Finite Group Databasebeta

← #31784116

Subgroups — #31784116

Order 17536.

1116 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×C22⋊D137C211yesmaximal, characteristic
2C2×#17984071C212yesmaximal, complemented
3#22577318C214yesmaximal, complemented
4#22577282C212yesmaximal, complemented
5#22577275C212yesmaximal, complemented
6#22577259C211yesmaximal, characteristic, complemented
7#22577258C212yesmaximal, complemented
8C22×C22⋊C4×C137C211yesmaximal, characteristic, complemented
9C24×D137C2211yescharacteristic
10C22×C22⋊D137—28no—
11C22×C22⋊D137—24no—
12C22×C22⋊D137C2212yes—
13C22×C22⋊D137C2212yes—
14C23×Dic137C2211yescharacteristic
15#17984071—28no—
16C2×#14555489C2214yescomplemented
17C2×C22⋊Dic137C2212yes—
18C2×C22⋊Dic137C2212yescomplemented
19C2×C22⋊Dic137C2212yescomplemented
20C2×#14555487C2214yescomplemented
21C2×#14555487C2214yescomplemented
22#17983995C2214yescomplemented
23C25×C137C2211yescharacteristic
24C23×C4×C137C2211yescharacteristic, complemented
25C23×C4×C137C2211yescharacteristic, complemented
26C2×C22⋊C4×C137—24no—
27C2×C22⋊C4×C137C2212yes—
28C2×C22⋊C4×C137C2212yescomplemented
29C23×D137—24no—
30C23×D137—22no—
31C23×D137—24no—
32C23×D137—42no—
33C23×D137C2312yes—
34C2×C22⋊D137—232no—
35C2×C22⋊D137—24no—
36C2×C22⋊D137—24no—
37C2×C22⋊D137—24no—
38C2×C22⋊D137—24no—
39C2×C22⋊D137—24no—
40C2×C22⋊D137D418yes—
41C22×Dic137—24no—
42C22×Dic137—22no—
43C22×Dic137C2314yes—
44C22×Dic137C2312yes—
45#14555489—28no—
46C22⋊Dic137—24no—
47C22⋊Dic137—24no—
48C22⋊Dic137—42no—
49#14555487—28no—
50#14555487—44no—
51C24×C137—24no—
52C24×C137—24no—
53C24×C137—42no—
54C24×C137D414yes—
55C24×C137C2311yescharacteristic
56C24×C137C2311yescharacteristic
57C24×C137C2311yescharacteristic
58C22×C4×C137—24no—
59C22×C4×C137—22no—
60C22×C4×C137—22no—
61C22×C4×C137—42no—
62C22×C4×C137C2312yes—
63C22×C4×C137C2312yes—
64C22⋊C4×C137—24no—
65C22⋊C4×C137—24no—
66C22×D137—28no—
67C22×D137—22no—
68C22×D137—22no—
69C22×D137—24no—
70C22×D137—24no—
71C22×D137—416no—
72C22×D137—44no—
73C22×D137—44no—
74C22×D137C2×D414yes—
75C22⋊D137—216no—
76C22⋊D137—216no—
77C2×Dic137—28no—
78C2×Dic137—22no—
79C2×Dic137—22no—
80C2×Dic137—24no—
81C2×Dic137—24no—
82C2×Dic137—24no—
83C2×Dic137C2×D414yes—
84C23×C137—22no—
85C23×C137—22no—
86C23×C137—22no—
87C23×C137—22no—
88C23×C137—28no—
89C23×C137—28no—
90C23×C137—22no—
91C23×C137—22no—
92C23×C137—22no—
93C23×C137—44no—
94C23×C137—416no—
95C23×C137C2×D414yes—
96C23×C137C2×D412yes—
97C23×C137C2×D412yes—
98C23×C137C2×D412yes—
99C23×C137#4114yes—
100C23×C137C2411yescharacteristic
101C2×C4×C137—22no—
102C2×C4×C137—22no—
103C2×C4×C137—24no—
104C2×C4×C137—24no—
105C2×C4×C137—44no—
106C2×C4×C137—44no—
107C22×C137—22no—
108C22×C137—22no—
109C22×C137—22no—
110C22×C137—22no—
111C22×C137—28no—
112C22×C137—28no—
113C22×C137—22no—
114C22×C137—22no—
115C22×C137—22no—
116C22×C137—416no—
117C22×C137—44no—
118C22×C137#12014yes—
119C22×C137#12314yescomplemented
120C22×C137C22×D412yes—
121C22×C137C22×D411yescharacteristic
122C22×C137C2×#4112yes—
123C22×C137H(5,𝔽2)12yes—
124C2×D137—24no—
125C2×D137—28no—
126C2×D137—416no—
127C2×D137—42no—
128C2×D137—42no—
129C2×D137—44no—
130C4×C137—42no—
131C4×C137—42no—
132Dic137—24no—
133Dic137—44no—
134C2×C137—24no—
135C2×C137—24no—
136C2×C137—42no—
137C2×C137C2×#12011yescharacteristic
138C2×C137C2×#12311yescharacteristic
139C2×C137#53411yescharacteristic
140C2×C137#54614yes—
141D137—44no—
142C137#240511yescharacteristic, complemented, Sylow
143#2405—1371nomaximal, Sylow
144C23×D4—1371no—
145C2×#123—1372no—
146C22×C22⋊C4D13711yescharacteristic, complemented
147#394—1374no—
148#386—1372no—
149#385—1372no—
150#379—1371no—
151#379—1372no—
152C25—1371no—
153C25C2×D13711yescharacteristic
154C22×D4—2744no—
155C22×D4—2748no—
156C22×D4—1372no—
157C22×D4—1372no—
158C23×C4—1371no—
159C23×C4C2×D13711yescharacteristic, complemented
160C23×C4C2×D13711yescharacteristic, complemented
161#123—2748no—
162C2×C4⋊C4—1374no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—1372no—
165C2×C22⋊C4—1372no—
166C2×C22⋊C4—1374no—
167C2×C22⋊C4—1372no—
168C2×C22⋊C4—1374no—
169C2×C22⋊C4C2×D13712yes—
170C2×C22⋊C4C2×D13712yescomplemented
171#95—1374no—
172C24—24no—
173C24—24no—
174C24—42no—
175C24—2744no—
176C24—2742no—
177C24—2744no—
178C24—5482no—
179C24—1372no—
180C24C22⋊D13714yes—
181C24C22×D13711yescharacteristic
182C24C22×D13711yescharacteristic
183C24C22×D13711yescharacteristic
184C2×D4—27432no—
185C2×D4—2744no—
186C2×D4—2744no—
187C2×D4—2744no—
188C2×D4—2744no—
189C2×D4—2744no—
190C2×D4—1378no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—2742no—
196C22×C4—2744no—
197C22×C4—1374no—
198C22×C4—1372no—
199C22×C4C22×D13712yes—
200C22×C4C22×D13712yes—
201C4⋊C4—2748no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—2748no—
205C22⋊C4—2744no—
206C22⋊C4—2744no—
207C22⋊C4—5484no—
208C22⋊C4—5482no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—44no—
219C23—416no—
220C23—2748no—
221C23—2742no—
222C23—2742no—
223C23—2744no—
224C23—2744no—
225C23—54816no—
226C23—5484no—
227C23—5484no—
228C23—1374no—
229C23D4×D13714yes—
230C23C2×C22⋊D13712yes—
231C23C2×C22⋊D13712yes—
232C23C2×C22⋊D13712yes—
233C23#1455551014yes—
234C23C23×D13711yescharacteristic
235D4—27416no—
236D4—27416no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—2748no—
244C2×C4—2742no—
245C2×C4—2742no—
246C2×C4—2744no—
247C2×C4—2744no—
248C2×C4—2744no—
249C2×C4—1374no—
250C22—22no—
251C22—22no—
252C22—28no—
253C22—28no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—416no—
260C22—44no—
261C22—2748no—
262C22—2744no—
263C22—54816no—
264C22—5482no—
265C22—5484no—
266C22—5482no—
267C22#1798405614yes—
268C22#1798407114yescomplemented
269C22C2×D4×D13712yes—
270C22C22×C22⋊D13711yescharacteristic
271C22C2×#1455551012yes—
272C22#1798413312yes—
273C4—42no—
274C4—42no—
275C4—2744no—
276C4—5484no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—5484no—
281C2C2×#1798405611yescharacteristic
282C2C2×#1798407111yescharacteristic
283C2#2257796211yescharacteristic
284C2#2257804314yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
FittingC22×C22⋊C4×C137C2
prime radicalC23×C22⋊D137C2
hypercenterC22×C22⋊C4D137
hypocenterC137#2405
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#17984071C2semidirect
3. #22577318C2semidirect
4. #22577282C2semidirect
5. #22577275C2semidirect
6. #22577259C2semidirect
7. #22577258C2semidirect
8. C22×C22⋊C4×C137C2semidirect
16. C2×#14555489C22semidirect
18. C2×C22⋊Dic137C22semidirect
19. C2×C22⋊Dic137C22semidirect
20. C2×#14555487C22semidirect
21. C2×#14555487C22semidirect
22. #17983995C22semidirect
24. C23×C4×C137C22semidirect
25. C23×C4×C137C22semidirect
28. C2×C22⋊C4×C137C22semidirect
119. C22×C137#123semidirect
142. C137#2405semidirect
146. C22×C22⋊C4D137semidirect
159. C23×C4C2×D137semidirect
160. C23×C4C2×D137semidirect
170. C2×C22⋊C4C2×D137semidirect
268. C22#17984071semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1372
C2×#179840712
#225773182
#225772822
#225772752
#225772592
#225772582
C22×C22⋊C4×C1372
#2405137