Finite Group Databasebeta

← #31790424

Subgroups — #31790424

Order 17536.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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×#17984094C211yesmaximal, characteristic, complemented
2#22577977C211yesmaximal, characteristic, complemented
3#22577970C211yesmaximal, characteristic
4#22577646C214yesmaximal, complemented
5#22577629C214yesmaximal, complemented
6#22577441C211yesmaximal, characteristic, complemented
7#22577412C211yesmaximal, characteristic, complemented
8#22577408C211yesmaximal, characteristic, complemented
9C22×SD16×C137C211yesmaximal, characteristic, complemented
10C2×#14555510C2211yescharacteristic
11#17984094—24no—
12C2×#14555494C2211yescharacteristic
13C2×#14555493C2212yescomplemented
14C2×C8×D137C2212yescomplemented
15#17984086—21no—
16#17984084—21no—
17#17984077—21no—
18#17984075—21no—
19#17984071—22no—
20#17984064C2214yescomplemented
21#17984059C2214yes—
22#17984057—22no—
23#17984047C2211yescharacteristic
24#17984047C2211yescharacteristic
25#17984022C2212yescomplemented
26#17984020C2212yescomplemented
27#17984012C2214yescomplemented
28#17984009C2214yes—
29C22×Q8×C137C2211yescharacteristic
30C22×D4×C137C2211yescharacteristic
31C2×SD16×C137—24no—
32C2×SD16×C137C2212yes—
33C2×SD16×C137C2212yes—
34C22×C8×C137C2211yescharacteristic
35#14555510—22no—
36#14555510—22no—
37C2×Dic274D411yescharacteristic
38C2×D548D411yescharacteristic
39C2×C22⋊D137—24no—
40C2×C22⋊D137D412yes—
41C2×C4×D137C2312yes—
42#14555494—22no—
43#14555493—24no—
44C2×#11764040C2312yes—
45C8×D137—24no—
46C4⋊Dic137C2312yes—
47C4⋊Dic137C2312yes—
48#14555489—24no—
49C22⋊Dic137—24no—
50C22⋊Dic137—21no—
51C22⋊Dic137—21no—
52#14555487—24no—
53C4×Dic137—21no—
54C4×Dic137—21no—
55C24×C137—21no—
56C2×Q8×C137—21no—
57C2×Q8×C137—22no—
58C2×Q8×C137—22no—
59C2×Q8×C137C2312yes—
60C2×D4×C137—22no—
61C2×D4×C137—21no—
62C2×D4×C137—22no—
63C2×D4×C137C2312yes—
64C22×C4×C137—21no—
65C22×C4×C137C2311yescharacteristic
66SD16×C137—24no—
67SD16×C137—24no—
68C2×C8×C137D414yescomplemented
69C2×C8×C137C2311yescharacteristic
70C2×C8×C137C2311yescharacteristic
71C22×D137C2×D412yes—
72Dic274—22no—
73D548—22no—
74C22⋊D137—24no—
75C22⋊D137—84no—
76C2×Dic137—22no—
77C2×Dic137—22no—
78C2×Dic137C2×D412yes—
79C4×D137—22no—
80C4×D137—22no—
81#11764040—22no—
82C23×C137—22no—
83C23×C137—42no—
84C23×C137—82no—
85C23×C137C2×D411yescharacteristic
86Q8×C137—22no—
87Q8×C137—22no—
88Q8×C137—42no—
89D4×C137—22no—
90D4×C137—22no—
91D4×C137—42no—
92C2×C4×C137—22no—
93C2×C4×C137—42no—
94C2×C4×C137C2×D411yescharacteristic
95C2×C4×C137C2×D412yes—
96C2×C4×C137#4112yes—
97C2×C4×C137C2411yescharacteristic
98C8×C137C2×D412yes—
99C8×C137C2×D412yes—
100C22×C137—22no—
101C22×C137—42no—
102C22×C137—44no—
103C22×C137—84no—
104C22×C137#12112yes—
105C22×C137C22×D411yescharacteristic
106C2×D137—22no—
107C2×D137—24no—
108C4×C137—42no—
109C4×C137#12112yes—
110C4×C137C22×D411yescharacteristic
111C4×C137C2×#4111yescharacteristic
112Dic137—22no—
113Dic137—42no—
114Dic137—42no—
115C2×C137—22no—
116C2×C137—82no—
117C2×C137C2×#12111yescharacteristic
118C2×C137#57712yes—
119D137—42no—
120C137#305711yescharacteristic, complemented, Sylow
121#3057—1371nomaximal, Sylow
122C22×SD16D13711yescharacteristic, complemented
123C2×#131—1371no—
124#536—1371no—
125#535—1371no—
126#468—1374no—
127#465—1374no—
128#427—1371no—
129#419—1371no—
130#418—1371no—
131C2×#41—1371no—
132C22×Q8C2×D13711yescharacteristic
133C22×D4C2×D13711yescharacteristic
134C2×SD16—24no—
135C2×SD16C2×D13712yes—
136C2×SD16C2×D13712yes—
137#131—2744no—
138C2×M16—1371no—
139C2×M16—1372no—
140C22×C8—1372no—
141C22×C8C2×D13711yescharacteristic
142C4⋊Q8—2741no—
143#127—2741no—
144#124—2741no—
145#124—2741no—
146#123—2742no—
147C22⋊Q8—1374no—
148#121—1374no—
149#120—2742no—
150#117—1371no—
151#117—1371no—
152#107—1372no—
153#106—1372no—
154#103—1374no—
155#102—1374no—
156C24—21no—
157#41—2742no—
158#41—2742no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—1371no—
163C2×Q8C22×D13712yes—
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—2744no—
168C2×D4—1372no—
169C2×D4—1371no—
170C2×D4C22×D13712yes—
171C22×C4—21no—
172C22×C4—1372no—
173C22×C4C22×D13711yescharacteristic
174SD16—24no—
175SD16—24no—
176M16—2742no—
177M16—2744no—
178C2×C8—2744no—
179C2×C8—1372no—
180C2×C8C22⋊D13714yescomplemented
181C2×C8C22×D13711yescharacteristic
182C2×C8C22×D13711yescharacteristic
183C4⋊C4—2744no—
184C4⋊C4—1372no—
185C4⋊C4—1372no—
186C22⋊C4—2744no—
187C22⋊C4—2741no—
188C22⋊C4—2741no—
189C22⋊C4—2744no—
190C42—2741no—
191C42—2741no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23—1372no—
196C23D4×D13711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—42no—
200Q8—2742no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—2744no—
205D4—2742no—
206D4—10964no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—2742no—
210C2×C4—2742no—
211C2×C4—2742no—
212C2×C4—2742no—
213C2×C4—1372no—
214C2×C4D4×D13711yescharacteristic
215C2×C4C2×C22⋊D13712yes—
216C2×C4#1455550912yes—
217C2×C4C23×D13711yescharacteristic
218C8—2742no—
219C8C2×C22⋊D13712yes—
220C8C2×C22⋊D13712yes—
221C22—22no—
222C22—44no—
223C22—42no—
224C22—84no—
225C22—2744no—
226C22—2742no—
227C22#1798405912yes—
228C22C2×D4×D13711yescharacteristic
229C4—42no—
230C4—2742no—
231C4—5482no—
232C4—5482no—
233C4#1798405912yes—
234C4C22×C22⋊D13711yescharacteristic
235C4C2×#1455550911yescharacteristic
236C2—22no—
237C2—82no—
238C2—5482no—
239C2C2×#1798405911yescharacteristic
240C2#2257820912yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC22×SD16×C137C2
socleC22×C137C22×D4
prime radical#22577970C2
hypercenterC22×SD16D137
hypocenterC137#3057
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#17984094C2semidirect
2. #22577977C2semidirect
4. #22577646C2semidirect
5. #22577629C2semidirect
6. #22577441C2semidirect
7. #22577412C2semidirect
8. #22577408C2semidirect
9. C22×SD16×C137C2semidirect
13. C2×#14555493C22semidirect
14. C2×C8×D137C22semidirect
20. #17984064C22semidirect
25. #17984022C22semidirect
26. #17984020C22semidirect
27. #17984012C22semidirect
68. C2×C8×C137D4semidirect
120. C137#3057semidirect
122. C22×SD16D137semidirect
180. C2×C8C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840942
#225779772
#225779702
#225776462
#225776292
#225774412
#225774122
#225774082
C22×SD16×C1372
#3057137