Finite Group Databasebeta

← #31790413

Subgroups — #31790413

Order 17536.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×M16×D137C211yesmaximal, characteristic, complemented
2#22577910C211yesmaximal, characteristic
3C2×#17984064C211yesmaximal, characteristic, complemented
4#22577658C212yesmaximal, complemented
5#22577646C212yesmaximal, complemented
6#22577641C212yesmaximal, complemented
7#22577629C212yesmaximal, complemented
8#22577445C211yesmaximal, characteristic, complemented
9#22577412C211yesmaximal, characteristic, complemented
10#22577404C211yesmaximal, characteristic, complemented
11C2×#137×C137C211yesmaximal, characteristic, complemented
12C22×C4×D137C2211yescharacteristic
13C2×#14555494C2211yescharacteristic
14C2×#14555493C2212yes—
15M16×D137—24no—
16C2×C8×D137C2212yes—
17#17984069—21no—
18#17984064—21no—
19#17984064—21no—
20#17984064C2212yes—
21#17984064C2211yescharacteristic
22#17984064C2212yes—
23#17984064C2211yescharacteristic
24#17984059C2211yescharacteristic
25#17984059C2211yescharacteristic
26#17984056—21no—
27C4×C22⋊D137—21no—
28C4×C22⋊D137—21no—
29#17984047C2211yescharacteristic
30C2×C4⋊Dic137C2211yescharacteristic
31C2×#14555489—21no—
32C2×#14555487—21no—
33#17984022C2212yescomplemented
34#17984020C2212yescomplemented
35#17984012C2212yescomplemented
36#17984012C2212yescomplemented
37#17984012C2212yescomplemented
38#17984009C2212yescomplemented
39C2×#41×C137C2211yescharacteristic
40C22×Q8×C137C2211yescharacteristic
41#137×C137—24no—
42C2×Q16×C137C2212yescomplemented
43C2×SD16×C137C2212yescomplemented
44C2×M16×C137C2211yescharacteristic
45C23×D137D411yescharacteristic
46C2×C22⋊D137—21no—
47C2×C22⋊D137—21no—
48C22×Dic137—21no—
49C22×Dic137D411yescharacteristic
50C2×C4×D137—22no—
51C2×C4×D137—22no—
52C2×C4×D137D411yescharacteristic
53C2×C4×D137D411yescharacteristic
54C2×C4×D137C2311yescharacteristic
55C2×C4×D137C2311yescharacteristic
56#14555494—22no—
57#14555493—24no—
58C2×#11764040C2312yes—
59C8×D137—24no—
60C4⋊Dic137—21no—
61C4⋊Dic137C2312yes—
62C4⋊Dic137C2311yescharacteristic
63C4⋊Dic137C2311yescharacteristic
64#14555489—22no—
65#14555489—22no—
66#14555489—21no—
67#14555489—21no—
68C22⋊Dic137—21no—
69C22⋊Dic137—21no—
70C22⋊Dic137—21no—
71#14555487—22no—
72#14555487—22no—
73#14555487—21no—
74#14555487—21no—
75C4×Dic137—21no—
76#41×C137—22no—
77#41×C137—22no—
78C2×Q8×C137—21no—
79C2×Q8×C137—22no—
80C2×Q8×C137—22no—
81C2×Q8×C137C2312yes—
82C2×Q8×C137C2311yescharacteristic
83C2×D4×C137—21no—
84C2×D4×C137C2311yescharacteristic
85C22×C4×C137—21no—
86C22×C4×C137—21no—
87C22×C4×C137C2311yescharacteristic
88Q16×C137—24no—
89SD16×C137—24no—
90M16×C137D414yescomplemented
91C2×C8×C137C2312yes—
92C22×D137—22no—
93C22×D137—22no—
94C22×D137—41no—
95C22×D137C2×D411yescharacteristic
96C22×D137C2×D411yescharacteristic
97C22⋊D137—42no—
98C22⋊D137—42no—
99C2×Dic137—22no—
100C2×Dic137—21no—
101C2×Dic137—21no—
102C2×Dic137—22no—
103C2×Dic137—42no—
104C2×Dic137C2×D411yescharacteristic
105C2×Dic137C2×D411yescharacteristic
106C4×D137—22no—
107C4×D137—22no—
108C4×D137—21no—
109C4×D137—21no—
110C4×D137—21no—
111C4×D137—21no—
112#11764040—22no—
113C23×C137—21no—
114C23×C137C2×D411yescharacteristic
115Q8×C137—22no—
116Q8×C137—21no—
117Q8×C137—22no—
118Q8×C137—21no—
119Q8×C137—42no—
120D4×C137—21no—
121D4×C137—21no—
122D4×C137—42no—
123C2×C4×C137—21no—
124C2×C4×C137—22no—
125C2×C4×C137—42no—
126C2×C4×C137—42no—
127C2×C4×C137C2×D411yescharacteristic
128C2×C4×C137C2×D412yes—
129C2×C4×C137#4112yes—
130C2×C4×C137C2411yescharacteristic
131C8×C137C2×D414yes—
132C22×C137—21no—
133C22×C137—21no—
134C22×C137—41no—
135C22×C137—41no—
136C22×C137—41no—
137C22×C137#12112yes—
138C22×C137C22×D411yescharacteristic
139C2×D137—21no—
140C2×D137—21no—
141C2×D137—21no—
142C2×D137—21no—
143C2×D137—21no—
144C2×D137—44no—
145C2×D137#13712yescomplemented
146C4×C137—41no—
147C4×C137—42no—
148C4×C137#12112yes—
149C4×C137C22×D411yescharacteristic
150C4×C137C2×#4111yescharacteristic
151Dic137—21no—
152Dic137—21no—
153Dic137—41no—
154Dic137—42no—
155Dic137—41no—
156C2×C137—22no—
157C2×C137—81no—
158C2×C137C2×#12111yescharacteristic
159C2×C137C2×#13711yescharacteristic
160C2×C137#57511yescharacteristic
161D137—22no—
162D137—41no—
163C137#305611yescharacteristic, complemented, Sylow
164#3056—1371nomaximal, Sylow
165C2×#137D13711yescharacteristic, complemented
166C22×M16—1371no—
167#525—1371no—
168C2×C22⋊Q8—1371no—
169#470—1372no—
170#468—1372no—
171#467—1372no—
172#465—1372no—
173#428—1371no—
174#419—1371no—
175#417—1371no—
176C2×#41C2×D13711yescharacteristic
177C22×Q8C2×D13711yescharacteristic
178C23×C4—1371no—
179#137—24no—
180C2×Q16C2×D13712yescomplemented
181C2×SD16C2×D13712yescomplemented
182C2×M16—2744no—
183C2×M16—1371no—
184C2×M16—1372no—
185C2×M16C2×D13711yescharacteristic
186C22×C8—1372no—
187#123—2741no—
188C22⋊Q8—2741no—
189C22⋊Q8—2741no—
190C22⋊Q8—1372no—
191C22⋊Q8—1372no—
192C22⋊Q8—1371no—
193C22⋊Q8—1371no—
194#121—1371no—
195#121—1371no—
196#120—2741no—
197C4×D4—2741no—
198C4×D4—2741no—
199#117—1371no—
200C2×C4⋊C4—2741no—
201C2×C4⋊C4—1371no—
202C2×C22⋊C4—2741no—
203#107—1372no—
204#106—1372no—
205#103—1372no—
206#103—1372no—
207#103—1372no—
208#102—1372no—
209C24—1371no—
210#41—22no—
211#41—22no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8C22×D13712yes—
216C2×Q8C22×D13711yescharacteristic
217C2×D4—21no—
218C2×D4—2741no—
219C2×D4—2741no—
220C2×D4C22×D13711yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—2742no—
224C22×C4—2742no—
225C22×C4—2741no—
226C22×C4—1371no—
227C22×C4—1371no—
228C22×C4—1371no—
229C22×C4—1371no—
230C22×C4—1371no—
231C22×C4C22×D13711yescharacteristic
232Q16—24no—
233SD16—24no—
234M16—2742no—
235M16—2744no—
236M16C22⋊D13714yescomplemented
237C2×C8—2744no—
238C2×C8—1372no—
239C2×C8C22×D13712yes—
240C4⋊C4—2742no—
241C4⋊C4—2742no—
242C4⋊C4—2741no—
243C4⋊C4—2741no—
244C4⋊C4—2741no—
245C4⋊C4—1372no—
246C4⋊C4—1371no—
247C4⋊C4—1371no—
248C22⋊C4—2742no—
249C22⋊C4—2742no—
250C22⋊C4—2741no—
251C22⋊C4—2741no—
252C22⋊C4—2741no—
253C22⋊C4—2741no—
254C22⋊C4—2741no—
255C42—2741no—
256C23—21no—
257C23—2742no—
258C23—2742no—
259C23—5481no—
260C23—1371no—
261C23—1371no—
262C23D4×D13711yescharacteristic
263Q8—22no—
264Q8—21no—
265Q8—22no—
266Q8—21no—
267Q8—42no—
268D4—21no—
269D4—21no—
270D4—42no—
271D4—5482no—
272D4—5482no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—2742no—
278C2×C4—2742no—
279C2×C4—2742no—
280C2×C4—2741no—
281C2×C4—2741no—
282C2×C4—2741no—
283C2×C4—2741no—
284C2×C4—2741no—
285C2×C4—2741no—
286C2×C4—2742no—
287C2×C4—5482no—
288C2×C4—1371no—
289C2×C4—1371no—
290C2×C4D4×D13711yescharacteristic
291C2×C4C2×C22⋊D13712yes—
292C2×C4#1455550912yes—
293C2×C4C23×D13711yescharacteristic
294C8—2742no—
295C8C2×C22⋊D13714yes—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—2741no—
302C22—2741no—
303C22—2741no—
304C22—2741no—
305C22—2741no—
306C22—5484no—
307C22—1372no—
308C22#1798405912yes—
309C22C2×D4×D13711yescharacteristic
310C4—41no—
311C4—42no—
312C4—2741no—
313C4—2741no—
314C4—5481no—
315C4—5482no—
316C4—5481no—
317C4#1798405912yes—
318C4C22×C22⋊D13711yescharacteristic
319C4C2×#1455550911yescharacteristic
320C2—22no—
321C2—81no—
322C2—2742no—
323C2—5481no—
324C2C2×#1798405911yescharacteristic
325C2#137×D13711yescharacteristic
326C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×#137×C137C2
socleC22×C137C22×D4
prime radical#22577910C2
hypercenterC2×#137D137
hypocenterC137#3056
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×M16×D137C2semidirect
3. C2×#17984064C2semidirect
4. #22577658C2semidirect
5. #22577646C2semidirect
6. #22577641C2semidirect
7. #22577629C2semidirect
8. #22577445C2semidirect
9. #22577412C2semidirect
10. #22577404C2semidirect
11. C2×#137×C137C2semidirect
33. #17984022C22semidirect
34. #17984020C22semidirect
35. #17984012C22semidirect
36. #17984012C22semidirect
37. #17984012C22semidirect
38. #17984009C22semidirect
42. C2×Q16×C137C22semidirect
43. C2×SD16×C137C22semidirect
90. M16×C137D4semidirect
145. C2×D137#137semidirect
163. C137#3056semidirect
165. C2×#137D137semidirect
180. C2×Q16C2×D137semidirect
181. C2×SD16C2×D137semidirect
236. M16C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1372
#225779102
C2×#179840642
#225776582
#225776462
#225776412
#225776292
#225774452
#225774122
#225774042
C2×#137×C1372
#3056137