Finite Group Databasebeta

← #44550811

Subgroups — #44550811

Order 32128.

358 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#30632198C211yesmaximal, characteristic
2C2×#22548674C211yesmaximal, characteristic, complemented
3C2×#22548641C212yesmaximal, complemented
4#30631448C212yesmaximal, complemented
5#407×C251C211yesmaximal, characteristic, complemented
6C2×#16860810C2211yescharacteristic
7#22548698—21no—
8#22548694C412yescomplemented
9#22548689C412yescomplemented
10#22548684—21no—
11C4×C22⋊D251—22no—
12#22548674—24no—
13#22548674C2211yescharacteristic
14C2×C4⋊Dic251—21no—
15C2×C22⋊Dic251—21no—
16C2×C4×Dic251C2212yes—
17#22548641—24no—
18C23×C4×C251C2211yescharacteristic
19C2×M16×C251C2212yes—
20C22⋊C8×C251—22no—
21#16860810—22no—
22#16860810—22no—
23C2×Dic502C2×C411yescharacteristic
24C2×D1004C2×C411yescharacteristic
25C2×C22⋊D251—22no—
26C2×C22⋊D251—21no—
27C22×Dic251—22no—
28C2×C4×D251—21no—
29C4⋊Dic251—24no—
30C4⋊Dic251C2×C412yes—
31Dic251⋊C4—22no—
32C22⋊Dic251—24no—
33C22⋊Dic251—22no—
34C22⋊Dic251—21no—
35(C2×D251)⋊C4—22no—
36C4×Dic251—22no—
37C4×Dic251—21no—
38C4×Dic251D414yescomplemented
39C24×C251D411yescharacteristic
40C22×C4×C251—22no—
41C22×C4×C251—22no—
42C22×C4×C251C2×C412yes—
43C22×C4×C251D412yes—
44C22×C4×C251D411yescharacteristic
45C22×C4×C251C2311yescharacteristic
46M16×C251—22no—
47M16×C251—22no—
48C2×C8×C251—22no—
49C22×D251—21no—
50Dic502—22no—
51D1004—22no—
52C22⋊D251—44no—
53C22⋊D251—41no—
54C22⋊D251—41no—
55C2×Dic251—22no—
56C2×Dic251—21no—
57C2×Dic251—24no—
58C2×Dic251—44no—
59C4×D251—42no—
60C23×C251—21no—
61C23×C251—21no—
62C23×C251—42no—
63C23×C251C22⋊C412yes—
64C23×C251C2×D411yescharacteristic
65C2×C4×C251—21no—
66C2×C4×C251—21no—
67C2×C4×C251—24no—
68C2×C4×C251—24no—
69C2×C4×C251C22⋊C412yes—
70C2×C4×C251C22×C411yescharacteristic
71C2×C4×C251C2×D412yes—
72C2×C4×C251C2×D411yescharacteristic
73C2×C4×C251#4112yes—
74C8×C251—42no—
75C22×C251—22no—
76C22×C251—22no—
77C22×C251—22no—
78C22×C251—22no—
79C22×C251—42no—
80C22×C251—42no—
81C22×C251C2×C22⋊C411yescharacteristic
82C22×C251#12011yescharacteristic
83C22×C251#12311yescharacteristic
84C2×D251—41no—
85C2×D251—42no—
86C4×C251—22no—
87C4×C251C4×D412yes—
88C4×C251#12112yes—
89Dic251—44no—
90Dic251—41no—
91Dic251—42no—
92C2×C251—21no—
93C2×C251—21no—
94C2×C251—42no—
95C2×C251#38611yescharacteristic
96C2×C251#42112yes—
97D251—81no—
98C251#189711yescharacteristic, complemented, Sylow
99#1897—2511nomaximal, Sylow
100#525—2511no—
101C2×#117—2511no—
102C2×#104—2512no—
103#407D25111yescharacteristic, complemented
104#339—2512no—
105C2×#41—2511no—
106C23×C4C2×D25111yescharacteristic
107C2×M16C2×D25112yes—
108#123—5021no—
109C22⋊Q8—2512no—
110#121—2512no—
111#120—5021no—
112C4×D4—5022no—
113#117—5024no—
114#117—2511no—
115C2×C4⋊C4—5021no—
116C2×C22⋊C4—5021no—
117C2×C42—2512no—
118#104—5024no—
119C22⋊C8—22no—
120C24D100411yescharacteristic
121#41—5022no—
122#41—5022no—
123C2×Q8—2511no—
124C2×D4—5022no—
125C2×D4—5021no—
126C2×D4—2511no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—5022no—
130C22×C4—5021no—
131C22×C4C4×D25112yes—
132C22×C4C22⋊D25112yes—
133C22×C4D100411yescharacteristic
134C22×C4C22×D25111yescharacteristic
135M16—22no—
136M16—22no—
137C2×C8—22no—
138C4⋊C4—5024no—
139C4⋊C4—5022no—
140C4⋊C4—2512no—
141C22⋊C4—5024no—
142C22⋊C4—5022no—
143C22⋊C4—5021no—
144C22⋊C4—5022no—
145C42—5022no—
146C42—5021no—
147C42—2514no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—5021no—
152C23(C2×D251)⋊C412yes—
153C23C2×D100411yescharacteristic
154Q8—5022no—
155D4—10044no—
156D4—10041no—
157D4—10041no—
158D4—5022no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—10044no—
164C2×C4—10042no—
165C2×C4—5022no—
166C2×C4—5021no—
167C2×C4—5024no—
168C2×C4(C2×D251)⋊C412yes—
169C2×C4C2×C4×D25111yescharacteristic
170C2×C4D4×D25112yes—
171C2×C4C2×C22⋊D25111yescharacteristic
172C2×C4#1686080912yes—
173C8—42no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—10042no—
181C22—10041no—
182C22C2×(C2×D251)⋊C411yescharacteristic
183C22#2254868511yescharacteristic
184C22#2254869911yescharacteristic
185C4—22no—
186C4—10044no—
187C4—10042no—
188C4—10041no—
189C4(C22⋊D251)⋊C412yes—
190C4#2254868712yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—20081no—
195C2#3063156411yescharacteristic
196C2#3063170312yes—
subgroupG/H
centerC2×C4C2×C22⋊D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#407×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632198C2
cosocleC22×C4×C251C23
hypercenter#407D251
hypocenterC251#1897
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#22548674C2semidirect
3. C2×#22548641C2semidirect
4. #30631448C2semidirect
5. #407×C251C2semidirect
8. #22548694C4semidirect
9. #22548689C4semidirect
38. C4×Dic251D4semidirect
98. C251#1897semidirect
103. #407D251semidirect

Maximal subgroups

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

subgroupindex
#306321982
C2×#225486742
C2×#225486412
#306314482
#407×C2512
#1897251