Finite Group Databasebeta

← #44401685

Subgroups — #44401685

Order 29056.

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#28409472C212yesmaximal, complemented
2#28409290C211yesmaximal, characteristic, complemented
3#28408842C212yesmaximal, complemented
4#28408814C211yesmaximal, characteristic, complemented
5#575×C227C211yesmaximal, characteristic
6C2×Dic908C2211yescharacteristic, complemented
7#20300267—21no—
8#20300234—22no—
9#20300232—22no—
10#20300228C2212yescomplemented
11#20300220C2211yescharacteristic, complemented
12#20300213—21no—
13C2×#41×C227—21no—
14#137×C227—22no—
15AGL(1,ℤ/8)×C227—22no—
16#135×C227—21no—
17#135×C227C2212yes—
18C2×M16×C227C2211yescharacteristic
19C2×Dic454—21no—
20Dic908—22no—
21C4⋊Dic227—21no—
22Dic227⋊C4—41no—
23C22⋊Dic227—41no—
24C227⋊C16—22no—
25#41×C227—22no—
26#41×C227—22no—
27#41×C227—41no—
28#41×C227—41no—
29C2×Q8×C227—21no—
30C2×D4×C227—21no—
31C2×D4×C227—41no—
32C22×C4×C227—41no—
33C22×C4×C227D411yescharacteristic
34Q16×C227—21no—
35Q16×C227D412yes—
36SD16×C227—22no—
37SD16×C227—22no—
38D8×C227—21no—
39D8×C227D412yes—
40M16×C227—22no—
41C2×C8×C227D411yescharacteristic
42C2×C8×C227C2311yescharacteristic
43Dic454—41no—
44Dic454—41no—
45C2×Dic227—41no—
46C2×Dic227—41no—
47C23×C227—41no—
48C23×C227D811yescharacteristic
49Q8×C227—22no—
50Q8×C227—41no—
51D4×C227—22no—
52D4×C227—41no—
53D4×C227—42no—
54D4×C227—42no—
55C2×C4×C227—21no—
56C2×C4×C227—21no—
57C2×C4×C227—42no—
58C2×C4×C227—41no—
59C2×C4×C227—42no—
60C2×C4×C227D811yescharacteristic
61C2×C4×C227C2×D411yescharacteristic
62C8×C227—21no—
63C8×C227C2×D412yes—
64C22×C227—21no—
65C22×C227—41no—
66C22×C227—42no—
67C22×C227—82no—
68C22×C227C2×D811yescharacteristic
69C4×C227—21no—
70C4×C227—42no—
71C4×C227#12011yescharacteristic
72C4×C227AGL(1,ℤ/8)11yescharacteristic
73Dic227—81no—
74Dic227—81no—
75C2×C227—21no—
76C2×C227—41no—
77C2×C227—82no—
78C2×C227#44711yescharacteristic
79C227#219111yescharacteristic, complemented, Sylow
80#2191—2271nomaximal, Sylow
81#575D22711yescharacteristic
82#510—2272no—
83#470—2271no—
84#360—2272no—
85#349—2271no—
86C2×#41—21no—
87#137—22no—
88AGL(1,ℤ/8)—22no—
89#135—21no—
90#135C2×D22712yes—
91C2×Q16—2271no—
92C2×M16C2×D22711yescharacteristic
93C22⋊Q8—4541no—
94Q32—4542no—
95SD32—4542no—
96M32—2272no—
97#106—2271no—
98#103—4541no—
99#41—22no—
100#41—22no—
101#41—41no—
102#41—41no—
103C2×Q8—21no—
104C2×Q8—4541no—
105C2×D4—21no—
106C2×D4—41no—
107C22×C4—41no—
108C22×C4C22⋊D22711yescharacteristic
109Q16—21no—
110Q16—4542no—
111Q16C22⋊D22712yes—
112SD16—22no—
113SD16—22no—
114D8—21no—
115D8C22⋊D22712yes—
116M16—22no—
117C2×C8C22⋊D22711yescharacteristic
118C2×C8C22×D22711yescharacteristic
119C4⋊C4—9081no—
120C4⋊C4—4541no—
121C22⋊C4—9081no—
122C16—4542no—
123C23—41no—
124C23#1681386911yescharacteristic
125Q8—22no—
126Q8—41no—
127Q8—9081no—
128Q8—9081no—
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—9081no—
139C2×C4—9081no—
140C2×C4#1681386911yescharacteristic
141C2×C4C2×C22⋊D22711yescharacteristic
142C8—21no—
143C8C2×C22⋊D22712yes—
144C22—21no—
145C22—41no—
146C22—42no—
147C22—82no—
148C22C2×#1681386911yescharacteristic
149C4—21no—
150C4—42no—
151C4—18161no—
152C4—18161no—
153C4#2030025711yescharacteristic
154C4#2030031411yescharacteristic
155C2—21no—
156C2—41no—
157C2—82no—
158C2#2840916611yescharacteristic
subgroupG/H
centerC2#28409166
commutator, cosocleC2×C8×C227C23
FrattiniC2×C8C22×D227
Fitting, prime radical#575×C227C2
socleC2×C227#447
hypercenter#575D227
hypocenterC227#2191
LCSG ▷ C2×C8×C227 ▷ C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #28409472C2semidirect
2. #28409290C2semidirect
3. #28408842C2semidirect
4. #28408814C2semidirect
6. C2×Dic908C22semidirect
10. #20300228C22semidirect
11. #20300220C22semidirect
79. C227#2191semidirect

Maximal subgroups

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

subgroupindex
#284094722
#284092902
#284088422
#284088142
#575×C2272
#2191227