Finite Group Databasebeta

← #1628×C251 #44546215

Subgroups — #1628×C251 #44546215

Order 32128.

216 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 68 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#556×C251C211yesmaximal, characteristic, complemented
2#486×C251C211yesmaximal, characteristic, complemented
3#460×C251C211yesmaximal, characteristic, complemented
4#459×C251C211yesmaximal, characteristic
5#332×C251C211yesmaximal, characteristic, complemented
6Q8⋊C8×C251C211yesmaximal, characteristic, complemented
7D4⋊C8×C251C211yesmaximal, characteristic, complemented
8C2×SD16×C251—21no—
9C2×D8×C251—21no—
10C4⋊Q8×C251C2211yescharacteristic
11#127×C251C2211yescharacteristic
12#126×C251—22no—
13#124×C251—21no—
14C22⋊Q8×C251—21no—
15C4×Q8×C251—21no—
16C4×Q8×C251C2211yescharacteristic
17C4×D4×C251C2211yescharacteristic
18#117×C251—21no—
19C4⋊C8×C251C2211yescharacteristic
20C4⋊C8×C251C2211yescharacteristic
21#102×C251—21no—
22#102×C251—21no—
23#102×C251—22no—
24C4×C8×C251C2211yescharacteristic
25C2×Q8×C251—21no—
26C2×Q8×C251D411yescharacteristic
27C2×D4×C251—21no—
28C2×D4×C251—21no—
29C2×D4×C251—21no—
30C2×D4×C251D411yescharacteristic
31C22×C4×C251—21no—
32SD16×C251—42no—
33D8×C251—42no—
34C2×C8×C251—21no—
35C2×C8×C251—21no—
36C2×C8×C251—21no—
37C4⋊C4×C251—22no—
38C4⋊C4×C251—21no—
39C4⋊C4×C251—21no—
40C4⋊C4×C251D411yescharacteristic
41C4⋊C4×C251D411yescharacteristic
42C4⋊C4×C251D412yes—
43C22⋊C4×C251—22no—
44C22⋊C4×C251—22no—
45C22⋊C4×C251—21no—
46C42×C251—22no—
47C42×C251—21no—
48C42×C251C2311yescharacteristic
49C23×C251—21no—
50C23×C251—41no—
51Q8×C251—21no—
52Q8×C251—21no—
53Q8×C251—41no—
54Q8×C251—41no—
55D4×C251—21no—
56D4×C251—21no—
57D4×C251—42no—
58D4×C251—41no—
59D4×C251—41no—
60D4×C251—41no—
61D4×C251—41no—
62C2×C4×C251—21no—
63C2×C4×C251—21no—
64C2×C4×C251—21no—
65C2×C4×C251—22no—
66C2×C4×C251—41no—
67C2×C4×C251—41no—
68C2×C4×C251C2×D411yescharacteristic
69C2×C4×C251C2×D411yescharacteristic
70C2×C4×C251C2×D411yescharacteristic
71C8×C251—41no—
72C8×C251—41no—
73C8×C251—41no—
74C22×C251—41no—
75C22×C251—41no—
76C22×C251—41no—
77C22×C251—81no—
78C22×C251—81no—
79C22×C251—81no—
80C22×C251#12011yescharacteristic
81C4×C251—21no—
82C4×C251—41no—
83C4×C251—41no—
84C4×C251—42no—
85C4×C251—41no—
86C4×C251#13511yescharacteristic
87C4×C251#13511yescharacteristic
88C4×C251AGL(1,ℤ/8)11yescharacteristic
89C4×C251AGL(1,ℤ/8)11yescharacteristic
90C2×C251—81no—
91C2×C251—161no—
92C2×C251#44911yescharacteristic
93C2×C251#44911yescharacteristic
94C2×C251#45411yescharacteristic
95C251#162811yescharacteristic, direct product factor, Sylow
96#1628C25111yesmaximal, characteristic, direct product factor, Sylow
97#556C2×C25111yescharacteristic, complemented
98#486C2×C25111yescharacteristic, complemented
99#460C2×C25111yescharacteristic, complemented
100#459C2×C25111yescharacteristic
101#332C2×C25111yescharacteristic, complemented
102Q8⋊C8C2×C25111yescharacteristic, complemented
103D4⋊C8C2×C25111yescharacteristic, complemented
104C2×SD16—21no—
105C2×D8—21no—
106C4⋊Q8C22×C25111yescharacteristic
107#127C22×C25111yescharacteristic
108#126—22no—
109#124—21no—
110C22⋊Q8—21no—
111C4×Q8—21no—
112C4×Q8C22×C25111yescharacteristic
113C4×D4C22×C25111yescharacteristic
114#117—21no—
115C4⋊C8C22×C25111yescharacteristic
116C4⋊C8C22×C25111yescharacteristic
117#102—21no—
118#102—21no—
119#102—22no—
120C4×C8C22×C25111yescharacteristic
121C2×Q8—21no—
122C2×Q8D4×C25111yescharacteristic
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4D4×C25111yescharacteristic
127C22×C4—21no—
128SD16—42no—
129D8—42no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4D4×C25111yescharacteristic
137C4⋊C4D4×C25111yescharacteristic
138C4⋊C4D4×C25112yes—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—21no—
142C42—22no—
143C42—21no—
144C42C23×C25111yescharacteristic
145C23—21no—
146C23—41no—
147Q8—21no—
148Q8—21no—
149Q8—41no—
150Q8—41no—
151D4—21no—
152D4—21no—
153D4—42no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4C2×D4×C25111yescharacteristic
165C2×C4C2×D4×C25111yescharacteristic
166C2×C4C2×D4×C25111yescharacteristic
167C8—41no—
168C8—41no—
169C8—41no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22#120×C25111yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—42no—
181C4—41no—
182C4#135×C25111yescharacteristic
183C4#135×C25111yescharacteristic
184C4AGL(1,ℤ/8)×C25111yescharacteristic
185C4AGL(1,ℤ/8)×C25111yescharacteristic
186C2—81no—
187C2—161no—
188C2#449×C25111yescharacteristic
189C2#449×C25111yescharacteristic
190C2#454×C25111yescharacteristic
subgroupG/H
center, socleC22×C251#120
commutator, Frattini, cosocleC42C23×C251
prime radical#459×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
1. #556×C251C2semidirect
2. #486×C251C2semidirect
3. #460×C251C2semidirect
5. #332×C251C2semidirect
6. Q8⋊C8×C251C2semidirect
7. D4⋊C8×C251C2semidirect
95. C251#1628direct (N × H)
96. #1628C251direct (N × H)
97. #556C2×C251semidirect
98. #486C2×C251semidirect
99. #460C2×C251semidirect
101. #332C2×C251semidirect
102. Q8⋊C8C2×C251semidirect
103. D4⋊C8C2×C251semidirect

Maximal subgroups

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

subgroupindex
#556×C2512
#486×C2512
#460×C2512
#459×C2512
#332×C2512
Q8⋊C8×C2512
D4⋊C8×C2512
#1628251