Finite Group Databasebeta

← #36363513

Subgroups — #36363513

Order 22144.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#19149659C211yesmaximal, characteristic
2#117×D173C211yesmaximal, characteristic
3#26011835C212yesmaximal, complemented
4#26011834C212yesmaximal, complemented
5#26011828C212yesmaximal, complemented
6#26011823C212yesmaximal, complemented
7#26011531C211yesmaximal, characteristic, complemented
8#26011501C211yesmaximal, characteristic, complemented
9#26011499C211yesmaximal, characteristic, complemented
10#26011454C211yesmaximal, characteristic
11#428×C173C211yesmaximal, characteristic, complemented
12C22×Dic346C2211yescharacteristic
13C22×C4×D173C2211yescharacteristic
14C2×#14583810C2211yescharacteristic
15#19149659—21no—
16#19149659—21no—
17#19149659C2212yes—
18#19149659C2212yes—
19#19149638C2211yescharacteristic
20#19149637C2211yescharacteristic
21#19149637C2211yescharacteristic
22#19149636—21no—
23#19149635—21no—
24C2×C4⋊Dic173—21no—
25C4⋊C4×D173C2211yescharacteristic
26C4⋊C4×D173C2211yescharacteristic
27C2×#14583803—21no—
28C22⋊C4×D173—21no—
29C42×D173—21no—
30#19149612C2212yescomplemented
31#19149610C2212yescomplemented
32#19149604C2212yescomplemented
33#19149604C2212yescomplemented
34#19149602C2212yescomplemented
35#19149602C2212yescomplemented
36#19149591C2212yes—
37#19149591C2212yes—
38C2×M16×C173C2211yescharacteristic
39#117×C173C2211yescharacteristic
40C2×C4⋊C4×C173C2211yescharacteristic
41#107×C173C2212yescomplemented
42#106×C173C2212yescomplemented
43C23×D173D411yescharacteristic
44C2×Dic346—21no—
45C2×Dic346—22no—
46C2×Dic346—22no—
47C2×Dic346C2312yes—
48C22×Dic173—21no—
49C22×Dic173D411yescharacteristic
50C2×C4×D173—21no—
51C2×C4×D173—21no—
52C2×C4×D173—22no—
53C2×C4×D173—22no—
54C2×C4×D173D411yescharacteristic
55C2×C4×D173D411yescharacteristic
56C2×C4×D173C2311yescharacteristic
57C2×C4×D173C2311yescharacteristic
58#14583810—22no—
59C2×#13063436C2312yes—
60C4⋊Dic173—22no—
61C4⋊Dic173—22no—
62C4⋊Dic173C2311yescharacteristic
63C4⋊Dic173C2311yescharacteristic
64#14583805—21no—
65#14583805—21no—
66C22⋊Dic173—21no—
67#14583803—21no—
68#14583803—21no—
69#14583803—22no—
70#14583803—22no—
71C4×Dic173—21no—
72C4×Dic173—21no—
73C4×Dic173—21no—
74C22×C4×C173—21no—
75C22×C4×C173C2311yescharacteristic
76M16×C173—22no—
77C2×C8×C173C2312yes—
78C4⋊C4×C173—21no—
79C4⋊C4×C173C2311yescharacteristic
80C4⋊C4×C173C2311yescharacteristic
81C4⋊C4×C173C2312yes—
82C22⋊C4×C173—21no—
83C42×C173—21no—
84C22×D173—22no—
85C22×D173—22no—
86C22×D173—41no—
87C22×D173C2×D411yescharacteristic
88C22×D173C2×D411yescharacteristic
89Dic346—22no—
90Dic346—22no—
91Dic346—42no—
92C2×Dic173—22no—
93C2×Dic173—21no—
94C2×Dic173—21no—
95C2×Dic173—22no—
96C2×Dic173—42no—
97C2×Dic173C2×D411yescharacteristic
98C2×Dic173C2×D411yescharacteristic
99C4×D173—22no—
100C4×D173—22no—
101C4×D173—21no—
102C4×D173—21no—
103C4×D173—21no—
104C4×D173—21no—
105C4×D173—42no—
106C4×D173—42no—
107#13063436—22no—
108C23×C173C2×D411yescharacteristic
109C2×C4×C173—21no—
110C2×C4×C173—21no—
111C2×C4×C173—22no—
112C2×C4×C173—42no—
113C2×C4×C173C2×D411yescharacteristic
114C2×C4×C173#4112yes—
115C2×C4×C173#4112yes—
116C2×C4×C173C2411yescharacteristic
117C8×C173—22no—
118C22×C173—21no—
119C22×C173—21no—
120C22×C173#12312yes—
121C22×C173C22×D411yescharacteristic
122C2×D173—21no—
123C2×D173—21no—
124C2×D173—21no—
125C2×D173—21no—
126C2×D173—21no—
127C2×D173—44no—
128C2×D173#13712yes—
129C4×C173—41no—
130C4×C173—42no—
131C4×C173—41no—
132C4×C173#12312yes—
133C4×C173C2×#4111yescharacteristic
134C4×C173C2×#4111yescharacteristic
135Dic173—21no—
136Dic173—21no—
137Dic173—42no—
138Dic173—41no—
139Dic173—41no—
140C2×C173—22no—
141C2×C173C2×#12311yescharacteristic
142C2×C173C2×#13711yescharacteristic
143C2×C173#57511yescharacteristic
144D173—22no—
145D173—41no—
146C173#309511yescharacteristic, complemented, Sylow
147#3095—1731nomaximal, Sylow
148C2×C22⋊Q8—1731no—
149C2×#117—1731no—
150#485—1732no—
151#485—1732no—
152#484—1732no—
153#483—1732no—
154#428—1731no—
155#428D17311yescharacteristic, complemented
156#419—1731no—
157#419—1731no—
158#407—1731no—
159C22×Q8—1731no—
160C23×C4—1731no—
161C2×M16—1731no—
162C2×M16C2×D17311yescharacteristic
163C22⋊Q8—3461no—
164C22⋊Q8—3461no—
165C22⋊Q8—1732no—
166C22⋊Q8—1732no—
167#117—3461no—
168#117—3461no—
169#117—1731no—
170#117—1731no—
171#117—1731no—
172#117C2×D17311yescharacteristic
173C2×C4⋊C4—3461no—
174C2×C4⋊C4—1731no—
175C2×C4⋊C4—1731no—
176C2×C4⋊C4C2×D17311yescharacteristic
177C2×C22⋊C4—3461no—
178C2×C22⋊C4—3461no—
179C2×C42—3461no—
180#107—1732no—
181#107C2×D17312yescomplemented
182#106—1732no—
183#106C2×D17312yescomplemented
184#103—1732no—
185#103—1732no—
186#103—1732no—
187#103—1732no—
188C22⋊C8—1732no—
189C22⋊C8—1732no—
190C24—1731no—
191C2×Q8—3461no—
192C2×Q8—3462no—
193C2×Q8—3462no—
194C2×Q8—1732no—
195C22×C4—21no—
196C22×C4—3461no—
197C22×C4—3461no—
198C22×C4—3462no—
199C22×C4—3462no—
200C22×C4—3461no—
201C22×C4—1731no—
202C22×C4—1731no—
203C22×C4—1731no—
204C22×C4—1731no—
205C22×C4—1731no—
206C22×C4C22×D17311yescharacteristic
207M16—22no—
208M16—3462no—
209C2×C8—1732no—
210C2×C8C22×D17312yes—
211C4⋊C4—21no—
212C4⋊C4—3462no—
213C4⋊C4—3462no—
214C4⋊C4—3461no—
215C4⋊C4—3461no—
216C4⋊C4—1731no—
217C4⋊C4—1731no—
218C4⋊C4C22×D17311yescharacteristic
219C4⋊C4C22×D17311yescharacteristic
220C4⋊C4C22×D17312yes—
221C22⋊C4—21no—
222C22⋊C4—3461no—
223C22⋊C4—3461no—
224C22⋊C4—3462no—
225C22⋊C4—3462no—
226C22⋊C4—3461no—
227C42—21no—
228C42—3461no—
229C42—3461no—
230C42—3461no—
231C23—3462no—
232C23—3462no—
233C23—1731no—
234C23—1731no—
235C23—6921no—
236C23D4×D17311yescharacteristic
237Q8—3462no—
238Q8—3462no—
239Q8—6922no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—3462no—
245C2×C4—3462no—
246C2×C4—3462no—
247C2×C4—3461no—
248C2×C4—3461no—
249C2×C4—3461no—
250C2×C4—3461no—
251C2×C4—3461no—
252C2×C4—3461no—
253C2×C4—3462no—
254C2×C4—1731no—
255C2×C4—1731no—
256C2×C4—6922no—
257C2×C4—6922no—
258C2×C4—6922no—
259C2×C4D4×D17311yescharacteristic
260C2×C4#1458382612yes—
261C2×C4#1458382712yes—
262C2×C4C23×D17311yescharacteristic
263C8—22no—
264C8—3462no—
265C22—21no—
266C22—21no—
267C22—3461no—
268C22—3461no—
269C22—3461no—
270C22—3461no—
271C22—3461no—
272C22—1732no—
273C22—6924no—
274C22#1914966412yes—
275C22C2×D4×D17311yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—3461no—
280C4—3461no—
281C4—6922no—
282C4—6921no—
283C4—6921no—
284C4#1914966412yes—
285C4C2×#1458382611yescharacteristic
286C4C2×#1458382711yescharacteristic
287C2—22no—
288C2—3462no—
289C2—6921no—
290C2C2×#1914966411yescharacteristic
291C2#137×D17311yescharacteristic
292C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#428×C173C2
socleC22×C173C22×D4
prime radicalC23×D173D4
hypercenter#428D173
hypocenterC173#3095
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #26011835C2semidirect
4. #26011834C2semidirect
5. #26011828C2semidirect
6. #26011823C2semidirect
7. #26011531C2semidirect
8. #26011501C2semidirect
9. #26011499C2semidirect
11. #428×C173C2semidirect
30. #19149612C22semidirect
31. #19149610C22semidirect
32. #19149604C22semidirect
33. #19149604C22semidirect
34. #19149602C22semidirect
35. #19149602C22semidirect
41. #107×C173C22semidirect
42. #106×C173C22semidirect
146. C173#3095semidirect
155. #428D173semidirect
181. #107C2×D173semidirect
183. #106C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496592
#117×D1732
#260118352
#260118342
#260118282
#260118232
#260115312
#260115012
#260114992
#260114542
#428×C1732
#3095173