Finite Group Databasebeta

← #36364376

Subgroups — #36364376

Order 22144.

466 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 113 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, complemented
2C2×#19149654C211yesmaximal, characteristic, complemented
3#26011821C211yesmaximal, characteristic, complemented
4#26011818C212yesmaximal, complemented
5#26011743C212yesmaximal, complemented
6#26011738C212yesmaximal, complemented
7#26011644C211yesmaximal, characteristic, complemented
8#26011637C212yesmaximal, complemented
9#26011624C211yesmaximal, characteristic, complemented
10#26011453C211yesmaximal, characteristic, complemented
11#480×C173C211yesmaximal, characteristic, complemented
12C2×#14583825C2211yescharacteristic, complemented
13C22×D692—21no—
14C22×D692C2211yescharacteristic
15C22×C4×D173C2211yescharacteristic
16C2×#14583815C2212yescomplemented
17C2×#14583811C2212yescomplemented
18C2×#14583809C2212yescomplemented
19#19149661—21no—
20#19149659C2211yescharacteristic, complemented
21#19149658C2211yescharacteristic
22#19149654—21no—
23#19149654—21no—
24#19149654C2212yes—
25#19149654C2212yes—
26#19149650C2211yescharacteristic, complemented
27#19149647—21no—
28D4×Dic173—21no—
29#19149639—21no—
30#19149637C2211yescharacteristic, complemented
31C2×#14583803—21no—
32#19149610C2212yescomplemented
33#19149602C2212yescomplemented
34#19149601C2212yescomplemented
35#19149599C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic
37#19149591C2212yescomplemented
38#121×C173C2211yescharacteristic
39C2×C4⋊C4×C173C2211yescharacteristic
40#107×C173C2212yescomplemented
41#102×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic
43C23×D173—21no—
44C23×D173D411yescharacteristic
45#14583825—42no—
46C2×Dic346C2311yescharacteristic
47C2×D692—22no—
48C2×D692—22no—
49C2×D692—21no—
50C2×D692—42no—
51C2×D692—44no—
52C2×D692C2312yes—
53C2×C22⋊D173—21no—
54C2×C22⋊D173—21no—
55C22×Dic173—21no—
56C22×Dic173D411yescharacteristic
57C2×C4×D173—42no—
58C2×C4×D173D411yescharacteristic
59C2×C4×D173D411yescharacteristic
60C2×C4×D173C2311yescharacteristic
61C2×C4×D173C2311yescharacteristic
62#14583815—42no—
63#14583811—42no—
64#14583809—42no—
65C2×#13063436C2312yes—
66C4⋊Dic173—21no—
67C4⋊Dic173C2311yescharacteristic
68#14583805—21no—
69C22⋊Dic173—21no—
70C22⋊Dic173—21no—
71#14583803—21no—
72#14583803—21no—
73#14583803—22no—
74#14583803—22no—
75C4×Dic173—21no—
76C2×D4×C173—21no—
77C2×D4×C173C2311yescharacteristic
78C22×C4×C173—21no—
79C22×C4×C173C2311yescharacteristic
80C2×C8×C173C2312yes—
81C4⋊C4×C173—21no—
82C4⋊C4×C173C2311yescharacteristic
83C4⋊C4×C173C2312yes—
84C22⋊C4×C173—21no—
85C22×D173—22no—
86C22×D173—22no—
87C22×D173—41no—
88C22×D173—41no—
89C22×D173—42no—
90C22×D173—42no—
91C22×D173—42no—
92C22×D173C2×D411yescharacteristic
93C22×D173C2×D411yescharacteristic
94Dic346—22no—
95D692—22no—
96D692—22no—
97D692—42no—
98D692—44no—
99D692—44no—
100C22⋊D173—42no—
101C22⋊D173—42no—
102C2×Dic173—21no—
103C2×Dic173—21no—
104C2×Dic173—41no—
105C2×Dic173—42no—
106C2×Dic173C2×D411yescharacteristic
107C2×Dic173C2×D411yescharacteristic
108C4×D173—22no—
109C4×D173—22no—
110C4×D173—41no—
111C4×D173—41no—
112#13063436—22no—
113C23×C173—21no—
114C23×C173C2×D411yescharacteristic
115D4×C173—22no—
116D4×C173—42no—
117C2×C4×C173—22no—
118C2×C4×C173—22no—
119C2×C4×C173—21no—
120C2×C4×C173—42no—
121C2×C4×C173C2×D411yescharacteristic
122C2×C4×C173C2411yescharacteristic
123C8×C173—22no—
124C22×C173—21no—
125C22×C173—21no—
126C22×C173—41no—
127C22×C173—41no—
128C22×C173—41no—
129C22×C173AGL(1,ℤ/8)12yes—
130C22×C173C22×D411yescharacteristic
131C2×D173—21no—
132C2×D173—21no—
133C2×D173—21no—
134C2×D173—21no—
135C2×D173—21no—
136C2×D173—21no—
137C2×D173—44no—
138C2×D173—42no—
139C2×D173—42no—
140C2×D173—42no—
141C2×D173—84no—
142C4×C173—21no—
143C4×C173—42no—
144C4×C173—41no—
145C4×C173H(5,𝔽2)12yes—
146Dic173—21no—
147Dic173—21no—
148Dic173—41no—
149Dic173—41no—
150C2×C173—22no—
151C2×C173—81no—
152C2×C173#53411yescharacteristic
153C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
154C2×C173#57511yescharacteristic
155D173—41no—
156D173—41no—
157D173—82no—
158C173#319411yescharacteristic, complemented, Sylow
159#3194—1731nomaximal, Sylow
160#525—1731no—
161C2×#121—1731no—
162#483—1731no—
163#482—1732no—
164#480D17311yescharacteristic, complemented
165#469—1732no—
166#468—1732no—
167#450—1731no—
168#449—1732no—
169#447—1731no—
170#407—1731no—
171C2×#41—1731no—
172C22×D4—3461no—
173C22×D4—1731no—
174C23×C4—1731no—
175C2×SD16—1732no—
176C2×D8—1732no—
177C2×M16—1732no—
178#123—3461no—
179C22⋊Q8—1731no—
180C22⋊Q8—1731no—
181#121—3461no—
182#121—3461no—
183#121—1731no—
184#121—1732no—
185#121—1732no—
186#121C2×D17311yescharacteristic
187#120—3461no—
188C4×D4—3461no—
189C4×D4—3461no—
190#117—1731no—
191C2×C4⋊C4C2×D17311yescharacteristic
192C2×C22⋊C4—3461no—
193#107C2×D17312yescomplemented
194#106—1732no—
195#103—1732no—
196#102—1732no—
197#102—1732no—
198#102C2×D17312yescomplemented
199C22⋊C8—1731no—
200C22⋊C8—1732no—
201C22⋊C8C2×D17311yescharacteristic
202C24—3461no—
203C24—1731no—
204#41—6922no—
205C2×Q8—1731no—
206C2×D4—21no—
207C2×D4—3462no—
208C2×D4—3462no—
209C2×D4—3461no—
210C2×D4—3461no—
211C2×D4—3461no—
212C2×D4—1732no—
213C2×D4—6922no—
214C2×D4—6924no—
215C2×D4C22×D17311yescharacteristic
216C22×C4—21no—
217C22×C4—3461no—
218C22×C4—1731no—
219C22×C4—1731no—
220C22×C4—1731no—
221C22×C4—1731no—
222C22×C4—1731no—
223C22×C4—6922no—
224C22×C4C22×D17311yescharacteristic
225SD16—6922no—
226D8—6922no—
227M16—6922no—
228C2×C8—1732no—
229C2×C8C22×D17312yes—
230C4⋊C4—21no—
231C4⋊C4—3461no—
232C4⋊C4—3461no—
233C4⋊C4—1731no—
234C4⋊C4C22×D17311yescharacteristic
235C4⋊C4C22×D17312yes—
236C22⋊C4—21no—
237C22⋊C4—3461no—
238C22⋊C4—3461no—
239C22⋊C4—3462no—
240C22⋊C4—3462no—
241C22⋊C4—3461no—
242C22⋊C4—3461no—
243C42—3461no—
244C23—21no—
245C23—3462no—
246C23—3462no—
247C23—1731no—
248C23—1731no—
249C23—6921no—
250C23—6921no—
251C23—6922no—
252C23—6922no—
253C23—6922no—
254C23D4×D17311yescharacteristic
255Q8—3462no—
256D4—22no—
257D4—42no—
258D4—3462no—
259D4—3462no—
260D4—6922no—
261D4—6924no—
262D4—6924no—
263D4—6922no—
264D4—6922no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—3462no—
270C2×C4—3462no—
271C2×C4—3461no—
272C2×C4—3461no—
273C2×C4—1731no—
274C2×C4—1731no—
275C2×C4—6921no—
276C2×C4—6921no—
277C2×C4—6921no—
278C2×C4—6922no—
279C2×C4D4×D17311yescharacteristic
280C2×C4C23×D17311yescharacteristic
281C8—22no—
282C8—3462no—
283C22—21no—
284C22—21no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—3461no—
289C22—3461no—
290C22—3461no—
291C22—3461no—
292C22—3461no—
293C22—3461no—
294C22—13844no—
295C22—6924no—
296C22—6922no—
297C22—6922no—
298C22—6922no—
299C22#1914970312yes—
300C22C2×D4×D17311yescharacteristic
301C4—21no—
302C4—42no—
303C4—41no—
304C4—3461no—
305C4—3461no—
306C4—6921no—
307C4—6921no—
308C4#1914972512yes—
309C2—22no—
310C2—81no—
311C2—13842no—
312C2—6921no—
313C2—6921no—
314C2#2601205211yescharacteristic
315C2C2×#1914970311yescharacteristic
316C2#2601229011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#480×C173C2
socleC22×C173C22×D4
hypercenter#480D173
hypocenterC173#3194
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
1. #26012001C2semidirect
2. C2×#19149654C2semidirect
3. #26011821C2semidirect
4. #26011818C2semidirect
5. #26011743C2semidirect
6. #26011738C2semidirect
7. #26011644C2semidirect
8. #26011637C2semidirect
9. #26011624C2semidirect
10. #26011453C2semidirect
11. #480×C173C2semidirect
12. C2×#14583825C22semidirect
16. C2×#14583815C22semidirect
17. C2×#14583811C22semidirect
18. C2×#14583809C22semidirect
20. #19149659C22semidirect
26. #19149650C22semidirect
30. #19149637C22semidirect
32. #19149610C22semidirect
33. #19149602C22semidirect
34. #19149601C22semidirect
35. #19149599C22semidirect
37. #19149591C22semidirect
40. #107×C173C22semidirect
41. #102×C173C22semidirect
158. C173#3194semidirect
164. #480D173semidirect
193. #107C2×D173semidirect
198. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120012
C2×#191496542
#260118212
#260118182
#260117432
#260117382
#260116442
#260116372
#260116242
#260114532
#480×C1732
#3194173