Finite Group Databasebeta

← #31782740

Subgroups — #31782740

Order 17536.

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#22577843C212yesmaximal, complemented
2#22577661C211yesmaximal, characteristic, complemented
3#22577213C212yesmaximal, complemented
4#22577185C211yesmaximal, characteristic, complemented
5#575×C137C211yesmaximal, characteristic
6C2×Dic548C2211yescharacteristic, complemented
7#17984067—21no—
8#17984034—22no—
9#17984032—22no—
10#17984028C2212yescomplemented
11#17984020C2211yescharacteristic, complemented
12#17984013—21no—
13C2×#41×C137—21no—
14#137×C137—22no—
15AGL(1,ℤ/8)×C137—22no—
16#135×C137—21no—
17#135×C137C2212yes—
18C2×M16×C137C2211yescharacteristic
19C2×Dic274—21no—
20Dic548—22no—
21C4⋊Dic137—21no—
22#14555489—41no—
23C22⋊Dic137—41no—
24#14555485—22no—
25#41×C137—22no—
26#41×C137—22no—
27#41×C137—41no—
28#41×C137—41no—
29C2×Q8×C137—21no—
30C2×D4×C137—21no—
31C2×D4×C137—41no—
32C22×C4×C137—41no—
33C22×C4×C137D411yescharacteristic
34Q16×C137—21no—
35Q16×C137D412yes—
36SD16×C137—22no—
37SD16×C137—22no—
38D8×C137—21no—
39D8×C137D412yes—
40M16×C137—22no—
41C2×C8×C137D411yescharacteristic
42C2×C8×C137C2311yescharacteristic
43Dic274—41no—
44Dic274—41no—
45C2×Dic137—41no—
46C2×Dic137—41no—
47C23×C137—41no—
48C23×C137D811yescharacteristic
49Q8×C137—22no—
50Q8×C137—41no—
51D4×C137—22no—
52D4×C137—41no—
53D4×C137—42no—
54D4×C137—42no—
55C2×C4×C137—21no—
56C2×C4×C137—21no—
57C2×C4×C137—42no—
58C2×C4×C137—41no—
59C2×C4×C137—42no—
60C2×C4×C137D811yescharacteristic
61C2×C4×C137C2×D411yescharacteristic
62C8×C137—21no—
63C8×C137C2×D412yes—
64C22×C137—21no—
65C22×C137—41no—
66C22×C137—42no—
67C22×C137—82no—
68C22×C137C2×D811yescharacteristic
69C4×C137—21no—
70C4×C137—42no—
71C4×C137#12011yescharacteristic
72C4×C137AGL(1,ℤ/8)11yescharacteristic
73Dic137—81no—
74Dic137—81no—
75C2×C137—21no—
76C2×C137—41no—
77C2×C137—82no—
78C2×C137#44711yescharacteristic
79C137#219111yescharacteristic, complemented, Sylow
80#2191—1371nomaximal, Sylow
81#575D13711yescharacteristic
82#510—1372no—
83#470—1371no—
84#360—1372no—
85#349—1371no—
86C2×#41—21no—
87#137—22no—
88AGL(1,ℤ/8)—22no—
89#135—21no—
90#135C2×D13712yes—
91C2×Q16—1371no—
92C2×M16C2×D13711yescharacteristic
93C22⋊Q8—2741no—
94Q32—2742no—
95SD32—2742no—
96M32—1372no—
97#106—1371no—
98#103—2741no—
99#41—22no—
100#41—22no—
101#41—41no—
102#41—41no—
103C2×Q8—21no—
104C2×Q8—2741no—
105C2×D4—21no—
106C2×D4—41no—
107C22×C4—41no—
108C22×C4C22⋊D13711yescharacteristic
109Q16—21no—
110Q16—2742no—
111Q16C22⋊D13712yes—
112SD16—22no—
113SD16—22no—
114D8—21no—
115D8C22⋊D13712yes—
116M16—22no—
117C2×C8C22⋊D13711yescharacteristic
118C2×C8C22×D13711yescharacteristic
119C4⋊C4—2741no—
120C4⋊C4—5481no—
121C22⋊C4—5481no—
122C16—2742no—
123C23—41no—
124C23#1455549511yescharacteristic
125Q8—22no—
126Q8—41no—
127Q8—5481no—
128Q8—5481no—
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—5481no—
139C2×C4—5481no—
140C2×C4#1455549511yescharacteristic
141C2×C4C2×C22⋊D13711yescharacteristic
142C8—21no—
143C8C2×C22⋊D13712yes—
144C22—21no—
145C22—41no—
146C22—42no—
147C22—82no—
148C22C2×#1455549511yescharacteristic
149C4—21no—
150C4—42no—
151C4—10961no—
152C4—10961no—
153C4#1798405711yescharacteristic
154C4#1798411411yescharacteristic
155C2—21no—
156C2—41no—
157C2—82no—
158C2#2257753711yescharacteristic
subgroupG/H
centerC2#22577537
commutator, cosocleC2×C8×C137C23
FrattiniC2×C8C22×D137
Fitting, prime radical#575×C137C2
socleC2×C137#447
hypercenter#575D137
hypocenterC137#2191
LCSG ▷ C2×C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #22577843C2semidirect
2. #22577661C2semidirect
3. #22577213C2semidirect
4. #22577185C2semidirect
6. C2×Dic548C22semidirect
10. #17984028C22semidirect
11. #17984020C22semidirect
79. C137#2191semidirect

Maximal subgroups

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

subgroupindex
#225778432
#225776612
#225772132
#225771852
#575×C1372
#2191137