Finite Group Databasebeta

← #31790708

Subgroups — #31790708

Order 17536.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#22577913C211yesmaximal, characteristic
2C2×C4⋊C4×D137C211yesmaximal, characteristic, complemented
3#22577740C212yesmaximal, complemented
4#22577734C211yesmaximal, characteristic, complemented
5#22577733C211yesmaximal, characteristic, complemented
6#22577720C212yesmaximal, complemented
7#22577714C211yesmaximal, characteristic, complemented
8#22577444C212yesmaximal, complemented
9#22577401C212yesmaximal, complemented
10#22577365C211yesmaximal, characteristic, complemented
11#480×C137C211yesmaximal, characteristic, complemented
12C2×#14555509C2211yescharacteristic
13C22×C4×D137—21no—
14C22×C4×D137C2211yescharacteristic
15C2×#14555493C2212yes—
16#17984070—21no—
17#17984068C2211yescharacteristic
18#17984067C2211yescharacteristic
19#17984059C2211yescharacteristic
20#17984056—21no—
21D4×Dic137—21no—
22#17984048—21no—
23#17984046C2211yescharacteristic
24C2×C4⋊Dic137C2211yescharacteristic
25C2×#14555489—21no—
26C4⋊C4×D137—21no—
27C4⋊C4×D137—21no—
28C4⋊C4×D137C2212yes—
29C4⋊C4×D137C2212yes—
30#17984021C2212yescomplemented
31#17984020C2212yescomplemented
32#17984019C2212yescomplemented
33#17984013C2212yescomplemented
34#17984011C2212yescomplemented
35#17984009C2212yescomplemented
36C22⋊#11764040C2211yescharacteristic
37#17984000C2212yes—
38#121×C137C2211yescharacteristic
39C2×C4⋊C4×C137C2211yescharacteristic
40#107×C137C2212yescomplemented
41#102×C137C2212yescomplemented
42C22⋊C8×C137C2211yescharacteristic
43C23×D137D411yescharacteristic
44#14555509—24no—
45C2×Dic274C2311yescharacteristic
46C2×C22⋊D137—21no—
47C2×C22⋊D137—21no—
48C22×Dic137—21no—
49C22×Dic137—21no—
50C22×Dic137D411yescharacteristic
51C2×C4×D137—24no—
52C2×C4×D137—22no—
53C2×C4×D137—22no—
54C2×C4×D137—44no—
55C2×C4×D137D411yescharacteristic
56C2×C4×D137D411yescharacteristic
57C2×C4×D137C2311yescharacteristic
58C2×C4×D137C2311yescharacteristic
59#14555493—24no—
60C2×#11764040C2312yes—
61C4⋊Dic137—21no—
62C4⋊Dic137—21no—
63C4⋊Dic137C2312yes—
64C4⋊Dic137C2311yescharacteristic
65#14555489—22no—
66#14555489—22no—
67#14555489—21no—
68C22⋊Dic137—21no—
69C22⋊Dic137—21no—
70#14555487—21no—
71#14555487—21no—
72C4×Dic137—21no—
73C2×D4×C137—21no—
74C2×D4×C137C2311yescharacteristic
75C22×C4×C137—21no—
76C22×C4×C137C2311yescharacteristic
77C2×C8×C137C2312yes—
78C4⋊C4×C137—21no—
79C4⋊C4×C137C2312yes—
80C4⋊C4×C137C2311yescharacteristic
81C22⋊C4×C137—21no—
82C22×D137—22no—
83C22×D137—22no—
84C22×D137—22no—
85C22×D137C2×D411yescharacteristic
86C22×D137C2×D411yescharacteristic
87Dic274—22no—
88C22⋊D137—42no—
89C22⋊D137—42no—
90C2×Dic137—22no—
91C2×Dic137—21no—
92C2×Dic137—21no—
93C2×Dic137—22no—
94C2×Dic137—42no—
95C2×Dic137—42no—
96C2×Dic137C2×D411yescharacteristic
97C2×Dic137C2×D411yescharacteristic
98C4×D137—22no—
99C4×D137—22no—
100C4×D137—22no—
101C4×D137—44no—
102C4×D137—44no—
103C4×D137#4114yes—
104#11764040—22no—
105C23×C137—21no—
106C23×C137C2×D411yescharacteristic
107D4×C137—22no—
108D4×C137—42no—
109C2×C4×C137—22no—
110C2×C4×C137—21no—
111C2×C4×C137—22no—
112C2×C4×C137—42no—
113C2×C4×C137C2×D411yescharacteristic
114C2×C4×C137C2411yescharacteristic
115C8×C137—22no—
116C22×C137—21no—
117C22×C137—21no—
118C22×C137—41no—
119C22×C137—41no—
120C22×C137—41no—
121C22×C137AGL(1,ℤ/8)12yes—
122C22×C137C22×D411yescharacteristic
123C2×D137—28no—
124C2×D137—21no—
125C2×D137—21no—
126C2×D137—21no—
127C2×D137—21no—
128C2×D137#12312yes—
129C2×D137#13712yes—
130C4×C137—21no—
131C4×C137—42no—
132C4×C137—41no—
133C4×C137C2×#4112yes—
134Dic137—21no—
135Dic137—41no—
136Dic137—42no—
137Dic137—41no—
138Dic137#12312yes—
139C2×C137—22no—
140C2×C137—81no—
141C2×C137C2×#12311yescharacteristic
142C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C137C2×#13711yescharacteristic
144D137—22no—
145D137—22no—
146C137#309211yescharacteristic, complemented, Sylow
147#3092—1371nomaximal, Sylow
148#525—1371no—
149C22×C4⋊C4—1371no—
150#484—1372no—
151#483—1371no—
152#483—1371no—
153#481—1372no—
154#480—1371no—
155#480D13711yescharacteristic, complemented
156#428—1372no—
157#417—1372no—
158#407—1371no—
159C2×#41—1371no—
160C23×C4—2741no—
161C23×C4—1371no—
162C2×M16—1372no—
163#123—2741no—
164C22⋊Q8—1371no—
165C22⋊Q8—1371no—
166#121—1371no—
167#121C2×D13711yescharacteristic
168#120—2741no—
169C4×D4—2741no—
170C4×D4—2741no—
171#117—1371no—
172C2×C4⋊C4—2741no—
173C2×C4⋊C4—2741no—
174C2×C4⋊C4—2741no—
175C2×C4⋊C4—1371no—
176C2×C4⋊C4—1372no—
177C2×C4⋊C4—1372no—
178C2×C4⋊C4C2×D13711yescharacteristic
179#107—1372no—
180#107C2×D13712yescomplemented
181#106—1372no—
182#106—1372no—
183#103—1372no—
184#103—1372no—
185#102—1372no—
186#102C2×D13712yescomplemented
187C22⋊C8—1371no—
188C22⋊C8—1372no—
189C22⋊C8C2×D13711yescharacteristic
190C24—1371no—
191#41—2744no—
192C2×Q8—1371no—
193C2×D4—21no—
194C2×D4—2741no—
195C2×D4—2741no—
196C2×D4C22×D13711yescharacteristic
197C22×C4—21no—
198C22×C4—2744no—
199C22×C4—2742no—
200C22×C4—2742no—
201C22×C4—2741no—
202C22×C4—2741no—
203C22×C4—5484no—
204C22×C4—1371no—
205C22×C4—1371no—
206C22×C4—1371no—
207C22×C4—1371no—
208C22×C4—1371no—
209C22×C4C22×D13711yescharacteristic
210M16—2744no—
211C2×C8—1372no—
212C2×C8C22×D13712yes—
213C4⋊C4—21no—
214C4⋊C4—2742no—
215C4⋊C4—2742no—
216C4⋊C4—2741no—
217C4⋊C4—2741no—
218C4⋊C4—2741no—
219C4⋊C4—1372no—
220C4⋊C4—1371no—
221C4⋊C4C22×D13712yes—
222C4⋊C4C22×D13711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—2741no—
225C22⋊C4—2741no—
226C22⋊C4—2741no—
227C22⋊C4—2741no—
228C42—2741no—
229C23—21no—
230C23—2742no—
231C23—2742no—
232C23—2742no—
233C23—1371no—
234C23—1371no—
235C23D4×D13711yescharacteristic
236Q8—2742no—
237D4—22no—
238D4—42no—
239D4—5482no—
240D4—5482no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—2742no—
246C2×C4—2742no—
247C2×C4—2742no—
248C2×C4—2742no—
249C2×C4—2741no—
250C2×C4—2741no—
251C2×C4—2742no—
252C2×C4—5484no—
253C2×C4—5484no—
254C2×C4—5482no—
255C2×C4—5482no—
256C2×C4—1371no—
257C2×C4—1371no—
258C2×C4—1374no—
259C2×C4D4×D13711yescharacteristic
260C2×C4C23×D13711yescharacteristic
261C8—22no—
262C8—2742no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—2748no—
269C22—2741no—
270C22—2741no—
271C22—2741no—
272C22—2741no—
273C22—1372no—
274C22—1372no—
275C22#1798411212yes—
276C22C2×D4×D13711yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—2741no—
281C4—5482no—
282C4—5481no—
283C4—5481no—
284C4—1372no—
285C4#41×D13712yes—
286C2—22no—
287C2—81no—
288C2—2742no—
289C2—2742no—
290C2#123×D13711yescharacteristic
291C2C2×#1798411211yescharacteristic
292C2#137×D13711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#480×C137C2
socleC22×C137C22×D4
prime radical#22577913C2
hypercenter#480D137
hypocenterC137#3092
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D137C2semidirect
3. #22577740C2semidirect
4. #22577734C2semidirect
5. #22577733C2semidirect
6. #22577720C2semidirect
7. #22577714C2semidirect
8. #22577444C2semidirect
9. #22577401C2semidirect
10. #22577365C2semidirect
11. #480×C137C2semidirect
30. #17984021C22semidirect
31. #17984020C22semidirect
32. #17984019C22semidirect
33. #17984013C22semidirect
34. #17984011C22semidirect
35. #17984009C22semidirect
40. #107×C137C22semidirect
41. #102×C137C22semidirect
146. C137#3092semidirect
155. #480D137semidirect
180. #107C2×D137semidirect
186. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
#225779132
C2×C4⋊C4×D1372
#225777402
#225777342
#225777332
#225777202
#225777142
#225774442
#225774012
#225773652
#480×C1372
#3092137