Finite Group Databasebeta

← #1605×C251 #44546192

Subgroups — #1605×C251 #44546192

Order 32128.

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

normal subgroups only

This group has 238 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#544×C251C211yesmaximal, characteristic
2#451×C251C212yesmaximal
3#395×C251C211yesmaximal, characteristic
4#328×C251C212yesmaximal
5#324×C251C211yesmaximal, characteristic
6C22×Q8×C251C2211yescharacteristic
7C2×Q16×C251—22no—
8C4⋊Q8×C251—22no—
9#125×C251—22no—
10C22⋊Q8×C251—21no—
11C22⋊Q8×C251C2212yes—
12#117×C251—21no—
13C2×C4⋊C4×C251—21no—
14C2×C4⋊C4×C251C2211yescharacteristic
15#103×C251—22no—
16C22⋊C8×C251C2212yes—
17#95×C251—21no—
18#95×C251C2211yescharacteristic
19C2×Q8×C251—21no—
20C2×Q8×C251—22no—
21C2×Q8×C251—42no—
22C2×Q8×C251D412yes—
23C22×C4×C251—21no—
24C22×C4×C251—21no—
25C22×C4×C251D411yescharacteristic
26C22×C4×C251D411yescharacteristic
27C22×C4×C251C2311yescharacteristic
28Q16×C251—44no—
29C2×C8×C251—22no—
30C4⋊C4×C251—21no—
31C4⋊C4×C251—21no—
32C4⋊C4×C251—21no—
33C4⋊C4×C251—22no—
34C4⋊C4×C251—22no—
35C4⋊C4×C251—22no—
36C4⋊C4×C251—42no—
37C4⋊C4×C251D412yes—
38C22⋊C4×C251—22no—
39C42×C251—22no—
40C23×C251C2×D411yescharacteristic
41Q8×C251—24no—
42Q8×C251—42no—
43Q8×C251—42no—
44Q8×C251—81no—
45C2×C4×C251—21no—
46C2×C4×C251—21no—
47C2×C4×C251—22no—
48C2×C4×C251—22no—
49C2×C4×C251—22no—
50C2×C4×C251—22no—
51C2×C4×C251—21no—
52C2×C4×C251—41no—
53C2×C4×C251—41no—
54C2×C4×C251—41no—
55C2×C4×C251—41no—
56C2×C4×C251—41no—
57C2×C4×C251—41no—
58C2×C4×C251Q1612yescomplemented
59C2×C4×C251C2×D412yes—
60C8×C251—42no—
61C22×C251—21no—
62C22×C251—21no—
63C22×C251#12011yescharacteristic
64C22×C251C2×Q1611yescharacteristic
65C22×C251#13711yescharacteristic
66C4×C251—22no—
67C4×C251—22no—
68C4×C251—42no—
69C4×C251—42no—
70C4×C251—42no—
71C4×C251—41no—
72C4×C251—42no—
73C2×C251—21no—
74C2×C251—21no—
75C2×C251#45111yescharacteristic
76C2×C251#45611yescharacteristic
77C2×C251B(4,𝔽2)11yescharacteristic
78C251#160511yescharacteristic, direct product factor, Sylow
79#1605C25111yesmaximal, characteristic, direct product factor, Sylow
80#544C2×C25111yescharacteristic
81#451C2×C25112yes—
82#395C2×C25111yescharacteristic
83#328C2×C25112yes—
84#324C2×C25111yescharacteristic
85C22×Q8C22×C25111yescharacteristic
86C2×Q16—22no—
87C4⋊Q8—22no—
88#125—22no—
89C22⋊Q8—21no—
90C22⋊Q8C22×C25112yes—
91#117—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4C22×C25111yescharacteristic
94#103—22no—
95C22⋊C8C22×C25112yes—
96#95—21no—
97#95C22×C25111yescharacteristic
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—42no—
101C2×Q8D4×C25112yes—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4D4×C25111yescharacteristic
105C22×C4D4×C25111yescharacteristic
106C22×C4C23×C25111yescharacteristic
107Q16—44no—
108C2×C8—22no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—42no—
116C4⋊C4D4×C25112yes—
117C22⋊C4—22no—
118C42—22no—
119C23C2×D4×C25111yescharacteristic
120Q8—24no—
121Q8—42no—
122Q8—42no—
123Q8—81no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4Q16×C25112yescomplemented
138C2×C4C2×D4×C25112yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#120×C25111yescharacteristic
143C22C2×Q16×C25111yescharacteristic
144C22#137×C25111yescharacteristic
145C4—22no—
146C4—22no—
147C4—42no—
148C4—42no—
149C4—42no—
150C4—41no—
151C4—42no—
152C2—21no—
153C2—21no—
154C2#451×C25111yescharacteristic
155C2#456×C25111yescharacteristic
156C2B(4,𝔽2)×C25111yescharacteristic
subgroupG/H
center, socleC22×C251#120
commutator, Frattini, cosocleC22×C4C23×C251
prime radicalC23×C251C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C251 ◁ C22×C4×C251 ◁ G

Complements

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

NH ≅ G/Ntype
58. C2×C4×C251Q16semidirect
78. C251#1605direct (N × H)
79. #1605C251direct (N × H)
137. C2×C4Q16×C251semidirect

Maximal subgroups

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

subgroupindex
#544×C2512
#451×C2512
#395×C2512
#328×C2512
#324×C2512
#1605251