Finite Group Databasebeta

← #45664710

Subgroups — #45664710

Order 32896.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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×#22551590C211yesmaximal, characteristic
2#30657214C211yesmaximal, characteristic
3#30656746C211yesmaximal, characteristic
4C2×#22551510C211yesmaximal, characteristic
5#30656498C211yesmaximal, characteristic
6#30656496C211yesmaximal, characteristic
7#419×C257C211yesmaximal, characteristic
8C22×Dic514C2211yescharacteristic
9C2×#16861602C2211yescharacteristic
10#22551590—22no—
11#22551590C414yescomplemented
12C4×Dic514—22no—
13#22551557—22no—
14#22551551C2211yescharacteristic
15#22551548—22no—
16C2×#16861597—22no—
17C2×C4×Dic257C2211yescharacteristic
18#22551516—22no—
19#22551510—22no—
20C22×Q8×C257C2211yescharacteristic
21C2×M16×C257C2211yescharacteristic
22#117×C257C2211yescharacteristic
23#103×C257—22no—
24C2×Dic514—21no—
25C2×Dic514—21no—
26C2×Dic514—21no—
27C2×Dic514—21no—
28C2×Dic514C2×C412yes—
29C2×Dic514D412yes—
30C22×Dic257—21no—
31C22×Dic257D412yes—
32#16861602—21no—
33#16861602—21no—
34C2×#14523339—21no—
35C4⋊Dic257D412yes—
36#16861597—24no—
37#16861597—28no—
38C22⋊Dic257—21no—
39C4×Dic257—24no—
40C4×Dic257—21no—
41C4×Dic257—21no—
42C4×Dic257C2×C412yes—
43C2×Q8×C257—21no—
44C2×Q8×C257—21no—
45C2×Q8×C257—21no—
46C2×Q8×C257—21no—
47C2×Q8×C257—21no—
48C2×Q8×C257C2×C412yes—
49C22×C4×C257—21no—
50C22×C4×C257C2311yescharacteristic
51M16×C257—21no—
52M16×C257—21no—
53C2×C8×C257—21no—
54C4⋊C4×C257D412yes—
55C22⋊C4×C257—21no—
56C42×C257—21no—
57Dic514—22no—
58Dic514—22no—
59Dic514—22no—
60Dic514—22no—
61C2×Dic257—24no—
62C2×Dic257—22no—
63C2×Dic257—22no—
64C2×Dic257—22no—
65C2×Dic257—22no—
66C2×Dic257C22⋊C414yes—
67#14523339—41no—
68C23×C257C2×D411yescharacteristic
69Q8×C257—22no—
70Q8×C257—22no—
71Q8×C257—41no—
72Q8×C257—41no—
73C2×C4×C257—22no—
74C2×C4×C257—22no—
75C2×C4×C257—41no—
76C2×C4×C257—41no—
77C2×C4×C257C22×C411yescharacteristic
78C2×C4×C257C2×D411yescharacteristic
79C2×C4×C257C2×D411yescharacteristic
80C2×C4×C257C2×D411yescharacteristic
81C2×C4×C257#4111yescharacteristic
82C2×C4×C257#4111yescharacteristic
83C8×C257—41no—
84C22×C257—22no—
85C22×C257C2×C22⋊C411yescharacteristic
86C22×C257#12011yescharacteristic
87C22×C257#12311yescharacteristic
88C4×C257—42no—
89C4×C257—42no—
90C4×C257C4×D411yescharacteristic
91C4×C257C4×D411yescharacteristic
92C4×C257#12111yescharacteristic
93C4×C257#12111yescharacteristic
94Dic257—24no—
95Dic257—24no—
96Dic257—42no—
97C2×C257—21no—
98C2×C257—21no—
99C2×C257#38611yescharacteristic
100C2×C257#45612yes—
101C257#188611yescharacteristic, complemented, Sylow
102#1886—2571nomaximal, Sylow
103C2×C4⋊Q8—2571no—
104#519—2571no—
105#419—2571no—
106#419D25711yescharacteristic
107C2×#101—2571no—
108#339—2571no—
109#339—2571no—
110C22×Q8—2571no—
111C22×Q8C2×D25711yescharacteristic
112C2×M16—2571no—
113C2×M16C2×D25711yescharacteristic
114C4⋊Q8—2574no—
115C4⋊Q8—5142no—
116C4×Q8—5142no—
117C4×Q8—5142no—
118#117—2571no—
119#117—5142no—
120#117C2×D25711yescharacteristic
121C2×C4⋊C4—5142no—
122C2×C42—2571no—
123#103—22no—
124#103—5142no—
125#101—5142no—
126C2×Q8—2572no—
127C2×Q8—2572no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—5141no—
134C2×Q8—5141no—
135C2×Q8—5141no—
136C2×Q8—5141no—
137C2×Q8C4×D25712yes—
138C22×C4—2572no—
139C22×C4—21no—
140C22×C4—5141no—
141C22×C4C22×D25711yescharacteristic
142M16—21no—
143M16—21no—
144M16—5141no—
145M16—5141no—
146C2×C8—21no—
147C2×C8—5141no—
148C4⋊C4—2572no—
149C4⋊C4—5148no—
150C4⋊C4—5144no—
151C4⋊C4C22⋊D25712yes—
152C22⋊C4—21no—
153C22⋊C4—5141no—
154C42—2572no—
155C42—21no—
156C42—5144no—
157C42—5141no—
158C42—5141no—
159C23D4×D25711yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—5142no—
163Q8—5142no—
164Q8—5142no—
165Q8—5142no—
166Q8—41no—
167Q8—41no—
168C2×C4—2574no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—5144no—
172C2×C4—5142no—
173C2×C4—5142no—
174C2×C4—5142no—
175C2×C4—5142no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4C2×C4×D25711yescharacteristic
179C2×C4D4×D25711yescharacteristic
180C2×C4D4×D25711yescharacteristic
181C2×C4C2×C22⋊D25711yescharacteristic
182C2×C4#1686161711yescharacteristic
183C2×C4#1686161811yescharacteristic
184C8—41no—
185C8—10281no—
186C22—22no—
187C22C22⋊C4×D25711yescharacteristic
188C22#2255156011yescharacteristic
189C22#2255157411yescharacteristic
190C4—5144no—
191C4—5144no—
192C4—42no—
193C4—42no—
194C4—10282no—
195C4#2255155211yescharacteristic
196C4C4×C22⋊D25711yescharacteristic
197C4#2255156311yescharacteristic
198C4#2255156511yescharacteristic
199C2—21no—
200C2—21no—
201C2#3065661311yescharacteristic
202C2#3065691812yes—
subgroupG/H
centerC22#22551560
commutatorC2×C4×C257C22×C4
FrattiniC22×C4C22×D257
Fitting#419×C257C2
socleC22×C257#120
prime radicalC23×C257C2×D4
cosocleC22×C4×C257C23
hypercenter#419D257
hypocenterC257#1886
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
11. #22551590C4semidirect
101. C257#1886semidirect

Maximal subgroups

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

subgroupindex
C2×#225515902
#306572142
#306567462
C2×#225515102
#306564982
#306564962
#419×C2572
#1886257