Finite Group Databasebeta

← #45667121

Subgroups — #45667121

Order 32896.

428 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 177 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#30656624C212yesmaximal
4#30656622C212yesmaximal
5#30656620C212yesmaximal
6#30656619C212yesmaximal
7C4×C4⋊Dic257C211yesmaximal, characteristic
8#30656588C212yesmaximal
9#383×C257C211yesmaximal, characteristic
10#22551584C418yescomplemented
11C2×C4⋊Dic257C2212yes—
12C2×C4⋊Dic257C2211yescharacteristic
13C2×C4⋊Dic257C2211yescharacteristic
14C2×#16861597C2212yes—
15C2×#16861597C2212yes—
16C2×#16861597C2212yes—
17C2×#16861597C2212yes—
18C2×C4×Dic257C2212yes—
19C2×C4×Dic257C2211yescharacteristic
20C2×C4×Dic257C2212yes—
21C2×C4×Dic257C2211yescharacteristic
22#22551499C2212yes—
23#22551499C2212yes—
24#22551499C2212yes—
25#22551499C2212yes—
26#22551499C2212yes—
27C2×C42×C257C2211yescharacteristic
28#95×C257C2212yes—
29#95×C257C2212yes—
30#95×C257C2211yescharacteristic
31#95×C257C2211yescharacteristic
32C22×Dic257C2312yes—
33C22×Dic257C2312yes—
34C22×Dic257C2312yes—
35C22×Dic257C2312yes—
36C4⋊Dic257—24no—
37C4⋊Dic257C2×C414yes—
38C4⋊Dic257C2×C414yes—
39#16861597—24no—
40#16861597—42no—
41#16861597C2×C418yes—
42#16861597C2×C418yes—
43C4×Dic257—24no—
44C4×Dic257—41no—
45C4×Dic257—41no—
46C4×Dic257—42no—
47C22×C4×C257C2312yes—
48C22×C4×C257C2312yes—
49C22×C4×C257C2311yescharacteristic
50C22×C4×C257C2311yescharacteristic
51C22×C4×C257C2311yescharacteristic
52C42×C257C2×C414yes—
53C2×Dic257—22no—
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×Dic257C22×C414yes—
63C2×Dic257C22×C414yes—
64C2×Dic257#4114yes—
65C23×C257C2411yescharacteristic
66C2×C4×C257—22no—
67C2×C4×C257—21no—
68C2×C4×C257—22no—
69C2×C4×C257—21no—
70C2×C4×C257—22no—
71C2×C4×C257—22no—
72C2×C4×C257—22no—
73C2×C4×C257—22no—
74C2×C4×C257—22no—
75C2×C4×C257—22no—
76C2×C4×C257C22×C412yes—
77C2×C4×C257C22×C412yes—
78C2×C4×C257C22×C412yes—
79C22×C257C23×C411yescharacteristic
80C22×C257C2×#4112yes—
81C22×C257H(5,𝔽2)11yescharacteristic
82C22×C2572−1+412yes—
83C22×C2572−1+411yescharacteristic
84C4×C257—22no—
85C4×C257—22no—
86C4×C257—22no—
87C4×C257—42no—
88C4×C257—42no—
89Dic257—24no—
90Dic257—24no—
91Dic257—24no—
92Dic257—42no—
93C2×C257C4×#4111yescharacteristic
94C2×C257#51911yescharacteristic
95C2×C257#52011yescharacteristic
96C2×C257#53912yes—
97C2×C257#54112yes—
98C257#233611yescharacteristic, complemented, Sylow
99#2336—2571nomaximal, Sylow
100C2×#125—2571no—
101#389—2572no—
102#387—2572no—
103#387—2572no—
104#387—2572no—
105#387—2572no—
106#383D25711yescharacteristic
107C4×C4⋊C4—2571no—
108C4×C4⋊C4—2572no—
109#125—2578no—
110C2×C4⋊C4—2572no—
111C2×C4⋊C4—2571no—
112C2×C4⋊C4—2571no—
113C2×C4⋊C4—2572no—
114C2×C4⋊C4—2572no—
115C2×C4⋊C4—2572no—
116C2×C4⋊C4—2572no—
117C2×C42—2572no—
118C2×C42—2571no—
119C2×C42—2571no—
120C2×C42—2572no—
121C2×C42C2×D25711yescharacteristic
122#95—2572no—
123#95—2572no—
124#95—2572no—
125#95—2572no—
126#95—2572no—
127#95C2×D25712yes—
128#95C2×D25711yescharacteristic
129#95C2×D25712yes—
130#95C2×D25711yescharacteristic
131C22×C4—2572no—
132C22×C4—2572no—
133C22×C4—2572no—
134C22×C4—2572no—
135C22×C4C22×D25712yes—
136C22×C4C22×D25712yes—
137C22×C4C22×D25711yescharacteristic
138C22×C4C22×D25711yescharacteristic
139C22×C4C22×D25711yescharacteristic
140C4⋊C4—2578no—
141C4⋊C4—2578no—
142C4⋊C4—2574no—
143C4⋊C4—2574no—
144C4⋊C4—5144no—
145C4⋊C4—5144no—
146C4⋊C4—10282no—
147C42—5144no—
148C42—10281no—
149C42—10281no—
150C42—10282no—
151C42C4×D25714yes—
152C23C23×D25711yescharacteristic
153C2×C4—2574no—
154C2×C4—2574no—
155C2×C4—2574no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—5142no—
167C2×C4—5142no—
168C2×C4—5142no—
169C2×C4—5142no—
170C2×C4—5142no—
171C2×C4—5142no—
172C2×C4—5142no—
173C2×C4—5142no—
174C2×C4—5142no—
175C2×C4C2×C4×D25712yes—
176C2×C4C2×C4×D25712yes—
177C2×C4C2×C4×D25712yes—
178C22C22×C4×D25711yescharacteristic
179C22#41×D25712yes—
180C22#2255163811yescharacteristic
181C22#2255163912yes—
182C22#2255163911yescharacteristic
183C4—22no—
184C4—22no—
185C4—22no—
186C4—5144no—
187C4—5144no—
188C4—5144no—
189C4—42no—
190C4—42no—
191C4—10282no—
192C2#3065720511yescharacteristic
193C2#3065721411yescharacteristic
194C2#3065721711yescharacteristic
195C2#3065732812yes—
196C2#3065734412yes—
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
Fitting#383×C257C2
socle, prime radical, cosocleC23×C257C24
hypercenter#383D257
hypocenterC257#2336
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
10. #22551584C4semidirect
98. C257#2336semidirect

Maximal subgroups

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

subgroupindex
C2×#225515842
#306566342
#306566242
#306566222
#306566202
#306566192
C4×C4⋊Dic2572
#306565882
#383×C2572
#2336257