Finite Group Databasebeta

← #1897×C257 #45660439

Subgroups — #1897×C257 #45660439

Order 32896.

358 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 112 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#525×C257C211yesmaximal, characteristic
2C2×#117×C257C211yesmaximal, characteristic, complemented
3C2×#104×C257C212yesmaximal, complemented
4#407×C257C211yesmaximal, characteristic, complemented
5#339×C257C212yesmaximal, complemented
6C2×#41×C257C2211yescharacteristic
7C23×C4×C257C2211yescharacteristic
8C2×M16×C257C2212yes—
9#123×C257—21no—
10C22⋊Q8×C257C412yescomplemented
11#121×C257C412yescomplemented
12#120×C257—21no—
13C4×D4×C257—22no—
14#117×C257—24no—
15#117×C257C2211yescharacteristic
16C2×C4⋊C4×C257—21no—
17C2×C22⋊C4×C257—21no—
18C2×C42×C257C2212yes—
19#104×C257—24no—
20C22⋊C8×C257—22no—
21C24×C257D411yescharacteristic
22#41×C257—22no—
23#41×C257—22no—
24C2×Q8×C257C2×C411yescharacteristic
25C2×D4×C257—22no—
26C2×D4×C257—21no—
27C2×D4×C257C2×C411yescharacteristic
28C22×C4×C257—22no—
29C22×C4×C257—22no—
30C22×C4×C257—22no—
31C22×C4×C257—21no—
32C22×C4×C257C2×C412yes—
33C22×C4×C257D411yescharacteristic
34C22×C4×C257D412yes—
35C22×C4×C257C2311yescharacteristic
36M16×C257—22no—
37M16×C257—22no—
38C2×C8×C257—22no—
39C4⋊C4×C257—24no—
40C4⋊C4×C257—22no—
41C4⋊C4×C257C2×C412yes—
42C22⋊C4×C257—24no—
43C22⋊C4×C257—22no—
44C22⋊C4×C257—21no—
45C22⋊C4×C257—22no—
46C42×C257—22no—
47C42×C257—21no—
48C42×C257D414yescomplemented
49C23×C257—21no—
50C23×C257—21no—
51C23×C257—21no—
52C23×C257—42no—
53C23×C257C22⋊C412yes—
54C23×C257C2×D411yescharacteristic
55Q8×C257—22no—
56D4×C257—22no—
57D4×C257—44no—
58D4×C257—41no—
59D4×C257—41no—
60C2×C4×C257—21no—
61C2×C4×C257—21no—
62C2×C4×C257—24no—
63C2×C4×C257—24no—
64C2×C4×C257—22no—
65C2×C4×C257—21no—
66C2×C4×C257—24no—
67C2×C4×C257—44no—
68C2×C4×C257—42no—
69C2×C4×C257C22⋊C412yes—
70C2×C4×C257C22×C411yescharacteristic
71C2×C4×C257C2×D412yes—
72C2×C4×C257C2×D411yescharacteristic
73C2×C4×C257#4112yes—
74C8×C257—42no—
75C22×C257—22no—
76C22×C257—22no—
77C22×C257—22no—
78C22×C257—22no—
79C22×C257—42no—
80C22×C257—42no—
81C22×C257—41no—
82C22×C257—42no—
83C22×C257C2×C22⋊C411yescharacteristic
84C22×C257#12011yescharacteristic
85C22×C257#12311yescharacteristic
86C4×C257—22no—
87C4×C257—44no—
88C4×C257—41no—
89C4×C257—42no—
90C4×C257C4×D412yes—
91C4×C257#12112yes—
92C2×C257—21no—
93C2×C257—21no—
94C2×C257—42no—
95C2×C257—81no—
96C2×C257#38611yescharacteristic
97C2×C257#42112yes—
98C257#189711yescharacteristic, direct product factor, Sylow
99#1897C25711yesmaximal, characteristic, direct product factor, Sylow
100#525C2×C25711yescharacteristic
101C2×#117C2×C25711yescharacteristic, complemented
102C2×#104C2×C25712yescomplemented
103#407C2×C25711yescharacteristic, complemented
104#339C2×C25712yescomplemented
105C2×#41C22×C25711yescharacteristic
106C23×C4C22×C25711yescharacteristic
107C2×M16C22×C25712yes—
108#123—21no—
109C22⋊Q8C4×C25712yescomplemented
110#121C4×C25712yescomplemented
111#120—21no—
112C4×D4—22no—
113#117—24no—
114#117C22×C25711yescharacteristic
115C2×C4⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C42C22×C25712yes—
118#104—24no—
119C22⋊C8—22no—
120C24D4×C25711yescharacteristic
121#41—22no—
122#41—22no—
123C2×Q8C2×C4×C25711yescharacteristic
124C2×D4—22no—
125C2×D4—21no—
126C2×D4C2×C4×C25711yescharacteristic
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4C2×C4×C25712yes—
132C22×C4D4×C25711yescharacteristic
133C22×C4D4×C25712yes—
134C22×C4C23×C25711yescharacteristic
135M16—22no—
136M16—22no—
137C2×C8—22no—
138C4⋊C4—24no—
139C4⋊C4—22no—
140C4⋊C4C2×C4×C25712yes—
141C22⋊C4—24no—
142C22⋊C4—22no—
143C22⋊C4—21no—
144C22⋊C4—22no—
145C42—22no—
146C42—21no—
147C42D4×C25714yescomplemented
148C23—21no—
149C23—21no—
150C23—21no—
151C23—42no—
152C23C22⋊C4×C25712yes—
153C23C2×D4×C25711yescharacteristic
154Q8—22no—
155D4—22no—
156D4—44no—
157D4—41no—
158D4—41no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—44no—
167C2×C4—42no—
168C2×C4C22⋊C4×C25712yes—
169C2×C4C22×C4×C25711yescharacteristic
170C2×C4C2×D4×C25712yes—
171C2×C4C2×D4×C25711yescharacteristic
172C2×C4#41×C25712yes—
173C8—42no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—41no—
182C22C2×C22⋊C4×C25711yescharacteristic
183C22#120×C25711yescharacteristic
184C22#123×C25711yescharacteristic
185C4—22no—
186C4—44no—
187C4—42no—
188C4—41no—
189C4C4×D4×C25712yes—
190C4#121×C25712yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—81no—
195C2#386×C25711yescharacteristic
196C2#421×C25712yes—
subgroupG/H
centerC2×C4×C257C2×D4
commutatorC2×C4C22×C4×C257
Frattini, cosocleC22×C4C23×C257
socleC22×C257C2×C22⋊C4
prime radical#525×C257C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C257 ◁ C23×C4×C257 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#117×C257C2semidirect
3. C2×#104×C257C2semidirect
4. #407×C257C2semidirect
5. #339×C257C2semidirect
10. C22⋊Q8×C257C4semidirect
11. #121×C257C4semidirect
48. C42×C257D4semidirect
98. C257#1897direct (N × H)
99. #1897C257direct (N × H)
101. C2×#117C2×C257semidirect
102. C2×#104C2×C257semidirect
103. #407C2×C257semidirect
104. #339C2×C257semidirect
109. C22⋊Q8C4×C257semidirect
110. #121C4×C257semidirect
147. C42D4×C257semidirect

Maximal subgroups

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

subgroupindex
#525×C2572
C2×#117×C2572
C2×#104×C2572
#407×C2572
#339×C2572
#1897257