Finite Group Databasebeta

← #31784099

Subgroups — #31784099

Order 17536.

1564 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 213 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×C22⋊D137C211yesmaximal, characteristic, complemented
2C2×#17984057C212yesmaximal, complemented
3C2×#17984056C212yesmaximal, complemented
4#22577311C214yesmaximal, complemented
5#22577283C212yesmaximal, complemented
6#22577282C212yesmaximal, complemented
7#22577259C211yesmaximal, characteristic, complemented
8C23×D4×C137C211yesmaximal, characteristic, complemented
9C24×D137C2211yescharacteristic, complemented
10C22×C22⋊D137—28no—
11C22×C22⋊D137—24no—
12C22×C22⋊D137C2212yes—
13C22×C22⋊D137C2212yescomplemented
14C22×C22⋊D137C2214yescomplemented
15C23×Dic137C2211yescharacteristic, complemented
16#17984057—28no—
17#17984056—28no—
18C2×C22⋊Dic137C2212yescomplemented
19C2×C22⋊Dic137C2212yescomplemented
20C2×C22⋊Dic137C2214yescomplemented
21C2×C22⋊Dic137C2212yescomplemented
22C2×#14555487C2214yescomplemented
23#17983995C2214yescomplemented
24C25×C137C2211yescharacteristic
25C25×C137C2211yescharacteristic, complemented
26C22×D4×C137—24no—
27C22×D4×C137—28no—
28C22×D4×C137C2212yes—
29C22×D4×C137C2212yescomplemented
30C23×C4×C137C2211yescharacteristic, complemented
31C23×D137—24no—
32C23×D137—22no—
33C23×D137—24no—
34C23×D137—42no—
35C23×D137C2312yes—
36C2×C22⋊D137—232no—
37C2×C22⋊D137—24no—
38C2×C22⋊D137—28no—
39C2×C22⋊D137—24no—
40C2×C22⋊D137—24no—
41C2×C22⋊D137—24no—
42C2×C22⋊D137—24no—
43C2×C22⋊D137—28no—
44C2×C22⋊D137—44no—
45C2×C22⋊D137D418yescomplemented
46C22×Dic137—24no—
47C22×Dic137—22no—
48C22×Dic137C2314yes—
49C22×Dic137C2312yes—
50C22⋊Dic137—28no—
51C22⋊Dic137—24no—
52C22⋊Dic137—24no—
53C22⋊Dic137—42no—
54#14555487—28no—
55C24×C137—24no—
56C24×C137—24no—
57C24×C137—22no—
58C24×C137—24no—
59C24×C137—24no—
60C24×C137—42no—
61C24×C137—42no—
62C24×C137D414yescomplemented
63C24×C137C2311yescharacteristic
64C24×C137C2312yes—
65C24×C137C2311yescharacteristic
66C24×C137C2311yescharacteristic
67C2×D4×C137—232no—
68C2×D4×C137—24no—
69C2×D4×C137—24no—
70C2×D4×C137—24no—
71C2×D4×C137—24no—
72C2×D4×C137—24no—
73C2×D4×C137D418yescomplemented
74C22×C4×C137—22no—
75C22×C4×C137—24no—
76C22×C4×C137C2312yes—
77C22×D137—28no—
78C22×D137—24no—
79C22×D137—22no—
80C22×D137—22no—
81C22×D137—24no—
82C22×D137—416no—
83C22×D137—44no—
84C22×D137—44no—
85C22×D137C2×D414yescomplemented
86C22⋊D137—216no—
87C22⋊D137—216no—
88C22⋊D137—416no—
89C2×Dic137—28no—
90C2×Dic137—22no—
91C2×Dic137—22no—
92C2×Dic137—24no—
93C2×Dic137—24no—
94C2×Dic137—24no—
95C2×Dic137C2×D414yescomplemented
96C23×C137—22no—
97C23×C137—22no—
98C23×C137—22no—
99C23×C137—22no—
100C23×C137—28no—
101C23×C137—28no—
102C23×C137—28no—
103C23×C137—22no—
104C23×C137—22no—
105C23×C137—22no—
106C23×C137—24no—
107C23×C137—22no—
108C23×C137—22no—
109C23×C137—24no—
110C23×C137—44no—
111C23×C137—416no—
112C23×C137—416no—
113C23×C137—44no—
114C23×C137—44no—
115C23×C137C2×D414yes—
116C23×C137C2×D412yes—
117C23×C137C2×D412yes—
118C23×C137C2×D412yes—
119C23×C137C2×D414yes—
120C23×C137C2×D414yescomplemented
121C23×C137C2411yescharacteristic
122D4×C137—216no—
123D4×C137—216no—
124C2×C4×C137—28no—
125C2×C4×C137—22no—
126C2×C4×C137—22no—
127C2×C4×C137C2×D414yescomplemented
128C22×C137—22no—
129C22×C137—22no—
130C22×C137—22no—
131C22×C137—22no—
132C22×C137—28no—
133C22×C137—28no—
134C22×C137—22no—
135C22×C137—24no—
136C22×C137—22no—
137C22×C137—22no—
138C22×C137—28no—
139C22×C137—416no—
140C22×C137—416no—
141C22×C137—42no—
142C22×C137—42no—
143C22×C137—44no—
144C22×C137—44no—
145C22×C137#12014yescomplemented
146C22×C137#12014yescomplemented
147C22×C137C22×D412yes—
148C22×C137C22×D411yescharacteristic
149C22×C137C22×D412yes—
150C22×C137H(5,𝔽2)12yes—
151C2×D137—24no—
152C2×D137—28no—
153C2×D137—416no—
154C2×D137—42no—
155C2×D137—42no—
156C2×D137—44no—
157C4×C137—24no—
158Dic137—24no—
159Dic137—44no—
160C2×C137—24no—
161C2×C137—24no—
162C2×C137—42no—
163C2×C137—44no—
164C2×C137C2×#12011yescharacteristic
165C2×C137C2×#12011yescharacteristic
166C2×C137#53411yescharacteristic
167C2×C137D4214yes—
168D137—44no—
169C137#240311yescharacteristic, complemented, Sylow
170#2403—1371nomaximal, Sylow
171C23×D4—1371no—
172C23×D4D13711yescharacteristic, complemented
173C2×#120—1372no—
174C2×#120—1372no—
175#392—1374no—
176#386—1372no—
177#386—1372no—
178#379—1371no—
179C25—1371no—
180C25C2×D13711yescharacteristic
181C25C2×D13711yescharacteristic, complemented
182C22×D4—24no—
183C22×D4—28no—
184C22×D4—2744no—
185C22×D4—2748no—
186C22×D4—1372no—
187C22×D4—1374no—
188C22×D4—1372no—
189C22×D4C2×D13712yes—
190C22×D4C2×D13712yescomplemented
191C23×C4—1371no—
192C23×C4C2×D13711yescharacteristic, complemented
193#120—2748no—
194#120—2748no—
195C2×C22⋊C4—1372no—
196C2×C22⋊C4—1372no—
197C2×C22⋊C4—1374no—
198C2×C22⋊C4—1374no—
199C2×C22⋊C4—1372no—
200#95—1374no—
201C24—24no—
202C24—24no—
203C24—22no—
204C24—24no—
205C24—24no—
206C24—42no—
207C24—42no—
208C24—2744no—
209C24—2742no—
210C24—2744no—
211C24—5482no—
212C24—1372no—
213C24C22⋊D13714yescomplemented
214C24C22×D13711yescharacteristic
215C24C22×D13712yes—
216C24C22×D13711yescharacteristic
217C24C22×D13711yescharacteristic
218C2×D4—232no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—27432no—
225C2×D4—2744no—
226C2×D4—2744no—
227C2×D4—2744no—
228C2×D4—2744no—
229C2×D4—2748no—
230C2×D4—2748no—
231C2×D4—2744no—
232C2×D4—5484no—
233C2×D4—1378no—
234C2×D4C22⋊D13718yescomplemented
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—2742no—
238C22×C4—2744no—
239C22×C4—1374no—
240C22×C4—1372no—
241C22×C4C22×D13712yes—
242C22⋊C4—2744no—
243C22⋊C4—2744no—
244C22⋊C4—2748no—
245C22⋊C4—2748no—
246C22⋊C4—5482no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—28no—
252C23—28no—
253C23—28no—
254C23—24no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—24no—
261C23—44no—
262C23—416no—
263C23—416no—
264C23—44no—
265C23—44no—
266C23—2748no—
267C23—2742no—
268C23—2744no—
269C23—2742no—
270C23—2744no—
271C23—54816no—
272C23—5484no—
273C23—5484no—
274C23—1374no—
275C23D4×D13714yes—
276C23C2×C22⋊D13712yes—
277C23C2×C22⋊D13712yes—
278C23C2×C22⋊D13714yes—
279C23C2×C22⋊D13712yes—
280C23C2×C22⋊D13714yescomplemented
281C23C23×D13711yescharacteristic
282D4—216no—
283D4—216no—
284D4—27416no—
285D4—27416no—
286D4—54816no—
287C2×C4—28no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—2748no—
291C2×C4—2742no—
292C2×C4—2742no—
293C2×C4—2744no—
294C2×C4—2744no—
295C2×C4—2744no—
296C2×C4—1374no—
297C2×C4C2×C22⋊D13714yescomplemented
298C22—22no—
299C22—22no—
300C22—28no—
301C22—28no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—28no—
306C22—22no—
307C22—22no—
308C22—24no—
309C22—416no—
310C22—416no—
311C22—42no—
312C22—44no—
313C22—44no—
314C22—42no—
315C22—2748no—
316C22—2744no—
317C22—54816no—
318C22—5482no—
319C22—5484no—
320C22—5482no—
321C22#1798405614yescomplemented
322C22#1798405714yescomplemented
323C22C2×D4×D13712yes—
324C22C22×C22⋊D13711yescharacteristic
325C22C22×C22⋊D13712yes—
326C22#1798413312yes—
327C4—24no—
328C4—2744no—
329C4—5484no—
330C2—24no—
331C2—24no—
332C2—42no—
333C2—44no—
334C2—5484no—
335C2C2×#1798405611yescharacteristic
336C2C2×#1798405711yescharacteristic
337C2#2257796211yescharacteristic
338C2D4×C22⋊D13714yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
FittingC23×D4×C137C2
hypercenterC23×D4D137
hypocenterC137#2403
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D137C2semidirect
2. C2×#17984057C2semidirect
3. C2×#17984056C2semidirect
4. #22577311C2semidirect
5. #22577283C2semidirect
6. #22577282C2semidirect
7. #22577259C2semidirect
8. C23×D4×C137C2semidirect
9. C24×D137C22semidirect
13. C22×C22⋊D137C22semidirect
14. C22×C22⋊D137C22semidirect
15. C23×Dic137C22semidirect
18. C2×C22⋊Dic137C22semidirect
19. C2×C22⋊Dic137C22semidirect
20. C2×C22⋊Dic137C22semidirect
21. C2×C22⋊Dic137C22semidirect
22. C2×#14555487C22semidirect
23. #17983995C22semidirect
25. C25×C137C22semidirect
29. C22×D4×C137C22semidirect
30. C23×C4×C137C22semidirect
45. C2×C22⋊D137D4semidirect
62. C24×C137D4semidirect
73. C2×D4×C137D4semidirect
85. C22×D137C2×D4semidirect
95. C2×Dic137C2×D4semidirect
120. C23×C137C2×D4semidirect
127. C2×C4×C137C2×D4semidirect
145. C22×C137#120semidirect
146. C22×C137#120semidirect
169. C137#2403semidirect
172. C23×D4D137semidirect
181. C25C2×D137semidirect
190. C22×D4C2×D137semidirect
192. C23×C4C2×D137semidirect
213. C24C22⋊D137semidirect
234. C2×D4C22⋊D137semidirect
280. C23C2×C22⋊D137semidirect
297. C2×C4C2×C22⋊D137semidirect
321. C22#17984056semidirect
322. C22#17984057semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1372
C2×#179840572
C2×#179840562
#225773112
#225772832
#225772822
#225772592
C23×D4×C1372
#2403137