Finite Group Databasebeta

← #24909436

Subgroups — #24909436

Order 10624.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#19128506C211yesmaximal, characteristic
2#19128144C211yesmaximal, characteristic, complemented
3C2×#14581737C211yesmaximal, characteristic, complemented
4#19127976C211yesmaximal, characteristic, complemented
5#19127515C211yesmaximal, characteristic, complemented
6#19127509C211yesmaximal, characteristic, complemented
7#350×C83C211yesmaximal, characteristic, complemented
8#14581838—21no—
9#14581822—21no—
10#14581820—22no—
11#14581820—21no—
12#14581814—21no—
13#14581814C2211yescharacteristic
14C2×Dic332C2211yescharacteristic, complemented
15C2×D664—21no—
16C2×D664C2211yescharacteristic
17Dic664—21no—
18#14581737—22no—
19#14581737—21no—
20#14581728C2211yescharacteristic, complemented
21#14581710—21no—
22#131×C83C2211yescharacteristic
23M32×C83C2211yescharacteristic, complemented
24C2×C16×C83C2211yescharacteristic, complemented
25#13050591—41no—
26#13050590—22no—
27#13050590—21no—
28C2×Dic166—21no—
29C2×D332—22no—
30C2×D332—21no—
31D4×D83—42no—
32D4×D83—41no—
33Dic332—21no—
34Dic332—21no—
35Dic332D411yescharacteristic
36#13050580—41no—
37#13050579—22no—
38#13050579—21no—
39#13050578—41no—
40D664—22no—
41D664—21no—
42D664D411yescharacteristic
43D664D412yes—
44#13050574—21no—
45#41×C83D411yescharacteristic
46M16×C83—21no—
47M16×C83D411yescharacteristic
48C2×C8×C83—21no—
49C2×C8×C83C2311yescharacteristic
50C16×C83—21no—
51C16×C83—21no—
52C22×D83—42no—
53C22×D83—41no—
54Q8×C83D811yescharacteristic
55D4×C83—21no—
56D4×C83D811yescharacteristic
57C2×C4×C83—21no—
58C2×C4×C83C2×D411yescharacteristic
59C22⋊D83—42no—
60C22⋊D83—41no—
61C2×Dic83—41no—
62D332—21no—
63D332—22no—
64D332—42no—
65D332—41no—
66C4×D83—42no—
67C4×D83—41no—
68Dic166—21no—
69Dic166—41no—
70Dic166—41no—
71C8×C83—21no—
72C8×C83C2×D411yescharacteristic
73C8×C83C2×D411yescharacteristic
74C83⋊C8—41no—
75C22×C83—21no—
76C22×C83AGL(1,ℤ/8)11yescharacteristic
77C2×D83—42no—
78C2×D83—41no—
79C2×D83—161no—
80C2×D83—162no—
81C4×C83—21no—
82C4×C83#12011yescharacteristic
83C4×C83C2×D811yescharacteristic
84Dic83—41no—
85Dic83—81no—
86C2×C83—21no—
87C2×C83—41no—
88C2×C83#44711yescharacteristic
89D83—82no—
90D83—81no—
91#2194—831nomaximal, Sylow
92C83#219411yescharacteristic, complemented, Sylow
93#576—831no—
94#510—831no—
95C2×SD32—831no—
96#473—831no—
97#361—831no—
98#359—831no—
99#350D8311yescharacteristic, complemented
100H(5,𝔽2)—1661no—
101#137—1661no—
102AGL(1,ℤ/8)—1662no—
103AGL(1,ℤ/8)—1661no—
104#135—831no—
105#135—1661no—
106C2×Q16—831no—
107C2×D8—831no—
108C2×D8—1661no—
109#131C2×D8311yescharacteristic
110Q32—1661no—
111SD32—1662no—
112SD32—1661no—
113M32C2×D8311yescharacteristic, complemented
114C2×C16C2×D8311yescharacteristic, complemented
115#108—831no—
116#101—1661no—
117#41—3321no—
118#41—1662no—
119#41—1661no—
120#41D33211yescharacteristic
121C2×Q8—1661no—
122C2×D4—3322no—
123C2×D4—3321no—
124C2×D4—1662no—
125C2×D4—1661no—
126Q16—3321no—
127Q16—831no—
128Q16—1661no—
129Q16—1661no—
130SD16—3321no—
131SD16—1662no—
132SD16—1661no—
133D8—832no—
134D8—831no—
135D8—1662no—
136D8—1661no—
137M16—21no—
138M16—1661no—
139M16D33211yescharacteristic
140C2×C8—21no—
141C2×C8C22×D8311yescharacteristic
142C16—21no—
143C16—21no—
144C23—3322no—
145C23—3321no—
146Q8—3321no—
147Q8—3321no—
148Q8—1661no—
149Q8D66411yescharacteristic
150D4—21no—
151D4—3322no—
152D4—3322no—
153D4—3321no—
154D4—3321no—
155D4—1662no—
156D4—1661no—
157D4D66411yescharacteristic
158C2×C4—21no—
159C2×C4—3322no—
160C2×C4—3321no—
161C2×C4—3321no—
162C2×C4C2×D33211yescharacteristic
163C8—21no—
164C8—3321no—
165C8D4×D8311yescharacteristic
166C8D4×D8311yescharacteristic
167C22—21no—
168C22—13281no—
169C22—13282no—
170C22—3322no—
171C22—3321no—
172C22#1458182011yescharacteristic
173C4—21no—
174C4—3321no—
175C4—6641no—
176C4#1458176211yescharacteristic
177C4C2×D66411yescharacteristic
178C2—21no—
179C2—41no—
180C2—6642no—
181C2—6641no—
182C2#1912783711yescharacteristic
subgroupG/H
centerC2#19127837
commutator, cosocleC2×C8×C83C23
FrattiniC2×C8C22×D83
Fitting#350×C83C2
socleC2×C83#447
prime radical#19128506C2
hypercenter#350D83
hypocenterC83#2194
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #350

Complements

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

NH ≅ G/Ntype
2. #19128144C2semidirect
3. C2×#14581737C2semidirect
4. #19127976C2semidirect
5. #19127515C2semidirect
6. #19127509C2semidirect
7. #350×C83C2semidirect
14. C2×Dic332C22semidirect
20. #14581728C22semidirect
23. M32×C83C22semidirect
24. C2×C16×C83C22semidirect
92. C83#2194semidirect
99. #350D83semidirect
113. M32C2×D83semidirect
114. C2×C16C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285062
#191281442
C2×#145817372
#191279762
#191275152
#191275092
#350×C832
#219483