Finite Group Databasebeta

← #2192×C251 #44546779

Subgroups — #2192×C251 #44546779

Order 32128.

220 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 64 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#576×C251C211yesmaximal, characteristic, complemented
2#509×C251C211yesmaximal, characteristic, complemented
3C2×D16×C251C211yesmaximal, characteristic, complemented
4#472×C251C211yesmaximal, characteristic, complemented
5#361×C251C211yesmaximal, characteristic, complemented
6#359×C251C211yesmaximal, characteristic, complemented
7#350×C251C211yesmaximal, characteristic, complemented
8H(5,𝔽2)×C251—21no—
9AGL(1,ℤ/8)×C251—22no—
10AGL(1,ℤ/8)×C251—21no—
11AGL(1,ℤ/8)×C251—21no—
12#135×C251—21no—
13#135×C251C2211yescharacteristic
14C2×D8×C251—21no—
15C2×D8×C251C2211yescharacteristic
16C2×D8×C251C2211yescharacteristic, complemented
17#131×C251C2211yescharacteristic
18SD32×C251—21no—
19D16×C251—22no—
20D16×C251—21no—
21M32×C251C2211yescharacteristic, complemented
22C2×C16×C251C2211yescharacteristic, complemented
23#108×C251C2211yescharacteristic, complemented
24#100×C251—21no—
25#41×C251—22no—
26#41×C251—21no—
27#41×C251—41no—
28#41×C251D411yescharacteristic
29C2×D4×C251—22no—
30C2×D4×C251—21no—
31C2×D4×C251—21no—
32C2×D4×C251—42no—
33C2×D4×C251—41no—
34Q16×C251D411yescharacteristic
35SD16×C251—22no—
36SD16×C251—21no—
37SD16×C251—41no—
38D8×C251—22no—
39D8×C251—21no—
40D8×C251—21no—
41D8×C251—21no—
42D8×C251—41no—
43D8×C251D411yescharacteristic
44D8×C251D412yes—
45M16×C251—21no—
46M16×C251—21no—
47M16×C251D411yescharacteristic
48C2×C8×C251—21no—
49C2×C8×C251C2311yescharacteristic
50C16×C251—21no—
51C16×C251—21no—
52C23×C251—42no—
53C23×C251—41no—
54C23×C251—41no—
55Q8×C251—21no—
56Q8×C251D811yescharacteristic
57D4×C251—21no—
58D4×C251—21no—
59D4×C251—22no—
60D4×C251—42no—
61D4×C251—42no—
62D4×C251—41no—
63D4×C251—41no—
64D4×C251—41no—
65D4×C251—41no—
66D4×C251D811yescharacteristic
67C2×C4×C251—21no—
68C2×C4×C251—42no—
69C2×C4×C251—41no—
70C2×C4×C251C2×D411yescharacteristic
71C8×C251—21no—
72C8×C251—41no—
73C8×C251C2×D411yescharacteristic
74C8×C251C2×D411yescharacteristic
75C22×C251—21no—
76C22×C251—42no—
77C22×C251—41no—
78C22×C251—81no—
79C22×C251—161no—
80C22×C251—161no—
81C22×C251—162no—
82C22×C251AGL(1,ℤ/8)11yescharacteristic
83C4×C251—21no—
84C4×C251—41no—
85C4×C251#12011yescharacteristic
86C4×C251C2×D811yescharacteristic
87C2×C251—21no—
88C2×C251—41no—
89C2×C251—82no—
90C2×C251—81no—
91C2×C251—161no—
92C2×C251#44711yescharacteristic
93C251#219211yescharacteristic, direct product factor, Sylow
94#2192C25111yesmaximal, characteristic, direct product factor, Sylow
95#576C2×C25111yescharacteristic, complemented
96#509C2×C25111yescharacteristic, complemented
97C2×D16C2×C25111yescharacteristic, complemented
98#472C2×C25111yescharacteristic, complemented
99#361C2×C25111yescharacteristic, complemented
100#359C2×C25111yescharacteristic, complemented
101#350C2×C25111yescharacteristic, complemented
102H(5,𝔽2)—21no—
103AGL(1,ℤ/8)—22no—
104AGL(1,ℤ/8)—21no—
105AGL(1,ℤ/8)—21no—
106#135—21no—
107#135C22×C25111yescharacteristic
108C2×D8—21no—
109C2×D8C22×C25111yescharacteristic
110C2×D8C22×C25111yescharacteristic, complemented
111#131C22×C25111yescharacteristic
112SD32—21no—
113D16—22no—
114D16—21no—
115M32C22×C25111yescharacteristic, complemented
116C2×C16C22×C25111yescharacteristic, complemented
117#108C22×C25111yescharacteristic, complemented
118#100—21no—
119#41—22no—
120#41—21no—
121#41—41no—
122#41D4×C25111yescharacteristic
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—42no—
127C2×D4—41no—
128Q16D4×C25111yescharacteristic
129SD16—22no—
130SD16—21no—
131SD16—41no—
132D8—22no—
133D8—21no—
134D8—21no—
135D8—21no—
136D8—41no—
137D8D4×C25111yescharacteristic
138D8D4×C25112yes—
139M16—21no—
140M16—21no—
141M16D4×C25111yescharacteristic
142C2×C8—21no—
143C2×C8C23×C25111yescharacteristic
144C16—21no—
145C16—21no—
146C23—42no—
147C23—41no—
148C23—41no—
149Q8—21no—
150Q8D8×C25111yescharacteristic
151D4—21no—
152D4—21no—
153D4—22no—
154D4—42no—
155D4—42no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4D8×C25111yescharacteristic
161C2×C4—21no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4C2×D4×C25111yescharacteristic
165C8—21no—
166C8—41no—
167C8C2×D4×C25111yescharacteristic
168C8C2×D4×C25111yescharacteristic
169C22—21no—
170C22—42no—
171C22—41no—
172C22—81no—
173C22—161no—
174C22—161no—
175C22—162no—
176C22AGL(1,ℤ/8)×C25111yescharacteristic
177C4—21no—
178C4—41no—
179C4#120×C25111yescharacteristic
180C4C2×D8×C25111yescharacteristic
181C2—21no—
182C2—41no—
183C2—82no—
184C2—81no—
185C2—161no—
186C2#447×C25111yescharacteristic
subgroupG/H
center, socleC2×C251#447
commutator, Frattini, cosocleC2×C8C23×C251
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C251 ◁ C2×C4×C251 ◁ #131×C251 ◁ G

Complements

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

NH ≅ G/Ntype
1. #576×C251C2semidirect
2. #509×C251C2semidirect
3. C2×D16×C251C2semidirect
4. #472×C251C2semidirect
5. #361×C251C2semidirect
6. #359×C251C2semidirect
7. #350×C251C2semidirect
16. C2×D8×C251C22semidirect
21. M32×C251C22semidirect
22. C2×C16×C251C22semidirect
23. #108×C251C22semidirect
93. C251#2192direct (N × H)
94. #2192C251direct (N × H)
95. #576C2×C251semidirect
96. #509C2×C251semidirect
97. C2×D16C2×C251semidirect
98. #472C2×C251semidirect
99. #361C2×C251semidirect
100. #359C2×C251semidirect
101. #350C2×C251semidirect
110. C2×D8C22×C251semidirect
115. M32C22×C251semidirect
116. C2×C16C22×C251semidirect
117. #108C22×C251semidirect

Maximal subgroups

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

subgroupindex
#576×C2512
#509×C2512
C2×D16×C2512
#472×C2512
#361×C2512
#359×C2512
#350×C2512
#2192251