Finite Group Databasebeta

← #24907985

Subgroups — #24907985

Order 10624.

312 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 89 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#19128272C211yesmaximal, characteristic
2#19127711C212yesmaximal, complemented
3#19127675C211yesmaximal, characteristic
4#19127458C212yesmaximal, complemented
5#568×C83C211yesmaximal, characteristic
6C22×Dic166C2211yescharacteristic
7#14581790C412yescomplemented
8#14581779C412yescomplemented
9#14581773—22no—
10#14581771—22no—
11#14581751C2212yes—
12#14581751C2211yescharacteristic
13#14581717—24no—
14#14581707—22no—
15C2×#41×C83C2211yescharacteristic
16#131×C83—24no—
17C2×M16×C83—22no—
18C2×M16×C83C2212yes—
19C2×Dic166—22no—
20C2×Dic166—22no—
21C2×Dic166—21no—
22C2×Dic166C2×C412yes—
23C22×Dic83—21no—
24C4⋊Dic83—21no—
25C4⋊Dic83—22no—
26Dic83⋊C4—24no—
27C22⋊Dic83—24no—
28C22⋊Dic83—21no—
29C22⋊Dic83—22no—
30C4×Dic83—22no—
31C4×Dic83C2×C412yes—
32#41×C83—22no—
33#41×C83—22no—
34C2×Q8×C83C2×C411yescharacteristic
35C2×D4×C83C2×C411yescharacteristic
36C2×D4×C83D412yes—
37C22×C4×C83D412yes—
38C22×C4×C83C2311yescharacteristic
39M16×C83—24no—
40M16×C83—22no—
41M16×C83—22no—
42C2×C8×C83—24no—
43C2×C8×C83D414yes—
44C23×C83C22⋊C412yes—
45C23×C83C2×D411yescharacteristic
46Q8×C83—22no—
47D4×C83—24no—
48D4×C83—22no—
49C2×C4×C83—22no—
50C2×C4×C83—22no—
51C2×C4×C83C22⋊C412yes—
52C2×C4×C83C22×C411yescharacteristic
53C2×C4×C83C2×D412yes—
54C2×C4×C83C2×D411yescharacteristic
55C2×C4×C83#4112yes—
56C2×Dic83—22no—
57C2×Dic83—22no—
58C2×Dic83—42no—
59C2×Dic83—42no—
60Dic166—24no—
61Dic166—41no—
62Dic166—41no—
63C8×C83—24no—
64C22×C83—22no—
65C22×C83—41no—
66C22×C83—42no—
67C22×C83C2×C22⋊C411yescharacteristic
68C22×C83#12112yes—
69C4×C83—22no—
70C4×C83C4×D412yes—
71C4×C83#12011yescharacteristic
72C4×C83#12311yescharacteristic
73Dic83—42no—
74Dic83—42no—
75Dic83—82no—
76C2×C83—21no—
77C2×C83—22no—
78C2×C83—42no—
79C2×C83#38611yescharacteristic
80#1909—831nomaximal, Sylow
81C83#190911yescharacteristic, complemented, Sylow
82#568D8311yescharacteristic
83#536—831no—
84#419—832no—
85#410—831no—
86H(3,ℤ/4)—832no—
87C2×#41C2×D8311yescharacteristic
88C22×Q8—831no—
89#131—24no—
90C2×M16—22no—
91C2×M16C2×D8312yes—
92C4⋊Q8—832no—
93#124—832no—
94#123—1662no—
95C22⋊Q8—1662no—
96#117—831no—
97#117—832no—
98#103—1664no—
99#99—1662no—
100#41—22no—
101#41—22no—
102C2×Q8—832no—
103C2×Q8—1662no—
104C2×Q8—1662no—
105C2×Q8—1661no—
106C2×Q8C4×D8311yescharacteristic
107C2×D4C4×D8311yescharacteristic
108C2×D4D33212yes—
109C22×C4—1661no—
110C22×C4C22⋊D8312yes—
111C22×C4C22×D8311yescharacteristic
112M16—24no—
113M16—22no—
114M16—22no—
115C2×C8—24no—
116C2×C8C22⋊D8314yes—
117C4⋊C4—1662no—
118C4⋊C4—1664no—
119C4⋊C4—1661no—
120C22⋊C4—1662no—
121C22⋊C4—1664no—
122C22⋊C4—1661no—
123C42—832no—
124C42—1662no—
125C23(C2×D83)⋊C412yes—
126C23C2×D33211yescharacteristic
127Q8—22no—
128Q8—3321no—
129Q8—3321no—
130Q8—1664no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—3322no—
136C2×C4—3322no—
137C2×C4—1662no—
138C2×C4—1662no—
139C2×C4(C2×D83)⋊C412yes—
140C2×C4C2×C4×D8311yescharacteristic
141C2×C4C2×C22⋊D8311yescharacteristic
142C2×C4C2×C22⋊D8312yes—
143C2×C4#1305059012yes—
144C8—24no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×(C2×D83)⋊C411yescharacteristic
149C22#1458176612yes—
150C4—22no—
151C4—3322no—
152C4—3322no—
153C4—6642no—
154C4C4×C22⋊D8312yes—
155C4#1458176111yescharacteristic
156C4#1458177511yescharacteristic
157C2—21no—
158C2—22no—
159C2—42no—
160C2#1912758011yescharacteristic
subgroupG/H
centerC2#19127580
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#568×C83C2
socleC2×C83#386
prime radicalC2×#41×C83C22
cosocleC22×C4×C83C23
hypercenter#568D83
hypocenterC83#1909
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #19127711C2semidirect
4. #19127458C2semidirect
7. #14581790C4semidirect
8. #14581779C4semidirect
81. C83#1909semidirect

Maximal subgroups

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

subgroupindex
#191282722
#191277112
#191276752
#191274582
#568×C832
#190983