Finite Group Databasebeta

← #44548913

Subgroups — #44548913

Order 32128.

252 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 69 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#30632237C211yesmaximal, characteristic
2#30631555C211yesmaximal, characteristic, complemented
3#30631422C214yesmaximal, complemented
4#432×C251C211yesmaximal, characteristic, complemented
5C2×#16860810—21no—
6#22548715C411yescharacteristic, complemented
7#22548712C411yescharacteristic, complemented
8#22548703—21no—
9#22548689—21no—
10#22548689C2212yes—
11C4×D1004—21no—
12C4×D1004C412yescomplemented
13C251⋊#95—21no—
14C251⋊#95—21no—
15C251⋊#95C2212yes—
16C2×C42×C251C2211yescharacteristic
17C4⋊C8×C251—21no—
18C22⋊C8×C251C2212yes—
19#97×C251—21no—
20#16860810—44no—
21C2×Dic502—21no—
22C2×D1004—21no—
23C2×D1004—21no—
24C2×D1004C2×C412yes—
25C2×C22⋊D251—22no—
26C22×Dic251—22no—
27C2×C4×D251—22no—
28C4⋊Dic251—21no—
29C4⋊Dic251C2×C412yes—
30(C2×D251)⋊C4—22no—
31(C2×D251)⋊C4—22no—
32C22×C4×C251D411yescharacteristic
33C22×C4×C251D411yescharacteristic
34C22×C4×C251C2311yescharacteristic
35C2×C8×C251—22no—
36C42×C251C2×C411yescharacteristic
37C42×C251C2×C411yescharacteristic
38C42×C251D411yescharacteristic
39C42×C251D411yescharacteristic
40C22×D251—22no—
41Dic502—42no—
42D1004—42no—
43D1004—42no—
44D1004—42no—
45C22⋊D251—44no—
46C2×Dic251—22no—
47C2×Dic251—42no—
48C2×Dic251—42no—
49C2×Dic251—42no—
50C4×D251—44no—
51C23×C251C2×D411yescharacteristic
52C2×C4×C251—22no—
53C2×C4×C251—41no—
54C2×C4×C251—41no—
55C2×C4×C251C22⋊C411yescharacteristic
56C2×C4×C251C22⋊C411yescharacteristic
57C2×C4×C251C22⋊C411yescharacteristic
58C2×C4×C251C22⋊C411yescharacteristic
59C2×C4×C251C22×C411yescharacteristic
60C2×C4×C251C2×D411yescharacteristic
61C8×C251—42no—
62C22×C251—21no—
63C22×C251—21no—
64C22×C251C2×C22⋊C411yescharacteristic
65C22×C251AGL(1,ℤ/8)12yes—
66C2×D251—42no—
67C2×D251—42no—
68C2×D251—42no—
69C4×C251—21no—
70C4×C251—21no—
71C4×C251—21no—
72C4×C251—21no—
73C4×C251—21no—
74C4×C251—21no—
75Dic251—42no—
76Dic251—82no—
77C2×C251—22no—
78C2×C251#41011yescharacteristic
79C2×C251#41811yescharacteristic
80C2×C251#42111yescharacteristic
81D251—82no—
82C251#151411yescharacteristic, complemented, Sylow
83#1514—2511nomaximal, Sylow
84#532—2511no—
85#432D25111yescharacteristic, complemented
86#383—2511no—
87#327—2514no—
88C2×#41—5021no—
89C4⋊Q8—2511no—
90#127—2511no—
91#124—5021no—
92#121—5021no—
93#121—2512no—
94C4×D4—5021no—
95C4×D4—2512no—
96C2×C42C2×D25111yescharacteristic
97C4⋊C8—21no—
98C22⋊C8C2×D25112yes—
99#97—21no—
100#95—5021no—
101#95—5021no—
102#95—2512no—
103#41—10044no—
104C2×Q8—5021no—
105C2×D4—5022no—
106C2×D4—5021no—
107C2×D4—5021no—
108C2×D4—2512no—
109C22×C4—5022no—
110C22×C4—5022no—
111C22×C4D100411yescharacteristic
112C22×C4D100411yescharacteristic
113C22×C4C22×D25111yescharacteristic
114C2×C8—22no—
115C4⋊C4—5021no—
116C4⋊C4—2512no—
117C22⋊C4—5022no—
118C22⋊C4—5022no—
119C42C4×D25111yescharacteristic
120C42C4×D25111yescharacteristic
121C42C22⋊D25111yescharacteristic
122C42C22⋊D25111yescharacteristic
123C23—5022no—
124C23C2×D100411yescharacteristic
125Q8—10042no—
126D4—10042no—
127D4—10042no—
128D4—10042no—
129D4—10044no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—10044no—
134C2×C4—10042no—
135C2×C4—10042no—
136C2×C4—10042no—
137C2×C4—5022no—
138C2×C4(C2×D251)⋊C411yescharacteristic
139C2×C4(C2×D251)⋊C411yescharacteristic
140C2×C4(C2×D251)⋊C411yescharacteristic
141C2×C4(C2×D251)⋊C411yescharacteristic
142C2×C4C2×C4×D25111yescharacteristic
143C2×C4C2×C22⋊D25111yescharacteristic
144C8—42no—
145C22—21no—
146C22—21no—
147C22—10042no—
148C22—10042no—
149C22—10042no—
150C22C2×(C2×D251)⋊C411yescharacteristic
151C22#2254874312yes—
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—20082no—
159C4—10042no—
160C2—22no—
161C2—20082no—
162C2#3063166011yescharacteristic
163C2#3063169311yescharacteristic
164C2#3063170311yescharacteristic
subgroupG/H
centerC22C2×(C2×D251)⋊C4
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#432×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632237C2
cosocleC22×C4×C251C23
hypercenter#432D251
hypocenterC251#1514
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #30631555C2semidirect
3. #30631422C2semidirect
4. #432×C251C2semidirect
6. #22548715C4semidirect
7. #22548712C4semidirect
12. C4×D1004C4semidirect
82. C251#1514semidirect
85. #432D251semidirect

Maximal subgroups

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

subgroupindex
#306322372
#306315552
#306314222
#432×C2512
#1514251