Finite Group Databasebeta

← #31782742

Subgroups — #31782742

Order 17536.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#22578203C211yesmaximal, characteristic
2#22577846C212yesmaximal, complemented
3#22577661C211yesmaximal, characteristic, complemented
4#22577214C212yesmaximal, complemented
5#349×C137C211yesmaximal, characteristic, complemented
6C2×#14555510—21no—
7#17984121—22no—
8#17984116—22no—
9#17984110—21no—
10#17984110C2212yes—
11C2×Dic548C2211yescharacteristic, complemented
12#17984067—21no—
13Dic1096—22no—
14#17984033—22no—
15#17984020C2211yescharacteristic, complemented
16#17984013—21no—
17C2×M16×C137C2211yescharacteristic
18M32×C137C2212yescomplemented
19#14555510—22no—
20#14555510—22no—
21#14555510—41no—
22#14555510—41no—
23C2×Dic274—21no—
24C2×Dic274—21no—
25C2×D548—21no—
26C2×C22⋊D137—41no—
27C2×C4×D137—41no—
28Dic548—21no—
29Dic548—22no—
30Dic548D412yes—
31#14555499—22no—
32#14555499—22no—
33D1096—21no—
34D1096D412yes—
35C4⋊Dic137—21no—
36#14555489—41no—
37C22⋊Dic137—41no—
38C22×C4×C137D411yescharacteristic
39M16×C137—22no—
40C2×C8×C137D411yescharacteristic
41C2×C8×C137C2311yescharacteristic
42C16×C137—22no—
43C22×D137—41no—
44Dic274—22no—
45Dic274—41no—
46Dic274—41no—
47Dic274—41no—
48D548—22no—
49D548—41no—
50C22⋊D137—42no—
51C22⋊D137—42no—
52C2×Dic137—41no—
53C2×Dic137—41no—
54C2×Dic137—41no—
55C4×D137—42no—
56C4×D137—42no—
57C23×C137D811yescharacteristic
58C2×C4×C137—21no—
59C2×C4×C137—21no—
60C2×C4×C137D811yescharacteristic
61C2×C4×C137C2×D411yescharacteristic
62C8×C137—21no—
63C8×C137C2×D412yes—
64C22×C137—21no—
65C22×C137—41no—
66C22×C137C2×D811yescharacteristic
67C2×D137—42no—
68C2×D137—82no—
69C4×C137—21no—
70C4×C137#12011yescharacteristic
71C4×C137AGL(1,ℤ/8)11yescharacteristic
72Dic137—42no—
73Dic137—81no—
74Dic137—81no—
75C2×C137—21no—
76C2×C137—41no—
77C2×C137#44711yescharacteristic
78D137—82no—
79C137#219111yescharacteristic, complemented, Sylow
80#2191—1371nomaximal, Sylow
81#575—1371no—
82#510—1372no—
83#470—1371no—
84#360—1372no—
85#349D13711yescharacteristic, complemented
86C2×#41—2741no—
87#137—2742no—
88AGL(1,ℤ/8)—2742no—
89#135—2741no—
90#135—1372no—
91C2×Q16—1371no—
92C2×M16C2×D13711yescharacteristic
93C22⋊Q8—2741no—
94Q32—2742no—
95SD32—2742no—
96M32C2×D13712yescomplemented
97#106—1371no—
98#103—2741no—
99#41—2742no—
100#41—2742no—
101#41—5481no—
102#41—5481no—
103C2×Q8—2741no—
104C2×Q8—2741no—
105C2×D4—2741no—
106C2×D4—5481no—
107C22×C4—5481no—
108C22×C4D54811yescharacteristic
109Q16—2741no—
110Q16—2742no—
111Q16—1372no—
112SD16—2742no—
113SD16—2742no—
114D8—2741no—
115D8—1372no—
116M16—22no—
117C2×C8D54811yescharacteristic
118C2×C8C22×D13711yescharacteristic
119C4⋊C4—2741no—
120C4⋊C4—5481no—
121C22⋊C4—5481no—
122C16—22no—
123C23—5481no—
124C23D109611yescharacteristic
125Q8—2742no—
126Q8—5481no—
127Q8—5481no—
128Q8—5481no—
129D4—2742no—
130D4—5481no—
131D4—5482no—
132D4—5482no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—5482no—
136C2×C4—5481no—
137C2×C4—5482no—
138C2×C4—5481no—
139C2×C4—5481no—
140C2×C4D109611yescharacteristic
141C2×C4C2×D54811yescharacteristic
142C8—21no—
143C8D4×D13712yes—
144C22—21no—
145C22—41no—
146C22—5482no—
147C22—10962no—
148C22C2×D109611yescharacteristic
149C4—21no—
150C4—5482no—
151C4—10961no—
152C4—10961no—
153C4#1798405811yescharacteristic
154C4#1798411611yescharacteristic
155C2—21no—
156C2—41no—
157C2—10962no—
158C2#2257753911yescharacteristic
subgroupG/H
centerC2#22577539
commutator, cosocleC2×C8×C137C23
FrattiniC2×C8C22×D137
Fitting#349×C137C2
socleC2×C137#447
prime radical#22578203C2
hypercenter#349D137
hypocenterC137#2191
LCSG ▷ C2×C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
2. #22577846C2semidirect
3. #22577661C2semidirect
4. #22577214C2semidirect
5. #349×C137C2semidirect
11. C2×Dic548C22semidirect
15. #17984020C22semidirect
18. M32×C137C22semidirect
79. C137#2191semidirect
85. #349D137semidirect
96. M32C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225782032
#225778462
#225776612
#225772142
#349×C1372
#2191137