Finite Group Databasebeta

← #45666271

Subgroups — #45666271

Order 32896.

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

normal subgroups only

This group has 216 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#30657177C212yesmaximal, complemented
2#30656995C211yesmaximal, characteristic, complemented
3#30656547C212yesmaximal, complemented
4#30656519C211yesmaximal, characteristic, complemented
5#575×C257C211yesmaximal, characteristic
6C2×Dic1028C2211yescharacteristic, complemented
7#22551571—21no—
8#22551538—22no—
9#22551536—22no—
10#22551532C2212yescomplemented
11#22551524C2211yescharacteristic, complemented
12#22551517—21no—
13C2×#41×C257—21no—
14#137×C257—22no—
15AGL(1,ℤ/8)×C257—22no—
16#135×C257—21no—
17#135×C257C2212yes—
18C2×M16×C257C2211yescharacteristic
19C2×Dic514—21no—
20Dic1028—22no—
21C4⋊Dic257—21no—
22#16861597—41no—
23C22⋊Dic257—41no—
24#16861593—22no—
25#41×C257—22no—
26#41×C257—22no—
27#41×C257—41no—
28#41×C257—41no—
29C2×Q8×C257—21no—
30C2×D4×C257—21no—
31C2×D4×C257—41no—
32C22×C4×C257—41no—
33C22×C4×C257D411yescharacteristic
34Q16×C257—21no—
35Q16×C257D412yes—
36SD16×C257—22no—
37SD16×C257—22no—
38D8×C257—21no—
39D8×C257D412yes—
40M16×C257—22no—
41C2×C8×C257D411yescharacteristic
42C2×C8×C257C2311yescharacteristic
43Dic514—41no—
44Dic514—41no—
45C2×Dic257—41no—
46C2×Dic257—41no—
47C23×C257—41no—
48C23×C257D811yescharacteristic
49Q8×C257—22no—
50Q8×C257—41no—
51D4×C257—22no—
52D4×C257—41no—
53D4×C257—42no—
54D4×C257—42no—
55C2×C4×C257—21no—
56C2×C4×C257—21no—
57C2×C4×C257—42no—
58C2×C4×C257—41no—
59C2×C4×C257—42no—
60C2×C4×C257D811yescharacteristic
61C2×C4×C257C2×D411yescharacteristic
62C8×C257—21no—
63C8×C257C2×D412yes—
64C22×C257—21no—
65C22×C257—41no—
66C22×C257—42no—
67C22×C257—82no—
68C22×C257C2×D811yescharacteristic
69C4×C257—21no—
70C4×C257—42no—
71C4×C257#12011yescharacteristic
72C4×C257AGL(1,ℤ/8)11yescharacteristic
73Dic257—81no—
74Dic257—81no—
75C2×C257—21no—
76C2×C257—41no—
77C2×C257—82no—
78C2×C257#44711yescharacteristic
79C257#219111yescharacteristic, complemented, Sylow
80#2191—2571nomaximal, Sylow
81#575D25711yescharacteristic
82#510—2572no—
83#470—2571no—
84#360—2572no—
85#349—2571no—
86C2×#41—21no—
87#137—22no—
88AGL(1,ℤ/8)—22no—
89#135—21no—
90#135C2×D25712yes—
91C2×Q16—2571no—
92C2×M16C2×D25711yescharacteristic
93C22⋊Q8—5141no—
94Q32—5142no—
95SD32—5142no—
96M32—2572no—
97#106—2571no—
98#103—5141no—
99#41—22no—
100#41—22no—
101#41—41no—
102#41—41no—
103C2×Q8—21no—
104C2×Q8—5141no—
105C2×D4—21no—
106C2×D4—41no—
107C22×C4—41no—
108C22×C4C22⋊D25711yescharacteristic
109Q16—21no—
110Q16—5142no—
111Q16C22⋊D25712yes—
112SD16—22no—
113SD16—22no—
114D8—21no—
115D8C22⋊D25712yes—
116M16—22no—
117C2×C8C22⋊D25711yescharacteristic
118C2×C8C22×D25711yescharacteristic
119C4⋊C4—5141no—
120C4⋊C4—10281no—
121C22⋊C4—10281no—
122C16—5142no—
123C23—41no—
124C23#1686160311yescharacteristic
125Q8—22no—
126Q8—41no—
127Q8—10281no—
128Q8—10281no—
129D4—22no—
130D4—41no—
131D4—42no—
132D4—42no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—10281no—
139C2×C4—10281no—
140C2×C4#1686160311yescharacteristic
141C2×C4C2×C22⋊D25711yescharacteristic
142C8—21no—
143C8C2×C22⋊D25712yes—
144C22—21no—
145C22—41no—
146C22—42no—
147C22—82no—
148C22C2×#1686160311yescharacteristic
149C4—21no—
150C4—42no—
151C4—20561no—
152C4—20561no—
153C4#2255156111yescharacteristic
154C4#2255161811yescharacteristic
155C2—21no—
156C2—41no—
157C2—82no—
158C2#3065687111yescharacteristic
subgroupG/H
centerC2#30656871
commutator, cosocleC2×C8×C257C23
FrattiniC2×C8C22×D257
Fitting, prime radical#575×C257C2
socleC2×C257#447
hypercenter#575D257
hypocenterC257#2191
LCSG ▷ C2×C8×C257 ▷ C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #30657177C2semidirect
2. #30656995C2semidirect
3. #30656547C2semidirect
4. #30656519C2semidirect
6. C2×Dic1028C22semidirect
10. #22551532C22semidirect
11. #22551524C22semidirect
79. C257#2191semidirect

Maximal subgroups

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

subgroupindex
#306571772
#306569952
#306565472
#306565192
#575×C2572
#2191257