Finite Group Databasebeta

← #27169670

Subgroups — #27169670

Order 12928.

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

normal subgroups only

This group has 222 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#20265234C211yesmaximal, characteristic, complemented
2#20265043C211yesmaximal, characteristic, complemented
3#20264914C212yesmaximal, complemented
4#20264911C212yesmaximal
5#529×C101C211yesmaximal, characteristic
6C2×C4⋊Dic101—21no—
7C4⋊#11717579—21no—
8C22⋊#11717579C2212yes—
9#15692166—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—
25C4⋊Dic101—42no—
26C2×Q8×C101C2×C411yescharacteristic
27C2×D4×C101—21no—
28C2×D4×C101C2×C411yescharacteristic
29C22×C4×C101—21no—
30C22×C4×C101D411yescharacteristic
31C22×C4×C101D411yescharacteristic
32C22×C4×C101C2311yescharacteristic
33C4⋊C4×C101—21no—
34C4⋊C4×C101—21no—
35C4⋊C4×C101—21no—
36C4⋊C4×C101—21no—
37C4⋊C4×C101C2×C411yescharacteristic
38C4⋊C4×C101C2×C411yescharacteristic
39C22⋊C4×C101—21no—
40C22⋊C4×C101—21no—
41C22⋊C4×C101—21no—
42C22⋊C4×C101—21no—
43C22⋊C4×C101—21no—
44C42×C101—21no—
45C42×C101C2×C411yescharacteristic
46C42×C101C2×C411yescharacteristic
47C42×C101D411yescharacteristic
48C42×C101D411yescharacteristic
49C23×C101—21no—
50C23×C101C2×D411yescharacteristic
51Q8×C101—41no—
52D4×C101—41no—
53D4×C101—41no—
54D4×C101—41no—
55C2×C4×C101—21no—
56C2×C4×C101—21no—
57C2×C4×C101—21no—
58C2×C4×C101—21no—
59C2×C4×C101—21no—
60C2×C4×C101—41no—
61C2×C4×C101—41no—
62C2×C4×C101—42no—
63C2×C4×C101C22⋊C411yescharacteristic
64C2×C4×C101C22⋊C411yescharacteristic
65C2×C4×C101C22⋊C411yescharacteristic
66C2×C4×C101C22⋊C411yescharacteristic
67C2×C4×C101C22×C411yescharacteristic
68C2×C4×C101C2×D411yescharacteristic
69C2×Dic101—42no—
70C2×Dic101—42no—
71C2×Dic101—42no—
72#11717579—42no—
73C22×C101—21no—
74C22×C101—21no—
75C22×C101—41no—
76C22×C101—41no—
77C22×C101—41no—
78C22×C101C2×C22⋊C411yescharacteristic
79C22×C101AGL(1,ℤ/8)11yescharacteristic
80C22×C101#13711yescharacteristic
81C4×C101—21no—
82C4×C101—21no—
83C4×C101—21no—
84C4×C101—21no—
85C4×C101—21no—
86C4×C101—21no—
87C4×C101—41no—
88C4×C101—41no—
89C4×C101—41no—
90Dic101—82no—
91C2×C101—21no—
92C2×C101—21no—
93C2×C101—81no—
94C2×C101#41011yescharacteristic
95C2×C101#41711yescharacteristic
96C2×C101#42111yescharacteristic
97#1512—1011nomaximal, Sylow
98C101#151211yescharacteristic, complemented, Sylow
99#529D10111yescharacteristic
100#432—1011no—
101#382—1011no—
102#328—1012no—
103#327—1012no—
104#126—21no—
105#125Dic10111yescharacteristic, complemented
106#124Dic10111yescharacteristic, complemented
107#123—21no—
108C22⋊Q8C2×D10111yescharacteristic
109#121C2×D10111yescharacteristic
110C4×Q8Dic10111yescharacteristic, complemented
111C4×D4—21no—
112C4×D4Dic10111yescharacteristic, complemented
113#117—21no—
114C2×C4⋊C4—2021no—
115C2×C42C2×D10111yescharacteristic
116C4⋊C8—2021no—
117C22⋊C8—1012no—
118#97—2021no—
119#95—1012no—
120#95—2021no—
121C2×Q8C2×Dic10111yescharacteristic
122C2×D4—21no—
123C2×D4C2×Dic10111yescharacteristic
124C22×C4—21no—
125C22×C4—2022no—
126C22×C4C22⋊D10111yescharacteristic
127C22×C4C22⋊D10111yescharacteristic
128C22×C4C22×D10111yescharacteristic
129C2×C8—2022no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—4042no—
135C4⋊C4C2×Dic10111yescharacteristic
136C4⋊C4C2×Dic10111yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C2×Dic10111yescharacteristic
144C42C2×Dic10111yescharacteristic
145C42C22⋊D10111yescharacteristic
146C42C22⋊D10111yescharacteristic
147C23—21no—
148C23C2×C22⋊D10111yescharacteristic
149Q8—41no—
150D4—41no—
151D4—41no—
152D4—41no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—4042no—
162C2×C4—4042no—
163C2×C4—4042no—
164C2×C4C22⋊Dic10111yescharacteristic
165C2×C4C22⋊Dic10111yescharacteristic
166C2×C4C22⋊Dic10111yescharacteristic
167C2×C4C22⋊Dic10111yescharacteristic
168C2×C4C22×Dic10111yescharacteristic
169C2×C4C2×C22⋊D10111yescharacteristic
170C8—4042no—
171C22—21no—
172C22—21no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22C2×C22⋊Dic10111yescharacteristic
177C22#1569228011yescharacteristic
178C22#1569228511yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—8082no—
189C2—21no—
190C2—21no—
191C2—81no—
192C2#2026515111yescharacteristic
193C2#2026517811yescharacteristic
194C2#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#1512
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. #20265234C2semidirect
2. #20265043C2semidirect
3. #20264914C2semidirect
13. #125×C101C4semidirect
14. #124×C101C4semidirect
18. C4×Q8×C101C4semidirect
20. C4×D4×C101C4semidirect
98. C101#1512semidirect
105. #125Dic101semidirect
106. #124Dic101semidirect
110. C4×Q8Dic101semidirect
112. C4×D4Dic101semidirect

Maximal subgroups

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

subgroupindex
#202652342
#202650432
#202649142
#202649112
#529×C1012
#1512101