Finite Group Databasebeta

← #24906402

Subgroups — #24906402

Order 10624.

266 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 53 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#19127852C212yesmaximal
3#19127437C212yesmaximal, complemented
4#19127427C211yesmaximal, characteristic, complemented
5#394×C83C211yesmaximal, characteristic
6C2×#13050578—22no—
7#14581791—22no—
8#14581783—22no—
9#14581771—21no—
10#14581771C2212yes—
11#14581751—21no—
12#14581713—22no—
13C22⋊(C83⋊C8)C2212yes—
14C22×D4×C83C2211yescharacteristic
15C2×C4⋊C4×C83C2211yescharacteristic
16C2×C22⋊C4×C83—21no—
17C2×C22⋊C4×C83—21no—
18#95×C83C2211yescharacteristic
19C2×Dic166—21no—
20C2×Dic166D412yes—
21#13050578—44no—
22C2×C83⋊C8—22no—
23C4⋊Dic83—21no—
24C4⋊Dic83—21no—
25C4⋊Dic83—21no—
26C4⋊Dic83D412yes—
27Dic83⋊C4—22no—
28Dic83⋊C4—22no—
29C22⋊Dic83—22no—
30C4×Dic83—22no—
31C24×C83—21no—
32C2×D4×C83—22no—
33C2×D4×C83—42no—
34C22×C4×C83—21no—
35C22×C4×C83D411yescharacteristic
36C22×C4×C83D411yescharacteristic
37C22×C4×C83C2311yescharacteristic
38C4⋊C4×C83—22no—
39C22⋊C4×C83—42no—
40C22⋊C4×C83—42no—
41C23×C83—22no—
42C23×C83—41no—
43C23×C83—41no—
44C23×C83—41no—
45C23×C83—41no—
46C23×C83—41no—
47C23×C83—41no—
48C23×C83C2×D411yescharacteristic
49D4×C83—24no—
50D4×C83—42no—
51C2×C4×C83—21no—
52C2×C4×C83—21no—
53C2×C4×C83—21no—
54C2×C4×C83—41no—
55C2×C4×C83—41no—
56C2×C4×C83—41no—
57C2×C4×C83—41no—
58C2×C4×C83—41no—
59C2×C4×C83SD1612yescomplemented
60C2×C4×C83C2×D412yes—
61C2×Dic83—22no—
62C2×Dic83—22no—
63Dic166—42no—
64Dic166—81no—
65C83⋊C8—42no—
66C22×C83—21no—
67C22×C83—21no—
68C22×C83—42no—
69C22×C83—42no—
70C22×C83—42no—
71C22×C83—81no—
72C22×C83—81no—
73C22×C83—81no—
74C22×C83—81no—
75C22×C83#12011yescharacteristic
76C22×C83C2×SD1611yescharacteristic
77C22×C83#13711yescharacteristic
78C4×C83—22no—
79C4×C83—22no—
80C4×C83—41no—
81C4×C83—81no—
82Dic83—42no—
83Dic83—42no—
84C2×C83—21no—
85C2×C83—21no—
86C2×C83—82no—
87C2×C83#44811yescharacteristic
88C2×C83#45411yescharacteristic
89C2×C83B(4,𝔽2)11yescharacteristic
90#1599—831nomaximal, Sylow
91C83#159911yescharacteristic, complemented, Sylow
92#544—831no—
93#450—832no—
94#394D8311yescharacteristic
95#328—832no—
96#324—831no—
97C22×D4C2×D8311yescharacteristic
98C2×SD16—1662no—
99C4⋊Q8—1662no—
100#125—1662no—
101C22⋊Q8—832no—
102C22⋊Q8—1661no—
103#117—1661no—
104C2×C4⋊C4C2×D8311yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107#102—1662no—
108C22⋊C8—832no—
109#95C2×D8311yescharacteristic
110C24—21no—
111C2×Q8—832no—
112C2×Q8—1661no—
113C2×D4—22no—
114C2×D4—42no—
115C22×C4—21no—
116C22×C4C22⋊D8311yescharacteristic
117C22×C4C22⋊D8311yescharacteristic
118C22×C4C22×D8311yescharacteristic
119SD16—3324no—
120C2×C8—1662no—
121C4⋊C4—22no—
122C4⋊C4—832no—
123C4⋊C4—1661no—
124C4⋊C4—1661no—
125C4⋊C4—1661no—
126C4⋊C4—1662no—
127C4⋊C4—1662no—
128C22⋊C4—42no—
129C22⋊C4—42no—
130C22⋊C4—1662no—
131C42—1662no—
132C23—22no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23C2×C22⋊D8311yescharacteristic
140Q8—3322no—
141Q8—6641no—
142D4—24no—
143D4—42no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—1662no—
153C2×C4—1662no—
154C2×C4#1305057812yescomplemented
155C2×C4D4×D8312yes—
156C8—3322no—
157C22—21no—
158C22—21no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22—81no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22#1458176011yescharacteristic
167C22C2×#1305057811yescharacteristic
168C22#1458182211yescharacteristic
169C4—22no—
170C4—22no—
171C4—41no—
172C4—81no—
173C4—3322no—
174C4—3322no—
175C2—21no—
176C2—21no—
177C2—82no—
178C2#1912783911yescharacteristic
179C2#1912787311yescharacteristic
180C2#1912788711yescharacteristic
subgroupG/H
centerC22#14581760
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting#394×C83C2
socleC22×C83#120
prime radicalC22×D4×C83C22
hypercenter#394D83
hypocenterC83#1599
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #19128332C2semidirect
3. #19127437C2semidirect
4. #19127427C2semidirect
59. C2×C4×C83SD16semidirect
91. C83#1599semidirect
154. C2×C4#13050578semidirect

Maximal subgroups

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

subgroupindex
#191283322
#191278522
#191274372
#191274272
#394×C832
#159983