Finite Group Databasebeta

← #44549338

Subgroups — #44549338

Order 32128.

238 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 53 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#30632318C211yesmaximal, characteristic
2#30631843C212yesmaximal
3#30631423C212yesmaximal
4#30631413C211yesmaximal, characteristic
5#395×C251C211yesmaximal, characteristic
6C2×#16860800—22no—
7#22548714—22no—
8#22548706—22no—
9#22548694—21no—
10#22548694C2212yes—
11#22548674—21no—
12#22548639—22no—
13C22⋊(C251⋊C8)C2212yes—
14C22×Q8×C251C2211yescharacteristic
15C2×C4⋊C4×C251—21no—
16C2×C4⋊C4×C251C2211yescharacteristic
17#95×C251—21no—
18#95×C251C2211yescharacteristic
19C2×Dic502—21no—
20C2×Dic502D412yes—
21#16860800—44no—
22C2×C251⋊C8—22no—
23C4⋊Dic251—21no—
24C4⋊Dic251—21no—
25C4⋊Dic251—21no—
26C4⋊Dic251D412yes—
27Dic251⋊C4—22no—
28Dic251⋊C4—22no—
29C22⋊Dic251—22no—
30C4×Dic251—22no—
31C2×Q8×C251—22no—
32C2×Q8×C251—42no—
33C22×C4×C251—21no—
34C22×C4×C251—21no—
35C22×C4×C251D411yescharacteristic
36C22×C4×C251D411yescharacteristic
37C22×C4×C251C2311yescharacteristic
38C4⋊C4×C251—22no—
39C4⋊C4×C251—42no—
40Dic502—42no—
41Dic502—81no—
42C2×Dic251—22no—
43C2×Dic251—22no—
44C251⋊C8—42no—
45C23×C251C2×D411yescharacteristic
46Q8×C251—24no—
47Q8×C251—42no—
48C2×C4×C251—21no—
49C2×C4×C251—21no—
50C2×C4×C251—22no—
51C2×C4×C251—22no—
52C2×C4×C251—21no—
53C2×C4×C251—41no—
54C2×C4×C251—41no—
55C2×C4×C251—41no—
56C2×C4×C251—41no—
57C2×C4×C251—41no—
58C2×C4×C251—41no—
59C2×C4×C251Q1612yescomplemented
60C2×C4×C251C2×D412yes—
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—41no—
71Dic251—42no—
72Dic251—42no—
73C2×C251—21no—
74C2×C251—21no—
75C2×C251#45111yescharacteristic
76C2×C251#45611yescharacteristic
77C2×C251B(4,𝔽2)11yescharacteristic
78C251#160511yescharacteristic, complemented, Sylow
79#1605—2511nomaximal, Sylow
80#544—2511no—
81#451—2512no—
82#395D25111yescharacteristic
83#328—2512no—
84#324—2511no—
85C22×Q8C2×D25111yescharacteristic
86C2×Q16—5022no—
87C4⋊Q8—5022no—
88#125—5022no—
89C22⋊Q8—5021no—
90C22⋊Q8—2512no—
91#117—5021no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4C2×D25111yescharacteristic
94#103—5022no—
95C22⋊C8—2512no—
96#95—21no—
97#95C2×D25111yescharacteristic
98C2×Q8—22no—
99C2×Q8—42no—
100C2×Q8—5021no—
101C2×Q8—2512no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4C22⋊D25111yescharacteristic
105C22×C4C22⋊D25111yescharacteristic
106C22×C4C22×D25111yescharacteristic
107Q16—10044no—
108C2×C8—5022no—
109C4⋊C4—22no—
110C4⋊C4—42no—
111C4⋊C4—5021no—
112C4⋊C4—5021no—
113C4⋊C4—5022no—
114C4⋊C4—5021no—
115C4⋊C4—5022no—
116C4⋊C4—2512no—
117C22⋊C4—5022no—
118C42—5022no—
119C23C2×C22⋊D25111yescharacteristic
120Q8—24no—
121Q8—42no—
122Q8—20081no—
123Q8—10042no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—5022no—
136C2×C4—5022no—
137C2×C4#1686080012yescomplemented
138C2×C4D4×D25112yes—
139C8—10042no—
140C22—21no—
141C22—21no—
142C22#2254868311yescharacteristic
143C22C2×#1686080011yescharacteristic
144C22#2254874511yescharacteristic
145C4—22no—
146C4—22no—
147C4—42no—
148C4—42no—
149C4—41no—
150C4—10042no—
151C4—10042no—
152C2—21no—
153C2—21no—
154C2#3063184211yescharacteristic
155C2#3063186911yescharacteristic
156C2#3063187311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#395×C251C2
socleC22×C251#120
prime radicalC23×C251C2×D4
hypercenter#395D251
hypocenterC251#1605
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #395

Complements

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

NH ≅ G/Ntype
59. C2×C4×C251Q16semidirect
78. C251#1605semidirect
137. C2×C4#16860800semidirect

Maximal subgroups

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

subgroupindex
#306323182
#306318432
#306314232
#306314132
#395×C2512
#1605251