Finite Group Databasebeta

← #24909417

Subgroups — #24909417

Order 10624.

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#19128503C211yesmaximal, characteristic, complemented
2C2×#14581733C211yesmaximal, characteristic, complemented
3#19127968C211yesmaximal, characteristic, complemented
4#19127515C211yesmaximal, characteristic, complemented
5#19127508C211yesmaximal, characteristic, complemented
6#19127486C211yesmaximal, characteristic, complemented
7#509×C83C211yesmaximal, characteristic, complemented
8#14581838—21no—
9#14581819—22no—
10#14581818—21no—
11#14581817—21no—
12#14581815—21no—
13C2×D664C2211yescharacteristic
14D8×D83—21no—
15#14581798C2211yescharacteristic
16#14581733—22no—
17C2×C83⋊C16C2211yescharacteristic, complemented
18#14581728C2211yescharacteristic, complemented
19#14581709—21no—
20#135×C83C2211yescharacteristic
21C2×D8×C83C2211yescharacteristic, complemented
22SD32×C83—21no—
23D16×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#13050578—41no—
33#13050577—22no—
34D664—21no—
35D664D412yes—
36#13050575—22no—
37#13050575—41no—
38#13050574—21no—
39#13050574D411yescharacteristic
40#13050573—21no—
41C8×D83—21no—
42C83⋊C16—21no—
43#41×C83—21no—
44C2×D4×C83—21no—
45Q16×C83D411yescharacteristic
46SD16×C83—21no—
47D8×C83—21no—
48D8×C83—21no—
49D8×C83D411yescharacteristic
50C2×C8×C83C2311yescharacteristic
51C16×C83—21no—
52C23×C83—41no—
53C22×D83—42no—
54C22×D83—41no—
55Q8×C83—21no—
56D4×C83—21no—
57D4×C83—41no—
58D4×C83—41no—
59D4×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×C83—81no—
77C22×C83—161no—
78C22×C83AGL(1,ℤ/8)11yescharacteristic
79C2×D83—21no—
80C2×D83—42no—
81C2×D83—162no—
82C2×D83—161no—
83C4×C83—41no—
84C4×C83#12011yescharacteristic
85C4×C83C2×D811yescharacteristic
86Dic83—21no—
87C2×C83—21no—
88C2×C83—81no—
89C2×C83—161no—
90C2×C83#44711yescharacteristic
91D83—41no—
92D83—82no—
93#2192—831nomaximal, Sylow
94C83#219211yescharacteristic, complemented, Sylow
95#576—831no—
96#509D8311yescharacteristic, complemented
97C2×D16—831no—
98#472—831no—
99#361—831no—
100#359—831no—
101#350—831no—
102H(5,𝔽2)—1661no—
103AGL(1,ℤ/8)—1661no—
104AGL(1,ℤ/8)—1661no—
105AGL(1,ℤ/8)—1662no—
106#135—1661no—
107#135C2×D8311yescharacteristic
108C2×D8—831no—
109C2×D8—1661no—
110C2×D8C2×D8311yescharacteristic, complemented
111#131—831no—
112SD32—21no—
113D16—21no—
114D16—1662no—
115M32C2×D8311yescharacteristic, complemented
116C2×C16—831no—
117#108—831no—
118#100—1661no—
119#41—21no—
120#41—3321no—
121#41—831no—
122#41—1662no—
123C2×D4—21no—
124C2×D4—3321no—
125C2×D4—3322no—
126C2×D4—1662no—
127C2×D4—1661no—
128Q16C22⋊D8311yescharacteristic
129SD16—21no—
130SD16—3321no—
131SD16—1662no—
132D8—21no—
133D8—21no—
134D8—3321no—
135D8—832no—
136D8—1662no—
137D8—1661no—
138D8C22⋊D8311yescharacteristic
139M16—831no—
140M16—1661no—
141M16—1661no—
142C2×C8—1661no—
143C2×C8C22×D8311yescharacteristic
144C16—21no—
145C16—1661no—
146C23—41no—
147C23—3322no—
148C23—3321no—
149Q8—21no—
150Q8—831no—
151D4—21no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—3322no—
156D4—3322no—
157D4—3321no—
158D4—831no—
159D4—1661no—
160D4—1662no—
161C2×C4—41no—
162C2×C4—3322no—
163C2×C4—1661no—
164C2×C4D4×D8311yescharacteristic
165C8—3321no—
166C8—1661no—
167C8D4×D8311yescharacteristic
168C8C2×C22⋊D8311yescharacteristic
169C22—41no—
170C22—81no—
171C22—161no—
172C22—13281no—
173C22—13282no—
174C22—3322no—
175C22—1661no—
176C22#1458181611yescharacteristic
177C4—41no—
178C4—1661no—
179C4#1458176011yescharacteristic
180C4D8×D8311yescharacteristic
181C2—21no—
182C2—81no—
183C2—161no—
184C2—3321no—
185C2—6642no—
186C2#1912783311yescharacteristic
subgroupG/H
centerC2#19127833
commutator, cosocleC2×C8×C83C23
FrattiniC2×C8C22×D83
Fitting#509×C83C2
socleC2×C83#447
hypercenter#509D83
hypocenterC83#2192
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #19128503C2semidirect
2. C2×#14581733C2semidirect
3. #19127968C2semidirect
4. #19127515C2semidirect
5. #19127508C2semidirect
6. #19127486C2semidirect
7. #509×C83C2semidirect
17. C2×C83⋊C16C22semidirect
18. #14581728C22semidirect
21. C2×D8×C83C22semidirect
24. M32×C83C22semidirect
94. C83#2192semidirect
96. #509D83semidirect
110. C2×D8C2×D83semidirect
115. M32C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285032
C2×#145817332
#191279682
#191275152
#191275082
#191274862
#509×C832
#219283