Finite Group Databasebeta

← #31796312

Subgroups — #31796312

Order 17536.

242 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 58 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#22578426C211yesmaximal, characteristic, complemented
2#22578421C211yesmaximal, characteristic, complemented
3#22578262C212yesmaximal, complemented
4#22578261C212yesmaximal, complemented
5#22577950C211yesmaximal, characteristic
6#17984155C2212yes—
7#17984155C2212yes—
8#17984145—21no—
9#17984145—21no—
10#17984143—21no—
11C4×#11764047—21no—
12C2×#14555510—21no—
13#17984087C2211yescharacteristic
14#17984084C2211yescharacteristic
15#17984074—21no—
16#17984063C411yescharacteristic
17#17984061—21no—
18#17984059C411yescharacteristic
19C4×D548C411yescharacteristic
20C4×C22⋊D137—21no—
21C42×D137C2211yescharacteristic
22C4×D4×C137C411yescharacteristic
23C2×#11764047—22no—
24C2×#11764047—22no—
25#14555510—44no—
26C2×Dic274—21no—
27C2×D548—21no—
28C2×D548C2×C411yescharacteristic
29C2×C22⋊D137—21no—
30C2×C22⋊D137—21no—
31C2×C22⋊D137—21no—
32C2×C4×D137—21no—
33C2×C4×D137C2×C411yescharacteristic
34C2×C4×D137D411yescharacteristic
35C2×C4×D137D411yescharacteristic
36C4⋊Dic137C2×C411yescharacteristic
37#14555489—21no—
38C22⋊Dic137—21no—
39#14555487—21no—
40#14555487—21no—
41C4×Dic137D411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137C2311yescharacteristic
44C2×D4×C137C2×C411yescharacteristic
45C22×C4×C137—21no—
46C4⋊C4×C137C2×C411yescharacteristic
47C22⋊C4×C137—21no—
48C42×C137C2×C411yescharacteristic
49#11764047—42no—
50#11764047—42no—
51C22×D137—21no—
52C22×D137C22⋊C411yescharacteristic
53Dic274—42no—
54D548—41no—
55D548—42no—
56C22⋊D137—42no—
57C22⋊D137—42no—
58C22⋊D137—42no—
59C2×Dic137—21no—
60C2×Dic137C22⋊C411yescharacteristic
61C2×Dic137C2×D411yescharacteristic
62C2×Dic137C2×D411yescharacteristic
63C4×D137—22no—
64C4×D137—22no—
65C4×D137—41no—
66C4×D137—42no—
67C23×C137—21no—
68D4×C137—41no—
69C2×C4×C137—21no—
70C2×C4×C137—42no—
71C2×C4×C137C22⋊C411yescharacteristic
72C2×C4×C137C22⋊C411yescharacteristic
73C2×C4×C137C22×C411yescharacteristic
74C22×C137—41no—
75C22×C137—41no—
76C22×C137—41no—
77C22×C137C2×C22⋊C411yescharacteristic
78C2×D137—21no—
79C2×D137—21no—
80C2×D137—21no—
81C2×D137—41no—
82C2×D137—41no—
83C2×D137—41no—
84C4×C137—21no—
85C4×C137—21no—
86C4×C137—21no—
87C4×C137—41no—
88Dic137—21no—
89Dic137—41no—
90Dic137#13512yes—
91Dic137AGL(1,ℤ/8)12yes—
92C2×C137—81no—
93C2×C137#41311yescharacteristic
94C2×C137#41611yescharacteristic
95C2×C137#41811yescharacteristic
96D137—41no—
97D137—81no—
98C137#155311yescharacteristic, complemented, Sylow
99#1553—1371nomaximal, Sylow
100#532—1371no—
101#432—1371no—
102#431—1371no—
103#332—1372no—
104#331—1372no—
105C2×#41—2741no—
106C4⋊Q8—1371no—
107#127—1371no—
108#124—2741no—
109#121—2741no—
110#121—1371no—
111#121—1371no—
112C4×D4—2741no—
113C4×D4—1371no—
114C4×D4#1058773911yescharacteristic
115C2×C42—1371no—
116C4⋊C8—1372no—
117C4⋊C8—1372no—
118C22⋊C8—2741no—
119C22⋊C8—2741no—
120#97—2741no—
121C4×C8—2741no—
122#41—5484no—
123C2×Q8—2741no—
124C2×D4—2741no—
125C2×D4—2741no—
126C2×D4—2741no—
127C2×D4—2741no—
128C2×D4—1371no—
129C2×D4C2×#1058773911yescharacteristic
130C22×C4—21no—
131C22×C4—2741no—
132C22×C4—1371no—
133C22×C4—1371no—
134C22×C4—1371no—
135C2×C8—2742no—
136C2×C8—2742no—
137C4⋊C4—2741no—
138C4⋊C4—1371no—
139C4⋊C4C2×#1058773911yescharacteristic
140C22⋊C4—21no—
141C22⋊C4—2741no—
142C22⋊C4—2741no—
143C22⋊C4—2741no—
144C42—1371no—
145C42—1371no—
146C42—1371no—
147C42C2×#1058773911yescharacteristic
148C23—21no—
149C23—2741no—
150C23—1371no—
151Q8—5482no—
152D4—41no—
153D4—5481no—
154D4—5482no—
155D4—5482no—
156D4—5482no—
157D4—5482no—
158C2×C4—21no—
159C2×C4—42no—
160C2×C4—2742no—
161C2×C4—2742no—
162C2×C4—2741no—
163C2×C4—5481no—
164C2×C4—5482no—
165C2×C4—1371no—
166C2×C4—1371no—
167C2×C4—1371no—
168C2×C4C22⋊#1058773911yescharacteristic
169C2×C4C22⋊#1058773911yescharacteristic
170C2×C4C22×#1058773911yescharacteristic
171C8—5482no—
172C8—5482no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—2741no—
177C22—2741no—
178C22—2741no—
179C22—5481no—
180C22—5481no—
181C22—5481no—
182C22C2×C22⋊#1058773911yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—41no—
187C4—2741no—
188C4—5481no—
189C4—1372no—
190C4—1372no—
191C2—81no—
192C2—5481no—
193C2—10961no—
194C2#2257838911yescharacteristic
195C2#2257839311yescharacteristic
196C2#2257839511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4×D4×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577950C2
cosocleC4×Dic137C23
hypercenterC4×D4#10587739
hypocenterC137#1553
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #22578426C2semidirect
2. #22578421C2semidirect
3. #22578262C2semidirect
4. #22578261C2semidirect
98. C137#1553semidirect

Maximal subgroups

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

subgroupindex
#225784262
#225784212
#225782622
#225782612
#225779502
#1553137