Finite Group Databasebeta

← #1631×C251 #44546218

Subgroups — #1631×C251 #44546218

Order 32128.

256 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 72 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#550×C251C211yesmaximal, characteristic
2#492×C251C211yesmaximal, characteristic, complemented
3#477×C251C211yesmaximal, characteristic, complemented
4#476×C251C211yesmaximal, characteristic, complemented
5#331×C251C211yesmaximal, characteristic, complemented
6Q8⋊C8×C251C211yesmaximal, characteristic, complemented
7D4⋊C8×C251C211yesmaximal, characteristic, complemented
8C2×#41×C251—21no—
9C2×SD16×C251—22no—
10C4⋊Q8×C251C2211yescharacteristic
11#127×C251—21no—
12#127×C251C2211yescharacteristic
13#121×C251—22no—
14#121×C251—21no—
15C4×Q8×C251C2211yescharacteristic
16C4×D4×C251—21no—
17C4×D4×C251C2211yescharacteristic
18#107×C251—21no—
19#106×C251—21no—
20C4⋊C8×C251C2211yescharacteristic
21C4⋊C8×C251C2211yescharacteristic
22#103×C251—21no—
23#102×C251—21no—
24C4×C8×C251C2211yescharacteristic
25#41×C251—42no—
26#41×C251—42no—
27C2×Q8×C251—21no—
28C2×Q8×C251D411yescharacteristic
29C2×D4×C251—22no—
30C2×D4×C251—22no—
31C2×D4×C251—21no—
32C2×D4×C251—21no—
33C2×D4×C251D411yescharacteristic
34C2×D4×C251D412yes—
35C22×C4×C251—22no—
36C22×C4×C251—21no—
37SD16×C251—44no—
38C2×C8×C251—21no—
39C2×C8×C251—21no—
40C2×C8×C251—21no—
41C4⋊C4×C251—21no—
42C4⋊C4×C251—21no—
43C4⋊C4×C251—21no—
44C4⋊C4×C251D411yescharacteristic
45C4⋊C4×C251D411yescharacteristic
46C22⋊C4×C251—22no—
47C22⋊C4×C251—21no—
48C42×C251—21no—
49C42×C251C2311yescharacteristic
50C23×C251—21no—
51C23×C251—22no—
52Q8×C251—21no—
53Q8×C251—42no—
54Q8×C251SD1612yescomplemented
55D4×C251—21no—
56D4×C251—21no—
57D4×C251—42no—
58D4×C251—44no—
59D4×C251—41no—
60D4×C251—41no—
61D4×C251—41no—
62D4×C251—41no—
63D4×C251—82no—
64C2×C4×C251—21no—
65C2×C4×C251—21no—
66C2×C4×C251—21no—
67C2×C4×C251—44no—
68C2×C4×C251—41no—
69C2×C4×C251—41no—
70C2×C4×C251—41no—
71C2×C4×C251C2×D411yescharacteristic
72C2×C4×C251C2×D411yescharacteristic
73C2×C4×C251C2×D411yescharacteristic
74C8×C251—22no—
75C8×C251—41no—
76C8×C251—41no—
77C22×C251—42no—
78C22×C251—41no—
79C22×C251—41no—
80C22×C251—42no—
81C22×C251—42no—
82C22×C251—41no—
83C22×C251#12011yescharacteristic
84C4×C251—22no—
85C4×C251—21no—
86C4×C251—41no—
87C4×C251—41no—
88C4×C251—81no—
89C4×C251C2×SD1611yescharacteristic
90C4×C251#13511yescharacteristic
91C4×C251AGL(1,ℤ/8)11yescharacteristic
92C4×C251#13711yescharacteristic
93C2×C251—81no—
94C2×C251—82no—
95C2×C251#45011yescharacteristic
96C2×C251#45211yescharacteristic
97C2×C251#45311yescharacteristic
98C251#163111yescharacteristic, direct product factor, Sylow
99#1631C25111yesmaximal, characteristic, direct product factor, Sylow
100#550C2×C25111yescharacteristic
101#492C2×C25111yescharacteristic, complemented
102#477C2×C25111yescharacteristic, complemented
103#476C2×C25111yescharacteristic, complemented
104#331C2×C25111yescharacteristic, complemented
105Q8⋊C8C2×C25111yescharacteristic, complemented
106D4⋊C8C2×C25111yescharacteristic, complemented
107C2×#41—21no—
108C2×SD16—22no—
109C4⋊Q8C22×C25111yescharacteristic
110#127—21no—
111#127C22×C25111yescharacteristic
112#121—22no—
113#121—21no—
114C4×Q8C22×C25111yescharacteristic
115C4×D4—21no—
116C4×D4C22×C25111yescharacteristic
117#107—21no—
118#106—21no—
119C4⋊C8C22×C25111yescharacteristic
120C4⋊C8C22×C25111yescharacteristic
121#103—21no—
122#102—21no—
123C4×C8C22×C25111yescharacteristic
124#41—42no—
125#41—42no—
126C2×Q8—21no—
127C2×Q8D4×C25111yescharacteristic
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4D4×C25111yescharacteristic
133C2×D4D4×C25112yes—
134C22×C4—22no—
135C22×C4—21no—
136SD16—44no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4D4×C25111yescharacteristic
144C4⋊C4D4×C25111yescharacteristic
145C22⋊C4—22no—
146C22⋊C4—21no—
147C42—21no—
148C42C23×C25111yescharacteristic
149C23—21no—
150C23—22no—
151Q8—21no—
152Q8—42no—
153Q8SD16×C25112yescomplemented
154D4—21no—
155D4—21no—
156D4—42no—
157D4—44no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—82no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4C2×D4×C25111yescharacteristic
171C2×C4C2×D4×C25111yescharacteristic
172C2×C4C2×D4×C25111yescharacteristic
173C8—22no—
174C8—41no—
175C8—41no—
176C22—42no—
177C22—41no—
178C22—41no—
179C22—42no—
180C22—42no—
181C22—41no—
182C22#120×C25111yescharacteristic
183C4—22no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—81no—
188C4C2×SD16×C25111yescharacteristic
189C4#135×C25111yescharacteristic
190C4AGL(1,ℤ/8)×C25111yescharacteristic
191C4#137×C25111yescharacteristic
192C2—81no—
193C2—82no—
194C2#450×C25111yescharacteristic
195C2#452×C25111yescharacteristic
196C2#453×C25111yescharacteristic
subgroupG/H
center, socleC22×C251#120
commutator, Frattini, cosocleC42C23×C251
prime radical#550×C251C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C251 ◁ C42×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. #492×C251C2semidirect
3. #477×C251C2semidirect
4. #476×C251C2semidirect
5. #331×C251C2semidirect
6. Q8⋊C8×C251C2semidirect
7. D4⋊C8×C251C2semidirect
54. Q8×C251SD16semidirect
98. C251#1631direct (N × H)
99. #1631C251direct (N × H)
101. #492C2×C251semidirect
102. #477C2×C251semidirect
103. #476C2×C251semidirect
104. #331C2×C251semidirect
105. Q8⋊C8C2×C251semidirect
106. D4⋊C8C2×C251semidirect
153. Q8SD16×C251semidirect

Maximal subgroups

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

subgroupindex
#550×C2512
#492×C2512
#477×C2512
#476×C2512
#331×C2512
Q8⋊C8×C2512
D4⋊C8×C2512
#1631251