Finite Group Databasebeta

← #31795874

Subgroups — #31795874

Order 17536.

234 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 44 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
1C2×#17984148C211yesmaximal, characteristic
2C2×#17984146C211yesmaximal, characteristic
3#22577280C211yesmaximal, characteristic
4C2×#14555519—21no—
5C2×C22⋊#10587739—21no—
6#17984148—21no—
7#17984148C412yescomplemented
8#17984146—21no—
9#17984146C412yescomplemented
10C22×C22⋊D137C2211yescharacteristic
11C23×Dic137C411yescharacteristic, complemented
12C2×#14555487—21no—
13#17983995—21no—
14C2×C22⋊C4×C137C411yescharacteristic, complemented
15C22×#10587739—41no—
16#14555519—22no—
17#14555519—41no—
18C2×#11764047—41no—
19C22⋊#10587739—22no—
20C22⋊#10587739—41no—
21C23×D137D411yescharacteristic
22C2×C22⋊D137—41no—
23C2×C22⋊D137—41no—
24C2×C22⋊D137C2×C411yescharacteristic
25C2×C22⋊D137C2×C411yescharacteristic
26C2×C22⋊D137D411yescharacteristic
27C2×C22⋊D137Q811yescharacteristic
28C22×Dic137—21no—
29C22×Dic137—41no—
30C22×Dic137—42no—
31C22×Dic137D411yescharacteristic
32#14555487—42no—
33C24×C137C2×C411yescharacteristic
34C22×C4×C137—21no—
35C22⋊C4×C137—41no—
36C2×#10587739—42no—
37C2×#10587739—81no—
38C2×#10587739—81no—
39#11764047—42no—
40C22×D137—21no—
41C22×D137—21no—
42C22×D137—41no—
43C22×D137—41no—
44C22×D137C22⋊C411yescharacteristic
45C22×D137C4⋊C411yescharacteristic
46C22⋊D137—41no—
47C22⋊D137—41no—
48C22⋊D137—41no—
49C22⋊D137—41no—
50C2×Dic137—21no—
51C2×Dic137—21no—
52C2×Dic137—22no—
53C2×Dic137—41no—
54C2×Dic137—42no—
55C2×Dic137—41no—
56C2×Dic137—42no—
57C2×Dic137C22⋊C411yescharacteristic
58C2×Dic137C4⋊C411yescharacteristic
59C23×C137—21no—
60C23×C137—21no—
61C23×C137—41no—
62C23×C137—41no—
63C23×C137C4211yescharacteristic
64C23×C137C22⋊C411yescharacteristic
65C23×C137C4⋊C411yescharacteristic
66C2×C4×C137—41no—
67C2×C4×C137—41no—
68C2×C4×C137—41no—
69C22×C137—21no—
70C22×C137—21no—
71C22×C137—21no—
72C22×C137—21no—
73C22×C137—41no—
74C22×C137—41no—
75C22×C137—42no—
76C22×C137—41no—
77C22×C137—41no—
78C22×C137#9511yescharacteristic
79C22×C137#9911yescharacteristic
80C22×C137#9911yescharacteristic
81C2×D137—21no—
82C2×D137—21no—
83C2×D137—41no—
84C2×D137—41no—
85C2×D137—41no—
86C2×D137—41no—
87C2×D137—81no—
88#10587739—82no—
89C4×C137—81no—
90Dic137—21no—
91Dic137—21no—
92Dic137—42no—
93C2×C137—21no—
94C2×C137—21no—
95C2×C137—41no—
96C2×C137—41no—
97C2×C137#34211yescharacteristic
98C2×C137AΣL(1,𝔽16)11yescharacteristic
99C2×C137#35211yescharacteristic
100D137—41no—
101D137—41no—
102C137#139011yescharacteristic, complemented, Sylow
103#1390—1371nomaximal, Sylow
104C2×#100—1371no—
105C2×#99—1371no—
106#386—1371no—
107C22×D4—1371no—
108C23×C4—1371no—
109C2×M16—2741no—
110C2×C22⋊C4—2741no—
111C2×C22⋊C4—2741no—
112C2×C22⋊C4#1058773911yescharacteristic, complemented
113#100—2741no—
114#100—1372no—
115#99—2741no—
116#99—1372no—
117#95—2741no—
118C24—1371no—
119C24C2×#1058773911yescharacteristic
120C2×D4—5481no—
121C2×D4—5481no—
122C2×D4—1371no—
123C2×D4—1371no—
124C2×D4—1371no—
125C2×D4—1371no—
126C22×C4—21no—
127C22×C4—2741no—
128C22×C4—5482no—
129C22×C4—5481no—
130C22×C4—5481no—
131C22×C4—1371no—
132M16—2742no—
133M16—5481no—
134C2×C8—5481no—
135C22⋊C4—41no—
136C22⋊C4—2742no—
137C22⋊C4—5482no—
138C22⋊C4—5481no—
139C23—21no—
140C23—21no—
141C23—41no—
142C23—41no—
143C23—2741no—
144C23—2741no—
145C23—5481no—
146C23—5481no—
147C23—1371no—
148C23—1371no—
149C23C4×#1058773911yescharacteristic
150C23C22⋊#1058773911yescharacteristic
151C23C4⋊#1058773911yescharacteristic
152D4—5481no—
153D4—5481no—
154D4—5481no—
155D4—5481no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—2741no—
160C2×C4—2741no—
161C2×C4—2742no—
162C2×C4—5481no—
163C2×C4—5482no—
164C2×C4—5481no—
165C2×C4—5482no—
166C2×C4—5482no—
167C2×C4—10961no—
168C2×C4—10961no—
169C2×C4—1371no—
170C2×C4—1371no—
171C8—5482no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—41no—
177C22—41no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—2741no—
182C22—2741no—
183C22—5481no—
184C22—5481no—
185C22—5481no—
186C22—5481no—
187C22—10961no—
188C22(C2×#10587739)⋊C411yescharacteristic
189C22#1798414611yescharacteristic
190C22#1798414711yescharacteristic
191C4—81no—
192C4—2741no—
193C4—2741no—
194C4—5482no—
195C4—10962no—
196C2—21no—
197C2—21no—
198C2—41no—
199C2—41no—
200C2—5481no—
201C2—5481no—
202C2#2257828111yescharacteristic
203C2#2257830411yescharacteristic
204C2#2257830611yescharacteristic
subgroupG/H
centerC22#17984147
commutatorC23×C137C42
FrattiniC24C2×#10587739
FittingC2×C22⋊C4×C137C4
socleC22×C137#99
prime radical, cosocleC22×C22⋊D137C22
hypercenterC2×C22⋊C4#10587739
hypocenterC137#1390
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
7. #17984148C4semidirect
9. #17984146C4semidirect
11. C23×Dic137C4semidirect
14. C2×C22⋊C4×C137C4semidirect
102. C137#1390semidirect
112. C2×C22⋊C4#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×#179841482
C2×#179841462
#225772802
#1390137