Finite Group Databasebeta

← #44550688

Subgroups — #44550688

Order 32128.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 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×#22548636C212yesmaximal, complemented
3C2×C22⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
4#30631449C212yesmaximal, complemented
5C2×#121×C251C211yesmaximal, characteristic
6C22×C251⋊C8C2212yes—
7#22548674—24no—
8C2×C4⋊Dic251C2212yes—
9C2×C22⋊Dic251—21no—
10C2×C4×Dic251—21no—
11#22548636—24no—
12C22⋊(C251⋊C8)—22no—
13C22×D4×C251—21no—
14C22×D4×C251C2211yescharacteristic
15C23×C4×C251C2211yescharacteristic
16#121×C251—22no—
17#121×C251C414yescomplemented
18C2×C4⋊C4×C251C2211yescharacteristic
19C2×C22⋊C4×C251—21no—
20C22×Dic251—22no—
21C2×C251⋊C8—22no—
22C2×C251⋊C8—22no—
23C2×C251⋊C8—22no—
24C4⋊Dic251—22no—
25C4⋊Dic251D414yescomplemented
26C22⋊Dic251—24no—
27C4×Dic251—24no—
28C24×C251—21no—
29C24×C251D411yescharacteristic
30C2×D4×C251—24no—
31C2×D4×C251—22no—
32C2×D4×C251—21no—
33C2×D4×C251—22no—
34C2×D4×C251—44no—
35C2×D4×C251C2×C412yes—
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—21no—
44C4⋊C4×C251C2×C412yes—
45C22⋊C4×C251—24no—
46C2×Dic251—24no—
47C2×Dic251—42no—
48C2×Dic251—42no—
49C251⋊C8—42no—
50C23×C251—21no—
51C23×C251—21no—
52C23×C251—22no—
53C23×C251—22no—
54C23×C251—22no—
55C23×C251—41no—
56C23×C251—41no—
57C23×C251—42no—
58C23×C251—42no—
59C23×C251C22⋊C412yes—
60C23×C251C2×D411yescharacteristic
61D4×C251—24no—
62D4×C251—48no—
63D4×C251—42no—
64C2×C4×C251—21no—
65C2×C4×C251—21no—
66C2×C4×C251—24no—
67C2×C4×C251—24no—
68C2×C4×C251—22no—
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—42no—
79C22×C251—42no—
80C22×C251—42no—
81C22×C251—42no—
82C22×C251—42no—
83C22×C251—82no—
84C22×C251—82no—
85C22×C251C2×C22⋊C411yescharacteristic
86C22×C251#12011yescharacteristic
87C22×C251#12311yescharacteristic
88C22×C251#13512yes—
89C22×C251AGL(1,ℤ/8)12yes—
90C4×C251—22no—
91C4×C251—42no—
92C4×C251C4×D412yes—
93C4×C251#12112yes—
94Dic251—44no—
95C2×C251—42no—
96C2×C251—82no—
97C2×C251#38611yescharacteristic
98C2×C251#41611yescharacteristic
99C2×C251#41811yescharacteristic
100C2×C251#44912yes—
101C2×C251#48212yes—
102C251#187511yescharacteristic, complemented, Sylow
103#1875—2511nomaximal, Sylow
104C2×#121D25111yescharacteristic
105C2×#117—2511no—
106C2×#102—2512no—
107C2×C22⋊C8—2511no—
108#340—2512no—
109C22×D4—21no—
110C22×D4C2×D25111yescharacteristic
111C23×C4C2×D25111yescharacteristic
112C22×C8—2512no—
113#121—22no—
114#121Dic25114yescomplemented
115#117—5024no—
116C2×C4⋊C4—2512no—
117C2×C4⋊C4C2×D25111yescharacteristic
118C2×C22⋊C4—21no—
119C2×C22⋊C4—5021no—
120C2×C42—5021no—
121#102—5024no—
122C22⋊C8—5022no—
123C24—21no—
124C24C22⋊D25111yescharacteristic
125C2×D4—24no—
126C2×D4—22no—
127C2×D4—21no—
128C2×D4—22no—
129C2×D4—44no—
130C2×D4C2×Dic25112yes—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—5022no—
135C22×C4C2×Dic25112yes—
136C22×C4C22⋊D25111yescharacteristic
137C22×C4C22⋊D25112yes—
138C22×C4C22×D25111yescharacteristic
139C2×C8—5022no—
140C2×C8—5022no—
141C2×C8—5022no—
142C4⋊C4—21no—
143C4⋊C4—5022no—
144C4⋊C4—2514no—
145C4⋊C4C2×Dic25112yes—
146C22⋊C4—24no—
147C22⋊C4—5024no—
148C42—5024no—
149C23—22no—
150C23—21no—
151C23—21no—
152C23—22no—
153C23—22no—
154C23—42no—
155C23—41no—
156C23—41no—
157C23—42no—
158C23C22⋊Dic25112yes—
159C23C2×C22⋊D25111yescharacteristic
160D4—24no—
161D4—48no—
162D4—42no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—42no—
169C2×C4—10042no—
170C2×C4—10042no—
171C2×C4—5024no—
172C2×C4C22⋊Dic25112yes—
173C2×C4C22×Dic25111yescharacteristic
174C2×C4D4×D25112yes—
175C2×C4C2×C22⋊D25111yescharacteristic
176C2×C4#1686080912yes—
177C8—10042no—
178C22—22no—
179C22—22no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22—42no—
186C22—82no—
187C22—82no—
188C22C2×C22⋊Dic25111yescharacteristic
189C22#2254868311yescharacteristic
190C22#2254869611yescharacteristic
191C22#2254873512yes—
192C22#2254874112yes—
193C4—22no—
194C4—42no—
195C4—10044no—
196C4D4×Dic25112yes—
197C4#2254868712yes—
198C2—42no—
199C2—82no—
200C2#3063156711yescharacteristic
201C2#3063168311yescharacteristic
202C2#3063169211yescharacteristic
203C2#3063183012yes—
204C2#3063200812yes—
subgroupG/H
centerC23C2×C22⋊D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting, prime radicalC2×#121×C251C2
socleC23×C251C2×D4
cosocleC22×C4×C251C23
hypercenterC2×#121D251
hypocenterC251#1875
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#22548674C2semidirect
2. C2×#22548636C2semidirect
3. C2×C22⋊(C251⋊C8)C2semidirect
4. #30631449C2semidirect
17. #121×C251C4semidirect
25. C4⋊Dic251D4semidirect
102. C251#1875semidirect
114. #121Dic251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486742
C2×#225486362
C2×C22⋊(C251⋊C8)2
#306314492
C2×#121×C2512
#1875251