Finite Group Databasebeta

← #44550803

Subgroups — #44550803

Order 32128.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#22548689C211yesmaximal, characteristic
2C2×#22548637C212yesmaximal, complemented
3#30631547C211yesmaximal, characteristic, complemented
4#30631450C212yesmaximal, complemented
5C22×M16×C251C211yesmaximal, characteristic, complemented
6C22×D1004C2211yescharacteristic
7C22×C22⋊D251—21no—
8#22548689—22no—
9#22548689C414yescomplemented
10C2×C4⋊Dic251C2212yes—
11C2×C4⋊Dic251C2211yescharacteristic
12C2×C22⋊Dic251—21no—
13C2×(C2×D251)⋊C4—21no—
14#22548637—24no—
15C251⋊#95—21no—
16C23×C4×C251C2211yescharacteristic
17C2×M16×C251—22no—
18C2×M16×C251—24no—
19C22×C8×C251C2212yes—
20C23×D251—21no—
21C2×D1004—22no—
22C2×D1004—21no—
23C2×D1004—22no—
24C2×D1004C2×C412yes—
25C2×C22⋊D251—24no—
26C2×C22⋊D251—44no—
27C22×Dic251—21no—
28C22×Dic251—22no—
29C4⋊Dic251—21no—
30C4⋊Dic251—22no—
31C4⋊Dic251—22no—
32C4⋊Dic251C2×C412yes—
33C22⋊Dic251—42no—
34(C2×D251)⋊C4—24no—
35C24×C251D411yescharacteristic
36C22×C4×C251—22no—
37C22×C4×C251—22no—
38C22×C4×C251C2×C412yes—
39C22×C4×C251D412yes—
40C22×C4×C251D411yescharacteristic
41C22×C4×C251C2311yescharacteristic
42M16×C251—28no—
43C2×C8×C251—22no—
44C2×C8×C251—22no—
45C2×C8×C251D414yescomplemented
46C22×D251—22no—
47C22×D251—42no—
48C22×D251—42no—
49C22×D251—41no—
50C22×D251—41no—
51D1004—24no—
52D1004—41no—
53D1004—41no—
54C22⋊D251—48no—
55C2×Dic251—22no—
56C2×Dic251—42no—
57C2×Dic251—42no—
58C2×Dic251—42no—
59C2×Dic251—42no—
60C23×C251—22no—
61C23×C251—21no—
62C23×C251—22no—
63C23×C251—21no—
64C23×C251C22⋊C412yes—
65C23×C251C2×D411yescharacteristic
66C2×C4×C251—22no—
67C2×C4×C251—24no—
68C2×C4×C251—22no—
69C2×C4×C251—22no—
70C2×C4×C251C22⋊C412yes—
71C2×C4×C251C22×C411yescharacteristic
72C2×C4×C251C2×D412yes—
73C2×C4×C251C2×D411yescharacteristic
74C2×C4×C251#4112yes—
75C8×C251—24no—
76C22×C251—22no—
77C22×C251—22no—
78C22×C251—22no—
79C22×C251—42no—
80C22×C251—42no—
81C22×C251C2×C22⋊C411yescharacteristic
82C22×C251#12112yes—
83C22×C251AGL(1,ℤ/8)11yescharacteristic
84C22×C251AGL(1,ℤ/8)12yes—
85C22×C251#13711yescharacteristic
86C2×D251—42no—
87C2×D251—42no—
88C2×D251—42no—
89C2×D251—82no—
90C2×D251—82no—
91C4×C251—22no—
92C4×C251C4×D412yes—
93C4×C251#12011yescharacteristic
94C4×C251#12311yescharacteristic
95Dic251—42no—
96Dic251—82no—
97C2×C251—42no—
98C2×C251#38611yescharacteristic
99C2×C251#41711yescharacteristic
100C2×C251#41811yescharacteristic
101C2×C251#46812yes—
102C2×C251#46912yes—
103D251—82no—
104C251#189511yescharacteristic, complemented, Sylow
105#1895—2511nomaximal, Sylow
106C22×M16D25111yescharacteristic, complemented
107C2×#121—2511no—
108C2×#102—2512no—
109#380—2511no—
110#340—2512no—
111C22×D4—5021no—
112C22×D4—2511no—
113C23×C4C2×D25111yescharacteristic
114C2×M16—22no—
115C2×M16—24no—
116C22×C8C2×D25112yes—
117#121—5022no—
118#121—2514no—
119C2×C4⋊C4—2512no—
120C2×C4⋊C4—2511no—
121C2×C22⋊C4—5021no—
122C2×C22⋊C4—5021no—
123#102—5024no—
124#95—5021no—
125C24—5021no—
126C24D100411yescharacteristic
127C2×D4—10044no—
128C2×D4—5024no—
129C2×D4—5021no—
130C2×D4—5022no—
131C2×D4—5022no—
132C2×D4—2512no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—5022no—
136C22×C4—5021no—
137C22×C4C4×D25112yes—
138C22×C4C22⋊D25112yes—
139C22×C4D100411yescharacteristic
140C22×C4C22×D25111yescharacteristic
141M16—28no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8C22⋊D25114yescomplemented
145C4⋊C4—5022no—
146C4⋊C4—5022no—
147C4⋊C4—5021no—
148C4⋊C4—2512no—
149C22⋊C4—10042no—
150C22⋊C4—5024no—
151C23—21no—
152C23—22no—
153C23—21no—
154C23—22no—
155C23—10042no—
156C23—10041no—
157C23—10041no—
158C23—10042no—
159C23—5022no—
160C23(C2×D251)⋊C412yes—
161C23C2×D100411yescharacteristic
162D4—10048no—
163D4—10041no—
164D4—10041no—
165D4—5024no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—10042no—
171C2×C4—10042no—
172C2×C4—10042no—
173C2×C4—10042no—
174C2×C4—5022no—
175C2×C4(C2×D251)⋊C412yes—
176C2×C4C2×C4×D25111yescharacteristic
177C2×C4C2×C22⋊D25112yes—
178C2×C4C2×C22⋊D25111yescharacteristic
179C2×C4#1686081012yes—
180C8—24no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—20082no—
187C22—20082no—
188C22—10042no—
189C22—10042no—
190C22—10042no—
191C22C2×(C2×D251)⋊C411yescharacteristic
192C22#2254868912yes—
193C22#2254874311yescharacteristic
194C22#2254874312yes—
195C22#2254874811yescharacteristic
196C4—22no—
197C4—20082no—
198C4—10042no—
199C4C4×C22⋊D25112yes—
200C4#2254868411yescharacteristic
201C4#2254869811yescharacteristic
202C2—42no—
203C2—20082no—
204C2#3063156611yescharacteristic
205C2#3063168911yescharacteristic
206C2#3063169311yescharacteristic
207C2#3063193312yes—
208C2#3063194012yes—
subgroupG/H
centerC23C2×D1004
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
FittingC22×M16×C251C2
socleC23×C251C2×D4
prime radicalC2×#22548689C2
cosocleC22×C4×C251C23
hypercenterC22×M16D251
hypocenterC251#1895
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#22548637C2semidirect
3. #30631547C2semidirect
4. #30631450C2semidirect
5. C22×M16×C251C2semidirect
9. #22548689C4semidirect
45. C2×C8×C251D4semidirect
104. C251#1895semidirect
106. C22×M16D251semidirect
144. C2×C8C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486892
C2×#225486372
#306315472
#306314502
C22×M16×C2512
#1895251