Finite Group Databasebeta

← #24907993

Subgroups — #24907993

Order 10624.

362 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 85 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#19128268C211yesmaximal, characteristic
2#19128176C211yesmaximal, characteristic, complemented
3#19127717C212yesmaximal, complemented
4#19127458C212yesmaximal, complemented
5C2×#100×C83C211yesmaximal, characteristic, complemented
6C2×#13050590C2211yescharacteristic
7#14581788C412yescomplemented
8#14581779C412yescomplemented
9#14581763—22no—
10#14581761—22no—
11D4×Dic83—24no—
12#14581751C2211yescharacteristic
13#14581751C2212yes—
14C2×C22⋊Dic83—22no—
15#14581718—24no—
16C22×D4×C83C2211yescharacteristic
17C2×M16×C83C2212yes—
18#100×C83—22no—
19#13050590—22no—
20#13050590—22no—
21C2×Dic166C2×C411yescharacteristic, complemented
22C2×D332C2×C411yescharacteristic, complemented
23C2×C22⋊D83—21no—
24C2×C22⋊D83—24no—
25C22×Dic83—24no—
26C2×C4×D83—21no—
27C4⋊Dic83—21no—
28C4⋊Dic83—22no—
29C22⋊Dic83—24no—
30C22⋊Dic83—24no—
31C22⋊Dic83—21no—
32C22⋊Dic83—22no—
33C4×Dic83C2×C412yescomplemented
34C4×Dic83D414yescomplemented
35C24×C83D412yes—
36C2×D4×C83—22no—
37C2×D4×C83—22no—
38C2×D4×C83C2×C412yes—
39C2×D4×C83D412yes—
40C22×C4×C83C2311yescharacteristic
41M16×C83—22no—
42M16×C83—22no—
43C2×C8×C83—22no—
44C23×C83—22no—
45C23×C83—22no—
46C23×C83—82no—
47C23×C83C22⋊C414yes—
48C23×C83C2×D411yescharacteristic
49C22×D83—21no—
50D4×C83—24no—
51D4×C83—24no—
52C2×C4×C83C22×C411yescharacteristic
53C2×C4×C83C2×D412yes—
54C2×C4×C83C2×D411yescharacteristic
55C2×C4×C83#4112yes—
56C22⋊D83—41no—
57C22⋊D83—41no—
58C22⋊D83—84no—
59C2×Dic83—22no—
60C2×Dic83—21no—
61C2×Dic83—24no—
62C2×Dic83—84no—
63D332—22no—
64C4×D83—42no—
65Dic166—22no—
66C8×C83—42no—
67C22×C83—24no—
68C22×C83—41no—
69C22×C83—44no—
70C22×C83—44no—
71C22×C83—82no—
72C22×C83C2×C22⋊C411yescharacteristic
73C22×C83#12011yescharacteristic
74C22×C83#12311yescharacteristic
75C2×D83—42no—
76C2×D83—41no—
77C4×C83C4×D412yes—
78C4×C83#12112yes—
79Dic83—44no—
80Dic83—42no—
81Dic83—41no—
82C2×C83—21no—
83C2×C83—21no—
84C2×C83—21no—
85C2×C83—44no—
86C2×C83#38611yescharacteristic
87D83—81no—
88#1911—831nomaximal, Sylow
89C83#191111yescharacteristic, complemented, Sylow
90#535—831no—
91#518—831no—
92#421—832no—
93C2×#100D8311yescharacteristic, complemented
94H(3,ℤ/4)—832no—
95C2×#41—831no—
96C22×D4C2×D8311yescharacteristic
97C2×M16C2×D8312yes—
98#127—832no—
99#124—832no—
100#121—1662no—
101#120—1662no—
102C4×D4—1664no—
103#117—832no—
104#117—831no—
105C2×C22⋊C4—1662no—
106#104—1664no—
107#100—22no—
108C24D33212yes—
109#41—1662no—
110#41—1662no—
111C2×Q8—831no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—831no—
115C2×D4—1661no—
116C2×D4—1664no—
117C2×D4C4×D8312yes—
118C2×D4C22⋊D8312yes—
119C22×C4—1661no—
120C22×C4—1664no—
121C22×C4C22×D8311yescharacteristic
122M16—22no—
123M16—22no—
124C2×C8—22no—
125C4⋊C4—1662no—
126C4⋊C4—1661no—
127C22⋊C4—1664no—
128C22⋊C4—1664no—
129C22⋊C4—1662no—
130C22⋊C4—1661no—
131C42—832no—
132C42—834no—
133C23—22no—
134C23—22no—
135C23—82no—
136C23—1661no—
137C23(C2×D83)⋊C414yes—
138C23C2×D33211yescharacteristic
139Q8—1662no—
140D4—24no—
141D4—24no—
142D4—3321no—
143D4—3321no—
144D4—6644no—
145D4—1662no—
146C2×C4—3322no—
147C2×C4—6644no—
148C2×C4—1664no—
149C2×C4—1662no—
150C2×C4—1661no—
151C2×C4C2×C4×D8311yescharacteristic
152C2×C4D4×D8312yes—
153C2×C4C2×C22⋊D8311yescharacteristic
154C2×C4#1305058912yes—
155C8—42no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—82no—
161C22—3322no—
162C22—3321no—
163C22C2×(C2×D83)⋊C411yescharacteristic
164C22#1458176211yescharacteristic
165C22#1458177611yescharacteristic
166C4—3324no—
167C4—3322no—
168C4—3321no—
169C4(C22⋊D83)⋊C412yes—
170C4#1458176412yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—6641no—
176C2#1912757811yescharacteristic
subgroupG/H
centerC2#19127578
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
FittingC2×#100×C83C2
socleC2×C83#386
prime radical#19128268C2
cosocleC22×C4×C83C23
hypercenterC2×#100D83
hypocenterC83#1911
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #19128176C2semidirect
3. #19127717C2semidirect
4. #19127458C2semidirect
5. C2×#100×C83C2semidirect
7. #14581788C4semidirect
8. #14581779C4semidirect
21. C2×Dic166C2×C4semidirect
22. C2×D332C2×C4semidirect
33. C4×Dic83C2×C4semidirect
34. C4×Dic83D4semidirect
89. C83#1911semidirect
93. C2×#100D83semidirect

Maximal subgroups

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

subgroupindex
#191282682
#191281762
#191277172
#191274582
C2×#100×C832
#191183