Finite Group Databasebeta

← #31784146

Subgroups — #31784146

Order 17536.

1232 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 189 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×D548C211yesmaximal, characteristic, complemented
2C2×#17984063C212yesmaximal, complemented
3C2×#17984058C212yesmaximal, complemented
4C2×C4×C22⋊D137C211yesmaximal, characteristic, complemented
5#22577321C212yesmaximal, complemented
6#22577313C212yesmaximal, complemented
7#22577304C212yesmaximal, complemented
8#22577258C212yesmaximal, complemented
9#380×C137C211yesmaximal, characteristic, complemented
10C24×D137C2212yes—
11C22×D548—28no—
12C22×D548—24no—
13C22×D548C2212yes—
14C22×D548C2212yes—
15C22×D548C2214yescomplemented
16C22×C22⋊D137C2211yescharacteristic
17C22×C4×D137C2211yescharacteristic, complemented
18#17984063—28no—
19#17984058—28no—
20C4×C22⋊D137—24no—
21C2×#14555489C2211yescharacteristic, complemented
22C2×C22⋊Dic137C2211yescharacteristic
23C2×#14555487C2214yescomplemented
24C2×#14555487C2214yescomplemented
25C2×#14555487C2211yescharacteristic, complemented
26C2×#14555487C2214yescomplemented
27C2×C4×Dic137C2211yescharacteristic, complemented
28C23×C4×C137C2211yescharacteristic
29C2×C4⋊C4×C137C2212yescomplemented
30C2×C22⋊C4×C137C2212yes—
31#95×C137C2212yescomplemented
32C23×D137—28no—
33C23×D137—24no—
34C23×D137—24no—
35C23×D137—24no—
36C23×D137—44no—
37C23×D137C2314yes—
38C23×D137C2311yescharacteristic
39C2×D548—232no—
40C2×D548—28no—
41C2×D548—28no—
42C2×D548—28no—
43C2×D548—24no—
44C2×D548—24no—
45C2×D548—24no—
46C2×D548—44no—
47C2×D548D418yescomplemented
48C2×C22⋊D137—22no—
49C2×C22⋊D137—22no—
50C2×C22⋊D137D414yescomplemented
51C22×Dic137C2311yescharacteristic
52C22×Dic137C2311yescharacteristic
53C22×Dic137C2311yescharacteristic
54C2×C4×D137—22no—
55C2×C4×D137—24no—
56#14555489—22no—
57C22⋊Dic137—22no—
58#14555487—28no—
59#14555487—28no—
60#14555487—22no—
61#14555487—44no—
62C4×Dic137—22no—
63C24×C137C2311yescharacteristic
64C22×C4×C137—24no—
65C22×C4×C137D414yescomplemented
66C22×C4×C137C2314yes—
67C22×C4×C137C2311yescharacteristic
68C22×C4×C137C2311yescharacteristic
69C4⋊C4×C137—24no—
70C22⋊C4×C137—24no—
71C22×D137—216no—
72C22×D137—21no—
73C22×D137—24no—
74C22×D137—22no—
75C22×D137—24no—
76C22×D137—24no—
77C22×D137—21no—
78C22×D137—24no—
79C22×D137—24no—
80C22×D137—22no—
81C22×D137—24no—
82C22×D137—416no—
83C22×D137—416no—
84C22×D137—48no—
85C22×D137—48no—
86C22×D137C2×D418yes—
87C22×D137C2×D412yescomplemented
88D548—216no—
89D548—216no—
90D548—416no—
91C22⋊D137—28no—
92C2×Dic137—21no—
93C2×Dic137—21no—
94C2×Dic137—21no—
95C2×Dic137—22no—
96C2×Dic137—21no—
97C2×Dic137—22no—
98C2×Dic137C2×D412yescomplemented
99C4×D137—44no—
100C23×C137—22no—
101C23×C137—22no—
102C23×C137C2×D412yes—
103C23×C137C2×D412yes—
104C23×C137#4112yes—
105C23×C137C2411yescharacteristic
106C2×C4×C137—22no—
107C2×C4×C137—28no—
108C2×C4×C137—24no—
109C2×C4×C137—24no—
110C2×C4×C137—24no—
111C2×C4×C137C2×D414yes—
112C2×C4×C137C2×D412yes—
113C2×C4×C137C2×D412yes—
114C22×C137—22no—
115C22×C137—22no—
116C22×C137—24no—
117C22×C137—24no—
118C22×C137#12114yescomplemented
119C22×C137C22×D412yes—
120C22×C137C22×D411yescharacteristic
121C22×C137C22×D411yescharacteristic
122C22×C137C2×#4111yescharacteristic
123C22×C137H(5,𝔽2)12yes—
124C2×D137—28no—
125C2×D137—22no—
126C2×D137—24no—
127C2×D137—28no—
128C2×D137—28no—
129C2×D137—432no—
130C2×D137—41no—
131C2×D137—44no—
132C2×D137—41no—
133C2×D137—44no—
134C2×D137—44no—
135C2×D137—42no—
136C2×D137—44no—
137C4×C137—22no—
138C4×C137—44no—
139C4×C137#12014yescomplemented
140Dic137—22no—
141Dic137—41no—
142Dic137—41no—
143C2×C137—24no—
144C2×C137C2×#12011yescharacteristic
145C2×C137C2×#12111yescharacteristic
146C2×C137#53511yescharacteristic
147C2×C137D4212yes—
148C2×C137#54612yes—
149D137—42no—
150D137—48no—
151C137#240811yescharacteristic, complemented, Sylow
152#2408—1371nomaximal, Sylow
153C23×D4—1371no—
154C2×#121—1372no—
155C2×#120—1372no—
156C2×C4×D4—1371no—
157#394—1372no—
158#392—1372no—
159#390—1372no—
160#380D13711yescharacteristic, complemented
161#379—1372no—
162C25—1372no—
163C22×D4—2748no—
164C22×D4—2744no—
165C22×D4—1371no—
166C22×D4—1372no—
167C22×D4—1374no—
168C22×D4—1372no—
169C23×C4—1371no—
170C23×C4C2×D13711yescharacteristic
171#121—2748no—
172#120—2748no—
173C4×D4—2744no—
174C2×C4⋊C4—1371no—
175C2×C4⋊C4C2×D13712yescomplemented
176C2×C22⋊C4—1371no—
177C2×C22⋊C4—1371no—
178C2×C22⋊C4—1374no—
179C2×C22⋊C4—1374no—
180C2×C22⋊C4—1374no—
181C2×C22⋊C4C2×D13712yes—
182C2×C42—1371no—
183#95C2×D13712yescomplemented
184C24—2748no—
185C24—2744no—
186C24—2744no—
187C24—2744no—
188C24—5484no—
189C24—1374no—
190C24—1371no—
191C24C22×D13711yescharacteristic
192C2×D4—27432no—
193C2×D4—2748no—
194C2×D4—2748no—
195C2×D4—2748no—
196C2×D4—2742no—
197C2×D4—2742no—
198C2×D4—2744no—
199C2×D4—2744no—
200C2×D4—2744no—
201C2×D4—5484no—
202C2×D4—1374no—
203C2×D4—1378no—
204C22×C4—24no—
205C22×C4—2744no—
206C22×C4—2742no—
207C22×C4—1371no—
208C22×C4—1371no—
209C22×C4—1371no—
210C22×C4D54814yescomplemented
211C22×C4C22×D13714yes—
212C22×C4C22×D13711yescharacteristic
213C22×C4C22×D13711yescharacteristic
214C4⋊C4—24no—
215C4⋊C4—2742no—
216C22⋊C4—24no—
217C22⋊C4—2748no—
218C22⋊C4—2748no—
219C22⋊C4—2742no—
220C22⋊C4—2742no—
221C22⋊C4—5484no—
222C42—2742no—
223C23—22no—
224C23—22no—
225C23—27416no—
226C23—2744no—
227C23—2742no—
228C23—2744no—
229C23—2741no—
230C23—2744no—
231C23—2741no—
232C23—2744no—
233C23—2744no—
234C23—2742no—
235C23—2744no—
236C23—54816no—
237C23—54816no—
238C23—5488no—
239C23—5488no—
240C23—1378no—
241C23—1372no—
242C23D4×D13712yes—
243C23C2×D54812yes—
244C23#1455551112yes—
245C23C23×D13711yescharacteristic
246D4—27416no—
247D4—2748no—
248D4—27416no—
249D4—54816no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—2741no—
256C2×C4—2741no—
257C2×C4—2742no—
258C2×C4—2741no—
259C2×C4—2741no—
260C2×C4—2742no—
261C2×C4—5484no—
262C2×C4—1372no—
263C2×C4D4×D13714yes—
264C2×C4C2×D54812yes—
265C2×C4C2×D54812yes—
266C22—22no—
267C22—24no—
268C22—24no—
269C22—22no—
270C22—2744no—
271C22—2748no—
272C22—2748no—
273C22—2748no—
274C22—2742no—
275C22—54832no—
276C22—5484no—
277C22—5481no—
278C22—5484no—
279C22—5484no—
280C22—5482no—
281C22—5481no—
282C22—5484no—
283C22#1798406314yescomplemented
284C22C2×D4×D13711yescharacteristic
285C22C2×D4×D13712yes—
286C22C22×D54811yescharacteristic
287C22C2×#1455551111yescharacteristic
288C22#1798413412yes—
289C4—22no—
290C4—44no—
291C4—2742no—
292C4—5481no—
293C4—5481no—
294C4#1798405814yescomplemented
295C2—24no—
296C2—5488no—
297C2—5482no—
298C2C2×#1798405811yescharacteristic
299C2C2×#1798406311yescharacteristic
300C2#2257797211yescharacteristic
301C2#2257803712yes—
302C2#2257804412yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#380×C137C2
hypercenter#380D137
hypocenterC137#2408
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×D548C2semidirect
2. C2×#17984063C2semidirect
3. C2×#17984058C2semidirect
4. C2×C4×C22⋊D137C2semidirect
5. #22577321C2semidirect
6. #22577313C2semidirect
7. #22577304C2semidirect
8. #22577258C2semidirect
9. #380×C137C2semidirect
15. C22×D548C22semidirect
17. C22×C4×D137C22semidirect
21. C2×#14555489C22semidirect
23. C2×#14555487C22semidirect
24. C2×#14555487C22semidirect
25. C2×#14555487C22semidirect
26. C2×#14555487C22semidirect
27. C2×C4×Dic137C22semidirect
29. C2×C4⋊C4×C137C22semidirect
31. #95×C137C22semidirect
47. C2×D548D4semidirect
50. C2×C22⋊D137D4semidirect
65. C22×C4×C137D4semidirect
87. C22×D137C2×D4semidirect
98. C2×Dic137C2×D4semidirect
118. C22×C137#121semidirect
139. C4×C137#120semidirect
151. C137#2408semidirect
160. #380D137semidirect
175. C2×C4⋊C4C2×D137semidirect
183. #95C2×D137semidirect
210. C22×C4D548semidirect
283. C22#17984063semidirect
294. C4#17984058semidirect

Maximal subgroups

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

subgroupindex
C23×D5482
C2×#179840632
C2×#179840582
C2×C4×C22⋊D1372
#225773212
#225773132
#225773042
#225772582
#380×C1372
#2408137