Finite Group Databasebeta

← #36364331

Subgroups — #36364331

Order 22144.

472 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 115 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#26012004C211yesmaximal, characteristic, complemented
2C2×#19149650C211yesmaximal, characteristic, complemented
3#26011824C211yesmaximal, characteristic, complemented
4#26011814C212yesmaximal, complemented
5#26011810C211yesmaximal, characteristic, complemented
6#26011726C212yesmaximal, complemented
7#26011717C212yesmaximal, complemented
8#26011635C212yesmaximal, complemented
9#26011625C211yesmaximal, characteristic, complemented
10C22⋊C8×D173C211yesmaximal, characteristic, complemented
11#450×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C2211yescharacteristic, complemented
13C22×C22⋊D173—21no—
14C22×C4×D173C2211yescharacteristic
15C2×#14583811C2212yescomplemented
16C2×C8×D173C2212yescomplemented
17#19149664—21no—
18#19149655C2211yescharacteristic, complemented
19#19149654C2211yescharacteristic, complemented
20#19149653C2211yescharacteristic
21#19149650—21no—
22#19149650—21no—
23#19149650C2212yes—
24#19149650C2212yes—
25#19149649—21no—
26#19149641—21no—
27#19149639—21no—
28#19149637C2211yescharacteristic, complemented
29C2×C4⋊Dic173C2211yescharacteristic
30C2×C22⋊Dic173—21no—
31#19149612C2212yescomplemented
32#19149611C2212yescomplemented
33#19149603C2212yescomplemented
34#19149601C2212yescomplemented
35#19149600C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic
37#19149591C2212yescomplemented
38C22×D4×C173C2211yescharacteristic
39C2×SD16×C173C2212yescomplemented
40C22⋊Q8×C173C2211yescharacteristic
41#102×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic
43C23×D173D411yescharacteristic
44#14583827—42no—
45C2×D692—21no—
46C2×D692—21no—
47C2×D692C2311yescharacteristic
48C2×C22⋊D173—22no—
49C2×C22⋊D173—22no—
50C2×C22⋊D173—21no—
51C2×C22⋊D173—44no—
52C22×Dic173—21no—
53C22×Dic173D411yescharacteristic
54C2×C4×D173—21no—
55C2×C4×D173—42no—
56C2×C4×D173D411yescharacteristic
57C2×C4×D173D411yescharacteristic
58C2×C4×D173C2311yescharacteristic
59C2×C4×D173C2311yescharacteristic
60#14583811—42no—
61C2×#13063436C2312yes—
62C8×D173—42no—
63C4⋊Dic173—21no—
64C4⋊Dic173C2311yescharacteristic
65C4⋊Dic173C2312yes—
66#14583805—21no—
67C22⋊Dic173—22no—
68C22⋊Dic173—22no—
69#14583803—21no—
70#14583803—21no—
71#14583803—21no—
72#14583803—21no—
73C4×Dic173—21no—
74C24×C173—21no—
75C2×Q8×C173C2311yescharacteristic
76C2×D4×C173—21no—
77C2×D4×C173—42no—
78C2×D4×C173C2312yes—
79C22×C4×C173C2311yescharacteristic
80SD16×C173—42no—
81C2×C8×C173C2312yes—
82C4⋊C4×C173—21no—
83C4⋊C4×C173C2311yescharacteristic
84C22⋊C4×C173—21no—
85C22×D173—22no—
86C22×D173—22no—
87C22×D173—21no—
88C22×D173—41no—
89C22×D173C2×D411yescharacteristic
90C22×D173C2×D411yescharacteristic
91D692—22no—
92D692—42no—
93D692—42no—
94C22⋊D173—44no—
95C22⋊D173—42no—
96C22⋊D173—44no—
97C2×Dic173—22no—
98C2×Dic173—21no—
99C2×Dic173—41no—
100C2×Dic173—42no—
101C2×Dic173C2×D411yescharacteristic
102C2×Dic173C2×D411yescharacteristic
103C4×D173—22no—
104C4×D173—22no—
105C4×D173—41no—
106C4×D173—41no—
107C4×D173—42no—
108#13063436—22no—
109C23×C173—22no—
110C23×C173—42no—
111C23×C173—42no—
112C23×C173—42no—
113C23×C173C2×D411yescharacteristic
114Q8×C173—22no—
115D4×C173—22no—
116D4×C173—22no—
117D4×C173—42no—
118C2×C4×C173—22no—
119C2×C4×C173—21no—
120C2×C4×C173—21no—
121C2×C4×C173C2×D411yescharacteristic
122C2×C4×C173C2411yescharacteristic
123C8×C173—22no—
124C22×C173—21no—
125C22×C173—21no—
126C22×C173—42no—
127C22×C173—42no—
128C22×C173—42no—
129C22×C173—84no—
130C22×C173#13512yes—
131C22×C173C22×D411yescharacteristic
132C2×D173—21no—
133C2×D173—21no—
134C2×D173—21no—
135C2×D173—21no—
136C2×D173—21no—
137C2×D173—44no—
138C2×D173—41no—
139C2×D173—41no—
140C2×D173—41no—
141C2×D173AGL(1,ℤ/8)12yes—
142C4×C173—21no—
143C4×C173—41no—
144C4×C173—41no—
145C4×C173H(5,𝔽2)12yes—
146Dic173—21no—
147Dic173—21no—
148Dic173—41no—
149Dic173—42no—
150C2×C173—22no—
151C2×C173—82no—
152C2×C173#53411yescharacteristic
153C2×C173C2×#13511yescharacteristic
154C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
155D173—22no—
156D173—41no—
157D173—81no—
158C173#318811yescharacteristic, complemented, Sylow
159#3188—1731nomaximal, Sylow
160#525—1731no—
161C2×#121—1731no—
162#484—1731no—
163#482—1732no—
164#481—1731no—
165#466—1732no—
166#465—1732no—
167#450D17311yescharacteristic, complemented
168#449—1732no—
169#447—1731no—
170C2×C22⋊C8—1731no—
171C2×#41—1731no—
172C22×D4—3461no—
173C22×D4C2×D17311yescharacteristic
174C23×C4—1731no—
175C2×SD16C2×D17312yescomplemented
176C2×D8—1732no—
177C22×C8—1732no—
178#123—3461no—
179C22⋊Q8—1731no—
180C22⋊Q8C2×D17311yescharacteristic
181#121—3461no—
182#121—3461no—
183#121—1732no—
184#121—1732no—
185#121—1731no—
186#121—1731no—
187#120—3461no—
188C4×D4—3461no—
189C4×D4—3461no—
190#117—1731no—
191C2×C4⋊C4—1731no—
192C2×C22⋊C4—3461no—
193#107—1732no—
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—21no—
203C24—1731no—
204#41—6922no—
205C2×Q8C22×D17311yescharacteristic
206C2×D4—21no—
207C2×D4—42no—
208C2×D4—3461no—
209C2×D4—3461no—
210C2×D4—3462no—
211C2×D4—3462no—
212C2×D4—3461no—
213C2×D4—1731no—
214C2×D4—6924no—
215C2×D4C22×D17312yes—
216C22×C4—3461no—
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—42no—
226D8—6922no—
227C2×C8—1732no—
228C2×C8—6922no—
229C2×C8C22×D17312yes—
230C4⋊C4—21no—
231C4⋊C4—3461no—
232C4⋊C4—3461no—
233C4⋊C4—1731no—
234C4⋊C4—1732no—
235C4⋊C4C22×D17311yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—3461no—
238C22⋊C4—3461no—
239C22⋊C4—3461no—
240C22⋊C4—3462no—
241C22⋊C4—3462no—
242C22⋊C4—3461no—
243C42—3461no—
244C23—22no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—3462no—
249C23—3462no—
250C23—3461no—
251C23—1731no—
252C23—1731no—
253C23—6921no—
254C23D4×D17311yescharacteristic
255Q8—22no—
256D4—22no—
257D4—22no—
258D4—42no—
259D4—3462no—
260D4—6922no—
261D4—6922no—
262D4—6924no—
263D4—6924no—
264D4—6922no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—3462no—
269C2×C4—3462no—
270C2×C4—3461no—
271C2×C4—3462no—
272C2×C4—1731no—
273C2×C4—1731no—
274C2×C4—6921no—
275C2×C4—6921no—
276C2×C4—6922no—
277C2×C4—6921no—
278C2×C4—6922no—
279C2×C4D4×D17311yescharacteristic
280C2×C4C23×D17311yescharacteristic
281C8—22no—
282C8—3462no—
283C22—21no—
284C22—21no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—3461no—
290C22—3461no—
291C22—3461no—
292C22—3461no—
293C22—3461no—
294C22—1732no—
295C22—6924no—
296C22—6921no—
297C22—6921no—
298C22—6921no—
299C22#1914970012yes—
300C22C2×D4×D17311yescharacteristic
301C4—21no—
302C4—41no—
303C4—41no—
304C4—3461no—
305C4—3461no—
306C4—6921no—
307C4—6922no—
308C4#1914972412yes—
309C2—22no—
310C2—82no—
311C2—3462no—
312C2—13841no—
313C2—6921no—
314C2#2601205111yescharacteristic
315C2C2×#1914970011yescharacteristic
316C2AGL(1,ℤ/8)×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
hypercenter#450D173
hypocenterC173#3188
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #26012004C2semidirect
2. C2×#19149650C2semidirect
3. #26011824C2semidirect
4. #26011814C2semidirect
5. #26011810C2semidirect
6. #26011726C2semidirect
7. #26011717C2semidirect
8. #26011635C2semidirect
9. #26011625C2semidirect
10. C22⋊C8×D173C2semidirect
11. #450×C173C2semidirect
12. C2×#14583827C22semidirect
15. C2×#14583811C22semidirect
16. C2×C8×D173C22semidirect
18. #19149655C22semidirect
19. #19149654C22semidirect
28. #19149637C22semidirect
31. #19149612C22semidirect
32. #19149611C22semidirect
33. #19149603C22semidirect
34. #19149601C22semidirect
35. #19149600C22semidirect
37. #19149591C22semidirect
39. C2×SD16×C173C22semidirect
41. #102×C173C22semidirect
158. C173#3188semidirect
167. #450D173semidirect
175. C2×SD16C2×D173semidirect
198. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120042
C2×#191496502
#260118242
#260118142
#260118102
#260117262
#260117172
#260116352
#260116252
C22⋊C8×D1732
#450×C1732
#3188173