Finite Group Databasebeta

← #44548903

Subgroups — #44548903

Order 32128.

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#30631744C211yesmaximal, characteristic, complemented
2#30631553C211yesmaximal, characteristic, complemented
3#30631424C212yesmaximal, complemented
4#30631421C212yesmaximal
5#529×C251C211yesmaximal, characteristic
6C2×C4⋊Dic251—21no—
7C4⋊(C251⋊C8)—21no—
8C22⋊(C251⋊C8)C2212yes—
9#22548626—21no—
10C251⋊#95—21no—
11C251⋊#95C2212yes—
12#126×C251—21no—
13#125×C251C411yescharacteristic, complemented
14#124×C251C411yescharacteristic, complemented
15#123×C251—21no—
16C22⋊Q8×C251C2211yescharacteristic
17#121×C251C2211yescharacteristic
18C4×Q8×C251C411yescharacteristic, complemented
19C4×D4×C251—21no—
20C4×D4×C251C411yescharacteristic, complemented
21#117×C251—21no—
22C2×C42×C251C2211yescharacteristic
23C22×Dic251—22no—
24C2×C251⋊C8—22no—
25C4⋊Dic251—42no—
26C2×Q8×C251C2×C411yescharacteristic
27C2×D4×C251—21no—
28C2×D4×C251C2×C411yescharacteristic
29C22×C4×C251—21no—
30C22×C4×C251D411yescharacteristic
31C22×C4×C251D411yescharacteristic
32C22×C4×C251C2311yescharacteristic
33C4⋊C4×C251—21no—
34C4⋊C4×C251—21no—
35C4⋊C4×C251—21no—
36C4⋊C4×C251—21no—
37C4⋊C4×C251C2×C411yescharacteristic
38C4⋊C4×C251C2×C411yescharacteristic
39C22⋊C4×C251—21no—
40C22⋊C4×C251—21no—
41C22⋊C4×C251—21no—
42C22⋊C4×C251—21no—
43C22⋊C4×C251—21no—
44C42×C251—21no—
45C42×C251C2×C411yescharacteristic
46C42×C251C2×C411yescharacteristic
47C42×C251D411yescharacteristic
48C42×C251D411yescharacteristic
49C2×Dic251—42no—
50C2×Dic251—42no—
51C2×Dic251—42no—
52C251⋊C8—42no—
53C23×C251—21no—
54C23×C251C2×D411yescharacteristic
55Q8×C251—41no—
56D4×C251—41no—
57D4×C251—41no—
58D4×C251—41no—
59C2×C4×C251—21no—
60C2×C4×C251—21no—
61C2×C4×C251—21no—
62C2×C4×C251—21no—
63C2×C4×C251—21no—
64C2×C4×C251—41no—
65C2×C4×C251—41no—
66C2×C4×C251—42no—
67C2×C4×C251C22⋊C411yescharacteristic
68C2×C4×C251C22⋊C411yescharacteristic
69C2×C4×C251C22⋊C411yescharacteristic
70C2×C4×C251C22⋊C411yescharacteristic
71C2×C4×C251C22×C411yescharacteristic
72C2×C4×C251C2×D411yescharacteristic
73C22×C251—21no—
74C22×C251—21no—
75C22×C251—41no—
76C22×C251—41no—
77C22×C251—41no—
78C22×C251C2×C22⋊C411yescharacteristic
79C22×C251AGL(1,ℤ/8)11yescharacteristic
80C22×C251#13711yescharacteristic
81C4×C251—21no—
82C4×C251—21no—
83C4×C251—21no—
84C4×C251—21no—
85C4×C251—21no—
86C4×C251—21no—
87C4×C251—41no—
88C4×C251—41no—
89C4×C251—41no—
90Dic251—82no—
91C2×C251—21no—
92C2×C251—21no—
93C2×C251—81no—
94C2×C251#41011yescharacteristic
95C2×C251#41711yescharacteristic
96C2×C251#42111yescharacteristic
97C251#151211yescharacteristic, complemented, Sylow
98#1512—2511nomaximal, Sylow
99#529D25111yescharacteristic
100#432—2511no—
101#382—2511no—
102#328—2512no—
103#327—2512no—
104#126—21no—
105#125Dic25111yescharacteristic, complemented
106#124Dic25111yescharacteristic, complemented
107#123—21no—
108C22⋊Q8C2×D25111yescharacteristic
109#121C2×D25111yescharacteristic
110C4×Q8Dic25111yescharacteristic, complemented
111C4×D4—21no—
112C4×D4Dic25111yescharacteristic, complemented
113#117—21no—
114C2×C4⋊C4—5021no—
115C2×C42C2×D25111yescharacteristic
116C4⋊C8—5021no—
117C22⋊C8—2512no—
118#97—5021no—
119#95—5021no—
120#95—2512no—
121C2×Q8C2×Dic25111yescharacteristic
122C2×D4—21no—
123C2×D4C2×Dic25111yescharacteristic
124C22×C4—21no—
125C22×C4—5022no—
126C22×C4C22⋊D25111yescharacteristic
127C22×C4C22⋊D25111yescharacteristic
128C22×C4C22×D25111yescharacteristic
129C2×C8—5022no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—10042no—
135C4⋊C4C2×Dic25111yescharacteristic
136C4⋊C4C2×Dic25111yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C2×Dic25111yescharacteristic
144C42C2×Dic25111yescharacteristic
145C42C22⋊D25111yescharacteristic
146C42C22⋊D25111yescharacteristic
147C23—21no—
148C23C2×C22⋊D25111yescharacteristic
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—10042no—
162C2×C4—10042no—
163C2×C4—10042no—
164C2×C4C22⋊Dic25111yescharacteristic
165C2×C4C22⋊Dic25111yescharacteristic
166C2×C4C22⋊Dic25111yescharacteristic
167C2×C4C22⋊Dic25111yescharacteristic
168C2×C4C22×Dic25111yescharacteristic
169C2×C4C2×C22⋊D25111yescharacteristic
170C8—10042no—
171C22—21no—
172C22—21no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22C2×C22⋊Dic25111yescharacteristic
177C22#2254874011yescharacteristic
178C22#2254874511yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—20082no—
189C2—21no—
190C2—21no—
191C2—81no—
192C2#3063166111yescharacteristic
193C2#3063168811yescharacteristic
194C2#3063170411yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#529×C251C2
socleC22×C251C2×C22⋊C4
prime radical#121×C251C22
cosocleC22×C4×C251C23
hypercenter#529D251
hypocenterC251#1512
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #30631744C2semidirect
2. #30631553C2semidirect
3. #30631424C2semidirect
13. #125×C251C4semidirect
14. #124×C251C4semidirect
18. C4×Q8×C251C4semidirect
20. C4×D4×C251C4semidirect
97. C251#1512semidirect
105. #125Dic251semidirect
106. #124Dic251semidirect
110. C4×Q8Dic251semidirect
112. C4×D4Dic251semidirect

Maximal subgroups

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

subgroupindex
#306317442
#306315532
#306314242
#306314212
#529×C2512
#1512251