Finite Group Databasebeta

← #36362838

Subgroups — #36362838

Order 22144.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19149725C211yesmaximal, characteristic
2C2×#19149710C211yesmaximal, characteristic, complemented
3C22×#14583813C211yesmaximal, characteristic, complemented
4#26011651C214yesmaximal, complemented
5#26011641C214yesmaximal, complemented
6#26011495C211yesmaximal, characteristic, complemented
7#26011485C211yesmaximal, characteristic, complemented
8#26011458C211yesmaximal, characteristic, complemented
9#536×C173C211yesmaximal, characteristic, complemented
10#19149725—24no—
11#19149725—24no—
12C2×#14583827—21no—
13C2×#14583826C2212yes—
14C2×#14583826C2211yescharacteristic
15C22×D692C2212yes—
16C22×D692C2211yescharacteristic
17C2×D4×D173—22no—
18C2×D4×D173—21no—
19#19149710—24no—
20C2×#14583816C2212yescomplemented
21C2×#14583813—24no—
22C2×#14583813C2214yescomplemented
23C2×#14583813C2212yescomplemented
24C2×#14583810C2211yescharacteristic, complemented
25C22×#13063436C2211yescharacteristic, complemented
26#19149602C2212yescomplemented
27#19149599C2214yescomplemented
28#19149599C2212yescomplemented
29C22⋊#13063436C2214yescomplemented
30C2×#41×C173C2211yescharacteristic
31C22×Q8×C173C2211yescharacteristic, complemented
32C4⋊Q8×C173—21no—
33#124×C173—21no—
34#123×C173—22no—
35C22⋊Q8×C173C2214yescomplemented
36#117×C173C2211yescharacteristic, complemented
37C23×D173—22no—
38C23×D173—21no—
39#14583827—28no—
40#14583826—24no—
41#14583826—24no—
42#14583826—22no—
43#14583826D414yes—
44C2×Dic346C2311yescharacteristic
45C2×D692—24no—
46C2×D692—24no—
47C2×D692—22no—
48C2×D692—21no—
49C2×D692—22no—
50C2×D692D414yes—
51C2×D692C2311yescharacteristic
52C2×D692C2312yes—
53C2×C22⋊D173—22no—
54C2×C22⋊D173—21no—
55D4×D173—216no—
56D4×D173—28no—
57C2×C4×D173—22no—
58C2×C4×D173—21no—
59#14583816—24no—
60#14583813—28no—
61#14583813—24no—
62#14583810—22no—
63C2×#13063436—22no—
64C2×#13063436C2312yes—
65C2×#13063436C2312yes—
66#41×C173—22no—
67#41×C173—22no—
68C2×Q8×C173—21no—
69C2×Q8×C173—22no—
70C2×Q8×C173—22no—
71C2×Q8×C173D411yescharacteristic
72C2×Q8×C173C2312yes—
73C2×D4×C173D412yes—
74C2×D4×C173D411yescharacteristic
75C22×C4×C173—21no—
76C22×C4×C173C2312yes—
77C22×C4×C173C2311yescharacteristic
78C4⋊C4×C173—24no—
79C4⋊C4×C173C2312yes—
80C22⋊C4×C173—21no—
81C22⋊C4×C173—24no—
82C42×C173—21no—
83C22×D173—22no—
84C22×D173—21no—
85C22×D173—28no—
86C22×D173—24no—
87C22×D173—82no—
88C22×D173—84no—
89Dic346—21no—
90Dic346C2×D412yes—
91D692—28no—
92D692—24no—
93D692—22no—
94D692—21no—
95D692C2×D412yes—
96D692C2×D414yes—
97C22⋊D173—28no—
98C22⋊D173—24no—
99C2×Dic173—21no—
100C4×D173—28no—
101C4×D173—24no—
102#13063436—22no—
103#13063436—22no—
104C23×C173C2×D412yes—
105C23×C173C2×D411yescharacteristic
106Q8×C173—22no—
107Q8×C173—22no—
108Q8×C173—22no—
109Q8×C173—42no—
110D4×C173—24no—
111D4×C173—22no—
112C2×C4×C173—22no—
113C2×C4×C173—22no—
114C2×C4×C173—22no—
115C2×C4×C173—22no—
116C2×C4×C173—42no—
117C2×C4×C173C2×D412yes—
118C2×C4×C173C2×D412yes—
119C2×C4×C173C2×D412yes—
120C2×C4×C173C2×D411yescharacteristic
121C2×C4×C173C2411yescharacteristic
122C22×C173—22no—
123C22×C173—22no—
124C22×C173—24no—
125C22×C173#12012yes—
126C22×C173C22×D411yescharacteristic
127C2×D173—24no—
128C2×D173—22no—
129C2×D173—44no—
130C2×D173—42no—
131C2×D173—84no—
132C2×D173—88no—
133C4×C173—22no—
134C4×C173—42no—
135C4×C173—42no—
136C4×C173#12012yes—
137C4×C173C22×D411yescharacteristic
138C4×C173C22×D411yescharacteristic
139Dic173—22no—
140C2×C173—22no—
141C2×C173—42no—
142C2×C173C2×#12011yescharacteristic
143C2×C173#57712yes—
144D173—44no—
145D173—42no—
146C173#300911yescharacteristic, complemented, Sylow
147#3009—1731nomaximal, Sylow
148C2×H(5,𝔽2)—1731no—
149C2×#137—1731no—
150C22×SD16—1731no—
151#536D17311yescharacteristic, complemented
152#452—1734no—
153#450—1734no—
154#418—1731no—
155#416—1731no—
156#408—1731no—
157H(5,𝔽2)—3464no—
158H(5,𝔽2)—3464no—
159C2×#41—3461no—
160C2×#41—1732no—
161C2×#41—1731no—
162C2×#41C2×D17311yescharacteristic
163C22×Q8C2×D17311yescharacteristic, complemented
164C22×D4—3462no—
165C22×D4—3461no—
166C22×D4—1732no—
167C22×D4—1731no—
168#137—3464no—
169C2×Q16—1732no—
170C2×SD16—3464no—
171C2×SD16—1734no—
172C2×SD16—1732no—
173C2×M16—1731no—
174C22×C8—1731no—
175C4⋊Q8—21no—
176#124—21no—
177#123—22no—
178C22⋊Q8C2×D17314yescomplemented
179#117C2×D17311yescharacteristic, complemented
180#103—1732no—
181#102—1732no—
182#102—1734no—
183C22⋊C8—1734no—
184C24—3462no—
185C24—3461no—
186#41—22no—
187#41—22no—
188#41—3464no—
189#41—3464no—
190#41—3468no—
191#41—3462no—
192#41—1734no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—1731no—
197C2×Q8C22⋊D17311yescharacteristic
198C2×Q8C22×D17312yes—
199C2×D4—3464no—
200C2×D4—3464no—
201C2×D4—34616no—
202C2×D4—3468no—
203C2×D4—3462no—
204C2×D4—3461no—
205C2×D4—3462no—
206C2×D4—3462no—
207C2×D4—3461no—
208C2×D4—1734no—
209C2×D4—1731no—
210C2×D4—1732no—
211C2×D4C22⋊D17312yes—
212C2×D4C22⋊D17311yescharacteristic
213C22×C4—21no—
214C22×C4—3462no—
215C22×C4—3461no—
216C22×C4C22×D17312yes—
217C22×C4C22×D17311yescharacteristic
218Q16—3464no—
219SD16—3468no—
220SD16—3464no—
221M16—3462no—
222C2×C8—3462no—
223C2×C8—1732no—
224C2×C8—1732no—
225C4⋊C4—24no—
226C4⋊C4C22×D17312yes—
227C22⋊C4—21no—
228C22⋊C4—24no—
229C42—21no—
230C23—3468no—
231C23—3464no—
232C23—3462no—
233C23—3461no—
234C23—13842no—
235C23—13844no—
236C23C2×C22⋊D17312yes—
237C23C2×C22⋊D17311yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—3461no—
243Q8—1732no—
244D4—24no—
245D4—22no—
246D4—3468no—
247D4—3468no—
248D4—3464no—
249D4—3464no—
250D4—3462no—
251D4—3461no—
252D4—1732no—
253D4—1734no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—3468no—
260C2×C4—3461no—
261C2×C4—3464no—
262C2×C4D4×D17312yes—
263C2×C4D4×D17312yes—
264C2×C4C2×C22⋊D17312yes—
265C2×C4C2×C22⋊D17311yescharacteristic
266C2×C4C23×D17311yescharacteristic
267C8—3462no—
268C8—3462no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—3464no—
273C22—3462no—
274C22—13844no—
275C22—13848no—
276C22—6924no—
277C22—6922no—
278C22#1914964712yes—
279C22C22×C22⋊D17311yescharacteristic
280C4—22no—
281C4—42no—
282C4—42no—
283C4—3462no—
284C4#1914964712yes—
285C4C2×D4×D17311yescharacteristic
286C4C2×D4×D17311yescharacteristic
287C2—22no—
288C2—42no—
289C2—6924no—
290C2—6922no—
291C2C2×#1914964711yescharacteristic
292C2#2601229912yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#536×C173C2
socleC22×C173C22×D4
prime radicalC2×#19149725C2
hypercenter#536D173
hypocenterC173#3009
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#19149710C2semidirect
3. C22×#14583813C2semidirect
4. #26011651C2semidirect
5. #26011641C2semidirect
6. #26011495C2semidirect
7. #26011485C2semidirect
8. #26011458C2semidirect
9. #536×C173C2semidirect
20. C2×#14583816C22semidirect
22. C2×#14583813C22semidirect
23. C2×#14583813C22semidirect
24. C2×#14583810C22semidirect
25. C22×#13063436C22semidirect
26. #19149602C22semidirect
27. #19149599C22semidirect
28. #19149599C22semidirect
29. C22⋊#13063436C22semidirect
31. C22×Q8×C173C22semidirect
35. C22⋊Q8×C173C22semidirect
36. #117×C173C22semidirect
146. C173#3009semidirect
151. #536D173semidirect
163. C22×Q8C2×D173semidirect
178. C22⋊Q8C2×D173semidirect
179. #117C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497252
C2×#191497102
C22×#145838132
#260116512
#260116412
#260114952
#260114852
#260114582
#536×C1732
#3009173