Finite Group Databasebeta

← #45666288

Subgroups — #45666288

Order 32896.

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

normal subgroups only

This group has 216 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
2C2×#22551535C211yesmaximal, characteristic, complemented
3#30657009C211yesmaximal, characteristic, complemented
4#30656551C211yesmaximal, characteristic, complemented
5#30656544C211yesmaximal, characteristic, complemented
6#30656522C211yesmaximal, characteristic, complemented
7#510×C257C211yesmaximal, characteristic, complemented
8#22551638—21no—
9#22551623—21no—
10#22551619—22no—
11#22551617—21no—
12#22551615—21no—
13C2×D2056C2211yescharacteristic
14D8×D257—21no—
15#22551598C2211yescharacteristic
16#22551535—22no—
17C2×#16861593C2211yescharacteristic, complemented
18#22551528C2211yescharacteristic, complemented
19#22551511—21no—
20#135×C257C2211yescharacteristic
21C2×Q16×C257C2211yescharacteristic, complemented
22Q32×C257—21no—
23SD32×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#16861608—41no—
33#16861605—22no—
34#16861605—41no—
35D2056—21no—
36D2056D412yes—
37#16861603—22no—
38#16861602—21no—
39#16861602D411yescharacteristic
40#16861601—21no—
41C8×D257—21no—
42#16861593—21no—
43#41×C257—21no—
44C2×Q8×C257—21no—
45Q16×C257—21no—
46Q16×C257—21no—
47Q16×C257D411yescharacteristic
48SD16×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—
65Q8×C257—21no—
66Q8×C257—41no—
67Q8×C257—41no—
68D4×C257—21no—
69D4×C257—41no—
70C2×C4×C257—41no—
71C2×C4×C257—41no—
72C2×C4×C257C2×D411yescharacteristic
73C8×C257C2×D411yescharacteristic
74C8×C257C2×D411yescharacteristic
75C22×C257—41no—
76C22×C257AGL(1,ℤ/8)11yescharacteristic
77C2×D257—21no—
78C2×D257—42no—
79C2×D257—162no—
80C2×D257—161no—
81C4×C257—41no—
82C4×C257—81no—
83C4×C257#12011yescharacteristic
84C4×C257C2×D811yescharacteristic
85Dic257—21no—
86C2×C257—21no—
87C2×C257—81no—
88C2×C257#44711yescharacteristic
89D257—41no—
90D257—82no—
91C257#219411yescharacteristic, complemented, Sylow
92#2194—2571nomaximal, Sylow
93#576—2571no—
94#510D25711yescharacteristic, complemented
95C2×SD32—2571no—
96#473—2571no—
97#361—2571no—
98#359—2571no—
99#350—2571no—
100H(5,𝔽2)—5141no—
101#137—5141no—
102AGL(1,ℤ/8)—5141no—
103AGL(1,ℤ/8)—5142no—
104#135—5141no—
105#135C2×D25711yescharacteristic
106C2×Q16C2×D25711yescharacteristic, complemented
107C2×D8—2571no—
108C2×D8—5141no—
109#131—2571no—
110Q32—21no—
111SD32—21no—
112SD32—5142no—
113M32C2×D25711yescharacteristic, complemented
114C2×C16—2571no—
115#108—2571no—
116#101—5141no—
117#41—2571no—
118#41—21no—
119#41—5142no—
120#41—10281no—
121C2×Q8—21no—
122C2×D4—5142no—
123C2×D4—5141no—
124C2×D4—10281no—
125C2×D4—10282no—
126Q16—21no—
127Q16—21no—
128Q16—10281no—
129Q16C22⋊D25711yescharacteristic
130SD16—21no—
131SD16—5142no—
132SD16—10281no—
133D8—2572no—
134D8—5142no—
135D8—5141no—
136D8C22⋊D25711yescharacteristic
137M16—2571no—
138M16—5141no—
139M16—5141no—
140C2×C8—5141no—
141C2×C8C22×D25711yescharacteristic
142C16—21no—
143C16—5141no—
144C23—10282no—
145C23—10281no—
146Q8—2571no—
147Q8—21no—
148Q8—41no—
149Q8—41no—
150D4—2571no—
151D4—21no—
152D4—5141no—
153D4—5142no—
154D4—41no—
155D4—10282no—
156D4—10282no—
157D4—10281no—
158C2×C4—5141no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—10282no—
162C2×C4D4×D25711yescharacteristic
163C8—5141no—
164C8—10281no—
165C8D4×D25711yescharacteristic
166C8C2×C22⋊D25711yescharacteristic
167C22—5141no—
168C22—41no—
169C22—10282no—
170C22—41121no—
171C22—41122no—
172C22#2255161611yescharacteristic
173C4—5141no—
174C4—41no—
175C4—81no—
176C4#2255156011yescharacteristic
177C4D8×D25711yescharacteristic
178C2—21no—
179C2—10281no—
180C2—81no—
181C2—20562no—
182C2#3065686911yescharacteristic
subgroupG/H
centerC2#30656869
commutator, cosocleC2×C8×C257C23
FrattiniC2×C8C22×D257
Fitting#510×C257C2
socleC2×C257#447
prime radical#30657539C2
hypercenter#510D257
hypocenterC257#2194
LCSG ▷ C2×C8×C257 ▷ C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. C2×#22551535C2semidirect
3. #30657009C2semidirect
4. #30656551C2semidirect
5. #30656544C2semidirect
6. #30656522C2semidirect
7. #510×C257C2semidirect
17. C2×#16861593C22semidirect
18. #22551528C22semidirect
21. C2×Q16×C257C22semidirect
24. M32×C257C22semidirect
91. C257#2194semidirect
94. #510D257semidirect
106. C2×Q16C2×D257semidirect
113. M32C2×D257semidirect

Maximal subgroups

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

subgroupindex
#306575392
C2×#225515352
#306570092
#306565512
#306565442
#306565222
#510×C2572
#2194257