Finite Group Databasebeta

← #44550808

Subgroups — #44550808

Order 32128.

442 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 101 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#30632198C211yesmaximal, characteristic
2C2×#22548643C212yesmaximal, complemented
3#30631651C211yesmaximal, characteristic, complemented
4#30631447C212yesmaximal, complemented
5C22×C42×C251C211yesmaximal, characteristic, complemented
6C2×#16860810C2211yescharacteristic
7C2×#16860794C2212yes—
8#22548698—21no—
9#22548694C412yescomplemented
10#22548689C412yescomplemented
11#22548684—21no—
12C4×C22⋊D251—22no—
13#22548674C2211yescharacteristic
14#22548643—24no—
15C22⋊(C251⋊C8)—22no—
16C23×C4×C251—21no—
17C23×C4×C251C2211yescharacteristic
18C2×C42×C251—21no—
19C2×C42×C251—24no—
20C2×C42×C251C2212yes—
21#16860810—22no—
22#16860810—22no—
23C2×Dic502C2×C411yescharacteristic, complemented
24C2×D1004C2×C411yescharacteristic, complemented
25C2×C22⋊D251—22no—
26C2×C22⋊D251—21no—
27C2×C4×D251—21no—
28#16860794—22no—
29#16860794—22no—
30C2×C251⋊C8—22no—
31C4⋊Dic251C2×C412yescomplemented
32Dic251⋊C4—22no—
33C22⋊Dic251—22no—
34C22⋊Dic251—21no—
35(C2×D251)⋊C4—22no—
36C4×Dic251—21no—
37C24×C251D411yescharacteristic
38C22×C4×C251—24no—
39C22×C4×C251—28no—
40C22×C4×C251—22no—
41C22×C4×C251—22no—
42C22×C4×C251—22no—
43C22×C4×C251C2×C412yes—
44C22×C4×C251D411yescharacteristic
45C22×C4×C251D412yes—
46C22×C4×C251C2311yescharacteristic
47C42×C251—24no—
48C42×C251—22no—
49C42×C251D414yescomplemented
50C22×D251—21no—
51Dic502—22no—
52D1004—22no—
53C22⋊D251—44no—
54C22⋊D251—41no—
55C22⋊D251—41no—
56C2×Dic251—22no—
57C2×Dic251—21no—
58C4×D251—42no—
59C251⋊C8—42no—
60C23×C251—24no—
61C23×C251—21no—
62C23×C251—21no—
63C23×C251C22⋊C412yes—
64C23×C251C2×D411yescharacteristic
65C2×C4×C251—216no—
66C2×C4×C251—28no—
67C2×C4×C251—21no—
68C2×C4×C251—24no—
69C2×C4×C251—21no—
70C2×C4×C251—24no—
71C2×C4×C251—24no—
72C2×C4×C251C22⋊C412yes—
73C2×C4×C251C22×C411yescharacteristic
74C2×C4×C251C2×D412yes—
75C2×C4×C251C2×D411yescharacteristic
76C2×C4×C251#4112yes—
77C22×C251—24no—
78C22×C251—24no—
79C22×C251—22no—
80C22×C251—22no—
81C22×C251—22no—
82C22×C251#10414yescomplemented
83C22×C251C2×C22⋊C411yescharacteristic
84C22×C251#12011yescharacteristic
85C22×C251#12311yescharacteristic
86C2×D251—41no—
87C2×D251—42no—
88C4×C251—28no—
89C4×C251—22no—
90C4×C251C4×D412yes—
91C4×C251#12112yes—
92Dic251—41no—
93Dic251—42no—
94C2×C251—21no—
95C2×C251—24no—
96C2×C251—21no—
97C2×C251#38611yescharacteristic
98C2×C251C2×#10412yes—
99D251—81no—
100C251#189611yescharacteristic, complemented, Sylow
101#1896—2511nomaximal, Sylow
102#525—2511no—
103C22×C42D25111yescharacteristic, complemented
104C2×#104—2512no—
105#407—2511no—
106#339—2512no—
107C2×#41—2511no—
108C23×C4—21no—
109C23×C4C2×D25111yescharacteristic
110C2×M16—2512no—
111#123—5021no—
112C22⋊Q8—2512no—
113#121—2512no—
114#120—5021no—
115C4×D4—5022no—
116#117—2511no—
117C2×C42—21no—
118C2×C42—24no—
119C2×C42C2×D25112yes—
120#104—5024no—
121C22⋊C8—5022no—
122C24C22⋊D25111yescharacteristic
123#41—5022no—
124#41—5022no—
125C2×Q8—2511no—
126C2×D4—5022no—
127C2×D4—5021no—
128C2×D4—2511no—
129C22×C4—24no—
130C22×C4—28no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—5021no—
135C22×C4C4×D25112yes—
136C22×C4C22⋊D25111yescharacteristic
137C22×C4D100412yes—
138C22×C4C22×D25111yescharacteristic
139M16—5022no—
140M16—5022no—
141C2×C8—5022no—
142C4⋊C4—5022no—
143C4⋊C4—2512no—
144C22⋊C4—5022no—
145C22⋊C4—5021no—
146C22⋊C4—5022no—
147C42—24no—
148C42—22no—
149C42—5021no—
150C42C22⋊D25114yescomplemented
151C23—24no—
152C23—21no—
153C23—21no—
154C23—5021no—
155C23(C2×D251)⋊C412yes—
156C23C2×C22⋊D25111yescharacteristic
157Q8—5022no—
158D4—10044no—
159D4—10041no—
160D4—10041no—
161D4—5022no—
162C2×C4—216no—
163C2×C4—28no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—10042no—
170C2×C4—5022no—
171C2×C4—5021no—
172C2×C4(C2×D251)⋊C412yes—
173C2×C4C2×C4×D25111yescharacteristic
174C2×C4C2×C22⋊D25112yes—
175C2×C4C2×D100411yescharacteristic
176C2×C4#1686081012yes—
177C8—10042no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—10042no—
184C22—10041no—
185C22#2254864314yescomplemented
186C22C2×(C2×D251)⋊C411yescharacteristic
187C22#2254868411yescharacteristic
188C22#2254869811yescharacteristic
189C4—28no—
190C4—22no—
191C4—10042no—
192C4—10041no—
193C4C4×C22⋊D25112yes—
194C4#2254868912yes—
195C2—21no—
196C2—24no—
197C2—21no—
198C2—20081no—
199C2#3063156611yescharacteristic
200C2C2×#2254864312yes—
subgroupG/H
centerC2×C4C2×D1004
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
FittingC22×C42×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632198C2
cosocleC22×C4×C251C23
hypercenterC22×C42D251
hypocenterC251#1896
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. C2×#22548643C2semidirect
3. #30631651C2semidirect
4. #30631447C2semidirect
5. C22×C42×C251C2semidirect
9. #22548694C4semidirect
10. #22548689C4semidirect
23. C2×Dic502C2×C4semidirect
24. C2×D1004C2×C4semidirect
31. C4⋊Dic251C2×C4semidirect
49. C42×C251D4semidirect
82. C22×C251#104semidirect
100. C251#1896semidirect
103. C22×C42D251semidirect
150. C42C22⋊D251semidirect
185. C22#22548643semidirect

Maximal subgroups

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

subgroupindex
#306321982
C2×#225486432
#306316512
#306314472
C22×C42×C2512
#1896251