Finite Group Databasebeta

← #44550814

Subgroups — #44550814

Order 32128.

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

normal subgroups only

This group has 358 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×#22548642C212yesmaximal, complemented
3#30631651C211yesmaximal, characteristic, complemented
4#30631446C212yesmaximal, complemented
5#525×C251C211yesmaximal, characteristic
6C2×#16860794C2212yes—
7#22548674—24no—
8C2×C4⋊Dic251—21no—
9C2×C22⋊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—22no—
22#16860794—22no—
23#16860794—22no—
24C2×C251⋊C8—22no—
25C4⋊Dic251—24no—
26C22⋊Dic251—24no—
27C4×Dic251—22no—
28C4×Dic251D414yescomplemented
29C24×C251D411yescharacteristic
30#41×C251—22no—
31#41×C251—22no—
32C2×Q8×C251C2×C411yescharacteristic
33C2×D4×C251—22no—
34C2×D4×C251—21no—
35C2×D4×C251C2×C411yescharacteristic
36C22×C4×C251—22no—
37C22×C4×C251—22no—
38C22×C4×C251—21no—
39C22×C4×C251C2×C412yes—
40C22×C4×C251D411yescharacteristic
41C22×C4×C251D412yes—
42C22×C4×C251C2311yescharacteristic
43C4⋊C4×C251—22no—
44C4⋊C4×C251C2×C412yes—
45C22⋊C4×C251—22no—
46C22⋊C4×C251—21no—
47C22⋊C4×C251—22no—
48C42×C251—21no—
49C2×Dic251—24no—
50C2×Dic251—44no—
51C251⋊C8—42no—
52C23×C251—21no—
53C23×C251—21no—
54C23×C251—21no—
55C23×C251—42no—
56C23×C251C22⋊C412yes—
57C23×C251C2×D411yescharacteristic
58Q8×C251—22no—
59D4×C251—22no—
60D4×C251—44no—
61D4×C251—41no—
62D4×C251—41no—
63C2×C4×C251—21no—
64C2×C4×C251—21no—
65C2×C4×C251—24no—
66C2×C4×C251—24no—
67C2×C4×C251—22no—
68C2×C4×C251—21no—
69C2×C4×C251—42no—
70C2×C4×C251C22⋊C412yes—
71C2×C4×C251C22×C411yescharacteristic
72C2×C4×C251C2×D412yes—
73C2×C4×C251C2×D411yescharacteristic
74C2×C4×C251#4112yes—
75C22×C251—22no—
76C22×C251—22no—
77C22×C251—22no—
78C22×C251—22no—
79C22×C251—42no—
80C22×C251—42no—
81C22×C251—41no—
82C22×C251—42no—
83C22×C251C2×C22⋊C411yescharacteristic
84C22×C251#12011yescharacteristic
85C22×C251#12311yescharacteristic
86C4×C251—22no—
87C4×C251—41no—
88C4×C251—42no—
89C4×C251C4×D412yes—
90C4×C251#12112yes—
91Dic251—44no—
92C2×C251—21no—
93C2×C251—21no—
94C2×C251—42no—
95C2×C251—81no—
96C2×C251#38611yescharacteristic
97C2×C251#42112yes—
98C251#189711yescharacteristic, complemented, Sylow
99#1897—2511nomaximal, Sylow
100#525D25111yescharacteristic
101C2×#117—2511no—
102C2×#104—2512no—
103#407—2511no—
104#339—2512no—
105C2×#41C2×D25111yescharacteristic
106C23×C4C2×D25111yescharacteristic
107C2×M16—2512no—
108#123—21no—
109C22⋊Q8Dic25112yescomplemented
110#121Dic25112yescomplemented
111#120—21no—
112C4×D4—22no—
113#117—5024no—
114#117C2×D25111yescharacteristic
115C2×C4⋊C4—5021no—
116C2×C22⋊C4—5021no—
117C2×C42—2512no—
118#104—5024no—
119C22⋊C8—5022no—
120C24C22⋊D25111yescharacteristic
121#41—22no—
122#41—22no—
123C2×Q8C2×Dic25111yescharacteristic
124C2×D4—22no—
125C2×D4—21no—
126C2×D4C2×Dic25111yescharacteristic
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—5022no—
131C22×C4C2×Dic25112yes—
132C22×C4C22⋊D25111yescharacteristic
133C22×C4C22⋊D25112yes—
134C22×C4C22×D25111yescharacteristic
135M16—5022no—
136M16—5022no—
137C2×C8—5022no—
138C4⋊C4—22no—
139C4⋊C4—5024no—
140C4⋊C4C2×Dic25112yes—
141C22⋊C4—22no—
142C22⋊C4—21no—
143C22⋊C4—22no—
144C22⋊C4—5024no—
145C42—21no—
146C42—5022no—
147C42—2514no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23—42no—
152C23C22⋊Dic25112yes—
153C23C2×C22⋊D25111yescharacteristic
154Q8—22no—
155D4—22no—
156D4—44no—
157D4—41no—
158D4—41no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—42no—
166C2×C4—10044no—
167C2×C4—5024no—
168C2×C4C22⋊Dic25112yes—
169C2×C4C22×Dic25111yescharacteristic
170C2×C4D4×D25112yes—
171C2×C4C2×C22⋊D25111yescharacteristic
172C2×C4#1686080912yes—
173C8—10042no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—41no—
182C22C2×C22⋊Dic25111yescharacteristic
183C22#2254868311yescharacteristic
184C22#2254869611yescharacteristic
185C4—22no—
186C4—42no—
187C4—41no—
188C4—10044no—
189C4D4×Dic25112yes—
190C4#2254868712yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—81no—
195C2#3063156711yescharacteristic
196C2#3063170412yes—
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#1897
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. C2×#22548674C2semidirect
2. C2×#22548642C2semidirect
3. #30631651C2semidirect
4. #30631446C2semidirect
16. C22⋊Q8×C251C4semidirect
17. #121×C251C4semidirect
28. C4×Dic251D4semidirect
98. C251#1897semidirect
109. C22⋊Q8Dic251semidirect
110. #121Dic251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486742
C2×#225486422
#306316512
#306314462
#525×C2512
#1897251