Finite Group Databasebeta

← #36363482

Subgroups — #36363482

Order 22144.

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#26012001C211yesmaximal, characteristic
2C2×C4⋊C4×D173C211yesmaximal, characteristic, complemented
3#26011828C212yesmaximal, complemented
4#26011822C211yesmaximal, characteristic, complemented
5#26011821C211yesmaximal, characteristic, complemented
6#26011808C212yesmaximal, complemented
7#26011802C211yesmaximal, characteristic, complemented
8#26011532C212yesmaximal, complemented
9#26011489C212yesmaximal, complemented
10#26011453C211yesmaximal, characteristic, complemented
11#480×C173C211yesmaximal, characteristic, complemented
12C2×#14583825C2211yescharacteristic
13C22×C4×D173—21no—
14C22×C4×D173C2211yescharacteristic
15C2×#14583809C2212yes—
16#19149661—21no—
17#19149659C2211yescharacteristic
18#19149658C2211yescharacteristic
19#19149650C2211yescharacteristic
20#19149647—21no—
21D4×Dic173—21no—
22#19149639—21no—
23#19149637C2211yescharacteristic
24C2×C4⋊Dic173C2211yescharacteristic
25C2×#14583805—21no—
26C4⋊C4×D173—21no—
27C4⋊C4×D173—21no—
28C4⋊C4×D173C2212yes—
29C4⋊C4×D173C2212yes—
30#19149612C2212yescomplemented
31#19149611C2212yescomplemented
32#19149610C2212yescomplemented
33#19149604C2212yescomplemented
34#19149602C2212yescomplemented
35#19149600C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic
37#19149591C2212yes—
38#121×C173C2211yescharacteristic
39C2×C4⋊C4×C173C2211yescharacteristic
40#107×C173C2212yescomplemented
41#102×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic
43C23×D173D411yescharacteristic
44#14583825—24no—
45C2×Dic346C2311yescharacteristic
46C2×C22⋊D173—21no—
47C2×C22⋊D173—21no—
48C22×Dic173—21no—
49C22×Dic173—21no—
50C22×Dic173D411yescharacteristic
51C2×C4×D173—24no—
52C2×C4×D173—22no—
53C2×C4×D173—22no—
54C2×C4×D173—44no—
55C2×C4×D173D411yescharacteristic
56C2×C4×D173D411yescharacteristic
57C2×C4×D173C2311yescharacteristic
58C2×C4×D173C2311yescharacteristic
59#14583809—24no—
60C2×#13063436C2312yes—
61C4⋊Dic173—21no—
62C4⋊Dic173—21no—
63C4⋊Dic173C2312yes—
64C4⋊Dic173C2311yescharacteristic
65#14583805—22no—
66#14583805—22no—
67#14583805—21no—
68C22⋊Dic173—21no—
69C22⋊Dic173—21no—
70#14583803—21no—
71#14583803—21no—
72C4×Dic173—21no—
73C2×D4×C173—21no—
74C2×D4×C173C2311yescharacteristic
75C22×C4×C173—21no—
76C22×C4×C173C2311yescharacteristic
77C2×C8×C173C2312yes—
78C4⋊C4×C173—21no—
79C4⋊C4×C173C2312yes—
80C4⋊C4×C173C2311yescharacteristic
81C22⋊C4×C173—21no—
82C22×D173—22no—
83C22×D173—22no—
84C22×D173—22no—
85C22×D173C2×D411yescharacteristic
86C22×D173C2×D411yescharacteristic
87Dic346—22no—
88C22⋊D173—42no—
89C22⋊D173—42no—
90C2×Dic173—22no—
91C2×Dic173—22no—
92C2×Dic173—21no—
93C2×Dic173—21no—
94C2×Dic173—42no—
95C2×Dic173—42no—
96C2×Dic173C2×D411yescharacteristic
97C2×Dic173C2×D411yescharacteristic
98C4×D173—22no—
99C4×D173—22no—
100C4×D173—22no—
101C4×D173—44no—
102C4×D173—44no—
103C4×D173#4114yes—
104#13063436—22no—
105C23×C173—21no—
106C23×C173C2×D411yescharacteristic
107D4×C173—22no—
108D4×C173—42no—
109C2×C4×C173—22no—
110C2×C4×C173—22no—
111C2×C4×C173—21no—
112C2×C4×C173—42no—
113C2×C4×C173C2×D411yescharacteristic
114C2×C4×C173C2411yescharacteristic
115C8×C173—22no—
116C22×C173—21no—
117C22×C173—21no—
118C22×C173—41no—
119C22×C173—41no—
120C22×C173—41no—
121C22×C173AGL(1,ℤ/8)12yes—
122C22×C173C22×D411yescharacteristic
123C2×D173—28no—
124C2×D173—21no—
125C2×D173—21no—
126C2×D173—21no—
127C2×D173—21no—
128C2×D173#12312yes—
129C2×D173#13712yes—
130C4×C173—21no—
131C4×C173—42no—
132C4×C173—41no—
133C4×C173C2×#4112yes—
134Dic173—21no—
135Dic173—41no—
136Dic173—42no—
137Dic173—41no—
138Dic173#12312yes—
139C2×C173—22no—
140C2×C173—81no—
141C2×C173C2×#12311yescharacteristic
142C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C173C2×#13711yescharacteristic
144D173—22no—
145D173—22no—
146C173#309211yescharacteristic, complemented, Sylow
147#3092—1731nomaximal, Sylow
148#525—1731no—
149C22×C4⋊C4—1731no—
150#484—1732no—
151#483—1731no—
152#483—1731no—
153#481—1732no—
154#480—1731no—
155#480D17311yescharacteristic, complemented
156#428—1732no—
157#417—1732no—
158#407—1731no—
159C2×#41—1731no—
160C23×C4—3461no—
161C23×C4—1731no—
162C2×M16—1732no—
163#123—3461no—
164C22⋊Q8—1731no—
165C22⋊Q8—1731no—
166#121—1731no—
167#121C2×D17311yescharacteristic
168#120—3461no—
169C4×D4—3461no—
170C4×D4—3461no—
171#117—1731no—
172C2×C4⋊C4—3461no—
173C2×C4⋊C4—3461no—
174C2×C4⋊C4—3461no—
175C2×C4⋊C4—1731no—
176C2×C4⋊C4—1732no—
177C2×C4⋊C4—1732no—
178C2×C4⋊C4C2×D17311yescharacteristic
179#107—1732no—
180#107C2×D17312yescomplemented
181#106—1732no—
182#106—1732no—
183#103—1732no—
184#103—1732no—
185#102—1732no—
186#102C2×D17312yescomplemented
187C22⋊C8—1731no—
188C22⋊C8—1732no—
189C22⋊C8C2×D17311yescharacteristic
190C24—1731no—
191#41—3464no—
192C2×Q8—1731no—
193C2×D4—21no—
194C2×D4—3461no—
195C2×D4—3461no—
196C2×D4C22×D17311yescharacteristic
197C22×C4—21no—
198C22×C4—3464no—
199C22×C4—3462no—
200C22×C4—3462no—
201C22×C4—3461no—
202C22×C4—3461no—
203C22×C4—1731no—
204C22×C4—1731no—
205C22×C4—1731no—
206C22×C4—1731no—
207C22×C4—1731no—
208C22×C4—6924no—
209C22×C4C22×D17311yescharacteristic
210M16—3464no—
211C2×C8—1732no—
212C2×C8C22×D17312yes—
213C4⋊C4—21no—
214C4⋊C4—3462no—
215C4⋊C4—3462no—
216C4⋊C4—3461no—
217C4⋊C4—3461no—
218C4⋊C4—3461no—
219C4⋊C4—1732no—
220C4⋊C4—1731no—
221C4⋊C4C22×D17312yes—
222C4⋊C4C22×D17311yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—3461no—
225C22⋊C4—3461no—
226C22⋊C4—3461no—
227C22⋊C4—3461no—
228C42—3461no—
229C23—21no—
230C23—3462no—
231C23—3462no—
232C23—3462no—
233C23—1731no—
234C23—1731no—
235C23D4×D17311yescharacteristic
236Q8—3462no—
237D4—22no—
238D4—42no—
239D4—6922no—
240D4—6922no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—3462no—
246C2×C4—3462no—
247C2×C4—3462no—
248C2×C4—3462no—
249C2×C4—3461no—
250C2×C4—3461no—
251C2×C4—3462no—
252C2×C4—1731no—
253C2×C4—1731no—
254C2×C4—1734no—
255C2×C4—6924no—
256C2×C4—6924no—
257C2×C4—6922no—
258C2×C4—6922no—
259C2×C4D4×D17311yescharacteristic
260C2×C4C23×D17311yescharacteristic
261C8—22no—
262C8—3462no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—3468no—
269C22—3461no—
270C22—3461no—
271C22—3461no—
272C22—3461no—
273C22—1732no—
274C22—1732no—
275C22#1914970312yes—
276C22C2×D4×D17311yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—3461no—
281C4—1732no—
282C4—6921no—
283C4—6922no—
284C4—6921no—
285C4#41×D17312yes—
286C2—22no—
287C2—81no—
288C2—3462no—
289C2—3462no—
290C2#123×D17311yescharacteristic
291C2C2×#1914970311yescharacteristic
292C2#137×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#480×C173C2
socleC22×C173C22×D4
prime radical#26012001C2
hypercenter#480D173
hypocenterC173#3092
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D173C2semidirect
3. #26011828C2semidirect
4. #26011822C2semidirect
5. #26011821C2semidirect
6. #26011808C2semidirect
7. #26011802C2semidirect
8. #26011532C2semidirect
9. #26011489C2semidirect
10. #26011453C2semidirect
11. #480×C173C2semidirect
30. #19149612C22semidirect
31. #19149611C22semidirect
32. #19149610C22semidirect
33. #19149604C22semidirect
34. #19149602C22semidirect
35. #19149600C22semidirect
40. #107×C173C22semidirect
41. #102×C173C22semidirect
146. C173#3092semidirect
155. #480D173semidirect
180. #107C2×D173semidirect
186. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120012
C2×C4⋊C4×D1732
#260118282
#260118222
#260118212
#260118082
#260118022
#260115322
#260114892
#260114532
#480×C1732
#3092173