Finite Group Databasebeta

← #45667505

Subgroups — #45667505

Order 32896.

408 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 173 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×#22551584C211yesmaximal, characteristic
2#30656634C212yesmaximal
3#30656632C212yesmaximal
4#30656624C212yesmaximal
5#30656622C212yesmaximal
6#30656620C212yesmaximal
7#30656618C212yesmaximal
8#30656603C211yesmaximal, characteristic
9#382×C257C211yesmaximal, characteristic
10#22551584C418yescomplemented
11C2×C4⋊Dic257C2211yescharacteristic
12C2×C4⋊Dic257C2211yescharacteristic
13C2×C4⋊Dic257C2211yescharacteristic
14C2×C4⋊Dic257C2211yescharacteristic
15C2×#16861597C2212yes—
16C2×#16861597C2212yes—
17C2×#16861597C2212yes—
18C2×#16861597C2212yes—
19C2×C4×Dic257C2212yes—
20C2×C4×Dic257C2212yes—
21#22551499C2212yes—
22#22551499C2212yes—
23#22551499C2212yes—
24#22551499C2212yes—
25#22551499C2212yes—
26#22551499C2212yes—
27C2×C4⋊C4×C257C2211yescharacteristic
28C2×C4⋊C4×C257C2211yescharacteristic
29C2×C42×C257C2211yescharacteristic
30#95×C257C2212yes—
31#95×C257C2212yes—
32C22×Dic257C2312yes—
33C22×Dic257C2312yes—
34C22×Dic257C2312yes—
35C22×Dic257C2312yes—
36C4⋊Dic257—41no—
37C4⋊Dic257—41no—
38C4⋊Dic257C2×C414yes—
39C4⋊Dic257C2×C414yes—
40#16861597—42no—
41#16861597—42no—
42#16861597C2×C418yes—
43#16861597C2×C418yes—
44C4×Dic257—42no—
45C4×Dic257—42no—
46C22×C4×C257C2312yes—
47C22×C4×C257C2312yes—
48C22×C4×C257C2311yescharacteristic
49C22×C4×C257C2311yescharacteristic
50C22×C4×C257C2311yescharacteristic
51C4⋊C4×C257—41no—
52C4⋊C4×C257—41no—
53C42×C257C2×C414yes—
54C2×Dic257—22no—
55C2×Dic257—22no—
56C2×Dic257—22no—
57C2×Dic257—22no—
58C2×Dic257—22no—
59C2×Dic257—22no—
60C2×Dic257—22no—
61C2×Dic257—22no—
62C2×Dic257—22no—
63C2×Dic257—22no—
64C2×Dic257C22×C414yes—
65C2×Dic257C22×C414yes—
66C23×C257C2411yescharacteristic
67C2×C4×C257—21no—
68C2×C4×C257—21no—
69C2×C4×C257—21no—
70C2×C4×C257—21no—
71C2×C4×C257—21no—
72C2×C4×C257—21no—
73C2×C4×C257—22no—
74C2×C4×C257—22no—
75C2×C4×C257—22no—
76C2×C4×C257—22no—
77C2×C4×C257—22no—
78C2×C4×C257—22no—
79C2×C4×C257C22×C412yes—
80C2×C4×C257C22×C412yes—
81C2×C4×C257C22×C412yes—
82C22×C257C23×C411yescharacteristic
83C22×C257H(5,𝔽2)11yescharacteristic
84C22×C257H(5,𝔽2)11yescharacteristic
85C22×C2572−1+411yescharacteristic
86C22×C2572−1+411yescharacteristic
87C22×C2572−1+411yescharacteristic
88C22×C2572−1+411yescharacteristic
89C4×C257—22no—
90C4×C257—22no—
91C4×C257—22no—
92C4×C257—42no—
93C4×C257—42no—
94Dic257—24no—
95Dic257—24no—
96Dic257—42no—
97Dic257—42no—
98C2×C257#51911yescharacteristic
99C2×C257#52011yescharacteristic
100C2×C257#52011yescharacteristic
101C2×C257#56211yescharacteristic
102C2×C257#56311yescharacteristic
103C2×C257#56311yescharacteristic
104C2×C257#56411yescharacteristic
105C257#238211yescharacteristic, complemented, Sylow
106#2382—2571nomaximal, Sylow
107C2×#125—2571no—
108#389—2572no—
109#389—2572no—
110#387—2572no—
111#387—2572no—
112#387—2572no—
113#387—2572no—
114#382—2571no—
115#382D25711yescharacteristic
116#125—2578no—
117C2×C4⋊C4—2571no—
118C2×C4⋊C4—2571no—
119C2×C4⋊C4—2572no—
120C2×C4⋊C4—2572no—
121C2×C4⋊C4—2571no—
122C2×C4⋊C4—2571no—
123C2×C4⋊C4—2572no—
124C2×C4⋊C4—2572no—
125C2×C4⋊C4C2×D25711yescharacteristic
126C2×C4⋊C4C2×D25711yescharacteristic
127C2×C42—2572no—
128C2×C42—2572no—
129C2×C42C2×D25711yescharacteristic
130#95—2572no—
131#95—2572no—
132#95—2572no—
133#95—2572no—
134#95—2572no—
135#95—2572no—
136#95C2×D25712yes—
137#95C2×D25712yes—
138C22×C4—2572no—
139C22×C4—2572no—
140C22×C4—2572no—
141C22×C4—2572no—
142C22×C4C22×D25712yes—
143C22×C4C22×D25712yes—
144C22×C4C22×D25711yescharacteristic
145C22×C4C22×D25711yescharacteristic
146C22×C4C22×D25711yescharacteristic
147C4⋊C4—2578no—
148C4⋊C4—2578no—
149C4⋊C4—2574no—
150C4⋊C4—2574no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—10282no—
154C4⋊C4—10282no—
155C4⋊C4—10281no—
156C4⋊C4—10281no—
157C42—10282no—
158C42—10282no—
159C42C4×D25714yes—
160C23C23×D25711yescharacteristic
161C2×C4—2574no—
162C2×C4—2574no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—5142no—
176C2×C4—5142no—
177C2×C4—5142no—
178C2×C4—5142no—
179C2×C4—5142no—
180C2×C4—5142no—
181C2×C4—5142no—
182C2×C4—5142no—
183C2×C4—5142no—
184C2×C4—5142no—
185C2×C4C2×C4×D25712yes—
186C2×C4C2×C4×D25712yes—
187C2×C4C2×C4×D25712yes—
188C22C22×C4×D25711yescharacteristic
189C22#2255163711yescharacteristic
190C22#2255163811yescharacteristic
191C22#2255163911yescharacteristic
192C22#2255163911yescharacteristic
193C22#2255163911yescharacteristic
194C22#2255164011yescharacteristic
195C4—22no—
196C4—22no—
197C4—22no—
198C4—5144no—
199C4—5144no—
200C4—42no—
201C4—42no—
202C4—10282no—
203C4—10282no—
204C2#3065721411yescharacteristic
205C2#3065721711yescharacteristic
206C2#3065721811yescharacteristic
207C2#3065748111yescharacteristic
208C2#3065748711yescharacteristic
209C2#3065749111yescharacteristic
210C2C257⋊#56411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#382×C257C2
socle, prime radical, cosocleC23×C257C24
hypercenter#382D257
hypocenterC257#2382
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
10. #22551584C4semidirect
105. C257#2382semidirect

Maximal subgroups

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

subgroupindex
C2×#225515842
#306566342
#306566322
#306566242
#306566222
#306566202
#306566182
#306566032
#382×C2572
#2382257