Finite Group Databasebeta

← #44486499

Subgroups — #44486499

Order 30592.

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#29505212C212yesmaximal, complemented
2#29505030C211yesmaximal, characteristic, complemented
3#29504582C212yesmaximal, complemented
4#29504554C211yesmaximal, characteristic, complemented
5#575×C239C211yesmaximal, characteristic
6C2×Dic956C2211yescharacteristic, complemented
7#21439939—21no—
8#21439906—22no—
9#21439904—22no—
10#21439900C2212yescomplemented
11#21439892C2211yescharacteristic, complemented
12#21439885—21no—
13C2×#41×C239—21no—
14#137×C239—22no—
15AGL(1,ℤ/8)×C239—22no—
16#135×C239—21no—
17#135×C239C2212yes—
18C2×M16×C239C2211yescharacteristic
19C2×Dic478—21no—
20Dic956—22no—
21C4⋊Dic239—21no—
22Dic239⋊C4—41no—
23C22⋊Dic239—41no—
24C239⋊C16—22no—
25#41×C239—22no—
26#41×C239—22no—
27#41×C239—41no—
28#41×C239—41no—
29C2×Q8×C239—21no—
30C2×D4×C239—21no—
31C2×D4×C239—41no—
32C22×C4×C239—41no—
33C22×C4×C239D411yescharacteristic
34Q16×C239—21no—
35Q16×C239D412yes—
36SD16×C239—22no—
37SD16×C239—22no—
38D8×C239—21no—
39D8×C239D412yes—
40M16×C239—22no—
41C2×C8×C239D411yescharacteristic
42C2×C8×C239C2311yescharacteristic
43Dic478—41no—
44Dic478—41no—
45C2×Dic239—41no—
46C2×Dic239—41no—
47C23×C239—41no—
48C23×C239D811yescharacteristic
49Q8×C239—22no—
50Q8×C239—41no—
51D4×C239—22no—
52D4×C239—41no—
53D4×C239—42no—
54D4×C239—42no—
55C2×C4×C239—21no—
56C2×C4×C239—21no—
57C2×C4×C239—42no—
58C2×C4×C239—41no—
59C2×C4×C239—42no—
60C2×C4×C239D811yescharacteristic
61C2×C4×C239C2×D411yescharacteristic
62C8×C239—21no—
63C8×C239C2×D412yes—
64C22×C239—21no—
65C22×C239—41no—
66C22×C239—42no—
67C22×C239—82no—
68C22×C239C2×D811yescharacteristic
69C4×C239—21no—
70C4×C239—42no—
71C4×C239#12011yescharacteristic
72C4×C239AGL(1,ℤ/8)11yescharacteristic
73Dic239—81no—
74Dic239—81no—
75C2×C239—21no—
76C2×C239—41no—
77C2×C239—82no—
78C2×C239#44711yescharacteristic
79C239#219111yescharacteristic, complemented, Sylow
80#2191—2391nomaximal, Sylow
81#575D23911yescharacteristic
82#510—2392no—
83#470—2391no—
84#360—2392no—
85#349—2391no—
86C2×#41—21no—
87#137—22no—
88AGL(1,ℤ/8)—22no—
89#135—21no—
90#135C2×D23912yes—
91C2×Q16—2391no—
92C2×M16C2×D23911yescharacteristic
93C22⋊Q8—4781no—
94Q32—4782no—
95SD32—4782no—
96M32—2392no—
97#106—2391no—
98#103—4781no—
99#41—22no—
100#41—22no—
101#41—41no—
102#41—41no—
103C2×Q8—21no—
104C2×Q8—4781no—
105C2×D4—21no—
106C2×D4—41no—
107C22×C4—41no—
108C22×C4C22⋊D23911yescharacteristic
109Q16—21no—
110Q16—4782no—
111Q16C22⋊D23912yes—
112SD16—22no—
113SD16—22no—
114D8—21no—
115D8C22⋊D23912yes—
116M16—22no—
117C2×C8C22⋊D23911yescharacteristic
118C2×C8C22×D23911yescharacteristic
119C4⋊C4—9561no—
120C4⋊C4—4781no—
121C22⋊C4—9561no—
122C16—4782no—
123C23—41no—
124C23#1683853311yescharacteristic
125Q8—22no—
126Q8—41no—
127Q8—9561no—
128Q8—9561no—
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—9561no—
139C2×C4—9561no—
140C2×C4#1683853311yescharacteristic
141C2×C4C2×C22⋊D23911yescharacteristic
142C8—21no—
143C8C2×C22⋊D23912yes—
144C22—21no—
145C22—41no—
146C22—42no—
147C22—82no—
148C22C2×#1683853311yescharacteristic
149C4—21no—
150C4—42no—
151C4—19121no—
152C4—19121no—
153C4#2143992911yescharacteristic
154C4#2143998611yescharacteristic
155C2—21no—
156C2—41no—
157C2—82no—
158C2#2950490611yescharacteristic
subgroupG/H
centerC2#29504906
commutator, cosocleC2×C8×C239C23
FrattiniC2×C8C22×D239
Fitting, prime radical#575×C239C2
socleC2×C239#447
hypercenter#575D239
hypocenterC239#2191
LCSG ▷ C2×C8×C239 ▷ C4×C239 ▷ C2×C239 ▷ C239
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #29505212C2semidirect
2. #29505030C2semidirect
3. #29504582C2semidirect
4. #29504554C2semidirect
6. C2×Dic956C22semidirect
10. #21439900C22semidirect
11. #21439892C22semidirect
79. C239#2191semidirect

Maximal subgroups

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

subgroupindex
#295052122
#295050302
#295045822
#295045542
#575×C2392
#2191239