Finite Group Databasebeta

← #45664761

Subgroups — #45664761

Order 32896.

442 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 97 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#30657248C211yesmaximal, characteristic
2C22×C4×Dic257C211yesmaximal, characteristic, complemented
3C2×#22551518C212yesmaximal, complemented
4#30656498C212yesmaximal, complemented
5#407×C257C211yesmaximal, characteristic, complemented
6C2×#16861618C2211yescharacteristic
7C23×Dic257—21no—
8#22551575—21no—
9#22551571C412yescomplemented
10#22551566C412yescomplemented
11#22551561—21no—
12C4×C22⋊D257—22no—
13#22551551C2211yescharacteristic
14C2×C4×Dic257—24no—
15C2×C4×Dic257—21no—
16C2×C4×Dic257C2212yes—
17#22551518—24no—
18C23×C4×C257C2211yescharacteristic
19C2×M16×C257C2212yes—
20C22⋊C8×C257—22no—
21#16861618—22no—
22#16861618—22no—
23C2×Dic514C2×C411yescharacteristic, complemented
24C2×D1028C2×C411yescharacteristic, complemented
25C2×C22⋊D257—22no—
26C2×C22⋊D257—21no—
27C22×Dic257—28no—
28C22×Dic257—24no—
29C22×Dic257—22no—
30C2×C4×D257—21no—
31C4⋊Dic257C2×C412yescomplemented
32#16861597—22no—
33C22⋊Dic257—22no—
34C22⋊Dic257—21no—
35#16861595—22no—
36C4×Dic257—24no—
37C4×Dic257—21no—
38C4×Dic257—22no—
39C4×Dic257D414yescomplemented
40C24×C257D411yescharacteristic
41C22×C4×C257—22no—
42C22×C4×C257—22no—
43C22×C4×C257C2×C412yes—
44C22×C4×C257D412yes—
45C22×C4×C257D411yescharacteristic
46C22×C4×C257C2311yescharacteristic
47M16×C257—22no—
48M16×C257—22no—
49C2×C8×C257—22no—
50C22×D257—21no—
51Dic514—22no—
52D1028—22no—
53C22⋊D257—44no—
54C22⋊D257—41no—
55C22⋊D257—41no—
56C2×Dic257—216no—
57C2×Dic257—28no—
58C2×Dic257—24no—
59C2×Dic257—22no—
60C2×Dic257—21no—
61C4×D257—42no—
62C23×C257—24no—
63C23×C257—21no—
64C23×C257—21no—
65C23×C257C22⋊C412yes—
66C23×C257C2×D411yescharacteristic
67C2×C4×C257—21no—
68C2×C4×C257—21no—
69C2×C4×C257—24no—
70C2×C4×C257—24no—
71C2×C4×C257C22⋊C412yes—
72C2×C4×C257C22×C411yescharacteristic
73C2×C4×C257C2×D412yes—
74C2×C4×C257C2×D411yescharacteristic
75C2×C4×C257#4112yes—
76C8×C257—42no—
77C22×C257—24no—
78C22×C257—24no—
79C22×C257—22no—
80C22×C257—22no—
81C22×C257—22no—
82C22×C257#10414yescomplemented
83C22×C257C2×C22⋊C411yescharacteristic
84C22×C257#12011yescharacteristic
85C22×C257#12311yescharacteristic
86C2×D257—41no—
87C2×D257—42no—
88C4×C257—22no—
89C4×C257C4×D412yes—
90C4×C257#12112yes—
91Dic257—28no—
92Dic257—41no—
93Dic257—42no—
94C2×C257—21no—
95C2×C257—24no—
96C2×C257—21no—
97C2×C257#38611yescharacteristic
98C2×C257C2×#10412yes—
99D257—81no—
100C257#189611yescharacteristic, complemented, Sylow
101#1896—2571nomaximal, Sylow
102#525—2571no—
103C22×C42—2571no—
104C2×#104—2572no—
105#407D25711yescharacteristic, complemented
106#339—2572no—
107C2×#41—2571no—
108C23×C4—5141no—
109C23×C4C2×D25711yescharacteristic
110C2×M16C2×D25712yes—
111#123—5141no—
112C22⋊Q8—2572no—
113#121—2572no—
114#120—5141no—
115C4×D4—5142no—
116#117—2571no—
117C2×C42—2572no—
118C2×C42—5144no—
119C2×C42—5141no—
120#104—5144no—
121C22⋊C8—22no—
122C24D102811yescharacteristic
123#41—5142no—
124#41—5142no—
125C2×Q8—2571no—
126C2×D4—2571no—
127C2×D4—5142no—
128C2×D4—5141no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—5144no—
132C22×C4—5148no—
133C22×C4—5142no—
134C22×C4—5141no—
135C22×C4C4×D25712yes—
136C22×C4C22⋊D25712yes—
137C22×C4D102811yescharacteristic
138C22×C4C22×D25711yescharacteristic
139M16—22no—
140M16—22no—
141C2×C8—22no—
142C4⋊C4—2572no—
143C4⋊C4—5142no—
144C22⋊C4—5142no—
145C22⋊C4—5141no—
146C22⋊C4—5142no—
147C42—2574no—
148C42—5144no—
149C42—5142no—
150C42—5141no—
151C23—21no—
152C23—21no—
153C23—24no—
154C23—5141no—
155C23#1686159512yes—
156C23C2×D102811yescharacteristic
157Q8—5142no—
158D4—5142no—
159D4—10281no—
160D4—10281no—
161D4—10284no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—51416no—
167C2×C4—5148no—
168C2×C4—5144no—
169C2×C4—5142no—
170C2×C4—5141no—
171C2×C4—10282no—
172C2×C4#1686159512yes—
173C2×C4C2×C4×D25711yescharacteristic
174C2×C4D4×D25712yes—
175C2×C4C2×C22⋊D25711yescharacteristic
176C2×C4#1686161712yes—
177C8—42no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—10281no—
184C22—10282no—
185C22#2255151814yescomplemented
186C22C2×#1686159511yescharacteristic
187C22#2255156211yescharacteristic
188C22#2255157611yescharacteristic
189C4—22no—
190C4—5148no—
191C4—10281no—
192C4—10282no—
193C4#2255155212yes—
194C4#2255156412yes—
195C2—21no—
196C2—21no—
197C2—24no—
198C2—20561no—
199C2#3065661411yescharacteristic
200C2C2×#2255151812yes—
subgroupG/H
centerC2×C4C2×C22⋊D257
commutatorC2×C4×C257C22×C4
FrattiniC22×C4C22×D257
Fitting#407×C257C2
socleC22×C257C2×C22⋊C4
prime radical#30657248C2
cosocleC22×C4×C257C23
hypercenter#407D257
hypocenterC257#1896
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic257C2semidirect
3. C2×#22551518C2semidirect
4. #30656498C2semidirect
5. #407×C257C2semidirect
9. #22551571C4semidirect
10. #22551566C4semidirect
23. C2×Dic514C2×C4semidirect
24. C2×D1028C2×C4semidirect
31. C4⋊Dic257C2×C4semidirect
39. C4×Dic257D4semidirect
82. C22×C257#104semidirect
100. C257#1896semidirect
105. #407D257semidirect
185. C22#22551518semidirect

Maximal subgroups

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

subgroupindex
#306572482
C22×C4×Dic2572
C2×#225515182
#306564982
#407×C2572
#1896257