Finite Group Databasebeta

← #31780042

Subgroups — #31780042

Order 17536.

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

normal subgroups only

This group has 200 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#22578127C211yesmaximal, characteristic
2#22577685C211yesmaximal, characteristic, complemented
3#22577605C211yesmaximal, characteristic
4#22577149C211yesmaximal, characteristic, complemented
5#22577135C211yesmaximal, characteristic, complemented
6#22577132C211yesmaximal, characteristic
7#500×C137C211yesmaximal, characteristic, complemented
8C2×#14555497—21no—
9#17984083—22no—
10#17984077—21no—
11#17984068—21no—
12Q8×Dic137—21no—
13C4×Dic274C2211yescharacteristic
14C4×D548C2211yescharacteristic
15#17984046—21no—
16#17984019—21no—
17C4⋊#11764040C2211yescharacteristic
18C4⋊#11764040C2211yescharacteristic
19#17984012—21no—
20#17984011—21no—
21C4⋊Q8×C137C2211yescharacteristic
22C4⋊Q8×C137C2211yescharacteristic
23#107×C137—22no—
24C4×C8×C137C2211yescharacteristic
25C2×Dic274D411yescharacteristic
26C2×D548D411yescharacteristic
27C2×C4×D137—21no—
28#14555497—42no—
29C2×#11764040—21no—
30C2×#11764040—21no—
31C4⋊Dic137—22no—
32C4⋊Dic137D411yescharacteristic
33C4⋊Dic137D411yescharacteristic
34#14555489—21no—
35#14555487—22no—
36#14555487—22no—
37#14555487—21no—
38C4×Dic137—22no—
39C4×Dic137—21no—
40C2×Q8×C137—21no—
41C2×Q8×C137—21no—
42C2×C8×C137—21no—
43C4⋊C4×C137—21no—
44C4⋊C4×C137—21no—
45C4⋊C4×C137—21no—
46C4⋊C4×C137D412yes—
47C42×C137C2311yescharacteristic
48C22×D137—21no—
49Dic274—21no—
50Dic274—21no—
51D548—21no—
52D548—21no—
53C2×Dic137—21no—
54C2×Dic137—21no—
55C2×Dic137—21no—
56C4×D137—41no—
57C4×D137—41no—
58#11764040—41no—
59#11764040—41no—
60Q8×C137—41no—
61Q8×C137—41no—
62Q8×C137—41no—
63Q8×C137—41no—
64C2×C4×C137—22no—
65C2×C4×C137—41no—
66C2×C4×C137C2×D411yescharacteristic
67C2×C4×C137C2×D411yescharacteristic
68C2×C4×C137C2×D411yescharacteristic
69C8×C137—22no—
70C22×C137#12011yescharacteristic
71C2×D137—41no—
72C2×D137—41no—
73C2×D137—41no—
74C4×C137—21no—
75C4×C137—42no—
76C4×C137—81no—
77C4×C137#13511yescharacteristic
78C4×C137#13511yescharacteristic
79C4×C137AGL(1,ℤ/8)11yescharacteristic
80C4×C137#13711yescharacteristic
81Dic137—41no—
82Dic137—41no—
83Dic137—41no—
84C2×C137#44911yescharacteristic
85C2×C137#45211yescharacteristic
86C2×C137#45611yescharacteristic
87D137—81no—
88C137#165011yescharacteristic, complemented, Sylow
89#1650—1371nomaximal, Sylow
90#556—1371no—
91#500D13711yescharacteristic, complemented
92#475—1371no—
93#461—1371no—
94#332—1371no—
95Q8⋊C8—1371no—
96D4⋊C8—1371no—
97C2×SD16—2741no—
98C4⋊Q8C2×D13711yescharacteristic
99C4⋊Q8C2×D13711yescharacteristic
100#126—2742no—
101#124—2741no—
102C22⋊Q8—2741no—
103C4×Q8—2741no—
104C4×Q8—1371no—
105C4×D4—1371no—
106#117—2741no—
107#107—22no—
108#106—2741no—
109C4⋊C8—1371no—
110C4⋊C8—1371no—
111#103—2741no—
112#103—2741no—
113C4×C8C2×D13711yescharacteristic
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—1371no—
117C2×D4—1371no—
118C22×C4—2741no—
119SD16—5482no—
120C2×C8—21no—
121C2×C8—2741no—
122C2×C8—2741no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—2742no—
127C4⋊C4—2741no—
128C4⋊C4—1371no—
129C4⋊C4—1371no—
130C4⋊C4C22⋊D13712yes—
131C22⋊C4—2742no—
132C22⋊C4—2742no—
133C22⋊C4—2741no—
134C42—2742no—
135C42—2741no—
136C42C22×D13711yescharacteristic
137C23—2741no—
138Q8—41no—
139Q8—41no—
140Q8—41no—
141Q8—41no—
142Q8—2741no—
143Q8—2741no—
144D4—2741no—
145D4—2741no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—2741no—
149C2×C4—2741no—
150C2×C4—2741no—
151C2×C4—5481no—
152C2×C4—5481no—
153C2×C4D4×D13711yescharacteristic
154C2×C4D4×D13711yescharacteristic
155C2×C4C2×C22⋊D13711yescharacteristic
156C8—22no—
157C8—5481no—
158C8—5481no—
159C22—5481no—
160C22—5481no—
161C22—5481no—
162C22#1798405611yescharacteristic
163C4—21no—
164C4—42no—
165C4—81no—
166C4—5481no—
167C4—5481no—
168C4—5481no—
169C4#1798410711yescharacteristic
170C4#1798411111yescharacteristic
171C4#1798411211yescharacteristic
172C4#1798412011yescharacteristic
173C2—10961no—
174C2#2257754511yescharacteristic
175C2#2257756411yescharacteristic
176C2#2257758511yescharacteristic
subgroupG/H
centerC22#17984056
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#500×C137C2
socleC22×C137#120
prime radicalC2×D548D4
hypercenter#500D137
hypocenterC137#1650
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. #22577685C2semidirect
4. #22577149C2semidirect
5. #22577135C2semidirect
7. #500×C137C2semidirect
88. C137#1650semidirect
91. #500D137semidirect

Maximal subgroups

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

subgroupindex
#225781272
#225776852
#225776052
#225771492
#225771352
#225771322
#500×C1372
#1650137