Finite Group Databasebeta

← #31790729

Subgroups — #31790729

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
2#22577727C214yesmaximal, complemented
3#22577721C212yesmaximal, complemented
4#22577716C212yesmaximal, complemented
5#22577446C212yesmaximal, complemented
6#22577407C212yesmaximal, complemented
7#22577367C211yesmaximal, characteristic, complemented
8C2×#117×C137C211yesmaximal, characteristic, complemented
9C22×D548C2211yescharacteristic
10C22×C22⋊D137—21no—
11C2×#14555494C2212yes—
12#17984062—21no—
13#17984062—21no—
14#17984062C2212yes—
15#17984062C2212yes—
16C2×C4⋊Dic137C2211yescharacteristic
17C2×#14555487—21no—
18#17984021C2214yescomplemented
19#17984019C2214yescomplemented
20#17984008C2214yescomplemented
21#17984008C2214yescomplemented
22C22⋊#11764040C2212yescomplemented
23C22⋊#11764040C2212yescomplemented
24C23×C4×C137C2211yescharacteristic
25#117×C137—21no—
26#117×C137—21no—
27#117×C137C2212yes—
28#117×C137C2212yescomplemented
29C2×C4⋊C4×C137C2211yescharacteristic, complemented
30C2×C4⋊C4×C137C2211yescharacteristic, complemented
31C2×C22⋊C4×C137—21no—
32C2×C42×C137—21no—
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—
41#14555494—24no—
42C2×#11764040C2314yes—
43C4⋊Dic137—21no—
44C4⋊Dic137C2312yes—
45#14555487—22no—
46#14555487—22no—
47C24×C137D411yescharacteristic
48C22×C4×C137—24no—
49C22×C4×C137—21no—
50C22×C4×C137—21no—
51C22×C4×C137D411yescharacteristic
52C22×C4×C137D411yescharacteristic
53C22×C4×C137D411yescharacteristic
54C22×C4×C137C2311yescharacteristic
55C22×C4×C137C2311yescharacteristic
56C22×C4×C137C2311yescharacteristic
57C4⋊C4×C137—21no—
58C4⋊C4×C137—21no—
59C4⋊C4×C137C2312yes—
60C4⋊C4×C137C2312yes—
61C22⋊C4×C137—22no—
62C22⋊C4×C137—22no—
63C42×C137—22no—
64C42×C137—22no—
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—42no—
75#11764040—24no—
76C23×C137—22no—
77C23×C137—22no—
78C23×C137—41no—
79C23×C137C2×D411yescharacteristic
80C23×C137C2×D411yescharacteristic
81C23×C137C2×D411yescharacteristic
82C2×C4×C137—22no—
83C2×C4×C137—22no—
84C2×C4×C137—22no—
85C2×C4×C137—22no—
86C2×C4×C137—22no—
87C2×C4×C137—22no—
88C2×C4×C137—22no—
89C2×C4×C137—42no—
90C2×C4×C137—42no—
91C2×C4×C137C2×D411yescharacteristic
92C2×C4×C137C2×D411yescharacteristic
93C2×C4×C137C2×D411yescharacteristic
94C2×C4×C137#4114yes—
95C2×C4×C137C2411yescharacteristic
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—42no—
114C4×C137—42no—
115C4×C137#12312yes—
116C4×C137C2×#4112yes—
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×#117D13711yescharacteristic, complemented
129#482—1374no—
130#481—1372no—
131#480—1372no—
132#428—1372no—
133#418—1372no—
134#407—1371no—
135C22×D4—2741no—
136C22×D4—1371no—
137C23×C4C2×D13711yescharacteristic
138C2×M16—1372no—
139#121—2741no—
140#121—2741no—
141#121—1372no—
142#121—1372no—
143#117—21no—
144#117—21no—
145#117C2×D13712yes—
146#117C2×D13712yescomplemented
147C2×C4⋊C4—1371no—
148C2×C4⋊C4C2×D13711yescharacteristic, complemented
149C2×C4⋊C4C2×D13711yescharacteristic, complemented
150C2×C22⋊C4—21no—
151C2×C22⋊C4—2741no—
152C2×C42—21no—
153#107—1374no—
154#106—1374no—
155#102—1374no—
156#102—1374no—
157C22⋊C8—1372no—
158C22⋊C8—1372no—
159C24—2741no—
160C24C22⋊D13711yescharacteristic
161C2×D4—2742no—
162C2×D4—2742no—
163C2×D4—2741no—
164C2×D4—2744no—
165C2×D4—5484no—
166C2×D4—1372no—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—2741no—
171C22×C4C22⋊D13711yescharacteristic
172C22×C4C22⋊D13711yescharacteristic
173C22×C4C22⋊D13711yescharacteristic
174C22×C4C22×D13711yescharacteristic
175C22×C4C22×D13711yescharacteristic
176C22×C4C22×D13711yescharacteristic
177M16—2744no—
178C2×C8—1374no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—2741no—
182C4⋊C4—1372no—
183C4⋊C4C22×D13712yes—
184C4⋊C4C22×D13712yes—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—2742no—
188C22⋊C4—2742no—
189C42—22no—
190C42—22no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—2742no—
195C23—5482no—
196C23—5482no—
197C23—5482no—
198C23C2×C22⋊D13711yescharacteristic
199C23C2×C22⋊D13711yescharacteristic
200C23C2×C22⋊D13711yescharacteristic
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—22no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—2742no—
215C2×C4—5482no—
216C2×C4C2×C22⋊D13711yescharacteristic
217C2×C4C2×C22⋊D13711yescharacteristic
218C2×C4C2×C22⋊D13711yescharacteristic
219C2×C4#1455551014yes—
220C2×C4C23×D13711yescharacteristic
221C8—2744no—
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#1798407112yes—
235C22#1798411312yes—
236C22C22×C22⋊D13711yescharacteristic
237C4—21no—
238C4—21no—
239C4—42no—
240C4—42no—
241C4—5482no—
242C4#1798407112yes—
243C4C2×#1455551012yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—10962no—
248C2C2×#1798407111yescharacteristic
249C2C2×#1798411311yescharacteristic
250C2#2257820111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#117×C137C2
socleC22×C137C22×D4
prime radicalC2×#17984062C2
hypercenterC2×#117D137
hypocenterC137#3094
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22577727C2semidirect
3. #22577721C2semidirect
4. #22577716C2semidirect
5. #22577446C2semidirect
6. #22577407C2semidirect
7. #22577367C2semidirect
8. C2×#117×C137C2semidirect
18. #17984021C22semidirect
19. #17984019C22semidirect
20. #17984008C22semidirect
21. #17984008C22semidirect
22. C22⋊#11764040C22semidirect
23. C22⋊#11764040C22semidirect
28. #117×C137C22semidirect
29. C2×C4⋊C4×C137C22semidirect
30. C2×C4⋊C4×C137C22semidirect
125. C137#3094semidirect
128. C2×#117D137semidirect
146. #117C2×D137semidirect
148. C2×C4⋊C4C2×D137semidirect
149. C2×C4⋊C4C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179840622
#225777272
#225777212
#225777162
#225774462
#225774072
#225773672
C2×#117×C1372
#3094137