Finite Group Databasebeta

← #24908688

Subgroups — #24908688

Order 10624.

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

normal subgroups only

This group has 306 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#19127570C211yesmaximal, characteristic, complemented
5#343×C83C211yesmaximal, characteristic, complemented
6C2×#13050590C2212yes—
7#14581772—22no—
8#14581767—22no—
9#14581766—22no—
10C2×C4⋊Dic83C2211yescharacteristic
11C2×Dic83⋊C4—22no—
12C2×C4×Dic83C2211yescharacteristic
13#14581718—24no—
14#14581717—22no—
15#14581714—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—
31Dic83⋊C4—44no—
32(C2×D83)⋊C4—24no—
33C4×Dic83—21no—
34C4×Dic83D412yescomplemented
35C22×C4×C83D412yes—
36C22×C4×C83C2311yescharacteristic
37M16×C83—22no—
38M16×C83—22no—
39C2×C8×C83—22no—
40C4⋊C4×C83—22no—
41C23×C83C2×D411yescharacteristic
42C22×D83—22no—
43C2×C4×C83—22no—
44C2×C4×C83—42no—
45C2×C4×C83C2×D411yescharacteristic
46C2×C4×C83C2×D412yes—
47C2×C4×C83#4112yes—
48C2×C4×C83#4111yescharacteristic
49C22⋊D83—42no—
50C22⋊D83—42no—
51C2×Dic83—22no—
52C2×Dic83—22no—
53C2×Dic83—22no—
54C2×Dic83—22no—
55C2×Dic83—42no—
56C2×Dic83—41no—
57D332—22no—
58D332—22no—
59D332—82no—
60C4×D83—42no—
61C4×D83—42no—
62Dic166—22no—
63Dic166—22no—
64C8×C83—42no—
65C22×C83—21no—
66C22×C83—21no—
67C22×C83#12011yescharacteristic
68C22×C83#12111yescharacteristic
69C22×C83#12411yescharacteristic
70C2×D83—42no—
71C2×D83—42no—
72C2×D83—42no—
73C4×C83—42no—
74C4×C83#12112yes—
75C4×C83#12312yes—
76Dic83—24no—
77Dic83—42no—
78Dic83—81no—
79C2×C83—21no—
80C2×C83—21no—
81C2×C83#39411yescharacteristic
82C2×C83#45311yescharacteristic
83C2×C83#45611yescharacteristic
84D83—82no—
85#2039—831nomaximal, Sylow
86C83#203911yescharacteristic, complemented, Sylow
87#537—831no—
88C2×#104—832no—
89#417—832no—
90#384—831no—
91#343D8311yescharacteristic, complemented
92C2×#41—832no—
93C2×M16C2×D8312yes—
94C22⋊Q8—1662no—
95#121—1662no—
96#121—1662no—
97C2×C4⋊C4—831no—
98C2×C4⋊C4—1662no—
99C2×C4⋊C4C2×D8311yescharacteristic
100C2×C42—831no—
101#104—1664no—
102#103—1662no—
103#102—1662no—
104#41—1662no—
105#41—1662no—
106#41—1662no—
107#41—1662no—
108C2×Q8—832no—
109C2×D4—832no—
110C2×D4—1662no—
111C2×D4—1662no—
112C22×C4—1661no—
113C22×C4—1662no—
114C22×C4—1661no—
115C22×C4D33212yes—
116C22×C4C22×D8311yescharacteristic
117M16—22no—
118M16—22no—
119C2×C8—22no—
120C4⋊C4—22no—
121C4⋊C4—3324no—
122C4⋊C4—1662no—
123C4⋊C4—1662no—
124C22⋊C4—1664no—
125C42—832no—
126C42—1661no—
127C23—1662no—
128C23C2×D33211yescharacteristic
129Q8—1662no—
130Q8—1662no—
131D4—3322no—
132D4—3322no—
133D4—6642no—
134D4—1662no—
135D4—1662no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—3322no—
139C2×C4—3322no—
140C2×C4—3322no—
141C2×C4—3321no—
142C2×C4—1662no—
143C2×C4—1662no—
144C2×C4—1662no—
145C2×C4—1662no—
146C2×C4D4×D8311yescharacteristic
147C2×C4D4×D8312yes—
148C2×C4#1305059012yes—
149C2×C4#1305059111yescharacteristic
150C8—42no—
151C22—21no—
152C22—21no—
153C22—3322no—
154C22—3322no—
155C22—3322no—
156C22#1458176211yescharacteristic
157C22#1458176711yescharacteristic
158C22#1458178011yescharacteristic
159C4—42no—
160C4—3322no—
161C4—6641no—
162C4—1664no—
163C4#1458176512yes—
164C4#1458177712yes—
165C2—21no—
166C2—21no—
167C2—6642no—
168C2#1912761911yescharacteristic
169C2#1912787111yescharacteristic
170C2#1912788611yescharacteristic
subgroupG/H
centerC22#14581762
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting#343×C83C2
socleC22×C83#120
prime radical#19128282C2
hypercenter#343D83
hypocenterC83#2039
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. #19127570C2semidirect
5. #343×C83C2semidirect
34. C4×Dic83D4semidirect
86. C83#2039semidirect
91. #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
#191275702
#343×C832
#203983