Finite Group Databasebeta

← #27169639

Subgroups — #27169639

Order 12928.

272 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 73 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#20265727C211yesmaximal, characteristic
2#20265234C211yesmaximal, characteristic, complemented
3#20264910C214yesmaximal, complemented
4#376×C101C211yesmaximal, characteristic, complemented
5C2×#13108074—21no—
6#15692255C411yescharacteristic, complemented
7#15692252C411yescharacteristic, complemented
8#15692243—21no—
9#15692229—21no—
10#15692229C2212yes—
11C4×D404—21no—
12C4×D404C412yescomplemented
13C4⋊#11717579—21no—
14C22⋊#11717579C2212yes—
15#15692166—21no—
16C2×C42×C101—21no—
17C2×C42×C101C2211yescharacteristic
18#95×C101—21no—
19#95×C101C2212yes—
20#13108074—44no—
21C2×Dic202—21no—
22C2×D404—21no—
23C2×D404—21no—
24C2×D404C2×C412yes—
25C2×C22⋊D101—22no—
26C2×C4×D101—22no—
27C2×#11717579—22no—
28C4⋊Dic101—21no—
29C4⋊Dic101C2×C412yes—
30#13108051—22no—
31#13108051—22no—
32C22×C4×C101—22no—
33C22×C4×C101D411yescharacteristic
34C22×C4×C101D411yescharacteristic
35C22×C4×C101C2311yescharacteristic
36C42×C101—24no—
37C42×C101C2×C411yescharacteristic
38C42×C101C2×C411yescharacteristic
39C42×C101D411yescharacteristic
40C42×C101D411yescharacteristic
41C23×C101C2×D411yescharacteristic
42C22×D101—22no—
43C2×C4×C101—22no—
44C2×C4×C101—22no—
45C2×C4×C101—24no—
46C2×C4×C101—41no—
47C2×C4×C101—42no—
48C2×C4×C101—42no—
49C2×C4×C101C22⋊C411yescharacteristic
50C2×C4×C101C22⋊C411yescharacteristic
51C2×C4×C101C22⋊C411yescharacteristic
52C2×C4×C101C22⋊C411yescharacteristic
53C2×C4×C101C22×C411yescharacteristic
54C2×C4×C101C2×D411yescharacteristic
55C22⋊D101—44no—
56C2×Dic101—22no—
57D404—42no—
58D404—42no—
59D404—42no—
60C4×D101—44no—
61Dic202—42no—
62#11717579—42no—
63C22×C101—21no—
64C22×C101—21no—
65C22×C101C2×C22⋊C411yescharacteristic
66C22×C101AGL(1,ℤ/8)12yes—
67C2×D101—42no—
68C2×D101—42no—
69C2×D101—42no—
70C4×C101—21no—
71C4×C101—21no—
72C4×C101—21no—
73C4×C101—21no—
74C4×C101—21no—
75C4×C101—44no—
76C4×C101#10412yes—
77Dic101—42no—
78C2×C101—22no—
79C2×C101#41011yescharacteristic
80C2×C101#41811yescharacteristic
81C2×C101C2×#10411yescharacteristic
82D101—82no—
83#1505—1011nomaximal, Sylow
84C101#150511yescharacteristic, complemented, Sylow
85#532—1011no—
86#432—1011no—
87#376D10111yescharacteristic, complemented
88#327—1014no—
89C2×#41—2021no—
90C4⋊Q8—1011no—
91#127—1011no—
92#124—2021no—
93#121—1012no—
94#121—2021no—
95C4×D4—1012no—
96C4×D4—2021no—
97C2×C42—21no—
98C2×C42C2×D10111yescharacteristic
99C4⋊C8—2021no—
100C22⋊C8—1012no—
101#97—2021no—
102#95—21no—
103#95C2×D10112yes—
104#41—4044no—
105C2×Q8—2021no—
106C2×D4—1012no—
107C2×D4—2022no—
108C2×D4—2021no—
109C2×D4—2021no—
110C22×C4—22no—
111C22×C4—2022no—
112C22×C4C22⋊D10111yescharacteristic
113C22×C4C22⋊D10111yescharacteristic
114C22×C4C22×D10111yescharacteristic
115C2×C8—2022no—
116C4⋊C4—1012no—
117C4⋊C4—2021no—
118C22⋊C4—2022no—
119C22⋊C4—2022no—
120C42—24no—
121C42C4×D10111yescharacteristic
122C42C4×D10111yescharacteristic
123C42D40411yescharacteristic
124C42D40411yescharacteristic
125C23—2022no—
126C23C2×C22⋊D10111yescharacteristic
127Q8—4042no—
128D4—4042no—
129D4—4042no—
130D4—4044no—
131D4—4042no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—24no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—4044no—
139C2×C4—2022no—
140C2×C4#1310805111yescharacteristic
141C2×C4#1310805111yescharacteristic
142C2×C4#1310805111yescharacteristic
143C2×C4#1310805111yescharacteristic
144C2×C4C2×C4×D10111yescharacteristic
145C2×C4C2×D40411yescharacteristic
146C8—4042no—
147C22—21no—
148C22—21no—
149C22—4042no—
150C22—4042no—
151C22—4042no—
152C22C2×#1310805111yescharacteristic
153C22#1569228012yes—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—44no—
160C4—4042no—
161C4#1569218312yes—
162C2—22no—
163C2—8082no—
164C2#2026515011yescharacteristic
165C2#2026518111yescharacteristic
166C2C2×#1569218311yescharacteristic
subgroupG/H
centerC22C2×#13108051
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#376×C101C2
socleC22×C101C2×C22⋊C4
prime radical#20265727C2
cosocleC22×C4×C101C23
hypercenter#376D101
hypocenterC101#1505
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C42 ◁ #376

Complements

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

NH ≅ G/Ntype
2. #20265234C2semidirect
3. #20264910C2semidirect
4. #376×C101C2semidirect
6. #15692255C4semidirect
7. #15692252C4semidirect
12. C4×D404C4semidirect
84. C101#1505semidirect
87. #376D101semidirect

Maximal subgroups

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

subgroupindex
#202657272
#202652342
#202649102
#376×C1012
#1505101