Finite Group Databasebeta

← #31790282

Subgroups — #31790282

Order 17536.

500 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 133 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×#17984121C211yesmaximal, characteristic, complemented
2C2×#17984114C211yesmaximal, characteristic
3C2×#17984075C211yesmaximal, characteristic, complemented
4#22577871C211yesmaximal, characteristic, complemented
5#22577624C214yesmaximal, complemented
6#22577617C214yesmaximal, complemented
7#22577426C211yesmaximal, characteristic, complemented
8#22577411C211yesmaximal, characteristic, complemented
9#418×C137C211yesmaximal, characteristic, complemented
10C2×#14555510C2211yescharacteristic
11C22×Dic274C2211yescharacteristic
12#17984121—24no—
13#17984114—24no—
14C2×Dic548C2212yescomplemented
15C2×#14555499C2212yescomplemented
16C2×#14555498C2212yes—
17C2×#14555495C2212yes—
18C2×#14555494C2211yescharacteristic
19#17984075—21no—
20#17984075—21no—
21#17984075C2214yes—
22#17984053C2212yescomplemented
23#17984050C2212yescomplemented
24#17984048—22no—
25#17984045—21no—
26C2×C22⋊Dic137—22no—
27C2×C4×Dic137C2211yescharacteristic
28C42×D137—21no—
29#17984017C2214yescomplemented
30#17984011C2214yescomplemented
31C22×D4×C137C2211yescharacteristic
32C2×M16×C137C2211yescharacteristic
33#117×C137C2211yescharacteristic
34#102×C137C2214yes—
35#14555510—22no—
36#14555510D414yes—
37C2×Dic274—21no—
38C2×Dic274—22no—
39C2×Dic274—22no—
40C2×Dic274C2312yes—
41C2×Dic274C2311yescharacteristic
42C2×D548C2311yescharacteristic
43C2×C22⋊D137—21no—
44C22×Dic137—21no—
45C22×Dic137D412yes—
46C2×C4×D137—22no—
47C2×C4×D137—21no—
48Dic548—24no—
49#14555499—24no—
50#14555498—24no—
51#14555495—24no—
52#14555494—22no—
53C2×#11764040C2312yes—
54#14555489—22no—
55C22⋊Dic137—24no—
56C22⋊Dic137—24no—
57#14555487—22no—
58C4×Dic137—22no—
59C4×Dic137D411yescharacteristic
60C4×Dic137D411yescharacteristic
61C4×Dic137C2312yes—
62C24×C137—21no—
63C2×D4×C137—22no—
64C2×D4×C137—21no—
65C2×D4×C137—22no—
66C2×D4×C137C2312yes—
67C22×C4×C137C2311yescharacteristic
68M16×C137—22no—
69C2×C8×C137C2312yes—
70C4⋊C4×C137C2312yes—
71C22⋊C4×C137—21no—
72C42×C137—21no—
73C22×D137—21no—
74Dic274—21no—
75Dic274—22no—
76Dic274—22no—
77Dic274—42no—
78Dic274C2×D412yes—
79D548—21no—
80D548C2×D412yes—
81C22⋊D137—24no—
82C2×Dic137—21no—
83C2×Dic137—22no—
84C2×Dic137—42no—
85C2×Dic137—42no—
86C2×Dic137C2×D412yes—
87C2×Dic137C2×D412yes—
88C4×D137—24no—
89C4×D137—44no—
90#11764040—22no—
91C23×C137—22no—
92C23×C137—44no—
93C23×C137—42no—
94C23×C137C2×D411yescharacteristic
95D4×C137—22no—
96D4×C137—22no—
97D4×C137—42no—
98C2×C4×C137—22no—
99C2×C4×C137C2×D411yescharacteristic
100C2×C4×C137C2×D412yes—
101C2×C4×C137#4112yes—
102C2×C4×C137C2411yescharacteristic
103C8×C137—22no—
104C22×C137—22no—
105C22×C137—42no—
106C22×C137—44no—
107C22×C137—84no—
108C22×C137#12112yes—
109C22×C137C22×D411yescharacteristic
110C2×D137—22no—
111C2×D137—42no—
112C4×C137—42no—
113C4×C137#12112yes—
114C4×C137C22×D411yescharacteristic
115C4×C137C2×#4111yescharacteristic
116Dic137—22no—
117Dic137—22no—
118Dic137—22no—
119Dic137—42no—
120C2×C137—22no—
121C2×C137—82no—
122C2×C137C2×#12111yescharacteristic
123C2×C137#57512yes—
124D137—42no—
125C137#304011yescharacteristic, complemented, Sylow
126#3040—1371nomaximal, Sylow
127C2×#137—1371no—
128C2×AGL(1,ℤ/8)—1371no—
129C2×#124—1371no—
130C4×#41—1371no—
131#464—1374no—
132#463—1374no—
133#423—1371no—
134#419—1371no—
135#418D13711yescharacteristic, complemented
136C2×#41—1371no—
137C22×Q8—1371no—
138C22×D4C2×D13711yescharacteristic
139#137—2744no—
140AGL(1,ℤ/8)—2744no—
141C2×Q16—1372no—
142C2×SD16—1372no—
143C2×SD16—1372no—
144C2×D8—1372no—
145C2×M16—1371no—
146C2×M16C2×D13711yescharacteristic
147#124—2741no—
148#124—2741no—
149#124—1374no—
150C4×Q8—1372no—
151C4×D4—2742no—
152C4×D4—1372no—
153#117—2741no—
154#117C2×D13711yescharacteristic
155C2×C22⋊C4—2742no—
156C2×C42—2741no—
157C2×C42—1371no—
158C4⋊C8—1374no—
159#103—1374no—
160#102C2×D13714yes—
161C24—21no—
162#41—2742no—
163#41—1374no—
164C2×Q8—2741no—
165C2×Q8—2742no—
166C2×Q8—2742no—
167C2×Q8—1372no—
168C2×Q8—1371no—
169C2×D4—22no—
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—2741no—
173C2×D4—1371no—
174C2×D4C22×D13712yes—
175C22×C4—2742no—
176C22×C4—2741no—
177C22×C4—2741no—
178C22×C4—1372no—
179C22×C4C22×D13711yescharacteristic
180Q16—2744no—
181SD16—2744no—
182SD16—2744no—
183D8—2744no—
184M16—22no—
185M16—2742no—
186C2×C8—1372no—
187C2×C8C22×D13712yes—
188C4⋊C4—2742no—
189C4⋊C4C22×D13712yes—
190C22⋊C4—21no—
191C22⋊C4—2742no—
192C22⋊C4—2744no—
193C22⋊C4—2744no—
194C42—21no—
195C42—2742no—
196C42—1371no—
197C42—1371no—
198C42—1372no—
199C23—22no—
200C23—44no—
201C23—42no—
202C23—2741no—
203C23D4×D13711yescharacteristic
204Q8—2741no—
205Q8—2742no—
206Q8—2742no—
207Q8—5482no—
208Q8—1372no—
209D4—22no—
210D4—22no—
211D4—42no—
212D4—2744no—
213D4—2741no—
214D4—1372no—
215C2×C4—22no—
216C2×C4—2741no—
217C2×C4—2744no—
218C2×C4—2742no—
219C2×C4—5484no—
220C2×C4—5482no—
221C2×C4—5482no—
222C2×C4—1372no—
223C2×C4—1372no—
224C2×C4D4×D13711yescharacteristic
225C2×C4D4×D13712yes—
226C2×C4#1455551012yes—
227C2×C4C23×D13711yescharacteristic
228C8—22no—
229C8—2742no—
230C22—22no—
231C22—44no—
232C22—42no—
233C22—84no—
234C22—2742no—
235C22—5482no—
236C22#1798406112yes—
237C22C2×D4×D13711yescharacteristic
238C4—42no—
239C4—2742no—
240C4—2742no—
241C4—2742no—
242C4—5482no—
243C4#1798406112yes—
244C4C2×D4×D13711yescharacteristic
245C4C2×#1455551011yescharacteristic
246C2—22no—
247C2—82no—
248C2—5482no—
249C2C2×#1798406111yescharacteristic
250C2#2257819912yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#418×C137C2
socleC22×C137C22×D4
prime radicalC2×#17984114C2
hypercenter#418D137
hypocenterC137#3040
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#17984121C2semidirect
3. C2×#17984075C2semidirect
4. #22577871C2semidirect
5. #22577624C2semidirect
6. #22577617C2semidirect
7. #22577426C2semidirect
8. #22577411C2semidirect
9. #418×C137C2semidirect
14. C2×Dic548C22semidirect
15. C2×#14555499C22semidirect
22. #17984053C22semidirect
23. #17984050C22semidirect
29. #17984017C22semidirect
30. #17984011C22semidirect
125. C137#3040semidirect
135. #418D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841212
C2×#179841142
C2×#179840752
#225778712
#225776242
#225776172
#225774262
#225774112
#418×C1372
#3040137