Finite Group Databasebeta

← #44548926

Subgroups — #44548926

Order 32128.

264 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 85 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#30631553C211yesmaximal, characteristic, complemented
3#30631422C214yesmaximal, complemented
4#431×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
13C2×C4⋊Dic251—21no—
14C251⋊#95—21no—
15C251⋊#95C2212yes—
16C2×C42×C251C2211yescharacteristic
17C4⋊C8×C251—21no—
18C22⋊C8×C251C2212yes—
19C4×C8×C251—21no—
20#16860810—42no—
21#16860810—42no—
22C2×Dic502—21no—
23C2×D1004—21no—
24C2×D1004—21no—
25C2×D1004C2×C412yes—
26C2×C22⋊D251—22no—
27C22×Dic251—22no—
28C2×C4×D251—22no—
29C4⋊Dic251—21no—
30C4⋊Dic251—42no—
31C4⋊Dic251C2×C412yes—
32(C2×D251)⋊C4—22no—
33(C2×D251)⋊C4—22no—
34C22×C4×C251D411yescharacteristic
35C22×C4×C251D411yescharacteristic
36C22×C4×C251C2311yescharacteristic
37C2×C8×C251—22no—
38C42×C251C2×C411yescharacteristic
39C42×C251C2×C411yescharacteristic
40C42×C251D411yescharacteristic
41C42×C251D411yescharacteristic
42C22×D251—22no—
43Dic502—42no—
44D1004—42no—
45D1004—42no—
46D1004—42no—
47C22⋊D251—42no—
48C22⋊D251—42no—
49C2×Dic251—22no—
50C2×Dic251—42no—
51C2×Dic251—42no—
52C2×Dic251—42no—
53C4×D251—44no—
54C23×C251C2×D411yescharacteristic
55C2×C4×C251—41no—
56C2×C4×C251—41no—
57C2×C4×C251C22⋊C411yescharacteristic
58C2×C4×C251C22⋊C411yescharacteristic
59C2×C4×C251C22⋊C411yescharacteristic
60C2×C4×C251C22⋊C411yescharacteristic
61C2×C4×C251D812yes—
62C2×C4×C251SD1612yes—
63C2×C4×C251C22×C411yescharacteristic
64C2×C4×C251C2×D411yescharacteristic
65C8×C251—42no—
66C22×C251—21no—
67C22×C251—21no—
68C22×C251C2×C22⋊C411yescharacteristic
69C22×C251C2×D811yescharacteristic
70C22×C251C2×SD1611yescharacteristic
71C2×D251—42no—
72C2×D251—42no—
73C2×D251—42no—
74C4×C251—21no—
75C4×C251—21no—
76C4×C251—21no—
77C4×C251—21no—
78C4×C251#10212yes—
79C4×C251#10212yes—
80Dic251—42no—
81Dic251—82no—
82C2×C251—21no—
83C2×C251—21no—
84C2×C251#41011yescharacteristic
85C2×C251C2×#10211yescharacteristic
86C2×C251#42111yescharacteristic
87D251—82no—
88C251#151711yescharacteristic, complemented, Sylow
89#1517—2511nomaximal, Sylow
90#532—2511no—
91#431D25111yescharacteristic, complemented
92#382—2511no—
93#327—2514no—
94C2×#41—5021no—
95C4⋊Q8—2511no—
96#127—2511no—
97#124—5021no—
98#121—5021no—
99#121—2512no—
100C4×D4—5021no—
101C4×D4—2512no—
102C2×C4⋊C4—5021no—
103C2×C42C2×D25111yescharacteristic
104C4⋊C8—21no—
105C22⋊C8C2×D25112yes—
106C4×C8—21no—
107#95—5021no—
108#95—2512no—
109#41—10042no—
110#41—10042no—
111C2×Q8—5021no—
112C2×D4—5022no—
113C2×D4—5021no—
114C2×D4—5021no—
115C2×D4—2512no—
116C22×C4—5022no—
117C22×C4—5022no—
118C22×C4D100411yescharacteristic
119C22×C4D100411yescharacteristic
120C22×C4C22×D25111yescharacteristic
121C2×C8—22no—
122C4⋊C4—10042no—
123C4⋊C4—5021no—
124C4⋊C4—2512no—
125C22⋊C4—5022no—
126C22⋊C4—5022no—
127C42C4×D25111yescharacteristic
128C42C4×D25111yescharacteristic
129C42C22⋊D25111yescharacteristic
130C42C22⋊D25111yescharacteristic
131C23—5022no—
132C23C2×D100411yescharacteristic
133Q8—10042no—
134D4—10042no—
135D4—10042no—
136D4—10042no—
137D4—10042no—
138D4—10042no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—10042no—
142C2×C4—10042no—
143C2×C4—10042no—
144C2×C4—10044no—
145C2×C4—5022no—
146C2×C4(C2×D251)⋊C411yescharacteristic
147C2×C4(C2×D251)⋊C411yescharacteristic
148C2×C4(C2×D251)⋊C411yescharacteristic
149C2×C4(C2×D251)⋊C411yescharacteristic
150C2×C4D200812yes—
151C2×C4#1686079912yes—
152C2×C4C2×C4×D25111yescharacteristic
153C2×C4C2×C22⋊D25111yescharacteristic
154C8—42no—
155C22—21no—
156C22—21no—
157C22—10042no—
158C22—10042no—
159C22—10042no—
160C22C2×(C2×D251)⋊C411yescharacteristic
161C22C2×D200811yescharacteristic
162C22C2×#1686079911yescharacteristic
163C4—21no—
164C4—21no—
165C4—21no—
166C4—21no—
167C4—20082no—
168C4—10042no—
169C4#2254863712yes—
170C4#2254863712yes—
171C2—21no—
172C2—21no—
173C2—20082no—
174C2#3063166011yescharacteristic
175C2C2×#2254863711yescharacteristic
176C2#3063170311yescharacteristic
subgroupG/H
centerC22C2×(C2×D251)⋊C4
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#431×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632237C2
cosocleC22×C4×C251C23
hypercenter#431D251
hypocenterC251#1517
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. #30631553C2semidirect
3. #30631422C2semidirect
4. #431×C251C2semidirect
6. #22548715C4semidirect
7. #22548712C4semidirect
12. C4×D1004C4semidirect
88. C251#1517semidirect
91. #431D251semidirect

Maximal subgroups

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

subgroupindex
#306322372
#306315532
#306314222
#431×C2512
#1517251