Finite Group Databasebeta

← #31784118

Subgroups — #31784118

Order 17536.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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
2C2×#17984073C212yesmaximal, complemented
3C2×C22⋊C4×D137C211yesmaximal, characteristic, complemented
4#22577321C212yesmaximal, complemented
5#22577320C212yesmaximal, complemented
6#22577281C212yesmaximal, complemented
7#22577276C211yesmaximal, characteristic, complemented
8#22577258C211yesmaximal, characteristic, complemented
9#22577258C211yesmaximal, characteristic, complemented
10#22577258C211yesmaximal, characteristic, complemented
11#385×C137C211yesmaximal, characteristic, complemented
12C24×D137C2211yescharacteristic
13C24×D137C2211yescharacteristic
14C22×D548—24no—
15C22×D548—28no—
16C22×D548C2212yes—
17C22×D548C2212yes—
18C22×C4×D137C2211yescharacteristic, complemented
19C22×C4×D137C2211yescharacteristic, complemented
20#17984073—28no—
21C2×#14555489C2212yescomplemented
22C2×C22⋊Dic137C2211yescharacteristic
23C2×C22⋊Dic137C2211yescharacteristic
24C2×#14555487C2212yescomplemented
25C2×#14555487C2212yescomplemented
26C2×#14555487C2212yescomplemented
27C2×#14555487C2211yescharacteristic, complemented
28C2×#14555487C2211yescharacteristic, complemented
29C2×#14555487C2212yescomplemented
30C2×#14555487C2212yescomplemented
31C2×#14555487C2212yescomplemented
32C22⋊C4×D137—24no—
33#17983995C2212yescomplemented
34C23×C4×C137C2211yescharacteristic
35C2×C4⋊C4×C137C2212yescomplemented
36C2×C22⋊C4×C137C2211yescharacteristic
37C2×C22⋊C4×C137C2211yescharacteristic
38#95×C137C2212yescomplemented
39C23×D137—24no—
40C23×D137—24no—
41C23×D137—22no—
42C23×D137—22no—
43C23×D137—22no—
44C23×D137—22no—
45C23×D137—22no—
46C23×D137—42no—
47C23×D137—42no—
48C23×D137D414yes—
49C23×D137C2311yescharacteristic
50C23×D137C2312yes—
51C23×D137C2311yescharacteristic
52C2×D548—232no—
53C2×D548—24no—
54C2×D548—24no—
55C2×D548—24no—
56C2×D548—24no—
57C2×D548—24no—
58C2×D548D418yes—
59C22×Dic137C2312yes—
60C22×Dic137C2311yescharacteristic
61C22×Dic137C2311yescharacteristic
62C2×C4×D137—22no—
63C2×C4×D137—22no—
64C2×C4×D137—24no—
65C2×C4×D137—42no—
66#14555489—24no—
67C22⋊Dic137—22no—
68C22⋊Dic137—22no—
69#14555487—22no—
70#14555487—22no—
71#14555487—24no—
72#14555487—24no—
73#14555487—24no—
74#14555487—42no—
75#14555487—42no—
76#14555487—42no—
77C24×C137C2311yescharacteristic
78C22×C4×C137—22no—
79C22×C4×C137—24no—
80C22×C4×C137C2312yes—
81C22×C4×C137C2311yescharacteristic
82C22×C4×C137C2311yescharacteristic
83C22×C4×C137C2312yes—
84C4⋊C4×C137—24no—
85C22⋊C4×C137—22no—
86C22⋊C4×C137—22no—
87C22×D137—21no—
88C22×D137—21no—
89C22×D137—22no—
90C22×D137—28no—
91C22×D137—28no—
92C22×D137—22no—
93C22×D137—22no—
94C22×D137—22no—
95C22×D137—21no—
96C22×D137—22no—
97C22×D137—21no—
98C22×D137—22no—
99C22×D137—22no—
100C22×D137—22no—
101C22×D137—22no—
102C22×D137—28no—
103C22×D137—44no—
104C22×D137—48no—
105C22×D137—48no—
106C22×D137—48no—
107C22×D137—48no—
108C22×D137—44no—
109C22×D137—44no—
110C22×D137C2×D412yes—
111C22×D137C2×D414yes—
112C22×D137C2×D414yes—
113C22×D137C2×D412yes—
114D548—216no—
115D548—216no—
116C2×Dic137—22no—
117C2×Dic137—21no—
118C2×Dic137—21no—
119C2×Dic137—22no—
120C2×Dic137—22no—
121C2×Dic137—22no—
122C2×Dic137—22no—
123C4×D137—44no—
124C4×D137—44no—
125C23×C137—22no—
126C23×C137—22no—
127C23×C137C2×D412yes—
128C23×C137#4112yes—
129C23×C137#4112yes—
130C23×C137C2411yescharacteristic
131C2×C4×C137—28no—
132C2×C4×C137—22no—
133C2×C4×C137—22no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
136C2×C4×C137—22no—
137C2×C4×C137—22no—
138C2×C4×C137—22no—
139C2×C4×C137—22no—
140C2×C4×C137—22no—
141C2×C4×C137C2×D414yes—
142C22×C137—22no—
143C22×C137—22no—
144C22×C137—24no—
145C22×C137—24no—
146C22×C137#12314yescomplemented
147C22×C137C22×D411yescharacteristic
148C22×C137C22×D412yes—
149C22×C137C2×#4111yescharacteristic
150C22×C137C2×#4111yescharacteristic
151C22×C137H(5,𝔽2)12yes—
152C2×D137—22no—
153C2×D137—22no—
154C2×D137—24no—
155C2×D137—24no—
156C2×D137—22no—
157C2×D137—24no—
158C2×D137—22no—
159C2×D137—22no—
160C2×D137—24no—
161C2×D137—24no—
162C2×D137—416no—
163C2×D137—416no—
164C2×D137—42no—
165C2×D137—42no—
166C2×D137—42no—
167C2×D137—42no—
168C2×D137—42no—
169C2×D137—42no—
170C2×D137#12014yes—
171C4×C137—24no—
172C4×C137—41no—
173C4×C137—41no—
174C4×C137—42no—
175Dic137—41no—
176Dic137—41no—
177Dic137—42no—
178C2×C137—24no—
179C2×C137C2×#12011yescharacteristic
180C2×C137C2×#12311yescharacteristic
181C2×C137#53411yescharacteristic
182C2×C137#54612yes—
183C2×C137#54612yes—
184D137—24no—
185D137—42no—
186D137—44no—
187C137#240511yescharacteristic, complemented, Sylow
188#2405—1371nomaximal, Sylow
189C23×D4—1371no—
190C2×#123—1372no—
191C22×C22⋊C4—1371no—
192#394—1372no—
193#394—1372no—
194#386—1372no—
195#385—1371no—
196#385D13711yescharacteristic, complemented
197#379—1371no—
198#379—1371no—
199#379—1371no—
200C25—1371no—
201C25—1371no—
202C22×D4—2744no—
203C22×D4—2748no—
204C22×D4—1372no—
205C22×D4—1372no—
206C23×C4—1371no—
207C23×C4—1371no—
208C23×C4C2×D13711yescharacteristic
209#123—2748no—
210C2×C4⋊C4—1372no—
211C2×C4⋊C4C2×D13712yescomplemented
212C2×C22⋊C4—2744no—
213C2×C22⋊C4—1371no—
214C2×C22⋊C4—1371no—
215C2×C22⋊C4—1371no—
216C2×C22⋊C4—1371no—
217C2×C22⋊C4—1372no—
218C2×C22⋊C4—1372no—
219C2×C22⋊C4—1372no—
220C2×C22⋊C4—1372no—
221C2×C22⋊C4—1372no—
222C2×C22⋊C4—1372no—
223C2×C22⋊C4C2×D13711yescharacteristic
224C2×C22⋊C4C2×D13711yescharacteristic
225#95—1372no—
226#95C2×D13712yescomplemented
227C24—2744no—
228C24—2744no—
229C24—2742no—
230C24—2742no—
231C24—2742no—
232C24—2742no—
233C24—2742no—
234C24—5482no—
235C24—5482no—
236C24—1371no—
237C24—1372no—
238C24—1374no—
239C24—1371no—
240C24C22×D13711yescharacteristic
241C2×D4—27432no—
242C2×D4—2744no—
243C2×D4—2744no—
244C2×D4—2744no—
245C2×D4—2744no—
246C2×D4—2744no—
247C2×D4—1378no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—2742no—
251C22×C4—2742no—
252C22×C4—2744no—
253C22×C4—5482no—
254C22×C4—1371no—
255C22×C4—1372no—
256C22×C4—1371no—
257C22×C4C22×D13711yescharacteristic
258C22×C4C22×D13712yes—
259C22×C4C22×D13711yescharacteristic
260C22×C4C22×D13712yes—
261C4⋊C4—24no—
262C4⋊C4—2744no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—2742no—
266C22⋊C4—2742no—
267C22⋊C4—2744no—
268C22⋊C4—2744no—
269C22⋊C4—2744no—
270C22⋊C4—2742no—
271C22⋊C4—2742no—
272C22⋊C4—5482no—
273C22⋊C4—5482no—
274C22⋊C4—5482no—
275C23—22no—
276C23—22no—
277C23—2748no—
278C23—2748no—
279C23—2741no—
280C23—2741no—
281C23—2742no—
282C23—2748no—
283C23—2742no—
284C23—2742no—
285C23—2742no—
286C23—2742no—
287C23—2741no—
288C23—2742no—
289C23—2741no—
290C23—2742no—
291C23—2742no—
292C23—2742no—
293C23—5484no—
294C23—5488no—
295C23—5488no—
296C23—5488no—
297C23—5488no—
298C23—5484no—
299C23—5484no—
300C23—1372no—
301C23—1374no—
302C23—1374no—
303C23—1372no—
304C23D4×D13712yes—
305C23#1455551012yes—
306C23#1455551112yes—
307C23C23×D13711yescharacteristic
308D4—27416no—
309D4—27416no—
310C2×C4—28no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—2742no—
321C2×C4—2741no—
322C2×C4—2741no—
323C2×C4—2742no—
324C2×C4—2742no—
325C2×C4—2742no—
326C2×C4—2742no—
327C2×C4—5484no—
328C2×C4—5484no—
329C2×C4D4×D13714yes—
330C22—22no—
331C22—24no—
332C22—24no—
333C22—22no—
334C22—2742no—
335C22—2744no—
336C22—2744no—
337C22—2742no—
338C22—2742no—
339C22—2744no—
340C22—2744no—
341C22—2742no—
342C22—2744no—
343C22—2742no—
344C22—54816no—
345C22—54816no—
346C22—5482no—
347C22—5482no—
348C22—5482no—
349C22—5482no—
350C22—5482no—
351C22—5482no—
352C22—1374no—
353C22#1798407314yescomplemented
354C22C2×D4×D13711yescharacteristic
355C22C2×D4×D13712yes—
356C22C2×#1455551011yescharacteristic
357C22C2×#1455551111yescharacteristic
358C22#1798413412yes—
359C4—24no—
360C4—41no—
361C4—42no—
362C4—41no—
363C4—5481no—
364C4—5482no—
365C4—5481no—
366C2—24no—
367C2—2744no—
368C2—5482no—
369C2—5484no—
370C2#120×D13711yescharacteristic
371C2C2×#1798407311yescharacteristic
372C2#2257796411yescharacteristic
373C2#2257804412yes—
374C2#2257804912yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#385×C137C2
prime radicalC23×D548C2
hypercenter#385D137
hypocenterC137#2405
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#17984073C2semidirect
3. C2×C22⋊C4×D137C2semidirect
4. #22577321C2semidirect
5. #22577320C2semidirect
6. #22577281C2semidirect
7. #22577276C2semidirect
8. #22577258C2semidirect
9. #22577258C2semidirect
10. #22577258C2semidirect
11. #385×C137C2semidirect
18. C22×C4×D137C22semidirect
19. C22×C4×D137C22semidirect
21. C2×#14555489C22semidirect
24. C2×#14555487C22semidirect
25. C2×#14555487C22semidirect
26. C2×#14555487C22semidirect
27. C2×#14555487C22semidirect
28. C2×#14555487C22semidirect
29. C2×#14555487C22semidirect
30. C2×#14555487C22semidirect
31. C2×#14555487C22semidirect
33. #17983995C22semidirect
35. C2×C4⋊C4×C137C22semidirect
38. #95×C137C22semidirect
146. C22×C137#123semidirect
187. C137#2405semidirect
196. #385D137semidirect
211. C2×C4⋊C4C2×D137semidirect
226. #95C2×D137semidirect
353. C22#17984073semidirect

Maximal subgroups

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

subgroupindex
C23×D5482
C2×#179840732
C2×C22⋊C4×D1372
#225773212
#225773202
#225772812
#225772762
#225772582
#385×C1372
#2405137