Finite Group Databasebeta

← #31790730

Subgroups — #31790730

Order 17536.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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×#17984062C211yesmaximal, characteristic
2C2×#17984047C211yesmaximal, characteristic, complemented
3#22577728C214yesmaximal, complemented
4#22577722C212yesmaximal, complemented
5#22577717C212yesmaximal, complemented
6#22577445C212yesmaximal, complemented
7#22577409C212yesmaximal, complemented
8#407×C137C211yesmaximal, characteristic, complemented
9C22×D548C2211yescharacteristic
10C22×C22⋊D137—21no—
11#17984062—21no—
12#17984062—21no—
13#17984062C2212yes—
14#17984062C2212yes—
15#17984047—21no—
16#17984047—21no—
17#17984047C2212yes—
18#17984047C2212yescomplemented
19C2×C4⋊Dic137C2211yescharacteristic
20C2×C4⋊Dic137C2211yescharacteristic, complemented
21C2×C4⋊Dic137C2211yescharacteristic, complemented
22C2×C22⋊Dic137—21no—
23C2×#14555487—21no—
24C2×C4×Dic137—21no—
25#17984022C2214yescomplemented
26#17984020C2214yescomplemented
27#17984010C2214yescomplemented
28#17984010C2214yescomplemented
29C23×C4×C137C2211yescharacteristic
30C2×M16×C137C2212yes—
31C22⋊C8×C137C2212yescomplemented
32C22⋊C8×C137C2212yescomplemented
33C23×D137—21no—
34C2×D548—21no—
35C2×D548—24no—
36C2×D548C2312yes—
37C2×C22⋊D137—22no—
38C2×C22⋊D137—22no—
39C2×C22⋊D137—44no—
40C22×Dic137—21no—
41C22×Dic137—21no—
42C22×Dic137—21no—
43C4⋊Dic137—21no—
44C4⋊Dic137—21no—
45C4⋊Dic137—21no—
46C4⋊Dic137C2312yes—
47C4⋊Dic137C2312yes—
48C4⋊Dic137C2312yes—
49C22⋊Dic137—22no—
50C22⋊Dic137—22no—
51#14555487—22no—
52#14555487—22no—
53C4×Dic137—22no—
54C4×Dic137—22no—
55C24×C137D411yescharacteristic
56C22×C4×C137—24no—
57C22×C4×C137D411yescharacteristic
58C22×C4×C137D411yescharacteristic
59C22×C4×C137D411yescharacteristic
60C22×C4×C137C2311yescharacteristic
61C22×C4×C137C2311yescharacteristic
62C22×C4×C137C2311yescharacteristic
63M16×C137—24no—
64C2×C8×C137C2314yes—
65C22×D137—22no—
66C22×D137—42no—
67C22×D137—42no—
68C22×D137—42no—
69D548—24no—
70D548—42no—
71C22⋊D137—44no—
72C22⋊D137—44no—
73C2×Dic137—22no—
74C2×Dic137—22no—
75C2×Dic137—22no—
76C2×Dic137—42no—
77C2×Dic137—42no—
78C2×Dic137—42no—
79C23×C137—22no—
80C23×C137—22no—
81C23×C137—41no—
82C23×C137C2×D411yescharacteristic
83C23×C137C2×D411yescharacteristic
84C23×C137C2×D411yescharacteristic
85C2×C4×C137—22no—
86C2×C4×C137—22no—
87C2×C4×C137—22no—
88C2×C4×C137—22no—
89C2×C4×C137—22no—
90C2×C4×C137C2×D411yescharacteristic
91C2×C4×C137C2×D411yescharacteristic
92C2×C4×C137C2×D411yescharacteristic
93C2×C4×C137#4114yes—
94C2×C4×C137C2411yescharacteristic
95C8×C137—24no—
96C22×C137—21no—
97C22×C137—21no—
98C22×C137—21no—
99C22×C137—21no—
100C22×C137—21no—
101C22×C137—21no—
102C22×C137—21no—
103C22×C137—44no—
104C22×C137#12312yes—
105C22×C137AGL(1,ℤ/8)12yes—
106C22×C137C22×D411yescharacteristic
107C2×D137—42no—
108C2×D137—42no—
109C2×D137—42no—
110C2×D137—84no—
111C4×C137—21no—
112C4×C137—21no—
113C4×C137#12312yes—
114C4×C137C2×#4112yes—
115Dic137—42no—
116Dic137—42no—
117Dic137—42no—
118C2×C137—22no—
119C2×C137—22no—
120C2×C137—41no—
121C2×C137C2×#12311yescharacteristic
122C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
123C2×C137#57511yescharacteristic
124D137—82no—
125C137#309411yescharacteristic, complemented, Sylow
126#3094—1371nomaximal, Sylow
127C2×#121—1371no—
128C2×#117—1371no—
129#482—1374no—
130#481—1372no—
131#480—1372no—
132#428—1372no—
133#418—1372no—
134#407D13711yescharacteristic, complemented
135C22×D4—2741no—
136C22×D4—1371no—
137C23×C4C2×D13711yescharacteristic
138C2×M16C2×D13712yes—
139#121—2741no—
140#121—2741no—
141#121—1372no—
142#121—1372no—
143#117—2741no—
144#117—2741no—
145#117—1372no—
146#117—1372no—
147C2×C4⋊C4—1371no—
148C2×C4⋊C4—1371no—
149C2×C4⋊C4—1371no—
150C2×C22⋊C4—2741no—
151C2×C22⋊C4—2741no—
152C2×C42—2741no—
153#107—1374no—
154#106—1374no—
155#102—1374no—
156#102—1374no—
157C22⋊C8C2×D13712yescomplemented
158C22⋊C8C2×D13712yescomplemented
159C24—2741no—
160C24D54811yescharacteristic
161C2×D4—2742no—
162C2×D4—2742no—
163C2×D4—2741no—
164C2×D4—2744no—
165C2×D4—5484no—
166C2×D4—1372no—
167C22×C4—24no—
168C22×C4—2741no—
169C22×C4—2741no—
170C22×C4—2741no—
171C22×C4D54811yescharacteristic
172C22×C4D54811yescharacteristic
173C22×C4D54811yescharacteristic
174C22×C4C22×D13711yescharacteristic
175C22×C4C22×D13711yescharacteristic
176C22×C4C22×D13711yescharacteristic
177M16—24no—
178C2×C8C22×D13714yes—
179C4⋊C4—2741no—
180C4⋊C4—2741no—
181C4⋊C4—2741no—
182C4⋊C4—1372no—
183C4⋊C4—1372no—
184C4⋊C4—1372no—
185C22⋊C4—2742no—
186C22⋊C4—2742no—
187C22⋊C4—2742no—
188C22⋊C4—2742no—
189C42—2742no—
190C42—2742no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—2742no—
195C23—5482no—
196C23—5482no—
197C23—5482no—
198C23C2×D54811yescharacteristic
199C23C2×D54811yescharacteristic
200C23C2×D54811yescharacteristic
201D4—2744no—
202D4—5484no—
203D4—5484no—
204D4—5482no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—2742no—
211C2×C4—2742no—
212C2×C4—2742no—
213C2×C4—5482no—
214C2×C4—5482no—
215C2×C4—5482no—
216C2×C4C2×D54811yescharacteristic
217C2×C4C2×D54811yescharacteristic
218C2×C4C2×D54811yescharacteristic
219C2×C4#1455550914yes—
220C2×C4C23×D13711yescharacteristic
221C8—24no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—5482no—
231C22—5482no—
232C22—5482no—
233C22—10964no—
234C22#1798407212yes—
235C22#1798411612yes—
236C22C22×D54811yescharacteristic
237C4—21no—
238C4—21no—
239C4—5482no—
240C4—5482no—
241C4—5482no—
242C4#1798407212yes—
243C4C2×#1455550912yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—10962no—
248C2C2×#1798407211yescharacteristic
249C2C2×#1798411611yescharacteristic
250C2#2257820311yescharacteristic
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#407×C137C2
socleC22×C137C22×D4
prime radicalC2×#17984062C2
hypercenter#407D137
hypocenterC137#3094
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#17984047C2semidirect
3. #22577728C2semidirect
4. #22577722C2semidirect
5. #22577717C2semidirect
6. #22577445C2semidirect
7. #22577409C2semidirect
8. #407×C137C2semidirect
18. #17984047C22semidirect
20. C2×C4⋊Dic137C22semidirect
21. C2×C4⋊Dic137C22semidirect
25. #17984022C22semidirect
26. #17984020C22semidirect
27. #17984010C22semidirect
28. #17984010C22semidirect
31. C22⋊C8×C137C22semidirect
32. C22⋊C8×C137C22semidirect
125. C137#3094semidirect
134. #407D137semidirect
157. C22⋊C8C2×D137semidirect
158. C22⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840622
C2×#179840472
#225777282
#225777222
#225777172
#225774452
#225774092
#407×C1372
#3094137