Finite Group Databasebeta

← #44550809

Subgroups — #44550809

Order 32128.

442 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×C4×Dic251C211yesmaximal, characteristic, complemented
2C2×#22548642C212yesmaximal, complemented
3#30631651C211yesmaximal, characteristic, complemented
4#30631446C212yesmaximal, complemented
5#525×C251C211yesmaximal, characteristic
6C23×Dic251—21no—
7C2×#16860794C2212yes—
8C2×C4×Dic251—24no—
9C2×C4×Dic251—21no—
10C2×C4×Dic251C2212yes—
11#22548642—24no—
12C22⋊(C251⋊C8)—22no—
13C2×#41×C251C2211yescharacteristic
14C23×C4×C251C2211yescharacteristic
15#123×C251—21no—
16C22⋊Q8×C251C412yescomplemented
17#121×C251C412yescomplemented
18#120×C251—21no—
19C4×D4×C251—22no—
20#117×C251C2211yescharacteristic
21C22×Dic251—28no—
22C22×Dic251—24no—
23C22×Dic251—22no—
24#16860794—22no—
25#16860794—22no—
26C2×C251⋊C8—22no—
27C4×Dic251—24no—
28C4×Dic251—22no—
29C4×Dic251D414yescomplemented
30C24×C251D411yescharacteristic
31#41×C251—22no—
32#41×C251—22no—
33C2×Q8×C251C2×C411yescharacteristic, complemented
34C2×D4×C251—22no—
35C2×D4×C251—21no—
36C2×D4×C251C2×C411yescharacteristic, complemented
37C22×C4×C251—22no—
38C22×C4×C251—22no—
39C22×C4×C251—21no—
40C22×C4×C251C2×C412yes—
41C22×C4×C251D411yescharacteristic
42C22×C4×C251D412yes—
43C22×C4×C251C2311yescharacteristic
44C4⋊C4×C251—22no—
45C4⋊C4×C251C2×C412yescomplemented
46C22⋊C4×C251—22no—
47C22⋊C4×C251—21no—
48C22⋊C4×C251—22no—
49C42×C251—21no—
50C2×Dic251—216no—
51C2×Dic251—28no—
52C2×Dic251—24no—
53C251⋊C8—42no—
54C23×C251—24no—
55C23×C251—21no—
56C23×C251—21no—
57C23×C251—21no—
58C23×C251C22⋊C412yes—
59C23×C251C2×D411yescharacteristic
60Q8×C251—22no—
61D4×C251—22no—
62D4×C251—44no—
63D4×C251—41no—
64D4×C251—41no—
65C2×C4×C251—21no—
66C2×C4×C251—21no—
67C2×C4×C251—24no—
68C2×C4×C251—24no—
69C2×C4×C251—22no—
70C2×C4×C251—21no—
71C2×C4×C251—42no—
72C2×C4×C251C22⋊C412yes—
73C2×C4×C251C22×C411yescharacteristic
74C2×C4×C251C2×D412yes—
75C2×C4×C251C2×D411yescharacteristic
76C2×C4×C251#4112yes—
77C22×C251—24no—
78C22×C251—24no—
79C22×C251—22no—
80C22×C251—22no—
81C22×C251—22no—
82C22×C251—41no—
83C22×C251—42no—
84C22×C251#10414yescomplemented
85C22×C251C2×C22⋊C411yescharacteristic
86C22×C251#12011yescharacteristic
87C22×C251#12311yescharacteristic
88C4×C251—22no—
89C4×C251—41no—
90C4×C251—42no—
91C4×C251C4×D412yes—
92C4×C251#12112yes—
93Dic251—28no—
94C2×C251—21no—
95C2×C251—24no—
96C2×C251—21no—
97C2×C251—81no—
98C2×C251#38611yescharacteristic
99C2×C251C2×#10412yes—
100C251#189611yescharacteristic, complemented, Sylow
101#1896—2511nomaximal, Sylow
102#525D25111yescharacteristic
103C22×C42—2511no—
104C2×#104—2512no—
105#407—2511no—
106#339—2512no—
107C2×#41C2×D25111yescharacteristic
108C23×C4—5021no—
109C23×C4C2×D25111yescharacteristic
110C2×M16—2512no—
111#123—21no—
112C22⋊Q8Dic25112yescomplemented
113#121Dic25112yescomplemented
114#120—21no—
115C4×D4—22no—
116#117C2×D25111yescharacteristic
117C2×C42—5021no—
118C2×C42—5024no—
119C2×C42—2512no—
120#104—5024no—
121C22⋊C8—5022no—
122C24C22⋊D25111yescharacteristic
123#41—22no—
124#41—22no—
125C2×Q8C2×Dic25111yescharacteristic, complemented
126C2×D4—22no—
127C2×D4—21no—
128C2×D4C2×Dic25111yescharacteristic, complemented
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—5024no—
133C22×C4—5028no—
134C22×C4—5022no—
135C22×C4C2×Dic25112yes—
136C22×C4C22⋊D25111yescharacteristic
137C22×C4C22⋊D25112yes—
138C22×C4C22×D25111yescharacteristic
139M16—5022no—
140M16—5022no—
141C2×C8—5022no—
142C4⋊C4—22no—
143C4⋊C4C2×Dic25112yescomplemented
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—22no—
147C42—21no—
148C42—5024no—
149C42—5022no—
150C42—2514no—
151C23—24no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23C22⋊Dic25112yes—
156C23C2×C22⋊D25111yescharacteristic
157Q8—22no—
158D4—22no—
159D4—44no—
160D4—41no—
161D4—41no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—50216no—
170C2×C4—5028no—
171C2×C4—5024no—
172C2×C4C22⋊Dic25112yes—
173C2×C4C22×Dic25111yescharacteristic
174C2×C4D4×D25112yes—
175C2×C4C2×C22⋊D25111yescharacteristic
176C2×C4#1686080912yes—
177C8—10042no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—42no—
184C22—41no—
185C22#2254864214yescomplemented
186C22C2×C22⋊Dic25111yescharacteristic
187C22#2254868311yescharacteristic
188C22#2254869611yescharacteristic
189C4—22no—
190C4—42no—
191C4—41no—
192C4—5028no—
193C4D4×Dic25112yes—
194C4#2254868712yes—
195C2—21no—
196C2—24no—
197C2—21no—
198C2—81no—
199C2#3063156711yescharacteristic
200C2C2×#2254864212yes—
subgroupG/H
centerC2×C4C2×C22⋊D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting, prime radical#525×C251C2
socleC22×C251C2×C22⋊C4
cosocleC22×C4×C251C23
hypercenter#525D251
hypocenterC251#1896
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic251C2semidirect
2. C2×#22548642C2semidirect
3. #30631651C2semidirect
4. #30631446C2semidirect
16. C22⋊Q8×C251C4semidirect
17. #121×C251C4semidirect
29. C4×Dic251D4semidirect
33. C2×Q8×C251C2×C4semidirect
36. C2×D4×C251C2×C4semidirect
45. C4⋊C4×C251C2×C4semidirect
84. C22×C251#104semidirect
100. C251#1896semidirect
112. C22⋊Q8Dic251semidirect
113. #121Dic251semidirect
125. C2×Q8C2×Dic251semidirect
128. C2×D4C2×Dic251semidirect
143. C4⋊C4C2×Dic251semidirect
185. C22#22548642semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic2512
C2×#225486422
#306316512
#306314462
#525×C2512
#1896251