Finite Group Databasebeta

← #24909423

Subgroups — #24909423

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#19128502C211yesmaximal, characteristic, complemented
2#19128139C211yesmaximal, characteristic, complemented
3C2×#14581733C211yesmaximal, characteristic, complemented
4#19127972C211yesmaximal, characteristic, complemented
5#19127507C211yesmaximal, characteristic, complemented
6#19127486C211yesmaximal, characteristic, complemented
7#361×C83C211yesmaximal, characteristic, complemented
8#14581837—21no—
9#14581820—21no—
10#14581818—21no—
11#14581816—22no—
12#14581814C2211yescharacteristic
13#14581811—21no—
14C2×D664C2211yescharacteristic, complemented
15D8×D83—21no—
16#14581798C2211yescharacteristic
17#14581737—21no—
18D1328—21no—
19#14581733—22no—
20C2×C83⋊C16C2211yescharacteristic, complemented
21#14581708—21no—
22C2×D8×C83C2211yescharacteristic
23M32×C83C2211yescharacteristic, complemented
24#108×C83C2211yescharacteristic, complemented
25#13050590—21no—
26#13050590D411yescharacteristic
27#13050589—22no—
28#13050589—41no—
29C2×D332—21no—
30C2×C22⋊D83—42no—
31D4×D83—22no—
32D4×D83—41no—
33Dic332D411yescharacteristic
34#13050579—21no—
35#13050579—41no—
36#13050578—22no—
37D664—21no—
38D664—21no—
39D664—41no—
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—
54C22×D83—41no—
55D4×C83—22no—
56D4×C83—41no—
57C2×C4×C83C2×D411yescharacteristic
58C22⋊D83—21no—
59C22⋊D83—42no—
60C22⋊D83—42no—
61C22⋊D83—41no—
62C2×Dic83—42no—
63D332—21no—
64D332—41no—
65D332—41no—
66D332D811yescharacteristic
67C4×D83—21no—
68C4×D83—41no—
69Dic166—21no—
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—81no—
81C2×D83—162no—
82C2×D83—161no—
83C4×C83#12011yescharacteristic
84C4×C83C2×D811yescharacteristic
85Dic83—21no—
86Dic83—41no—
87C2×C83—21no—
88C2×C83—82no—
89C2×C83#44711yescharacteristic
90D83—41no—
91D83—81no—
92D83—161no—
93#2192—831nomaximal, Sylow
94C83#219211yescharacteristic, complemented, Sylow
95#576—831no—
96#509—831no—
97C2×D16—831no—
98#472—831no—
99#361D8311yescharacteristic, complemented
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—831no—
107#135—1661no—
108C2×D8—831no—
109C2×D8—1661no—
110C2×D8C2×D8311yescharacteristic
111#131—831no—
112SD32—1661no—
113D16—1661no—
114D16—1662no—
115M32C2×D8311yescharacteristic, complemented
116C2×C16—831no—
117#108C2×D8311yescharacteristic, complemented
118#100—1661no—
119#41—3321no—
120#41—831no—
121#41—1662no—
122#41—1661no—
123C2×D4—21no—
124C2×D4—3321no—
125C2×D4—3322no—
126C2×D4—1662no—
127C2×D4—1661no—
128Q16—831no—
129SD16—3321no—
130SD16—1662no—
131SD16—1661no—
132D8—21no—
133D8—3321no—
134D8—831no—
135D8—1662no—
136D8—1661no—
137D8—1661no—
138D8D33212yes—
139M16—21no—
140M16—831no—
141M16—1661no—
142C2×C8—1661no—
143C2×C8C22×D8311yescharacteristic
144C16—21no—
145C16—1661no—
146C23—41no—
147C23—3322no—
148C23—3321no—
149Q8—831no—
150Q8—1661no—
151D4—22no—
152D4—41no—
153D4—3322no—
154D4—3321no—
155D4—3322no—
156D4—3321no—
157D4—3321no—
158D4—831no—
159D4—1661no—
160D4—1661no—
161C2×C4—3322no—
162C2×C4—3321no—
163C2×C4—1661no—
164C2×C4D4×D8311yescharacteristic
165C8—41no—
166C8—1661no—
167C8D4×D8311yescharacteristic
168C8C2×D33211yescharacteristic
169C22—42no—
170C22—161no—
171C22—13281no—
172C22—13282no—
173C22—3321no—
174C22—6641no—
175C22—1661no—
176C22#1458181911yescharacteristic
177C4—3321no—
178C4—1661no—
179C4#1458176211yescharacteristic
180C4D8×D8311yescharacteristic
181C2—21no—
182C2—82no—
183C2—13281no—
184C2—3321no—
185C2—6641no—
186C2#1912783611yescharacteristic
subgroupG/H
centerC2#19127836
commutator, cosocleC2×C8×C83C23
FrattiniC2×C8C22×D83
Fitting#361×C83C2
socleC2×C83#447
hypercenter#361D83
hypocenterC83#2192
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
1. #19128502C2semidirect
2. #19128139C2semidirect
3. C2×#14581733C2semidirect
4. #19127972C2semidirect
5. #19127507C2semidirect
6. #19127486C2semidirect
7. #361×C83C2semidirect
14. C2×D664C22semidirect
20. C2×C83⋊C16C22semidirect
23. M32×C83C22semidirect
24. #108×C83C22semidirect
94. C83#2192semidirect
99. #361D83semidirect
115. M32C2×D83semidirect
117. #108C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285022
#191281392
C2×#145817332
#191279722
#191275072
#191274862
#361×C832
#219283