Finite Group Databasebeta

← #24909437

Subgroups — #24909437

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#19128502C211yesmaximal, characteristic
2#19128144C211yesmaximal, characteristic, complemented
3C2×#14581736C211yesmaximal, characteristic, complemented
4#19127977C211yesmaximal, characteristic, complemented
5#19127507C211yesmaximal, characteristic, complemented
6#19127486C211yesmaximal, characteristic, complemented
7#361×C83C211yesmaximal, characteristic, complemented
8#14581837—21no—
9#14581825—21no—
10#14581818—21no—
11#14581816—22no—
12#14581814C2211yescharacteristic
13#14581811—21no—
14C2×Dic332C2211yescharacteristic, complemented
15D8×D83—21no—
16#14581798C2211yescharacteristic
17Dic664—21no—
18#14581737—21no—
19#14581736—22no—
20C2×C83⋊C16C2211yescharacteristic, complemented
21#14581710—21no—
22C2×D8×C83C2211yescharacteristic
23M32×C83C2211yescharacteristic, complemented
24#108×C83C2211yescharacteristic, complemented
25#13050590—21no—
26#13050590D411yescharacteristic
27#13050589—22no—
28#13050589—41no—
29C2×Dic166—21no—
30C2×C22⋊D83—42no—
31D4×D83—22no—
32D4×D83—41no—
33Dic332—21no—
34Dic332—21no—
35Dic332—41no—
36Dic332D411yescharacteristic
37#13050579—21no—
38#13050579—41no—
39#13050578—22no—
40D664D411yescharacteristic
41#13050575—22no—
42#13050574D411yescharacteristic
43#13050573—21no—
44C8×D83—21no—
45C83⋊C16—21no—
46C2×D4×C83—21no—
47D8×C83—21no—
48D8×C83D412yes—
49M16×C83—21no—
50C2×C8×C83C2311yescharacteristic
51C16×C83—21no—
52C23×C83—41no—
53C22×D83—42no—
54D4×C83—22no—
55D4×C83—41no—
56C2×C4×C83C2×D411yescharacteristic
57C22⋊D83—21no—
58C22⋊D83—42no—
59C22⋊D83—42no—
60C22⋊D83—41no—
61C2×Dic83—42no—
62C2×Dic83—41no—
63D332—21no—
64D332D811yescharacteristic
65C4×D83—21no—
66C4×D83—41no—
67Dic166—21no—
68Dic166—41no—
69Dic166—41no—
70Dic166D811yescharacteristic
71C8×C83—41no—
72C8×C83C2×D411yescharacteristic
73C8×C83C2×D411yescharacteristic
74C83⋊C8—21no—
75C22×C83—42no—
76C22×C83—161no—
77C22×C83AGL(1,ℤ/8)11yescharacteristic
78C2×D83—21no—
79C2×D83—41no—
80C2×D83—162no—
81C4×C83#12011yescharacteristic
82C4×C83C2×D811yescharacteristic
83Dic83—21no—
84Dic83—41no—
85Dic83—81no—
86C2×C83—21no—
87C2×C83—82no—
88C2×C83#44711yescharacteristic
89D83—41no—
90D83—81no—
91#2194—831nomaximal, Sylow
92C83#219411yescharacteristic, complemented, Sylow
93#576—831no—
94#510—831no—
95C2×SD32—831no—
96#473—831no—
97#361D8311yescharacteristic, complemented
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—831no—
105#135—1661no—
106C2×Q16—831no—
107C2×D8—1661no—
108C2×D8C2×D8311yescharacteristic
109#131—831no—
110Q32—1661no—
111SD32—1661no—
112SD32—1662no—
113M32C2×D8311yescharacteristic, complemented
114C2×C16—831no—
115#108C2×D8311yescharacteristic, complemented
116#101—1661no—
117#41—3321no—
118#41—831no—
119#41—1662no—
120#41—1661no—
121C2×Q8—1661no—
122C2×D4—21no—
123C2×D4—3321no—
124C2×D4—3322no—
125C2×D4—1662no—
126Q16—3321no—
127Q16—831no—
128Q16—1661no—
129Q16—1661no—
130SD16—3321no—
131SD16—1662no—
132SD16—1661no—
133D8—21no—
134D8—831no—
135D8—1662no—
136D8D33212yes—
137M16—21no—
138M16—831no—
139M16—1661no—
140C2×C8—1661no—
141C2×C8C22×D8311yescharacteristic
142C16—21no—
143C16—1661no—
144C23—41no—
145C23—3322no—
146Q8—3321no—
147Q8—3321no—
148Q8—831no—
149Q8—1661no—
150D4—22no—
151D4—41no—
152D4—3322no—
153D4—3321no—
154D4—3322no—
155D4—831no—
156D4—1661no—
157D4—1661no—
158C2×C4—3322no—
159C2×C4—3321no—
160C2×C4—3321no—
161C2×C4—1661no—
162C2×C4D4×D8311yescharacteristic
163C8—41no—
164C8—1661no—
165C8D4×D8311yescharacteristic
166C8C2×D33211yescharacteristic
167C22—42no—
168C22—161no—
169C22—13282no—
170C22—3321no—
171C22—1661no—
172C22#1458181911yescharacteristic
173C4—3321no—
174C4—6641no—
175C4—1661no—
176C4#1458176211yescharacteristic
177C4D8×D8311yescharacteristic
178C2—21no—
179C2—82no—
180C2—3321no—
181C2—6641no—
182C2#1912783611yescharacteristic
subgroupG/H
centerC2#19127836
commutator, cosocleC2×C8×C83C23
FrattiniC2×C8C22×D83
Fitting#361×C83C2
socleC2×C83#447
prime radical#19128502C2
hypercenter#361D83
hypocenterC83#2194
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
2. #19128144C2semidirect
3. C2×#14581736C2semidirect
4. #19127977C2semidirect
5. #19127507C2semidirect
6. #19127486C2semidirect
7. #361×C83C2semidirect
14. C2×Dic332C22semidirect
20. C2×C83⋊C16C22semidirect
23. M32×C83C22semidirect
24. #108×C83C22semidirect
92. C83#2194semidirect
97. #361D83semidirect
113. M32C2×D83semidirect
115. #108C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285022
#191281442
C2×#145817362
#191279772
#191275072
#191274862
#361×C832
#219483