Finite Group Databasebeta

← #31796068

Subgroups — #31796068

Order 17536.

270 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 64 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#22578373C211yesmaximal, characteristic
2C2×C22⋊#11764047C211yesmaximal, characteristic
3#22578249C212yesmaximal, complemented
4#22578248C212yesmaximal, complemented
5C2×#17984065C211yesmaximal, characteristic
6C2×#14555519—21no—
7C22×#11764047—21no—
8C22⋊#11764047—21no—
9C22⋊#11764047—21no—
10C22⋊#11764047C2212yes—
11C22⋊#11764047C2212yes—
12C22×Dic274C411yescharacteristic
13C23×Dic137C2211yescharacteristic
14#17984065—21no—
15#17984065—21no—
16#17984065—21no—
17#17984065—21no—
18C2×C4⋊Dic137C2211yescharacteristic
19C2×#14555489C411yescharacteristic
20C2×#14555489C2211yescharacteristic
21C2×C22⋊Dic137C411yescharacteristic
22C2×C22⋊C4×C137C411yescharacteristic
23#14555519—42no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C2×#11764047—42no—
27C2×Dic274—21no—
28C2×Dic274—21no—
29C2×Dic274—42no—
30C22×Dic137—42no—
31C22×Dic137C2×C411yescharacteristic
32C22×Dic137C2×C411yescharacteristic
33C22×Dic137C2×C411yescharacteristic
34C22×Dic137D411yescharacteristic
35C22×Dic137D411yescharacteristic
36C22×Dic137D411yescharacteristic
37C22×Dic137D411yescharacteristic
38C22×Dic137C2311yescharacteristic
39C4⋊Dic137—21no—
40C4⋊Dic137—21no—
41#14555489—21no—
42#14555489—21no—
43#14555489—21no—
44#14555489—21no—
45C22⋊Dic137—21no—
46C22⋊Dic137—21no—
47C24×C137C2×C411yescharacteristic
48C22×C4×C137C2×C411yescharacteristic
49C22×C4×C137C2×C411yescharacteristic
50C22⋊C4×C137—21no—
51C22⋊C4×C137—21no—
52#11764047—42no—
53#11764047—42no—
54Dic274—42no—
55Dic274—42no—
56C2×Dic137—22no—
57C2×Dic137—22no—
58C2×Dic137—22no—
59C2×Dic137—21no—
60C2×Dic137—21no—
61C2×Dic137—22no—
62C2×Dic137—41no—
63C2×Dic137—41no—
64C2×Dic137—41no—
65C2×Dic137—41no—
66C2×Dic137C22⋊C411yescharacteristic
67C2×Dic137C22⋊C411yescharacteristic
68C2×Dic137C2×D411yescharacteristic
69C2×Dic137C2×D411yescharacteristic
70C23×C137—22no—
71C23×C137—22no—
72C23×C137—41no—
73C23×C137C22⋊C411yescharacteristic
74C23×C137C22⋊C411yescharacteristic
75C23×C137C22×C411yescharacteristic
76C2×C4×C137—21no—
77C2×C4×C137—21no—
78C2×C4×C137—22no—
79C2×C4×C137—41no—
80C22×C137—21no—
81C22×C137—21no—
82C22×C137—21no—
83C22×C137—21no—
84C22×C137—21no—
85C22×C137—21no—
86C22×C137—21no—
87C22×C137—44no—
88C22×C137#10112yes—
89C22×C137C2×C22⋊C411yescharacteristic
90C22×C137#13112yes—
91C4×C137—41no—
92C4×C137—41no—
93Dic137—21no—
94Dic137—21no—
95Dic137—21no—
96Dic137—41no—
97Dic137—41no—
98Dic137#9912yes—
99C2×C137—22no—
100C2×C137—22no—
101C2×C137—41no—
102C2×C137#40811yescharacteristic
103C2×C137C2×#9911yescharacteristic
104C2×C137C2×#10111yescharacteristic
105C137#147211yescharacteristic, complemented, Sylow
106#1472—1371nomaximal, Sylow
107C2×C22⋊Q8—1371no—
108#407—1371no—
109C2×C22⋊C8—1371no—
110#324—1372no—
111#324—1372no—
112C22×Q8—1371no—
113C23×C4—1371no—
114C2×M16—2741no—
115C22×C8—2741no—
116C22⋊Q8—2741no—
117C22⋊Q8—2741no—
118C22⋊Q8—2741no—
119C22⋊Q8—2741no—
120C2×C4⋊C4—1371no—
121C2×C4⋊C4—1371no—
122C2×C4⋊C4—1371no—
123C2×C22⋊C4—1371no—
124C2×C22⋊C4#1058773911yescharacteristic
125C22⋊C8—2741no—
126C22⋊C8—2741no—
127C22⋊C8—1372no—
128C22⋊C8—1372no—
129C24C2×#1058773911yescharacteristic
130C2×Q8—2741no—
131C2×Q8—2741no—
132C2×Q8—5482no—
133C22×C4—5482no—
134C22×C4—1371no—
135C22×C4—1371no—
136C22×C4—1371no—
137C22×C4—1371no—
138C22×C4—1371no—
139C22×C4—1371no—
140C22×C4—1371no—
141C22×C4—1371no—
142C22×C4C2×#1058773911yescharacteristic
143C22×C4C2×#1058773911yescharacteristic
144M16—5482no—
145C2×C8—2742no—
146C2×C8—2742no—
147C2×C8—5482no—
148C4⋊C4—2741no—
149C4⋊C4—2741no—
150C4⋊C4—2741no—
151C4⋊C4—2741no—
152C4⋊C4—2741no—
153C4⋊C4—2741no—
154C22⋊C4—21no—
155C22⋊C4—21no—
156C22⋊C4—2741no—
157C22⋊C4—2741no—
158C23—22no—
159C23—22no—
160C23—41no—
161C23C22⋊#1058773911yescharacteristic
162C23C22⋊#1058773911yescharacteristic
163C23C22×#1058773911yescharacteristic
164Q8—5482no—
165Q8—5482no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—2742no—
171C2×C4—2742no—
172C2×C4—2742no—
173C2×C4—2741no—
174C2×C4—2741no—
175C2×C4—2742no—
176C2×C4—5481no—
177C2×C4—5481no—
178C2×C4—5481no—
179C2×C4—5481no—
180C2×C4—1371no—
181C2×C4—1371no—
182C2×C4—1371no—
183C2×C4—1371no—
184C8—5482no—
185C8—5482no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—44no—
194C22#1798414912yes—
195C22C2×C22⋊#1058773911yescharacteristic
196C22#1798417512yes—
197C4—41no—
198C4—41no—
199C4—2741no—
200C4—2741no—
201C4—2741no—
202C4—5481no—
203C4—5481no—
204C4—1372no—
205C2—22no—
206C2—22no—
207C2—41no—
208C2#2257837611yescharacteristic
209C2D137⋊#9911yescharacteristic
210C2C2×#1798414911yescharacteristic
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#1472
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #22578249C2semidirect
4. #22578248C2semidirect
105. C137#1472semidirect

Maximal subgroups

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

subgroupindex
#225783732
C2×C22⋊#117640472
#225782492
#225782482
C2×#179840652
#1472137