Finite Group Databasebeta

← #31796115

Subgroups — #31796115

Order 17536.

312 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 76 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#22578422C211yesmaximal, characteristic, complemented
2#22578407C211yesmaximal, characteristic, complemented
3#22578251C212yesmaximal, complemented
4D4⋊#11764047C212yesmaximal, complemented
5C4×D4×D137C211yesmaximal, characteristic
6C2×#14555520—21no—
7C4⋊#11764047—21no—
8C4⋊#11764047—21no—
9C4⋊#11764047C2212yescomplemented
10#17984145—21no—
11C4×#11764047C2212yescomplemented
12C2×D4×D137C411yescharacteristic
13C22×C4×D137—21no—
14C4×D548C411yescharacteristic
15D4×Dic137C2211yescharacteristic
16#17984050C2211yescharacteristic
17C4×C22⋊D137—21no—
18#17984048—21no—
19C4⋊C4×D137C411yescharacteristic
20C22⋊C4×D137—21no—
21C42×D137C2211yescharacteristic
22C4×D4×C137C411yescharacteristic
23#14555520—42no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C23×D137—21no—
27C2×D548C2×C411yescharacteristic
28C2×C22⋊D137—21no—
29D4×D137—22no—
30D4×D137—22no—
31C22×Dic137—21no—
32C2×C4×D137—21no—
33C2×C4×D137—21no—
34C2×C4×D137—21no—
35C2×C4×D137—42no—
36C2×C4×D137—42no—
37C2×C4×D137C2×C411yescharacteristic
38C2×C4×D137D411yescharacteristic
39C2×C4×D137D411yescharacteristic
40C4⋊Dic137C2×C411yescharacteristic
41#14555489—21no—
42C22⋊Dic137—21no—
43#14555487—21no—
44#14555487—21no—
45C4×Dic137D411yescharacteristic
46C4×Dic137D411yescharacteristic
47C4×Dic137C2311yescharacteristic
48C2×D4×C137C2×C411yescharacteristic
49C22×C4×C137—21no—
50C4⋊C4×C137C2×C411yescharacteristic
51C22⋊C4×C137—21no—
52C42×C137C2×C411yescharacteristic
53#11764047—42no—
54#11764047—42no—
55C22×D137—24no—
56C22×D137—21no—
57C22×D137—42no—
58C22×D137—42no—
59C22×D137C22⋊C411yescharacteristic
60D548—21no—
61D548M1612yescomplemented
62C22⋊D137—24no—
63C2×Dic137—21no—
64C2×Dic137—41no—
65C2×Dic137—41no—
66C2×Dic137C22⋊C411yescharacteristic
67C2×Dic137C2×D411yescharacteristic
68C2×Dic137C2×D411yescharacteristic
69C4×D137—21no—
70C4×D137—21no—
71C4×D137—41no—
72C4×D137—42no—
73C4×D137—42no—
74C4×D137—41no—
75C4×D137—41no—
76C4×D137D812yes—
77C4×D137SD1612yes—
78C23×C137—21no—
79D4×C137—21no—
80D4×C137M1612yescomplemented
81C2×C4×C137—21no—
82C2×C4×C137—42no—
83C2×C4×C137C22⋊C411yescharacteristic
84C2×C4×C137C22⋊C411yescharacteristic
85C2×C4×C137C22×C411yescharacteristic
86C22×C137—22no—
87C22×C137—41no—
88C22×C137—41no—
89C22×C137C2×C22⋊C411yescharacteristic
90C2×D137—21no—
91C2×D137—21no—
92C2×D137—22no—
93C2×D137—48no—
94C2×D137—41no—
95C2×D137—41no—
96C2×D137#10212yes—
97C4×C137—21no—
98C4×C137—41no—
99C4×C137#10212yes—
100C4×C137C2×M1611yescharacteristic
101C4×C137C2×M1611yescharacteristic
102Dic137—21no—
103Dic137—21no—
104Dic137—41no—
105Dic137C2×D811yescharacteristic
106Dic137C2×SD1611yescharacteristic
107C2×C137—42no—
108C2×C137#40711yescharacteristic
109C2×C137C2×#10211yescharacteristic
110C2×C137#42111yescharacteristic
111D137—22no—
112D137—42no—
113C137#148611yescharacteristic, complemented, Sylow
114#1486—1371nomaximal, Sylow
115C2×C4×D4—1371no—
116#431—1371no—
117#423—1371no—
118D4⋊C8—1372no—
119D4⋊C8—1372no—
120C22×D4—1371no—
121C23×C4—2741no—
122C2×M16—2741no—
123C4×D4—2741no—
124C4×D4—2741no—
125C4×D4—1371no—
126C4×D4—1371no—
127C4×D4—1371no—
128C4×D4#1058773911yescharacteristic
129C2×C4⋊C4—1371no—
130C2×C22⋊C4—2741no—
131C2×C42—1371no—
132C4⋊C8—2741no—
133C4⋊C8—2741no—
134C4⋊C8—1372no—
135C22⋊C8—2741no—
136C4×C8—1372no—
137C24—2741no—
138C2×D4—2742no—
139C2×D4—2742no—
140C2×D4—2741no—
141C2×D4—1371no—
142C2×D4C2×#1058773911yescharacteristic
143C22×C4—21no—
144C22×C4—2741no—
145C22×C4—2741no—
146C22×C4—2741no—
147C22×C4—2741no—
148C22×C4—5482no—
149C22×C4—5482no—
150C22×C4—1371no—
151C22×C4—1371no—
152C22×C4—1371no—
153M16—5482no—
154C2×C8—2742no—
155C2×C8—2742no—
156C4⋊C4—2741no—
157C4⋊C4—1371no—
158C4⋊C4C2×#1058773911yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—2741no—
161C22⋊C4—2741no—
162C22⋊C4—2741no—
163C42—1371no—
164C42—1371no—
165C42—1371no—
166C42C2×#1058773911yescharacteristic
167C23—21no—
168C23—2744no—
169C23—2741no—
170C23—5482no—
171C23—5482no—
172C23—1371no—
173D4—21no—
174D4—2744no—
175D4—2741no—
176D4—1372no—
177D4#1455552012yescomplemented
178C2×C4—21no—
179C2×C4—42no—
180C2×C4—2741no—
181C2×C4—2741no—
182C2×C4—2741no—
183C2×C4—5481no—
184C2×C4—5482no—
185C2×C4—5481no—
186C2×C4—5481no—
187C2×C4—5481no—
188C2×C4—5481no—
189C2×C4—5482no—
190C2×C4—1371no—
191C2×C4—1371no—
192C2×C4—1371no—
193C2×C4—1372no—
194C2×C4—1372no—
195C2×C4C22⋊#1058773911yescharacteristic
196C2×C4C22⋊#1058773911yescharacteristic
197C2×C4C22×#1058773911yescharacteristic
198C8—5482no—
199C8—5482no—
200C22—22no—
201C22—41no—
202C22—41no—
203C22—2741no—
204C22—2741no—
205C22—2742no—
206C22—5488no—
207C22—5481no—
208C22—5481no—
209C22—1372no—
210C22C2×C22⋊#1058773911yescharacteristic
211C4—21no—
212C4—41no—
213C4—2741no—
214C4—2741no—
215C4—5481no—
216C4—1371no—
217C4—1371no—
218C4#1798415012yes—
219C4D137⋊M1611yescharacteristic
220C4C2×#1455552011yescharacteristic
221C2—42no—
222C2—2742no—
223C2—5482no—
224C2#2257837411yescharacteristic
225C2C2×#1798415011yescharacteristic
226C2#2257840011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4×D4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×D4×D137C4
cosocleC4×Dic137C23
hypercenterC4×D4#10587739
hypocenterC137#1486
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #22578422C2semidirect
2. #22578407C2semidirect
3. #22578251C2semidirect
4. D4⋊#11764047C2semidirect
9. C4⋊#11764047C22semidirect
11. C4×#11764047C22semidirect
61. D548M16semidirect
80. D4×C137M16semidirect
113. C137#1486semidirect
177. D4#14555520semidirect

Maximal subgroups

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

subgroupindex
#225784222
#225784072
#225782512
D4⋊#117640472
C4×D4×D1372
#1486137