Finite Group Databasebeta

← #36364377

Subgroups — #36364377

Order 22144.

466 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 121 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#26012002C211yesmaximal, characteristic, complemented
2#26011821C211yesmaximal, characteristic, complemented
3#26011812C212yesmaximal, complemented
4#26011804C211yesmaximal, characteristic, complemented
5#26011742C212yesmaximal, complemented
6#26011735C212yesmaximal, complemented
7#26011642C211yesmaximal, characteristic, complemented
8#26011634C212yesmaximal, complemented
9#26011625C211yesmaximal, characteristic, complemented
10#26011455C211yesmaximal, characteristic, complemented
11C2×#121×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2211yescharacteristic, complemented
13C2×#14583814C2212yescomplemented
14C2×#14583811C2212yescomplemented
15C2×#14583810C2212yescomplemented
16#19149662—21no—
17#19149658C2211yescharacteristic
18#19149658C2211yescharacteristic, complemented
19#19149653C2211yescharacteristic
20#19149653C2211yescharacteristic, complemented
21#19149648—21no—
22C4×C22⋊D173—21no—
23C4×C22⋊D173—21no—
24#19149638C2211yescharacteristic, complemented
25#19149612C2212yescomplemented
26#19149610C2212yescomplemented
27#19149602C2212yescomplemented
28#19149600C2212yescomplemented
29#19149600C2212yescomplemented
30#19149599C2212yescomplemented
31C22⋊#13063436C2211yescharacteristic
32C22⋊#13063436C2211yescharacteristic
33C22⋊#13063436C2212yescomplemented
34C22×D4×C173—21no—
35C22×D4×C173C2211yescharacteristic
36C23×C4×C173C2211yescharacteristic
37#121×C173—21no—
38#121×C173—21no—
39#121×C173C2212yes—
40#121×C173C2212yes—
41C2×C4⋊C4×C173C2211yescharacteristic
42C2×C22⋊C4×C173—21no—
43#14583826—42no—
44C2×Dic346C2311yescharacteristic
45C2×D692C2311yescharacteristic
46C2×C22⋊D173—21no—
47C2×C22⋊D173—21no—
48C2×C22⋊D173—21no—
49C2×C4×D173—21no—
50#14583814—42no—
51#14583811—42no—
52#14583810—42no—
53C2×#13063436C2312yes—
54C2×#13063436C2312yes—
55C4⋊Dic173C2311yescharacteristic
56C4⋊Dic173C2311yescharacteristic
57#14583805—21no—
58#14583805—21no—
59C22⋊Dic173—21no—
60C22⋊Dic173—21no—
61C22⋊Dic173—21no—
62#14583803—21no—
63#14583803—21no—
64C4×Dic173—21no—
65C24×C173—21no—
66C24×C173D411yescharacteristic
67C2×D4×C173—22no—
68C2×D4×C173—22no—
69C2×D4×C173—21no—
70C2×D4×C173—42no—
71C2×D4×C173—44no—
72C2×D4×C173C2312yes—
73C22×C4×C173—21no—
74C22×C4×C173—42no—
75C22×C4×C173D411yescharacteristic
76C22×C4×C173D411yescharacteristic
77C22×C4×C173D411yescharacteristic
78C22×C4×C173C2311yescharacteristic
79C22×C4×C173C2311yescharacteristic
80C22×C4×C173C2311yescharacteristic
81C4⋊C4×C173—21no—
82C4⋊C4×C173C2312yes—
83C22⋊C4×C173—22no—
84C22⋊C4×C173—22no—
85C22×D173—21no—
86Dic346—22no—
87D692—22no—
88C22⋊D173—42no—
89C22⋊D173—42no—
90C22⋊D173—42no—
91C2×Dic173—21no—
92C2×Dic173—21no—
93C2×Dic173—21no—
94C4×D173—42no—
95#13063436—22no—
96#13063436—22no—
97C23×C173—22no—
98C23×C173—22no—
99C23×C173—41no—
100C23×C173—41no—
101C23×C173—42no—
102C23×C173—42no—
103C23×C173—42no—
104C23×C173C2×D411yescharacteristic
105C23×C173C2×D411yescharacteristic
106C23×C173C2×D411yescharacteristic
107D4×C173—22no—
108D4×C173—22no—
109D4×C173—42no—
110D4×C173—44no—
111D4×C173—44no—
112C2×C4×C173—22no—
113C2×C4×C173—22no—
114C2×C4×C173—22no—
115C2×C4×C173—22no—
116C2×C4×C173—41no—
117C2×C4×C173—41no—
118C2×C4×C173—41no—
119C2×C4×C173—42no—
120C2×C4×C173C2×D411yescharacteristic
121C2×C4×C173C2×D411yescharacteristic
122C2×C4×C173C2×D411yescharacteristic
123C2×C4×C173C2411yescharacteristic
124C22×C173—21no—
125C22×C173—21no—
126C22×C173—21no—
127C22×C173—21no—
128C22×C173—21no—
129C22×C173—21no—
130C22×C173—21no—
131C22×C173—21no—
132C22×C173—44no—
133C22×C173—42no—
134C22×C173—42no—
135C22×C173—42no—
136C22×C173—84no—
137C22×C173AGL(1,ℤ/8)12yes—
138C22×C173C22×D411yescharacteristic
139C2×D173—41no—
140C2×D173—41no—
141C2×D173—41no—
142C4×C173—21no—
143C4×C173—21no—
144C4×C173—21no—
145C4×C173—42no—
146C4×C173H(5,𝔽2)12yes—
147Dic173—41no—
148Dic173—41no—
149Dic173—41no—
150C2×C173—22no—
151C2×C173—41no—
152C2×C173—41no—
153C2×C173—82no—
154C2×C173#53411yescharacteristic
155C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
156C2×C173#57511yescharacteristic
157D173—81no—
158C173#319411yescharacteristic, complemented, Sylow
159#3194—1731nomaximal, Sylow
160#525—1731no—
161C2×#121D17311yescharacteristic, complemented
162#483—1731no—
163#482—1732no—
164#480—1731no—
165#469—1732no—
166#468—1732no—
167#450—1731no—
168#449—1732no—
169#447—1731no—
170#407—1731no—
171C2×#41—1731no—
172C22×D4—21no—
173C22×D4C2×D17311yescharacteristic
174C23×C4C2×D17311yescharacteristic
175C2×SD16—1732no—
176C2×D8—1732no—
177C2×M16—1732no—
178#123—3461no—
179C22⋊Q8—1731no—
180C22⋊Q8—1731no—
181#121—21no—
182#121—21no—
183#121—1731no—
184#121—1731no—
185#121C2×D17312yes—
186#121C2×D17312yes—
187#120—3461no—
188C4×D4—3461no—
189C4×D4—3461no—
190#117—1731no—
191C2×C4⋊C4C2×D17311yescharacteristic
192C2×C22⋊C4—21no—
193#107—1732no—
194#106—1732no—
195#103—1732no—
196#102—1732no—
197#102—1732no—
198#102—1732no—
199C22⋊C8—1731no—
200C22⋊C8—1731no—
201C22⋊C8—1732no—
202C24—21no—
203C24C22⋊D17311yescharacteristic
204#41—6922no—
205C2×Q8—1731no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—42no—
210C2×D4—44no—
211C2×D4—3461no—
212C2×D4—3461no—
213C2×D4—3461no—
214C2×D4—1731no—
215C2×D4C22×D17312yes—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—3461no—
219C22×C4C22⋊D17311yescharacteristic
220C22×C4C22⋊D17311yescharacteristic
221C22×C4C22⋊D17311yescharacteristic
222C22×C4C22×D17311yescharacteristic
223C22×C4C22×D17311yescharacteristic
224C22×C4C22×D17311yescharacteristic
225SD16—6922no—
226D8—6922no—
227M16—6922no—
228C2×C8—1732no—
229C2×C8—1732no—
230C4⋊C4—21no—
231C4⋊C4—3461no—
232C4⋊C4—3461no—
233C4⋊C4—1731no—
234C4⋊C4—1731no—
235C4⋊C4C22×D17312yes—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—3461no—
239C22⋊C4—3461no—
240C22⋊C4—3461no—
241C22⋊C4—3461no—
242C22⋊C4—3461no—
243C42—3461no—
244C23—22no—
245C23—22no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—3461no—
252C23C2×C22⋊D17311yescharacteristic
253C23C2×C22⋊D17311yescharacteristic
254C23C2×C22⋊D17311yescharacteristic
255Q8—3462no—
256D4—22no—
257D4—22no—
258D4—42no—
259D4—44no—
260D4—44no—
261D4—3462no—
262D4—6922no—
263D4—6922no—
264D4—6922no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—3461no—
274C2×C4—3461no—
275C2×C4—3461no—
276C2×C4—6922no—
277C2×C4C2×C22⋊D17311yescharacteristic
278C2×C4C2×C22⋊D17311yescharacteristic
279C2×C4C2×C22⋊D17311yescharacteristic
280C2×C4C23×D17311yescharacteristic
281C8—3462no—
282C8—3462no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—44no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—84no—
296C22—6921no—
297C22—6921no—
298C22—6921no—
299C22#1914970512yes—
300C22C22×C22⋊D17311yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—42no—
305C4—6921no—
306C4—6921no—
307C4—6921no—
308C4#1914972412yes—
309C2—22no—
310C2—41no—
311C2—41no—
312C2—82no—
313C2—13841no—
314C2#2601205011yescharacteristic
315C2C2×#1914970511yescharacteristic
316C2#2601228911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#121×C173C2
socleC22×C173C22×D4
hypercenterC2×#121D173
hypocenterC173#3194
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #26012002C2semidirect
2. #26011821C2semidirect
3. #26011812C2semidirect
4. #26011804C2semidirect
5. #26011742C2semidirect
6. #26011735C2semidirect
7. #26011642C2semidirect
8. #26011634C2semidirect
9. #26011625C2semidirect
10. #26011455C2semidirect
11. C2×#121×C173C2semidirect
12. C2×#14583826C22semidirect
13. C2×#14583814C22semidirect
14. C2×#14583811C22semidirect
15. C2×#14583810C22semidirect
18. #19149658C22semidirect
20. #19149653C22semidirect
24. #19149638C22semidirect
25. #19149612C22semidirect
26. #19149610C22semidirect
27. #19149602C22semidirect
28. #19149600C22semidirect
29. #19149600C22semidirect
30. #19149599C22semidirect
33. C22⋊#13063436C22semidirect
158. C173#3194semidirect
161. C2×#121D173semidirect

Maximal subgroups

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

subgroupindex
#260120022
#260118212
#260118122
#260118042
#260117422
#260117352
#260116422
#260116342
#260116252
#260114552
C2×#121×C1732
#3194173