Finite Group Databasebeta

← #24908693

Subgroups — #24908693

Order 10624.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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#19128282C211yesmaximal, characteristic
2C2×#14581718C212yesmaximal, complemented
3#19127703C212yesmaximal, complemented
4#19127566C211yesmaximal, characteristic, complemented
5#343×C83C211yesmaximal, characteristic, complemented
6C2×#13050590C2212yes—
7#14581772—22no—
8#14581767—22no—
9#14581766—22no—
10C2×C4⋊Dic83C2211yescharacteristic
11C2×C4×Dic83C2211yescharacteristic
12#14581718—24no—
13#14581717—22no—
14#14581714—22no—
15C83⋊#95—22no—
16C2×M16×C83C2212yes—
17C2×C4⋊C4×C83C2211yescharacteristic
18#13050590—22no—
19#13050590—22no—
20#13050590—22no—
21#13050590—22no—
22C2×Dic166D412yes—
23C2×D332—22no—
24C2×D332D412yes—
25C2×C22⋊D83—22no—
26C22×Dic83—21no—
27C22×Dic83—21no—
28C2×C4×D83—22no—
29C4⋊Dic83—22no—
30C4⋊Dic83—22no—
31(C2×D83)⋊C4—24no—
32C4×Dic83—21no—
33C4×Dic83D412yescomplemented
34C22×C4×C83D412yes—
35C22×C4×C83C2311yescharacteristic
36M16×C83—22no—
37M16×C83—22no—
38C2×C8×C83—22no—
39C4⋊C4×C83—22no—
40C23×C83C2×D411yescharacteristic
41C22×D83—22no—
42C2×C4×C83—22no—
43C2×C4×C83—42no—
44C2×C4×C83C2×D411yescharacteristic
45C2×C4×C83C2×D412yes—
46C2×C4×C83#4112yes—
47C2×C4×C83#4111yescharacteristic
48C22⋊D83—42no—
49C22⋊D83—42no—
50C2×Dic83—22no—
51C2×Dic83—22no—
52C2×Dic83—41no—
53C2×Dic83—41no—
54C2×Dic83—42no—
55C2×Dic83—41no—
56D332—22no—
57D332—22no—
58D332—82no—
59C4×D83—42no—
60C4×D83—42no—
61Dic166—22no—
62Dic166—22no—
63C8×C83—42no—
64C22×C83—21no—
65C22×C83—21no—
66C22×C83#12011yescharacteristic
67C22×C83#12111yescharacteristic
68C22×C83#12411yescharacteristic
69C2×D83—42no—
70C2×D83—42no—
71C2×D83—42no—
72C4×C83—42no—
73C4×C83#12112yes—
74C4×C83#12312yes—
75Dic83—42no—
76Dic83—42no—
77Dic83—81no—
78C2×C83—21no—
79C2×C83—21no—
80C2×C83#39411yescharacteristic
81C2×C83#45411yescharacteristic
82C2×C83#45511yescharacteristic
83D83—82no—
84#2040—831nomaximal, Sylow
85C83#204011yescharacteristic, complemented, Sylow
86#537—831no—
87C2×#104—832no—
88#417—832no—
89#382—831no—
90#343D8311yescharacteristic, complemented
91C2×#41—832no—
92C2×M16C2×D8312yes—
93C22⋊Q8—1662no—
94#121—1662no—
95#121—1662no—
96C2×C4⋊C4—831no—
97C2×C4⋊C4C2×D8311yescharacteristic
98C2×C42—831no—
99#104—1664no—
100#103—1662no—
101#102—1662no—
102#95—1662no—
103#41—1662no—
104#41—1662no—
105#41—1662no—
106#41—1662no—
107C2×Q8—832no—
108C2×D4—832no—
109C2×D4—1662no—
110C2×D4—1662no—
111C22×C4—1661no—
112C22×C4—1661no—
113C22×C4—1662no—
114C22×C4D33212yes—
115C22×C4C22×D8311yescharacteristic
116M16—22no—
117M16—22no—
118C2×C8—22no—
119C4⋊C4—22no—
120C4⋊C4—1662no—
121C4⋊C4—1662no—
122C22⋊C4—1664no—
123C42—832no—
124C42—1661no—
125C23—1662no—
126C23C2×D33211yescharacteristic
127Q8—1662no—
128Q8—1662no—
129D4—3322no—
130D4—3322no—
131D4—6642no—
132D4—1662no—
133D4—1662no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—3321no—
137C2×C4—3321no—
138C2×C4—3322no—
139C2×C4—3322no—
140C2×C4—3322no—
141C2×C4—3321no—
142C2×C4—1662no—
143C2×C4—1662no—
144C2×C4D4×D8311yescharacteristic
145C2×C4D4×D8312yes—
146C2×C4#1305059012yes—
147C2×C4#1305059111yescharacteristic
148C8—42no—
149C22—21no—
150C22—21no—
151C22—3322no—
152C22—3322no—
153C22—3322no—
154C22#1458176211yescharacteristic
155C22#1458176711yescharacteristic
156C22#1458178011yescharacteristic
157C4—42no—
158C4—3322no—
159C4—3322no—
160C4—6641no—
161C4#1458176512yes—
162C4#1458177712yes—
163C2—21no—
164C2—21no—
165C2—6642no—
166C2#1912761911yescharacteristic
167C2#1912787611yescharacteristic
168C2#1912788111yescharacteristic
subgroupG/H
centerC22#14581762
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting#343×C83C2
socleC22×C83#120
prime radical#19128282C2
hypercenter#343D83
hypocenterC83#2040
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
2. C2×#14581718C2semidirect
3. #19127703C2semidirect
4. #19127566C2semidirect
5. #343×C83C2semidirect
33. C4×Dic83D4semidirect
85. C83#2040semidirect
90. #343D83semidirect

Maximal subgroups

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

subgroupindex
#191282822
C2×#145817182
#191277032
#191275662
#343×C832
#204083