Finite Group Databasebeta

← #44550804

Subgroups — #44550804

Order 32128.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C22×#16860794C211yesmaximal, characteristic, complemented
2C2×#22548636C212yesmaximal, complemented
3#30631547C211yesmaximal, characteristic, complemented
4#30631449C212yesmaximal, complemented
5C2×#121×C251C211yesmaximal, characteristic
6C2×#16860794—24no—
7C2×#16860794—22no—
8C22×C251⋊C8C2212yes—
9C2×C4⋊Dic251C2212yes—
10C2×C22⋊Dic251—21no—
11#22548636—24no—
12C251⋊#95—21no—
13C22×D4×C251—21no—
14C22×D4×C251C2211yescharacteristic
15C23×C4×C251C2211yescharacteristic
16#121×C251—22no—
17#121×C251C414yescomplemented
18C2×C4⋊C4×C251C2211yescharacteristic
19C2×C22⋊C4×C251—21no—
20C22×Dic251—22no—
21#16860794—28no—
22C2×C251⋊C8—22no—
23C2×C251⋊C8—22no—
24C2×C251⋊C8D414yescomplemented
25C4⋊Dic251—22no—
26C4⋊Dic251—22no—
27C22⋊Dic251—42no—
28C24×C251—21no—
29C24×C251D411yescharacteristic
30C2×D4×C251—24no—
31C2×D4×C251—22no—
32C2×D4×C251—21no—
33C2×D4×C251—22no—
34C2×D4×C251—44no—
35C2×D4×C251C2×C412yes—
36C22×C4×C251—22no—
37C22×C4×C251—22no—
38C22×C4×C251—21no—
39C22×C4×C251C2×C412yes—
40C22×C4×C251D411yescharacteristic
41C22×C4×C251D412yes—
42C22×C4×C251C2311yescharacteristic
43C4⋊C4×C251—21no—
44C4⋊C4×C251C2×C412yes—
45C22⋊C4×C251—24no—
46C2×Dic251—42no—
47C2×Dic251—42no—
48C2×Dic251—42no—
49C251⋊C8—24no—
50C23×C251—22no—
51C23×C251—22no—
52C23×C251—21no—
53C23×C251—21no—
54C23×C251—22no—
55C23×C251—42no—
56C23×C251—42no—
57C23×C251—41no—
58C23×C251—41no—
59C23×C251C22⋊C412yes—
60C23×C251C2×D411yescharacteristic
61D4×C251—24no—
62D4×C251—48no—
63D4×C251—41no—
64D4×C251—41no—
65C2×C4×C251—22no—
66C2×C4×C251—24no—
67C2×C4×C251—22no—
68C2×C4×C251—22no—
69C2×C4×C251—22no—
70C2×C4×C251—42no—
71C2×C4×C251C22⋊C412yes—
72C2×C4×C251C22×C411yescharacteristic
73C2×C4×C251C2×D412yes—
74C2×C4×C251C2×D411yescharacteristic
75C2×C4×C251#4112yes—
76C22×C251—22no—
77C22×C251—22no—
78C22×C251—22no—
79C22×C251—42no—
80C22×C251—42no—
81C22×C251—42no—
82C22×C251—42no—
83C22×C251—42no—
84C22×C251—82no—
85C22×C251—82no—
86C22×C251C2×C22⋊C411yescharacteristic
87C22×C251#12112yes—
88C22×C251AGL(1,ℤ/8)11yescharacteristic
89C22×C251AGL(1,ℤ/8)12yes—
90C22×C251#13711yescharacteristic
91C4×C251—22no—
92C4×C251—42no—
93C4×C251C4×D412yes—
94C4×C251#12011yescharacteristic
95C4×C251#12311yescharacteristic
96Dic251—82no—
97C2×C251—42no—
98C2×C251—82no—
99C2×C251#38611yescharacteristic
100C2×C251#41711yescharacteristic
101C2×C251#41811yescharacteristic
102C2×C251#46812yes—
103C2×C251#46912yes—
104C251#189511yescharacteristic, complemented, Sylow
105#1895—2511nomaximal, Sylow
106C22×M16—2511no—
107C2×#121D25111yescharacteristic
108C2×#102—2512no—
109#380—2511no—
110#340—2512no—
111C22×D4—21no—
112C22×D4C2×D25111yescharacteristic
113C23×C4C2×D25111yescharacteristic
114C2×M16—5022no—
115C2×M16—5024no—
116C22×C8—2512no—
117#121—22no—
118#121Dic25114yescomplemented
119C2×C4⋊C4—2512no—
120C2×C4⋊C4C2×D25111yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—5021no—
123#102—5024no—
124#95—5021no—
125C24—21no—
126C24C22⋊D25111yescharacteristic
127C2×D4—24no—
128C2×D4—22no—
129C2×D4—21no—
130C2×D4—22no—
131C2×D4—44no—
132C2×D4C2×Dic25112yes—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—5022no—
137C22×C4C2×Dic25112yes—
138C22×C4C22⋊D25111yescharacteristic
139C22×C4C22⋊D25112yes—
140C22×C4C22×D25111yescharacteristic
141M16—5028no—
142C2×C8—5022no—
143C2×C8—5022no—
144C2×C8—2514no—
145C4⋊C4—21no—
146C4⋊C4—5022no—
147C4⋊C4—5022no—
148C4⋊C4C2×Dic25112yes—
149C22⋊C4—24no—
150C22⋊C4—10042no—
151C23—21no—
152C23—22no—
153C23—21no—
154C23—22no—
155C23—22no—
156C23—41no—
157C23—42no—
158C23—41no—
159C23—42no—
160C23C22⋊Dic25112yes—
161C23C2×C22⋊D25111yescharacteristic
162D4—24no—
163D4—48no—
164D4—41no—
165D4—41no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—42no—
172C2×C4—10042no—
173C2×C4—10042no—
174C2×C4—10042no—
175C2×C4C22⋊Dic25112yes—
176C2×C4C22×Dic25111yescharacteristic
177C2×C4D4×D25112yes—
178C2×C4C2×C22⋊D25111yescharacteristic
179C2×C4#1686080912yes—
180C8—5024no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—42no—
189C22—82no—
190C22—82no—
191C22C2×C22⋊Dic25111yescharacteristic
192C22#2254868712yes—
193C22#2254874011yescharacteristic
194C22#2254874112yes—
195C22#2254874511yescharacteristic
196C4—22no—
197C4—42no—
198C4—20082no—
199C4D4×Dic25112yes—
200C4#2254868311yescharacteristic
201C4#2254869611yescharacteristic
202C2—42no—
203C2—82no—
204C2#3063156711yescharacteristic
205C2#3063168811yescharacteristic
206C2#3063169211yescharacteristic
207C2#3063193112yes—
208C2#3063193812yes—
subgroupG/H
centerC23C2×C22⋊D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting, prime radicalC2×#121×C251C2
socleC23×C251C2×D4
cosocleC22×C4×C251C23
hypercenterC2×#121D251
hypocenterC251#1895
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#16860794C2semidirect
2. C2×#22548636C2semidirect
3. #30631547C2semidirect
4. #30631449C2semidirect
17. #121×C251C4semidirect
24. C2×C251⋊C8D4semidirect
104. C251#1895semidirect
118. #121Dic251semidirect

Maximal subgroups

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

subgroupindex
C22×#168607942
C2×#225486362
#306315472
#306314492
C2×#121×C2512
#1895251