Finite Group Databasebeta

← #31796070

Subgroups — #31796070

Order 17536.

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

normal subgroups only

This group has 214 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#22578424C211yesmaximal, characteristic, complemented
2#22578420C211yesmaximal, characteristic, complemented
3#22578248C212yesmaximal, complemented
4#22578246C212yesmaximal
5C2×#17984081C211yesmaximal, characteristic
6#17984155—21no—
7C4⋊#11764047—21no—
8C22⋊#11764047C2212yes—
9C22⋊#11764047C2212yes—
10#17984143—21no—
11C4×#11764047—21no—
12#17984081—21no—
13#17984081—21no—
14#17984081—21no—
15#17984081—21no—
16C2×C4⋊Dic137C2211yescharacteristic
17C2×#14555489C411yescharacteristic
18C2×#14555489C411yescharacteristic
19C2×C22⋊Dic137C411yescharacteristic
20C2×C22⋊Dic137C2211yescharacteristic
21C2×C4×Dic137C2211yescharacteristic
22C2×C22⋊C4×C137C411yescharacteristic
23C2×#11764047—22no—
24C2×#11764047—22no—
25C22×Dic137C2×C411yescharacteristic
26C22×Dic137C2×C411yescharacteristic
27C22×Dic137C2×C411yescharacteristic
28C22×Dic137C2311yescharacteristic
29C4⋊Dic137—21no—
30C4⋊Dic137—21no—
31#14555489—21no—
32#14555489—21no—
33#14555489—21no—
34#14555489—21no—
35C22⋊Dic137—21no—
36C22⋊Dic137—21no—
37C22⋊Dic137—21no—
38C22⋊Dic137—21no—
39C4×Dic137D411yescharacteristic
40C4×Dic137D411yescharacteristic
41C4×Dic137D411yescharacteristic
42C4×Dic137D411yescharacteristic
43C24×C137C2×C411yescharacteristic
44C22×C4×C137C2×C411yescharacteristic
45C22×C4×C137C2×C411yescharacteristic
46C22⋊C4×C137—21no—
47C22⋊C4×C137—21no—
48#11764047—42no—
49#11764047—42no—
50C2×Dic137—22no—
51C2×Dic137—21no—
52C2×Dic137—21no—
53C2×Dic137—41no—
54C2×Dic137—41no—
55C2×Dic137—41no—
56C2×Dic137C22⋊C411yescharacteristic
57C2×Dic137C22⋊C411yescharacteristic
58C2×Dic137C2×D411yescharacteristic
59C2×Dic137C2×D411yescharacteristic
60C23×C137—21no—
61C23×C137—22no—
62C23×C137—41no—
63C23×C137—41no—
64C23×C137C22×C411yescharacteristic
65C2×C4×C137—21no—
66C2×C4×C137—22no—
67C2×C4×C137—41no—
68C2×C4×C137C22⋊C411yescharacteristic
69C2×C4×C137C22⋊C411yescharacteristic
70C22×C137—21no—
71C22×C137—21no—
72C22×C137—41no—
73C22×C137—41no—
74C22×C137—41no—
75C22×C137—82no—
76C22×C137C2×C22⋊C411yescharacteristic
77C22×C137#13112yes—
78C4×C137—21no—
79C4×C137—21no—
80C4×C137—41no—
81Dic137—21no—
82Dic137—21no—
83Dic137—21no—
84Dic137—21no—
85Dic137—41no—
86Dic137—41no—
87C2×C137—22no—
88C2×C137—81no—
89C2×C137#40811yescharacteristic
90C2×C137#41011yescharacteristic
91C2×C137#41311yescharacteristic
92C137#147311yescharacteristic, complemented, Sylow
93#1473—1371nomaximal, Sylow
94C2×#126—1371no—
95#432—1371no—
96#431—1371no—
97#324—1372no—
98#323—1372no—
99#126—2741no—
100#126—2741no—
101#126—2741no—
102#126—2741no—
103C2×C4⋊C4—1371no—
104C2×C4⋊C4—1371no—
105C2×C4⋊C4—1371no—
106C2×C22⋊C4—1371no—
107C2×C22⋊C4—1371no—
108C2×C22⋊C4#1058773911yescharacteristic
109C2×C42—1371no—
110C4⋊C8—2741no—
111C4⋊C8—2741no—
112C22⋊C8—1372no—
113C22⋊C8—1372no—
114#97—2741no—
115C4×C8—2741no—
116C24C2×#1058773911yescharacteristic
117C22×C4—1371no—
118C22×C4—1371no—
119C22×C4—1371no—
120C22×C4—1371no—
121C22×C4C2×#1058773911yescharacteristic
122C22×C4C2×#1058773911yescharacteristic
123C2×C8—2742no—
124C2×C8—2742no—
125C4⋊C4—2741no—
126C4⋊C4—2741no—
127C4⋊C4—2741no—
128C4⋊C4—2741no—
129C4⋊C4—2741no—
130C4⋊C4—2741no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—2741no—
134C22⋊C4—2741no—
135C22⋊C4—2741no—
136C22⋊C4—2741no—
137C42—1371no—
138C42—1371no—
139C42—1371no—
140C42—1371no—
141C23—21no—
142C23—22no—
143C23—41no—
144C23—41no—
145C23C22×#1058773911yescharacteristic
146C2×C4—21no—
147C2×C4—22no—
148C2×C4—41no—
149C2×C4—2742no—
150C2×C4—2741no—
151C2×C4—2741no—
152C2×C4—5481no—
153C2×C4—5481no—
154C2×C4—5481no—
155C2×C4—1371no—
156C2×C4—1371no—
157C2×C4—1371no—
158C2×C4—1371no—
159C2×C4C22⋊#1058773911yescharacteristic
160C2×C4C22⋊#1058773911yescharacteristic
161C8—5482no—
162C8—5482no—
163C22—21no—
164C22—21no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—82no—
169C22C2×C22⋊#1058773911yescharacteristic
170C22#1798417512yes—
171C4—21no—
172C4—21no—
173C4—41no—
174C4—2741no—
175C4—2741no—
176C4—2741no—
177C4—2741no—
178C4—5481no—
179C4—5481no—
180C2—22no—
181C2—81no—
182C2#2257837711yescharacteristic
183C2#2257838211yescharacteristic
184C2#2257839011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C22⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC24×C137C2×C4
cosocleC22×Dic137C23
hypercenterC2×C22⋊C4#10587739
hypocenterC137#1473
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #22578424C2semidirect
2. #22578420C2semidirect
3. #22578248C2semidirect
92. C137#1473semidirect

Maximal subgroups

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

subgroupindex
#225784242
#225784202
#225782482
#225782462
C2×#179840812
#1473137