Finite Group Databasebeta

← #44550824

Subgroups — #44550824

Order 32128.

378 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 97 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×#22548674C211yesmaximal, characteristic, complemented
2C2×#22548630C212yesmaximal
3#30631550C211yesmaximal, characteristic, complemented
4#30631454C212yesmaximal, complemented
5C2×#123×C251C211yesmaximal, characteristic
6#22548674—24no—
7C2×C4⋊Dic251—21no—
8C2×C22⋊Dic251C2212yes—
9C2×C22⋊Dic251C2212yes—
10C2×C4×Dic251—21no—
11#22548630—24no—
12C251⋊#95—22no—
13C22×D4×C251C2211yescharacteristic
14C23×C4×C251C2211yescharacteristic
15#123×C251—22no—
16#123×C251—21no—
17#123×C251—21no—
18C2×C4⋊C4×C251C412yescomplemented
19C2×C22⋊C4×C251C412yescomplemented
20C2×C22⋊C4×C251C2211yescharacteristic
21C22×Dic251—22no—
22C22×Dic251—22no—
23C4⋊Dic251—24no—
24C22⋊Dic251—22no—
25C22⋊Dic251—22no—
26C22⋊Dic251—22no—
27C22⋊Dic251D414yes—
28C4×Dic251—24no—
29C24×C251C2×C411yescharacteristic
30C24×C251C2311yescharacteristic
31C2×D4×C251—21no—
32C2×D4×C251—21no—
33C2×D4×C251—22no—
34C2×D4×C251—22no—
35C22×C4×C251—22no—
36C22×C4×C251—22no—
37C22×C4×C251C2×C411yescharacteristic
38C22×C4×C251C2×C412yes—
39C22×C4×C251C2×C412yes—
40C22×C4×C251D412yes—
41C22×C4×C251D411yescharacteristic
42C22×C4×C251D411yescharacteristic
43C4⋊C4×C251—22no—
44C4⋊C4×C251—22no—
45C22⋊C4×C251—21no—
46C22⋊C4×C251—21no—
47C22⋊C4×C251—22no—
48C22⋊C4×C251—22no—
49C2×Dic251—24no—
50C2×Dic251—44no—
51C2×Dic251—44no—
52C2×Dic251—42no—
53C23×C251—21no—
54C23×C251—22no—
55C23×C251—42no—
56C23×C251—42no—
57C23×C251C22×C411yescharacteristic
58C23×C251C2×D412yes—
59C23×C251C2×D411yescharacteristic
60C23×C251C2×D411yescharacteristic
61C23×C251#4112yes—
62D4×C251—41no—
63D4×C251—42no—
64D4×C251—41no—
65C2×C4×C251—24no—
66C2×C4×C251—21no—
67C2×C4×C251—21no—
68C2×C4×C251—24no—
69C2×C4×C251—21no—
70C2×C4×C251—21no—
71C2×C4×C251—22no—
72C2×C4×C251—21no—
73C2×C4×C251—22no—
74C2×C4×C251—42no—
75C2×C4×C251C22⋊C412yes—
76C2×C4×C251C22⋊C412yes—
77C22×C251—22no—
78C22×C251—22no—
79C22×C251—22no—
80C22×C251—42no—
81C22×C251—42no—
82C22×C251—41no—
83C22×C251—42no—
84C22×C251—82no—
85C22×C251C2×C22⋊C411yescharacteristic
86C22×C251C4×D412yes—
87C22×C251#12011yescharacteristic
88C22×C251#12112yes—
89C22×C251#12311yescharacteristic
90C4×C251—22no—
91C4×C251—22no—
92C4×C251—41no—
93C4×C251—42no—
94Dic251—44no—
95Dic251—82no—
96C2×C251—21no—
97C2×C251—22no—
98C2×C251—22no—
99C2×C251—21no—
100C2×C251—81no—
101C2×C251#38611yescharacteristic
102C2×C251#41012yes—
103C251#189911yescharacteristic, complemented, Sylow
104#1899—2511nomaximal, Sylow
105C2×#123D25111yescharacteristic
106C2×#117—2511no—
107C2×#99—2512no—
108#381—2511no—
109#342—2512no—
110C22×D4C2×D25111yescharacteristic
111C23×C4C2×D25111yescharacteristic
112#123—22no—
113#123—21no—
114#123—21no—
115#117—5024no—
116C2×C4⋊C4—5021no—
117C2×C4⋊C4Dic25112yescomplemented
118C2×C22⋊C4—2512no—
119C2×C22⋊C4—2512no—
120C2×C22⋊C4Dic25112yescomplemented
121C2×C22⋊C4C2×D25111yescharacteristic
122C2×C42—5021no—
123#99—5024no—
124#95—5022no—
125C24C2×Dic25111yescharacteristic
126C24C22×D25111yescharacteristic
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—5022no—
134C22×C4—5022no—
135C22×C4C2×Dic25111yescharacteristic
136C22×C4C2×Dic25112yes—
137C22×C4C2×Dic25112yes—
138C22×C4C22⋊D25112yes—
139C22×C4C22⋊D25111yescharacteristic
140C22×C4C22⋊D25111yescharacteristic
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—5024no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—22no—
147C22⋊C4—22no—
148C22⋊C4—5022no—
149C22⋊C4—5022no—
150C22⋊C4—5022no—
151C22⋊C4—2514no—
152C42—5024no—
153C23—21no—
154C23—22no—
155C23—42no—
156C23—42no—
157C23C22×Dic25111yescharacteristic
158C23D4×D25112yes—
159C23C2×C22⋊D25111yescharacteristic
160C23C2×C22⋊D25111yescharacteristic
161C23#1686080912yes—
162D4—41no—
163D4—42no—
164D4—41no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—10044no—
176C2×C4—10042no—
177C2×C4—10044no—
178C2×C4—5024no—
179C2×C4C22⋊Dic25112yes—
180C2×C4C22⋊Dic25112yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—42no—
188C22—82no—
189C22C2×C22⋊Dic25111yescharacteristic
190C22D4×Dic25112yes—
191C22#2254868311yescharacteristic
192C22#2254868712yes—
193C22#2254869611yescharacteristic
194C4—22no—
195C4—22no—
196C4—41no—
197C4—42no—
198C4—20082no—
199C4—10044no—
200C2—22no—
201C2—21no—
202C2—22no—
203C2—21no—
204C2—81no—
205C2#3063156711yescharacteristic
206C2#3063166112yes—
subgroupG/H
centerC22C2×C22⋊Dic251
commutatorC23×C251C22×C4
FrattiniC24C22×D251
FittingC2×#123×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC22×D4×C251C22
cosocleC24×C251C23
hypercenterC2×#123D251
hypocenterC251#1899
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#22548674C2semidirect
3. #30631550C2semidirect
4. #30631454C2semidirect
18. C2×C4⋊C4×C251C4semidirect
19. C2×C22⋊C4×C251C4semidirect
103. C251#1899semidirect
117. C2×C4⋊C4Dic251semidirect
120. C2×C22⋊C4Dic251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486742
C2×#225486302
#306315502
#306314542
C2×#123×C2512
#1899251