Finite Group Databasebeta

← #31779972

Subgroups — #31779972

Order 17536.

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

normal subgroups only

This group has 212 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#22578139C211yesmaximal, characteristic
2#22577685C211yesmaximal, characteristic, complemented
3#22577610C211yesmaximal, characteristic, complemented
4#22577147C211yesmaximal, characteristic
5#22577135C211yesmaximal, characteristic, complemented
6#22577135C211yesmaximal, characteristic, complemented
7#486×C137C211yesmaximal, characteristic
8C2×#14555500—21no—
9#17984075—21no—
10#17984075—21no—
11#17984069—22no—
12C4×Dic274C2211yescharacteristic
13C4×Dic274C2211yescharacteristic
14D4×Dic137—21no—
15D4×Dic137—21no—
16#17984019—21no—
17C4⋊#11764040C2211yescharacteristic
18C4⋊#11764040C2211yescharacteristic
19#17984012—21no—
20#17984011—21no—
21C4⋊Q8×C137C2211yescharacteristic
22#127×C137C2211yescharacteristic
23#102×C137—22no—
24C4×C8×C137C2211yescharacteristic
25C2×Dic274D411yescharacteristic
26C2×Dic274D411yescharacteristic
27C22×Dic137—22no—
28#14555500—42no—
29C2×#11764040—21no—
30C2×#11764040—21no—
31C4⋊Dic137D411yescharacteristic
32C4⋊Dic137D411yescharacteristic
33#14555489—21no—
34#14555489—21no—
35C22⋊Dic137—22no—
36C22⋊Dic137—22no—
37C22⋊Dic137—22no—
38C4×Dic137—21no—
39C4×Dic137—21no—
40C2×Q8×C137—21no—
41C2×D4×C137—21no—
42C2×D4×C137—21no—
43C2×D4×C137D412yes—
44C2×C8×C137—21no—
45C4⋊C4×C137—21no—
46C4⋊C4×C137—21no—
47C42×C137C2311yescharacteristic
48Dic274—21no—
49Dic274—21no—
50Dic274—21no—
51Dic274—21no—
52C2×Dic137—21no—
53C2×Dic137—21no—
54C2×Dic137—21no—
55C2×Dic137—21no—
56C2×Dic137—82no—
57#11764040—41no—
58#11764040—41no—
59C23×C137—22no—
60Q8×C137—41no—
61Q8×C137—41no—
62D4×C137—42no—
63D4×C137—41no—
64D4×C137—41no—
65D4×C137—41no—
66D4×C137—41no—
67C2×C4×C137—41no—
68C2×C4×C137C2×D411yescharacteristic
69C2×C4×C137C2×D411yescharacteristic
70C2×C4×C137C2×D411yescharacteristic
71C8×C137—41no—
72C22×C137—42no—
73C22×C137—42no—
74C22×C137—42no—
75C22×C137#12011yescharacteristic
76C4×C137—21no—
77C4×C137—81no—
78C4×C137#13511yescharacteristic
79C4×C137#13511yescharacteristic
80C4×C137AGL(1,ℤ/8)11yescharacteristic
81C4×C137#13711yescharacteristic
82Dic137—41no—
83Dic137—41no—
84Dic137—41no—
85Dic137—41no—
86C2×C137—82no—
87C2×C137#44911yescharacteristic
88C2×C137#45211yescharacteristic
89C2×C137#45511yescharacteristic
90C137#164011yescharacteristic, complemented, Sylow
91#1640—1371nomaximal, Sylow
92#559—1371no—
93#486D13711yescharacteristic
94#475—1371no—
95#462—1371no—
96#331—1371no—
97Q8⋊C8—1371no—
98Q8⋊C8—1371no—
99C2×Q16—2741no—
100C4⋊Q8C2×D13711yescharacteristic
101#127C2×D13711yescharacteristic
102#124—2741no—
103#124—2741no—
104#123—2742no—
105C4×Q8—1371no—
106C4×Q8—1371no—
107C4×D4—2741no—
108C4×D4—2741no—
109#106—2741no—
110C4⋊C8—1371no—
111C4⋊C8—1371no—
112#103—2741no—
113#103—2741no—
114#102—22no—
115C4×C8C2×D13711yescharacteristic
116C2×Q8—21no—
117C2×Q8—1371no—
118C2×Q8—1371no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4C22⋊D13712yes—
122C22×C4—2742no—
123Q16—5482no—
124C2×C8—21no—
125C2×C8—2741no—
126C2×C8—2741no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—2741no—
130C4⋊C4—2741no—
131C4⋊C4—1371no—
132C4⋊C4—1371no—
133C22⋊C4—2742no—
134C22⋊C4—2742no—
135C22⋊C4—2742no—
136C42—2741no—
137C42—2741no—
138C42C22×D13711yescharacteristic
139C23—22no—
140Q8—41no—
141Q8—41no—
142Q8—2741no—
143Q8—2741no—
144Q8—2741no—
145Q8—2741no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—41no—
152C2×C4—2741no—
153C2×C4—2741no—
154C2×C4—2741no—
155C2×C4—2741no—
156C2×C4—10962no—
157C2×C4D4×D13711yescharacteristic
158C2×C4D4×D13711yescharacteristic
159C2×C4C2×C22⋊D13711yescharacteristic
160C8—41no—
161C8—5481no—
162C8—5481no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#1798405611yescharacteristic
167C4—21no—
168C4—81no—
169C4—5481no—
170C4—5481no—
171C4—5481no—
172C4—5481no—
173C4#1798410711yescharacteristic
174C4#1798410911yescharacteristic
175C4#1798411211yescharacteristic
176C4#1798411711yescharacteristic
177C2—82no—
178C2#2257754511yescharacteristic
179C2#2257756211yescharacteristic
180C2#2257758011yescharacteristic
subgroupG/H
centerC22#17984056
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#486×C137C2
socleC22×C137#120
prime radical#127×C137C22
hypercenter#486D137
hypocenterC137#1640
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #22577685C2semidirect
3. #22577610C2semidirect
5. #22577135C2semidirect
6. #22577135C2semidirect
90. C137#1640semidirect

Maximal subgroups

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

subgroupindex
#225781392
#225776852
#225776102
#225771472
#225771352
#486×C1372
#1640137