Finite Group Databasebeta

← #31790059

Subgroups — #31790059

Order 17536.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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×#17984134C211yesmaximal, characteristic, complemented
2C2×#17984116C211yesmaximal, characteristic, complemented
3C22×D1096C211yesmaximal, characteristic, complemented
4#22577967C211yesmaximal, characteristic, complemented
5#22577550C214yesmaximal, complemented
6#22577539C214yesmaximal, complemented
7#22577409C211yesmaximal, characteristic, complemented
8#22577400C211yesmaximal, characteristic, complemented
9#408×C137C211yesmaximal, characteristic, complemented
10#17984134—24no—
11#17984134—24no—
12C2×#14555511—21no—
13C2×#14555510C2211yescharacteristic
14C2×#14555510C2212yescomplemented
15C22×D548C2211yescharacteristic
16C22×D548C2211yescharacteristic, complemented
17C22×D548C2212yescomplemented
18C2×D4×D137—22no—
19C2×D4×D137—21no—
20#17984116—24no—
21C2×#14555499C2212yescomplemented
22C2×D1096—24no—
23C2×D1096C2212yescomplemented
24C2×D1096C2214yescomplemented
25#17984084—21no—
26#17984077—21no—
27#17984062C2214yescomplemented
28#17984056—22no—
29#17984047C2211yescharacteristic, complemented
30#17984013C2212yescomplemented
31#17984010C2214yescomplemented
32#17984010C2212yescomplemented
33C2×#41×C137C2211yescharacteristic
34C2×M16×C137C2211yescharacteristic, complemented
35C22×C8×C137C2211yescharacteristic, complemented
36C22⋊C8×C137C2214yescomplemented
37C23×D137—21no—
38C23×D137—22no—
39C23×D137—21no—
40#14555511—28no—
41#14555510—24no—
42#14555510—24no—
43#14555510—22no—
44#14555510D414yescomplemented
45C2×Dic274C2311yescharacteristic
46C2×D548—24no—
47C2×D548—24no—
48C2×D548—21no—
49C2×D548—22no—
50C2×D548—22no—
51C2×D548—22no—
52C2×D548—22no—
53C2×D548—21no—
54C2×D548D414yescomplemented
55C2×D548C2312yes—
56C2×D548C2312yes—
57C2×D548C2311yescharacteristic
58C2×C22⋊D137—21no—
59C2×C22⋊D137—22no—
60C2×C22⋊D137—24no—
61D4×D137—216no—
62D4×D137—28no—
63C2×C4×D137—21no—
64C2×C4×D137—22no—
65#14555499—24no—
66D1096—28no—
67D1096—24no—
68C4⋊Dic137C2312yes—
69C22⋊Dic137—21no—
70#14555487—24no—
71C4×Dic137—21no—
72#41×C137—22no—
73#41×C137—22no—
74C2×Q8×C137D411yescharacteristic
75C2×D4×C137D412yes—
76C2×D4×C137D411yescharacteristic
77C22×C4×C137C2312yes—
78C22×C4×C137C2311yescharacteristic
79M16×C137—22no—
80C2×C8×C137—22no—
81C2×C8×C137C2312yes—
82C2×C8×C137C2312yes—
83C22×D137—21no—
84C22×D137—22no—
85C22×D137—28no—
86C22×D137—24no—
87C22×D137—22no—
88C22×D137—42no—
89C22×D137—82no—
90C22×D137—84no—
91C22×D137—82no—
92Dic274—21no—
93Dic274C2×D412yes—
94D548—28no—
95D548—24no—
96D548—22no—
97D548—21no—
98D548—22no—
99D548—22no—
100D548—42no—
101D548C2×D412yes—
102D548C2×D414yes—
103C22⋊D137—28no—
104C22⋊D137—24no—
105C22⋊D137—84no—
106C2×Dic137—21no—
107C2×Dic137—22no—
108C4×D137—28no—
109C4×D137—24no—
110C23×C137C2×D412yes—
111C23×C137C2×D411yescharacteristic
112Q8×C137—22no—
113D4×C137—24no—
114D4×C137—22no—
115C2×C4×C137—22no—
116C2×C4×C137—22no—
117C2×C4×C137C2×D412yes—
118C2×C4×C137C2×D412yes—
119C2×C4×C137C2×D412yes—
120C2×C4×C137C2×D411yescharacteristic
121C2×C4×C137C2411yescharacteristic
122C8×C137—22no—
123C8×C137—22no—
124C22×C137—22no—
125C22×C137—22no—
126C22×C137—24no—
127C22×C137#12012yes—
128C22×C137C22×D411yescharacteristic
129C2×D137—24no—
130C2×D137—22no—
131C2×D137—44no—
132C2×D137—42no—
133C2×D137—42no—
134C2×D137—44no—
135C2×D137—84no—
136C2×D137—88no—
137C2×D137—84no—
138C4×C137—22no—
139C4×C137#12012yes—
140C4×C137C22×D411yescharacteristic
141C4×C137C22×D411yescharacteristic
142Dic137—22no—
143Dic137—42no—
144C2×C137—22no—
145C2×C137—42no—
146C2×C137C2×#12011yescharacteristic
147C2×C137#57612yes—
148D137—44no—
149D137—42no—
150D137—82no—
151C137#300811yescharacteristic, complemented, Sylow
152#3008—1371nomaximal, Sylow
153C2×H(5,𝔽2)—1371no—
154C2×AGL(1,ℤ/8)—1371no—
155C22×D8—1371no—
156#535—1371no—
157#449—1374no—
158#447—1374no—
159#418—1371no—
160#416—1371no—
161#408D13711yescharacteristic, complemented
162H(5,𝔽2)—2744no—
163H(5,𝔽2)—2744no—
164C2×#41—2741no—
165C2×#41—1372no—
166C2×#41—1371no—
167C2×#41C2×D13711yescharacteristic
168C22×D4—2742no—
169C22×D4—2741no—
170C22×D4—1371no—
171C22×D4—1372no—
172C22×D4—1371no—
173AGL(1,ℤ/8)—2744no—
174C2×SD16—1372no—
175C2×D8—2744no—
176C2×D8—1374no—
177C2×D8—1372no—
178C2×M16C2×D13711yescharacteristic, complemented
179C22×C8C2×D13711yescharacteristic, complemented
180#127—2741no—
181#124—2741no—
182#121—1374no—
183#120—2742no—
184#117—1371no—
185#103—1372no—
186#102—1372no—
187#102—1374no—
188C22⋊C8C2×D13714yescomplemented
189C24—2742no—
190C24—2741no—
191C24—2741no—
192#41—22no—
193#41—22no—
194#41—2744no—
195#41—2744no—
196#41—2748no—
197#41—2742no—
198#41—1374no—
199C2×Q8—1371no—
200C2×Q8D54811yescharacteristic
201C2×D4—2744no—
202C2×D4—2744no—
203C2×D4—27416no—
204C2×D4—2748no—
205C2×D4—2742no—
206C2×D4—2741no—
207C2×D4—2744no—
208C2×D4—2742no—
209C2×D4—2742no—
210C2×D4—2742no—
211C2×D4—2742no—
212C2×D4—2741no—
213C2×D4—2741no—
214C2×D4—1374no—
215C2×D4—1372no—
216C2×D4—1371no—
217C2×D4—1372no—
218C2×D4D54812yes—
219C2×D4D54811yescharacteristic
220C22×C4—2742no—
221C22×C4—2741no—
222C22×C4C22×D13712yes—
223C22×C4C22×D13711yescharacteristic
224SD16—2744no—
225D8—2748no—
226D8—2744no—
227M16—22no—
228C2×C8—22no—
229C2×C8C22×D13712yes—
230C2×C8C22×D13712yes—
231C4⋊C4—1372no—
232C22⋊C4—2741no—
233C22⋊C4—2744no—
234C42—2741no—
235C23—2748no—
236C23—2742no—
237C23—2741no—
238C23—2744no—
239C23—2742no—
240C23—5482no—
241C23—10962no—
242C23—10962no—
243C23—10964no—
244C23C2×D54812yes—
245C23C2×D54811yescharacteristic
246Q8—22no—
247Q8—2741no—
248Q8—1372no—
249D4—24no—
250D4—22no—
251D4—2748no—
252D4—2748no—
253D4—2744no—
254D4—2744no—
255D4—2742no—
256D4—2741no—
257D4—2742no—
258D4—2742no—
259D4—5482no—
260D4—10964no—
261D4—1372no—
262D4—1374no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—2748no—
266C2×C4—2741no—
267C2×C4—2744no—
268C2×C4—2742no—
269C2×C4D4×D13712yes—
270C2×C4D4×D13712yes—
271C2×C4C2×D54812yes—
272C2×C4C2×D54811yescharacteristic
273C2×C4C23×D13711yescharacteristic
274C8—22no—
275C8—22no—
276C22—22no—
277C22—24no—
278C22—22no—
279C22—2744no—
280C22—2742no—
281C22—5484no—
282C22—5482no—
283C22—5484no—
284C22—5482no—
285C22—10964no—
286C22—10964no—
287C22—10968no—
288C22#1798405812yes—
289C22C22×D54811yescharacteristic
290C4—22no—
291C4—2742no—
292C4—5482no—
293C4#1798405812yes—
294C4C2×D4×D13711yescharacteristic
295C4C2×D4×D13711yescharacteristic
296C2—22no—
297C2—42no—
298C2—5484no—
299C2—5482no—
300C2—10962no—
301C2C2×#1798405811yescharacteristic
302C2#2257820812yes—
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#408×C137C2
socleC22×C137C22×D4
hypercenter#408D137
hypocenterC137#3008
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#17984134C2semidirect
2. C2×#17984116C2semidirect
3. C22×D1096C2semidirect
4. #22577967C2semidirect
5. #22577550C2semidirect
6. #22577539C2semidirect
7. #22577409C2semidirect
8. #22577400C2semidirect
9. #408×C137C2semidirect
14. C2×#14555510C22semidirect
16. C22×D548C22semidirect
17. C22×D548C22semidirect
21. C2×#14555499C22semidirect
23. C2×D1096C22semidirect
24. C2×D1096C22semidirect
27. #17984062C22semidirect
29. #17984047C22semidirect
30. #17984013C22semidirect
31. #17984010C22semidirect
32. #17984010C22semidirect
34. C2×M16×C137C22semidirect
35. C22×C8×C137C22semidirect
36. C22⋊C8×C137C22semidirect
44. #14555510D4semidirect
54. C2×D548D4semidirect
151. C137#3008semidirect
161. #408D137semidirect
178. C2×M16C2×D137semidirect
179. C22×C8C2×D137semidirect
188. C22⋊C8C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841342
C2×#179841162
C22×D10962
#225779672
#225775502
#225775392
#225774092
#225774002
#408×C1372
#3008137