Finite Group Databasebeta

← #31790715

Subgroups — #31790715

Order 17536.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#22577915C211yesmaximal, characteristic
2C2×C4⋊C4×D137C211yesmaximal, characteristic, complemented
3#22577740C212yesmaximal, complemented
4#22577735C212yesmaximal, complemented
5#22577723C212yesmaximal, complemented
6#22577718C212yesmaximal, complemented
7#22577446C211yesmaximal, characteristic, complemented
8#22577405C211yesmaximal, characteristic, complemented
9#22577402C211yesmaximal, characteristic, complemented
10#22577366C211yesmaximal, characteristic, complemented
11#428×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13C22×C4×D137—21no—
14C22×C4×D137C2211yescharacteristic
15C2×#14555494C2211yescharacteristic
16#17984072—21no—
17#17984068C2211yescharacteristic
18#17984068C2211yescharacteristic
19#17984063C2211yescharacteristic
20#17984063C2211yescharacteristic
21#17984058—21no—
22C4×D548—21no—
23C4×D548—21no—
24C2×C4⋊Dic137C2211yescharacteristic
25C2×#14555489—21no—
26C4⋊C4×D137—21no—
27C4⋊C4×D137—21no—
28C4⋊C4×D137C2212yes—
29C4⋊C4×D137C2212yes—
30#17984021C2212yescomplemented
31#17984019C2212yescomplemented
32#17984013C2212yescomplemented
33#17984011C2212yescomplemented
34#17984010C2212yescomplemented
35#17984008C2212yescomplemented
36#17984000C2212yes—
37#17984000C2212yes—
38C2×M16×C137C2211yescharacteristic
39#117×C137C2211yescharacteristic
40C2×C4⋊C4×C137C2211yescharacteristic
41#107×C137C2212yescomplemented
42#106×C137C2212yescomplemented
43C23×D137D411yescharacteristic
44#14555510—22no—
45#14555510—22no—
46C2×Dic274C2311yescharacteristic
47C2×D548—21no—
48C2×D548—21no—
49C2×D548C2311yescharacteristic
50C2×C22⋊D137—21no—
51C22×Dic137—21no—
52C22×Dic137D411yescharacteristic
53C2×C4×D137—22no—
54C2×C4×D137—22no—
55C2×C4×D137—21no—
56C2×C4×D137—22no—
57C2×C4×D137—22no—
58C2×C4×D137—44no—
59C2×C4×D137D411yescharacteristic
60C2×C4×D137D411yescharacteristic
61C2×C4×D137C2311yescharacteristic
62C2×C4×D137C2311yescharacteristic
63#14555494—22no—
64C2×#11764040C2312yes—
65C4⋊Dic137—21no—
66C4⋊Dic137—21no—
67C4⋊Dic137—21no—
68C4⋊Dic137C2312yes—
69#14555489—22no—
70#14555489—22no—
71#14555487—21no—
72#14555487—21no—
73#14555487—21no—
74#14555487—21no—
75C22×C4×C137—21no—
76C22×C4×C137C2311yescharacteristic
77M16×C137—22no—
78C2×C8×C137C2312yes—
79C4⋊C4×C137—21no—
80C4⋊C4×C137C2312yes—
81C4⋊C4×C137C2311yescharacteristic
82C4⋊C4×C137C2311yescharacteristic
83C22⋊C4×C137—21no—
84C42×C137—21no—
85C22×D137—22no—
86C22×D137—22no—
87C22×D137—22no—
88C22×D137—21no—
89C22×D137C2×D411yescharacteristic
90C22×D137C2×D411yescharacteristic
91Dic274—21no—
92Dic274—21no—
93D548—21no—
94D548—21no—
95D548—42no—
96D548—42no—
97C22⋊D137—42no—
98C2×Dic137—22no—
99C2×Dic137—21no—
100C2×Dic137—22no—
101C2×Dic137—42no—
102C2×Dic137C2×D411yescharacteristic
103C2×Dic137C2×D411yescharacteristic
104C4×D137—22no—
105C4×D137—22no—
106C4×D137—21no—
107C4×D137—21no—
108C4×D137—21no—
109C4×D137—21no—
110C4×D137—44no—
111C4×D137—44no—
112C4×D137—42no—
113#11764040—22no—
114C23×C137C2×D411yescharacteristic
115C2×C4×C137—21no—
116C2×C4×C137—21no—
117C2×C4×C137—22no—
118C2×C4×C137—42no—
119C2×C4×C137C2×D411yescharacteristic
120C2×C4×C137#4112yes—
121C2×C4×C137#4112yes—
122C2×C4×C137C2411yescharacteristic
123C8×C137—22no—
124C22×C137—21no—
125C22×C137—21no—
126C22×C137#12312yes—
127C22×C137C22×D411yescharacteristic
128C2×D137—28no—
129C2×D137—21no—
130C2×D137—21no—
131C2×D137—21no—
132C2×D137—21no—
133C2×D137—41no—
134C2×D137—41no—
135C2×D137—41no—
136C2×D137AGL(1,ℤ/8)12yes—
137C2×D137#13712yes—
138C4×C137—42no—
139C4×C137—41no—
140C4×C137—41no—
141C4×C137#12312yes—
142C4×C137C2×#4111yescharacteristic
143C4×C137C2×#4111yescharacteristic
144Dic137—21no—
145Dic137—21no—
146Dic137—41no—
147Dic137—42no—
148C2×C137—22no—
149C2×C137C2×#12311yescharacteristic
150C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C137C2×#13711yescharacteristic
152D137—22no—
153D137—22no—
154D137—81no—
155C137#309211yescharacteristic, complemented, Sylow
156#3092—1371nomaximal, Sylow
157#525—1371no—
158C22×C4⋊C4—1371no—
159#484—1372no—
160#483—1372no—
161#481—1372no—
162#480—1372no—
163#428—1371no—
164#428D13711yescharacteristic, complemented
165#417—1371no—
166#417—1371no—
167#407—1371no—
168C2×#41—1371no—
169C23×C4—2741no—
170C23×C4—1371no—
171C2×M16—1371no—
172C2×M16C2×D13711yescharacteristic
173#123—2741no—
174C22⋊Q8—1371no—
175C22⋊Q8—1371no—
176#121—1371no—
177#121—1371no—
178#120—2741no—
179C4×D4—2741no—
180C4×D4—2741no—
181#117C2×D13711yescharacteristic
182C2×C4⋊C4—2741no—
183C2×C4⋊C4—2741no—
184C2×C4⋊C4—2741no—
185C2×C4⋊C4—1371no—
186C2×C4⋊C4—1372no—
187C2×C4⋊C4—1372no—
188C2×C4⋊C4C2×D13711yescharacteristic
189#107—1372no—
190#107C2×D13712yescomplemented
191#106—1372no—
192#106C2×D13712yescomplemented
193#103—1372no—
194#103—1372no—
195#102—1372no—
196#102—1372no—
197C22⋊C8—1372no—
198C22⋊C8—1372no—
199C24—1371no—
200#41—2742no—
201#41—2742no—
202C2×Q8—1371no—
203C2×D4—2741no—
204C2×D4—2741no—
205C2×D4—2741no—
206C2×D4—1371no—
207C22×C4—21no—
208C22×C4—2742no—
209C22×C4—2742no—
210C22×C4—2742no—
211C22×C4—2742no—
212C22×C4—2741no—
213C22×C4—2741no—
214C22×C4—5484no—
215C22×C4—1371no—
216C22×C4—1371no—
217C22×C4—1371no—
218C22×C4—1371no—
219C22×C4—1371no—
220C22×C4C22×D13711yescharacteristic
221M16—22no—
222M16—2742no—
223C2×C8—1372no—
224C2×C8C22×D13712yes—
225C4⋊C4—21no—
226C4⋊C4—2742no—
227C4⋊C4—2742no—
228C4⋊C4—2741no—
229C4⋊C4—2741no—
230C4⋊C4—2741no—
231C4⋊C4—1372no—
232C4⋊C4C22×D13712yes—
233C4⋊C4C22×D13711yescharacteristic
234C4⋊C4C22×D13711yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—2741no—
237C22⋊C4—2741no—
238C22⋊C4—2741no—
239C22⋊C4—2741no—
240C42—21no—
241C23—2742no—
242C23—2742no—
243C23—2742no—
244C23—2741no—
245C23—1371no—
246C23—1371no—
247C23D4×D13711yescharacteristic
248Q8—2741no—
249Q8—2741no—
250D4—2741no—
251D4—2741no—
252D4—5482no—
253D4—5482no—
254D4—5482no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—2742no—
260C2×C4—2742no—
261C2×C4—2742no—
262C2×C4—2741no—
263C2×C4—2741no—
264C2×C4—2741no—
265C2×C4—2741no—
266C2×C4—2741no—
267C2×C4—2742no—
268C2×C4—5484no—
269C2×C4—5484no—
270C2×C4—5482no—
271C2×C4—5482no—
272C2×C4—1371no—
273C2×C4—1371no—
274C2×C4D4×D13711yescharacteristic
275C2×C4#1455551012yes—
276C2×C4#1455551112yes—
277C2×C4C23×D13711yescharacteristic
278C8—22no—
279C8—2742no—
280C22—21no—
281C22—21no—
282C22—2748no—
283C22—2741no—
284C22—2741no—
285C22—2741no—
286C22—2741no—
287C22—5481no—
288C22—5481no—
289C22—5481no—
290C22—1372no—
291C22—1372no—
292C22#1798407312yes—
293C22C2×D4×D13711yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—2741no—
298C4—2741no—
299C4—5482no—
300C4—5481no—
301C4#1798407312yes—
302C4C2×#1455551011yescharacteristic
303C4C2×#1455551111yescharacteristic
304C2—22no—
305C2—2742no—
306C2—2742no—
307C2—10961no—
308C2C2×#1798407311yescharacteristic
309C2AGL(1,ℤ/8)×D13711yescharacteristic
310C2#137×D13711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#428×C137C2
socleC22×C137C22×D4
prime radical#22577915C2
hypercenter#428D137
hypocenterC137#3092
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D137C2semidirect
3. #22577740C2semidirect
4. #22577735C2semidirect
5. #22577723C2semidirect
6. #22577718C2semidirect
7. #22577446C2semidirect
8. #22577405C2semidirect
9. #22577402C2semidirect
10. #22577366C2semidirect
11. #428×C137C2semidirect
30. #17984021C22semidirect
31. #17984019C22semidirect
32. #17984013C22semidirect
33. #17984011C22semidirect
34. #17984010C22semidirect
35. #17984008C22semidirect
41. #107×C137C22semidirect
42. #106×C137C22semidirect
155. C137#3092semidirect
164. #428D137semidirect
190. #107C2×D137semidirect
192. #106C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779152
C2×C4⋊C4×D1372
#225777402
#225777352
#225777232
#225777182
#225774462
#225774052
#225774022
#225773662
#428×C1372
#3092137