Finite Group Databasebeta

← #45665387

Subgroups — #45665387

Order 32896.

394 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 69 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#30657537C211yesmaximal, characteristic, complemented
2#30657248C212yesmaximal, complemented
3#30656753C212yesmaximal, complemented
4#30656494C211yesmaximal, characteristic, complemented
5#407×C257C211yesmaximal, characteristic, complemented
6C2×#16861618C2212yes—
7#22551625—22no—
8#22551620—22no—
9#22551614—22no—
10#22551575—22no—
11#22551571—22no—
12#22551566—22no—
13#22551561—22no—
14C4×C22⋊D257—24no—
15#22551551C2212yes—
16#22551518—24no—
17C23×C4×C257C2211yescharacteristic
18C2×M16×C257C2211yescharacteristic
19C2×M16×C257C2211yescharacteristic
20C22⋊C8×C257—22no—
21#16861618—22no—
22#16861618—22no—
23#16861618—22no—
24#16861618—22no—
25C2×Dic514D412yescomplemented
26C2×D1028D412yescomplemented
27C2×C22⋊D257—24no—
28C2×C22⋊D257—22no—
29C2×C4×D257—22no—
30Dic1028—22no—
31#16861607—24no—
32D2056—22no—
33C4⋊Dic257—22no—
34#16861597—24no—
35C22⋊Dic257—24no—
36C22⋊Dic257—22no—
37#16861595—24no—
38C4×Dic257D414yescomplemented
39C24×C257D411yescharacteristic
40C22×C4×C257—22no—
41C22×C4×C257—22no—
42C22×C4×C257—21no—
43C22×C4×C257—21no—
44C22×C4×C257D411yescharacteristic
45C22×C4×C257C2311yescharacteristic
46M16×C257—22no—
47M16×C257—22no—
48C2×C8×C257—21no—
49C2×C8×C257D412yescomplemented
50C22×D257—22no—
51Dic514—22no—
52Dic514—22no—
53D1028—22no—
54D1028—22no—
55C22⋊D257—48no—
56C22⋊D257—42no—
57C22⋊D257—42no—
58C2×Dic257—24no—
59C2×Dic257—22no—
60C4×D257—42no—
61C4×D257—42no—
62C23×C257—22no—
63C23×C257—21no—
64C23×C257—42no—
65C23×C257C2×D411yescharacteristic
66C2×C4×C257—22no—
67C2×C4×C257—21no—
68C2×C4×C257—41no—
69C2×C4×C257—42no—
70C2×C4×C257—41no—
71C2×C4×C257C2×D412yes—
72C2×C4×C257C2×D412yes—
73C2×C4×C257C2×D411yescharacteristic
74C2×C4×C257#4111yescharacteristic
75C8×C257—22no—
76C8×C257—41no—
77C22×C257—22no—
78C22×C257—24no—
79C22×C257—41no—
80C22×C257—42no—
81C22×C257—41no—
82C22×C257—41no—
83C22×C257#12012yes—
84C22×C257#12111yescharacteristic
85C2×D257—44no—
86C2×D257—42no—
87C4×C257—41no—
88C4×C257#12011yescharacteristic
89C4×C257#12112yes—
90C4×C257#12711yescharacteristic
91Dic257—44no—
92Dic257—42no—
93C2×C257—22no—
94C2×C257—21no—
95C2×C257—42no—
96C2×C257#39211yescharacteristic
97D257—82no—
98C257#201011yescharacteristic, complemented, Sylow
99#2010—2571nomaximal, Sylow
100#575—2571no—
101#525—2572no—
102#421—2572no—
103#407D25711yescharacteristic, complemented
104H(3,ℤ/4)—2571no—
105C2×#41—2572no—
106C23×C4C2×D25711yescharacteristic
107#137—5142no—
108AGL(1,ℤ/8)—5142no—
109#135—5142no—
110C2×M16C2×D25711yescharacteristic
111C2×M16C2×D25711yescharacteristic
112#123—5142no—
113C22⋊Q8—5142no—
114#121—5142no—
115#120—5142no—
116C4×D4—5144no—
117#117—2572no—
118#104—5144no—
119C22⋊C8—22no—
120C24D102811yescharacteristic
121#41—5142no—
122#41—5142no—
123#41—5142no—
124#41—5142no—
125C2×Q8—2572no—
126C2×D4—2572no—
127C2×D4—5144no—
128C2×D4—5142no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—5142no—
134C22×C4D102811yescharacteristic
135C22×C4C22×D25711yescharacteristic
136Q16—5142no—
137SD16—5144no—
138D8—5142no—
139M16—22no—
140M16—22no—
141C2×C8—21no—
142C2×C8C22⋊D25712yescomplemented
143C4⋊C4—5144no—
144C4⋊C4—5142no—
145C22⋊C4—5144no—
146C22⋊C4—5144no—
147C22⋊C4—5142no—
148C42—2574no—
149C23—22no—
150C23—21no—
151C23—5142no—
152C23—42no—
153C23C2×D102811yescharacteristic
154Q8—5142no—
155Q8—5142no—
156D4—5142no—
157D4—5142no—
158D4—10288no—
159D4—10282no—
160D4—10282no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—5144no—
164C2×C4—5142no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—10282no—
169C2×C4—10282no—
170C2×C4D4×D25712yes—
171C2×C4D4×D25712yes—
172C2×C4C2×C22⋊D25711yescharacteristic
173C2×C4#1686161811yescharacteristic
174C8—22no—
175C8—41no—
176C22—24no—
177C22—22no—
178C22—41no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—10284no—
183C22—10282no—
184C22#2255156212yes—
185C22#2255156611yescharacteristic
186C4—41no—
187C4—10284no—
188C4—10282no—
189C4#2255156011yescharacteristic
190C4#2255156512yes—
191C4#2255158811yescharacteristic
192C2—22no—
193C2—21no—
194C2—42no—
195C2—20562no—
196C2#3065664611yescharacteristic
subgroupG/H
centerC4#22551588
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#407×C257C2
socleC2×C257#392
hypercenter#407D257
hypocenterC257#2010
LCSG ▷ C22×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #30657537C2semidirect
2. #30657248C2semidirect
3. #30656753C2semidirect
4. #30656494C2semidirect
5. #407×C257C2semidirect
25. C2×Dic514D4semidirect
26. C2×D1028D4semidirect
38. C4×Dic257D4semidirect
49. C2×C8×C257D4semidirect
98. C257#2010semidirect
103. #407D257semidirect
142. C2×C8C22⋊D257semidirect

Maximal subgroups

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

subgroupindex
#306575372
#306572482
#306567532
#306564942
#407×C2572
#2010257