Finite Group Databasebeta

← #44553169

Subgroups — #44553169

Order 32128.

428 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 181 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×#22548707C211yesmaximal, characteristic
2#30631582C212yesmaximal
3#30631572C212yesmaximal
4#30631571C212yesmaximal
5#30631570C212yesmaximal
6#30631568C212yesmaximal
7#30631555C211yesmaximal, characteristic
8C4×Dic251⋊C4C212yesmaximal
9C4×C4⋊C4×C251C211yesmaximal, characteristic
10#22548707C418yescomplemented
11C2×C4⋊Dic251C2211yescharacteristic
12C2×C4⋊Dic251C2211yescharacteristic
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2212yes—
16C2×Dic251⋊C4C2212yes—
17C2×C4×Dic251C2212yes—
18C2×C4×Dic251C2212yes—
19C251⋊#95C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2211yescharacteristic
22C251⋊#95C2212yes—
23C251⋊#95C2211yescharacteristic
24C251⋊#95C2212yes—
25C251⋊#95C2212yes—
26C251⋊#95C2212yes—
27C2×C4⋊C4×C251C2212yes—
28C2×C42×C251C2211yescharacteristic
29C2×C42×C251C2211yescharacteristic
30C2×C42×C251C2211yescharacteristic
31#95×C251C2212yes—
32C22×Dic251C2312yes—
33C22×Dic251C2312yes—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C4⋊Dic251C2×C414yes—
37C4⋊Dic251C2×C414yes—
38Dic251⋊C4—24no—
39Dic251⋊C4—42no—
40Dic251⋊C4C2×C418yes—
41Dic251⋊C4C2×C418yes—
42C4×Dic251—24no—
43C4×Dic251—42no—
44C22×C4×C251C2312yes—
45C22×C4×C251C2311yescharacteristic
46C22×C4×C251C2311yescharacteristic
47C22×C4×C251C2311yescharacteristic
48C22×C4×C251C2312yes—
49C4⋊C4×C251—24no—
50C42×C251—41no—
51C42×C251—41no—
52C42×C251C2×C414yes—
53C2×Dic251—22no—
54C2×Dic251—22no—
55C2×Dic251—22no—
56C2×Dic251—22no—
57C2×Dic251—22no—
58C2×Dic251—22no—
59C2×Dic251—22no—
60C2×Dic251—22no—
61C2×Dic251—22no—
62C2×Dic251—22no—
63C2×Dic251C22×C414yes—
64C2×Dic251C22×C414yes—
65C23×C251C2411yescharacteristic
66C2×C4×C251—22no—
67C2×C4×C251—21no—
68C2×C4×C251—22no—
69C2×C4×C251—22no—
70C2×C4×C251—22no—
71C2×C4×C251—21no—
72C2×C4×C251—22no—
73C2×C4×C251—22no—
74C2×C4×C251—22no—
75C2×C4×C251C22×C412yes—
76C2×C4×C251C22×C412yes—
77C2×C4×C251C22×C412yes—
78C2×C4×C251#4114yes—
79C22×C251C23×C411yescharacteristic
80C22×C251C2×#4112yes—
81C22×C251H(5,𝔽2)11yescharacteristic
82C22×C2512−1+412yes—
83C22×C2512−1+411yescharacteristic
84C4×C251—22no—
85C4×C251—22no—
86C4×C251—22no—
87C4×C251—24no—
88C4×C251—42no—
89Dic251—24no—
90Dic251—24no—
91Dic251—42no—
92Dic251—42no—
93C2×C251C4×#4111yescharacteristic
94C2×C251#51911yescharacteristic
95C2×C251#52011yescharacteristic
96C2×C251#53912yes—
97C2×C251#54112yes—
98C251#233611yescharacteristic, complemented, Sylow
99#2336—2511nomaximal, Sylow
100C2×#125—2511no—
101#389—2512no—
102#387—2512no—
103#387—2512no—
104#387—2512no—
105#387—2512no—
106#383—2511no—
107C4×C4⋊C4—2512no—
108C4×C4⋊C4D25111yescharacteristic
109#125—2518no—
110C2×C4⋊C4—2512no—
111C2×C4⋊C4—2512no—
112C2×C4⋊C4—2512no—
113C2×C4⋊C4—2511no—
114C2×C4⋊C4—2511no—
115C2×C4⋊C4—2512no—
116C2×C4⋊C4C2×D25112yes—
117C2×C42—2512no—
118C2×C42—2512no—
119C2×C42C2×D25111yescharacteristic
120C2×C42C2×D25111yescharacteristic
121C2×C42C2×D25111yescharacteristic
122#95—2512no—
123#95—2512no—
124#95—2511no—
125#95—2512no—
126#95—2511no—
127#95—2512no—
128#95—2512no—
129#95—2512no—
130#95C2×D25112yes—
131C22×C4—2512no—
132C22×C4—2512no—
133C22×C4—2512no—
134C22×C4—2512no—
135C22×C4C22×D25112yes—
136C22×C4C22×D25111yescharacteristic
137C22×C4C22×D25111yescharacteristic
138C22×C4C22×D25111yescharacteristic
139C22×C4C22×D25112yes—
140C4⋊C4—24no—
141C4⋊C4—10042no—
142C4⋊C4—5024no—
143C4⋊C4—2518no—
144C4⋊C4—2518no—
145C4⋊C4—2514no—
146C4⋊C4—2514no—
147C42—41no—
148C42—41no—
149C42—10042no—
150C42—5024no—
151C42C4×D25114yes—
152C23C23×D25111yescharacteristic
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—5022no—
163C2×C4—5022no—
164C2×C4—5022no—
165C2×C4—5022no—
166C2×C4—5022no—
167C2×C4—5022no—
168C2×C4—5022no—
169C2×C4—5022no—
170C2×C4—5022no—
171C2×C4—5022no—
172C2×C4—2514no—
173C2×C4—2514no—
174C2×C4C2×C4×D25112yes—
175C2×C4C2×C4×D25112yes—
176C2×C4C2×C4×D25112yes—
177C2×C4#1686081014yes—
178C22C22×C4×D25111yescharacteristic
179C22C2×#1686081012yes—
180C22#2254876011yescharacteristic
181C22#2254876212yes—
182C22#2254876311yescharacteristic
183C4—24no—
184C4—22no—
185C4—22no—
186C4—22no—
187C4—42no—
188C4—10042no—
189C4—10042no—
190C4—5024no—
191C4—5024no—
192C2C4×#1686081011yescharacteristic
193C2#3063216411yescharacteristic
194C2#3063216811yescharacteristic
195C2#3063227912yes—
196C2#3063229512yes—
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
FittingC4×C4⋊C4×C251C2
socle, prime radical, cosocleC23×C251C24
hypercenterC4×C4⋊C4D251
hypocenterC251#2336
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
10. #22548707C4semidirect
98. C251#2336semidirect

Maximal subgroups

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

subgroupindex
C2×#225487072
#306315822
#306315722
#306315712
#306315702
#306315682
#306315552
C4×Dic251⋊C42
C4×C4⋊C4×C2512
#2336251