Finite Group Databasebeta

← #24908620

Subgroups — #24908620

Order 10624.

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

normal subgroups only

This group has 282 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#19128332C211yesmaximal, characteristic, complemented
2#19127707C212yesmaximal
3#19127602C211yesmaximal, characteristic
4#19127462C212yesmaximal, complemented
5#343×C83C211yesmaximal, characteristic, complemented
6C22×D332C2211yescharacteristic
7#14581791—22no—
8#14581783—22no—
9#14581771—22no—
10#14581751C2212yes—
11C2×(C2×D83)⋊C4—22no—
12C2×C4×Dic83C2211yescharacteristic
13#14581714—22no—
14#14581714—22no—
15C2×M16×C83C2212yes—
16C2×C4⋊C4×C83C2211yescharacteristic
17C23×D83—21no—
18C2×Dic166—22no—
19C2×D332—22no—
20C2×D332—22no—
21C2×D332—21no—
22C2×D332Q812yescomplemented
23C22×Dic83—21no—
24C4⋊Dic83—22no—
25C4⋊Dic83D412yes—
26C4⋊Dic83D412yes—
27Dic83⋊C4—24no—
28C22⋊Dic83—22no—
29(C2×D83)⋊C4—44no—
30C4×Dic83—22no—
31C4×Dic83—22no—
32C22×C4×C83D412yes—
33C22×C4×C83C2311yescharacteristic
34M16×C83—22no—
35M16×C83—22no—
36C2×C8×C83—22no—
37C4⋊C4×C83—22no—
38C23×C83C2×D411yescharacteristic
39C22×D83—22no—
40C22×D83—42no—
41C22×D83—42no—
42C22×D83—41no—
43C22×D83—41no—
44C2×C4×C83—22no—
45C2×C4×C83—42no—
46C2×C4×C83C2×D412yes—
47C2×C4×C83C2×Q811yescharacteristic
48C2×C4×C83#4111yescharacteristic
49C2×C4×C83#4112yes—
50C2×Dic83—22no—
51C2×Dic83—22no—
52C2×Dic83—22no—
53C2×Dic83—41no—
54C2×Dic83—41no—
55D332—24no—
56D332—42no—
57Dic166—82no—
58C8×C83—42no—
59C22×C83—21no—
60C22×C83—21no—
61C22×C83#12011yescharacteristic
62C22×C83C22⋊Q811yescharacteristic
63C22×C83#12411yescharacteristic
64C2×D83—44no—
65C2×D83—42no—
66C2×D83—82no—
67C2×D83—81no—
68C2×D83—81no—
69C4×C83—42no—
70C4×C83C22⋊Q812yes—
71C4×C83#12412yes—
72Dic83—42no—
73Dic83—42no—
74Dic83—42no—
75C2×C83—21no—
76C2×C83—21no—
77C2×C83#39311yescharacteristic
78C2×C83#45311yescharacteristic
79C2×C83#45411yescharacteristic
80D83—82no—
81#2027—831nomaximal, Sylow
82C83#202711yescharacteristic, complemented, Sylow
83#544—831no—
84#418—832no—
85#390—831no—
86#343D8311yescharacteristic, complemented
87#339—832no—
88C22×D4—831no—
89C2×M16C2×D8312yes—
90C4⋊Q8—1662no—
91#125—1662no—
92C22⋊Q8—1662no—
93#117—832no—
94C2×C4⋊C4C2×D8311yescharacteristic
95C2×C22⋊C4—1662no—
96C2×C42—831no—
97#102—1662no—
98#102—1662no—
99C24—1661no—
100C2×Q8—1662no—
101C2×D4—832no—
102C2×D4—1662no—
103C2×D4—1661no—
104C2×D4—1662no—
105C22×C4—1661no—
106C22×C4D33212yes—
107C22×C4C22×D8311yescharacteristic
108M16—22no—
109M16—22no—
110C2×C8—22no—
111C4⋊C4—22no—
112C4⋊C4—832no—
113C4⋊C4—832no—
114C4⋊C4—1662no—
115C4⋊C4—1664no—
116C22⋊C4—3324no—
117C22⋊C4—1662no—
118C42—1662no—
119C42—1662no—
120C23—3322no—
121C23—3322no—
122C23—3321no—
123C23—3321no—
124C23—1662no—
125C23C2×D33211yescharacteristic
126Q8—6642no—
127D4—3322no—
128D4—1664no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—3321no—
132C2×C4—3321no—
133C2×C4—1662no—
134C2×C4—1662no—
135C2×C4—1662no—
136C2×C4D4×D8312yes—
137C2×C4Q8×D8311yescharacteristic
138C2×C4#1305058911yescharacteristic
139C2×C4#1305059012yes—
140C8—42no—
141C22—21no—
142C22—21no—
143C22—3324no—
144C22—3322no—
145C22—6642no—
146C22—6641no—
147C22—6641no—
148C22#1458176211yescharacteristic
149C22#1458177211yescharacteristic
150C22#1458178011yescharacteristic
151C4—42no—
152C4—3322no—
153C4—3322no—
154C4—3322no—
155C4#1458177012yes—
156C4#1458177812yes—
157C2—21no—
158C2—21no—
159C2—6642no—
160C2#1912761411yescharacteristic
161C2#1912787111yescharacteristic
162C2#1912787611yescharacteristic
subgroupG/H
centerC22#14581762
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting#343×C83C2
socleC22×C83#120
prime radicalC22×D332C22
hypercenter#343D83
hypocenterC83#2027
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
1. #19128332C2semidirect
4. #19127462C2semidirect
5. #343×C83C2semidirect
22. C2×D332Q8semidirect
82. C83#2027semidirect
86. #343D83semidirect

Maximal subgroups

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

subgroupindex
#191283322
#191277072
#191276022
#191274622
#343×C832
#202783