Finite Group Databasebeta

← #24906556

Subgroups — #24906556

Order 10624.

210 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 55 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#19128426C211yesmaximal, characteristic, complemented
2#19127898C211yesmaximal, characteristic
3#19127898C211yesmaximal, characteristic
4#19127447C211yesmaximal, characteristic, complemented
5#19127433C211yesmaximal, characteristic, complemented
6#19127433C211yesmaximal, characteristic, complemented
7#486×C83C211yesmaximal, characteristic
8C2×#13050578—21no—
9C2×#13050578—21no—
10#14581790—21no—
11#14581783—21no—
12#14581782—22no—
13Q8×Dic83—21no—
14C4×Dic166C2211yescharacteristic
15C4×Dic166C2211yescharacteristic
16#14581757—21no—
17C4⋊(C83⋊C8)C2211yescharacteristic
18C4⋊(C83⋊C8)C2211yescharacteristic
19#14581713—21no—
20#14581713—21no—
21C4⋊Q8×C83C2211yescharacteristic
22#127×C83C2211yescharacteristic
23#102×C83—22no—
24C4×C8×C83C2211yescharacteristic
25C2×Dic166D411yescharacteristic
26C2×Dic166D411yescharacteristic
27#13050578—42no—
28#13050578—42no—
29C2×C83⋊C8—21no—
30C2×C83⋊C8—21no—
31C4⋊Dic83—22no—
32C4⋊Dic83D411yescharacteristic
33C4⋊Dic83D411yescharacteristic
34Dic83⋊C4—22no—
35Dic83⋊C4—22no—
36Dic83⋊C4—21no—
37Dic83⋊C4—21no—
38C4×Dic83—22no—
39C4×Dic83—21no—
40C4×Dic83—21no—
41C2×Q8×C83—21no—
42C2×D4×C83—21no—
43C2×D4×C83—21no—
44C2×D4×C83—21no—
45C2×C8×C83—21no—
46C4⋊C4×C83—21no—
47C4⋊C4×C83D412yes—
48C42×C83C2311yescharacteristic
49C23×C83—41no—
50Q8×C83—41no—
51Q8×C83—41no—
52D4×C83—42no—
53D4×C83—41no—
54D4×C83—41no—
55D4×C83—41no—
56D4×C83—41no—
57C2×C4×C83—22no—
58C2×C4×C83C2×D411yescharacteristic
59C2×C4×C83C2×D411yescharacteristic
60C2×C4×C83C2×D411yescharacteristic
61C2×Dic83—21no—
62C2×Dic83—21no—
63C2×Dic83—21no—
64C2×Dic83—21no—
65Dic166—21no—
66Dic166—21no—
67Dic166—21no—
68Dic166SD1612yescomplemented
69C8×C83—41no—
70C83⋊C8—41no—
71C83⋊C8—41no—
72C22×C83—81no—
73C22×C83—81no—
74C22×C83—81no—
75C22×C83#12011yescharacteristic
76C4×C83—21no—
77C4×C83—42no—
78C4×C83C2×SD1611yescharacteristic
79C4×C83#13511yescharacteristic
80C4×C83AGL(1,ℤ/8)11yescharacteristic
81C4×C83#13711yescharacteristic
82Dic83—22no—
83Dic83—41no—
84Dic83—41no—
85Dic83—41no—
86C2×C83—161no—
87C2×C83#44811yescharacteristic
88C2×C83#44911yescharacteristic
89C2×C83#45411yescharacteristic
90#1626—831nomaximal, Sylow
91C83#162611yescharacteristic, complemented, Sylow
92#557—831no—
93#486D8311yescharacteristic
94#460—831no—
95#460—831no—
96#332—831no—
97Q8⋊C8—831no—
98Q8⋊C8—831no—
99C2×SD16—1661no—
100C2×SD16—1661no—
101C4⋊Q8—1661no—
102C4⋊Q8C2×D8311yescharacteristic
103#127C2×D8311yescharacteristic
104#125—1661no—
105#125—1662no—
106C4×Q8—831no—
107C4×Q8—831no—
108C4×Q8—1661no—
109C4×Q8—1661no—
110C4⋊C8—831no—
111C4⋊C8—831no—
112#102—22no—
113#102—1661no—
114#102—1661no—
115C4×C8C2×D8311yescharacteristic
116C2×Q8—21no—
117C2×Q8—831no—
118C2×Q8—831no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122SD16—3322no—
123SD16—3322no—
124C2×C8—21no—
125C2×C8—1661no—
126C2×C8—1661no—
127C4⋊C4—21no—
128C4⋊C4—831no—
129C4⋊C4—831no—
130C4⋊C4—1662no—
131C4⋊C4—1662no—
132C4⋊C4—1662no—
133C4⋊C4—1661no—
134C4⋊C4—1661no—
135C4⋊C4C22⋊D8312yes—
136C42—1662no—
137C42—1661no—
138C42—1661no—
139C42C22×D8311yescharacteristic
140C23—41no—
141Q8—41no—
142Q8—41no—
143Q8—832no—
144Q8—1661no—
145Q8—1661no—
146Q8—1661no—
147D4—42no—
148D4—41no—
149D4—41no—
150D4—41no—
151D4—41no—
152C2×C4—22no—
153C2×C4—1661no—
154C2×C4—1661no—
155C2×C4—1661no—
156C2×C4—1661no—
157C2×C4D4×D8311yescharacteristic
158C2×C4D4×D8311yescharacteristic
159C2×C4C2×C22⋊D8311yescharacteristic
160C8—41no—
161C8—3321no—
162C8—3321no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22#1458176011yescharacteristic
167C4—21no—
168C4—42no—
169C4—3321no—
170C4—3321no—
171C4—3321no—
172C4—1662no—
173C4SD16×D8311yescharacteristic
174C4#1458181111yescharacteristic
175C4#1458181611yescharacteristic
176C4#1458182111yescharacteristic
177C2—161no—
178C2#1912783811yescharacteristic
179C2#1912784311yescharacteristic
180C2#1912787311yescharacteristic
subgroupG/H
centerC22#14581760
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting#486×C83C2
socleC22×C83#120
prime radical#127×C83C22
hypercenter#486D83
hypocenterC83#1626
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #19128426C2semidirect
4. #19127447C2semidirect
5. #19127433C2semidirect
6. #19127433C2semidirect
68. Dic166SD16semidirect
91. C83#1626semidirect

Maximal subgroups

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

subgroupindex
#191284262
#191278982
#191274472
#191274332
#486×C832
#162683