Finite Group Databasebeta

← #45663255

Subgroups — #45663255

Order 32896.

310 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 57 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#30656870C212yesmaximal, complemented
2#30656654C211yesmaximal, characteristic, complemented
3#30656471C212yesmaximal, complemented
4#30656463C211yesmaximal, characteristic, complemented
5#537×C257C211yesmaximal, characteristic, complemented
6C22×D1028C2211yescharacteristic
7C2×#16861603—22no—
8C2×#16861595—21no—
9C2×#16861595—21no—
10#22551512—22no—
11C22⋊#14523339C2212yes—
12#22551499C2211yescharacteristic
13C2×#41×C257—21no—
14C22⋊Q8×C257—22no—
15#121×C257—22no—
16#121×C257—21no—
17#121×C257C2212yes—
18C2×C4⋊C4×C257C2211yescharacteristic
19C23×D257—21no—
20C2×D1028—22no—
21C2×D1028—42no—
22C22×Dic257—21no—
23#16861603—44no—
24C2×#14523339—22no—
25#16861595—42no—
26#16861595—42no—
27#41×C257—42no—
28#41×C257—42no—
29C2×Q8×C257—21no—
30C2×D4×C257—21no—
31C2×D4×C257—22no—
32C2×D4×C257—21no—
33C2×D4×C257D412yes—
34C22×C4×C257—22no—
35C22×C4×C257D411yescharacteristic
36C22×C4×C257D411yescharacteristic
37C22×C4×C257C2311yescharacteristic
38C4⋊C4×C257—21no—
39C4⋊C4×C257—22no—
40C4⋊C4×C257D412yes—
41C22⋊C4×C257—22no—
42C22⋊C4×C257—22no—
43C22×D257—22no—
44C22×D257—41no—
45C22×D257—41no—
46C22×D257—41no—
47C22×D257—41no—
48C22×D257—41no—
49C22×D257—41no—
50D1028—24no—
51D1028—42no—
52C2×Dic257—41no—
53C2×Dic257—41no—
54C2×Dic257—41no—
55#14523339—42no—
56C23×C257—22no—
57C23×C257C2×D411yescharacteristic
58Q8×C257—42no—
59D4×C257—42no—
60D4×C257—42no—
61D4×C257—42no—
62D4×C257—42no—
63D4×C257—81no—
64C2×C4×C257—21no—
65C2×C4×C257—21no—
66C2×C4×C257—21no—
67C2×C4×C257—22no—
68C2×C4×C257—41no—
69C2×C4×C257—44no—
70C2×C4×C257—41no—
71C2×C4×C257D812yescomplemented
72C2×C4×C257C2×D412yes—
73C22×C257—21no—
74C22×C257—21no—
75C22×C257—42no—
76C22×C257—42no—
77C22×C257—42no—
78C22×C257#12011yescharacteristic
79C22×C257C2×D811yescharacteristic
80C22×C257AGL(1,ℤ/8)11yescharacteristic
81C2×D257—42no—
82C2×D257—42no—
83C2×D257—42no—
84C2×D257—81no—
85C2×D257—81no—
86C2×D257—81no—
87C2×D257—81no—
88C4×C257—22no—
89C4×C257—22no—
90C4×C257—42no—
91C4×C257—41no—
92Dic257—81no—
93C2×C257—21no—
94C2×C257—21no—
95C2×C257—82no—
96C2×C257#44711yescharacteristic
97C2×C257#45411yescharacteristic
98C2×C257B(4,𝔽2)11yescharacteristic
99D257—82no—
100C257#159811yescharacteristic, complemented, Sylow
101#1598—2571nomaximal, Sylow
102#537D25711yescharacteristic, complemented
103#447—2572no—
104#394—2571no—
105#327—2572no—
106#324—2571no—
107C2×#41—21no—
108C22×D4—2571no—
109C2×D8—5142no—
110C22⋊Q8—22no—
111#121—22no—
112#121—21no—
113#121C2×D25712yes—
114C2×C4⋊C4C2×D25711yescharacteristic
115C2×C22⋊C4—5141no—
116C2×C22⋊C4—5141no—
117#102—5142no—
118C22⋊C8—2572no—
119#95—2571no—
120C24—5141no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—5142no—
128C2×D4—10282no—
129C2×D4C22⋊D25712yes—
130C22×C4—22no—
131C22×C4—5141no—
132C22×C4C22⋊D25711yescharacteristic
133C22×C4C22⋊D25711yescharacteristic
134C22×C4C22×D25711yescharacteristic
135D8—10284no—
136C2×C8—5142no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4C22⋊D25712yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—10282no—
143C22⋊C4—10282no—
144C23—22no—
145C23—5142no—
146C23—10281no—
147C23—10281no—
148C23—10281no—
149C23—10281no—
150C23—10281no—
151C23—10281no—
152C23C2×C22⋊D25711yescharacteristic
153Q8—42no—
154D4—5144no—
155D4—42no—
156D4—42no—
157D4—42no—
158D4—42no—
159D4—10282no—
160D4—81no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—10281no—
169C2×C4—10281no—
170C2×C4—10281no—
171C2×C4#1686160312yescomplemented
172C2×C4C2×C22⋊D25712yes—
173C8—10282no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—10282no—
180C22—10282no—
181C22—10282no—
182C22—20561no—
183C22—20561no—
184C22—20561no—
185C22—20561no—
186C22#2255156111yescharacteristic
187C22C2×#1686160311yescharacteristic
188C22#2255161811yescharacteristic
189C4—22no—
190C4—22no—
191C4—42no—
192C4—41no—
193C4—20561no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—20562no—
198C2#3065687111yescharacteristic
199C2#3065691011yescharacteristic
200C2#3065692411yescharacteristic
subgroupG/H
centerC22#22551561
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#537×C257C2
socleC22×C257#120
hypercenter#537D257
hypocenterC257#1598
LCSG ▷ C22×C4×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #30656870C2semidirect
2. #30656654C2semidirect
3. #30656471C2semidirect
4. #30656463C2semidirect
5. #537×C257C2semidirect
71. C2×C4×C257D8semidirect
100. C257#1598semidirect
102. #537D257semidirect
171. C2×C4#16861603semidirect

Maximal subgroups

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

subgroupindex
#306568702
#306566542
#306564712
#306564632
#537×C2572
#1598257