Finite Group Databasebeta

← #31784140

Subgroups — #31784140

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×#17984061C212yesmaximal, complemented
3C2×#17984058C212yesmaximal, complemented
4C2×#17984050C211yesmaximal, characteristic, complemented
5#22577320C211yesmaximal, characteristic, complemented
6#22577313C211yesmaximal, characteristic, complemented
7#22577311C211yesmaximal, characteristic, complemented
8#22577303C212yesmaximal, complemented
9#22577262C211yesmaximal, characteristic, complemented
10#22577258C212yesmaximal, complemented
11#394×C137C211yesmaximal, characteristic, complemented
12C24×D137C2212yes—
13C22×D548C2211yescharacteristic
14C22×D548C2211yescharacteristic
15C22×D548C2212yescomplemented
16C22×C22⋊D137C2212yes—
17C22×C22⋊D137C2212yescomplemented
18C2×D4×D137—28no—
19C2×D4×D137—24no—
20C22×C4×D137C2211yescharacteristic
21C22×C4×D137C2211yescharacteristic, complemented
22#17984061—28no—
23#17984058—28no—
24#17984050—24no—
25C2×#14555489C2212yescomplemented
26C2×C22⋊Dic137C2212yescomplemented
27C2×#14555487C2211yescharacteristic
28C2×#14555487C2212yes—
29C2×#14555487C2212yescomplemented
30C2×#14555487C2212yescomplemented
31C2×#14555487C2211yescharacteristic, complemented
32C2×#14555487C2212yescomplemented
33C2×C4×Dic137C2211yescharacteristic, complemented
34#17983995C2211yescharacteristic, complemented
35C22×D4×C137C2211yescharacteristic
36C2×C4⋊C4×C137C2211yescharacteristic, complemented
37C2×C22⋊C4×C137C2212yescomplemented
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
49C23×D137C2311yescharacteristic
50C2×D548—22no—
51C2×D548—24no—
52C2×D548—22no—
53C2×D548—24no—
54C2×D548—24no—
55C2×D548—22no—
56C2×D548—42no—
57C2×D548D414yescomplemented
58C2×C22⋊D137—24no—
59C2×C22⋊D137—24no—
60C2×C22⋊D137—24no—
61C2×C22⋊D137—24no—
62C2×C22⋊D137—42no—
63C2×C22⋊D137D418yescomplemented
64D4×D137—232no—
65C22×Dic137C2311yescharacteristic
66C22×Dic137C2312yes—
67C22×Dic137C2311yescharacteristic
68C2×C4×D137—24no—
69C2×C4×D137—22no—
70C2×C4×D137—24no—
71C2×C4×D137D414yescomplemented
72#14555489—24no—
73C22⋊Dic137—42no—
74#14555487—22no—
75#14555487—24no—
76#14555487—24no—
77#14555487—22no—
78#14555487—24no—
79#14555487—42no—
80C4×Dic137—22no—
81C24×C137C2312yes—
82C2×D4×C137—24no—
83C2×D4×C137—22no—
84C22×C4×C137C2311yescharacteristic
85C22×C4×C137C2312yes—
86C22×C4×C137C2311yescharacteristic
87C22×C4×C137C2311yescharacteristic
88C4⋊C4×C137—22no—
89C22⋊C4×C137—24no—
90C22⋊C4×C137—24no—
91C22×D137—22no—
92C22×D137—22no—
93C22×D137—216no—
94C22×D137—21no—
95C22×D137—22no—
96C22×D137—22no—
97C22×D137—22no—
98C22×D137—22no—
99C22×D137—21no—
100C22×D137—22no—
101C22×D137—22no—
102C22×D137—22no—
103C22×D137—22no—
104C22×D137—48no—
105C22×D137—48no—
106C22×D137—48no—
107C22×D137—48no—
108C22×D137—48no—
109C22×D137—48no—
110C22×D137C2×D414yes—
111C22×D137C2×D412yes—
112C22×D137C2×D412yes—
113C22×D137C2×D412yescomplemented
114C22×D137#4112yes—
115D548—28no—
116D548—28no—
117D548—48no—
118C22⋊D137—216no—
119C22⋊D137—48no—
120C2×Dic137—21no—
121C2×Dic137—22no—
122C2×Dic137—22no—
123C2×Dic137—22no—
124C2×Dic137—22no—
125C2×Dic137C2×D412yes—
126C2×Dic137C2×D414yes—
127C4×D137—28no—
128C4×D137—44no—
129C23×C137—22no—
130C23×C137—22no—
131C23×C137—22no—
132C23×C137—22no—
133C23×C137—22no—
134C23×C137—22no—
135C23×C137C2×D414yes—
136C23×C137C2411yescharacteristic
137D4×C137—28no—
138C2×C4×C137—21no—
139C2×C4×C137—22no—
140C2×C4×C137—21no—
141C2×C4×C137—22no—
142C2×C4×C137—22no—
143C2×C4×C137—22no—
144C2×C4×C137—21no—
145C2×C4×C137—22no—
146C2×C4×C137C2×D412yes—
147C2×C4×C137C2×D412yescomplemented
148C22×C137—24no—
149C22×C137—24no—
150C22×C137—24no—
151C22×C137—42no—
152C22×C137—42no—
153C22×C137—42no—
154C22×C137—42no—
155C22×C137C22×D411yescharacteristic
156C22×C137C22×D412yes—
157C22×C137C22×D411yescharacteristic
158C22×C137C2×#4111yescharacteristic
159C22×C137H(5,𝔽2)12yes—
160C2×D137—22no—
161C2×D137—22no—
162C2×D137—24no—
163C2×D137—24no—
164C2×D137—24no—
165C2×D137—22no—
166C2×D137—24no—
167C2×D137—24no—
168C2×D137—24no—
169C2×D137—432no—
170C2×D137—41no—
171C2×D137—42no—
172C2×D137—41no—
173C2×D137—42no—
174C2×D137—42no—
175C2×D137—42no—
176C2×D137—42no—
177C2×D137#12114yescomplemented
178C4×C137—22no—
179C4×C137—22no—
180C4×C137—41no—
181C4×C137—42no—
182Dic137—41no—
183Dic137—42no—
184Dic137#12014yescomplemented
185C2×C137—44no—
186C2×C137C2×#12011yescharacteristic
187C2×C137C2×#12111yescharacteristic
188C2×C137#53511yescharacteristic
189C2×C137D4212yes—
190C2×C137#54612yes—
191D137—24no—
192D137—42no—
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#394D13711yescharacteristic, complemented
202#392—1371no—
203#392—1371no—
204#390—1372no—
205#380—1371no—
206#379—1372no—
207C25—1372no—
208C22×D4—2748no—
209C22×D4—2744no—
210C22×D4—1372no—
211C22×D4—1372no—
212C22×D4—1371no—
213C22×D4—1372no—
214C22×D4—1371no—
215C22×D4C2×D13711yescharacteristic
216C23×C4—1371no—
217C23×C4—1371no—
218#121—2748no—
219#120—2748no—
220C4×D4—2744no—
221C2×C4⋊C4—1372no—
222C2×C4⋊C4C2×D13711yescharacteristic, complemented
223C2×C22⋊C4—1371no—
224C2×C22⋊C4—1371no—
225C2×C22⋊C4—1372no—
226C2×C22⋊C4—1372no—
227C2×C22⋊C4—1372no—
228C2×C22⋊C4—1372no—
229C2×C22⋊C4—1372no—
230C2×C22⋊C4C2×D13712yescomplemented
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—
244C24—1371no—
245C24C22×D13712yes—
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—2742no—
249C2×D4—2744no—
250C2×D4—2742no—
251C2×D4—2744no—
252C2×D4—2744no—
253C2×D4—27432no—
254C2×D4—2744no—
255C2×D4—2744no—
256C2×D4—2744no—
257C2×D4—2744no—
258C2×D4—2742no—
259C2×D4—5482no—
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—1374no—
269C22×C4—1371no—
270C22×C4C22×D13712yes—
271C22×C4C22×D13711yescharacteristic
272C22×C4C22×D13711yescharacteristic
273C22×C4C22×D13711yescharacteristic
274C4⋊C4—22no—
275C4⋊C4—2744no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—2742no—
279C22⋊C4—2744no—
280C22⋊C4—2744no—
281C22⋊C4—2742no—
282C22⋊C4—2744no—
283C22⋊C4—5482no—
284C22⋊C4—5482no—
285C42—2742no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—27416no—
293C23—2742no—
294C23—2742no—
295C23—2741no—
296C23—2742no—
297C23—2742no—
298C23—2742no—
299C23—2742no—
300C23—2741no—
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—
315C23—1372no—
316C23D4×D13714yes—
317C23C23×D13711yescharacteristic
318D4—28no—
319D4—2748no—
320D4—2748no—
321D4—27416no—
322D4—5488no—
323D4—5488no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—2748no—
333C2×C4—2741no—
334C2×C4—2742no—
335C2×C4—2742no—
336C2×C4—2742no—
337C2×C4—2742no—
338C2×C4—5484no—
339C2×C4—1372no—
340C2×C4—1374no—
341C2×C4D4×D13712yes—
342C2×C4D4×D13712yescomplemented
343C22—24no—
344C22—24no—
345C22—24no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—2742no—
351C22—2744no—
352C22—2744no—
353C22—2742no—
354C22—2744no—
355C22—2744no—
356C22—2744no—
357C22—2744no—
358C22—2742no—
359C22—54832no—
360C22—5482no—
361C22—5481no—
362C22—5482no—
363C22—5482no—
364C22—5482no—
365C22—5481no—
366C22—5482no—
367C22—1374no—
368C22C2×D4×D13711yescharacteristic
369C22C2×D4×D13711yescharacteristic
370C22C2×D4×D13712yes—
371C22#41×D13711yescharacteristic
372C22#1798413412yes—
373C4—22no—
374C4—22no—
375C4—42no—
376C4—41no—
377C4—5481no—
378C4—5482no—
379C4—1374no—
380C2—44no—
381C2—2744no—
382C2—5484no—
383C2—5482no—
384C2#120×D13711yescharacteristic
385C2#121×D13711yescharacteristic
386C2#2257797111yescharacteristic
387C2#2257803712yes—
388C2#2257804512yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#394×C137C2
hypercenter#394D137
hypocenterC137#2408
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×D4×D137C2semidirect
2. C2×#17984061C2semidirect
3. C2×#17984058C2semidirect
4. C2×#17984050C2semidirect
5. #22577320C2semidirect
6. #22577313C2semidirect
7. #22577311C2semidirect
8. #22577303C2semidirect
9. #22577262C2semidirect
10. #22577258C2semidirect
11. #394×C137C2semidirect
15. C22×D548C22semidirect
17. C22×C22⋊D137C22semidirect
21. C22×C4×D137C22semidirect
25. C2×#14555489C22semidirect
26. C2×C22⋊Dic137C22semidirect
29. C2×#14555487C22semidirect
30. C2×#14555487C22semidirect
31. C2×#14555487C22semidirect
32. C2×#14555487C22semidirect
33. C2×C4×Dic137C22semidirect
34. #17983995C22semidirect
36. C2×C4⋊C4×C137C22semidirect
37. C2×C22⋊C4×C137C22semidirect
38. C2×C22⋊C4×C137C22semidirect
39. #95×C137C22semidirect
57. C2×D548D4semidirect
63. C2×C22⋊D137D4semidirect
71. C2×C4×D137D4semidirect
113. C22×D137C2×D4semidirect
147. C2×C4×C137C2×D4semidirect
177. C2×D137#121semidirect
184. Dic137#120semidirect
194. C137#2408semidirect
201. #394D137semidirect
222. C2×C4⋊C4C2×D137semidirect
230. C2×C22⋊C4C2×D137semidirect
231. C2×C22⋊C4C2×D137semidirect
234. #95C2×D137semidirect
342. C2×C4D4×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×#179840612
C2×#179840582
C2×#179840502
#225773202
#225773132
#225773112
#225773032
#225772622
#225772582
#394×C1372
#2408137