Finite Group Databasebeta

← #481×D173 #36363459

Subgroups — #481×D173 #36363459

Order 22144.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 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#121×D173C211yesmaximal, characteristic
2C2×C4⋊C4×D173C211yesmaximal, characteristic, complemented
3#26011811C212yesmaximal, complemented
4#26011810C211yesmaximal, characteristic, complemented
5#26011809C211yesmaximal, characteristic, complemented
6#26011808C212yesmaximal, complemented
7#26011807C211yesmaximal, characteristic, complemented
8#106×D173C212yesmaximal, complemented
9#102×D173C212yesmaximal, complemented
10C22⋊C8×D173C211yesmaximal, characteristic, complemented
11#481×C173C211yesmaximal, characteristic, complemented
12C2×D4×D173—21no—
13C2×D4×D173C2211yescharacteristic
14C22×C4×D173—21no—
15C22×C4×D173C2211yescharacteristic
16C2×C8×D173C2212yes—
17#19149654C2211yescharacteristic
18#19149653C2211yescharacteristic
19#19149652—21no—
20#19149651—21no—
21#19149650C2211yescharacteristic
22C2×C4⋊Dic173C2211yescharacteristic, complemented
23C2×#14583805—21no—
24C4⋊C4×D173—21no—
25C4⋊C4×D173—21no—
26C4⋊C4×D173C2212yes—
27C4⋊C4×D173C2211yescharacteristic
28C4⋊C4×D173C2212yescomplemented
29C22⋊C4×D173—21no—
30#19149611C2212yescomplemented
31#19149610C2212yescomplemented
32#19149610C2212yescomplemented
33#19149601C2212yescomplemented
34#19149600C2212yescomplemented
35#19149599C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic, complemented
37#19149591C2212yescomplemented
38#121×C173C2211yescharacteristic
39C2×C4⋊C4×C173C2211yescharacteristic, complemented
40#106×C173C2212yescomplemented
41#102×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic, complemented
43C23×D173—21no—
44C23×D173D411yescharacteristic
45C2×D692—21no—
46C2×D692C2311yescharacteristic
47C2×C22⋊D173—21no—
48C2×C22⋊D173—21no—
49C2×C22⋊D173—21no—
50D4×D173—24no—
51D4×D173—44no—
52C22×Dic173—21no—
53C22×Dic173D411yescharacteristic
54C2×C4×D173—24no—
55C2×C4×D173—21no—
56C2×C4×D173—22no—
57C2×C4×D173—22no—
58C2×C4×D173—44no—
59C2×C4×D173D411yescharacteristic
60C2×C4×D173D411yescharacteristic
61C2×C4×D173C2311yescharacteristic
62C2×C4×D173C2311yescharacteristic
63C2×#13063436C2312yes—
64C8×D173—24no—
65C4⋊Dic173—21no—
66C4⋊Dic173C2312yes—
67C4⋊Dic173C2311yescharacteristic
68#14583805—22no—
69#14583805—22no—
70#14583805—21no—
71C22⋊Dic173—21no—
72#14583803—21no—
73#14583803—21no—
74C2×D4×C173—21no—
75C2×D4×C173C2311yescharacteristic
76C22×C4×C173—21no—
77C22×C4×C173C2311yescharacteristic
78C2×C8×C173C2312yes—
79C4⋊C4×C173—21no—
80C4⋊C4×C173C2312yes—
81C4⋊C4×C173C2311yescharacteristic
82C22⋊C4×C173—21no—
83C22×D173—22no—
84C22×D173—22no—
85C22×D173—21no—
86C22×D173—42no—
87C22×D173—42no—
88C22×D173—42no—
89C22×D173SD1614yes—
90C22×D173C2×D411yescharacteristic
91C22×D173C2×D411yescharacteristic
92D692—22no—
93D692—42no—
94C22⋊D173—42no—
95C22⋊D173—42no—
96C22⋊D173—42no—
97C2×Dic173—22no—
98C2×Dic173—22no—
99C2×Dic173—21no—
100C2×Dic173—42no—
101C2×Dic173C2×D411yescharacteristic
102C2×Dic173C2×D411yescharacteristic
103C4×D173—22no—
104C4×D173—22no—
105C4×D173—22no—
106C4×D173—44no—
107C4×D173—42no—
108C4×D173—44no—
109C4×D173#4114yes—
110#13063436—22no—
111C23×C173—21no—
112C23×C173C2×D411yescharacteristic
113D4×C173—22no—
114D4×C173—42no—
115C2×C4×C173—22no—
116C2×C4×C173—22no—
117C2×C4×C173—21no—
118C2×C4×C173—42no—
119C2×C4×C173C2×D411yescharacteristic
120C2×C4×C173C2411yescharacteristic
121C8×C173—22no—
122C22×C173—21no—
123C22×C173—21no—
124C22×C173—41no—
125C22×C173—41no—
126C22×C173—41no—
127C22×C173C2×SD1612yes—
128C22×C173C22×D411yescharacteristic
129C2×D173—21no—
130C2×D173—21no—
131C2×D173—28no—
132C2×D173—41no—
133C2×D173—41no—
134C2×D173—41no—
135C2×D173—84no—
136C2×D173#12312yes—
137C2×D173C2×SD1612yes—
138C2×D173C2×SD1612yes—
139C2×D173AGL(1,ℤ/8)12yes—
140C4×C173—21no—
141C4×C173—41no—
142C4×C173—42no—
143C4×C173C2×#4112yes—
144Dic173—21no—
145Dic173—41no—
146Dic173—42no—
147Dic173#12312yes—
148C2×C173—22no—
149C2×C173—81no—
150C2×C173C2×#12311yescharacteristic
151C2×C173C22×SD1611yescharacteristic
152C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
153D173—22no—
154D173—81no—
155D173#48114yesdirect product factor
156C173C2×#48111yescharacteristic, complemented, Sylow
157C2×#481—1731nomaximal, Sylow
158C2×#121—1731no—
159C22×C4⋊C4—1731no—
160#481—1731no—
161#481—1731no—
162#481—1732no—
163#481—1732no—
164#481—1731no—
165#481D17311yescharacteristic, direct product factor
166C2×#106—1732no—
167C2×#102—1732no—
168C2×C22⋊C8—1731no—
169C22×D4—3461no—
170C22×D4—1731no—
171C23×C4—3461no—
172C23×C4—1731no—
173C22×C8—1732no—
174#121—3461no—
175#121—3461no—
176#121—1731no—
177#121—1731no—
178#121—1731no—
179#121C2×D17311yescharacteristic
180C2×C4⋊C4—3461no—
181C2×C4⋊C4—3461no—
182C2×C4⋊C4—3461no—
183C2×C4⋊C4—1731no—
184C2×C4⋊C4—1732no—
185C2×C4⋊C4—1732no—
186C2×C4⋊C4—1731no—
187C2×C4⋊C4C2×D17311yescharacteristic, complemented
188C2×C22⋊C4—3461no—
189#106—1732no—
190#106—1732no—
191#106—1732no—
192#106C2×D17312yescomplemented
193#102—1732no—
194#102—1732no—
195#102—1732no—
196#102C2×D17312yescomplemented
197C22⋊C8—1731no—
198C22⋊C8—1732no—
199C22⋊C8C2×D17311yescharacteristic, complemented
200C24—3461no—
201C24—1731no—
202C2×D4—21no—
203C2×D4—3461no—
204C2×D4—3461no—
205C2×D4—3461no—
206C2×D4—3464no—
207C2×D4—3461no—
208C2×D4—1731no—
209C2×D4—6924no—
210C2×D4C22×D17311yescharacteristic
211C22×C4—21no—
212C22×C4—3464no—
213C22×C4—3461no—
214C22×C4—3462no—
215C22×C4—3462no—
216C22×C4—3461no—
217C22×C4—1731no—
218C22×C4—1731no—
219C22×C4—1731no—
220C22×C4—1731no—
221C22×C4—1731no—
222C22×C4—6924no—
223C22×C4C22×D17311yescharacteristic
224C2×C8—3464no—
225C2×C8—1732no—
226C2×C8C22×D17312yes—
227C4⋊C4—21no—
228C4⋊C4—3462no—
229C4⋊C4—3462no—
230C4⋊C4—3461no—
231C4⋊C4—3461no—
232C4⋊C4—1732no—
233C4⋊C4—1731no—
234C4⋊C4C22×D17312yes—
235C4⋊C4C22×D17311yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—3461no—
238C22⋊C4—3461no—
239C22⋊C4—3461no—
240C23—21no—
241C23—3462no—
242C23—3462no—
243C23—3461no—
244C23—1734no—
245C23—1731no—
246C23—1731no—
247C23—6922no—
248C23—6922no—
249C23—6922no—
250C23D4×D17311yescharacteristic
251D4—22no—
252D4—42no—
253D4—3462no—
254D4—6922no—
255D4—6922no—
256D4—6922no—
257D4—6922no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—3462no—
263C2×C4—3462no—
264C2×C4—3462no—
265C2×C4—3462no—
266C2×C4—3461no—
267C2×C4—3462no—
268C2×C4—1731no—
269C2×C4—1731no—
270C2×C4—1734no—
271C2×C4—6924no—
272C2×C4—6924no—
273C2×C4—6922no—
274C2×C4—6922no—
275C2×C4D4×D17311yescharacteristic
276C2×C4C23×D17311yescharacteristic
277C8—22no—
278C8—3462no—
279C22—21no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—3468no—
285C22—3461no—
286C22—3461no—
287C22—13844no—
288C22—1732no—
289C22—1732no—
290C22—1732no—
291C22—1732no—
292C22—6921no—
293C22—6921no—
294C22—6921no—
295C22SD16×D17312yes—
296C22C2×D4×D17311yescharacteristic
297C4—21no—
298C4—41no—
299C4—42no—
300C4—3461no—
301C4—1732no—
302C4—6921no—
303C4—6922no—
304C4#41×D17312yes—
305C2—22no—
306C2—81no—
307C2—3462no—
308C2—13841no—
309C2—1734no—
310C2#123×D17311yescharacteristic
311C2C2×SD16×D17311yescharacteristic
312C2AGL(1,ℤ/8)×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#481×C173C2
socleC22×C173C22×D4
prime radical#121×D173C2
hypercenter#481D173
hypocenterC173C2×#481
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D173C2semidirect
3. #26011811C2semidirect
4. #26011810C2semidirect
5. #26011809C2semidirect
6. #26011808C2semidirect
7. #26011807C2semidirect
8. #106×D173C2semidirect
9. #102×D173C2semidirect
10. C22⋊C8×D173C2semidirect
11. #481×C173C2semidirect
22. C2×C4⋊Dic173C22semidirect
28. C4⋊C4×D173C22semidirect
30. #19149611C22semidirect
31. #19149610C22semidirect
32. #19149610C22semidirect
33. #19149601C22semidirect
34. #19149600C22semidirect
35. #19149599C22semidirect
36. C22⋊#13063436C22semidirect
37. #19149591C22semidirect
39. C2×C4⋊C4×C173C22semidirect
40. #106×C173C22semidirect
41. #102×C173C22semidirect
42. C22⋊C8×C173C22semidirect
155. D173#481direct (N × H)
156. C173C2×#481semidirect
165. #481D173direct (N × H)
187. C2×C4⋊C4C2×D173semidirect
192. #106C2×D173semidirect
196. #102C2×D173semidirect
199. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#121×D1732
C2×C4⋊C4×D1732
#260118112
#260118102
#260118092
#260118082
#260118072
#106×D1732
#102×D1732
C22⋊C8×D1732
#481×C1732
C2×#481173