Finite Group Databasebeta

← #31797048

Subgroups — #31797048

Order 17536.

266 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 54 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#22578389C211yesmaximal, characteristic, complemented
2#22578382C211yesmaximal, characteristic, complemented
3#22578303C214yesmaximal, complemented
4#22577970C211yesmaximal, characteristic
5D137⋊M16—21no—
6#17984167—21no—
7#17984149—21no—
8#17984148C2212yes—
9#17984147—21no—
10#17984146C2212yes—
11C2×#14555510C2211yescharacteristic
12#17984084—21no—
13#17984075—21no—
14#17984059—21no—
15#17984059—21no—
16#17984057C412yescomplemented
17#17984057C2212yes—
18#17984047C411yescharacteristic, complemented
19C22×D4×C137C411yescharacteristic, complemented
20#14555520—41no—
21#14555519—22no—
22#14555518—41no—
23C4⋊#10587739—41no—
24C22⋊#10587739—22no—
25C4×#10587739—41no—
26#14555510—41no—
27#14555510—41no—
28C2×Dic274D411yescharacteristic
29C2×D548D411yescharacteristic
30C2×C22⋊D137—22no—
31C2×C22⋊D137—22no—
32C2×C22⋊D137C2×C411yescharacteristic
33C2×C22⋊D137C2311yescharacteristic
34C2×C4×D137D411yescharacteristic
35C2×C4×D137D411yescharacteristic
36C4⋊Dic137—21no—
37C22⋊Dic137—22no—
38C22⋊Dic137—22no—
39C22⋊Dic137C2×C412yes—
40C4×Dic137—21no—
41C24×C137C2×C412yes—
42C2×D4×C137—21no—
43C2×D4×C137—21no—
44C2×D4×C137—41no—
45C2×D4×C137—41no—
46C22×C4×C137C2×C411yescharacteristic
47C2×#10587739—42no—
48#11764047—42no—
49C22×D137C22⋊C411yescharacteristic
50C22×D137C2×D411yescharacteristic
51Dic274—41no—
52D548—41no—
53C22⋊D137—41no—
54C22⋊D137—41no—
55C22⋊D137—82no—
56C22⋊D137—82no—
57C2×Dic137—22no—
58C2×Dic137C22⋊C411yescharacteristic
59C2×Dic137C2×D411yescharacteristic
60C4×D137—21no—
61C4×D137—21no—
62C4×D137—21no—
63C4×D137—21no—
64C23×C137—22no—
65C23×C137—42no—
66C23×C137—82no—
67C23×C137C22×C411yescharacteristic
68D4×C137—42no—
69D4×C137—41no—
70D4×C137—41no—
71C2×C4×C137—21no—
72C2×C4×C137—21no—
73C2×C4×C137C22⋊C411yescharacteristic
74C2×C4×C137C22⋊C411yescharacteristic
75C22×C137—21no—
76C22×C137—41no—
77C22×C137—42no—
78C22×C137—42no—
79C22×C137—42no—
80C22×C137—84no—
81C22×C137C2×C22⋊C411yescharacteristic
82C2×D137—21no—
83C2×D137—21no—
84C2×D137—41no—
85C2×D137—41no—
86#10587739—82no—
87C4×C137—21no—
88C4×C137#9911yescharacteristic
89C4×C137#9911yescharacteristic
90Dic137—21no—
91Dic137—21no—
92Dic137—42no—
93C2×C137—21no—
94C2×C137—41no—
95C2×C137—82no—
96C2×C137C2×#9911yescharacteristic
97D137—41no—
98D137—41no—
99C137#212211yescharacteristic, complemented, Sylow
100#2122—1371nomaximal, Sylow
101#535—1371no—
102#413—1371no—
103#410—1371no—
104AΣL(1,𝔽16)—1374no—
105C2×#41—1371no—
106C22×D4#1058773911yescharacteristic, complemented
107C2×M16—2741no—
108#127—2741no—
109#124—2741no—
110#121—2741no—
111#121—2741no—
112#120—1372no—
113#120—1372no—
114#117—2741no—
115#117—1371no—
116#101—2741no—
117#100—1372no—
118#99—2741no—
119#99—1372no—
120C24C2×#1058773912yes—
121#41—5481no—
122#41—5481no—
123C2×Q8—1371no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—2742no—
129C2×D4—2742no—
130C2×D4—1371no—
131C2×D4—1371no—
132C2×D4—1371no—
133C22×C4—1371no—
134C22×C4—1371no—
135C22×C4C2×#1058773911yescharacteristic
136M16—2742no—
137M16—5481no—
138C2×C8—5481no—
139C4⋊C4—2741no—
140C4⋊C4—5481no—
141C22⋊C4—2742no—
142C22⋊C4—2742no—
143C22⋊C4—2742no—
144C22⋊C4—1372no—
145C42—2741no—
146C42—5481no—
147C23—22no—
148C23—42no—
149C23—82no—
150C23—1371no—
151C23—1371no—
152C23C22×#1058773911yescharacteristic
153Q8—5481no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—5481no—
158D4—5481no—
159D4—5481no—
160D4—10962no—
161D4—10962no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—2741no—
165C2×C4—2741no—
166C2×C4—2741no—
167C2×C4—2741no—
168C2×C4—2742no—
169C2×C4—5482no—
170C2×C4—1371no—
171C2×C4—1371no—
172C2×C4C22⋊#1058773911yescharacteristic
173C2×C4C22⋊#1058773911yescharacteristic
174C8—5482no—
175C22—21no—
176C22—41no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—84no—
181C22—2741no—
182C22—2741no—
183C22—5481no—
184C22—5481no—
185C22C2×C22⋊#1058773911yescharacteristic
186C4—21no—
187C4—2741no—
188C4—2741no—
189C4—5482no—
190C4—10962no—
191C4#1798414611yescharacteristic
192C4#1798414611yescharacteristic
193C2—21no—
194C2—41no—
195C2—82no—
196C2—5481no—
197C2—5481no—
198C2C2×#1798414611yescharacteristic
subgroupG/H
centerC2C2×#17984146
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC22×D4×C137C4
socleC2×C137C2×#99
prime radical#22577970C2
cosocleC2×C22⋊D137C23
hypercenterC22×D4#10587739
hypocenterC137#2122
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #22578389C2semidirect
2. #22578382C2semidirect
3. #22578303C2semidirect
16. #17984057C4semidirect
18. #17984047C4semidirect
19. C22×D4×C137C4semidirect
99. C137#2122semidirect
106. C22×D4#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783892
#225783822
#225783032
#225779702
#2122137