Finite Group Databasebeta

← #31784109

Subgroups — #31784109

Order 17536.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 159 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×#14555511C211yesmaximal, characteristic, complemented
2C2×#17984058C212yesmaximal, complemented
3C2×C22⋊C4×D137C211yesmaximal, characteristic, complemented
4#22577314C211yesmaximal, characteristic, complemented
5#22577313C211yesmaximal, characteristic, complemented
6#22577312C211yesmaximal, characteristic, complemented
7#22577281C212yesmaximal, complemented
8#22577281C212yesmaximal, complemented
9#22577279C212yesmaximal, complemented
10#22577265C211yesmaximal, characteristic, complemented
11#393×C137C211yesmaximal, characteristic, complemented
12C24×D137C2211yescharacteristic
13C2×#14555511—28no—
14C2×#14555511—24no—
15C22×D548C2212yes—
16C22×D548C2212yes—
17C22×D548C2211yescharacteristic, complemented
18C22×C22⋊D137C2212yes—
19C22×C4×D137C2211yescharacteristic
20C22×C4×D137C2211yescharacteristic
21C22×C4×D137C2212yescomplemented
22#17984058—28no—
23C2×C22⋊Dic137C2211yescharacteristic
24C2×#14555487C2212yes—
25C2×#14555487C2211yescharacteristic
26C2×#14555487C2212yes—
27C2×#14555487C2211yescharacteristic
28C2×#14555487C2212yescomplemented
29C2×#14555487C2212yescomplemented
30C22⋊C4×D137—24no—
31#17983995C2212yescomplemented
32#17983995C2211yescharacteristic, complemented
33#17983995C2212yescomplemented
34C22×Q8×C137C2211yescharacteristic, complemented
35C2×C22⋊C4×C137C2212yes—
36C2×C22⋊C4×C137C2211yescharacteristic
37#95×C137C2211yescharacteristic, complemented
38#95×C137C2212yescomplemented
39C23×D137—22no—
40C23×D137—22no—
41C23×D137—24no—
42C23×D137—22no—
43C23×D137—42no—
44C23×D137C2312yes—
45C23×D137C2311yescharacteristic
46C23×D137C2311yescharacteristic
47#14555511—232no—
48C2×D548—24no—
49C2×D548—24no—
50C2×D548—24no—
51C2×D548—24no—
52C2×D548—24no—
53C2×D548—22no—
54C2×D548—42no—
55C2×D548D418yescomplemented
56C2×C22⋊D137—24no—
57C2×C22⋊D137—24no—
58C2×C22⋊D137—42no—
59C22×Dic137C2312yes—
60C22×Dic137C2311yescharacteristic
61C22×Dic137C2311yescharacteristic
62C2×C4×D137—24no—
63C2×C4×D137—24no—
64C2×C4×D137—24no—
65C2×C4×D137—22no—
66C2×C4×D137—24no—
67C2×C4×D137—42no—
68C2×C4×D137D414yescomplemented
69C22⋊Dic137—22no—
70#14555487—24no—
71#14555487—22no—
72#14555487—22no—
73#14555487—24no—
74#14555487—42no—
75#14555487—42no—
76C24×C137C2311yescharacteristic
77C2×Q8×C137—24no—
78C2×Q8×C137—22no—
79C22×C4×C137C2311yescharacteristic
80C22×C4×C137C2312yes—
81C22×C4×C137C2311yescharacteristic
82C22×C4×C137C2312yes—
83C22⋊C4×C137—22no—
84C22⋊C4×C137—24no—
85C22×D137—21no—
86C22×D137—21no—
87C22×D137—22no—
88C22×D137—28no—
89C22×D137—21no—
90C22×D137—22no—
91C22×D137—22no—
92C22×D137—22no—
93C22×D137—22no—
94C22×D137—22no—
95C22×D137—21no—
96C22×D137—22no—
97C22×D137—22no—
98C22×D137—22no—
99C22×D137—48no—
100C22×D137—48no—
101C22×D137—44no—
102C22×D137—44no—
103C22×D137C2×D414yes—
104C22×D137C2×D414yes—
105C22×D137C2×D412yes—
106C22×D137#4112yescomplemented
107D548—28no—
108D548—216no—
109D548—48no—
110C22⋊D137—48no—
111C2×Dic137—22no—
112C2×Dic137—22no—
113C2×Dic137—21no—
114C2×Dic137—22no—
115C2×Dic137—22no—
116C2×Dic137—22no—
117C2×Dic137C2×D412yes—
118C4×D137—216no—
119C4×D137—28no—
120C4×D137—44no—
121C23×C137—21no—
122C23×C137—22no—
123C23×C137—21no—
124C23×C137—22no—
125C23×C137#4112yescomplemented
126C23×C137C2411yescharacteristic
127Q8×C137—28no—
128C2×C4×C137—22no—
129C2×C4×C137—22no—
130C2×C4×C137—22no—
131C2×C4×C137—21no—
132C2×C4×C137—22no—
133C2×C4×C137—22no—
134C2×C4×C137—22no—
135C2×C4×C137—22no—
136C2×C4×C137C2×D414yes—
137C2×C4×C137C2×D412yes—
138C22×C137—22no—
139C22×C137—24no—
140C22×C137—41no—
141C22×C137—41no—
142C22×C137—42no—
143C22×C137C22×D411yescharacteristic
144C22×C137C22×D412yes—
145C22×C137C2×#4111yescharacteristic
146C22×C137C2×#4111yescharacteristic
147C22×C137H(5,𝔽2)12yes—
148C2×D137—22no—
149C2×D137—22no—
150C2×D137—24no—
151C2×D137—24no—
152C2×D137—24no—
153C2×D137—22no—
154C2×D137—24no—
155C2×D137—24no—
156C2×D137—24no—
157C2×D137—416no—
158C2×D137—41no—
159C2×D137—42no—
160C2×D137—42no—
161C2×D137—41no—
162C2×D137—42no—
163C2×D137—42no—
164C2×D137—42no—
165C2×D137#12014yes—
166C4×C137—24no—
167C4×C137—22no—
168C4×C137—41no—
169C4×C137—42no—
170Dic137—22no—
171Dic137—41no—
172Dic137—42no—
173C2×C137—42no—
174C2×C137C2×#12011yescharacteristic
175C2×C137#52511yescharacteristic
176C2×C137#53511yescharacteristic
177C2×C137#54612yes—
178C2×C137#54612yes—
179D137—24no—
180D137—44no—
181D137—42no—
182C137#240411yescharacteristic, complemented, Sylow
183#2404—1371nomaximal, Sylow
184C22×#41—1371no—
185C2×#120—1372no—
186C22×C22⋊C4—1371no—
187#393—1371no—
188#393D13711yescharacteristic, complemented
189#392—1371no—
190#392—1371no—
191#386—1372no—
192#386—1372no—
193#386—1372no—
194#381—1371no—
195C25—1371no—
196C2×#41—2748no—
197C2×#41—2744no—
198C22×Q8C2×D13711yescharacteristic, complemented
199C22×D4—1372no—
200C22×D4—1371no—
201C22×D4—1372no—
202C22×D4—1372no—
203C23×C4—1372no—
204C23×C4—1371no—
205C23×C4—1371no—
206#120—2748no—
207C2×C22⋊C4—2744no—
208C2×C22⋊C4—1371no—
209C2×C22⋊C4—1371no—
210C2×C22⋊C4—1371no—
211C2×C22⋊C4—1372no—
212C2×C22⋊C4—1372no—
213C2×C22⋊C4—1372no—
214C2×C22⋊C4—1372no—
215C2×C22⋊C4C2×D13711yescharacteristic
216C2×C22⋊C4C2×D13712yes—
217#95—1372no—
218#95—1371no—
219#95—1372no—
220#95C2×D13711yescharacteristic, complemented
221#95C2×D13712yescomplemented
222C24—2742no—
223C24—2742no—
224C24—2744no—
225C24—2742no—
226C24—5482no—
227C24—1371no—
228C24—1372no—
229C24—1371no—
230C24C22×D13711yescharacteristic
231#41—27432no—
232C2×Q8—24no—
233C2×Q8—22no—
234C2×D4—2744no—
235C2×D4—2744no—
236C2×D4—2744no—
237C2×D4—2744no—
238C2×D4—2744no—
239C2×D4—2744no—
240C2×D4—2744no—
241C2×D4—2742no—
242C2×D4—5482no—
243C2×D4—5482no—
244C2×D4—1378no—
245C22×C4—2744no—
246C22×C4—2744no—
247C22×C4—2744no—
248C22×C4—2742no—
249C22×C4—2744no—
250C22×C4—5482no—
251C22×C4—1372no—
252C22×C4—1371no—
253C22×C4—1371no—
254C22×C4—1374no—
255C22×C4C22×D13712yes—
256C22×C4C22×D13711yescharacteristic
257C22×C4C22×D13712yes—
258C22×C4C22×D13711yescharacteristic
259C22⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—2744no—
262C22⋊C4—2742no—
263C22⋊C4—2742no—
264C22⋊C4—2742no—
265C22⋊C4—2744no—
266C22⋊C4—5482no—
267C22⋊C4—5482no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23—22no—
272C23—2748no—
273C23—2741no—
274C23—2741no—
275C23—2742no—
276C23—2741no—
277C23—2742no—
278C23—2742no—
279C23—2742no—
280C23—2742no—
281C23—2742no—
282C23—2741no—
283C23—2742no—
284C23—2742no—
285C23—2742no—
286C23—5488no—
287C23—5488no—
288C23—5484no—
289C23—5484no—
290C23—1374no—
291C23—1372no—
292C23—1374no—
293C23—1372no—
294C23#1455551112yescomplemented
295C23C23×D13711yescharacteristic
296Q8—28no—
297D4—2748no—
298D4—27416no—
299D4—5488no—
300D4—5488no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—27416no—
310C2×C4—2748no—
311C2×C4—2742no—
312C2×C4—2741no—
313C2×C4—2742no—
314C2×C4—2742no—
315C2×C4—2742no—
316C2×C4—2742no—
317C2×C4—5484no—
318C2×C4—1372no—
319C2×C4D4×D13714yes—
320C2×C4D4×D13712yes—
321C22—24no—
322C22—22no—
323C22—41no—
324C22—42no—
325C22—41no—
326C22—2742no—
327C22—2744no—
328C22—2744no—
329C22—2742no—
330C22—2744no—
331C22—2744no—
332C22—2744no—
333C22—2744no—
334C22—2742no—
335C22—54816no—
336C22—5481no—
337C22—5482no—
338C22—5482no—
339C22—5482no—
340C22—5482no—
341C22—5481no—
342C22—5482no—
343C22—1374no—
344C22C2×D4×D13711yescharacteristic
345C22C2×D4×D13712yes—
346C22#41×D13711yescharacteristic
347C22C2×#1455551111yescharacteristic
348C22#1798413412yes—
349C4—24no—
350C4—22no—
351C4—41no—
352C4—42no—
353C4—2742no—
354C4—5481no—
355C4—5482no—
356C2—42no—
357C2—2744no—
358C2—5482no—
359C2—5484no—
360C2#120×D13711yescharacteristic
361C2#2257791611yescharacteristic
362C2#2257797111yescharacteristic
363C2#2257804412yes—
364C2#2257804612yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#393×C137C2
hypercenter#393D137
hypocenterC137#2404
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×#14555511C2semidirect
2. C2×#17984058C2semidirect
3. C2×C22⋊C4×D137C2semidirect
4. #22577314C2semidirect
5. #22577313C2semidirect
6. #22577312C2semidirect
7. #22577281C2semidirect
8. #22577281C2semidirect
9. #22577279C2semidirect
10. #22577265C2semidirect
11. #393×C137C2semidirect
17. C22×D548C22semidirect
21. C22×C4×D137C22semidirect
28. C2×#14555487C22semidirect
29. C2×#14555487C22semidirect
31. #17983995C22semidirect
32. #17983995C22semidirect
33. #17983995C22semidirect
34. C22×Q8×C137C22semidirect
37. #95×C137C22semidirect
38. #95×C137C22semidirect
55. C2×D548D4semidirect
68. C2×C4×D137D4semidirect
106. C22×D137#41semidirect
125. C23×C137#41semidirect
182. C137#2404semidirect
188. #393D137semidirect
198. C22×Q8C2×D137semidirect
220. #95C2×D137semidirect
221. #95C2×D137semidirect
294. C23#14555511semidirect

Maximal subgroups

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

subgroupindex
C22×#145555112
C2×#179840582
C2×C22⋊C4×D1372
#225773142
#225773132
#225773122
#225772812
#225772792
#225772652
#393×C1372
#2404137