Finite Group Databasebeta

← #27169709

Subgroups — #27169709

Order 12928.

218 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 77 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#20265230C211yesmaximal, characteristic, complemented
2#20265045C211yesmaximal, characteristic, complemented
3#20264914C212yesmaximal, complemented
4#20264911C212yesmaximal
5#529×C101C211yesmaximal, characteristic
6C4⋊#11717579—21no—
7C22⋊#11717579C2212yes—
8C4×#11717579—21no—
9#15692162—21no—
10#15692162—21no—
11#15692162C2212yes—
12#126×C101—21no—
13#125×C101C411yescharacteristic, complemented
14#124×C101C411yescharacteristic, complemented
15#123×C101—21no—
16C22⋊Q8×C101C2211yescharacteristic
17#121×C101C2211yescharacteristic
18C4×Q8×C101C411yescharacteristic, complemented
19C4×D4×C101—21no—
20C4×D4×C101C411yescharacteristic, complemented
21#117×C101—21no—
22C2×C42×C101C2211yescharacteristic
23C22×Dic101—22no—
24C2×#11717579—22no—
25C2×Q8×C101C2×C411yescharacteristic
26C2×D4×C101—21no—
27C2×D4×C101C2×C411yescharacteristic
28C22×C4×C101—21no—
29C22×C4×C101D411yescharacteristic
30C22×C4×C101D411yescharacteristic
31C22×C4×C101C2311yescharacteristic
32C4⋊C4×C101—21no—
33C4⋊C4×C101—21no—
34C4⋊C4×C101—21no—
35C4⋊C4×C101—21no—
36C4⋊C4×C101C2×C411yescharacteristic
37C4⋊C4×C101C2×C411yescharacteristic
38C22⋊C4×C101—21no—
39C22⋊C4×C101—21no—
40C22⋊C4×C101—21no—
41C22⋊C4×C101—21no—
42C22⋊C4×C101—21no—
43C42×C101—21no—
44C42×C101C2×C411yescharacteristic
45C42×C101C2×C411yescharacteristic
46C42×C101D411yescharacteristic
47C42×C101D411yescharacteristic
48C23×C101—21no—
49C23×C101C2×D411yescharacteristic
50Q8×C101—41no—
51D4×C101—41no—
52D4×C101—42no—
53C2×C4×C101—22no—
54C2×C4×C101—21no—
55C2×C4×C101—21no—
56C2×C4×C101—21no—
57C2×C4×C101—41no—
58C2×C4×C101—41no—
59C2×C4×C101—42no—
60C2×C4×C101C22⋊C411yescharacteristic
61C2×C4×C101C22⋊C411yescharacteristic
62C2×C4×C101C22⋊C411yescharacteristic
63C2×C4×C101C22⋊C411yescharacteristic
64C2×C4×C101C22×C411yescharacteristic
65C2×C4×C101C2×D411yescharacteristic
66C2×Dic101—42no—
67C2×Dic101—42no—
68C2×Dic101—42no—
69#11717579—42no—
70C22×C101—21no—
71C22×C101—21no—
72C22×C101—41no—
73C22×C101—41no—
74C22×C101—41no—
75C22×C101C2×C22⋊C411yescharacteristic
76C22×C101#13512yes—
77C4×C101—21no—
78C4×C101—21no—
79C4×C101—21no—
80C4×C101—21no—
81C4×C101—21no—
82C4×C101—21no—
83C4×C101—41no—
84C4×C101—41no—
85C4×C101—41no—
86Dic101—82no—
87C2×C101—22no—
88C2×C101—81no—
89C2×C101#41011yescharacteristic
90C2×C101#41611yescharacteristic
91C2×C101#42111yescharacteristic
92#1521—1011nomaximal, Sylow
93C101#152111yescharacteristic, complemented, Sylow
94#529D10111yescharacteristic
95#431—1011no—
96#383—1011no—
97#328—1012no—
98#327—1012no—
99#126—21no—
100#125Dic10111yescharacteristic, complemented
101#124Dic10111yescharacteristic, complemented
102#123—21no—
103C22⋊Q8C2×D10111yescharacteristic
104#121C2×D10111yescharacteristic
105C4×Q8Dic10111yescharacteristic, complemented
106C4×D4—21no—
107C4×D4Dic10111yescharacteristic, complemented
108#117—21no—
109C2×C42C2×D10111yescharacteristic
110C4⋊C8—2021no—
111C22⋊C8—1012no—
112C4×C8—2021no—
113#95—1012no—
114#95—2021no—
115#95—2021no—
116C2×Q8C2×Dic10111yescharacteristic
117C2×D4—21no—
118C2×D4C2×Dic10111yescharacteristic
119C22×C4—21no—
120C22×C4—2022no—
121C22×C4C22⋊D10111yescharacteristic
122C22×C4C22⋊D10111yescharacteristic
123C22×C4C22×D10111yescharacteristic
124C2×C8—2022no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4C2×Dic10111yescharacteristic
130C4⋊C4C2×Dic10111yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C42—21no—
137C42C2×Dic10111yescharacteristic
138C42C2×Dic10111yescharacteristic
139C42C22⋊D10111yescharacteristic
140C42C22⋊D10111yescharacteristic
141C23—21no—
142C23C2×C22⋊D10111yescharacteristic
143Q8—41no—
144D4—41no—
145D4—42no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—4042no—
154C2×C4—4042no—
155C2×C4—4042no—
156C2×C4C22⋊Dic10111yescharacteristic
157C2×C4C22⋊Dic10111yescharacteristic
158C2×C4C22⋊Dic10111yescharacteristic
159C2×C4C22⋊Dic10111yescharacteristic
160C2×C4C22×Dic10111yescharacteristic
161C2×C4C2×C22⋊D10111yescharacteristic
162C8—4042no—
163C22—21no—
164C22—21no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22C2×C22⋊Dic10111yescharacteristic
169C22#1569227512yes—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—41no—
177C4—41no—
178C4—41no—
179C4—8082no—
180C2—22no—
181C2—81no—
182C2#2026515111yescharacteristic
183C2#2026517311yescharacteristic
184C2#2026519411yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#529×C101C2
socleC22×C101C2×C22⋊C4
prime radical#121×C101C22
cosocleC22×C4×C101C23
hypercenter#529D101
hypocenterC101#1521
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #20265230C2semidirect
2. #20265045C2semidirect
3. #20264914C2semidirect
13. #125×C101C4semidirect
14. #124×C101C4semidirect
18. C4×Q8×C101C4semidirect
20. C4×D4×C101C4semidirect
93. C101#1521semidirect
100. #125Dic101semidirect
101. #124Dic101semidirect
105. C4×Q8Dic101semidirect
107. C4×D4Dic101semidirect

Maximal subgroups

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

subgroupindex
#202652302
#202650452
#202649142
#202649112
#529×C1012
#1521101