Finite Group Databasebeta

← #31790268

Subgroups — #31790268

Order 17536.

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

normal subgroups only

This group has 584 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×#17984116C211yesmaximal, characteristic, complemented
2C2×#17984114C211yesmaximal, characteristic, complemented
3C2×#17984084C211yesmaximal, characteristic, complemented
4#22577871C211yesmaximal, characteristic, complemented
5#22577602C214yesmaximal, complemented
6#22577597C214yesmaximal, complemented
7#22577426C211yesmaximal, characteristic, complemented
8#22577407C211yesmaximal, characteristic, complemented
9#418×C137C211yesmaximal, characteristic, complemented
10C2×#14555510C2211yescharacteristic
11C22×D548C2211yescharacteristic
12C22×C22⋊D137—22no—
13#17984116—24no—
14#17984114—24no—
15C2×#14555499C2212yescomplemented
16C2×#14555498C2212yescomplemented
17C2×D1096C2212yescomplemented
18C2×#14555495C2212yescomplemented
19C2×#14555494C2211yescharacteristic
20#17984084—21no—
21#17984084—21no—
22#17984084C2214yes—
23#17984053C2212yescomplemented
24#17984050C2212yescomplemented
25#17984048—22no—
26#17984045—21no—
27C2×C4×Dic137C2211yescharacteristic
28C42×D137—21no—
29#17984017C2214yescomplemented
30#17984008C2214yescomplemented
31C22×D4×C137C2211yescharacteristic
32C2×M16×C137C2211yescharacteristic
33#117×C137C2211yescharacteristic
34#102×C137C2214yescomplemented
35C23×D137—21no—
36#14555510—22no—
37#14555510D414yescomplemented
38C2×Dic274C2311yescharacteristic
39C2×D548—22no—
40C2×D548—21no—
41C2×D548—22no—
42C2×D548C2312yes—
43C2×D548C2311yescharacteristic
44C2×C22⋊D137—24no—
45C2×C22⋊D137—24no—
46C2×C22⋊D137—21no—
47C2×C22⋊D137—48no—
48C22×Dic137D412yes—
49C2×C4×D137—22no—
50C2×C4×D137—21no—
51#14555499—24no—
52#14555498—24no—
53D1096—24no—
54#14555495—24no—
55#14555494—22no—
56C2×#11764040C2312yes—
57#14555489—22no—
58#14555487—22no—
59C4×Dic137—22no—
60C4×Dic137D411yescharacteristic
61C4×Dic137D411yescharacteristic
62C4×Dic137C2312yes—
63C24×C137—21no—
64C2×D4×C137—22no—
65C2×D4×C137—21no—
66C2×D4×C137—22no—
67C2×D4×C137C2312yes—
68C22×C4×C137C2311yescharacteristic
69M16×C137—22no—
70C2×C8×C137C2312yes—
71C4⋊C4×C137C2312yes—
72C22⋊C4×C137—21no—
73C42×C137—21no—
74C22×D137—21no—
75C22×D137—22no—
76C22×D137—44no—
77C22×D137—42no—
78Dic274—21no—
79Dic274C2×D412yescomplemented
80D548—22no—
81D548—21no—
82D548—22no—
83D548—42no—
84D548C2×D412yescomplemented
85C22⋊D137—24no—
86C22⋊D137—48no—
87C22⋊D137—48no—
88C2×Dic137—21no—
89C2×Dic137—24no—
90C2×Dic137C2×D412yes—
91C2×Dic137C2×D412yes—
92C4×D137—24no—
93C4×D137—44no—
94#11764040—22no—
95C23×C137—22no—
96C23×C137—44no—
97C23×C137—42no—
98C23×C137C2×D411yescharacteristic
99D4×C137—22no—
100D4×C137—22no—
101D4×C137—42no—
102C2×C4×C137—22no—
103C2×C4×C137C2×D411yescharacteristic
104C2×C4×C137C2×D412yes—
105C2×C4×C137#4112yes—
106C2×C4×C137C2411yescharacteristic
107C8×C137—22no—
108C22×C137—22no—
109C22×C137—42no—
110C22×C137—44no—
111C22×C137—84no—
112C22×C137#12112yes—
113C22×C137C22×D411yescharacteristic
114C2×D137—22no—
115C2×D137—42no—
116C2×D137—42no—
117C2×D137—44no—
118C2×D137—84no—
119C4×C137—42no—
120C4×C137#12112yes—
121C4×C137C22×D411yescharacteristic
122C4×C137C2×#4111yescharacteristic
123Dic137—22no—
124Dic137—22no—
125Dic137AGL(1,ℤ/8)14yescomplemented
126C2×C137—22no—
127C2×C137—82no—
128C2×C137C2×#12111yescharacteristic
129C2×C137C2×AGL(1,ℤ/8)12yes—
130D137—42no—
131D137—82no—
132C137#303811yescharacteristic, complemented, Sylow
133#3038—1371nomaximal, Sylow
134C2×AGL(1,ℤ/8)—1371no—
135C2×AGL(1,ℤ/8)—1371no—
136C2×#127—1371no—
137C4×#41—1371no—
138#460—1374no—
139#459—1374no—
140#423—1371no—
141#418—1371no—
142#418D13711yescharacteristic, complemented
143C2×#41—1371no—
144C22×D4—2742no—
145C22×D4—1371no—
146C22×D4C2×D13711yescharacteristic
147AGL(1,ℤ/8)—2744no—
148AGL(1,ℤ/8)—2744no—
149C2×SD16—1372no—
150C2×SD16—1372no—
151C2×D8—1372no—
152C2×D8—1372no—
153C2×M16—1371no—
154C2×M16C2×D13711yescharacteristic
155#127—2741no—
156#127—2741no—
157#127—1374no—
158C4×Q8—1372no—
159C4×D4—2742no—
160C4×D4—1372no—
161#117—2741no—
162#117C2×D13711yescharacteristic
163C2×C42—2741no—
164C2×C42—1371no—
165C4⋊C8—1374no—
166#102—1374no—
167#102C2×D13714yescomplemented
168C24—21no—
169C24—2741no—
170#41—2742no—
171#41—1374no—
172C2×Q8—1371no—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—2741no—
177C2×D4—2744no—
178C2×D4—2744no—
179C2×D4—2742no—
180C2×D4—2741no—
181C2×D4—2742no—
182C2×D4—5488no—
183C2×D4—1372no—
184C2×D4—1371no—
185C2×D4C22×D13712yes—
186C22×C4—2742no—
187C22×C4—2741no—
188C22×C4—1372no—
189C22×C4C22×D13711yescharacteristic
190SD16—2744no—
191SD16—2744no—
192D8—2744no—
193D8—2744no—
194M16—22no—
195M16—2742no—
196C2×C8—1372no—
197C2×C8C22×D13712yes—
198C4⋊C4—2742no—
199C4⋊C4C22×D13712yes—
200C22⋊C4—21no—
201C22⋊C4—2742no—
202C42—21no—
203C42—2742no—
204C42—1371no—
205C42—1371no—
206C42—1372no—
207C23—22no—
208C23—44no—
209C23—42no—
210C23—2741no—
211C23—2742no—
212C23—5484no—
213C23—5482no—
214C23D4×D13711yescharacteristic
215Q8—2741no—
216Q8—1372no—
217D4—22no—
218D4—22no—
219D4—42no—
220D4—2744no—
221D4—2742no—
222D4—2741no—
223D4—2742no—
224D4—5488no—
225D4—5488no—
226D4—5482no—
227D4—1372no—
228C2×C4—22no—
229C2×C4—2741no—
230C2×C4—2744no—
231C2×C4—2744no—
232C2×C4—5484no—
233C2×C4—1372no—
234C2×C4—1372no—
235C2×C4D4×D13711yescharacteristic
236C2×C4D4×D13712yes—
237C2×C4#1455551012yes—
238C2×C4C23×D13711yescharacteristic
239C8—22no—
240C8—2742no—
241C22—22no—
242C22—44no—
243C22—42no—
244C22—84no—
245C22—2742no—
246C22—5482no—
247C22—5484no—
248C22—5482no—
249C22—10964no—
250C22#1798406112yes—
251C22C2×D4×D13711yescharacteristic
252C4—42no—
253C4—2742no—
254C4—2742no—
255C4—1374no—
256C4#1798406112yes—
257C4C2×D4×D13711yescharacteristic
258C4C2×#1455551011yescharacteristic
259C2—22no—
260C2—82no—
261C2—5482no—
262C2—10962no—
263C2C2×#1798406111yescharacteristic
264C2AGL(1,ℤ/8)×D13712yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#418×C137C2
socleC22×C137C22×D4
hypercenter#418D137
hypocenterC137#3038
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×#17984116C2semidirect
2. C2×#17984114C2semidirect
3. C2×#17984084C2semidirect
4. #22577871C2semidirect
5. #22577602C2semidirect
6. #22577597C2semidirect
7. #22577426C2semidirect
8. #22577407C2semidirect
9. #418×C137C2semidirect
15. C2×#14555499C22semidirect
16. C2×#14555498C22semidirect
17. C2×D1096C22semidirect
18. C2×#14555495C22semidirect
23. #17984053C22semidirect
24. #17984050C22semidirect
29. #17984017C22semidirect
30. #17984008C22semidirect
34. #102×C137C22semidirect
37. #14555510D4semidirect
79. Dic274C2×D4semidirect
84. D548C2×D4semidirect
125. Dic137AGL(1,ℤ/8)semidirect
132. C137#3038semidirect
142. #418D137semidirect
167. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841162
C2×#179841142
C2×#179840842
#225778712
#225776022
#225775972
#225774262
#225774072
#418×C1372
#3038137