Finite Group Databasebeta

← #24908684

Subgroups — #24908684

Order 10624.

306 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#14581719C212yesmaximal, complemented
2#19127702C212yesmaximal, complemented
3#19127570C211yesmaximal, characteristic, complemented
4#19127470C211yesmaximal, characteristic, complemented
5#537×C83C211yesmaximal, characteristic
6C2×#13050574C2212yes—
7C2×C4⋊Dic83C2211yescharacteristic
8C2×Dic83⋊C4—22no—
9C2×C4×Dic83C2211yescharacteristic
10#14581719—24no—
11#14581716—22no—
12#14581713—22no—
13C2×#41×C83C2212yes—
14C22⋊Q8×C83—22no—
15#121×C83—22no—
16#121×C83—22no—
17C2×C4⋊C4×C83C2211yescharacteristic
18C22×Dic83—21no—
19C22×Dic83—21no—
20#13050574—22no—
21#13050574—22no—
22C2×C83⋊C8—22no—
23C4⋊Dic83—22no—
24Dic83⋊C4—44no—
25C4×Dic83—21no—
26C4×Dic83D412yescomplemented
27#41×C83—22no—
28#41×C83—22no—
29#41×C83—22no—
30#41×C83—22no—
31C2×Q8×C83D412yes—
32C2×D4×C83—22no—
33C2×D4×C83—22no—
34C2×D4×C83D412yes—
35C22×C4×C83—22no—
36C22×C4×C83D412yes—
37C22×C4×C83C2311yescharacteristic
38C4⋊C4×C83—22no—
39C4⋊C4×C83—22no—
40C22⋊C4×C83—24no—
41C23×C83—22no—
42C23×C83C2×D411yescharacteristic
43Q8×C83—22no—
44Q8×C83—22no—
45D4×C83—22no—
46D4×C83—22no—
47D4×C83—42no—
48D4×C83—42no—
49D4×C83—82no—
50C2×C4×C83—22no—
51C2×C4×C83—22no—
52C2×C4×C83—42no—
53C2×C4×C83—42no—
54C2×C4×C83—42no—
55C2×C4×C83C2×D411yescharacteristic
56C2×C4×C83C2×D412yes—
57C2×C4×C83#4112yes—
58C2×C4×C83#4111yescharacteristic
59C2×Dic83—22no—
60C2×Dic83—22no—
61C2×Dic83—22no—
62C2×Dic83—42no—
63C2×Dic83—41no—
64C83⋊C8—42no—
65C22×C83—21no—
66C22×C83—21no—
67C22×C83—42no—
68C22×C83—42no—
69C22×C83—42no—
70C22×C83#12011yescharacteristic
71C22×C83#12111yescharacteristic
72C22×C83#12411yescharacteristic
73C4×C83—42no—
74C4×C83—42no—
75C4×C83#12112yes—
76C4×C83#12312yes—
77Dic83—24no—
78Dic83—81no—
79C2×C83—21no—
80C2×C83—21no—
81C2×C83—82no—
82C2×C83#39411yescharacteristic
83C2×C83#45311yescharacteristic
84C2×C83#45611yescharacteristic
85#2039—831nomaximal, Sylow
86C83#203911yescharacteristic, complemented, Sylow
87#537D8311yescharacteristic
88C2×#104—832no—
89#417—832no—
90#384—831no—
91#343—831no—
92C2×#41C2×D8312yes—
93C2×M16—832no—
94C22⋊Q8—22no—
95#121—22no—
96#121—22no—
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—22no—
105#41—22no—
106#41—22no—
107#41—22no—
108C2×Q8C22⋊D8312yes—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4C22⋊D8312yes—
112C22×C4—22no—
113C22×C4—1661no—
114C22×C4—1661no—
115C22×C4C22⋊D8312yes—
116C22×C4C22×D8311yescharacteristic
117M16—1662no—
118M16—1662no—
119C2×C8—1662no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—3324no—
123C4⋊C4—1662no—
124C22⋊C4—24no—
125C42—832no—
126C42—1661no—
127C23—22no—
128C23C2×C22⋊D8311yescharacteristic
129Q8—22no—
130Q8—22no—
131D4—22no—
132D4—22no—
133D4—42no—
134D4—42no—
135D4—82no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—3322no—
142C2×C4—3321no—
143C2×C4—1662no—
144C2×C4—1662no—
145C2×C4—1662no—
146C2×C4D4×D8311yescharacteristic
147C2×C4C2×C22⋊D8312yes—
148C2×C4#1305058912yes—
149C2×C4#1305058911yescharacteristic
150C8—3322no—
151C22—21no—
152C22—21no—
153C22—42no—
154C22—42no—
155C22—42no—
156C22#1458176111yescharacteristic
157C22#1458176311yescharacteristic
158C22#1458177911yescharacteristic
159C4—42no—
160C4—42no—
161C4—6641no—
162C4—1664no—
163C4#1458176412yes—
164C4#1458177312yes—
165C2—21no—
166C2—21no—
167C2—82no—
168C2#1912761511yescharacteristic
169C2#1912786911yescharacteristic
170C2#1912788411yescharacteristic
subgroupG/H
centerC22#14581761
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting, prime radical#537×C83C2
socleC22×C83#120
hypercenter#537D83
hypocenterC83#2039
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#14581719C2semidirect
2. #19127702C2semidirect
3. #19127570C2semidirect
4. #19127470C2semidirect
26. C4×Dic83D4semidirect
86. C83#2039semidirect

Maximal subgroups

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

subgroupindex
C2×#145817192
#191277022
#191275702
#191274702
#537×C832
#203983