C8C22×D25711yescharacteristic
144C16—21no—
145C16—5141no—
146C23—41no—
147C23—10282no—
148C23—10281no—
149Q8—2571no—
150Q8—21no—
151D4—2571no—
152D4—21no—
153D4—5141no—
154D4—5142no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—10282no—
159D4—10282no—
160D4—10281no—
161C2×C4—5141no—
162C2×C4—41no—
163C2×C4—10282no—
164C2×C4D4×D25711yescharacteristic
165C8—5141no—
166C8—10281no—
167C8D4×D25711yescharacteristic
168C8C2×C22⋊D25711yescharacteristic
169C22—5141no—
170C22—41no—
171C22—10282no—
172C22—81no—
173C22—161no—
174C22—41121no—
175C22—41122no—
176C22#2255161611yescharacteristic
177C4—5141no—
178C4—41no—
179C4#2255156011yescharacteristic
180C4D8×D25711yescharacteristic
181C2—21no—
182C2—10281no—
183C2—81no—
184C2—20562no—
185C2—161no—
186C2#3065686911yescharacteristic
subgroupG/H
centerC2#30656869
commutator, cosocleC2×C8×C257C23
FrattiniC2×C8C22×D257
Fitting#509×C257C2
socleC2×C257#447
hypercenter#509D257
hypocenterC257#2192
LCSG ▷ C2×C8×C257 ▷ C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #30657539C2semidirect
2. C2×#22551533C2semidirect
3. #30657004C2semidirect
4. #30656551C2semidirect
5. #30656544C2semidirect
6. #30656522C2semidirect
7. #509×C257C2semidirect
17. C2×#16861593C22semidirect
18. #22551528C22semidirect
21. C2×D8×C257C22semidirect
24. M32×C257C22semidirect
93. C257#2192semidirect
96. #509D257semidirect
110. C2×D8C2×D257semidirect
115. M32C2×D257semidirect

Maximal subgroups

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

subgroupindex
#306575392
C2×#225515332
#306570042
#306565512
#306565442
#306565222
#509×C2572
#2192257