Finite Group Databasebeta

← #36362839

Subgroups — #36362839

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
2C22×#14583813C211yesmaximal, characteristic, complemented
3#26012065C211yesmaximal, characteristic, complemented
4#26011652C214yesmaximal, complemented
5#26011640C214yesmaximal, complemented
6#26011497C211yesmaximal, characteristic, complemented
7#26011487C211yesmaximal, characteristic, complemented
8#26011459C211yesmaximal, characteristic, complemented
9C2×#137×C173C211yesmaximal, characteristic, complemented
10#19149725—24no—
11#19149725—24no—
12C2×#14583827C2212yes—
13C2×#14583826—21no—
14C2×#14583826C2211yescharacteristic
15C22×D692—21no—
16C22×D692C2211yescharacteristic
17C2×D4×D173—22no—
18C2×D4×D173C2212yes—
19C2×#14583813—24no—
20C2×#14583813C2214yescomplemented
21C22×#13063436C2211yescharacteristic, complemented
22#19149677—21no—
23#19149668—21no—
24#19149662—22no—
25#19149655C2214yescomplemented
26#19149638C2211yescharacteristic, complemented
27#19149603C2212yescomplemented
28#19149601C2214yescomplemented
29#19149600C2212yescomplemented
30#19149591C2214yescomplemented
31C2×#41×C173C2211yescharacteristic
32C22×Q8×C173C2211yescharacteristic, complemented
33#137×C173—24no—
34C2×Q16×C173C2212yescomplemented
35C2×SD16×C173C2212yescomplemented
36C2×M16×C173C2211yescharacteristic, complemented
37C23×D173—22no—
38C23×D173—21no—
39#14583827—24no—
40#14583827—24no—
41#14583826—28no—
42#14583826—22no—
43#14583826—22no—
44C2×Dic346D411yescharacteristic
45C2×D692—28no—
46C2×D692—22no—
47C2×D692—21no—
48C2×D692—22no—
49C2×D692D411yescharacteristic
50C2×D692D414yes—
51C2×D692C2312yes—
52C2×C22⋊D173—22no—
53C2×C22⋊D173D412yes—
54D4×D173—216no—
55D4×D173—24no—
56D4×D173—24no—
57C2×C4×D173—22no—
58C2×C4×D173C2312yes—
59#14583813—28no—
60C2×#13063436—22no—
61C2×#13063436C2312yes—
62C4⋊Dic173C2312yes—
63#14583805—24no—
64C22⋊Dic173—21no—
65#14583803—24no—
66C4×Dic173—21no—
67#41×C173—22no—
68#41×C173D414yes—
69C2×Q8×C173—21no—
70C2×Q8×C173—22no—
71C2×Q8×C173—22no—
72C2×Q8×C173C2312yes—
73C2×Q8×C173C2311yescharacteristic
74C2×D4×C173—21no—
75C2×D4×C173C2311yescharacteristic
76C22×C4×C173—21no—
77C22×C4×C173—21no—
78C22×C4×C173C2311yescharacteristic
79Q16×C173—24no—
80SD16×C173—24no—
81M16×C173—22no—
82C2×C8×C173C2312yes—
83C22×D173—22no—
84C22×D173—28no—
85C22×D173—24no—
86C22×D173—84no—
87C22×D173—82no—
88C22×D173C2×D412yes—
89Dic346—22no—
90D692—28no—
91D692—22no—
92D692—24no—
93D692—22no—
94D692C2×D414yes—
95C22⋊D173—28no—
96C22⋊D173—24no—
97C2×Dic173—22no—
98C2×Dic173C2×D412yes—
99C4×D173—28no—
100C4×D173—22no—
101C4×D173—22no—
102#13063436—22no—
103C23×C173—21no—
104C23×C173C2×D411yescharacteristic
105Q8×C173—21no—
106Q8×C173—22no—
107Q8×C173—22no—
108Q8×C173—42no—
109Q8×C173C2×D412yes—
110D4×C173—24no—
111D4×C173—21no—
112D4×C173C2×D412yes—
113C2×C4×C173—21no—
114C2×C4×C173—24no—
115C2×C4×C173—22no—
116C2×C4×C173—42no—
117C2×C4×C173C2×D411yescharacteristic
118C2×C4×C173C2×D412yes—
119C2×C4×C173C2×D412yes—
120C2×C4×C173C2411yescharacteristic
121C8×C173—22no—
122C22×C173—22no—
123C22×C173—22no—
124C22×C173—42no—
125C22×C173#12012yes—
126C22×C173C22×D411yescharacteristic
127C2×D173—22no—
128C2×D173—24no—
129C2×D173—24no—
130C2×D173—44no—
131C2×D173—84no—
132C2×D173—88no—
133C4×C173—22no—
134C4×C173—42no—
135C4×C173#12012yes—
136C4×C173C22×D411yescharacteristic
137C4×C173C22×D411yescharacteristic
138Dic173—22no—
139Dic173—42no—
140C2×C173—22no—
141C2×C173—42no—
142C2×C173C2×#12011yescharacteristic
143C2×C173#57712yes—
144D173—42no—
145D173—44no—
146C173#300911yescharacteristic, complemented, Sylow
147#3009—1731nomaximal, Sylow
148C2×H(5,𝔽2)—1731no—
149C2×#137D17311yescharacteristic, complemented
150C22×SD16—1731no—
151#536—1731no—
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—3461no—
165C22×D4—3462no—
166C22×D4—1732no—
167C22×D4—1731no—
168#137—24no—
169C2×Q16C2×D17312yescomplemented
170C2×SD16—3464no—
171C2×SD16—1734no—
172C2×SD16C2×D17312yescomplemented
173C2×M16C2×D17311yescharacteristic, complemented
174C22×C8—1731no—
175C4⋊Q8—3461no—
176#124—3461no—
177#123—3462no—
178C22⋊Q8—1734no—
179#117—1731no—
180#103—1732no—
181#102—1732no—
182#102—1734no—
183C22⋊C8—1734no—
184C24—3462no—
185C24—3461no—
186#41—22no—
187#41—3468no—
188#41—3464no—
189#41—3464no—
190#41—3462no—
191#41—3462no—
192#41C22⋊D17314yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—1731no—
197C2×Q8C22×D17312yes—
198C2×Q8C22×D17311yescharacteristic
199C2×D4—21no—
200C2×D4—3468no—
201C2×D4—3462no—
202C2×D4—34616no—
203C2×D4—3464no—
204C2×D4—3464no—
205C2×D4—3462no—
206C2×D4—3461no—
207C2×D4—3462no—
208C2×D4—1732no—
209C2×D4—1731no—
210C2×D4—1734no—
211C2×D4—1732no—
212C2×D4C22×D17311yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—3462no—
216C22×C4—1732no—
217C22×C4C22×D17311yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—3468no—
221M16—22no—
222C2×C8—3462no—
223C2×C8—1732no—
224C2×C8C22×D17312yes—
225C4⋊C4—3464no—
226C4⋊C4—1732no—
227C22⋊C4—3464no—
228C22⋊C4—3461no—
229C42—3461no—
230C23—21no—
231C23—3468no—
232C23—3464no—
233C23—3462no—
234C23—13842no—
235C23—13844no—
236C23—1732no—
237C23D4×D17311yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—3462no—
243Q8C2×C22⋊D17312yes—
244D4—24no—
245D4—21no—
246D4—3468no—
247D4—3468no—
248D4—3464no—
249D4—3462no—
250D4—3464no—
251D4—3462no—
252D4—1734no—
253D4C2×C22⋊D17312yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—3468no—
259C2×C4—3462no—
260C2×C4—3462no—
261C2×C4—3462no—
262C2×C4—1732no—
263C2×C4D4×D17311yescharacteristic
264C2×C4D4×D17312yes—
265C2×C4C2×C22⋊D17312yes—
266C2×C4C23×D17311yescharacteristic
267C8—22no—
268C8—3462no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—3462no—
273C22—3464no—
274C22—3464no—
275C22—13844no—
276C22—13848no—
277C22—6924no—
278C22#1914964712yes—
279C22C2×D4×D17311yescharacteristic
280C4—22no—
281C4—42no—
282C4—3462no—
283C4—6922no—
284C4#1914964712yes—
285C4C2×D4×D17311yescharacteristic
286C4C22×C22⋊D17311yescharacteristic
287C2—22no—
288C2—42no—
289C2—6922no—
290C2—6924no—
291C2C2×#1914964711yescharacteristic
292C2#2601230012yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#137×C173C2
socleC22×C173C22×D4
prime radicalC2×#19149725C2
hypercenterC2×#137D173
hypocenterC173#3009
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#14583813C2semidirect
3. #26012065C2semidirect
4. #26011652C2semidirect
5. #26011640C2semidirect
6. #26011497C2semidirect
7. #26011487C2semidirect
8. #26011459C2semidirect
9. C2×#137×C173C2semidirect
20. C2×#14583813C22semidirect
21. C22×#13063436C22semidirect
25. #19149655C22semidirect
26. #19149638C22semidirect
27. #19149603C22semidirect
28. #19149601C22semidirect
29. #19149600C22semidirect
30. #19149591C22semidirect
32. C22×Q8×C173C22semidirect
34. C2×Q16×C173C22semidirect
35. C2×SD16×C173C22semidirect
36. C2×M16×C173C22semidirect
146. C173#3009semidirect
149. C2×#137D173semidirect
163. C22×Q8C2×D173semidirect
169. C2×Q16C2×D173semidirect
172. C2×SD16C2×D173semidirect
173. C2×M16C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497252
C22×#145838132
#260120652
#260116522
#260116402
#260114972
#260114872
#260114592
C2×#137×C1732
#3009173