Finite Group Databasebeta

← #2193×C251 #44546780

Subgroups — #2193×C251 #44546780

Order 32128.

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

normal subgroups only

This group has 210 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#578×C251C211yesmaximal, characteristic, complemented
2#509×C251C211yesmaximal, characteristic, complemented
3C2×SD32×C251C211yesmaximal, characteristic, complemented
4#472×C251C211yesmaximal, characteristic, complemented
5#362×C251C211yesmaximal, characteristic, complemented
6#359×C251C211yesmaximal, characteristic, complemented
7#350×C251C211yesmaximal, characteristic, complemented
82−1+4×C251—21no—
9#137×C251—22no—
10#137×C251—21no—
11AGL(1,ℤ/8)×C251—21no—
12#135×C251—21no—
13#135×C251C2211yescharacteristic
14C2×Q16×C251—21no—
15C2×Q16×C251C2211yescharacteristic
16C2×D8×C251C2211yescharacteristic
17#131×C251C2211yescharacteristic
18SD32×C251—22no—
19SD32×C251—21no—
20D16×C251—21no—
21M32×C251C2211yescharacteristic
22C2×C16×C251C2211yescharacteristic
23#108×C251C2211yescharacteristic
24#100×C251—21no—
25#41×C251—22no—
26#41×C251—21no—
27#41×C251—42no—
28#41×C251—41no—
29#41×C251D411yescharacteristic
30C2×Q8×C251—22no—
31C2×Q8×C251—21no—
32C2×Q8×C251—41no—
33C2×D4×C251—21no—
34Q16×C251—22no—
35Q16×C251—21no—
36Q16×C251D411yescharacteristic
37Q16×C251D412yes—
38SD16×C251—22no—
39SD16×C251—21no—
40SD16×C251—41no—
41D8×C251—21no—
42D8×C251—21no—
43D8×C251—41no—
44D8×C251D411yescharacteristic
45M16×C251—21no—
46M16×C251—21no—
47M16×C251D411yescharacteristic
48C2×C8×C251—21no—
49C2×C8×C251C2311yescharacteristic
50C16×C251—21no—
51C16×C251—21no—
52C23×C251—41no—
53Q8×C251—22no—
54Q8×C251—21no—
55Q8×C251—42no—
56Q8×C251—41no—
57Q8×C251D811yescharacteristic
58D4×C251—21no—
59D4×C251—21no—
60D4×C251—42no—
61D4×C251—41no—
62D4×C251—41no—
63D4×C251—41no—
64D4×C251D811yescharacteristic
65C2×C4×C251—21no—
66C2×C4×C251—42no—
67C2×C4×C251—42no—
68C2×C4×C251—41no—
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—41no—
77C22×C251—81no—
78C22×C251—161no—
79C22×C251AGL(1,ℤ/8)11yescharacteristic
80C4×C251—21no—
81C4×C251—42no—
82C4×C251—41no—
83C4×C251#12011yescharacteristic
84C4×C251C2×D811yescharacteristic
85C2×C251—21no—
86C2×C251—41no—
87C2×C251—81no—
88C2×C251—161no—
89C2×C251#44711yescharacteristic
90C251#219311yescharacteristic, direct product factor, Sylow
91#2193C25111yesmaximal, characteristic, direct product factor, Sylow
92#578C2×C25111yescharacteristic, complemented
93#509C2×C25111yescharacteristic, complemented
94C2×SD32C2×C25111yescharacteristic, complemented
95#472C2×C25111yescharacteristic, complemented
96#362C2×C25111yescharacteristic, complemented
97#359C2×C25111yescharacteristic, complemented
98#350C2×C25111yescharacteristic, complemented
992−1+4—21no—
100#137—22no—
101#137—21no—
102AGL(1,ℤ/8)—21no—
103#135—21no—
104#135C22×C25111yescharacteristic
105C2×Q16—21no—
106C2×Q16C22×C25111yescharacteristic
107C2×D8C22×C25111yescharacteristic
108#131C22×C25111yescharacteristic
109SD32—22no—
110SD32—21no—
111D16—21no—
112M32C22×C25111yescharacteristic
113C2×C16C22×C25111yescharacteristic
114#108C22×C25111yescharacteristic
115#100—21no—
116#41—22no—
117#41—21no—
118#41—42no—
119#41—41no—
120#41D4×C25111yescharacteristic
121C2×Q8—22no—
122C2×Q8—21no—
123C2×Q8—41no—
124C2×D4—21no—
125Q16—22no—
126Q16—21no—
127Q16D4×C25111yescharacteristic
128Q16D4×C25112yes—
129SD16—22no—
130SD16—21no—
131SD16—41no—
132D8—21no—
133D8—21no—
134D8—41no—
135D8D4×C25111yescharacteristic
136M16—21no—
137M16—21no—
138M16D4×C25111yescharacteristic
139C2×C8—21no—
140C2×C8C23×C25111yescharacteristic
141C16—21no—
142C16—21no—
143C23—41no—
144Q8—22no—
145Q8—21no—
146Q8—42no—
147Q8—41no—
148Q8D8×C25111yescharacteristic
149D4—21no—
150D4—21no—
151D4—42no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4D8×C25111yescharacteristic
156C2×C4—21no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4C2×D4×C25111yescharacteristic
162C8—21no—
163C8—41no—
164C8C2×D4×C25111yescharacteristic
165C8C2×D4×C25111yescharacteristic
166C22—21no—
167C22—41no—
168C22—81no—
169C22—161no—
170C22AGL(1,ℤ/8)×C25111yescharacteristic
171C4—21no—
172C4—42no—
173C4—41no—
174C4#120×C25111yescharacteristic
175C4C2×D8×C25111yescharacteristic
176C2—21no—
177C2—41no—
178C2—81no—
179C2—161no—
180C2#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. #578×C251C2semidirect
2. #509×C251C2semidirect
3. C2×SD32×C251C2semidirect
4. #472×C251C2semidirect
5. #362×C251C2semidirect
6. #359×C251C2semidirect
7. #350×C251C2semidirect
90. C251#2193direct (N × H)
91. #2193C251direct (N × H)
92. #578C2×C251semidirect
93. #509C2×C251semidirect
94. C2×SD32C2×C251semidirect
95. #472C2×C251semidirect
96. #362C2×C251semidirect
97. #359C2×C251semidirect
98. #350C2×C251semidirect

Maximal subgroups

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

subgroupindex
#578×C2512
#509×C2512
C2×SD32×C2512
#472×C2512
#362×C2512
#359×C2512
#350×C2512
#2193251