Finite Group Databasebeta

← #36364317

Subgroups — #36364317

Order 22144.

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

normal subgroups only

This group has 468 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
1C22⋊Q8×D173C211yesmaximal, characteristic, complemented
2C2×#19149650C211yesmaximal, characteristic
3#26011822C211yesmaximal, characteristic, complemented
4#26011819C211yesmaximal, characteristic, complemented
5#26011808C212yesmaximal, complemented
6#26011719C212yesmaximal, complemented
7#26011717C212yesmaximal, complemented
8#26011642C211yesmaximal, characteristic, complemented
9#26011628C212yesmaximal, complemented
10C22⋊C8×D173C211yesmaximal, characteristic, complemented
11#450×C173C211yesmaximal, characteristic, complemented
12C2×Q8×D173C2211yescharacteristic, complemented
13C22×C22⋊D173—21no—
14C22×C4×D173C2211yescharacteristic
15C2×#14583814C2212yescomplemented
16C2×C8×D173C2212yescomplemented
17#19149659C2211yescharacteristic, complemented
18#19149658C2211yescharacteristic
19#19149657—21no—
20#19149656—21no—
21#19149655C2211yescharacteristic, complemented
22#19149650—21no—
23#19149650—21no—
24#19149650C2212yes—
25#19149650C2212yes—
26C2×C4⋊Dic173C2211yescharacteristic
27C4⋊C4×D173—21no—
28C4⋊C4×D173C2211yescharacteristic, complemented
29C2×C22⋊Dic173—21no—
30C22⋊C4×D173—21no—
31#19149611C2212yescomplemented
32#19149610C2212yescomplemented
33#19149604C2212yescomplemented
34#19149603C2212yescomplemented
35#19149600C2212yescomplemented
36C22⋊#13063436C2211yescharacteristic
37#19149591C2212yescomplemented
38C22×D4×C173C2211yescharacteristic
39C2×SD16×C173C2212yescomplemented
40C22⋊Q8×C173C2211yescharacteristic
41#102×C173C2212yescomplemented
42C22⋊C8×C173C2211yescharacteristic
43C23×D173D411yescharacteristic
44C2×Dic346—21no—
45C2×Dic346C2311yescharacteristic
46Q8×D173—42no—
47C2×C22⋊D173—22no—
48C2×C22⋊D173—22no—
49C2×C22⋊D173—44no—
50C22×Dic173—21no—
51C22×Dic173D411yescharacteristic
52C2×C4×D173—21no—
53C2×C4×D173—21no—
54C2×C4×D173—42no—
55C2×C4×D173D411yescharacteristic
56C2×C4×D173D411yescharacteristic
57C2×C4×D173C2311yescharacteristic
58C2×C4×D173C2311yescharacteristic
59#14583814—42no—
60C2×#13063436C2312yes—
61C8×D173—42no—
62C4⋊Dic173—21no—
63C4⋊Dic173—21no—
64C4⋊Dic173C2311yescharacteristic
65C4⋊Dic173C2312yes—
66#14583805—21no—
67#14583805—21no—
68#14583805—21no—
69C22⋊Dic173—22no—
70C22⋊Dic173—22no—
71C22⋊Dic173—21no—
72#14583803—21no—
73#14583803—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—41no—
86C22×D173—41no—
87C22×D173SD1614yescomplemented
88C22×D173C2×D411yescharacteristic
89C22×D173C2×D411yescharacteristic
90Dic346—22no—
91Dic346—42no—
92C22⋊D173—44no—
93C22⋊D173—44no—
94C2×Dic173—21no—
95C2×Dic173—21no—
96C2×Dic173—22no—
97C2×Dic173—41no—
98C2×Dic173—42no—
99C2×Dic173C2×D411yescharacteristic
100C2×Dic173C2×D411yescharacteristic
101C4×D173—22no—
102C4×D173—22no—
103C4×D173—41no—
104C4×D173—41no—
105C4×D173—42no—
106C4×D173—42no—
107#13063436—22no—
108C23×C173—22no—
109C23×C173—42no—
110C23×C173—42no—
111C23×C173—42no—
112C23×C173C2×D411yescharacteristic
113Q8×C173—22no—
114D4×C173—22no—
115D4×C173—22no—
116D4×C173—42no—
117C2×C4×C173—22no—
118C2×C4×C173—21no—
119C2×C4×C173—21no—
120C2×C4×C173C2×D411yescharacteristic
121C2×C4×C173C2411yescharacteristic
122C8×C173—22no—
123C22×C173—21no—
124C22×C173—21no—
125C22×C173—42no—
126C22×C173—42no—
127C22×C173—42no—
128C22×C173—84no—
129C22×C173C2×SD1612yes—
130C22×C173C22×D411yescharacteristic
131C2×D173—21no—
132C2×D173—21no—
133C2×D173—21no—
134C2×D173—21no—
135C2×D173—44no—
136C2×D173C2×SD1612yes—
137C2×D173C2×SD1612yes—
138C4×C173—21no—
139C4×C173—41no—
140C4×C173—41no—
141C4×C173H(5,𝔽2)12yes—
142Dic173—21no—
143Dic173—21no—
144Dic173—41no—
145Dic173—42no—
146Dic173—41no—
147C2×C173—22no—
148C2×C173—82no—
149C2×C173#53411yescharacteristic
150C2×C173C22×SD1611yescharacteristic
151C2×C173#57511yescharacteristic
152D173—22no—
153D173—22no—
154C173#318711yescharacteristic, complemented, Sylow
155#3187—1731nomaximal, Sylow
156C2×C22⋊Q8—1731no—
157C2×#121—1731no—
158#483—1731no—
159#483—1731no—
160#481—1732no—
161#465—1732no—
162#465—1732no—
163#450—1731no—
164#450D17311yescharacteristic, complemented
165#448—1732no—
166C2×C22⋊C8—1731no—
167C22×Q8—1731no—
168C22×D4—3461no—
169C22×D4C2×D17311yescharacteristic
170C23×C4—1731no—
171C2×SD16—1732no—
172C2×SD16C2×D17312yescomplemented
173C22×C8—1732no—
174C22⋊Q8—3461no—
175C22⋊Q8—3461no—
176C22⋊Q8—1731no—
177C22⋊Q8—1731no—
178C22⋊Q8—1731no—
179C22⋊Q8C2×D17311yescharacteristic
180#121—3461no—
181#121—3461no—
182#121—1732no—
183#121—1732no—
184C2×C4⋊C4—3461no—
185C2×C4⋊C4—1731no—
186C2×C4⋊C4—1731no—
187C2×C22⋊C4—3461no—
188C2×C22⋊C4—3461no—
189#106—1732no—
190#106—1732no—
191#103—1732no—
192#103—1732no—
193#102—1732no—
194#102C2×D17312yescomplemented
195C22⋊C8—1731no—
196C22⋊C8—1732no—
197C22⋊C8C2×D17311yescharacteristic
198C24—21no—
199C24—1731no—
200C2×Q8—3461no—
201C2×Q8—1731no—
202C2×Q8—6922no—
203C2×Q8C22×D17311yescharacteristic
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—3462no—
207C2×D4—3462no—
208C2×D4—6924no—
209C2×D4C22×D17312yes—
210C22×C4—3461no—
211C22×C4—3461no—
212C22×C4—3461no—
213C22×C4—1731no—
214C22×C4—1731no—
215C22×C4—1731no—
216C22×C4—1731no—
217C22×C4—1731no—
218C22×C4—6922no—
219C22×C4C22×D17311yescharacteristic
220SD16—42no—
221SD16—6922no—
222C2×C8—1732no—
223C2×C8—6922no—
224C2×C8C22×D17312yes—
225C4⋊C4—21no—
226C4⋊C4—3461no—
227C4⋊C4—3461no—
228C4⋊C4—3461no—
229C4⋊C4—3461no—
230C4⋊C4—3461no—
231C4⋊C4—1731no—
232C4⋊C4—1732no—
233C4⋊C4C22×D17311yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—3461no—
236C22⋊C4—3461no—
237C22⋊C4—3461no—
238C22⋊C4—3462no—
239C22⋊C4—3462no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—1731no—
245C23—1731no—
246C23—1734no—
247C23—6921no—
248C23—6921no—
249C23D4×D17311yescharacteristic
250Q8—22no—
251Q8—3462no—
252Q8—6922no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—6924no—
257D4—6924no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—3462no—
262C2×C4—3462no—
263C2×C4—3461no—
264C2×C4—3461no—
265C2×C4—3462no—
266C2×C4—1731no—
267C2×C4—1731no—
268C2×C4—6921no—
269C2×C4—6921no—
270C2×C4—6922no—
271C2×C4—6922no—
272C2×C4—6921no—
273C2×C4—6922no—
274C2×C4D4×D17311yescharacteristic
275C2×C4C23×D17311yescharacteristic
276C8—22no—
277C8—3462no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—84no—
284C22—3461no—
285C22—3461no—
286C22—3461no—
287C22—3461no—
288C22—1732no—
289C22—1732no—
290C22—6924no—
291C22SD16×D17312yes—
292C22C2×D4×D17311yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—3461no—
297C4—3461no—
298C4—6922no—
299C4—6921no—
300C4—6921no—
301C4#1914972412yes—
302C2—22no—
303C2—82no—
304C2—3462no—
305C2—3462no—
306C2#2601205111yescharacteristic
307C2C2×SD16×D17311yescharacteristic
308C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#450×C173C2
socleC22×C173C22×D4
prime radicalC2×#19149650C2
hypercenter#450D173
hypocenterC173#3187
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. C22⋊Q8×D173C2semidirect
3. #26011822C2semidirect
4. #26011819C2semidirect
5. #26011808C2semidirect
6. #26011719C2semidirect
7. #26011717C2semidirect
8. #26011642C2semidirect
9. #26011628C2semidirect
10. C22⋊C8×D173C2semidirect
11. #450×C173C2semidirect
12. C2×Q8×D173C22semidirect
15. C2×#14583814C22semidirect
16. C2×C8×D173C22semidirect
17. #19149659C22semidirect
21. #19149655C22semidirect
28. C4⋊C4×D173C22semidirect
31. #19149611C22semidirect
32. #19149610C22semidirect
33. #19149604C22semidirect
34. #19149603C22semidirect
35. #19149600C22semidirect
37. #19149591C22semidirect
39. C2×SD16×C173C22semidirect
41. #102×C173C22semidirect
87. C22×D173SD16semidirect
154. C173#3187semidirect
164. #450D173semidirect
172. C2×SD16C2×D173semidirect
194. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1732
C2×#191496502
#260118222
#260118192
#260118082
#260117192
#260117172
#260116422
#260116282
C22⋊C8×D1732
#450×C1732
#3187173