Finite Group Databasebeta

← #31784138

Subgroups — #31784138

Order 17536.

1232 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×D4×D137C211yesmaximal, characteristic, complemented
2C2×#17984060C212yesmaximal, complemented
3C2×#17984056C212yesmaximal, complemented
4C2×#17984048C211yesmaximal, characteristic, complemented
5#22577321C211yesmaximal, characteristic, complemented
6#22577317C211yesmaximal, characteristic, complemented
7#22577312C211yesmaximal, characteristic, complemented
8#22577303C212yesmaximal, complemented
9#22577262C211yesmaximal, characteristic, complemented
10#22577258C212yesmaximal, complemented
11#392×C137C211yesmaximal, characteristic, complemented
12C24×D137C2212yes—
13C22×D548C2211yescharacteristic
14C22×C22⋊D137C2212yes—
15C22×C22⋊D137C2211yescharacteristic
16C22×C22⋊D137C2212yescomplemented
17C2×D4×D137—28no—
18C2×D4×D137—24no—
19C23×Dic137C2211yescharacteristic, complemented
20C22×C4×D137C2211yescharacteristic
21#17984060—28no—
22#17984056—28no—
23#17984048—24no—
24C2×#14555489C2212yescomplemented
25C2×#14555489C2211yescharacteristic, complemented
26C2×C22⋊Dic137C2212yescomplemented
27C2×C22⋊Dic137C2212yescomplemented
28C2×#14555487C2211yescharacteristic
29C2×#14555487C2212yes—
30C2×#14555487C2212yescomplemented
31C2×#14555487C2212yescomplemented
32C2×#14555487C2212yescomplemented
33C2×C4×Dic137C2211yescharacteristic, complemented
34#17983995C2211yescharacteristic, complemented
35C22×D4×C137C2211yescharacteristic
36C22×D4×C137C2212yescomplemented
37C2×C22⋊C4×C137C2211yescharacteristic, complemented
38C2×C22⋊C4×C137C2212yescomplemented
39#95×C137C2211yescharacteristic, complemented
40C23×D137—24no—
41C23×D137—24no—
42C23×D137—24no—
43C23×D137—24no—
44C23×D137—24no—
45C23×D137—42no—
46C23×D137—42no—
47C23×D137C2312yes—
48C23×D137C2311yescharacteristic
49C2×D548—24no—
50C2×D548—22no—
51C2×C22⋊D137—22no—
52C2×C22⋊D137—22no—
53C2×C22⋊D137—24no—
54C2×C22⋊D137—24no—
55C2×C22⋊D137—24no—
56C2×C22⋊D137—24no—
57C2×C22⋊D137—42no—
58C2×C22⋊D137D418yescomplemented
59C2×C22⋊D137D414yescomplemented
60D4×D137—232no—
61C22×Dic137—22no—
62C22×Dic137—24no—
63C22×Dic137C2311yescharacteristic
64C22×Dic137C2312yes—
65C22×Dic137C2311yescharacteristic
66C22×Dic137C2311yescharacteristic
67C2×C4×D137—24no—
68C2×C4×D137D414yescomplemented
69#14555489—24no—
70#14555489—22no—
71C22⋊Dic137—24no—
72C22⋊Dic137—24no—
73#14555487—22no—
74#14555487—24no—
75#14555487—24no—
76#14555487—24no—
77#14555487—42no—
78C4×Dic137—22no—
79C24×C137C2312yes—
80C24×C137C2311yescharacteristic
81C2×D4×C137—24no—
82C2×D4×C137—24no—
83C2×D4×C137—24no—
84C2×D4×C137—22no—
85C2×D4×C137—42no—
86C22×C4×C137C2311yescharacteristic
87C22×C4×C137C2312yes—
88C22×C4×C137C2311yescharacteristic
89C22⋊C4×C137—22no—
90C22⋊C4×C137—42no—
91C22×D137—22no—
92C22×D137—22no—
93C22×D137—216no—
94C22×D137—22no—
95C22×D137—22no—
96C22×D137—22no—
97C22×D137—22no—
98C22×D137—22no—
99C22×D137—22no—
100C22×D137—48no—
101C22×D137—48no—
102C22×D137—48no—
103C22×D137—48no—
104C22×D137—48no—
105C22×D137—48no—
106C22×D137C2×D414yes—
107C22×D137C2×D412yes—
108C22×D137C2×D412yes—
109C22×D137#4112yes—
110D548—28no—
111C22⋊D137—28no—
112C22⋊D137—216no—
113C22⋊D137—48no—
114C2×Dic137—21no—
115C2×Dic137—21no—
116C2×Dic137—22no—
117C2×Dic137—22no—
118C2×Dic137—21no—
119C2×Dic137—22no—
120C2×Dic137—22no—
121C2×Dic137—44no—
122C2×Dic137C2×D412yes—
123C2×Dic137C2×D414yes—
124C2×Dic137C2×D412yescomplemented
125C4×D137—28no—
126C23×C137—21no—
127C23×C137—22no—
128C23×C137—22no—
129C23×C137—22no—
130C23×C137—22no—
131C23×C137—21no—
132C23×C137—22no—
133C23×C137—22no—
134C23×C137—22no—
135C23×C137—22no—
136C23×C137C2×D414yes—
137C23×C137C2×D412yescomplemented
138C23×C137C2411yescharacteristic
139D4×C137—28no—
140D4×C137—48no—
141C2×C4×C137—22no—
142C2×C4×C137—21no—
143C2×C4×C137—22no—
144C2×C4×C137—22no—
145C2×C4×C137—22no—
146C2×C4×C137—22no—
147C2×C4×C137C2×D412yes—
148C22×C137—24no—
149C22×C137—22no—
150C22×C137—24no—
151C22×C137—24no—
152C22×C137—24no—
153C22×C137—41no—
154C22×C137—42no—
155C22×C137—41no—
156C22×C137—42no—
157C22×C137—42no—
158C22×C137—42no—
159C22×C137—42no—
160C22×C137C22×D411yescharacteristic
161C22×C137C22×D412yes—
162C22×C137C22×D411yescharacteristic
163C22×C137C2×#4111yescharacteristic
164C22×C137H(5,𝔽2)12yes—
165C2×D137—22no—
166C2×D137—22no—
167C2×D137—24no—
168C2×D137—24no—
169C2×D137—24no—
170C2×D137—24no—
171C2×D137—24no—
172C2×D137—432no—
173C2×D137—42no—
174C2×D137—42no—
175C2×D137—42no—
176C2×D137—42no—
177C2×D137#12114yescomplemented
178C4×C137—22no—
179C4×C137—41no—
180C4×C137—42no—
181Dic137—22no—
182Dic137—41no—
183Dic137—42no—
184Dic137#12014yescomplemented
185C2×C137—44no—
186C2×C137—42no—
187C2×C137C2×#12011yescharacteristic
188C2×C137C2×#12111yescharacteristic
189C2×C137#53511yescharacteristic
190C2×C137D4212yes—
191C2×C137#54612yes—
192D137—24no—
193D137—44no—
194C137#240811yescharacteristic, complemented, Sylow
195#2408—1371nomaximal, Sylow
196C23×D4—1371no—
197C2×#121—1372no—
198C2×#120—1372no—
199C2×C4×D4—1371no—
200#394—1371no—
201#394—1371no—
202#392—1371no—
203#392D13711yescharacteristic, complemented
204#390—1372no—
205#380—1371no—
206#379—1372no—
207C25—1372no—
208C22×D4—2748no—
209C22×D4—2744no—
210C22×D4—1372no—
211C22×D4—1371no—
212C22×D4—1372no—
213C22×D4—1371no—
214C22×D4C2×D13711yescharacteristic
215C22×D4C2×D13712yescomplemented
216C23×C4—1371no—
217C23×C4—1371no—
218#121—2748no—
219#120—2748no—
220C4×D4—2744no—
221C2×C4⋊C4—1371no—
222C2×C4⋊C4—1372no—
223C2×C22⋊C4—1372no—
224C2×C22⋊C4—1372no—
225C2×C22⋊C4—1371no—
226C2×C22⋊C4—1372no—
227C2×C22⋊C4—1372no—
228C2×C22⋊C4—1372no—
229C2×C22⋊C4—1372no—
230C2×C22⋊C4C2×D13711yescharacteristic, complemented
231C2×C22⋊C4C2×D13712yescomplemented
232C2×C42—1371no—
233#95—1371no—
234#95C2×D13711yescharacteristic, complemented
235C24—2744no—
236C24—2744no—
237C24—2744no—
238C24—2744no—
239C24—2744no—
240C24—5482no—
241C24—5482no—
242C24—1372no—
243C24—1371no—
244C24C22×D13712yes—
245C24C22×D13711yescharacteristic
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—42no—
251C2×D4—2744no—
252C2×D4—27432no—
253C2×D4—2742no—
254C2×D4—2742no—
255C2×D4—2744no—
256C2×D4—2744no—
257C2×D4—2744no—
258C2×D4—2744no—
259C2×D4—2742no—
260C2×D4—5482no—
261C2×D4—1378no—
262C2×D4—1374no—
263C22×C4—2744no—
264C22×C4—2742no—
265C22×C4—2744no—
266C22×C4—1372no—
267C22×C4—1371no—
268C22×C4—1371no—
269C22×C4—1374no—
270C22×C4—1371no—
271C22×C4C22×D13712yes—
272C22×C4C22×D13711yescharacteristic
273C22×C4C22×D13711yescharacteristic
274C4⋊C4—2744no—
275C4⋊C4—2742no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—2742no—
279C22⋊C4—2744no—
280C22⋊C4—2744no—
281C22⋊C4—2744no—
282C22⋊C4—2744no—
283C22⋊C4—2744no—
284C22⋊C4—5482no—
285C42—2742no—
286C23—21no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—21no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—27416no—
297C23—2742no—
298C23—2742no—
299C23—2742no—
300C23—2742no—
301C23—2742no—
302C23—2742no—
303C23—2742no—
304C23—2742no—
305C23—5488no—
306C23—5488no—
307C23—5488no—
308C23—5488no—
309C23—5488no—
310C23—5488no—
311C23—1374no—
312C23—1372no—
313C23—1372no—
314C23—1372no—
315C23D4×D13714yes—
316C23D4×D13712yescomplemented
317C23C23×D13711yescharacteristic
318D4—28no—
319D4—48no—
320D4—2748no—
321D4—2748no—
322D4—27416no—
323D4—5488no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—2748no—
331C2×C4—2741no—
332C2×C4—2741no—
333C2×C4—2742no—
334C2×C4—2742no—
335C2×C4—2741no—
336C2×C4—2742no—
337C2×C4—2742no—
338C2×C4—5484no—
339C2×C4—1372no—
340C2×C4—1372no—
341C2×C4—1374no—
342C2×C4D4×D13712yes—
343C22—24no—
344C22—24no—
345C22—24no—
346C22—24no—
347C22—22no—
348C22—42no—
349C22—41no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—42no—
355C22—2742no—
356C22—2744no—
357C22—2744no—
358C22—2742no—
359C22—2744no—
360C22—2744no—
361C22—2744no—
362C22—54832no—
363C22—5482no—
364C22—5482no—
365C22—5482no—
366C22—5482no—
367C22—1374no—
368C22C2×D4×D13711yescharacteristic
369C22C2×D4×D13711yescharacteristic
370C22C2×D4×D13712yes—
371C22#41×D13711yescharacteristic
372C22#1798413312yes—
373C4—22no—
374C4—41no—
375C4—42no—
376C4—2742no—
377C4—5482no—
378C4—5481no—
379C4—1374no—
380C2—44no—
381C2—42no—
382C2—2744no—
383C2—5484no—
384C2#120×D13711yescharacteristic
385C2#121×D13711yescharacteristic
386C2#2257796611yescharacteristic
387C2#2257803512yes—
388C2#2257804212yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#392×C137C2
hypercenter#392D137
hypocenterC137#2408
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D137C2semidirect
2. C2×#17984060C2semidirect
3. C2×#17984056C2semidirect
4. C2×#17984048C2semidirect
5. #22577321C2semidirect
6. #22577317C2semidirect
7. #22577312C2semidirect
8. #22577303C2semidirect
9. #22577262C2semidirect
10. #22577258C2semidirect
11. #392×C137C2semidirect
16. C22×C22⋊D137C22semidirect
19. C23×Dic137C22semidirect
24. C2×#14555489C22semidirect
25. C2×#14555489C22semidirect
26. C2×C22⋊Dic137C22semidirect
27. C2×C22⋊Dic137C22semidirect
30. C2×#14555487C22semidirect
31. C2×#14555487C22semidirect
32. C2×#14555487C22semidirect
33. C2×C4×Dic137C22semidirect
34. #17983995C22semidirect
36. C22×D4×C137C22semidirect
37. C2×C22⋊C4×C137C22semidirect
38. C2×C22⋊C4×C137C22semidirect
39. #95×C137C22semidirect
58. C2×C22⋊D137D4semidirect
59. C2×C22⋊D137D4semidirect
68. C2×C4×D137D4semidirect
124. C2×Dic137C2×D4semidirect
137. C23×C137C2×D4semidirect
177. C2×D137#121semidirect
184. Dic137#120semidirect
194. C137#2408semidirect
203. #392D137semidirect
215. C22×D4C2×D137semidirect
230. C2×C22⋊C4C2×D137semidirect
231. C2×C22⋊C4C2×D137semidirect
234. #95C2×D137semidirect
316. C23D4×D137semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1372
C2×#179840602
C2×#179840562
C2×#179840482
#225773212
#225773172
#225773122
#225773032
#225772622
#225772582
#392×C1372
#2408137