Finite Group Databasebeta

← #24906405

Subgroups — #24906405

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#19127854C212yesmaximal
3#19127618C211yesmaximal, characteristic
4#19127439C212yesmaximal, complemented
5#324×C83C211yesmaximal, characteristic, complemented
6C22×D332C2211yescharacteristic
7C2×#13050579—22no—
8#14581791—22no—
9#14581783—22no—
10#14581771—21no—
11#14581771C2212yes—
12#14581751—21no—
13C2×(C2×D83)⋊C4—21no—
14C2×(C2×D83)⋊C4—21no—
15#14581714—22no—
16C83⋊#95C2211yescharacteristic
17C2×C4⋊C4×C83C2211yescharacteristic
18C22⋊C8×C83C2212yes—
19C23×D83—21no—
20C2×Dic166—21no—
21C2×Dic166D412yes—
22C2×D332—22no—
23C2×D332—42no—
24C22×Dic83—21no—
25#13050579—44no—
26C4⋊Dic83—21no—
27C4⋊Dic83—21no—
28C4⋊Dic83—21no—
29C4⋊Dic83D412yes—
30Dic83⋊C4—22no—
31Dic83⋊C4—22no—
32C22⋊Dic83—22no—
33(C2×D83)⋊C4—42no—
34(C2×D83)⋊C4—42no—
35C4×Dic83—22no—
36C22×C4×C83D411yescharacteristic
37C22×C4×C83D411yescharacteristic
38C22×C4×C83C2311yescharacteristic
39C2×C8×C83—22no—
40C4⋊C4×C83—22no—
41C23×C83C2×D411yescharacteristic
42C22×D83—22no—
43C22×D83—41no—
44C22×D83—41no—
45C22×D83—41no—
46C22×D83—41no—
47C22×D83—41no—
48C22×D83—41no—
49C2×C4×C83—21no—
50C2×C4×C83—21no—
51C2×C4×C83—21no—
52C2×C4×C83—41no—
53C2×C4×C83—41no—
54C2×C4×C83SD1612yescomplemented
55C2×C4×C83C2×D412yes—
56C2×Dic83—22no—
57C2×Dic83—22no—
58C2×Dic83—41no—
59C2×Dic83—41no—
60C2×Dic83—41no—
61D332—24no—
62D332—42no—
63Dic166—42no—
64Dic166—81no—
65C8×C83—42no—
66C22×C83—21no—
67C22×C83—21no—
68C22×C83#12011yescharacteristic
69C22×C83C2×SD1611yescharacteristic
70C22×C83#13711yescharacteristic
71C2×D83—42no—
72C2×D83—42no—
73C2×D83—42no—
74C2×D83—81no—
75C2×D83—81no—
76C2×D83—81no—
77C2×D83—81no—
78C4×C83—22no—
79C4×C83—22no—
80C4×C83—41no—
81Dic83—42no—
82Dic83—42no—
83Dic83—81no—
84C2×C83—21no—
85C2×C83—21no—
86C2×C83#44811yescharacteristic
87C2×C83#45411yescharacteristic
88C2×C83B(4,𝔽2)11yescharacteristic
89D83—82no—
90#1599—831nomaximal, Sylow
91C83#159911yescharacteristic, complemented, Sylow
92#544—831no—
93#450—832no—
94#394—831no—
95#328—832no—
96#324D8311yescharacteristic, complemented
97C22×D4—831no—
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—1661no—
106C2×C22⋊C4—1661no—
107#102—1662no—
108C22⋊C8C2×D8312yes—
109#95—831no—
110C24—1661no—
111C2×Q8—832no—
112C2×Q8—1661no—
113C2×D4—3322no—
114C2×D4—1662no—
115C22×C4—1661no—
116C22×C4D33211yescharacteristic
117C22×C4D33211yescharacteristic
118C22×C4C22×D8311yescharacteristic
119SD16—3324no—
120C2×C8—22no—
121C4⋊C4—22no—
122C4⋊C4—832no—
123C4⋊C4—1661no—
124C4⋊C4—1661no—
125C4⋊C4—1661no—
126C4⋊C4—1662no—
127C4⋊C4—1662no—
128C22⋊C4—3322no—
129C22⋊C4—3322no—
130C22⋊C4—1662no—
131C42—1662no—
132C23—3321no—
133C23—3321no—
134C23—3321no—
135C23—3321no—
136C23—3321no—
137C23—3321no—
138C23—1662no—
139C23C2×D33211yescharacteristic
140Q8—3322no—
141Q8—6641no—
142D4—3322no—
143D4—1664no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—3321no—
150C2×C4—3321no—
151C2×C4—3321no—
152C2×C4—1662no—
153C2×C4—1662no—
154C2×C4#1305057912yescomplemented
155C2×C4D4×D8312yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—3322no—
160C22—3322no—
161C22—3322no—
162C22—6641no—
163C22—6641no—
164C22—6641no—
165C22—6641no—
166C22#1458176211yescharacteristic
167C22C2×#1305057911yescharacteristic
168C22#1458182511yescharacteristic
169C4—22no—
170C4—22no—
171C4—41no—
172C4—3322no—
173C4—3322no—
174C4—6641no—
175C2—21no—
176C2—21no—
177C2—6642no—
178C2#1912784211yescharacteristic
179C2#1912787611yescharacteristic
180C2#1912788911yescharacteristic
subgroupG/H
centerC22#14581762
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting#324×C83C2
socleC22×C83#120
prime radicalC22×D332C22
hypercenter#324D83
hypocenterC83#1599
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
1. #19128332C2semidirect
4. #19127439C2semidirect
5. #324×C83C2semidirect
54. C2×C4×C83SD16semidirect
91. C83#1599semidirect
96. #324D83semidirect
154. C2×C4#13050579semidirect

Maximal subgroups

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

subgroupindex
#191283322
#191278542
#191276182
#191274392
#324×C832
#159983