Finite Group Databasebeta

← #24909431

Subgroups — #24909431

Order 10624.

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#19128503C211yesmaximal, characteristic
2C2×#14581735C211yesmaximal, characteristic, complemented
3#19127973C211yesmaximal, characteristic, complemented
4#19127515C211yesmaximal, characteristic, complemented
5#19127508C211yesmaximal, characteristic, complemented
6#19127486C211yesmaximal, characteristic, complemented
7#510×C83C211yesmaximal, characteristic, complemented
8#14581838—21no—
9#14581823—21no—
10#14581819—22no—
11#14581817—21no—
12#14581815—21no—
13C2×D664C2211yescharacteristic
14D8×D83—21no—
15#14581798C2211yescharacteristic
16#14581735—22no—
17C2×C83⋊C16C2211yescharacteristic, complemented
18#14581728C2211yescharacteristic, complemented
19#14581711—21no—
20#135×C83C2211yescharacteristic
21C2×Q16×C83C2211yescharacteristic, complemented
22Q32×C83—21no—
23SD32×C83—21no—
24M32×C83C2211yescharacteristic, complemented
25#13050591—22no—
26#13050590—41no—
27#13050590D411yescharacteristic
28C2×D332—21no—
29C2×D332—41no—
30D4×D83—22no—
31D4×D83—42no—
32#13050580—41no—
33#13050577—22no—
34#13050577—41no—
35D664—21no—
36D664D412yes—
37#13050575—22no—
38#13050574—21no—
39#13050574D411yescharacteristic
40#13050573—21no—
41C8×D83—21no—
42C83⋊C16—21no—
43#41×C83—21no—
44C2×Q8×C83—21no—
45Q16×C83—21no—
46Q16×C83—21no—
47Q16×C83D411yescharacteristic
48SD16×C83—21no—
49D8×C83D411yescharacteristic
50C2×C8×C83C2311yescharacteristic
51C16×C83—21no—
52C22×D83—42no—
53C22×D83—41no—
54Q8×C83—21no—
55Q8×C83—41no—
56Q8×C83—41no—
57D4×C83—21no—
58D4×C83—41no—
59C2×C4×C83—41no—
60C2×C4×C83—41no—
61C2×C4×C83C2×D411yescharacteristic
62C22⋊D83—21no—
63C22⋊D83—42no—
64D332—22no—
65D332—42no—
66D332—41no—
67D332D811yescharacteristic
68C4×D83—21no—
69C4×D83—42no—
70Dic166D811yescharacteristic
71C8×C83C2×D411yescharacteristic
72C8×C83C2×D411yescharacteristic
73C83⋊C8—21no—
74C83⋊C8—41no—
75C22×C83—41no—
76C22×C83AGL(1,ℤ/8)11yescharacteristic
77C2×D83—21no—
78C2×D83—42no—
79C2×D83—162no—
80C2×D83—161no—
81C4×C83—41no—
82C4×C83—81no—
83C4×C83#12011yescharacteristic
84C4×C83C2×D811yescharacteristic
85Dic83—21no—
86C2×C83—21no—
87C2×C83—81no—
88C2×C83#44711yescharacteristic
89D83—41no—
90D83—82no—
91#2194—831nomaximal, Sylow
92C83#219411yescharacteristic, complemented, Sylow
93#576—831no—
94#510D8311yescharacteristic, complemented
95C2×SD32—831no—
96#473—831no—
97#361—831no—
98#359—831no—
99#350—831no—
100H(5,𝔽2)—1661no—
101#137—1661no—
102AGL(1,ℤ/8)—1661no—
103AGL(1,ℤ/8)—1662no—
104#135—1661no—
105#135C2×D8311yescharacteristic
106C2×Q16C2×D8311yescharacteristic, complemented
107C2×D8—831no—
108C2×D8—1661no—
109#131—831no—
110Q32—21no—
111SD32—21no—
112SD32—1662no—
113M32C2×D8311yescharacteristic, complemented
114C2×C16—831no—
115#108—831no—
116#101—1661no—
117#41—21no—
118#41—3321no—
119#41—831no—
120#41—1662no—
121C2×Q8—21no—
122C2×D4—3321no—
123C2×D4—3322no—
124C2×D4—1662no—
125C2×D4—1661no—
126Q16—21no—
127Q16—21no—
128Q16—3321no—
129Q16C22⋊D8311yescharacteristic
130SD16—21no—
131SD16—3321no—
132SD16—1662no—
133D8—832no—
134D8—1662no—
135D8—1661no—
136D8C22⋊D8311yescharacteristic
137M16—831no—
138M16—1661no—
139M16—1661no—
140C2×C8—1661no—
141C2×C8C22×D8311yescharacteristic
142C16—21no—
143C16—1661no—
144C23—3322no—
145C23—3321no—
146Q8—21no—
147Q8—41no—
148Q8—41no—
149Q8—831no—
150D4—21no—
151D4—41no—
152D4—3322no—
153D4—3322no—
154D4—3321no—
155D4—831no—
156D4—1661no—
157D4—1662no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—3322no—
161C2×C4—1661no—
162C2×C4D4×D8311yescharacteristic
163C8—3321no—
164C8—1661no—
165C8D4×D8311yescharacteristic
166C8C2×C22⋊D8311yescharacteristic
167C22—41no—
168C22—13281no—
169C22—13282no—
170C22—3322no—
171C22—1661no—
172C22#1458181611yescharacteristic
173C4—41no—
174C4—81no—
175C4—1661no—
176C4#1458176011yescharacteristic
177C4D8×D8311yescharacteristic
178C2—21no—
179C2—81no—
180C2—3321no—
181C2—6642no—
182C2#1912783311yescharacteristic
subgroupG/H
centerC2#19127833
commutator, cosocleC2×C8×C83C23
FrattiniC2×C8C22×D83
Fitting#510×C83C2
socleC2×C83#447
prime radical#19128503C2
hypercenter#510D83
hypocenterC83#2194
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. C2×#14581735C2semidirect
3. #19127973C2semidirect
4. #19127515C2semidirect
5. #19127508C2semidirect
6. #19127486C2semidirect
7. #510×C83C2semidirect
17. C2×C83⋊C16C22semidirect
18. #14581728C22semidirect
21. C2×Q16×C83C22semidirect
24. M32×C83C22semidirect
92. C83#2194semidirect
94. #510D83semidirect
106. C2×Q16C2×D83semidirect
113. M32C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285032
C2×#145817352
#191279732
#191275152
#191275082
#191274862
#510×C832
#219483