Finite Group Databasebeta

← #31790069

Subgroups — #31790069

Order 17536.

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

normal subgroups only

This group has 752 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
2C2×#17984118C211yesmaximal, characteristic, complemented
3C22×#14555499C211yesmaximal, characteristic, complemented
4#22577979C211yesmaximal, characteristic, complemented
5#22577568C214yesmaximal, complemented
6#22577555C214yesmaximal, complemented
7#22577407C211yesmaximal, characteristic, complemented
8#22577369C211yesmaximal, characteristic, complemented
9#416×C137C211yesmaximal, characteristic, complemented
10#17984134—24no—
11#17984134—24no—
12C2×#14555511C2212yes—
13C2×#14555510—21no—
14C2×#14555510C2211yescharacteristic
15C22×Dic274C2211yescharacteristic, complemented
16C22×D548—21no—
17C22×D548C2211yescharacteristic
18C2×D4×D137—22no—
19C2×D4×D137C2212yes—
20#17984118—24no—
21C2×#14555500C2212yescomplemented
22C2×#14555499—24no—
23C2×#14555499C2214yescomplemented
24C2×#14555498C2212yescomplemented
25C2×#14555494C2211yescharacteristic, complemented
26#17984088—21no—
27#17984076—21no—
28#17984072—22no—
29#17984068C2214yescomplemented
30#17984008C2214yescomplemented
31#17984000C2214yescomplemented
32C2×#41×C137C2211yescharacteristic
33C22×C8×C137C2211yescharacteristic, complemented
34#117×C137C2211yescharacteristic, complemented
35#103×C137C2212yescomplemented
36#102×C137C2212yescomplemented
37C23×D137—22no—
38C23×D137—21no—
39#14555511—24no—
40#14555511—24no—
41#14555510—28no—
42#14555510—22no—
43#14555510—22no—
44C2×Dic274—21no—
45C2×Dic274—22no—
46C2×Dic274—22no—
47C2×Dic274D411yescharacteristic
48C2×Dic274C2312yes—
49C2×D548—28no—
50C2×D548—22no—
51C2×D548—21no—
52C2×D548—22no—
53C2×D548D411yescharacteristic
54C2×D548D414yes—
55C2×D548C2312yes—
56C2×C22⋊D137—22no—
57C2×C22⋊D137D412yes—
58D4×D137—216no—
59D4×D137—24no—
60D4×D137—24no—
61C22×Dic137—21no—
62C2×C4×D137—22no—
63C2×C4×D137C2312yes—
64#14555500—24no—
65#14555499—28no—
66#14555498—24no—
67#14555494—22no—
68C2×#11764040C2312yes—
69C4⋊Dic137—24no—
70#14555487—24no—
71#41×C137—22no—
72#41×C137D414yes—
73C2×Q8×C137C2311yescharacteristic
74C2×D4×C137—21no—
75C2×D4×C137C2311yescharacteristic
76C22×C4×C137—21no—
77C22×C4×C137C2311yescharacteristic
78C2×C8×C137—22no—
79C2×C8×C137C2312yes—
80C4⋊C4×C137C2312yes—
81C22⋊C4×C137—21no—
82C42×C137—21no—
83C22×D137—22no—
84C22×D137—28no—
85C22×D137—24no—
86C22×D137—84no—
87C22×D137—82no—
88C22×D137C2×D412yes—
89Dic274—22no—
90Dic274—22no—
91Dic274—22no—
92Dic274—42no—
93D548—28no—
94D548—22no—
95D548—24no—
96D548—22no—
97D548C2×D414yes—
98C22⋊D137—28no—
99C22⋊D137—24no—
100C2×Dic137—22no—
101C2×Dic137—42no—
102C2×Dic137C2×D412yes—
103C4×D137—28no—
104C4×D137—22no—
105C4×D137—22no—
106#11764040—22no—
107C23×C137—21no—
108C23×C137C2×D411yescharacteristic
109Q8×C137—21no—
110Q8×C137C2×D412yes—
111D4×C137—24no—
112D4×C137—21no—
113D4×C137C2×D412yes—
114C2×C4×C137—21no—
115C2×C4×C137—24no—
116C2×C4×C137—22no—
117C2×C4×C137C2×D411yescharacteristic
118C2×C4×C137C2×D412yes—
119C2×C4×C137C2×D412yes—
120C2×C4×C137C2411yescharacteristic
121C8×C137—22no—
122C22×C137—22no—
123C22×C137—22no—
124C22×C137—42no—
125C22×C137#12012yes—
126C22×C137C22×D411yescharacteristic
127C2×D137—22no—
128C2×D137—24no—
129C2×D137—24no—
130C2×D137—44no—
131C2×D137—84no—
132C2×D137—88no—
133C4×C137—22no—
134C4×C137—42no—
135C4×C137#12012yes—
136C4×C137C22×D411yescharacteristic
137C4×C137C22×D411yescharacteristic
138Dic137—22no—
139Dic137—42no—
140C2×C137—22no—
141C2×C137—42no—
142C2×C137C2×#12011yescharacteristic
143C2×C137#57712yes—
144D137—42no—
145D137—44no—
146C137#300911yescharacteristic, complemented, Sylow
147#3009—1371nomaximal, Sylow
148C2×H(5,𝔽2)—1371no—
149C2×#137—1371no—
150C22×SD16—1371no—
151#536—1371no—
152#452—1374no—
153#450—1374no—
154#418—1371no—
155#416D13711yescharacteristic, complemented
156#408—1371no—
157H(5,𝔽2)—2744no—
158H(5,𝔽2)—2744no—
159C2×#41—2741no—
160C2×#41—1372no—
161C2×#41—1371no—
162C2×#41C2×D13711yescharacteristic
163C22×Q8—1371no—
164C22×D4—2741no—
165C22×D4—2742no—
166C22×D4—1372no—
167C22×D4—1371no—
168#137—2744no—
169C2×Q16—1372no—
170C2×SD16—2744no—
171C2×SD16—1372no—
172C2×SD16—1374no—
173C2×M16—1371no—
174C22×C8C2×D13711yescharacteristic, complemented
175C4⋊Q8—2741no—
176#124—2741no—
177#123—2742no—
178C22⋊Q8—1374no—
179#117C2×D13711yescharacteristic, complemented
180#103C2×D13712yescomplemented
181#102—1374no—
182#102C2×D13712yescomplemented
183C22⋊C8—1374no—
184C24—2742no—
185C24—2741no—
186#41—22no—
187#41—2748no—
188#41—2744no—
189#41—2744no—
190#41—2742no—
191#41—2742no—
192#41D54814yes—
193C2×Q8—2741no—
194C2×Q8—2742no—
195C2×Q8—2742no—
196C2×Q8—1372no—
197C2×Q8—1371no—
198C2×Q8C22×D13711yescharacteristic
199C2×D4—21no—
200C2×D4—2748no—
201C2×D4—2742no—
202C2×D4—27416no—
203C2×D4—2744no—
204C2×D4—2744no—
205C2×D4—2742no—
206C2×D4—2741no—
207C2×D4—2742no—
208C2×D4—1372no—
209C2×D4—1371no—
210C2×D4—1374no—
211C2×D4—1372no—
212C2×D4C22×D13711yescharacteristic
213C22×C4—21no—
214C22×C4—2742no—
215C22×C4—2741no—
216C22×C4—1372no—
217C22×C4C22×D13711yescharacteristic
218Q16—2744no—
219SD16—2748no—
220SD16—2744no—
221M16—2742no—
222C2×C8—22no—
223C2×C8—1372no—
224C2×C8C22×D13712yes—
225C4⋊C4—2744no—
226C4⋊C4C22×D13712yes—
227C22⋊C4—21no—
228C22⋊C4—2744no—
229C42—21no—
230C23—21no—
231C23—2748no—
232C23—2742no—
233C23—2744no—
234C23—10962no—
235C23—10964no—
236C23—1372no—
237C23D4×D13711yescharacteristic
238Q8—21no—
239Q8—2742no—
240Q8—2742no—
241Q8—2742no—
242Q8—5482no—
243Q8C2×D54812yes—
244D4—24no—
245D4—21no—
246D4—2748no—
247D4—2748no—
248D4—2744no—
249D4—2742no—
250D4—2744no—
251D4—2742no—
252D4—1374no—
253D4C2×D54812yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—2748no—
258C2×C4—2742no—
259C2×C4—2742no—
260C2×C4—2742no—
261C2×C4—5482no—
262C2×C4—1372no—
263C2×C4D4×D13711yescharacteristic
264C2×C4D4×D13712yes—
265C2×C4C2×D54812yes—
266C2×C4C23×D13711yescharacteristic
267C8—22no—
268C8—2742no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—2744no—
273C22—2742no—
274C22—2744no—
275C22—5484no—
276C22—10964no—
277C22—10968no—
278C22#1798405812yes—
279C22C2×D4×D13711yescharacteristic
280C4—22no—
281C4—42no—
282C4—2742no—
283C4—5482no—
284C4#1798405812yes—
285C4C2×D4×D13711yescharacteristic
286C4C22×D54811yescharacteristic
287C2—22no—
288C2—42no—
289C2—5482no—
290C2—5484no—
291C2C2×#1798405811yescharacteristic
292C2#2257821012yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#416×C137C2
socleC22×C137C22×D4
prime radicalC2×#17984134C2
hypercenter#416D137
hypocenterC137#3009
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C2×#17984118C2semidirect
3. C22×#14555499C2semidirect
4. #22577979C2semidirect
5. #22577568C2semidirect
6. #22577555C2semidirect
7. #22577407C2semidirect
8. #22577369C2semidirect
9. #416×C137C2semidirect
15. C22×Dic274C22semidirect
21. C2×#14555500C22semidirect
23. C2×#14555499C22semidirect
24. C2×#14555498C22semidirect
25. C2×#14555494C22semidirect
29. #17984068C22semidirect
30. #17984008C22semidirect
31. #17984000C22semidirect
33. C22×C8×C137C22semidirect
34. #117×C137C22semidirect
35. #103×C137C22semidirect
36. #102×C137C22semidirect
146. C137#3009semidirect
155. #416D137semidirect
174. C22×C8C2×D137semidirect
179. #117C2×D137semidirect
180. #103C2×D137semidirect
182. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841342
C2×#179841182
C22×#145554992
#225779792
#225775682
#225775552
#225774072
#225773692
#416×C1372
#3009137