Finite Group Databasebeta

← #31796769

Subgroups — #31796769

Order 17536.

408 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C22×#14555520C211yesmaximal, characteristic, complemented
2C2×#17984145C212yesmaximal, complemented
3C2×#17984145C211yesmaximal, characteristic, complemented
4#22578269C212yesmaximal, complemented
5C2×#17984050C211yesmaximal, characteristic
6C2×#14555520—24no—
7C2×#14555520—22no—
8C22×#11764047C2212yes—
9C22×#11764047C2212yes—
10#17984145—24no—
11#17984145—22no—
12C22×D548C411yescharacteristic
13C22×C4×D137C2211yescharacteristic
14C22×C4×D137C2211yescharacteristic
15#17984050—22no—
16#17984050—22no—
17C2×#14555487C411yescharacteristic
18C2×#14555487C411yescharacteristic
19C2×C4×Dic137C2211yescharacteristic
20C2×C4⋊C4×C137C411yescharacteristic
21#14555520—28no—
22C2×#11764047—22no—
23C2×#11764047—22no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C2×#11764047—22no—
27C2×#11764047D414yescomplemented
28C23×D137C2×C411yescharacteristic
29C23×D137C2×C411yescharacteristic
30C2×D548—22no—
31C2×D548—22no—
32C2×D548—22no—
33C22×Dic137C2×C411yescharacteristic
34C22×Dic137C2311yescharacteristic
35C2×C4×D137—22no—
36C2×C4×D137—22no—
37C2×C4×D137—21no—
38C2×C4×D137—21no—
39C2×C4×D137—22no—
40C2×C4×D137—22no—
41C2×C4×D137D412yes—
42C2×C4×D137D412yes—
43#14555487—22no—
44#14555487—22no—
45C4×Dic137—21no—
46C4×Dic137—21no—
47C22×C4×C137C2×C411yescharacteristic
48C22×C4×C137C2×C411yescharacteristic
49C22×C4×C137C2×C411yescharacteristic
50C4⋊C4×C137—22no—
51#11764047—24no—
52#11764047—42no—
53C22×D137—21no—
54C22×D137—22no—
55C22×D137—22no—
56C22×D137—21no—
57C22×D137—21no—
58C22×D137—21no—
59C22×D137—22no—
60C22×D137—22no—
61C22×D137C22⋊C412yes—
62C22×D137M1612yescomplemented
63D548—28no—
64C2×Dic137—21no—
65C2×Dic137—22no—
66C2×Dic137C2×D412yes—
67C2×Dic137C2×D411yescharacteristic
68C2×Dic137C2×D411yescharacteristic
69C2×Dic137#4112yes—
70C4×D137—22no—
71C4×D137—24no—
72C4×D137—22no—
73C4×D137—41no—
74C4×D137—42no—
75C4×D137—41no—
76C23×C137C22×C411yescharacteristic
77C2×C4×C137—21no—
78C2×C4×C137—22no—
79C2×C4×C137—21no—
80C2×C4×C137—21no—
81C2×C4×C137—22no—
82C2×C4×C137C22⋊C412yes—
83C2×C4×C137M1612yescomplemented
84C22×C137C2×C22⋊C411yescharacteristic
85C22×C137C4×D412yes—
86C22×C137C2×M1611yescharacteristic
87C22×C137C2×M1611yescharacteristic
88C22×C137#13112yes—
89C2×D137—22no—
90C2×D137—22no—
91C2×D137—22no—
92C2×D137—22no—
93C2×D137—24no—
94C2×D137—41no—
95C2×D137—41no—
96C2×D137—42no—
97C2×D137—42no—
98C2×D137—42no—
99C4×C137—22no—
100C4×C137—22no—
101C4×C137—41no—
102Dic137—41no—
103Dic137#12011yescharacteristic
104Dic137#12112yes—
105Dic137#12311yescharacteristic
106C2×C137#38611yescharacteristic
107C2×C137#40711yescharacteristic
108C2×C137#40811yescharacteristic
109C2×C137#43512yes—
110C2×C137#43612yes—
111D137—42no—
112D137—42no—
113C137#187011yescharacteristic, complemented, Sylow
114#1870—1371nomaximal, Sylow
115C22×M16—1371no—
116C2×C4×D4—1371no—
117C2×C22⋊C8—1371no—
118C2×C22⋊C8—1372no—
119#336—1372no—
120C22×D4—1371no—
121C23×C4—1371no—
122C23×C4—1371no—
123C2×M16—2742no—
124C2×M16—2744no—
125C22×C8—1372no—
126C22×C8—1372no—
127C4×D4—2742no—
128C4×D4—2742no—
129C2×C4⋊C4#1058773911yescharacteristic
130C2×C22⋊C4—1371no—
131C2×C22⋊C4—1371no—
132C2×C42—1371no—
133C22⋊C8—2742no—
134C22⋊C8—2744no—
135C24—1371no—
136C24—1371no—
137C2×D4—2742no—
138C2×D4—2742no—
139C2×D4—2742no—
140C22×C4—2742no—
141C22×C4—2742no—
142C22×C4—2742no—
143C22×C4—2741no—
144C22×C4—2741no—
145C22×C4—2742no—
146C22×C4—1371no—
147C22×C4—1372no—
148C22×C4—1372no—
149C22×C4—1371no—
150C22×C4C2×#1058773911yescharacteristic
151C22×C4C2×#1058773911yescharacteristic
152C22×C4C2×#1058773911yescharacteristic
153M16—2748no—
154C2×C8—2742no—
155C2×C8—2742no—
156C2×C8—2742no—
157C2×C8—2742no—
158C2×C8—2742no—
159C2×C8—1374no—
160C4⋊C4—22no—
161C22⋊C4—2742no—
162C22⋊C4—2742no—
163C42—2741no—
164C42—2741no—
165C23—2741no—
166C23—2742no—
167C23—2742no—
168C23—2741no—
169C23—2741no—
170C23—2741no—
171C23—2742no—
172C23—2742no—
173C23—1372no—
174C23—1372no—
175C23C22×#1058773911yescharacteristic
176D4—2748no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—2742no—
183C2×C4—2744no—
184C2×C4—2742no—
185C2×C4—2741no—
186C2×C4—2742no—
187C2×C4—5481no—
188C2×C4—5482no—
189C2×C4—5481no—
190C2×C4—1371no—
191C2×C4—1371no—
192C2×C4—1372no—
193C2×C4—1372no—
194C2×C4C22⋊#1058773912yes—
195C2×C4#1455552012yescomplemented
196C8—2744no—
197C8—5482no—
198C22—2744no—
199C22—2742no—
200C22—2742no—
201C22—2742no—
202C22—2742no—
203C22—5482no—
204C22—5482no—
205C22—5482no—
206C22—5481no—
207C22—5481no—
208C22C2×C22⋊#1058773911yescharacteristic
209C22D4×#1058773912yes—
210C22D137⋊M1611yescharacteristic
211C22C2×#1455552011yescharacteristic
212C22#1798417612yes—
213C4—22no—
214C4—22no—
215C4—41no—
216C4—5481no—
217C4—1371no—
218C4—1372no—
219C4—1371no—
220C2—5482no—
221C2—5482no—
222C2#2257835111yescharacteristic
223C2#2257837411yescharacteristic
224C2#2257837811yescharacteristic
225C2#2257843312yes—
226C2#2257843712yes—
subgroupG/H
center, FrattiniC23C22×#10587739
commutator, socleC23×C137C22×C4
FittingC2×C4⋊C4×C137C4
prime radicalC22×D548C4
cosocleC22×Dic137C23
hypercenterC2×C4⋊C4#10587739
hypocenterC137#1870
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#14555520C2semidirect
2. C2×#17984145C2semidirect
3. C2×#17984145C2semidirect
4. #22578269C2semidirect
27. C2×#11764047D4semidirect
62. C22×D137M16semidirect
83. C2×C4×C137M16semidirect
113. C137#1870semidirect
195. C2×C4#14555520semidirect

Maximal subgroups

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

subgroupindex
C22×#145555202
C2×#179841452
#225782692
C2×#179840502
#1870137