Finite Group Databasebeta

← #1644×C251 #44546231

Subgroups — #1644×C251 #44546231

Order 32128.

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

normal subgroups only

This group has 208 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#554×C251C211yesmaximal, characteristic
2#487×C251C211yesmaximal, characteristic, complemented
3#474×C251C211yesmaximal, characteristic
4#462×C251C211yesmaximal, characteristic, complemented
5#333×C251C211yesmaximal, characteristic, complemented
6Q8⋊C8×C251C211yesmaximal, characteristic, complemented
7D4⋊C8×C251C211yesmaximal, characteristic
8C2×Q16×C251—21no—
9C4⋊Q8×C251C2211yescharacteristic
10C4⋊Q8×C251C2211yescharacteristic
11#125×C251—21no—
12C22⋊Q8×C251—21no—
13C22⋊Q8×C251—22no—
14C4×Q8×C251C2211yescharacteristic
15C4×D4×C251—21no—
16C4×D4×C251C2211yescharacteristic
17C2×C4⋊C4×C251—21no—
18#107×C251—21no—
19C4⋊C8×C251C2211yescharacteristic
20C4⋊C8×C251C2211yescharacteristic
21#103×C251—22no—
22#103×C251—21no—
23#102×C251—21no—
24C4×C8×C251C2211yescharacteristic
25C2×Q8×C251—21no—
26C2×Q8×C251D411yescharacteristic
27C2×Q8×C251D412yes—
28C2×D4×C251D411yescharacteristic
29C22×C4×C251—22no—
30C22×C4×C251—21no—
31Q16×C251—42no—
32C2×C8×C251—21no—
33C2×C8×C251—21no—
34C2×C8×C251—21no—
35C4⋊C4×C251—22no—
36C4⋊C4×C251—22no—
37C4⋊C4×C251—21no—
38C4⋊C4×C251—21no—
39C4⋊C4×C251—21no—
40C4⋊C4×C251—21no—
41C4⋊C4×C251—21no—
42C4⋊C4×C251D411yescharacteristic
43C4⋊C4×C251D411yescharacteristic
44C22⋊C4×C251—22no—
45C22⋊C4×C251—21no—
46C42×C251—21no—
47C42×C251C2311yescharacteristic
48C23×C251—21no—
49Q8×C251—21no—
50Q8×C251—21no—
51Q8×C251—42no—
52Q8×C251—41no—
53Q8×C251—41no—
54D4×C251—21no—
55D4×C251Q1612yescomplemented
56C2×C4×C251—21no—
57C2×C4×C251—21no—
58C2×C4×C251—21no—
59C2×C4×C251—22no—
60C2×C4×C251—41no—
61C2×C4×C251—41no—
62C2×C4×C251—41no—
63C2×C4×C251—82no—
64C2×C4×C251C2×D411yescharacteristic
65C2×C4×C251C2×D411yescharacteristic
66C2×C4×C251C2×D411yescharacteristic
67C8×C251—41no—
68C8×C251—41no—
69C8×C251—41no—
70C22×C251—22no—
71C22×C251—41no—
72C22×C251—41no—
73C22×C251#12011yescharacteristic
74C4×C251—21no—
75C4×C251—41no—
76C4×C251—41no—
77C4×C251—41no—
78C4×C251—42no—
79C4×C251—81no—
80C4×C251C2×Q1611yescharacteristic
81C4×C251#13511yescharacteristic
82C4×C251AGL(1,ℤ/8)11yescharacteristic
83C4×C251#13711yescharacteristic
84C2×C251—42no—
85C2×C251#44911yescharacteristic
86C2×C251#45111yescharacteristic
87C2×C251#45511yescharacteristic
88C251#164411yescharacteristic, direct product factor, Sylow
89#1644C25111yesmaximal, characteristic, direct product factor, Sylow
90#554C2×C25111yescharacteristic
91#487C2×C25111yescharacteristic, complemented
92#474C2×C25111yescharacteristic
93#462C2×C25111yescharacteristic, complemented
94#333C2×C25111yescharacteristic, complemented
95Q8⋊C8C2×C25111yescharacteristic, complemented
96D4⋊C8C2×C25111yescharacteristic
97C2×Q16—21no—
98C4⋊Q8C22×C25111yescharacteristic
99C4⋊Q8C22×C25111yescharacteristic
100#125—21no—
101C22⋊Q8—21no—
102C22⋊Q8—22no—
103C4×Q8C22×C25111yescharacteristic
104C4×D4—21no—
105C4×D4C22×C25111yescharacteristic
106C2×C4⋊C4—21no—
107#107—21no—
108C4⋊C8C22×C25111yescharacteristic
109C4⋊C8C22×C25111yescharacteristic
110#103—22no—
111#103—21no—
112#102—21no—
113C4×C8C22×C25111yescharacteristic
114C2×Q8—21no—
115C2×Q8D4×C25111yescharacteristic
116C2×Q8D4×C25112yes—
117C2×D4D4×C25111yescharacteristic
118C22×C4—22no—
119C22×C4—21no—
120Q16—42no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4D4×C25111yescharacteristic
132C4⋊C4D4×C25111yescharacteristic
133C22⋊C4—22no—
134C22⋊C4—21no—
135C42—21no—
136C42C23×C25111yescharacteristic
137C23—21no—
138Q8—21no—
139Q8—21no—
140Q8—42no—
141Q8—41no—
142Q8—41no—
143D4—21no—
144D4Q16×C25112yescomplemented
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—82no—
153C2×C4C2×D4×C25111yescharacteristic
154C2×C4C2×D4×C25111yescharacteristic
155C2×C4C2×D4×C25111yescharacteristic
156C8—41no—
157C8—41no—
158C8—41no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22#120×C25111yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—41no—
167C4—42no—
168C4—81no—
169C4C2×Q16×C25111yescharacteristic
170C4#135×C25111yescharacteristic
171C4AGL(1,ℤ/8)×C25111yescharacteristic
172C4#137×C25111yescharacteristic
173C2—42no—
174C2#449×C25111yescharacteristic
175C2#451×C25111yescharacteristic
176C2#455×C25111yescharacteristic
subgroupG/H
center, socleC22×C251#120
commutator, Frattini, cosocleC42C23×C251
prime radicalC2×D4×C251D4
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. #487×C251C2semidirect
4. #462×C251C2semidirect
5. #333×C251C2semidirect
6. Q8⋊C8×C251C2semidirect
55. D4×C251Q16semidirect
88. C251#1644direct (N × H)
89. #1644C251direct (N × H)
91. #487C2×C251semidirect
93. #462C2×C251semidirect
94. #333C2×C251semidirect
95. Q8⋊C8C2×C251semidirect
144. D4Q16×C251semidirect

Maximal subgroups

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

subgroupindex
#554×C2512
#487×C2512
#474×C2512
#462×C2512
#333×C2512
Q8⋊C8×C2512
D4⋊C8×C2512
#1644251