Finite Group Databasebeta

← #44548873

Subgroups — #44548873

Order 32128.

272 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 81 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#30631744C211yesmaximal, characteristic, complemented
2#30631533C211yesmaximal, characteristic, complemented
3#30631421C214yesmaximal, complemented
4#532×C251C211yesmaximal, characteristic
5C2×C4×Dic251—21no—
6C4⋊(C251⋊C8)—21no—
7C22⋊(C251⋊C8)C2212yes—
8#22548626—21no—
9C251⋊#95—21no—
10C251⋊#95C2212yes—
11C2×#41×C251—21no—
12C4⋊Q8×C251C411yescharacteristic, complemented
13#127×C251C411yescharacteristic, complemented
14#124×C251—21no—
15#121×C251—21no—
16#121×C251C2212yes—
17C4×D4×C251—21no—
18C4×D4×C251C412yescomplemented
19C2×C42×C251C2211yescharacteristic
20C22×Dic251—22no—
21C2×C251⋊C8—22no—
22C4×Dic251—24no—
23#41×C251—44no—
24C2×Q8×C251—21no—
25C2×D4×C251—22no—
26C2×D4×C251—21no—
27C2×D4×C251—21no—
28C2×D4×C251C2×C412yes—
29C22×C4×C251—22no—
30C22×C4×C251D411yescharacteristic
31C22×C4×C251D411yescharacteristic
32C22×C4×C251C2311yescharacteristic
33C4⋊C4×C251—21no—
34C4⋊C4×C251C2×C412yes—
35C22⋊C4×C251—22no—
36C22⋊C4×C251—22no—
37C42×C251C2×C411yescharacteristic
38C42×C251C2×C411yescharacteristic
39C42×C251D411yescharacteristic
40C42×C251D411yescharacteristic
41C2×Dic251—24no—
42C2×Dic251—42no—
43C2×Dic251—42no—
44C251⋊C8—42no—
45C23×C251—22no—
46C23×C251C2×D411yescharacteristic
47Q8×C251—42no—
48D4×C251—42no—
49D4×C251—42no—
50D4×C251—44no—
51D4×C251—42no—
52C2×C4×C251—22no—
53C2×C4×C251—22no—
54C2×C4×C251—22no—
55C2×C4×C251—41no—
56C2×C4×C251—44no—
57C2×C4×C251C22⋊C411yescharacteristic
58C2×C4×C251C22⋊C411yescharacteristic
59C2×C4×C251C22⋊C411yescharacteristic
60C2×C4×C251C22⋊C411yescharacteristic
61C2×C4×C251C22×C411yescharacteristic
62C2×C4×C251C2×D411yescharacteristic
63C22×C251—21no—
64C22×C251—21no—
65C22×C251—42no—
66C22×C251—42no—
67C22×C251—42no—
68C22×C251C2×C22⋊C411yescharacteristic
69C22×C251AGL(1,ℤ/8)12yes—
70C4×C251—21no—
71C4×C251—21no—
72C4×C251—21no—
73C4×C251—21no—
74C4×C251—21no—
75C4×C251—42no—
76C4×C251#10412yes—
77Dic251—44no—
78C2×C251—22no—
79C2×C251—82no—
80C2×C251#41011yescharacteristic
81C2×C251#41811yescharacteristic
82C2×C251C2×#10411yescharacteristic
83C251#150511yescharacteristic, complemented, Sylow
84#1505—2511nomaximal, Sylow
85#532D25111yescharacteristic
86#432—2511no—
87#376—2511no—
88#327—2514no—
89C2×#41—21no—
90C4⋊Q8Dic25111yescharacteristic, complemented
91#127Dic25111yescharacteristic, complemented
92#124—21no—
93#121—21no—
94#121C2×D25112yes—
95C4×D4—21no—
96C4×D4Dic25112yescomplemented
97C2×C42—5021no—
98C2×C42C2×D25111yescharacteristic
99C4⋊C8—5021no—
100C22⋊C8—2512no—
101#97—5021no—
102#95—5021no—
103#95—2512no—
104#41—44no—
105C2×Q8—21no—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4C2×Dic25112yes—
110C22×C4—22no—
111C22×C4—5022no—
112C22×C4C22⋊D25111yescharacteristic
113C22×C4C22⋊D25111yescharacteristic
114C22×C4C22×D25111yescharacteristic
115C2×C8—5022no—
116C4⋊C4—21no—
117C4⋊C4C2×Dic25112yes—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—5024no—
121C42C2×Dic25111yescharacteristic
122C42C2×Dic25111yescharacteristic
123C42C22⋊D25111yescharacteristic
124C42C22⋊D25111yescharacteristic
125C23—22no—
126C23C2×C22⋊D25111yescharacteristic
127Q8—42no—
128D4—42no—
129D4—42no—
130D4—44no—
131D4—42no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—44no—
137C2×C4—10042no—
138C2×C4—10042no—
139C2×C4—5024no—
140C2×C4C22⋊Dic25111yescharacteristic
141C2×C4C22⋊Dic25111yescharacteristic
142C2×C4C22⋊Dic25111yescharacteristic
143C2×C4C22⋊Dic25111yescharacteristic
144C2×C4C22×Dic25111yescharacteristic
145C2×C4C2×C22⋊D25111yescharacteristic
146C8—10042no—
147C22—21no—
148C22—21no—
149C22—42no—
150C22—42no—
151C22—42no—
152C22C2×C22⋊Dic25111yescharacteristic
153C22#2254874112yes—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—42no—
160C4—10044no—
161C4#2254864212yes—
162C2—22no—
163C2—82no—
164C2#3063166111yescharacteristic
165C2#3063169211yescharacteristic
166C2C2×#2254864211yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting, prime radical#532×C251C2
socleC22×C251C2×C22⋊C4
cosocleC22×C4×C251C23
hypercenter#532D251
hypocenterC251#1505
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #30631744C2semidirect
2. #30631533C2semidirect
3. #30631421C2semidirect
12. C4⋊Q8×C251C4semidirect
13. #127×C251C4semidirect
18. C4×D4×C251C4semidirect
83. C251#1505semidirect
90. C4⋊Q8Dic251semidirect
91. #127Dic251semidirect
96. C4×D4Dic251semidirect

Maximal subgroups

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

subgroupindex
#306317442
#306315332
#306314212
#532×C2512
#1505251